The aim of this paper is to develop a deformation theory of monoid schemes, generalising the approach developed by Grillet. The core idea of this approach is to introduce the notion of a system of abelian groups, as the naive approach to exactness does not work for monoids. We first study the case of monoid sheaves (functors over a poset into the category of monoids) and prove a classification theorem in this setting, showing that the coextensions of a monoid functor with a system of abelian groups is a symmetric categorical group and equivalent to the one obtained by the abelian group homomorphism $[\mathcal{C}^0 \to \mathsf{ker}\partial^1]$, thereby linking with cohomology of certain types of complexes, as expected. We then move towards monoid schemes, which are a type of a monoid sheaf, but where localisations now allow us to develop our most noteworthy result: We show that coextensions can be seen in a natural way as a stack of symmetric categorical groups. We will mention a few mild implications of this, but leave the deeper uses of stack theory in this setting for later papers.

For a monic polynomial $p$, its filled unit lemniscate is the planar set ${z: |p(z)|<1}$. Let $\kappa_n(K,1)$ denote the least possible area of this set among monic polynomials of degree $n$ whose zeros lie in a compact set $K$. We prove that there are absolute constants $c,C>0$ such that $c/\log n \leq \kappa_n(\overline{\mathbb{D}},1) \leq \kappa_n(\mathbb{T},1) \leq C/\log n$. Thus the recently established lower bound has the correct order, even when all zeros are required to lie on the unit circle. The upper bound is obtained by combining a quantitative Faber-polynomial separator for a thin keyhole domain with an equal-weight midpoint discretization that preserves the degree exactly. We also deduce that the critical boundary-zero minimizers form a normal family in $\mathbb{D}$.

We study the differential inclusion $\text{d}z_t\in F(z_t)\text{d}x_t$, with initial condition $z_0=\xi$, where $F$ is a nonconvex-valued multifunction, and $x$ a path of bounded $q$-variation, for some $1\leqslant q<2$, extending the work of Bailleul, Brault, and Coutin (2020). We obtain existence of local and global solutions to this inclusion without assuming $F$ bounded. If $z(\xi,x)$ denotes such a solution, we obtain measurability of $z$ with respect to $x$ and $\xi$. To establish this, we introduce a Skorokhod-type distance and prove that Young integration is continuous with respect to it. By the way, we prove that a compact-valued $\gamma$-H{ö}lder map $F$ has, for any $p>1/\gamma$ and $\xi\in F(0)$, a selection $f(\xi)$ of bounded $p$-variation, started at $\xi$, such that $f$ is measurable in $\xi$.

We give a short proof of the quantitative isoperimetric inequality. Our argument is based on a notion of quantitative calibrations which induce a natural distance controlling both the Fraenkel asymmetry and the tilt excess. The proof of our key result which can be viewed as a nonlinear, geometric version of Fuglede's result in $BV$ is direct and self-contained. In particular, we do not make any use of regularity theory for almost minimizers.

For an amenable unimodular random rooted network, we show that the presence of a positive point mass in the expected spectral measure implies that, with positive probability, there exists an eigenfunction with finite support.

In this paper we consider minimal surfaces in hyperbolic space of arbitrary codimension that are critical for renormalized area. We give two criteria which imply that $Y$ must be a totally geodesic disk. One can be viewed as a 'gap' result for the renormalized area.

The main thrust of these notes is 3-fold: (1) An analysis of certain $K(\pi,1)$'s that arise from the connections between configuration spaces, braid groups, and mapping class groups, (2) a function space interpretation of these results, and (3) a homological analysis of the cohomology of some of these groups for genus zero, one, and two surfaces possibly with marked points, as well as the cohomology of certain associated function spaces. An example of the type of results given here is an analysis of the space k particles moving on a punctured torus up to equivalence by the natural $SL(2,\mathbb{Z})$ action.

Let $K$ be a $1$-separated set of diameter at most $R-1$, and let $\ell_m$ denote a collection of $m$ points on a line, with consecutive points of distance $1$ apart. Conlon and Fox (2019) demonstrated a coloring of $n$-dimensional Euclidean space avoiding red congruent copies of $\ell_2$ and blue congruent copies of $K$ for $|K| > 10000^n\log R$. We show here a stronger bound, that in fact $|K| > (11 + o(1))^n\ln R$ suffices for arbitrary $1$-separated $K$, while the improvement $|K| > (5 + o(1))^n\ln R$ holds in many cases, including when $K = \ell_m$, or more generally when $K$ is contained in a low-dimensional affine subspace. We also make a special study of the case when $n=2$, demonstrating a two-coloring of two-dimensional Euclidean space avoiding red copies of $\ell_2$ and blue copies of $\ell_{6330}$. This latter result addresses a question of Erdős and Graham.

The end space of an infinite graph arises naturally in many contexts as an important invariant and an interesting construction. It compactifies a locally finite graph and Diestel shows how to extend the end space to a larger space, called the tangle space, which is able to compactify any infinite graph. In both ends and tangles, it is the vertex-connectivity structure of the graph that is being studied. If we switch our attention to edge-connectivity, we can analogously define edge-ends. There is a space known as the edge-direction space which turns out to play an analogous role as the tangle space in its relationship with the end space: the edge-direction space provides a larger compact space in which the not necessarily compact space of edge-ends lives in. In this paper, we make this analogy precise, providing a natural edge analogue definition of tangles and proving they result in exactly the edge-directions. We also describe a combinatorial construction of certain Boolean algebras which give rise, via Stone duality, to the tangle and the edge-direction spaces. Finally, we pursue functorial definitions of the combinatorial constructions used in the paper, inspired by the famously functorial nature of Stone duality and by previous work by one of the authors and colleagues on trying to functorialize the end space construction. We hope our work will provide foundation and inspiration for further work on infinite graph theory that makes ample use of category theory and powerful algebraic constructions such as Boolean algebras and Stone duality.

Let $G$ be a finite group and let $\F$ be a finite field. We prove that any finite-dimensional $G$-graded associative algebra $A$ over $\F$ has a finite basis for its $G$-graded polynomial identities.

This paper revisits the strong duality in the linear quadratic regulator (LQR) for continuous-time and discrete-time systems, and explores its interconnection with typical assumptions and the uniqueness of primal-dual solutions. Using a linear operator $\Psi$, we formulate a common nonconvex LQR problem that captures both time domains. We then derive its Lagrange dual problem and establish the strong duality via a rank-constrained tight semidefinite program (SDP) relaxation. Further, we show that the primal-dual optimal solutions to the SDP relaxation, after dropping the rank constraint, recover the classical algebraic Riccati equations and optimal feedback gains in a constructive manner. The dual derivation and strong duality analysis rely on mild standard assumptions and exploit the properties of the linear operator and its adjoint, revealing a structural symmetry between the two time domains.

In this paper, we consider Newton-Puiseux parametrizations of irreducible curves in (n+1)-space, n greater or equal to 1, within a fixed semigroup under the action of Mather's group A. We establish criteria for eliminating parameters while preserving the Newton-Puiseux form, extending known results for plane curves.

We prove that target-avoidance sets for Manneville--Pomeau maps are strong winning for Schmidt's game. More precisely, for the class of nonuniformly expanding interval maps considered here, there exists a single parameter $\alpha>0$ such that for every target $p\in[0,1]$, the set of points whose forward orbit does not accumulate on $p$ is $\alpha$-strong winning. The proof induces on the uniformly expanding region $[r_1,1]$. The resulting first-return map has infinitely many branches, so we approximate it by finite-branch expanding maps, apply a theorem of Hu--Li--Yu to those finite approximants, and then transfer the resulting strategies first to the induced map and then to the original Manneville--Pomeau map.

Analog in-memory computing enables highly parallel matrix-vector multiplications with reduced data movement, but the resulting operations are noisy, quantized, and affected by device- and circuit-level non-idealities. This paper studies approximate inverse preconditioning for Krylov subspace methods in a hybrid digital-analog setting. The digital host performs sparse products with the coefficient matrix and the precision-sensitive Krylov operations, while preconditioner applications are performed through analog crossbar matrix-vector multiplications. Since the realized preconditioner is inexact and application-dependent, the outer iteration is formulated as the flexible GMRES method. We show that analog execution changes the usual preconditioner design problem in the sense that a stronger digital preconditioner may be less effective after analog scaling, write noise, input/output perturbations, quantization, and clipping are taken into account. We compare various block Jacobi preconditioning schemes including exact block inverses, sparse approximate inverses, Monte Carlo approximate inverses (MCAI), damping, and nested block Jacobi schemes. Numerical experiments with realistic analog matrix-vector simulations show that analog-aware choices of block size, damping, MCAI construction accuracy, and nesting are important for robust convergence.

We investigate the image of polynomials multilinear in graded variables evaluated on the algebra of upper triangular matrices endowed with a group grading. We show that, in general, such an image need not be a vector subspace. However, under the additional assumption that the identity component of the grading is commutative, we prove that the image is always a vector subspace. We further investigate the image of polynomials evaluated on inverse and direct limits of algebras. As a consequence, we prove that the image of a polynomial evaluated on a direct limit of upper triangular matrix algebras whose identity component is commutative is always a vector subspace.

We develop Hamiltonian dynamics-based algorithms for smooth convex optimization that achieve accelerated rates of convergence. By exploiting contraction of averaged Hamiltonian flow trajectories rather than requiring contraction at trajectory endpoints, we show that Hamiltonian dynamics-based optimization methods admit deterministic and accelerated convergence guarantees, extending prior work that is limited to quadratic objectives or holds only in expectation. We analyze an idealized continuous-time algorithm and derive practical discrete-time implementations with optimal first-order complexity, thereby establishing Hamiltonian dynamics as a useful algorithmic primitive for deterministic accelerated convex optimization.

For a finite poset $\mathcal{P}$, its height $h(\mathcal{P})$ is the number of cover relations in its longest chain. When $\mathcal{P}$ is a lattice $\mathcal{L}$, we label its elements $x$ with $h(x_\downarrow) = h([\hat{0},x])$ and its cover relations $x \lessdot y$ with $h(y_\downarrow) - h(x_\downarrow)$. When a lattice $\mathcal{L}'$ extends $\mathcal{L}$, $h(x_\downarrow)_\mathcal{L} \leq h(x_\downarrow)_{\mathcal{L}'}$. We study lattices $\mathcal{L}$ and $\mathcal{L}'$ such that $h(x_\downarrow)_\mathcal{L} = h(x_\downarrow)_{\mathcal{L}'}$. Cover relations labeled $1$ in $\mathcal{L}$ induce a poset that we call the (long) skeletal poset $\mathrm{SK}(\mathcal{L})$. Its Hasse diagram is the largest spanning subgraph that the Hasse diagrams of $\mathcal{L}$ and $\mathcal{L}'$ have in common. An example of lattices $\mathcal{L}$ and $\mathcal{L}'$ is the alt-Tamari lattices introduced by Chenevière, where every alt-Tamari lattice $\mathrm{alt}\text{-}\mathrm{Tam}_n$ extends the Tamari lattice $\mathrm{Tam}_n$/refines the Dyck lattice $\mathrm{Dyck}_n$ such that $h(x_\downarrow)_{\mathrm{Tam}_n} = h(x_\downarrow)_{\mathrm{alt}\text{-}\mathrm{Tam}_n}$. We study $\mathrm{SK}(\mathrm{Tam}_n)$ with another poset we introduce. We enumerate intervals in these posets. For a well-chosen distributive lattice, we introduce its altitude lattices, which generalize the alt-Tamari lattices $\mathrm{alt}\text{-}\mathrm{Tam}_n$. Altitude lattices within a family have the same number of linear intervals. They are related to each other via extensions, refinements, and embeddings of some skeletal posets. For a poset $\mathcal{P}$ with $\hat{0}$, we define its Kneser graphs $KG(k) := (V(k),E)$, where $V(k) := \{x: h(x_\downarrow) = k, 1 \leq k \leq h(\mathcal{P})\}$ and $E := \{(x,y): x_\downarrow \cap y_\downarrow =\hat{0}\}$. We give some observations about them in a reconstruction setting.

Let $p$ be a prime. For every field $F$ of characteristic $p$ we exhibit pairs of non-isomorphic finite-dimensional $p$-nilpotent restricted Lie algebras $L$ and $H$ over $F$ whose restricted enveloping algebras $u(L)$ and $u(H)$ are isomorphic as $F$-algebras. Such pairs exist in every dimension at least $p+5$, with $\dim L'=p$ and $\dim H'=p+1$. Thus, the restricted isomorphism problem has a negative answer over every field of positive characteristic, even for $p$-nilpotent algebras over perfect fields.

Many nonsmooth, nonconvex objectives in learning and signal recovery are $\rho$-weakly convex. We minimize such a function in deterministic and stochastic settings when the weak-convexity parameter $\rho$ is unknown. The objective is not required to be globally Lipschitz continuous or smooth. We propose the Adaptive Prox-Guided Scheme (APS), a one-trial proximal algorithm that adapts the proximal parameter online and bidirectionally through a descent test, allowing it to exploit favorable local structure. In the deterministic setting, APS obtains an $O(\varepsilon^{-2})$ iteration complexity for producing an $\varepsilon$-subgradient stationary point. In the stochastic setting, APS achieves a high-probability $O(\varepsilon^{-2})$ iteration bound for driving the Moreau-envelope gradient below $\varepsilon$. This result holds under deliberately weak oracle assumptions: the function-difference estimates may be biased and heavy-tailed, and the stochastic proximal oracle need only be sufficiently accurate with constant probability when the proximal parameter lies below $1/(2\rho)$ (unknown to the algorithm), and can be arbitrary otherwise.

In this article, we investigate the construction of linear codes over a finite ring $\mathcal{S}$, where $\mathcal{S}$ is taken to be an extension of a commutative non-unital ring $I$ of order $p^2$. Our approach is based on the defining set method. The defining sets considered in this work are derived from general simplicial complexes that may contain multiple maximal elements. We determine the parameters of these codes over $\mathcal{S}$ and study their Gray images. We also study the corresponding subfield-like codes. We show that these Gray image codes and subfield-like codes produce several families of divisible codes. Furthermore, we establish sufficient conditions under which these codes are minimal, optimal, and self-orthogonal. As applications of our results, we obtain several families of projective few-weight codes, and locally recoverable codes with small locality. We also study the minimal access structures of secret-sharing schemes associated with the duals of these minimal codes. Moreover, we construct several families of strongly regular graphs from projective two-weight codes and determine their parameters explicitly.

The irrationality of e+pi remains open, despite the separate transcendence of e and pi. This paper studies the problem from the viewpoint of finite irrationality certificates and gives a bounded no-go audit for low-complexity Apéry-type proof mechanisms. First, we prove exact equivalences between the hypothesis e+pi in Q and eventual factorial-arithmetic phenomena: a ceiling recurrence, a factorial-Cantor digit condition, and a divisibility criterion. These criteria identify what rationality would force, while showing why tail conditions are not finite obstructions. Second, we formulate an Apéry-type certificate framework based on integer linear forms L_n = A_n(e+pi)+B_n with A_n,B_n in Z, L_n nonzero, and |L_n| tending to zero. A mixed integration-by-parts identity produces such forms from integer polynomials. We then audit several low-complexity constructions, including mixed Padé approximation, crossed separate approximations to e and pi, simple J-fractions, holonomic ansatzes, Rodrigues-type families, and an integer kernel-lattice search. The main contribution is a rigid boundary probe: no-go filters marking a tested zone where analytic smallness is destroyed by denominator clearing, coefficient growth, primitive reduction, or continued-fraction shadows. In the final kernel-lattice audit, 145 raw candidates reduce to 133 primitive records; the best signals are dominated by continued-fraction shadows, while non-CF candidates do not form a degree-continuing family. Thus, within the tested low-complexity families, no non-circular Apéry-type mechanism for e+pi is found.

In this paper we study the structure of the $\infty$-category of spectral Lie algebras. We show that this $\infty$-category admits an interesting symmetric monoidal structure, defined by an analog of the smash product of pointed spaces, and that the free Lie algebra functor $\mathrm{Sp} \to \mathrm{Lie}(\mathrm{Sp})$ is symmetric monoidal with respect to it. Moreover, this property of the free functor essentially characterizes the spectral Lie operad (among nonunital operads in spectra). This result may be thought of as Koszul dual to the more familiar fact that the free commutative algebra functor takes direct sums to tensor products. One of the key ideas is that the $\infty$-category of spectral Lie algebras behaves in many ways like the $\infty$-category of pointed spaces. More precisely, we deduce structural facts about spectral Lie algebras from familiar statements about spaces by differentiating, in the sense of Goodwillie calculus. The tool to do this is the highly structured generalization of Arone-Ching's chain rule established by Blans-Blom. Numerous other features of spectral Lie algebras follow as well, such as a version of Mather's second cube lemma, the relation between the James construction and loop-suspensions, the Hilton-Milnor splitting, and a version of the EHP sequence.

We study Schubert varieties of the affine Grassmannian for the general linear supergroup $GL_{n|m}$. An explicit computational study is conducted in low-dimensional cases, namely for dimensions $n|m = 1|1$ and $2|1$. We describe the supervariety structures that arise in these settings, providing coordinate descriptions, equations, and morphisms.

We associate a profinite group to every group \(G\) acting on a set \(\Omega\) with finite orbits. For each finite \(G\)-stable subset \(A\subseteq\Omega\), let \(G_A\leq\operatorname{Sym}(A)\) be the induced finite permutation group. The groups \(G_A\), with the natural restriction maps, form an inverse system, and we define $\Gamma_\Omega:=\varprojlim_A G_A$. We show that \(\Gamma_\Omega\) acts naturally on \(\Omega\) and is canonically topologically isomorphic to the closure of the image of \(G\) in \(\operatorname{Sym}(\Omega)\), endowed with the topology of pointwise convergence. We introduce the finite-level exactness property \(\textup{FLEP}\), under which subgroups of \(\Gamma_\Omega\) are recovered up to closure from their fixed-point sets, and closed subgroups are recovered exactly. We prove several equivalent formulations of \(\textup{FLEP}\). Under this condition, the fixed-point set construction gives an inclusion-reversing bijection between closed subgroups of \(\Gamma_\Omega\) and the fixed subsets of \(\Omega\) arising from closed subgroups. We apply the theory in two directions. First, every profinite group \(\Gamma\) is recovered from its normal finite-quotient action on $\coprod_{N}\Gamma/N$, where \(N\) ranges over the open normal subgroups of \(\Gamma\). For this action, \(\textup{FLEP}\) holds precisely when every finite quotient \(\Gamma/N\), with \(N\) open and normal, is a Dedekind group. Second, if \(G\leq\Aut(E)\) acts on a field \(E\) with finite orbits and \(F=E^G\), then \(E/F\) is Galois and the construction yields a canonical topological isomorphism $\Gamma_E \cong_{\mathrm{top}} \operatorname{Gal}(E/F)$, where \(\operatorname{Gal}(E/F)\) has the Krull topology. Thus the Krull Galois group is recovered from finite-orbit data.

Mazur's knot exterior in $S^1\times S^2$ admits a geometric description using a single regular ideal octahedron. The resulting hyperbolic structure is closely related to the Whitehead link exterior through Adams' theorem on thrice-punctured spheres. The same octahedral framework applies to the family of Jester manifolds introduced by Sparks. Using hyperbolic geometry, hyperbolic Dehn filling, and recent results on systolic geodesics, we prove that the boundaries of all Mazur and Jester manifolds are pairwise nonhomeomorphic, regardless of orientation. Consequently, the corresponding compact, contractible $4$-manifolds are pairwise nonhomeomorphic.

We provide several new characterizations of Property A for bounded degree graphs. In particular, we show that $(X,d)$ has Property A if and only if there is a proper gauge $\omega$ such that the Lipschitz free space $\operatorname{LF}(X,\omega\circ d)$ is isomorphic to $\ell_1$. As a consequence, all finitely generated groups with Property A admit proper uniformly Lipschitz affine actions on $\ell_1$. Moreover, for groups with finite Nagata dimension, we obtain actions with compression exponent 1. This result applies to higher rank lattices, such as $\operatorname{SL}(3,\mathbb{Z})$. We also show that a countable discrete group coarsely embeds into $L_1$ if and only if it admits a proper uniformly Lipschitz affine action on a subspace of $L_1$.

We consider mean curvature flow with free boundary through cylindrical or half-cylindrical singularities, namely singularities of the types $\mathbb{R}^k\times S^{n-k}$, $\mathbb{R}^k_+\times S^{n-k}$ or $\mathbb{R}^k\times S^{n-k}_+$. Using the foundational results for free boundary Brakke flows by Edelen and the first author, and the recent classification of ancient asymptotically cylindrical flows by Bamler-Lai, we prove that all these singularities have a mean-convex neighborhood. Moreover, generalizing work of Hershkovits-White to the free boundary setting we show that the free boundary level set flow is nonfattening provided all singularities have a mean-convex neighborhood. We conclude that free boundary flow through singularities is well-posed as long as all singularities are of cylindrical or half-cylindrical type.

A general framework is developed for deriving sharp inequalities on simple graphs from majorization and Schur-convexity. After establishing majorization relations between the spectrum of an arbitrary graph and the spectra of the complete, complete bipartite, and matching graphs, it is shown that every positive Schur-convex spectral functional yields several sharp inequalities relating $\lambda_1$, $|\lambda_n|$, and $\|G\|_\ast$. This reduces the problem of proving graph inequalities to the choice of a suitable Schur-convex function. This optimization problem is then studied within the family of random vector norms, whose moment and cumulant expansions connect the framework to the numbers of closed walks. This yields new sharp results, recovers classical inequalities from a unified viewpoint, and produces further bounds in settings such as triangle-free and square-free graphs.

We develop a non-overlapping domain decomposition method for the numerical solution of the Stokes-Biot model of fluid-poroelastic structure interaction in a mixed form. The model is based on a velocity-pressure formulation for the free fluid, a three-field stress-displacement-rotation formulation with weakly symmetric stress for the solid deformation, and a Darcy velocity-pressure formulation for the fluid in the poroelastic media. Mass conservation, balance of stress, and the Beavers-Joseph-Saffman slip with friction condition are imposed on the interface. The interface conditions are incorporated through Lagrange multipliers modeling the traces of the displacement and the Darcy pressure. The system is discretized using stable mixed finite element spaces for Stokes flow, elasticity, and Darcy flow. The domain is decomposed into a union of subdomains of either Stokes or Biot type with three types of interfaces: Stokes-Stokes, Biot-Biot, and Stokes-Biot. On the Stokes-Stokes interfaces, a normal stress Lagrange multiplier is introduced to impose weakly velocity continuity, while the Biot-Biot and Stokes-Biot interfaces are equipped with displacement and pressure Lagrange multipliers to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced via Schur complement to an interface problem for the Lagrange multipliers, which is solved by GMRES. Each iteration requires the solution of local Stokes or Biot problems, which can be performed in parallel. We show that the resulting interface operator is positive definite and analyze the convergence of the GMRES iteration through fields-of-value analysis. Numerical experiments are presented to illustrate the performance of the method.

Adaptive optimizers combining preconditioning, momentum, and weight decay (Adam and AdamW) are, under Polyak-Ruppert averaging, candidate engines for one-pass inference. Does the averaged iterate keep the classical Polyak-Ruppert central limit theorem (CLT), with sandwich covariance $H^{-1}SH^{-1}$ (Hessian $H$, gradient covariance $S$), under momentum and non-convergent preconditioning? The preconditioner-only analysis does not carry over: with momentum the canonical decomposition collapses to a tautology. Treating the augmented state (iterate, momentum buffer) as a time-varying linear stochastic approximation (SA), we prove (under local stabilization) positive drift stability, a non-autonomous Polyak-Ruppert CLT, and a projection identity. The upshot: the iterate-marginal covariance is exactly the plain stochastic gradient descent (SGD) sandwich $H^{-1}SH^{-1}$, so the adaptivity is asymptotically invisible. This holds for SA-Adam (sub-linearly vanishing momentum gain, $\gamma\in(\alpha,1)$; the sub-linear regime is essential), not constant-$\beta$ deployed Adam. Coupled $L_2$ weight decay yields the ridge-penalized sandwich, extending one-pass inference to regularized problems.

We ask whether knotting can be recognised using persistent homology. Starting from a point-cloud representation of a curve, we compute one-dimensional persistent homology, extract cycle representatives, and assign a hypergraph curvature-based score to these cycles. Motivated by proteins but tested more broadly, the method reveals systematic differences between knotted and unknotted structures in both protein families and synthetic examples. This suggests that knotting leaves a detectable persistent-homology-based signature.

For datasets with unknown but stationary serial dependence, a robust long run variance estimator is essential to handle diverse scenarios. Spectral variance estimators are commonly used but tend to exhibit significant negative bias in the presence of positive correlation. To overcome this, zero lugsail estimators have been introduced, offering zero asymptotic bias regardless of the correlation structure. However, there are currently no guidelines for selecting the optimal bandwidth for lugsail estimators, a critical component in the estimation process. We propose an inference optimal bandwidth rule for lugsail estimators, based on nonstandard fixed-smoothing limiting distributions developed in our study. This approach significantly improves bias correction, accounts for variability, and provides an estimator optimized for robust inference. Our theoretical findings are supported by a simulation study.

Relative symplectic cohomology is an invariant of compact subsets of a closed symplectic manifold, introduced by Varolgunes. There are many examples of computations of this invariant over the Novikov field, but the collection of computed examples over the Novikov ring is still quite limited. One reason for this is that such computations require determining the relevant Floer complexes for Hamiltonians that are not necessarily $C^2$-small Morse functions. In this work, we present a computation of relative symplectic cohomology over the Novikov ring for balls and their complements in $\mathbb{C}P^n$. Our computation relies on explicit descriptions of Floer complexes, in the Morse--Bott setting with cascades, for J-shaped Hamiltonians on $\mathbb{C}P^n$. This allows us to deduce new estimates for the stable displacement energy of the boundaries of balls in $\mathbb{C}P^n$.

In this paper we consider the friendship-bias of the vertices in an infinite rooted Galton--Watson tree. The friendship-bias of a vertex is the difference between the average degree of the neighbours of the vertex and the degree of the vertex itself. A vertex is said to be of type $\chi \in S$, with $S = \{-,0,+\}$, when its friendship-bias is, respectively, strictly negative, zero or strictly positive. We consider the fractions $f_l^\chi$ of vertices of type $\chi \in S$ along a random downward path up to branching depth $l \in \mathbb{N}$ and derive a large deviation principle (LDP) for the triple $(f_l^\chi)_{\chi \in S}$ as $l\to\infty$. The branching depth of a vertex counts the number of branchings that occur along the path that connects the vertex to the root of the tree. The rate in the LDP is $l$, while the rate function in the LDP is identified in terms of a variational formula minimising a relative entropy under a linear constraint. We focus on the case of binary branching, for which the rate function is already quite involved. We identify the qualitative properties of the rate function and show how it can be computed numerically. We briefly indicate how to proceed for more general branching and for vertex types along a tree consisting of a finite number of random downward paths. Our paper is the first to consider large deviations of vertex types.

We use perfect matchings and labelled Fano planes to construct and study the $2025$ orthogonal bases of positive roots in the $E_8$ root system. The set of these bases forms a highly structured, Bruhat-like graded poset $(\Omega, \leq_Q)$ whose rank function can be computed from the cardinalities of so-called generalized Rothe diagrams. We give combinatorial characterizations of these diagrams in terms of matchings and Fano planes, and we explain how to compute the ranks of the elements of $\Omega$ using suitable combinatorial statistics such as the weights of perfect matchings. We establish simple formulas for the rank generating functions of $\Omega$ and of its 50 congruence classes under a natural order congruence relation. Our derivation of the generating functions contains some intermediate results on general perfect matchings and labelled Fano planes that can be stated without mentioning root systems and may be of independent interest.

A central operator schedules each vehicle's passage time through a bottleneck to achieve a dynamic system optimum (DSO). The assignment depends on each vehicle's preferred arrival time, which is private and must be elicited from each vehicle. Mechanisms that elicit exact preferences, such as the Vickrey-Clarke-Groves (VCG) mechanism, can achieve strategyproofness but involve relatively complex rules and a computational burden on the operator. We focus instead on coarse reporting, in which each vehicle selects from a finite menu of time slots of a common width. This discrete interface already structures reservation and appointment systems in practice, including managed lanes for automated vehicles, airport slot allocation, and delivery appointment windows. We design a slot-based DSO mechanism on this coarse interface, in which the operator implements DSO assignment based on the reported slots and charges a capacity shadow price as a toll, and evaluate its performance. We prove that both the worst-case misreporting gain and the expected efficiency loss decrease quadratically in the slot width. The efficiency loss decays in this way under binding capacity, while the worst-case misreporting gain requires an additional condition on the preferred arrival time distribution and the schedule cost function. Analyzing the no-toll case, we find that the misreporting incentive persists, however finely the slots are refined, indicating that the toll also serves to elicit truthful reports. Numerical experiments support these theoretical results and show that they continue to hold in parameter regions outside the sufficient conditions.

This paper is concerned with a class of the long time Kähler-Ricci flow on a compact Kähler manifold. It is shown that the uniform $\mu$-entropy or uniform Sobolev inequality along the normalized Kähler-Ricci flow with semiample canonical bundle. As a consequence, we prove that the scalar curvature of the Kähler metrics along the normalized Kähler-Ricci flow converge to negative Kodaira dimension of the compact Kähler manifold.

In this paper, we establish a general comparison theorem for the order statistics of the edge eigenvectors for generalized Wigner matrices. Consequently, we derive the Gumbel law for the maximal edge eigenvector component and prove the universality of the Gaussian fluctuations of the order statistics in an intermediate regime close to the maximum. In addition, our comparison result also implies a quantitative first order estimate for moderately small order statistics.

This work analyzes revises the interior penalty (IP) discontinuous Galerkin (DG) method imposed in [Chen, G., Qiu, W., \& Xu, L. (2021). Analysis of an interior penalty DG method for the quad-curl problem. IMA Journal of Numerical Analysis, 41(4), 2990-3023.] for the quad-curl problem in a nonconvex polyhedral domain, while introducing a piecewise constant coefficient matrix. We derive two main results: Under minimal regularity assumptions, we prove that the numerical solutions converge strongly to the true solution in the $ H(\text{curl}) \times H^1(\Omega)$ norm. Under slightly higher regularity, we establish the optimal estimate of the convergence rate depending on the regularity of the solution. These two results, serving as a complement to the existing literature, completely answer how the concerned IP DG method performs on quad-curl problems with low regularity.

We study extension of Ulrich bundles from a smooth nondegenerate subvariety $X$ of $\mathbb{P}^n$. If $X$ is a complete intersection of dimension $\geq 2$, we show that the extension is not possible except in the trivial case. For an arbitrary $X$, we characterize when the extension is possible, assuming some condition on the extended vector bundle. As an application, we generalize previous results of L{ó}pez and Zamora. We also give several classes of examples of Ulrich bundles on curves that extend to the ambient projective space.

Let $L =-\Delta +V$ be a Schrödinger operator on $\mathbb{R}^n$, $n \geq 3$, with the potential $V$ being nonnegative and belonging to the reverse Hölder class $RH_q$ for some $q >n/2$. For $0< \alpha <2$, the Lipschitz space $\Lambda_L^\alpha(\mathbb{R}^n)$ adapted to $L$ is defined as the space of all measurable functions $f$ on $\mathbb{R}^n$ such that \[ \|f\|_{\Lambda_L^\alpha}:= \|\rho(\cdot)^{-\alpha}f(\cdot)\|_{L^\infty}+ \sup_{z \in \mathbb{R}^n \backslash \{0\}} \frac{\|f(\cdot + z) + f(\cdot -z) -2 f(\cdot)\|_{L^\infty}}{|z|^\alpha} <\infty, \] where $\rho$ is the critical radius function related to $L$. In this paper, we provide characterizations of $\Lambda^\alpha_L(\mathbb{R}^n)$ in terms of Littlewood-Paley-type decompositions and Carleson measures, for $0< \alpha < 2 -(n /q)$.

In recent work of Dombos, the set of integer partitions of $n$ wherein the parts are either divisible by 4 or congruent to $\pm 1 \pmod{6}$ arose in a natural way. In this work, we will denote the function which counts the number of such partitions of $n$ by $dp(n)$. Using elementary generating function manipulations and classical $q$--series results, we prove several congruences satisfied by $dp(n)$. As an example, we prove that, for all $\alpha \geq 1$ and all $n \geq 0$, \begin{equation*} dp \left( 3^{2\alpha + 1}n + \frac{7 \cdot 9^\alpha + 1}{4} \right) \equiv 0 \pmod{3}. \end{equation*}

The study of principal symmetric ideals (PSIs) in ambient polynomial rings was complicated by the combinatorial instability of minimal generators for ideal powers. We resolve this instability in the two variable case by translating the problem into the arithmetic geometry of symmetric affine plane curves. By working topdown within the Dedekind domain of a symmetric coordinate ring, we establish a precise geometric dictionary for PSIs. We prove that the prime factorization of a PSI is strictly determined by the $S_2$-orbits of its symmetric intersection locus, and that ramification corresponds exactly to tangential intersections, which are detected globally by a novel Symmetric Discriminant ideal. Crucially, we demonstrate that the ideal class of any PSI is a $2$-torsion element in the Ideal Class Group. This establishes that the powers of a PSI exhibit strict periodicity, alternating between being principal and requiring exactly two generators. Finally, we localize this arithmetic obstruction to the axis of symmetry, culminating in a Parity Criterion that determines principality based on intersection multiplicities along the diagonal.

For every integer $g\ge 2$, we determine exactly which integers occur in the greedy 3-sumfree sequence that starts with $1$, $g$, and $g+1$. This gives a direct proof of a conjecture of Bosma, Bruin, Fokkink, Grube, Reuijl, and Tromp. We also obtain an explicit eventual periodic description, including both the preperiod and the repeating block.

Superhedging supermartingales, introduced by the authors in previous work, are non-probabilistic processes defined via subadditive outer integrals that carry a purely financial interpretation in terms of superhedging cost. Building on the Leinert-König theory of non-lattice integration, the present paper establishes several results that are classical in probability theory but whose non-probabilistic proofs require fundamentally new arguments: (i) a tower inequality for the conditional outer integral \overline{\sigma}_j applied at stopping times, reducing to equality when the integrand is conditionally integrable; (ii) three versions of Doob's optional stopping theorem, organised by the class of supermartingale and the range of the stopping times; and (iii) Dubins' upcrossing inequality in both finite- and infinite-time horizons. A key structural result, property (K)-a.e., identifies conditions under which the two superhedging operators \overline{\sigma}_j and \overline{I}_j coincide on non-negative functions, extending the scope of all preceding results to the positive operator \overline{I}_j. None of the proofs invoke classical measure-theoretic tools; in particular, (classical) integrability and measurability are not assumed. The analogues of classical stochastic results acquire a purely financial interpretation and, in this way, gain depth and generality by providing a context that is independent of any a priori probabilistic structure.

The rapid growth of AI workloads is substantially increasing data center electricity demand and carbon emissions, motivating the development of carbon-aware scheduling methods. However, effective scheduling is challenging because renewable generation and AI workloads are subject to forecast errors, while training and inference workloads exhibit heterogeneity in computational characteristics. This paper proposes a contextual robust optimization framework for AI data center operation. The proposed model explicitly captures the heterogeneous computational characteristics of AI training and inference workloads. To deal with renewable generation and workload forecast errors, we develop loss-based uncertainty learning models that directly map contextual features to covariate-dependent uncertainty sets. The resulting contextual joint chance-constrained scheduling problem is reformulated into a tractable robust optimization problem, and a calibration algorithm is developed to provide finite-sample probabilistic feasibility guarantees for multiple joint chance constraints. Numerical experiments based on real-world AI workload traces and renewable generation data show that the proposed method reduces operating costs by an average of 5.57% compared to benchmark methods while maintaining reliable feasibility and strong computational scalability.

Tanaka's formula is a classical identity for Brownian motion, and Tsukada (2018) extended it to Lévy processes not necessarily symmetric. From a potential-theoretic point of view, this formula shows that the invariant function for the process killed upon hitting a singleton can be decomposed into the sum of a martingale part and a local time. In this paper, we generalize this singleton setting and derive a Tanaka-type formula for a compact set $B$. To this end, we introduce the equilibrium measure, defined as the rescaled limit of the $q$-capacity measures, and show that the invariant function for the process killed upon hitting $B$ can be represented as the integral, with respect to the equilibrium measure, of the invariant functions associated with processes killed upon hitting singletons, up to an additive constant called the Robin constant. Moreover, when $B$ is an interval, we obtain explicit representations of the equilibrium measure, the Robin constant, and the martingale part for recurrent stable processes as well as for recurrent spectrally negative Lévy processes. Finally, we discuss how an analogous Tanaka-type formula can also be established for transient Lévy processes.

We study the Dirichlet problem for the $p$-Poisson equation in the metric measure space setting equipped with a doubling measure and supporting a $(p,p)$-Poincaré inequality. We prove the existence of the solutions by using a variational approach. We prove the stability and uniqueness of the solutions, when the space is also geodesic.

We develop a new combinatorial large sieve method for sets with bounded algebraic multiplicities. The method exploits algebraic splitting modulo many small primes: local congruence branching produces many modular collisions, while global bounded-multiplicity hypotheses force these collisions to be rare. As a first application, we prove that every Sidon subset $A\subset\{1^2,\ldots,N^2\}$ satisfies \[ |A| \le N\exp\left( -c\frac{\log N}{\log\log N} \right) \] for some absolute constant $c>0$. This gives the first super-polylogarithmic saving for a classical problem of Alon and Erdős. As a second application, we establish new upper bounds for two grid-distance problems. We show that the largest subset of $[N]^2$ with no repeated distance has size at most $N\exp\left(-c\log N/\log\log N\right)$, giving the first progress in over thirty years on a problem of Erdős and Guy. The same method also gives a super-polylogarithmic saving for subsets of $[N]^2$ with no isosceles triangles, a problem recently popularized by Ellenberg and by the PatternBoost work of Charton, Ellenberg, Wagner, and Williamson. We then develop an entropic version of the method. This gives bounds for $B_2[g]$-sets in the squares and for analogous bounded-multiplicity problems associated with norm forms over arbitrary number fields. Moreover, we prove the first nontrivial bounds for $B_3[g]$-sets in the cubes and $B_4[g]$-sets in the fourth powers.

This work introduces and investigates the function \( \mathcal{V}(G) = \frac{\text{Simp}(G)}{|L(G)|} \), where \( \text{Simp}(G) \) denotes the number of simple subgroups and \( |L(G)| \) the total number of subgroups of a finite group \( G \). The function \( \mathcal{V}(G) \), defined on the interval \( [0,1] \), represents the proportion of simple subgroups relative to the total number of subgroups. It serves as a tool for analyzing structural patterns in finite groups, particularly in p-groups and other families.

The purpose of this note is to study spectral properties of the horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution. We show that the horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold of the Riemannian submersion. We give an application of this interpretation to the asymptotic behavior of the scaled first nonzero eigenvalue of the canonical variations introduced by Berard-Bergery and Bourguignon. Our approach enables us to compare the horizontal Laplacian with the usual Laplacian on a Riemannian covering over the base manifold, and, when the holonomy group is infinite and amenable, we prove a coincidence of the essential spectrum, which strengthen, in our special setup, a result due to Kordyukov in the context of geometric analysis on foliated manifolds.

It is a well-known problem to identify the nontrivial zeros of the Riemann zeta function in terms of an eigenvalue problem. We here find such an eigenvalue problem for second order differential operators on the half-line. In a sense, our analysis pushesthe analysis of the zeta function over to the study of the Jacobi theta function, which may be thought of as the fundamental solution of the heat (or Schrödinger) equation on the unit circle (or the semi-infinite cylinder, if time is added). The eigenvalue problem takes the form $LD u+\alpha Lu=0$, where $L$ and $D$ are first-order differential operators, of which only $L$ involves the theta function. In a formal sense, then, $\alpha$ is an eigenvalue of the twisted operator $-LDL^{-1}$. Based on this formal thinking, we develop the notion of self-adjointness of the pair $(LD,L)$, to adapt the Hilbert-Pólya idea to the spectral problem at hand.

We investigate data-driven finite-volume discretizations of the linear advection equation in one dimension. Neural networks for use as numerical advection schemes are constructed adhering to first principles of numerical analysis, allowing us to examine how normalization, training data, and architectural choices influence stability, accuracy, and shape preservation. (i) We show that reconstruction based solely on cell averages leads to a multi-valued learning problem, explaining limited generalization when training data includes widely different curvature regimes. (ii) Numerical stability and good generalization can be achieved by enforcing semilinearity (Lin and Rood 1998) through local stencil-scale normalization, which ensures invariance under affine transformations of the inputs. (iii) A new data-driven flux limiter is introduced that outperforms the classical 'OSTVD3' (Arora and Roe, 1997) scheme in shape preservation by introducing mild antidiffusion in near-linear regimes, while higher-order reconstruction in non-monotonic regions provides limited benefit. (iv) We show that training on polynomial profiles yields stable, high-order accurate discretizations, with the polynomial degree controlling the formal order of accuracy. Together, these results illustrate how the representational, architectural, and training choices govern the stability and generalization of data-driven finite-volume schemes for linear advection.

In this paper, we study the Schrödinger-type operator associated with geometric stable processes on $\mathbb{R}^{d}$, especially the differentiability of spectral function. Let $\mathcal{H}$ be the generator of the geometric stable process and $\mu$ a smooth measure on $\mathbb{R}^{d}$. Then the spectral function $C(\theta)$ is defined as $C(\theta) = -\inf \sigma(-\mathcal{H} - \theta \mu)$, where $\sigma(\mathcal{A})$ denotes the spectrum of $\mathcal{A}$ and $\theta$ is a real parameter. Since the geometric stable process exhibits severe local singularities in its Lévy measure, its transition semigroup lacks ultracontractivity, which invalidates classical methods for proving the differentiability. To overcome this obstacle, we use the compact embedding of the extended Dirichlet space into $L^2(\mu)$. As a primary application of this differentiability, we establish a large deviation principle for a positive continuous additive functional associated with the smooth measure $\mu$.

In this article, we study the image of the Hitchin morphism for some classical groups over an algebraic surface. The Hitchin morphism is a map from the moduli stack of $G$-Higgs bundles $\mathscr{M}_{X,G}$ to the Hitchin base $\mathscr{A}_{X,G}$, where $X$ is a smooth projective variety. In general, this morphism is not surjective when the dimension of $X$ is greater than one. Chen and Ng{ô} showed that the Hitchin morphism factors through a closed subscheme $\mathscr{B}_{X,G}$ of the Hitchin base, which is called the spectral base. They conjectured that the image of the Hitchin morphism is exactly the spectral base. When $X$ is a smooth projective surface, we prove that this conjecture holds for the special linear algebraic group of odd rank. We also confirm this conjecture for the classical groups ${\rm SL}_n$ and ${\rm Sp}_{2n}$ when $X$ is a product of smooth curves.

We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the positive quadrant with nonsmooth max-type reward \(G(x_1,x_2)=x_1\vee \alpha x_2\). The paper develops a conditional measure-theoretic framework for the associated reflected obstacle problem. The main innovation is to show that the stopping gain \(\Gamma=c+rG-\mathcal LG\) is a signed measure, not a function: the kink of \(G\) generates an explicit negative surface measure on \(\Delta=\{x_1=\alpha x_2\}\). We then prove that the correct potential representation uses the resolvent of the reflected diffusion killed on first entry into the stopping set, rather than the unrestricted reflected resolvent. Under explicit monotonicity, regularity, and measure-superharmonicity assumptions, we derive an epigraph representation, a continuation-side boundary-trace condition, and a candidate verification theorem. The framework clarifies hidden regularity and uniqueness assumptions in multidimensional nonsmooth optimal stopping.

Information-Geometric Optimization (IGO) provides a unified framework for black-box optimization by interpreting the adaptation of a search distribution as a natural gradient update. Despite its conceptual importance, the convergence theory of IGO remains limited: most existing results concern continuous-time idealizations such as the IGO flow, rather than discrete-time updates with non-infinitesimal learning rates. In this paper, we study discrete-time IGO in continuous spaces, formulated as natural gradient updates in the expectation-parameter coordinates of an exponential family. In particular, we analyze IGO over the multivariate Gaussian family on strongly convex quadratic objective functions. Our analysis covers a setting that simultaneously incorporates full covariance adaptation, a fixed positive learning rate, and quantile-based weights. In this setting, we prove that the covariance matrix converges to the zero matrix. We further show that the mean vector converges to the global optimum, provided that the condition number of the appropriately scaled covariance matrix is bounded at sufficiently frequent iterations. These results advance the convergence theory of IGO and help bridge the gap between the mathematical theory of IGO and practical covariance-adaptive search methods such as CMA-ES.

The Fomin local rules for Schensted column insertion can be seen as a two-lane box-ball system, in which a carrier moves particles forward or laterally. Running such two-lane dynamics in parallel on a periodic lattice gives rise to a two-dimensional generalization of the box-ball system, which we call the \emph{skew column RSK dynamics}. Equivalently, this is a deterministic dynamics on pairs of skew semistandard Young tableaux $(P_t,Q_t)_{t \in \mathbb{Z}}$. We prove that this dynamics exhibits solitonic behavior and construct an explicit bijection $(P,Q) \leftrightarrow (H_1,H_2,\kappa,\nu)$ that linearizes the time evolution. The resulting coordinates consist of two horizontally weak tableaux $H_1,H_2$ recording the asymptotic soliton data, integer riggings $\kappa$, and a weakly decreasing sequence of integers $\nu$. A key feature of the construction is an explicit projection from the skew column RSK dynamics to the classical box-ball system; under this projection, the rigging $\kappa$ is precisely the Kerov--Kirillov--Reshetikhin rigging of the associated box-ball configuration. Our proof uses two commuting affine crystal structures on pairs of skew tableaux and a novel connectivity theorem for distinguished subgraphs of tensor products of Kirillov--Reshetikhin crystals. We also derive Greene-type formulas for the soliton lengths in terms of last-passage percolation on the associated cylindrical environment. Finally, by taking generating functions in the linearizing coordinates, we obtain bijective proofs of Cauchy and Kawanaka--Littlewood-type identities for transformed Hall--Littlewood polynomials.

An upper homogeneous (upho) poset is a poset whose every principal filter is isomorphic to the whole poset. Fu--Peng--Zhang conjectured that every finitary upho poset admits a compatible left-cancellative, invertible-free monoid structure whose left-divisibility order coincides with the given order. We disprove this conjecture. For every vertex-transitive graph $G$, we construct a finitary upho poset $P(G,v_0)$ from walks starting at a fixed vertex $v_0$. Applying this construction to the Petersen graph, we show that multiplicability of $P(G,v_0)$ would force the automorphism group of $G$ to contain a regular subgroup. This would imply that $G$ is a Cayley graph, contradicting the fact that the Petersen graph is not Cayley. Hence $P(G,v_0)$ is a non-multiplicable finitary upho poset. We also show that the analogous poset associated with the line graph of the Petersen graph is multiplicable, demonstrating that non-Cayleyness of the underlying graph alone does not determine multiplicability.

Linear-multi-parametric optimization problems are a widely studied class of optimization problems. The objective function in such a problem is affine linear dependent on a parameter vector, and the goal is to compute a set of solutions that contains an optimal solution for every fixed parameter vector. However, this is known to be computationally challenging: The underlying non-parametric problem might be NP-hard, and, in addition, optimal solution sets might have exponential cardinality. Parametric approximation aims at providing polynomial-time algorithms that overcome these challenges. Instead of computing an optimal solution set, the goal is to compute an approximation set that contains only an approximate solution for every fixed parameter vector. Several new parametric approximation algorithms have been developed in recent literature. However, all of these share a common set of assumptions, which limits the class of parametric optimization problems that can be approximated. Namely, they do not allow negative parameter dependencies and have their parameter sets fixed to the positive orthant. We present a new adaptive approximation (and, also, exact) algorithm that can be applied to a wider class of linear-multi-parametric optimization problems. Our algorithm builds upon existing algorithms from both the fields of parametric and multi-objective optimization and generalizes these algorithms. In addition, we provide structural results for the transformation of parameter sets, and demonstrate that, for linear-multi-parametric maximization problems, the assumption of non-negative optimal objective values over the whole parameter set is not sufficient to ensure approximability.

In this paper, we introduce the power mean transform $P_{\lambda}(T)$ of an operator $T$ on a Hilbert space, which is a convex combination of some classical operator transforms such as the mean transform $M(T)$, the Aluthge transform $\Delta(T)$, and the Duggal transform $T^D$. In particular, when $T$ is invertible, this transform coincides with the induced Aluthge transform $\Delta_{\mathsf{m}_{f}}(T)$ recently defined by Yamazaki \cite{yamazaki-laa-2021} with $f(x)=(\lambda+(1-\lambda)\sqrt{x})^2$ for $x\in(0,\infty)$ and $\lambda\in(0,1)$. We study basic properties of $P_{\lambda}(T)$ including its spectrum, norm and numerical radius. Moreover, we use the power mean transform to give new characterizations of normal, quasinormal and binormal operators. The questions of Golla et al. \cite{yamazaki-laa-2023} and some new results on the Duggal transform are also mentioned. We obtain a result close to the recent one of Osaka and Yamazaki \cite[Theorem 3.3]{yamazaki-tams-2025} on the iteration of the induced Aluthge transform for centered operators. Finally, we describe the form of bijective maps commuting with the power mean transform of the product of matrices.

In this paper, we provide a complete characterization of the weighted Hardy inequalities involving the supremum operator, restricted to the cone of non-increasing sequences, for all positive parameters. We reduce such inequalities to equivalent ones on the cone of non-negative sequences. The latter setting provides a broader framework for analysis and significantly expands the range of proofs that can be established.

We study spectral intertwining operators between spectral Eisenstein series $\operatorname{Eis}_{P^\vee}$, $\operatorname{Eis}_{Q^\vee}$ for two parabolic subgroups $P, Q$ of a $p$-adic reductive group $G$ with the same Levi subgroup $M$, inspired by the analogy with the classical intertwining operators between parabolic induced representations of $p$-adic reductive groups. In particular, we construct the normalized (canonical) intertwining operator that satisfies the transitivity and an unnormalized intertwining operator that is adjoint to a rational section of an analog of the Bruhat-Mackey's filtration. Moreover, the normalized and unnormalized intertwining operators differ by a ratio of L-functions, analogously to the Langlands conjecture about classical ones up to units. Finally we prove that the spectral intertwining operators correspond to classical ones up to units under any conjectural categorical local Langlands correspondence.

In this study, the set of orthomorphisms of a finite group G of order 3M with a normal subgroup of order M has been partitioned into classes. Additionally, a number of relationships between the orders of these classes have been established. Ultimately, this approach has been used to theoretically determine all 3840 orthomorphisms of alternating groups on four symbols by just counting 30 orthomorphisms, a problem that had been unsolved since 1992.

In this paper, we develop a framework based on differential forms that enables us to deal with the Hall problem on domains in any dimension $n\ge2$. In the case of smooth bounded domains of $R^n$, we prove local existence of mild solutions in subcritical spaces.

Many fractals $G$ admit a harmonic immersion into $\R^n$, i.e. an immersion which minimises a natural energy under fixed boundary conditions; we look for harmonic immersions of the Sierpinski gasket into the hyperbolic plane. We show that, given any three points $\tilde A$, $\tilde B$, $\tilde C$ in the hyperbolic plane there is a harmonic map bringing the three points $A$, $B$, $C$ of the boundary of the gasket to $\tilde A$, $\tilde B$, $\tilde C$ respectively. Moreover, if the points $\tilde A$, $\tilde B$, $\tilde C$ are sufficiently close in the hyperbolic distance, then the harmonic map is unique and depends differentiably on $\tilde A$, $\tilde B$, $\tilde C$. Lastly, we show that, if the harmonic map $\phi$ is injective, then it brings geodesics of the gasket $G$ into geodesics of $\phi(G)$.

For a manifold $\overline{M}$ with boundary $\partial M$ and interior $M$, we introduce and study a weakening of the concept of projective compactness for torsion-free linear connections on $M$, which we call projective pre-compactness. Via the Levi-Civita connection, this concept applies to pseudo-Riemannian metrics on $M$. This is motivated by scattering theory and general relativity (GR), via asymptotic forms of metrics used in these areas. In the general setting of a projectively pre-compact connection $\nabla$ we show that, assuming weak asymptotic conditions on the Ricci curvature, there is an induced projective structure on the boundary. Under a slightly stronger condition on Ricci, we show that the standard tractor bundle and its normal tractor connection arise naturally on this boundary structure. The key ingredient to this is that $\nabla$ admits a smooth extension to the boundary as a linear connection on the tensor product of Melrose's b-tangent bundle with a density bundle, which then restricts to the boundary tractor bundle. A projectively pre-compact pseudo-Riemannian metric (satisfying the conditions on the Ricci curvature) is then shown to induce a holonomy reduction of the boundary projective structure to an indefinite orthogonal group. This endows the boundary with a decomposition into so-called curved orbits, which are either open or embedded hypersurfaces, representing space-like, time-like and light-like infinities in a GR context. We introduce and study a new asymptotic form for such metrics which is available near any boundary point and relate it to an asymptotic form used in general relativity, which is only available near boundary points in the open curved orbits. We show that, in that region, projective pre-compactness essentially is equivalent to the asymptotic form from GR, and projective compactness is equivalent to vanishing of the mass aspect.

Menger's theorem is one of the most fundamental results in graph theory. It states that if a graph $G$ does not contain $k$ disjoint paths between two given sets $X$ and $Y$ of vertices in $G$, then there is a set of at most $k-1$ vertices that intersects every path between $X$ and $Y$. Nguyen, Scott, and Seymour gave a counterexample to the conjectured natural coarse variant in which the paths are required to be pairwise at distance at least $d$, and, conversely, there is a set of at most $k-1$ bounded-radius balls intersecting every path between $X$ and $Y$. In other words, the coarse Menger property does not hold in general. We prove that graphs whose cycles space is generated by cycles of bounded length do have the coarse Menger property. As a corollary, we show that many natural graphs and geodesic metric spaces have the coarse Menger property. These include hyperbolic graphs, Cayley graphs of finitely presented groups, planar graphs with bounded face size, and complete Riemannian planes.

The classical Frenet frame is defined by a concrete construction from the tangent, principal normal, and binormal vectors of a regular space curve. However, this construction breaks down at singular points and at points where the curvature vanishes. Motivated by this observation, we reconsider the Frenet frame from an axiomatic viewpoint and identify the fundamental properties that characterize it independently of its classical construction. Based on the theory of frontals on the unit sphere and Legendre duality, we introduce a generalized Frenet frame for singular space curves. Furthermore, we introduce the notion of a frame sequence, which gives rise to an integer-indexed family of Frenet frames together with the corresponding curvatures and torsions. This viewpoint provides a unified framework encompassing both the Frenet and Bishop frames of space curves and the evolute-involute correspondence for spherical frontals. Moreover, explicit recursive formulas are derived, revealing that the curvatures and torsions at each level encode, respectively, the magnitude and rotational behavior of the invariants arising at the preceding level.

In response to the growing demand for sustainable and efficient urban deliveries, this study introduces a two-echelon vehicle sharing and repositioning problem for containerized delivery operations within a hyperconnected urban logistics system. We leverage a Physical Internet (PI)-enabled three-tier logistic hub network, comprising gateway, local, and access hubs, to facilitate efficient flows. By adopting containerized delivery, vehicles can rapidly swap standardized modular containers at hubs to reduce handling time. Moreover, inspired by the PI concept of open resource sharing, we determine optimal service routes within a two-echelon structure that jointly utilizes heterogeneous vehicle fleets and enables dynamic vehicle relocation across hubs. We formulate this problem as a multi-period integer program that integrates path-based service vehicle planning with arc-based container routing. To address real-world large-scale instances, we propose a decomposition-based heuristic with capacity-aware flow assignment, which partitions the problem into subproblems structured by echelons and regions. A case study on the Atlanta metropolitan area demonstrates the effectiveness of the proposed model and solution approach. Experimental results show that the two-echelon hyperconnected delivery system reduces CO2-eq emissions by 45.0% and total costs by 16.8% at full market share compared to a traditional single-echelon alternative, while enabling vehicle repositioning further lowers costs by up to 17.7%.

We investigate the impact of large synaptic delays on the emergence of periodic dynamics in inhibitory neuronal networks, within the framework of the NNLIF model. Inspired by the work of [11] where the notion of pseudo-equilibria was introduced and developed, and by our earlier analysis in [14], we show that, as the delay tends to infinity, solutions of sufficitently inhibitory networks oscillate between distinct pseudo-equilibria over any finite time interval. Employing the Doeblin-Harris method, we rigorously establish a local convergence in the Ces{à}ro mean toward a limit function determined solely by these pseudo-equilibria.

We study the vanishing viscosity limit for the incompressible Navier-Stokes equations with anisotropic viscosity in bounded domains, analyzing certain classes of symmetric flows: plane parallel, pipe parallel and circularly symmetric. By anisotropic viscosity, it is meant here that the viscosity coefficient in the direction normal to the wall is different than that in the direction tangential to the wall. Using boundary layer theory and semigroup techniques, we establish the validity of the vanishing viscosity limit in the energy norm, that is, in $L^2$ in space uniformly in time, for all three classes of flows, with explicit convergence rates. We further obtain higher-order estimates under suitable assumptions on the anisotropic viscosity coefficients. In particular, we consider both the case in which the tangential viscosity coefficient goes to zero faster than the normal one and, conversely, the case when the normal coefficient vanishes faster then the tangential one. Our results extend previous works on isotropic viscosity and provide new examples where the vanishing viscosity limit can be rigorously justified in the anisotropic setting.

We show that the topological structures, such as loops, voids, and higher-dimensional holes of unknown objects (of flow of an object in space-time) can be recovered from noisy and indirect measurements. More precisely, we describe how the part of the persistent homology of a space can be determined from a noise-prone and discretized model space when there is a quasi-isometry between the original space and the space modeling indirect measurements. The result not only guarantees the existence of the structures but also provides size bounds for them. The structure is studied using persistent homology, and the results assume the existence of a quasi-isometry between a model space and the noisy measurements. We explore imaging problems, particularly X-ray imaging and EIT, that are well-suited to this framework.

In kinetic theory, interactions between particles are typically assumed to be "all-to-all", meaning that any pair of randomly selected particles may, in principle, interact. This assumption originates from the theory of colliding gas molecules; however, it may be less appropriate for describing other forms of interaction, such as social interactions. These are more naturally characterised as "some-to-some", reflecting the existence of preferential connections between agents. In this paper, we consider homogeneous Boltzmann-type equations on finite graphs that model such networks of preferential interactions, and we rigorously derive their dense graph limit as the number of agents tends to infinity. We also investigate the long-time behaviour of the limiting equation in the case of linear pairwise interactions, characterising the emergent equilibrium distributions and relating them to their counterparts in the classical "all-to-all" setting.

We study variational inequalities governed by a point-to-set maximal monotone operator in a real Hilbert space and constrained by a convex inequality \(C=\{x\in\Hi:c(x)\le0\}\), where the defining function \(c\) is continuous and not necessarily differentiable. The proposed method uses only projections onto intersections of half-spaces and avoids the metric projection onto \(C\). Feasibility is handled by subgradient cuts and, when a trial operator point is infeasible, by a Slater correction based on a fixed strictly feasible point. The variational inequality is represented by Minty-type separating half-spaces generated at feasible graph points of the operator, and a Haugazeau half-space is added to obtain best-approximation convergence. Under a Slater-corrected feasible-separation condition, together with explicit exact, approximate and finite-candidate oracle realisations, the whole sequence converges strongly to \(P_{S^*}(x^0)\), the projection of the initial point onto the solution set. We also derive best-iterate \(O(N^{-1/2})\) residual estimates for the step residual, feasibility violation and Minty gap. The analysis is stated directly for point-to-set maximal monotone operators, while the concrete oracle realisations include finite-dimensional single-valued models. We record the consequences of strong monotonicity in the point-to-set setting and provide numerical comparisons on nonsmooth and large-scale constraints, including maxima of convex quadratics, a discretised optimal-control problem, mixed-norm sparse recovery, a Cournot--Nash capacity equilibrium, and a genuine point-to-set \(\ell_1\)-subdifferential example.

We study a continuous-time robust Bayesian portfolio optimization problem under drift uncertainty of risky assets. The investor learns unknown asset drifts through Bayesian filtering while considering uncertainty around posterior estimates via discrepancy-based ambiguity sets, including Wasserstein and $L^p$ distances. To address the resulting time inconsistency, we introduce a feedback-type ambiguity framework that reformulates ambiguity conditionally on observable states. This leads to a modified Hamilton--Jacobi--Bellman--Isaacs (HJBI) equation characterizing the value function and the optimal strategy. For a semi-explicit solution example, we use the exponential utility to derive a reduced semilinear parabolic PDE and establish existence of classical solutions via a verification theorem.

Motivated by the recent constructions of Golay Complementary Array (GCA), we extend the "Three-stage construction" proposed by Fiedler and Jedwab to a more general form. All the known 2-phase Golay Complementary Sequences (GCSs) of length $2^a10^b26^c,a,b,c\ge 0$ and 4-phase GCSs of length $2^{a+u}3^b5^c11^d13^e$ where $a,b,c,d,e,u\ge0,b+c+d+e\le a+2u+1,u\le c+e$ can be constructed in a uniform method from five 2-phase seeds of length 2,10,10,20,26 and five 4-phase seeds of length 3,5,8,11,13. Furthermore, additional 2-phase and 4-phase GCSs of many lengths are produced and a closed form for numbers of 2-phase and 4-phase GCSs is given.

This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale (KSS) random field on the sphere, and applies it to two problems: Spiked Tensor PCA and the landscape of the spherical $k$-spin model. For Tensor PCA, we study the non-asymptotic statistical limits of estimating a rank-$R$ symmetric signal tensor of order~$k\ge 3$ and dimension~$d\ge 3$ from a single Gaussian observation at signal-to-noise ratio~$\lambda$, through the \emph{profile maximum likelihood estimator}, the MLE restricted to normalized rank-$R$ tensors of coherence at least~$\kappa$. Our analysis uses a single reduction: a deterministic geometric inequality (the Tube Method) and a rank-reduction step bound the estimation error by the supremum of the canonical KSS field, which the Kac--Rice formula turns into a Gaussian integral against the expected absolute characteristic polynomial of a shifted Gaussian Orthogonal Ensemble, controlled in turn by the four explicit tail bounds of our hierarchy (three from a Mehta--Fyodorov representation, one from a Ben Arous--Dembo--Guionnet large deviation). The same reduction yields two results, each with explicit constants. For estimation, a finite-$(k,d)$ error bound recovers the asymptotically optimal rate~$\sqrt{d\log k}$ of Perry, Wein and Bandeira, with explicit dependence on the rank~$R$ and the coherence~$\kappa$. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical $k$-spin Hamiltonian recovers the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit.

Learning kernels in operators from data is a fundamental task that arises in nonlocal continuum mechanics, operator learning, and interacting particle systems. A central question is how the roughness of input data impacts the accuracy of kernel recovery. We quantify the roughness of the input data via its spectral decay exponent and analyze how it determines the degree of ill-posedness of the inverse problem and, consequently, the convergence rates of the Tikhonov-regularized estimator in the small-noise limit. Within this framework, we identify a phase transition between an under-rough regime, in which rougher data improves recovery, and an over-rough regime, in which further roughening leads to slower rates. These theoretical findings are supported by numerical experiments ranging from idealized settings to more realistic configurations, with quantitative agreement in the former and broad consistency of the main trends in the latter.

We study the counting problem for non-uniform intersecting families in extremal set theory. Let $J(n,k)$ denote the number of intersecting families $\mathcal{F}\subset 2^{[n]}$ such that every member of $\mathcal{F}$ has size at most $k$. Extending recent counting results for uniform intersecting families, we prove that for $n\ge 2k+2+2\sqrt{k \log k}$ and $k \rightarrow +\infty$, \[ J(n,k) =(n+o(1)) 2^{\sum_{i=1}^{k} \binom{n-1}{i-1}}. \] This result reveals that typical non-uniform intersecting families of bounded size are trivial, i.e., almost all such families share a common fixed element.

We develop a semiclassical theory of tunnelling of Dirac fermions through an n-p-n junction in monolayer graphene subjected to a perpendicular magnetic field. Electrostatic and magnetic fields are assumed to be smooth functions of a single spatial coordinate, supported on a finite interval, vanishing outside it, and thus ensuring asymptotically free states. In contrast to earlier studies restricted to constant magnetic fields and exactly solvable electrostatic potential profiles, we consider a general electrostatic potential forming an n-p-n junction and an arbitrary magnetic field, and formulate the corresponding scattering problem. Within the semiclassical approximation, and under an additional assumption on the incidence angle, the problem reduces to a connection problem for a pair of degeneracy points, treated using results from our earlier work. We obtain explicit expressions for the reflection and transmission coefficients, including their phases, as functions of energy and incidence angle. Furthermore, we derive semiclassical conditions for Fabry-Pérot resonances and "magic" angles, and analyse the resulting interference pattern. Numerical results demonstrate the angular dependence of transmission induced by the magnetic field.

We study monoids in an infinite-dimensional setting that are invariant under the action of the infinite symmetric group Sym. Our main result establishes a local--global principle characterizing equivariant finite generation for arbitrary Sym-invariant monoids, extending earlier results that required additional assumptions. We further analyze local--global phenomena for other fundamental properties, including positivity, normality, seminormality, and simplicity. In addition, we obtain structural results for symmetric monoids, including characterizations of positivity and non-positivity, a description of their groups of units, and explicit formulas for the ranks of local symmetric monoids and stabilizing Sym-invariant chains.

We develop an ABP approach to Sobolev and Michael-Simon type inequalities under volume noncollapsing assumptions. The main new observation is a refinement of Brendle's contact-set argument: the ABP image contains the full geodesic ball centered at the minimum point of the Neumann potential, with radius equal to the ABP parameter. This allows one to use lower bounds for the volumes of geodesic balls, either at a fixed scale or under prescribed volume-growth assumptions, rather than positive asymptotic volume ratio. The central application is a Michael-Simon type inequality for immersed submanifolds of ambient manifolds with nonnegative sectional curvature and volume noncollapsing. The resulting inequality contains a lower-order term determined by the noncollapsing scale and applies to submanifolds with controlled mean curvature. In the intrinsic case, the same method gives an ABP proof of Varopoulos' $L^{1}$-Sobolev inequality with lower-order term, identifying the optimal constant in front of the gradient term, as well as explicit lower bounds for the isoperimetric profile in terms of lower bounds on the volumes of geodesic balls. Further geometric applications include Topping-type diameter estimates for submanifolds involving the $L^{n-1}$-norm of the mean curvature and various heat kernel and spectral estimates in the minimal case.

Blind recognition of polar-coded transmissions is an important task in non-cooperative wireless forensics and security-oriented signal analysis. When the code length is known or has been estimated, recovering the frozen/information bit-position pattern is a key step in identifying the underlying polar-code structure and enabling subsequent information recovery from intercepted observations. In this paper, blind recognition of polar codes is investigated from a hypothesis-testing perspective under the successive cancellation (SC)-based synthetic bit-channel representation. First, under an ideal SC-consistent condition, we formulate position-wise recognition as a binary hypothesis test between frozen-position and information-position models, which provides a theoretical benchmark for analyzing their intrinsic distinguishability. Second, we show that the adopted soft recognition metric admits an exact shifted log-likelihood-ratio interpretation. This justifies ln 2 as the neutral threshold under equal priors and costs, while unequal priors or costs lead to the corresponding Bayesian threshold shift. Third, under the ideal SC-consistent model and this neutral setting, we derive upper and lower bounds on the position-wise and sequence-level recognition error probabilities with multiple independent observations. The resulting overlap coefficient is further related to the classical Bhattacharyya parameter, establishing an interpretable link between blind-recognition difficulty and polar synthetic-channel reliability. Simulation results show that the derived bounds characterize the recognition performance under the ideal SC-consistent model and capture the effects of code length, the number of intercepted observations, and SNR. Further paired comparisons in the tested settings indicate that the SC-consistent recursion provides a good sequence-level match to the realistic SC-recursive procedure.

Lossless data compression remains central to computer science, with direct impact on storage, communication bandwidth, computational cost, and energy consumption. It is also closely related to Algorithmic Information Theory, where compressibility provides an operational measure of structure and non-randomness. This paper presents the 2026 Algorithmic Information Theory Data Compression Challenge, a benchmark for evaluating general-purpose lossless compressors under realistic constraints. Submissions were encouraged to use arithmetic or range coding, limited to at most 8 GB of memory, and required to include a decompressor no larger than 1 MB. The benchmark comprised sixteen heterogeneous files, split into public training and hidden testing datasets. In total, 117 valid submitted compressors were evaluated alongside established reference compressors using compression ratio, compression and decompression time, Weissman score, and Pareto-frontier analysis. The results show that performance depends strongly on the optimization criterion: fast compressors achieved the best speed-oriented scores, whereas modelling-intensive compressors produced smaller outputs at higher computational cost. A Normalized Compression Distance analysis further revealed clusters of related submissions and distinguished incremental variants from more independent implementations. Selected submissions were described for their methodological novelty or competitive performance and further tested on four large external datasets, where several achieved competitive or superior results relative to established compressors. Overall, the challenge confirms the importance of probabilistic modelling, hidden testing, and external datasets for assessing compression performance and generalization. Benchmark resources, leaderboard data, binaries, and selected source code are publicly available at this https URL.

Let (R,m) be a Noetherian local ring and M1,...,Mn finitely generated R-modules. Set M is the direct sum of Mi. The main purpose of this paper is to extend the results on the sequential Cohen-Macaulayness of the direct sum of modules (Taniguchi et al, 2018), the sequential generalized Cohen-Macaulayness of the direct sum of modules (Cuong and Nhan, 2003) We first describe the largest submodule of M of dimension less than dimM by using the largest submodule so f its component modules. Then we give a necessary and sufficient condition for M being 1-Cohen-Macaulay. The purpose of this paper is to characterize the sequential 1-Cohen-Macaulayness of the direct sum M. We show that M is sequentially 1-Cohen-Macaulay if and only if Mi is sequentially 1-Cohen-Macaulay for all i <=n. We provide an example to clarify the results. We employ inductive methods as well as the dimension filtration of a finitely generated module.

We study duality for monoids in an infinite-dimensional setting that are invariant under the action of the infinite symmetric group Sym. Our main result is an equivariant Minkowski--Weyl theorem for monoids. More precisely, we analyze the evolution of dual monoids along stabilizing Sym-invariant chains and describe the eventual behavior of their equivariant Hilbert bases. In addition, we develop a systematic study of structural properties of dual symmetric monoids, including a characterization of the duals of positive and non-positive monoids.

We give a constructive proof for existence of absolutely continuous curves connecting each pair $\mu,\nu \in \mathrm{PL}_\infty^p(\mathbb{R}^n)$, for every $1\leq p\leq \infty$, where $(\mathrm{PL}_\infty^p(\mathbb{R}^n),\mathfrak{d}_\infty^p)$ is the complete metric space of absolutely continuous measures with density in $L^p(\mathbb{R}^n)$ and bounded support introduced in [1].

The Hyperfibonacci sequence of the $r$th generation is defined recursively as a generalization of Fibonacci numbers, where each term is obtained by summing the terms of the Hyperfibonacci sequence of the preceding generation. We introduce the transformation matrix for Hyperfibonacci numbers, which enables us to determine the next term in a given generation. We explore the algebraic structure of that matrix, and its power of $n$, similarity transformations between these matrices and their Jordan canonical forms. Finally, we analyze the powers of these matrices using their Jordan forms, obtaining compact and elegant formulas for expressing $r$-generation Hyperfibonacci numbers in terms of Fibonacci numbers.

This paper considers sequential gray-box optimization where the objective function is given as the composition of a loss function and a parametric model. Crucially, the parameters of the model are unknown and need to be iteratively estimated from noisy observations of the model outputs. This problem setup generalizes the parametric black-box optimization problem known as (contextual) stochastic linear bandit. To address the sequential gray-box optimization problem, we propose a structure-exploiting method that leverages known problem structure given in terms of the loss function and an a priori set of admissible parameters. The method is based on the principle of optimism in the face of uncertainty and trades off exploration and exploitation by minimizing a lower confidence bound on the true objective function. We provide a detailed regret analysis of the novel method, improving on state-of-the-art results for the special case of linear stochastic bandits due to the use of a recently published bound for the parameter confidence sets arising in multi-output linear least-squares estimation. Numerical examples illustrate the superior performance of structure-exploiting methods compared to structure-agnostic approaches.

We prove Sobolev interpolation inequalities on extension domains that have a form reminiscent of the corresponding whole-space inequalities. This form is crucial in certain applications, which we discuss as well. The technical key ingredient is the notion of a Lebesgue $W^{1,p}$-extension domain, which we introduce here, and our proof that, for $1<p<\infty$, any $W^{1,p}$-extension domain is a Lebesgue $W^{1,p}$-extension domain.

Let $H$ be a reproducing kernel Hilbert space over the unit disk $\mathbb{D}$, where analytic monomials span a dense subset. Given $\mathcal{N} \subseteq\mathbb{Z}_+$ and $\Lambda \subseteq \mathbb{D}$ we say that $(\Lambda,\mathcal{N})$ is a uniqueness pair for $H$ if $\Lambda$ is a uniqueness set for the subspace of $H$ spanned by $\{z^n:\;n\in\mathcal{N}\}$. We examine uniqueness pairs in the Dirichlet-type spaces $\mathbb{D}_\alpha$, $0\leq\alpha\leq1$. We prove two complementary results. First, if $\mathcal{N}$ contains sufficiently long finite arithmetic progressions with fixed gap size, then no sequence $\Lambda$ tending sufficiently rapidly to the boundary forms a uniqueness pair with $\mathcal{N}$. Second, if $\mathcal{N}$ satisfies a suitable arithmetic sparsity condition then one can construct uniqueness pairs $(\Lambda,\mathcal{N})$ with the points of $\Lambda$ tending to the boundary arbitrarily fast.

In this paper, we study the information-theoretic characterization of simultaneous signalling and control over channels modeled by partially observable Markov decision processes (POMDPs). The problem is formulated as an optimization over randomized control strategies that maximize the directed information from actions to observations, subject to an average-cost constraint. We derive a novel dynamic programming framework in which the state is defined on the space of conditional probability distributions, leading to a high-level ``meta'' dynamic program. Specifically, we show that two coupled information states, namely, the posterior distribution of the system state and a distribution over such posteriors, satisfy Markov recursions and provide sufficient statistics for optimal control. This structure enables the decomposition of optimal strategies into separated randomized policies that depend only on these information states. Our results establish necessary and sufficient conditions for optimality and unify classical stochastic control and information-theoretic formulations. In particular, we show that in the absence of signalling, the proposed framework reduces to the standard dynamic programming equations for POMDPs. The developed approach provides a principled foundation for analyzing and designing control systems with intrinsic information constraints.

We explore the numerical solution of large-scale non-autonomous Differential Riccati Equations (DREs). While we assume to discretize the differential operator using a Backward Differentiation Formula (BDF) of order s, we solve the generalized Algebraic Riccati Equation (gARE) resulting at each time step by different state-of-the-art methods. In particular, we compare the performance of the inexact Newton- Kleinman method with line search and the low-rank RADI iteration, considering for both methods two different initialization strategies: zero initialization and warm-start. A comprehensive panel of numerical results illustrate the potential and limitations of these methods when employed within a numerical pipeline for the solution of DREs, rather than for the isolated solution of a single gARE, as commonly considered in the existing literature.

The problem of computing the volume of tubes in riemannian manifolds goes back to Weyl and Hotelling. Here we find explicit Taylor series for the volume of a tube in a Lie group equipped with a bi-invariant metric. The coefficients are smooth valuations, given by the convolution powers of the surface area valuation. We show that the tube coefficients can be naturally described as the unique valuations given by universal formulas through the formalism of differential graded Lie and Gerstenhaber algebras; in fact, they are generated by the gauge action on the Maurer--Cartan cone in the free differential graded Lie algebra on one generator. Moreover, we introduce a new convolution product on the corresponding free Gerstenhaber algebra which is compatible with the convolution of valuations and differential forms. To complete the picture, we show that a Lie group -- not necessarily connected -- admits a smooth bi-invariant valuation, beyond the Euler characteristic and the Haar measure, if and only if it admits a bi-invariant riemannian metric.

We study the optimal placement and tuning of a small number of pointwise viscous dampers for a vibrating string. Starting from a finite element discretization of the damped wave equation, the system is transformed into a first-order phase-space formulation, which enables a unified Lyapunov trace framework. Three optimization criteria are considered: average total energy, average total displacement, and energy for a fixed initial state. For all criteria, explicit gradient formulas with respect to damper positions and viscosities are derived, requiring only one primal and one dual Lyapunov solve. Due to the strong non-convexity of the problem, a simple heuristic based on an explicit single-damper formula is proposed to generate effective initial guesses. Numerical examples illustrate the influence of spectral selection and discretization on the optimal damping configuration.

Hypergraph polytopes (or nestohedra) form a broad class of polytopes obtained by truncating faces of a simplex according to a hypergraph. In earlier work, the authors constructed q-tridendriform algebras on the set of faces of certain families of hypergraph polytopes, including associahedra and permutohedra. The well-definedness of these structures relied on a connectedness property on the hypergraphs involved, called strictness. Nevertheless, notable examples of hypergraph polytopes such as cyclohedra fell outside this setting. We introduce a new connectedness condition, called anti-strictness, which goes opposite to strictness and captures a different class of hypergraph polytopes, including associahedra, permutohedra and cyclohedra. Our main result produces natural (-1)-tridendriform algebras in the anti-strict framework, which match previously introduced tridendriform algebras in the overlap of the two frameworks, thereby extending the range of hypergraph polytopes admitting such algebraic structures.

A dynamical system is said to be surjunctive if every injective endomorphism of the system is surjective and it is said to be injunctive if every surjective endomorphism is injective. An endomorphism of a dynamical system is called pre-injective if its restriction to every homoclinicity class of the phase space is injective. One says that a dynamical system has the Moore property if every surjective endomorphism of the system is pre-injective and that it has the Myhill property if every pre-injective endomorphism is surjective. We give characterisations of surjunctivity and injunctivity for $\Z$-subshifts of finite type in terms of their irreducible components and their Cantor-Bendixson decomposition. We also prove that a $\Z$-subshift of finite type is surjunctive if and only if it has the Moore property and that every injunctive $\Z$-subshift of finite type is surjunctive. This implies in particular that a $\Z$-subshift of finite type has the Moore property whenever it has the Myhill property.

Discretization-aware LES yields an exact expression for the discrete target flux in finite-volume LES by recognizing that a coarse finite difference is a top-hat-filtered exact derivative (the "filter-swap" property). That argument is purely spatial; here we observe that the forward-Euler time difference is itself a (one-sided) top-hat-filtered exact time derivative, and repeat the construction in space-time. The resulting exact discrete flux decomposition extends the spatial one with a single temporal term: a flux-quadrature error that shrinks with the quadrature order of the time integrator. In a Burgers experiment this term grows with the CFL number while the spatial terms do not, and a Smagorinsky closure augmented with its leading order - a Lax-Wendroff-type diffusion - stays accurate at coarse time steps where space-only closures degrade.

We study horizon-uniform local branches of finite-horizon discrete-time Pontryagin boundary value systems after smooth control elimination. The central input is a two-point endpoint inverse for the linearization. We verify this inverse from scaled stable--unstable boundary transversality, prove the associated endpoint-corrected Green estimate, and combine it with weighted contractions to obtain existence, uniqueness, Lipschitz dependence, and first-order expansions with constants independent of the horizon. The framework covers smooth nonlinear endpoint maps, including the original Pontryagin rows that fix the initial state and couple the terminal costate to the terminal state. Symplectic and Riccati criteria verify the inverse hypothesis at the level of the matrix data; in particular, every stabilizable linear-quadratic system with invertible dynamics and definite weights is covered, including noncommuting coupled data. A numerical section illustrates the certificates and the horizon-uniform first-order expansion.

This paper develops a learning-based boundary control framework for stabilizing a parabolic equation defined on time-varying spatial domain. Although the partial differential equation (PDE) backstepping method provides a systematic theoretical framework for such moving-boundary systems, its real-time implementation is hindered by the need to repeatedly solve time-varying kernel PDEs on evolving domains. To overcome this limitation, we first formulate the time-varying backstepping design as an operator that maps the moving-boundary trajectory to the corresponding backstepping kernel. By mapping the time-varying domain of the backstepping kernel equation onto a fixed reference domain, we establish the continuous dependence of the kernel on the moving-boundary trajectory, which provides the theoretical basis for approximating the backstepping design operator by a neural operator. Based on the approximate kernel operator, we construct the corresponding boundary feedback controller to stabilize the system. It is shown that the closed-loop system admits an exponential decay estimate on any prescribed finite time interval. For numerical implementation, DeepONet is employed to learn the time-varying kernel operator from offline-generated numerical kernel solutions and is subsequently deployed online to generate the required time-varying kernels without repeatedly solving the kernel PDE. Numerical benchmarks demonstrate that the proposed neural-operator-based implementation bypasses repeated online solution of the time-varying kernel PDE, achieves a significant acceleration of close to three orders of magnitude compared with conventional numerical kernel solvers, and thus enables real-time stabilization of the system on time-varying spatial domain.

We consider piecewise linear polygonal chains connecting two given points $A, B \in \mathbb{R}^3$ and consisting of exactly $n+1$ segments (i.e., having $n$ turning points). The absolute value of the turning angle at each interior point is bounded by a given number $\varphi \in (0,\pi)$. Under the condition $n\varphi \leq \pi$, we describe the set to which all interior vertices of such a polygonal chain belong (Theorem 1). It is proved that for any point $B^{(1)}$ from this set, there exists a polygonal chain with the specified parameters (Lemma 1). Based on these results, we obtain an explicit formula describing the set of all admissible sequences $(B^{(1)}, \ldots, B^{(n)})$ of the angular points of the polygonal chain. The obtained description can serve as a basis for constructing algorithms to enumerate admissible polygonal chains and to solve optimization problems for an objective function that accounts for the cost of traversing the segments and the cost of turns.

The increasing penetration of distributed energy resources (DERs) is transforming low-voltage (LV) networks into active systems, including energy communities, whose generation, storage, and energy exchange activities require enhanced coordination with upstream medium-voltage (MV) networks. In the proposed Stackelberg structure, a local market operator aggregates multiple LV communities and acts as a single leader, determining DER operations and boundary energy exchanges, while the MV network serves as the follower, ensuring efficient system feasibility through an economic dispatch that includes both conventional and utility-scale PV generation. The proposed bottom-up coordination scheme concentrates discrete DER scheduling at the LV level while the MV level retains a convex continuous formulation, enabling an exact single-level reformulation via the Karush-Kuhn-Tucker (KKT) conditions. In addition, a distributed coordination algorithm that combines Lagrangian Dual Decomposition (LDD) with the Alternating Direction Method of Multipliers (ADMM) is developed to coordinate LV communities in parallel while preserving data confidentiality. The framework is validated using the IEEE 33-bus system at the MV level and six European 206-bus LV test feeders. Results indicate that the LDD-ADMM algorithm closely matches the exact reformulation, with an average relative deviation of 1.7e-4, with deviations confined to periods of scarcity for the cheap resource. Furthermore, leaders' decisions can induce operating conditions that increase followers' costs relative to their independently optimal dispatch, a pattern reinforced by comparison with a feasibility-based single-level relaxation that satisfies the required energy exchanges but fails to achieve a cost-efficient allocation of MV resources.

In this work we establish that the recently introduced fractional Sobolev spaces based on the Riesz fractional gradient of Musielak-Orlicz functions by one of the authors, coincide with the space of Bessel potentials of functions on such generalized Orlicz setting. Moreover, we identify them as complex interpolation spaces, and exploiting the well known properties for interpolation of operators we obtain several structural properties for those spaces.

The tacnode process is a universal determinantal point process arising in non-intersecting particle systems and random tiling models. In this paper, we study the generating function for the counting functions of the tacnode process on a union of $m$ intervals, $m\in\mathbb{N}^{+}$. Our first result provides an integral representation for the $m$-point generating function in terms of the Hamiltonian governing a system of $8m+4$ coupled differential equations. Combined with several differential identities for this Hamiltonian, the representation yields the large gap asymptotics, up to and including the constant term. As further applications, we obtain asymptotic formulae for the expectations, variances, and covariances of the counting functions, and establish a central limit theorem for their joint fluctuations. These results extend the previously known $1$-point theory for the tacnode process to the multi-interval setting with multiple discontinuities.

This study presents a high-order compact finite difference gas-kinetic scheme (FD-CGKS) that introduces a novel spatial discretization strategy for the efficient implementation of space-time coupled high-order schemes on structured grids. A conservative nonlinear compact discretization is achieved by formulating numerical fluxes from physical fluxes at both nodal and interfacial locations. To simplify the multidimensional spatial reconstruction required for the GKS flux evaluation, we propose a dual-grid approach that updates conservative variables on both a primary grid and an identical dual grid, offset by half the mesh spacing. By leveraging the time-accurate interface solutions from the gas-kinetic evolution model, the scheme explicitly updates averaged spatial derivatives between virtual interfaces, naturally enabling compact high-order reconstruction. Furthermore, a nonlinear GENO method is incorporated to capture flow discontinuities with high resolution and robustness, effectively suppressing spurious oscillations. The proposed framework, which also offers new perspectives for designing schemes based on space-time decoupled Riemann solvers, is systematically validated. Comprehensive benchmark computations of inviscid and viscous flows demonstrate the scheme's high accuracy in resolving a wide spectrum of flow features, from smooth multiscale structures to strong shock discontinuities.

We investigate the relationship between volumetric and surface growth within a recently proposed optimization-driven framework for linearly elastic solids. In this approach, growth is not prescribed through an evolution law; instead, the growth distribution is determined as the solution of a constrained optimization problem. Focusing on processes driven by the minimization of the work performed by external loads in one-dimensional and axisymmetric settings, we derive explicit analytical solutions for the resulting growth distributions. Although growth is initially formulated as a volumetric process through a distributed growth strain tensor, we show that the optimal growth distributions are singular and concentrate on boundaries or internal interfaces. These results provide a variational mechanism through which, under certain conditions, surface growth is selected as the optimal realization of volumetric growth.

The manuscript establishes sharp bound of the third-order Toeplitz determinant for a subclass of starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n.$

We prove that the Hilbert scheme of conics on $X = G/P$, a projective rational homogeneous space with Picard rank one, is $G$-spherical if and only if $P$ is associated to a long root of the Dynkin diagram.

In this article, we describe and analyse a novel mechanism for symmetry-breaking in minimal symmetrically coupled identical slow/fast oscillator networks with strong nonlinear mutually inhibitory coupling. We show that the symmetry-breaking, surprisingly, originates from the canard dynamics of a folded node that lies on the axis of symmetry. By applying geometric singular perturbation theory and the blow-up technique to a normal form, we determine the geometric mechanisms by which the {\em symmetric folded node} induces symmetry-breaking. More specifically, we show that (i) the fold curve of the coupled system is orthogonal to the axis of symmetry at the symmetric folded node; (ii) there is only one primary maximal canard (either strong or weak, depending on parameters), which always lies on the axis of symmetry and is the axis of rotation for the twisting of solutions; and (iii) the number of rotations is the key local diagnostic feature that breaks the symmetry. Our work is closely related to that of Kristiansen and Pedersen [SIAM J. Appl. Dyn. Syst., {\bf 22} (2023)] on symmetrically coupled FitzHugh-Nagumo oscillators with strong linear inhibitory gap junctional coupling, however, we consider nonlinear coupling and we identify and study multiple sub-types of their `cusped singularities'. We demonstrate our theoretical results by applying them to a model of the eukaryotic cell cycle in which the symmetric folded node plays a key role in rhythmogenesis. More specifically, we study periodic and quasi-periodic symmetry-breaking mixed-mode oscillatory attractors of the cell cycle model. We show that the local twisting induced by the symmetric folded node is the local mechanism that both breaks the symmetry and generates the small-amplitude oscillations in the mixed-mode dynamics.

We study coefficient identification problems for elliptic partial differential equations with total variation regularization and control constraints. Existing related literature relies on continuity and differentiability properties of the control-to-state operator with respect to the $L^\infty$-norm. While this is sufficient for deriving optimality conditions, it is not well-suited for numerical algorithms, as it neglects the spatial extent of perturbations and leads to a qualitative discrepancy compared to $L^q$-norms with $q < \infty$. In this work, we address this gap by exploiting $W^{1,s}$-regularity results to establish differentiability properties of the control-to-state operator with respect to $L^q$-norms for finite $q$. Based on this framework, we derive first- and second-order differentiability results for the reduced objective functional and establish first-order optimality conditions involving a restricted subdifferential characterization of the total variation seminorm and corresponding regularity of the associated multipliers. Building on this, we analyze a nonsmooth trust-region method based on an $L^r$-trust region for $r > 0.5 q$ and prove its convergence to first-order stationary points.

What is the effect of randomly removing material from an infinite stretched membrane? Under what conditions can the membrane still sustain tension? This problem was introduced by Robert Connelly in connection with applications of rigidity theory in the natural sciences, and was later studied in M. V. Menshikov, K. A. Rybnikov, and S. E. Volkov, "The loss of tension in an infinite membrane with holes distributed according to a Poisson law" (2002); a discrete version was also considered in Robert Connelly, Konstantin Rybnikov, and Stanislav Volkov, "Percolation and the Loss of Tension in an Infinite Triangular Lattice" (2001). We study a mathematical framework based on a non-homogeneous Poisson point process whose intensity $\lambda$ tends to zero at infinity. The hole shapes are i.i.d.\ and independent of their locations. We show that if the intensity does not decay too quickly, then tension is still lost throughout the whole plane, as in the homogeneous model studied in 2002. Conversely, we give sufficient conditions under which complete loss of tension does not occur. Thus, both destruction and non-destruction regimes are possible even when the intensity tends to zero, indicating a phase transition in the model. The processes studied here are closely related to bootstrap percolation.

We prove that if a holomorphic diffeomorphism of a compact complex manifold is bi-Lipschitz conjugate to an ergodic affine automorphism $A$ on $\Gamma\backslash G$, then the conjugacy is $C^\infty$. Moreover, if $A$ is weakly mixing, then the induced complex structure on $\Gamma\backslash G$ is left-invariant. As applications, we establish a regularity bootstrap result for holomorphic Anosov diffeomorphisms bi-Lipschitz conjugate to affine models, as well as a holomorphic analogue of the rigidity theorem for higher-rank abelian Anosov actions by Hertz--Wang. The key observation is that the condition for a diffeomorphism to preserve a complex structure has the same form as the cocycle compatibility relation appearing in the study of centralizers. This places invariant complex structures and centralizers within a common $\mathbb Z^2$-cocycle framework. From this viewpoint, our main result may be regarded as a holomorphic counterpart of the Lipschitz centralizer rigidity theorem of Damjanović--Wilkinson--Wu--Xu for affine automorphisms.

In this article we examine the cross-connections of the normal category of finite dimensional subspaces of a Hilbert space and it's dual space. Further, we describe the cross-connection semigroup of cones, which is a normed algebra isomrphic to the normed algebra of finite rank bounded operators on a Hilbert space. We also characterize compact operators and their spectrum by the normal cones in the normal category of proper subspaces of a Hilbert space.

Jointly inferring latent node states and unknown network topology in nonlinear graphical dynamical systems is a fundamental yet largely unsolved problem, where the mutual entanglement of continuous states and discrete structure renders accurate recovery of either quantity critically dependent on the other. We propose \textbf{Koopman-GKFA} (Koopman Group-sparse Kalman Filter--ADMM), a unified framework that lifts nonlinear network dynamics into an approximately linear system via Koopman operator embedding with a separable node-wise dictionary, enabling optimal linear filtering for state estimation and provably convergent convex optimization for topology inference. Three theoretical contributions underpin the framework: (i)~a \emph{structural homomorphism lemma} proving that, under a separable-dictionary condition, block sparsity of the lifted coupling operator is isomorphic to the graph topology, providing the rigorous foundation for group-sparse regularization; (ii)~a block-structured group-sparse ADMM topology subproblem with certified linear convergence, extended by an exponential forgetting factor to track time-varying topologies; and (iii)~a \emph{three-term certified mean-squared error bound} that decomposes total estimation error into Koopman truncation, observation noise, and topology residual components, with monotone consistency established as the dictionary dimension grows. Extensive experiments on synthetic benchmarks (Kuramoto oscillators, Hill-kinetics gene-regulatory networks) and real-world datasets (NGSIM US-101, DREAM4) demonstrate that Koopman-GKFA consistently outperforms EKF-, UKF-, and particle-filter-based joint estimators in both state estimation and topology recovery, while exhibiting polynomial computational scaling and strong robustness in high-dimensional nonlinear settings.

We study observability and controllability for constant-coefficient first-order hyperbolic systems on the real line when only part of the state is observed or controlled. Even when the Kalman rank condition holds, the usual \(L^2(\mathbb R)^N\)-observability estimate may fail because some components are detected only through the dynamics. We show that the Kalman structure determines the appropriate observability estimate. A component that becomes visible after \(k\) algebraic steps is measured at low Fourier frequencies with a weight of order \(|\xi|^{2k}\) in the Fourier variable. This yields a natural Kalman-adapted observation space for the system. We also prove localized observability for diagonalizable systems on observation sets with uniformly bounded gaps and, separately, extend the whole-line construction to systems with real spectrum and Jordan blocks.

The Seidel energy is defined as the sum of the absolute values of the eigenvalues of the Seidel matrix of a hypergraph. We first characterize the k-uniform hypergraphs of fixed order n with minimum and maximum Frobenius norms of Seidel matrices and then derive bounds for the Seidel energy. Building on these results, we obtain a negative answer to the hypergraph analogue of Haemers Conjecture by showing that the complete k-uniform hypergraph does not, in general, minimize Seidel energy. Motivated by the theory of hypoenergetic and non-hypoenergetic graphs, we define Seidel hypoenergetic and Seidel non-hypoenergetic hypergraphs and prove that almost all k-uniform hypergraphs are Seidel non-hypoenergetic.

The shifted Jack Littlewood-Richardson coefficients $g^\lambda_{\mu\nu}(\alpha)$, first studied by Alexandersson-Féray, are Laurent polynomials in the Jack parameter $\alpha$ attached to triples of partitions, which generalize the classical Jack Littlewood-Richardson coefficients investigated by Stanley, et al. In a previous work of the author's, it was conjectured that the Littlewood-Richardson coefficients for two triples, in which one of the partitions differ by a single box move, are congruent modulo the $\alpha$-hook length of the pivot box for that move. In this note we prove that conjecture. We also investigate the extension of that conjecture to shifted Macdonald functions, which remains open pending two properies of Lassalle's shift map in that case.

Let $G$ be a graph with vertex set $\{v_1,\dots,v_n\}$. The Seidel matrix of $G$ is an $ n\times n$ matrix whose diagonal entries are zero, $ij$-th entry is $-1$ if $v_i$ and $v_j$ are adjacent, and otherwise is $1$. The $p$-Seidel energy of the graph $G$ is defined as the sum of the absolute values of the $p$-th powers of all eigenvalues of the Seidel matrix of $G$ and introduced in [European Journal of Combinatorics, (86) (2020), 103078]. In this article, we characterize the graph that minimizes the $p$-Seidel energy among all graphs with fixed order $n$, for $p>2$. We also characterize the graph that maximizes the $p$-Seidel energy among all graphs with fixed order $n$, for $0<p<2$. In addition, for every $p>2$, we characterize the graph that minimizes the $p$-Seidel energy among all $r$-regular graphs with fixed order $n$, where $n$ is a prime power with $n\equiv 1\pmod 4$, $r=\frac{n-1}{2}$. For every $p>2$, we also characterize the graph that maximizes the $p$-Seidel energy among all $r$-regular graphs with fixed order $n=2r$. Finally, we pose several open problems concerning the $p$-Seidel energy for different values of $p$.

We introduce and study a synthetic notion of timelike total curvature for curves in Lorentzian length spaces with upper curvature bounds. In particular, we prove that our notion agrees with its smooth counterpart, and we show that timelike curves of finite total curvature are rectifiable. As the main application, we provide a sharp lower bound for the length of timelike curves solely in terms of the time separation between their endpoints and their total curvature.

Two well-known decoding algorithms for BCH codes are conventional decoding, based on the Berlekamp-Massey algorithm in combination with Chien search, and direct decoding, which uses direct solutions to find the error locator polynomial and its roots. We introduce hardware architectures for conventional and direct decoding of extended BCH codes. Both architectures support implementation for any blocklength. Our conventional decoder supports any error-correction capability, whereas direct decoding is supported up to error correcting capability t = 4. To the best of our knowledge, our work is the first to implement a direct BCH decoder with an error-correction capability 4. We synthesize for the Xilinx Ultrascale+ XCZU48DR field-programmable gate-array and 16 nm FinFET for blocklengths up to 1024 bits and t = 4. We show that the direct decoder outperforms the conventional decoder in area efficiency for t = 2, t = 3, and for t = 4 for blocklengths longer than 256. Post-synthesis results for 16 nm FinFET show codeword per clock-cycle throughput at 1 GHz, achieving 239 Gb/s for the (256, 239) eBCH code and 223 Gb/s for (256, 223) eBCH code at 2 ns and 8 ns latency, respectively.

We investigate the uniqueness of radial bound state solutions to the sublinear elliptic equation \[ \begin{cases} -\Delta u - u + |u|^{q-2}u = 0 & \text{in } \mathbb{R}^n, \\ u(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] where $q\in(1,2)$ and $n\geq 2$. We prove that for any prescribed integer $k\geq 1$, the equation admits exactly one radial bound state solution with $k$ simple zeros. Furthermore, we consider the superlinear equation \[ \begin{cases} -\Delta u + u -|u|^{p-1}u = 0 & \text{in } \mathbb{R}^n, \\ u(x) \to 0 & \text{as } |x| \to \infty. \end{cases} \] While the uniqueness of radial bound state solutions for this equation was established by Tang (2026) for $n\geq 3$ and $1<p<\frac{n+2}{n-2}$, we provide the necessary arguments to show that this uniqueness result remains valid for the case $n=2$ with $p>1$.

This paper initiates a mathematical investigation of a PDE model for the transport of high voltage direct current via overhead lines. We prove the existence of infinitely many solutions, give necessary conditions for existence, explicitly compute the continuum of all radial solutions, and develop a new numerical algorithm for this problem.

This article surveys the mathematical contributions of Manfred Denker, with a focus on themes that connect ergodic theory, probability theory, dynamical systems, fractal geometry, and statistics. Denker's highly influential work includes a systematic study of the statistical properties of dynamical systems, the development of limit theorems for dependent processes, and the use of thermodynamic formalism to relate geometric and measure-theoretic properties. Particular emphasis is placed on the emergence of probabilistic behavior in deterministic systems, including central limit theorems, invariance principles or local limit theorems, under weak dependence assumptions or in infinite measure. Further topics include equilibrium states and transfer operator methods, the role of conformal measures in fractal geometry, and the asymptotic theory of statistical procedures for dependent data, such as rank statistics and U-statistics. In addition to these theoretical developments, the survey highlights contributions connecting rigorous analysis with computational and statistical methods. Taken together, these works illustrate a unifying perspective in which ergodic, probabilistic, geometric, and statistical methods interact in the study of dynamical systems.

Consider the diffusion process \begin{equation*} dX_{\epsilon}(t) = \mss b(X_{\epsilon}(t)) \, dt + \sqrt{2\, \epsilon\, \mss a(X_\epsilon(t))} \, dW_{t}, \end{equation*} on the one-dimensional torus $\bb T = [0,1)$. Here $\epsilon$ is the temperature, $W_{t}$ a Brownian motion on $\bb T$ and $\mss a$, $\mss b$ functions of class $C^{2}(\bb T)$ satisfying further conditions. Denote by $\mss P(\bb T)$ the set of probability measures on $\bb T$ equipped with the weak topology, and by $\ms I_{\epsilon}\colon \mss P(\bb T)\to [0,+\infty)$ the level two large deviation rate functional of the diffusion $X_{\epsilon}(\cdot)$. We derive a full $\Gamma-$expansion of $\ms I_{\epsilon}$, as $\epsilon \to 0$, expressing it as \begin{equation*} \ms I_{\epsilon} = \frac{1}{\epsilon} \;\ms J^{(-1)} \; +\; \ms J^{(0)} \;+\; \sum_{p=1}^{\widehat{\mf q}}\frac{1}{\theta^{(p)}_{\epsilon}}\;\ms J^{(p)}\,, \end{equation*} where $\ms J^{(-1)}$, $\ms J^{(0)}$, $\ms J^{(p)} \colon \mss P(\bb T)\to [0,+\infty]$ represent rate functionals, independent of $\epsilon$, and $\theta^{(p)}_{\epsilon}$ are the time-scales at which the Markov process $X_{\epsilon}(\cdot)$ exhibits a metastable behaviour.

In this article, we develop a multi-dimensional two-layer thin film model extending the thin film model proposed in \cite{barthwal2025hyperbolic}. The model considered in \cite{barthwal2025hyperbolic} considered a very specific Marangoni scale by choosing Marangoni numbers in both layers to be $1$. We relax this condition here and prove that the obtained system possesses a full set of Riemann invariants. Based on these findings, we develop a Riemann Invariant-based Local Characteristic Decomposition WENO (RI-WENO) method for the two-layer thin film model in one and two dimensions. The method is built upon a specially designed variable transformation constructed from the derived Riemann invariants of the system. This transformation partially diagonalizes the governing equations and yields a sparse structure in the transformed eigenvector matrices. As a result, the proposed RI-WENO framework significantly reduces the computational cost of the standard Local Characteristic Decomposition WENO approach while retaining its strong capability to suppress spurious oscillations. Numerical experiments, including new benchmark test cases, demonstrate that the RI-WENO method achieves an effective balance between accuracy and computational efficiency, making it a promising and practical choice for solving the two-layer thin film model.

In this paper, we study the global dynamics of the 3D ionic Euler-Poisson equations with the parameter of Mach number $\varepsilon$. We first establish the global well-posedness and scattering for the high Mach number regime $0<\varepsilon\leq1$ and pressureless case $\varepsilon=0$. Moreover, we prove the high Mach number limit, showing that the profile of the solution for ionic Euler-Poisson equations converged to that of the pressureless equation as $\varepsilon\rightarrow0$. Our approach combines energy estimates, dispersive estimates and the normal form method. The major difficulty lies in establishing the uniform estimates with respect to the parameter, as the dispersive or resonance structure degenerates when $\varepsilon$ tends to 0. A crucial observation is that despite the disappearance of the pressure ($\varepsilon\rightarrow0$), dispersive phase function always remains a wave-type structure in zero frequencies, which enables us to derive linear and bilinear multiplier estimates adapted to the uniformity of Mach number parameter.

For $0<q<1$ we compare two natural $q$-analogs of the Laguerre--Pólya class. The first one is a coefficient-side class, defined as the inverse image of the classical Laguerre--Pólya class under the normalized $q$-Borel transform \[ \Bq\left(\sum_{k\ge 0}a_k\frac{z^k}{k!}\right) =\sum_{k\ge 0}a_k\frac{q^{k(k-1)/2}(1-q)^k}{(q;q)_k}z^k . \] The second one is a zero-side class, defined as the locally uniform closure of real polynomials whose nonzero zeros are logarithmically $q$-separated on each side of the origin. We prove that the normalized $q$-Borel transform maps the classical Laguerre--Pólya class, and its type-I subclass, into themselves. This yields a $q$-Jensen-polynomial criterion and shows that the coefficient-side class strictly contains the classical Laguerre--Pólya class. On the zero side, we prove a genus-zero product representation. The logarithmic separation condition prevents zeros escaping to infinity from producing a residual exponential factor; consequently no nonconstant exponential factor can occur. For every $q\in(0,1)$ we obtain the strict chains \[ \qLPs\subsetneq \LP\subsetneq \qLPw, \qquad \qLPIs\subsetneq \LPI\subsetneq \qLPIw . \]

We propose a continuous time dynamical system model for tracing the evolution of confidence in a small decision making group by consensus with the possibility that a forcing factor is exerted onto the confidence of the participants. We experimentally check whether a forcing factor appears, where it begins and ends and how it affects the evolution. We find that the equilibrium value of the confidence is lower in the model with the forcing factor than without it, and that the forcing factor can be identified and induces additional oscillations of the confidence level. This is probably one of the first times when a mathematical model is able to speak about visible effects on the confidence process tracing under alterations of its levels. Pragmatically we find a model that captures influence of participants' confidence level by observing oscillations and equilibrium and we experimentally test it with measures of individual "confidence that the decision is correct" throughout the group decision-making.

We study the pair correlation statistics of orbits generated by maps on the interval. We show that under suitable mixing and multifractal assumptions, the pair correlation statistics of an orbit will almost surely exhibit the same asymptotic behaviour as a suitable sequence of i.i.d. random variables. We will also show that, under suitable hypotheses, the pair correlation statistics defined by two orbits will almost surely exhibit the same behaviour as two suitable sequences of i.i.d. random variables. Specific dynamical systems to which our results apply to include Gibbs-Markov maps and the Gauss map. We also give an example of a slowly mixing system for which the pair correlation statistics of an orbit almost surely behave distinctly to an i.i.d. sequence.

We define naive atomic decompositions of smooth projective varieties. We show that they satisfy a naive version of Iritani's blowup formula in several examples that are complicated enough to show most interesting features of the general theory while being simple enough to be computable by elementary methods.

A cyclic square matrix (and its conjugacy class) C over a field K is called (m,k)-cyclic if it has a decomposition $C = A \oplus B$, where $\dim A = m, \dim B = k$ and $m, k \ne 0$. It is shown that the product of two nonsingular (m,k)-cyclic conjugacy classes $\Omega$ and $\Psi$ of GL(m+k,K) contains all nonscalar matrices P in GL(m+k,K) with determinant $\det P = \det \Omega \Psi$.

Let us consider a random infinite lower triangular matrix, where the entries on and below the diagonal are i.i.d. uniform random elements of a fixed finite field. We investigate the evolution of the span of the first $n$ rows of this matrix as $n$ grows. Many properties of this evolving subspace can be captured with the help of the verbose persistence diagram, which is a standard tool in stochastic topology and topological data analysis. We give an explicit formula for the distribution of the persistence diagram. We prove a law of large numbers for the distribution of lifetimes. We also describe the fluctuations of the persistent Betti numbers.

It is well established that the wave operators $W_{\pm}(H,-\Delta)$ for the one-dimensional Schrödinger operator $H=-\Delta+V(x)$ are bounded on $L^p(\mathbb{R})$ for all $1<p<\infty$ in both generic and exceptional cases. They are also bounded on $L^1(\mathbb{R})$ and $L^{\infty}(\mathbb{R})$ in the exceptional case with $\lim\limits_{x\rightarrow-\infty}f_+(0,x)=1$. For the remaining endpoint cases, it has long been expected that they are generally unbounded at the endpoints $p=1,\infty$ due to the presence of the Hilbert transform in the low energy part, yet a rigorous proof has been missing. In this paper, we show that even for a bounded and compactly supported non-zero potential $V$, the wave operators $W_{\pm}(H,-\Delta)$ are unbounded on $L^1(\mathbb{R})$ and $L^{\infty}(\mathbb{R})$ in the generic case, as well as in the exceptional case with the condition $\lim\limits_{x\rightarrow-\infty}f_+(0,x)\neq1$. Moreover, in the latter case, they are even unbounded from $L^{\infty}(\mathbb{R})$ to ${\rm BMO}(\mathbb{R})$ (Bounded Mean Oscillation space). Hence together with those known results, our counterexamples complete the picture of the $L^{p}$ boundedness of one-dimensional wave operators.

For a graph $G$ put $h_r(G)=\max{\chi(H):H\subseteq G,\operatorname{girth}(H)\ge r}.$ Erdős and Hajnal asked whether $h_r(G)\to\infty$ as $\chi(G)\to\infty$, for every fixed $r\ge4$. We prove this in every fixed polynomial edge-density regime: for all $r\ge4$, $k\ge2$, $P,C>0$, there is $M=M_{r,k}(P,C)$ such that $\chi(G)\ge M,\ e(G)\le C\chi(G)^P\Longrightarrow h_r(G)\ge k.$ Quantitatively, after replacing $P$ by $P\vee2$ and $C$ by $C\vee2$, $M_{r,k}(P,C)\le \exp!\left(O_{r,k}\bigl((P+2+\log(C\vee2))^2\bigr)\right),$ and consequently the same conclusion holds throughout the quasi-polynomial range $e(G)\le \exp\bigl(C_0(\log\chi(G))^a\bigr),\ 1 < a < 3/2,$ for all sufficiently large $\chi(G)$. In each fixed polynomial-density regime we also obtain $f_{P,C}(k,r)\le k^{O_{r,P,C}(1)}.$ The proof combines a chromatic-defect random extraction lemma, compact and near-quadratic sparse-core bases, and a peeling/thinning bootstrap increasing the admissible edge exponent by $1/(r-1)$. We also prove structural saturation results for possible counterexamples, including Moore-strength exact-cycle packings and quadratic saturation in projected colour-pair space. Finally, writing $h_r^{\mathrm f}(G)=\max{\chi_{\mathrm f}(H):H\subseteq G,\operatorname{girth}(H)\ge r},$ we develop a fractional random-extraction framework based on Mohar-Wu preservation. We prove sufficient cheap-cycle-killing criteria and verify them for several structured families, including clique-organised families, line graphs of incidence graphs of equal-order generalized quadrangles and generalized hexagons, and the Bohman-Keevash tracking-time triangle-free-process graph. We also isolate a density-free obstruction that any proof using this fractional surgery route must overcome.

We study the asymptotic enumeration of labelled connected multitype graphs in the sparse regime, where both the number of vertices and edges grow linearly and the excess is proportional to the size of the graph. Extending the classical theory of connected graph enumeration to the multitype setting, we consider graphs with prescribed numbers of vertices of each type and prescribed edge counts between each pair of types. Our approach is probabilistic and relies on the theory of inhomogeneous random graphs. In particular, we exploit large-deviation principles and asymptotic estimates for connectedness probabilities to relate the counting problem to the emergence of giant components in suitably tuned supercritical random graphs. From large deviation asymptotics of connected components of inhomogeneous random graphs, we recognize that a connected graph with a given edge statistics corresponds to the (unique) giant component of larger inhomogeneous random graph with a suitably chosen connection kernel. This correspondence allows us to derive the leading exponential asymptotics for the number of connected multitype graphs with fixed type profile and edge matrix. The resulting formula generalizes the asymptotic enumeration results of Bender, Canfield, and McKay for connected sparse graphs to the multitype framework. More broadly, the paper illustrates how probabilistic techniques can provide transparent and effective tools for addressing new combinatorial enumeration problems.

We develop and analyze a conforming Virtual Element Method (VEM) for the fourth-order Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The proposed scheme is based on compatible discretizations of each component: an \(H^2\)-conforming VEM for a fourth-order electrostatic potential equation, an \(H^1\)-conforming VEM for the Nernst--Planck equations, and a \textit{divergence-free} and \textit{pressure-robust} VEM for the Navier--Stokes equations. Time integration is performed using a backward Euler scheme to ensure stability. We establish the well-posedness of the continuous problem up to three-dimensions and also establish existence and uniqueness of the fully discrete solution via a fixed-point argument. Further, we derive a priori error estimates showing that the electrostatic potential, concentration and velocity converge optimally in Bochner norms \(L^\infty\bigl(0,T; H^2(\Omega)\bigr)\), \(L^2\bigl(0,T; H^1(\Omega)\bigr)\), and \(L^2\bigl(0,T; \bm{H}^1(\Omega)\bigr)\), respectively. The analysis requires a sophisticated argument to avoid any restrictive assumptions on the coercivity and continuity constants and to handle the trilinear form involving three different variables. The pressure-robust design permits the use of lowest-order pressure approximations without compromising convergence rates of the other variables, reducing computational cost. Numerical experiments confirm the theoretical convergence rates for various polynomial orders and demonstrate the scheme's robustness, including in low-viscosity regimes.

In this short note, we prove that if $A \in \mathbb C^{N \times M}$ with $N=4M-5$ has i.i.d.\ standard complex Gaussian entries, then the probability that the phase retrieval map generated by $A$ is not injective is positive. This proves Part (1) of a conjecture of Cynthia Vinzant, which was later restated by Afonso S. Bandeira in \cite{BDL+26}. The main result of this paper was obtained using generative AI, in particular the Rethlas system.

Let $T(n)=n+\tau(n)$, where $\tau$ is the divisor function. We study the Erdos-Graham coalescence problem by encoding finite-level obstructions in the divisor-successor graph and in square-annular transfer maps. Coalescence is equivalent both to connectedness of this graph and to synchronization along an infinite non-autonomous sequence of finite annular systems. The basic identities are \[ \operatorname{im}(\mathcal A_k)=E_{k+1}, \qquad \mathcal F_{k^2}=k^2+E_k, \] where $E_k$ is the set of square-crossing overshoots from below $k^2$. We prove a transfer parity law, dynamic frontier bounds for the widths $W_{k,s}$, and the criterion that $\liminf_k|\mathcal A_k(E_k)|=1$ would imply connectedness. Unconditionally, \[ R(X)\le \log X+2\gamma+O(X^{-1/4}), \] and the exit sets are residue-universal, satisfy $|E_k|\le k^{o(1)}$, and obey \[ \frac94K+O(1)\le \sum_{k\le K}|E_k|\ll K(\log K)^3. \] Using the shifted-square estimate HST, obtained from the corrected Henriot--Nair--Tenenbaum theorem in the specialized form of Proposition 8.4 and from separate square-shift estimates, we obtain fixed-moment bounds \[ \sum_{k\le K}|E_k|^m\ll_m K(\log K)^{C_m}\quad(m\ge2). \] A further first-moment refinement to $K(\log K)^2$ is conditional on the additional, currently unproved, uniform quadratic Euler-product mean-value hypothesis HQE. We also prove quantitative large-jump and lower-runner race theorems, isolate interval filling, and formulate a square-gated two-branch criterion. No proof of the full Erdos-Graham problem is claimed.

Fang, Lan, and Xiao established the geometric Green's formula as a categorical isomorphism for arbitrary semisimple complexes. In this short note, we generalize their work to multi-step compositions. Specifically, we establish the iterated geometric Green's formulas for the composition of an $(n-1)$-fold restriction and an induction, as well as its dual.

For PDEs naturally posed in the de Rham complex, structure-preserving mixed and saddle-point finite element discretizations typically produce indefinite linear systems. We propose a multigrid preconditioning framework that combines mass-lumped (explicitly invertible) FEEC mass matrices with transforming smoothers that map the operator to a block form with positive definite diagonal blocks, enabling Gauss-Seidel-type relaxation on the transformed system. Under mild h-uniform norm-equivalence assumptions (and for trivial topology), we prove stability of the mass-lumped systems, and by extension spectral equivalence between the mass-lumped and original FEEC operators, which motivates using multigrid cycles designed for the mass-lumped operators as preconditioners for the consistent FEEC systems. While our primary focus is on algorithmic design rather than formal convergence theory, extensive numerical experiments on the Hodge-Dirac operator, mixed Hodge-Laplacians, and a magnetostatics saddle-point system in 2D and 3D demonstrate the robustness of the approach.

In joint source-channel coding (JSCC)-based semantic communication systems, achieving stable and reliable image semantic transmission under channel constraints remains a key challenge. In most channel adaptation modules, the signal-to-noise ratio (SNR) is often injected into each layer of a channel-adaptation model in an independent and layer-wise manner, which undermines global coordination across layers. Therefore, consistent noise-robust representations may fail to be learned throughout the model. To address this problem, we propose SA-RA-JSCC, a novel channel-adaptive JSCC model. SA-RA-JSCC maps SNR into a unified semantic vector in the feature space and then applies a one-shot global reweighting to the encoded features, thereby enabling globally consistent and learnable channel adaptation. Moreover, in order to further enhance the anti-channel capability of semantic information, a semantic-rate-aware module is introduced, enabling the adaptive policy to respond simultaneously to fluctuations in channel quality and changes in semantic-rate constraints, thereby enhancing global network coordination and channel adaptivity. Extensive experiment results across multiple channels and datasets demonstrate that SA-RA-JSCC significantly outperforms existing semantic communication models in terms of reconstruction metrics such as PSNR and MS-SSIM, exhibiting stronger robustness across a broad range of SNR regimes.

Effective pricing and promotion planning constitutes a central pillar of strategic revenue management for firms operating in highly competitive and dynamic markets. These planning activities require the simultaneous consideration of demand elasticity, competitor actions, channel and market specific constraints, and financial objectives. As the dimensionality and interdependencies inherent in these problems increase, manual or traditional approaches become suboptimal and insufficient. In this context, Operations Research provides a robust methodological foundation for scalable, data-driven decision support systems that can optimize complex planning processes across large product and customer portfolios. This paper presents two large-scale optimization systems developed and deployed at PepsiCo to support Revenue Growth Management initiatives: PromoAI and PricingAI. PromoAI couples machine learning-based promotional forecasts with a mixed-integer linear programming model to optimize promotional calendars across trade channels, searching millions of product-promotion-timing combinations for the one that maximizes PepsiCo and retailer revenues subject to customizable business constraints. PricingAI optimizes base prices across product portfolios over multi-period horizons, using Bayesian hierarchical models to estimate own- and cross-price elasticities and competitive interactions, then feeding these into a nonlinear programming engine that recommends price changes aligned with revenue and margin targets under operational constraints. Together, these systems demonstrate the feasibility and scalability of advanced optimization in large-scale enterprise environments. They highlight the value of integrating statistical learning with mathematical programming to enable enterprise-level, automated decision-making that is both data-informed and aligned with strategic business objectives.

We develop a parsimonious stochastic growth model in which state-dependent saving behavior generates endogenous business-cycle-like dynamics. The model consists of three coupled equations: a Solow-type capital accumulation equation, a linear filtering equation for the saving rate, and a bounded stochastic adjustment process. Saving is modeled as a logistic function of deviations from a balanced growth path, introducing nonlinear feedback controlled by a gain parameter. In the deterministic limit, increasing feedback strength produces a supercritical pitchfork bifurcation, splitting the balanced-growth equilibrium into two locally attracting regimes corresponding to expansion and contraction. When stochastic perturbations are introduced, these equilibria become metastable states, and the economy undergoes rare noise-induced transitions between them. The resulting dynamics exhibit persistent regimes, bimodal stationary densities, and right-skewed dwell-time distributions with approximately exponential survival tails. A discrete-time approximation is estimated using U.S. real GDP data, and Monte Carlo simulations are used to compute stationary distributions and regime persistence statistics. The results demonstrate that nonlinear state dependence, bounded multiplicative noise, and time-scale separation are sufficient to generate realistic business-cycle behavior within a low-dimensional framework.

For $n\ge 2$, let $\mathcal{M}(n)$ be the supremum of $\sum_{a\in A}1/(n-a)$ over pairwise coprime sets $A\subset [1,n)$. Erdős asked whether $\mathcal{M}(n)\le \sum_{p<n}1/p+O(1)$ uniformly in $n$. We prove the quantitative average-order formula $$ \sum_{n\le N}\mathcal{M}(n) = e^{-\gamma}N\log\log N+O(N). $$ The lower bound comes from the self-rough construction $\{n-d:P^{-}(n-d)>d\}$, while the upper bound uses bounded-cost dual certificates and Buchstab--de Bruijn estimates for rough numbers. We also prove that $$ \mathcal{M}(n)=(e^{-\gamma}+o(1))\log\log n $$ for almost all $n$, with a quantitative exceptional-set bound, and hence Erdős's inequality holds for almost all $n$. The almost-all proof uses a long-interval two-dimensional beta-sieve estimate for two moving forbidden residue classes, together with an exact finite singular-series cancellation. Finally, we prove the pointwise bound $\mathcal{M}(n)\le (2+\varepsilon)\log\log n+O_{\varepsilon}(1)$, explain the linear-sieve barrier behind the constant $2$, and record structural certificates, conditional window-packing reductions, numerical examples, and CRT sharpness constructions.

Random states play a foundational role in different branches of modern quantum science. In this work, we study a recently proposed random state ensemble induced from von Neumann entropy through the Bogoliubov-Kubo-Mori (BKM) metric. In particular, we derive an exact yet explicit formula of average entanglement entropy over BKM ensemble. In obtaining the formula, we only make use of properties of normalization constant of the ensemble in the absence of its correlation kernel, contrary to average entropy computation of other ensembles. This new framework paves the way for calculating higher-order cumulants of BKM ensemble beyond the average.

We analyze Allen-Cahn functionals with stationary ergodic coefficients in the regime where the length scale $\delta$ of the heterogeneities is much smaller (microscopic) than the interface width $\epsilon$ (mesoscopic). In a companion paper, we show that if the ratio $\epsilon^{-1} \delta$ vanishes fast enough as $\epsilon \to 0$, then the functionals converge to an effective surface energy where the energy density is determined by homogenization effects originating at microscopic scales. Here we prove that if the ratio $\epsilon^{-1} \delta $ vanishes too slowly, the limit of the functional may actually be smaller than this homogenized energy. We refer to this as the rare events regime. In the case of the random checkerboard in dimension one, we use large deviations techniques to give a complete description of the rare events regime, showing that the limiting energy depends in a nontrivial way on the limit of $\epsilon^{-1} \delta | \log \epsilon |$. We further construct, in any dimension, examples of random media in which rare events become relevant at algebraic scales $\delta \approx \epsilon^{1 + \alpha}$ for an arbitrary $\alpha > 0$, as well as almost periodic examples in which atypical configurations play the same role as rare events.

Communication-centric ISAC is a promising paradigm for future 6G networks, in which data payload signals are expected to be reused for sensing to enhance time-frequency resource efficiency. For random payload signals, existing studies have mainly characterized the expected sidelobe level (ESL) of the periodic auto-correlation function (P-ACF). However, ESL only captures the average sidelobe behavior and does not control large spurious sidelobe peaks in individual payload realizations, which may deteriorate weak-target detection performance. This motivates the design of information-bearing signals whose random P-ACF satisfies stringent peak sidelobe level (PSL) constraints. In this paper, we formulate a mutual information maximization problem under PSL constraints and a power budget. For quasi-static frequency-flat channels, we show that a continuous auto-correlation function (ACF)-domain uniform construction provides an asymptotically optimal high-SNR design principle. Motivated by this principle, we propose auto-correlation function keying (ACFK), a finite-constellation modulation architecture that embeds data symbols directly onto the ACF-domain sidelobes. ACFK enables exact control of the nominal P-ACF, which coincides with the actual P-ACF when a spectral non-negativity constraint is met. When this is not the case, we quantify the non-negativity violation probability and bound the resulting peak sidelobe level ratio (PSLR) degradation. We further provide a reference ISAC transceiver design for ACFK over quasi-static multipath channels, together with high-SNR approximate BER analysis. Numerical results validate the theoretical analysis and show that, compared with a generalized probabilistic amplitude shaping (PAS) baseline, ACFK provides substantially stronger PSLR control and improved weak-target detection performance under comparable sensing and communication settings.

The inversion of a set $X$ of vertices in an oriented graph reverses every arc with both endpoints in $X$. The inversion graph $I(G)$ of a graph $G$ has the labelled orientations of $G$ as its vertices, two orientations being adjacent when a single inversion transforms one into the other, and the inversion diameter $\diam(I(G))$ is its diameter. Answering a question of Havet, Hörsch and Rambaud, we prove the bound in terms of edge number $\diam(I(G)) \le 2\sqrt{|E(G)|}$, and we complement it with a lower bound $\diam(I(G)) \ge \frac{|E(G)|}{|V(G)|}$ obtained by viewing $I(G)$ as a Cayley graph on $\F_2^{E(G)}$. We further refine the upper bound for bipartite graphs $G$ by showing $ \diam(I(G))\le \max\left\{\rho, \left\lceil\log_2\bigl(2+\sigma(2^{\rho-1}-1)\bigr)\right\rceil\right\}$ where the two parts of $G$ have maximum degrees $\sigma$ and $\rho$, respectively.

In this paper we develop a general framework for potentials on Lauritzen manifolds, namely smooth manifolds equipped with a pseudo-Riemannian metric and a pair of conjugate affine connections that may have non-vanishing torsion. We show how the theory of bi-forms accommodates torsion-full statistical structures and unifies contrast and pre-contrast functions in a cohomological framework. Within this formalism, we construct a canonical contrast bi-form on dually curvature-free Lauritzen manifolds and establish its principal structural properties. Several illustrative examples are analysed.

We determine the precise asymptotic number of unlabelled cubic planar graphs with $n$ vertices. Our approach blends generating series methods with computational bounds and probabilistic local large deviation theorems.

In this paper, we study the Radon transform associated with the unipotent radical subgroups of $GL_n(\mathbb{F}_q)$. We analyze properties of the Radon transform with a specific emphasis on its eigenvalues. We provide a description of its eigenvalues for the cases of $GL_2(\mathbb{F}_q)$ and $GL_3(\mathbb{F}_q)$.

We consider the Neumann initial boundary value problem associated to the chemotaxis system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = \big((u+1)^{m-1} u_x - u(u+1)^m v_x\big)_x & \text{in $(0, 1) \times (0, \infty)$}, \\ v_t = v_{xx} - v + u, &\text{in $(0, 1) \times (0, \infty)$}, \end{cases} \end{align} where $m \in \mathbb R$ is a given parameter. The relation between diffusion and taxis sensitivity is critical since the ratio $u(u+1)^m/(u+1)^{m-1}$ grows like $u^{2/n}$ for large $u$ with $n = \dim((0, 1)) = 1$. Nonetheless, we show that there is no critical mass phenomenon if $m \le -1$; that is, in that case all solutions emanating from suitably regular initial data are globally bounded. For certain parabolic-elliptic simplifications of \eqref{prob:abstract}, we obtain the same conclusion for all $m \in (-\infty, -1] \cup (0, \infty)$ and even for all $m \in \mathbb R$ if the initial datum is additionally assumed to be monotone. This stands in contrast to critical mass phenomena known to occur for critical quasilinear Keller-Segel systems considered in higher-dimensional domains. Accordingly, we make use of several special features of the one-dimensional setting such as the boundedness of the energy functional from below, the embedding $W^{1, n} \hookrightarrow L^\infty$, and the fact that the mass accumulation function solves a spatially non-degenerate parabolic equation.

Whether the multi-dimensional Euler-Poisson system admits global smooth solutions remains a challenging open problem. In this paper, we construct a class of large-data global smooth solutions to the one-fluid Euler-Poisson system in $\mathbb{R}^d$ ($1\leq d\leq 5$) by using the relaxation dissipation mechanism. Precisely, assuming that the initial density is far from vacuum and $\varepsilon E_0$ is sufficiently small, where $E_0$ denotes the initial energy and $\varepsilon$ is the relaxation time, we establish the global well-posedness of smooth solutions to the Cauchy problem. In particular, the size of the initial perturbation may be arbitrarily large, provided that the relaxation time is sufficiently small. Furthermore, we introduce an effective unknown motivated by Darcy's law to derive quantitative error estimates at the rate $\mathcal O(e^{-\lambda t}\varepsilon)$ between the rescaled Euler-Poisson system and the limiting drift-diffusion system for ill-prepared data. The new ingredient lies in developing the maximum principle for the nonlinear drift-diffusion system with nonlocal effect, which leads to the large-data global existence.

The paper deals with new formulations of a Lagrange interpolant polynomial based on the nodes of the well-known anti-Gauss rule. A first representation is given in terms of the classical Christoffel-Darboux kernel appropriately modified. The second one closely follows the barycentric form of the classical Lagrange polynomial, while the third formulation represents the interpolant as a combination of an orthonormal family of polynomials with respect to the discrete anti-Gauss inner product. A numerical test shows the performance of the explored forms.

A fourth-order conservative adaptive multiresolution average-interpolating wavelet upwind scheme is proposed for compressible flows governed by hyperbolic conservation laws. A family of asymmetric average-interpolating wavelets with upwind properties is constructed for conservative finite volume discretization, while symmetric average-interpolating wavelets are employed for multiresolution decomposition and reconstruction of physical variables in the adaptive procedure. Since both the conservative discretization and the adaptive multiresolution representation are constructed from cell-average quantities, the proposed scheme preserves strict conservation during both numerical evolution and adaptive cell redistribution. Unlike hybrid adaptive wavelet methods that use wavelets mainly for data compression and mesh adaptation, the present adaptive wavelet upwind scheme utilizes average-interpolating wavelet multiresolution approximation to reconstruct the interface values directly for numerical flux evaluation, thereby avoiding additional ghost-cell marking and reconstruction near coarse--fine mesh interfaces. The boundary variation diminishing reconstruction is incorporated at the finest resolution level to achieve non-oscillatory shock-capturing capability. Numerical tests demonstrate that the proposed scheme achieves the expected fourth-order accuracy, maintains conservation errors close to machine precision, and controls numerical errors around the prescribed threshold. The proposed method also sharply captures shock waves and contact discontinuities without spurious oscillations and resolves multiscale smooth structures through a sparse adaptive representation. These results indicate that the proposed scheme provides an efficient, conservative, and reliable approach for high-resolution simulations of compressible flows.

Let $\mathbb{N}$ be the set of positive integers. For subsets $\mathcal{A},\mathcal{M}\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $p(n,\mathcal{A},\mathcal{M})$ denote the number of representations of $n$ in the form $$ n=\sum_{a\in \mathcal{A}}m_a a, $$ where $m_a\in \mathcal{M}\cup \{0\}$ for all $a\in \mathcal{A}$, and only finitely many $m_a$ are nonzero. We prove that there exist two infinite sets $\mathcal{A}=\{a_n\}_{n=1}^{\infty}$ and $\mathcal{M}$ of positive integers such that $$ \lim_{n\to\infty}\frac{\log a_{n+1}-\log a_n}{\log n}=+\infty, $$ $p(n,\mathcal{A},\mathcal{M})>0$ for every $n\in\mathbb{N}$, and $p$ has polynomial growth. More generally, we prove a construction that associates restricted partition functions of polynomial growth with additive complements satisfying a simple counting condition. This answers a 2016 question of Dai and Chen in the affirmative.

We propose a new weak-form Physics-Informed Neural Network approach (named INI-VPINN). INI-VPINN naturally incorporates Neumann boundary and interface conditions into the variational formulation. It removes the need for additional loss terms or multiple subdomain networks. This framework employs compact support weighting functions and integration by parts to implicitly impose flux and continuity constraints. In this way, it implicitly ensures physical consistency across material boundaries. The proposed method is tested on Poisson and Laplace problems with sharp interfaces and complex geometries. Results show that, compared with several other Physics Informed Neural Networks-based formulations, the INI-VPINN consistently achieves higher accuracy, smoother and faster convergence. The proposed framework provides a general approach for solving multimaterial problems with complex geometries and mixed Neumann-Dirichlet boundary conditions using neural networks. The implementation is publicly available in a GitHub repository.

We study $\mathrm{PD}^3$ groups which admit an unbounded quasimorphism to $\mathbb{R}$ with coarsely-connected quasikernel. We show that such a group must either arise as the fundamental group of a torus or Klein-bottle bundle over $S^1$, or be quasiisometric to a Riemannian manifold $(\mathbb{R}^3,g)$, with the quasikernel being coarsely equivalent to $\mathbb{H}^2$. If $G$ is moreover hyperbolic, it admits a faithful action on $S^1$ by quasisymmetric homeomorphisms. Our approach features a coarse generalisation of Shapiro's lemma, and the development of a theory of homological isoperimetric inequalities for metric spaces; these tools make use of Margolis's framework for coarse homological algebra.

A mechanical shuffler consists of $m$ shelves. A deck of $n$ cards, arranged in increasing order, is dealt from the bottom sequentially. Each card is assigned a shelf uniformly at random and placed on the top (bottom) of the existing pile with probability $p$ ($1-p$) independently. We refer to this as asymmetric shelf-shuffle. We find the law $\nu_{n, m}^{(p)}$ of the permutation induced by the asymmetric shelf-shuffle and show that the pair consisting of the number of descents and the number of valleys is a sufficient statistic. This generalizes a result of Diaconis, Fulman, and Holmes (Ann. Appl. Prob., 2013) corresponding to the case $p=1/2$. For $p=1/2$, Chen and Ottolini (ECP, 2025) established the cutoff in the total variation distance near $\lfloor n^{5/4}\rfloor$. We establish the cutoff for the asymmetric shelf shuffle. Let $\nu_n$ be the uniform measure on the set of all permutations $S_n$ of $\{1, \ldots, n\}$. For a fixed $p\neq 1/2$ and $c>0$, we show that \[\TV\left(\nu_{n, \lfloor cn^{3/2}\rfloor }^{(p)}, \nu_n\right)=1-2\Phi\left(-\frac{|2p-1|}{4\sqrt{3}c}\right)+O_{c, p}(n^{-1/2})\;.\] We also establish the cutoff in the separation distance near $m\approx n^{2}$ and in the relative entropy near $m=n^{3/2}$. In both cases, we also obtain the cutoff profile explicitly.

Let $a\ge2$, let $p\in\mathbb{Z}[n]$ be eventually increasing and eventually non-negative, and let $\lambda_n=a^{p(n)}+f(n)>0$, where $f(n)\in\mathbb{Z}$. We prove, using Bourgain's bounded entropy criterion, that if $\log(1+f(n)a^{-p(n)})$ is eventually non-zero and decays geometrically in absolute value, then $(\lambda_n)$ is neither $L^\infty$-Khintchin nor $L^1$-Khintchin. In particular, for every $c\in\mathbb{Z}\setminus\{0\}$, every positive tail of $(a^{p(n)}+c)_{n\ge1}$ is non-Khintchin. The same conclusion applies to the standard examples $a^n+c$, $a^n+b^n$, and, whenever eventually positive, $a^n-b^n$, with $a\neq b$. Thus these perturbations of geometric powers lie on the unstable side of the Khintchin problem. This gives a negative answer, in the translated-power case, to the question of Fan--Fan--Queffélec--Queffélec on the stability of translated powers.

In an asymmetric shelf shuffle, a deck of $n$ cards is dealt sequentially from the bottom and assigned one of the $m$ shelves uniformly at random. The card is placed at the top of the assigned shelf with probability $p$, and at the bottom of the assigned shelf with probability $(1-p)$. Analysis of the shelf shuffle has gained much attention recently, and the case $p=1/2$ was first treated by Diaconis--Fulman--Holmes [Ann. Appl. Prob. 23 (2013), no. 4, 1692--1720]. In this paper, we extend the analysis of the shelf shuffle to general $p\in (0, 1)$. In particular, we study the distribution of cycles, cycle lengths, number of descents, number of valleys, number of inversions, and the RSK shape of a permutation obtained from an asymmetric shelf shuffle. Our results extend the analysis of Diaconis--Fulman--Holmes to arbitrary $p$. Furthermore, our analysis of the distribution of descents and inversions is new even for $p=1/2$.

A foundational theorem of Ajtai, Komlós, Pintz, Spencer, and Szemerédi asserts that every $n$-vertex $k$-uniform uncrowded hypergraph with maximum degree $\Delta$ contains an independent set of size $c_k n{\left(\frac{\log \Delta}{\Delta}\right)^{\frac{1}{k-1}}}$, for some constant $c_k>0$. Determining the optimal leading constant $c_k$ in this bound is a major open problem. A natural target is the so-called shattering-threshold constant $\left(\frac{1}{k-1}\right)^{\frac{1}{k-1}}$, which appears in the solution-space geometry of random constraint satisfaction problems, in average-case complexity theory, and in statistical physics, among other areas. We prove that uncrowded hypergraphs attain this threshold. More precisely, for every $\epsilon>0$ and $k\geq 2$, every $n$-vertex $k$-uniform uncrowded hypergraph of sufficiently large maximum degree $\Delta$ contains an independent set of size at least $(1-\epsilon) n {\left(\frac{1}{k-1}\frac{\log \Delta}{\Delta}\right)^{\frac{1}{k-1}}}$. Consequently, we obtain the first pseudorandom class of hypergraphs whose guaranteed independence number matches the shattering threshold, resolving a folklore conjecture. Moreover, as another direct consequence, we resolve a conjecture of Verstraëte and Wilson by proving that there exists a constant $c_k=1-o_k(1)$ such that every $n$-vertex $k$-uniform linear hypergraph of maximum degree $\Delta$ has independence number at least $c_k n\left(\frac{\log \Delta}{\Delta}\right)^{\frac{1}{k-1}}$. Our techniques are constructive yielding efficient algorithms for both static and distributed settings. Specifically, we provide an $\tilde O(n\Delta)$-time randomized static algorithm and an $\tilde O(1)$-round randomized $\textsf{LOCAL}$ algorithm to find an independent set in uncrowded hypergraphs at the shattering threshold. These results extend seamlessly to the the setting of linear hypergraphs.

Let $G$ be a finite connected weighted graph and let $R$ be its effective-resistance matrix. For every nonempty vertex set $S$, we factor the cofactor sum and determinant of the principal resistance submatrix $R[S]$ into an enumerative term and a boundary potential-theoretic term. If $\tau(G)$ is the weighted spanning tree enumerator and $\kappa_G(S)$ is the weighted enumerator of $S$-rooted spanning forests, then \[ \cof R[S]=(-2)^{|S|-1}\kappa_G(S)/\tau(G). \] After Kron reduction to $S$, with reduced Laplacian $K=L^S$, $Q=K^+$, and $q=\diag(Q)$, the remaining normalized factor is \[ \det R[S]/\cof R[S] =\frac{2}{|S|}\tr Q+\frac12 q^{\mathsf T}Kq. \] Equivalently, this factor is the maximum of $u^{\mathsf T}R[S]u$ over all $u\in\R^S$ satisfying $\one^{\mathsf T}u=1$. This optimization viewpoint yields monotonicity under enlargement of $S$, an exact one-point update formula, and a support criterion for equality. Small star examples show that the resulting set function is neither submodular nor supermodular in general.

The generalized Krylov subspace (GKS) method is an effective projection technique for large-scale Tikhonov regularization with a general regularization matrix. As the subspace expands, however, two computational bottlenecks limit scalability: the thin QR factorizations of the tall projected matrices formed by the forward operator and the regularization matrix applied to the basis, and the full reorthogonalization of each new basis vector against all previous columns. We propose a sketched variant, named sGKS, that addresses both bottlenecks. The QR factorizations are performed on compressed matrices of much smaller row dimension, maintained incrementally via rank-one updates. Moreover, we observe that explicit reorthogonalization can be skipped entirely without compromising the quality of the approximation subspace, since no step of GKS relies intrinsically on the orthogonality of the basis. The resulting algorithm is independent of the choice of sketching operator and preserves the approximation quality of the original method: we show that, in the absence of sketching in the projected solve, sGKS produces iterates identical to those of standard GKS, and that the sketched projected solve delivers quasi-optimal residual norms controlled by the embedding quality. For more challenging problems where the loss of basis orthogonality becomes significant, we show that incorporating a small number of iterative refinement steps in the projected solve restores the spectral properties of the basis and recovers the full accuracy of the unsketched method. Numerical experiments on image deblurring, X-ray computerized tomography, seismic travel-time tomography, and dynamic computerized tomography demonstrate that sGKS matches the reconstruction quality of standard GKS while significantly reducing per-iteration costs and overall wall-clock time.

Given a hyperbolic group $G$ and a quasiconvex subgroup $Q$, we define a \emph{branched cover of $G$ along $g$}, which is a hyperbolic group $H$ with a certain map into $G$. This builds on recent work on drilling hyperbolic groups and generalizes the case where $G$ is the fundamental group of a closed hyperbolic $3$-manifold $M$, $Q \cong \mathbb{Z}$ is represented by an embedded geodesic loop $\gamma$, and $H$ is the fundamental group of a branched cover of $M$ with branching locus $\gamma$. We show that certain deepness assumptions on Dehn fillings induce branched covers, providing many examples of such branched covers. Some additional assumptions imply these branched covers have boundary $S^2$, which may hold interest for the Cannon Conjecture.

We characterise the density of the positive rank one subspace of a masa-bimodule in terms of its support. We prove that strongly reflexive masa-bimodules form a Boolean lattice under naturally defined operations. We examine the lattice-theoretic properties of the class of strongly reflexive masa-bimodules that are also operator systems and some natural subclasses thereof, and provide a topological description of the lattice operations in the case the masa-bimodules arise from closed subsets of a locally compact group.

In this short note, using the Sarkisov link between a prime Fano threefold $V_{22}$ of degree $22$ and the quintic del Pezzo threefold $V_5$, we prove that the Hilbert scheme of rational quartic curves in $V_{22}$ admits a generically $2$-to-$1$ rational map onto the projective space $\mathbb{P}^4$.

We establish weighted recursions for the coefficients of Ramanujan's third order mock theta functions $f$ and $\omega$. Specifically, we apply a holomorphic projection operator to vector-valued Rankin-Cohen brackets of completed mock theta series and their shadows. By employing a vector-valued framework, we exploit the vanishing of certain spaces of vector-valued cusp forms. Our proof is AI-assisted and prioritizes accessibility, allowing for straightforward customization and replication within the broader research community.

Inspired by \cite{CCP}, we investigate a delayed leader-follower Cucker-Smale model describing the collective dynamics of interacting agents subject to communication lags. We first study the particle dynamics and establish sufficient conditions ensuring the emergence of asymptotic flocking. Our analysis shows that velocity alignment and bounded spatial dispersion persist despite the presence of delays and heterogeneous interactions between leaders and followers. We then derive and analyze two continuum descriptions of the system. In the first regime, the number of leaders is kept fixed while the number of followers tends to infinity, leading to a hybrid particle-kinetic model. In the second regime, both populations become infinitely large, yielding a fully kinetic delayed leader-follower model. For both mean-field formulations, we prove global existence, uniqueness, and Wasserstein stability of measure-valued solutions. These results provide a rigorous mathematical framework for the study of collective dynamics with leadership and memory effects and establish a bridge between delayed flocking models and their continuum counterparts.

The Chebfun software system is a Matlab extension essentially based on representations of (piece-wise) smooth one-variable functions by expansions in Chebyshev polynomials. One of Chebfun's attractive features is the ability to provide solutions to nonlinear boundary value problems (BVP) with accuracy close to the machine precision. This is done by the chebop class which provides automatic solutions by performing linearizations with a Newton method in function spaces of the nonlinear BVP, automatic differentiation, and using Fast Fourier Transform computations for the coefficients of the Chebyshev polynomials. A drawback of chebop automatic approach is the possible lack of convergence of the Newton method if the initial guess is not close enough to the exact solution. An explicit functional linearization done for each particular shape of the differential operator (i.e. without automatic differentiation) proves to be more robust than the chebop class and allows an enlargement of the range of convergence. Another alternative is the implicit L-scheme (quasi-Newton approach with derivatives replaced by suitable positive constants L), with a much simpler implementation and globally convergent. While chebop is the easiest way to solve the BVP, provided that it converges, the last two approaches largely overcome the convergence issues, yielding accurate solutions to a wide class of steady-state one-dimensional problems governed by Richards' equation. Chebfun2 and Chebfun3, which at the current stage cannot solve BVPs, provide efficient tools for accuracy and convergence assessments of the non-steady solutions in one or two spatial dimensions obtained by classical discretization schemes.

We compute explicitly the infimum of the norms of $W^{1,2}$-extension operators for planar sectors and exhibit an extension operator attaining this infimum. This solves an open problem posed by Maz'ya.

A maximal eventually different family is called $n$-productive if the product family $\mathcal{F}^n$ is still maximal. We construct closed $n$-productive families separating these strengthenings of maximality at every $n \geq 1$. Furthermore, we show how to force and construct an even stronger type of $\mathcal{I}_0$-productive family and discuss the relation of productivity to Van Douwen families.

We study the global-in-time continuation, past the singularity, of the smooth, non-isentropic, radially symmetric imploding solutions of the compressible Euler equations recently constructed by Chen, Shkoller, and Vicol. In three space dimensions, for all physically relevant adiabatic exponents $\gamma>1$, we consider the Euler solution that evolves smoothly until an implosion singularity forms at the origin at time $t=0$. We then prove that this solution can be uniquely continued for $t>0$ as a reflected outward-propagating shock, sometimes called a reflected blast wave. For $t>0$, the continuation is a globally forward self-similar weak solution of the Euler equations, selected by the Rankine--Hugoniot conditions and the Lax entropy inequality; it is smooth away from the expanding shock sphere and the spatial origin. The structure at the center of symmetry distinguishes these explosions from the classical Guderley reflected shock. In Guderley's continuation, the reflected blast wave leaves a point vacuum at the origin, where the density vanishes. The solutions constructed here exhibit the opposite behavior: for every fixed $t>0$ the density is unbounded at $r=0$ (though it remains locally integrable), while the pressure stays bounded and the temperature vanishes there.

We prove a weak version of the classical Hodge theorem on piecewise-algebraic spaces, a class of spaces introduced by Kontsevich and Soibelman in [KS00]. Precisely, we first prove the Poincare lemma that computes singular cohomology as a variant of de Rham cohomology. Then, as a weak Hodge theorem, we naturally embed the singular cohomology into the space of harmonic forms, instead of establishing an isomorphism (which does not hold for those spaces). Our approach in the latter is classical: Sobolev space theory. In addition, we give more detailed proofs for the claims in the appendix to [KS00]. This work is part of a program of extending arithmetic intersection theory to singular spaces. In particular, a type of currents in this singular setup is introduced.

We show that the von Neumann algebra generated by position operators in a 2-periodic interacting Fock space is a type $II_1$ factor. On the probabilistic side, we prove that the squared position operators have a Marchenko-Pastur distribution with respect to the vacuum state, yielding a natural realization of free Poisson laws within this framework.

We prove an average-degree lower bound on the independence number of uncrowded uniform hypergraphs. For every fixed integer $r\geq 2$ and every $\eta>0$, there exists $d_*=d_*(r,\eta)$ such that for every $d\geq d_*$, any uncrowded $(r+1)$-uniform hypergraph $G$ with $n$ vertices and average degree $d$ satisfies \[ \alpha(G)\geq (1-\eta)r^{-1/r}\left(\frac{\log d}{d}\right)^{1/r}n. \] The proof combines a cleaning procedure, which reduces the maximum $r$-degree to the average scale, with a random nibble that repeatedly extracts independent vertices while controlling all lower-order degrees created by the process. After an initial top-layer cleaning, we run a trace nibble. Since the residual hypergraph contains traces of all sizes $2,\ldots,r+1$, we track the maximum degrees in every layer. A binomial-type recurrence for this degree profile yields the stated leading constant.

We consider the relationships between the domination number of graph, denoted $\gamma$, and the distance-$2$ domination number, denoted $\gamma_2$, and three parameters that lie between them: the packing number, denoted $\rho$, the lower packing number, denoted $\rho_L$, and the isolation number, denoted $\iota$. There has been recent attention on the question of whether $\gamma/\rho$ is bounded or unbounded for various families of graphs. We consider similar questions for the ratios of the five parameters. In particular we show that, while $\gamma/\rho_L$ is unbounded in trees, it holds that $\iota/\gamma_2$ is less than $2$ for all trees. Further, $\gamma/\rho_L$ is at most $3$ in interval graphs, at most~$4$ in permutation graphs, and at most $5$ in general asteroidal-triple-free graphs. We also show that every tree has a set of vertices that is both isolating and a packing, and characterize trees where $\rho=\rho_L$.

The Loewner order on Hermitian matrices is a partial order that compares matrices in terms of positive semidefiniteness. The Loewner order plays a key role in many fields such as optimization, numerical linear algebra, control theory, operator theory, and quantum information. A fundamental difficulty is that two or more Hermitian matrices do not necessarily have a unique minimal upper bound (or maximal lower bound). In this paper, we propose an iterative method to exactly compute a minimal upper bound for any finite collection of $n\times n$ Hermitian matrices. It is shown that the algorithm terminates in at most $n$ iterations. The exactitude of the algorithm is proved using standard results from finite-dimensional linear algebra. A self-contained proof of an algebraic characterization of minimality originally explored by Stott is provided. We illustrate the algorithm in examples and also provide an implementation of the algorithm in Python.

This paper studies the probability of substructures occurring in random Latin squares. Our main result states that if $\alpha,\beta>0$ are such that $2\alpha+\beta<1$, then there are positive constants $\delta = \delta(\alpha, \beta)$ and $\Delta = \Delta(\alpha, \beta)$ such that if $P$ is a partial Latin square of order $n$ with $k = k(n)$ non-empty cells occupying at most $\alpha n$ rows and $\beta n$ columns, the probability that a random Latin square of order $n$ contains $P$ lies between $(\delta/n)^k$ and $(\Delta/n)^k$. We apply this result to subsquares in random Latin squares to obtain the first proof of the fact that the expected number of subsquares of order $3$ in a random Latin square of order $n$ is non-vanishing as $n \to \infty$. We are also able to provide the best known asymptotics for the expected number of subsquares of order $a$ in a random Latin square of order $n$ when $2<a=o(n^{1/2})$. Finally, we discuss the implications of our result on other configurations in random Latin squares as well as on completions of partial Latin squares.

We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs) in which Bellman-Kalaba quasilinearization reduces the nonlinear problem to a sequence of linear subproblems, each discretized by collocation onto a trial space that is linear in its parameters and solved by a single direct linear least-squares QR factorization. The trial space, which we term Linear-in-Learnables (LiL), comprises representations whose trainable parameters enter linearly, including random-feature extreme learning machines, spectral polynomial bases, and trigonometric expansions, each implemented as a physics-informed neural network. The method thus replaces the nonconvex gradient-based training that limits standard PINNs with a convex per-step solve. We establish local Newton-Kantorovich convergence of the outer iteration to a residual-limited neighborhood under an explicit smallness condition, with the limiting accuracy governed by the best-approximation residual of the trial space rather than by an optimization tolerance. The method, denoted LiL-Q, is assessed on seven benchmarks spanning scalar nonlinear PDEs (Bratu, viscous Burgers, Buckley-Leverett), coupled systems (plane-strain elasticity and the incompressible Navier-Stokes equations in two and three spatial dimensions), and steady-state Darcy flow with heterogeneous permeability. Across these problems, LiL-Q converges in single-digit outer iterations in most cases, even at the coarsest basis sizes and independent of the parameter count. When the exact solution lies in the span of the trial space, the method recovers it to machine precision in a single solve. On the Navier-Stokes benchmarks, it matches or exceeds published PINN solvers with up to two orders of magnitude fewer trainable parameters, without gradient-based optimization.

We develop a minimizing-movement framework for parametric finite element approximations of geometric gradient flows with admissible tangential motion. At each time step, the discrete variational problem combines a metric dissipation term for the normal displacement with a surface Dirichlet energy. The metric determines the normal geometric evolution: the $L^2(\Gamma)$ metric gives mean curvature flow, while the $H^{-1}(\Gamma)$ metric gives surface diffusion flow. Tangential velocity is selected independently through weak constraints on the deformation map. The central structural condition is admissibility, namely, that the identity map satisfies the constraint. This condition keeps the identity map available as a comparison function and yields the natural stability estimate. The framework recovers the classical Barrett--Garcke--Nürnberg (BGN) scheme from the unconstrained formulation and the dual minimal-deformation-rate (MDR) scheme from the MDR constraint. We further introduce two new admissible variants: an admissible BGN scheme and a relaxed MDR scheme. For the resulting fully discrete schemes, we prove existence and uniqueness under natural nondegeneracy assumptions and establish unconditional energy stability. Numerical experiments compare the admissible and classical schemes and illustrate their stability properties and mesh-quality behavior.

We prove a Trudinger-Moser type inequality in fractional Sobolev spaces with singularities on smooth compact sets of codimension $k$, where $1 < k < d$ and $sp = d$. The singular term is given by the inverse $d$-th power of the distance to the submanifold. The proof is based on a fractional Hardy inequality adapted to smooth submanifolds, and we show the sharpness of the constant. We also establish the equivalence of two natural fractional Sobolev spaces vanishing on the singular set.

We propose a machine-learning framework for detecting whether a given matrix is a perturbation of a matrix with a large Jordan block. The proposed model achieves high classification accuracy on synthetically generated, robustly perturbed data and outperforms a classical numerical baseline. Moreover, we demonstrate that the learned model generalizes to several classes of matrices not seen during training. These results suggest that the architecture captures structural properties associated with matrix defectiveness.

This paper extends classical conformal frameworks for constructing prediction intervals with global marginal coverage $1-\alpha$ to intervals that provide explicitly calibrated guarantees for the upper and lower tails separately. Focusing on split conformal prediction, we first construct lower and upper one-sided conformal intervals that achieve marginal validity, and then derive the induced two-sided interval by intersection. Theoretical results prove both tail-specific and global marginal coverage of the induced two-sided interval. Results are presented first for the exchangeable setting, where coverage has finite-sample guarantees, and then for non-exchangeable data, where guarantees are asymptotic. Simulation studies show that the proposed approach achieves improved directional calibration relative to classical two-sided intervals, especially relevant in skewed data. Finally, the benefit of the proposed framework is showcased in a financial application, where one aims for return maximization while seeking strict control on the left tail.

For a self-similar graph $(G, E)$, we find a distinguished subgroupoid of the associated path groupoid $\mathcal{G}_{G,E}$ -- the symmetric cycline subgroupoid $\mathcal{S}_{\text{sym}}$. If the acting group $G$ is abelian, we show that $\mathcal{S}_{\text{sym}}$ is open, abelian, and normal. For $G=\mathbb{Z}$, we describe the dual bundle $\hat{\mathcal{S}}_{\text{sym}}$ of $\mathcal{S}_{\text{sym}}$ which can be used to provide a different groupoid model for the self-similar graph $C^*$-algebra $\mathcal{O}_{\mathbb{Z}, E}\cong C^*_r(\mathcal{G}_{\mathbb{Z},E})$. For a large class of self-similar graphs $(\mathbb{Z}, E)$, we further prove that $\mathcal{S}_{\text{sym}}$ is maximal among open abelian subgroupoids of $\mathrm{Iso}(\mathcal{G}_{\mathbb{Z},E})^{\circ}$ and closed in $\mathcal{G}_{\mathbb{Z},E}$, so that it gives rise to a Cartan subalgebra of $\mathcal{O}_{\mathbb{Z}, E}$. This result seems new even for genuine actions. Our proofs heavily rely on careful studies of dynamical behaviours of cycline triples of $(\mathbb{Z}, E)$ and on a dynamical-flavour classification for the vertices of $E$. Some results hold in more general settings and may be of independent interest.

We review the current state of the Cherlin-Zilber Algebraicity Conjecture on simple groups of finite Morley rank, which states that every such group is the group of $K$-rational points of an algebraic group for some algebraically closed field $K$. We will explain the relevance of sharply 2-transitive groups as a potential source of counterexamples and how the Burnside problem necessarily comes into the picture.

The Beltrami-Vekua normal form assigns to every smooth first-order real planar elliptic system a complex equation $w_{\bar z}-\mu w_z+\mathcal{A}w+\mathcal{B}\bar w=\mathcal{F}$ by an explicit pipeline. A companion paper showed that the density $\Theta=|\mathcal{B}|^2/(1-|\mu|^2)\,dx\,dy$ and its total mass are invariants under multiplicative gauges $w\mapsto\phi w$ and orientation-preserving diffeomorphisms. The real system carries a larger symmetry: its unknowns may be recombined by any pointwise invertible real-linear substitution $w=\varphi v'+\psi\bar v'$, the complex gauges being the case $\psi\equiv0$. We prove the absorption theorem: re-normalizing through the pipeline after any such substitution returns to the gauge orbit of the original equation, with a universal explicit gauge $\tilde\varphi=-i\lambda/(\varphi-\psi)$, where $\lambda$ is the spectral root of the structure polynomial.

We study two inverse problems for a nonlinear dynamical Schrödinger operator with magnetic and electric potentials. Under suitable analyticity assumptions, we show that the Dirichlet-to-Neumann map uniquely determines time-dependent magnetic and electric potentials. We establish the uniqueness of these potentials from both full data and partial data. In particular, for the partial data problem, the desired uniqueness is established by assuming that the potentials are known near the boundary, and the Neumann data is measured on arbitrarily small open subsets of the boundary. In addition, we establish the well-posedness of the forward problem, where we obtain the optimal Sobolev regularity for solutions.

We study infinite-horizon optimal stopping for normally reflected two-dimensional diffusions in the positive quadrant with max payoff \(G(x_1,x_2)=x_1\vee\alpha x_2\). The non-smooth payoff produces a singular stopping-gain measure on the kink set \(\Delta=\{x_1=\alpha x_2\}\). We prove $\displaystyle \Gamma^\Delta(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+\alpha^2}}\,\sigma_\Delta(dx)$, with $n=(1,-\alpha)$, so the diagonal component is non-positive and strictly negative under local ellipticity. This implies that every interior kink point lies in the continuation region. We further show that the correct value representation uses the resolvent killed at first entry into the stopping set, $\displaystyle V=G-R_r^{\mathcal C}\Gamma$, and give a closed-form reflected Brownian counter-example showing that the unrestricted reflected resolvent is generally wrong. A reflected Brownian benchmark and numerical experiments illustrate the local-time, resolvent-gap, and diagonal-avoidance mechanisms.

We construct non-rigid tensor-triangulated categories with non-distributive lattice of thick tensor-ideals.

We study finite-horizon queue peaks in generalized switches, a standard stochastic-network model in which many queues share constrained service resources. Arrivals may be dependent, time-varying, and adapted to the past; the standing load condition is uniform interior slack, meaning the conditional mean arrival vector stays in a fixed contraction of the capacity region. We show that this slack reshapes the finite-time peak law for drift-minimizing scheduling policies such as MaxWeight. The square-root envelope that is sharp without slack persists only up to a geometry-dependent threshold; beyond that threshold, the running maximum grows only logarithmically with the horizon, both with high probability and in expectation. The mechanism is self-normalization: in the current queue direction, the projected fluctuation scale is normalized by the stabilizing drift scale. This removes capacity geometry from the logarithmic coefficient, while geometry remains in the threshold. Matching lower bounds show that both the logarithmic term and a geometric threshold are unavoidable. When finite-time state-space collapse is available, the threshold can be sharpened using local bottleneck geometry. For generalized input-queued switches, we obtain finite-time peak bounds with tight logarithmic coefficients. Simulations illustrate the two-phase envelope, local geometric refinements, and variance-sensitive improvements predicted by the theory.

Solutions of hyperbolic conservation laws exhibit both smooth structures across large scales and sharp localized features such as shocks and contact discontinuities, making them difficult to approximate accurately with existing neural operators. The Fourier Neural Operator (FNO) captures long-range interactions well but tends to smear localized structures through excessive numerical dissipation. To address this, we propose a Local-Global Neural Operator (LGNO) that learns a one-step discrete flow map by combining a global FNO branch for representing smooth dynamics at large scales with a local multiresolution branch for enhancing localized discontinuities and nonsmooth features. The model is trained with a one-step loss that combines a physical space prediction term and a spectral penalty on high frequencies to suppress spurious oscillations near steep fronts. On a large collection of benchmarks in one and two dimensions, LGNO consistently outperforms FNO baselines with matched parameter counts, reducing one-step errors by factors of 2-5 and remaining significantly more accurate over long autoregressive rollouts. Most strikingly, although it is trained only on short-time data from a high-order WENO-Z scheme, the long-time rollout of LGNO on a coarse $256^2$ grid exhibits lower numerical dissipation than the same WENO-Z scheme run on a finer $512^2$ grid, while being orders of magnitude cheaper to evaluate. These results suggest that, with an appropriate architecture and training objective, learned operators can effectively learn discrete flow maps. They further suggest that such learned operators have the potential to control long-time numerical dissipation better than the conventional shock-capturing schemes that generate the training data.

In this paper, we mainly investigate zero-sum problems over $\mathbb Z/n\mathbb Z$ (with $n>1$) of two new types. Let $s_1(n)$ (resp. $t_1(n)$) be the least positive integer $k$ such that for any integers $a_1,\ldots,a_k$ not divisible by $n$ (resp., relatively prime to $n$), there is an $I\subseteq\{1,\ldots,k\}$ with $|I|=n$ for which the sum $\sum_{i\in I}a_i$ is divisible by $n$ but not divisible by $n^2$. For $n\geqslant 4$, we prove that $2n+1\leqslant s_1(n)\leqslant n^2-2n+2$ and $2n-(-1)^n\leqslant t_1(n)\leqslant (n-1)\varphi(n)+1$. We conjecture that $s_1(n)=2n+1$ and $t_1(n)=2n-(-1)^n$ for any integer $n>2$.

We study derived categories associated with log del Pezzo surfaces whose singularities are of type $\frac{1}{3}(1,1)$. For such a surface $X$, we consider the canonical smooth Deligne--Mumford stack $\pi:\mathcal X\to X$ and compare it with the singular coarse surface $X$. Our main result proves that, if $X$ is a complex log del Pezzo surface whose singularities are all of type $\frac{1}{3}(1,1)$, then $D^b(\operatorname{coh}\mathcal X)$ admits a full exceptional collection. The proof combines rationality of log del Pezzo surfaces, Orlov's blow-up formula, and the special McKay correspondence of Ishii--Ueda. We then specialize to a general degree $10$ hypersurface $X_{10}\subset \mathbb P(1,2,3,5)$. The Corti--Heuberger cascade identifies its minimal resolution as $\widetilde{X}_{10}\cong \operatorname{Bl}8\mathbb F_3$, and therefore the canonical stack $\mathcal X_{10}$ has a full exceptional collection of length $13$. We also discuss the singular coarse category through the approach of Karmazyn--Kuznetsov--Shinder.

Let $(A,\mathfrak{m},k)$ be a Gorenstein ring of dimension $d\ge 1$, $N$ a perfect module of dimension $t\ge 1$ and $I$ an ideal of definition of $N$. For a non-free maximal Cohen-Macaulay (=MCM) $A$-module $M$ and an integer $i\ge 1$, it is well known that the functions $n \mapsto \ell(Tor_i^A(M,N/I^{n+1}N))$ and $n \mapsto \ell(Ext^i_A(M,N/I^{n+1}N))$ are of polynomial types of degrees $r_i^{I,N}(M)$ and $s_{I,N}^i(M)$, respectively. We prove that $r_i^{I,N}(M)\le t-1$ and $s^i_{I,N}(M)\le t-1$ and when $I$ is the maximal ideal $\mathfrak{m}$, both the inequalities become equalities. We also show that $r_i^{I,N}(M)\le r_1^{I,N}(\Omega^dk)$, $s^i_{I,N}(M)\le s^1_{I,N}(\Omega^dk)$ and $r_i^{I,N}(\Omega^dk)=r_1^{I,N}(\Omega^dk)=s^1_{I,N}(\Omega^dk)=s^i_{I,N}(\Omega^dk)$. \end

We study the capacity of a quantum channel for retrocausal communication, where messages are transmitted backward in time, from a sender in the future to a receiver in the past, through a noisy postselected closed timelike curve mathematically represented by the channel. We completely characterize the one-shot retrocausal quantum and classical capacities, and we show that the corresponding asymptotic capacities are equal to the average and sum, respectively, of the channel's max-information and its regularized Doeblin information. This endows these information measures with a novel operational interpretation. Furthermore, our characterization can be generalized beyond quantum channels to all completely positive maps. This imposes information-theoretic limits on transmitting messages via postselected-teleportation-like mechanisms with arbitrary initial- and final-state boundary conditions, including those considered in various black-hole final-state models.

We revisit the variational principle for relativistic perfect fluids in a manifestly covariant formulation based on differential forms, with particular attention to the boundary data required for a well-posed action principle. For timelike flows, the formalism is largely a geometric reformulation of the Schutz action principle for perfect fluids. We then analyse the extension of the same variational principle to null flows. In that case, the system is not a generic perfect fluid: the equations of motion force the enthalpy density to vanish, $\rho+P=0$. The resulting stress-energy tensor decomposes into a vacuum energy-like term with variable pressure and a null dust contribution. This shows that the obstruction to the naive fluid extension is dynamical rather than kinematical. Since the matter action is formulated independently of any gravitational field equations, the construction can be generalised to first-order or non-metric theories of gravity.

Highly accurate stability boundary values for the ADER-DG method are obtained for arbitrary degrees $N$ of basis polynomials. In the linear case, stability is violated precisely when one of the matrix eigenvalues reaches $\lambda = -1$, regardless of the phase $\theta$. A rigorous mathematical framework for the stability is developed. The stability condition is significantly simplified, reducing it to the problem of calculating the roots of polynomials in the Courant number $\mathrm{CFL}$. The maximum of the Courant numbers $\mathrm{CFL}_{\rm max}(N)$ are calculated. These results are new and very convenient for practical use. A comparison of the obtained results with existing results reveals differences that may be significant for the selection of calculation parameters, especially for high degrees $N$. It is shown that widely used existing estimates $\mathrm{CFL}_{\rm max}(N) \propto 1/(2N+1)$ are overestimated. An interesting qualitative asymptotic $\mathrm{CFL}_{\rm max}(N) \propto (N+1)^{2}$ is obtained. A rigorous direct proof of the approximation is presented. Approximation orders $p = N+1$ for arbitrary degrees $N$ are rigorously derived. A set of numerical experiments is carried out to apply the ADER-DG method to solving both a linear advection equation and an Euler system of equations. The results obtained in these calculations confirm the theoretical results well. In particular, an excess of the Courant number over the $\mathrm{CFL}_{\rm max}(N)$ by even 1% in the linear case immediately leads to significant instability of the numerical solution. The obtained estimates of the boundary Courant number in the nonlinear case are somewhat underestimated -- by no more than 5%, which is due to the diffusivity and stability of the approximate Riemann solver. Empirical convergence orders are obtained, which are in good agreement with the theoretical results.

We describe two self-replicating mechanisms of energy accumulation in the seismic focus, which modify the Gutenberg-Richter law in the region of the big earthquakes. The first mechanism acts for magnitudes smaller than a narrow region of large critical magnitudes; it slows down the energy accumulation and may produce precursors. The second mechanism acts above that region, and accelerates the energy accumulation; the precursors may be absent. Both mechanisms reduce the Gutenberg-Richter excedence distribution. On the left of the critical region the Gutenberg-Richter magnitude probability density is unchanged, while on the right the probability density is reduced. The procedure described in this paper introduces a critical-magnitude region (range) as an additional fitting parameter. The results may bear relevance upon the recent concepts of "self-arresting" and "dragon-king" earthquakes. The two self-replicating mechanisms may introduce a magnitude gap between the two types of big earthquakes, and two branches in the excedence law, in the vicinity of the critical region.

Over-the-air federated learning (FL) leverages the superposition property of multiple-access channels to enable communication-efficient distributed model training. Existing integrated sensing, communication, and computation (ISCC)-enabled over-the-air FL systems typically require dedicated resources for the sensing module, inevitably compromising FL performance due to resource competition. In this paper, we propose a sensing-native over-the-air FL framework that explores built-in distributed wireless sensing capability with zero overhead per model aggregation. Specifically, the high-dimensional local gradient signals possessing favorable autocorrelation property are concurrently leveraged for target distance estimation, while the gradient statistics already required for over-the-air FL serve as a ready-made gateway to deliver locally-sensed results to the edge server for cooperative localization. To combat inter-device interference, channel fading, and communication noise, we put forth a robust trilateration-based target positioning method building upon an efficient matched-filtering-based distance estimation. Then, by explicitly characterizing the impact of imperfect model aggregation and noisy gradient-statistics transmission on the sensing-native over-the-air FL convergence, we develop a statistics-aware communication-learning co-design approach. We first derive the closed-form optimal power budgets allocated to local gradients and their statistics, based on which an efficient successive convex approximation method is proposed for receiver beamforming optimization. Simulation results show that the proposed framework simultaneously achieves superior learning and sensing performance compared to representative baselines.

We formulate and analyze a sex- and stage-structured model for Anopheles dynamics under the sterile insect technique (SIT), motivated by the need for tools robust to insecticide resistance and outdoor transmission. The model tracks aquatic stages, adult males, unmated females, and females mated with wild or sterile males; includes egg-laying capacity and larval competition; and uses a refractory period followed by density-dependent mate search. The resulting Holling type-II mating term generates a mate-finding Allee effect. After establishing well-posedness, we prove that this Allee effect makes the mosquito-free equilibrium locally stable for all admissible parameters and globally asymptotically stable when a quick-mate-search reproduction number $R_0^q$ is below one. When $R_0^q>1$, habitat capacity is large, and larval competition is weak, two positive equilibria arise through a saddle-node bifurcation: a stable natural equilibrium and an unstable Allee equilibrium separating persistence from extinction. For a reduced model, a Goh-Volterra Lyapunov functional estimates the persistence basin. We then show how constant and population-responsive sterile-male releases reshape this bistability. Sufficiently large releases annihilate the positive equilibria in a second saddle-node bifurcation, while a sufficiently large constant release drives local elimination from every admissible initial state. Thus SIT need only push the population across the Allee separatrix, after which mate-finding failure can complete extinction. In a free-horizon optimization framework with an Allee-threshold stopping rule, a hybrid release strategy reduces the sterile-male requirement by about $5\%$ relative to the best constant-only strategy and $39\%$ relative to the best population-responsive-only strategy. These results recast the Allee effect as a control lever for vector suppression.

Organizations and researchers show increasing interest in using large language models (LLMs) in place of human participants in A/B tests, in the hope of experimenting faster and at lower cost. We study when a treatment effect estimated on LLM outcomes recovers the effect that would have been measured on the human population of interest. Distributional equivalence between LLM and human outcomes would make any standard estimator valid but is unrealistic. We therefore develop a statistical framework that adapts surrogate endpoint theory to LLMs. The framework shows that calibrating LLM outcomes to human outcomes identifies the average treatment effect under surrogacy and comparability conditions that are jointly weaker than distributional equivalence. When these conditions fail, the effect of interest is only partially identified, and we provide diagnostics that can falsify surrogacy on historical experiments together with a bound on the worst-case bias from limited overlap. We further show that the stochasticity inherent to LLMs introduces both bias and variance, but using an average of multiple draws as the surrogate mitigates both. We illustrate the methods and theory in simulations and an application to A/B tests on Upworthy headlines. A central takeaway from our work is that the validity of LLM outcomes as surrogates can only be falsified for past treatments and never verified for new ones, so human experiments remain indispensable for novel interventions. We discuss the role of LLM choice, prompting, and temperature as design variables, and how to size human experiments for validation.

We investigate the thermal partition and one-point functions of the three-dimensional conformal field theory dual to a massive interacting scalar field in AdS$_4$. Using thermal inversion formulas, we determine the asymptotic behaviour of the spectral density and OPE coefficients involving heavy operators at fixed spin. We first analyse these CFT data for the generalised free field, corresponding to the non-interacting bulk theory. Then we compute the first-order perturbative corrections induced by the cubic and quartic bulk interactions. The thermal observables considered here probe a sector associated with operators of large dimension and, in the bulk description, a regime dominated by states with large particle number. This regime remains comparatively unexplored even in generalised free field theory. Remarkably, the asymptotic formulas obtained from thermal inversion remain quantitatively accurate far from the asymptotic regime, describing CFT data reliably already at intermediate conformal weights. Our results show that this feature survives the inclusion of bulk interactions and provide new analytic control over heavy-state data in conformal field theories.

Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.

Farrow structures based on linear--phase FIR subfilters provide an efficient realization of variable fractional--delay (VFD) filters with reduced implementation complexity. While the all--linear--phase configuration admits a decoupled least--squares (LS) formulation with an analytical solution, this decoupling fails when branches of mixed types, linear--phase and general FIR, are required, as occurs when a group--delay constraint is imposed. This letter presents a unified LS design for Farrow structures via a nullspace parameterization of the per--branch symmetry constraints, yielding an analytical solution that accommodates arbitrary per--branch types. Numerical results demonstrate that the proposed framework satisfies group--delay constraints that the all linear--phase approach cannot meet, while substantially reducing the number of free parameters relative to the unconstrained general FIR baseline.

We investigate the nonlinear dynamics of a single-component plasma confined in a time-dependent harmonic trap regarding aspects of symmetry and invariant functions. The system is described as a fluid in an isentropic adiabatic regime by a system of partial differential equations. A convenient change of variables, with a Gaussian ansatz for the number density distribution, allows a consistent mathematical description in terms of ordinary differential equations, from which we follow up with an analysis concerning the corresponding differential operators algebraic structure and Noether symmetries in specific physical regimes. For each studied case, proper invariants are identified. The obtained conserved quantities capture an interplay between the internal plasma dynamics and the time modulation of the trap, resulting in a sharp restriction for the system evolution.

Starting from a spectral problem posed in a Clifford algebra with $d$ generators and Euclidean signature, we study an integrable, coupled system of PDEs that can be viewed as a vector perturbation of the Camassa--Holm equation with residual orthogonal symmetry. In the two-component case $d=2$, we show that the travelling wave solutions correspond to a Liouville integrable Hamiltonian system with two degrees of freedom, making use of a reciprocal transformation linking the coupled PDEs to a symmetry of the Hirota--Satsuma system. We also present a symmetry classification of all integrable two-component perturbations of Camassa--Holm, and find that besides the $d=2$ system analyzed here, the coupled 2CH system studied by Olver and Rosenau (as well as by Chen, Liu and Zhang, and Falqui), and equations related to either of those systems by Miura transformations, we also obtain a new system that (to the best of our knowledge) has not been reported previously. For the case of an arbitrary number of components $d$, we additionally investigate the short-pulse (high-frequency) regime, in which the limiting dynamics are governed by a vector-valued Hunter-Saxton type system. Furthermore, we provide a detailed analysis of the corresponding measure-valued (weak) solutions associated with this system.

We study sparse semi-continuous tensor data with excess zeros, heavy right tails, and slice-specific dispersion. Such features arise naturally in monetary-valued multi-way data, such as international trade, where most exporter--importer--product--year cells are zero while positive values are continuous and highly variable. To model these data, we propose a Bayesian hierarchical tensor factorization model that places a low-rank CP structure on a latent Poisson rate tensor and couples it with a conditional Gamma model for positive outcomes, with rate parameters that can vary across slices within a mode. The model therefore separates the occurrence and magnitude of positive observations while borrowing strength across all tensor dimensions through a shared low-rank latent structure. To scale posterior inference to large arrays, we develop a hybrid variational--Monte Carlo algorithm that combines efficient coordinate ascent updates with a partially collapsed augmented-data sampler. Applied to approximately 60 million trade flows, the method surfaces multiway dependence across exporters, importers, products, and years that is difficult to recover from gravity-type or pairwise network analyses, which do not jointly model the product and temporal dimensions.

Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes, including weight-6 codes with parameters $[[48,8,6]]$, $[[96,8,10]]$, and $[[224,12,16]]$, as well as weight-8 codes with parameters $[[84,16,8]]$, $[[112,16,10]]$, $[[128,16,12]]$, and $[[168,16,15]]$. Furthermore, we introduce a maximally packed syndrome extraction schedule of depth $w+2$, including initialization and measurement steps, for any code with a maximum stabilizer weight of $w$ from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of $\approx0.65\%$ for the weight-6 family and $\approx0.35\%$ for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.

In this paper, a novel optimal soft-landing guidance law with inequality approach-angle path constraints is analytically derived. The proposed guidance law prevents ground collision and enables approach-angle control by constraining the optimal trajectory to remain within a convex inverted pyramid originating at the landing point. A 3D point-mass linear kinematic model in a constant gravitational field is employed, together with a quadratic control-effort cost and terminal constraints on position and velocity. Analytical open-loop and closed-loop solutions, together with the optimal final time, are derived using Pontryagin's Minimum Principle and the optimality conditions at the transitions between unconstrained and constrained arcs. It is additionally shown that the optimal final time decreases when the path constraints become active. The resulting guidance law is continuous, piecewise linear in time, and nonlinear in the states in closed-loop. When a constraint becomes active, the controller cancels the gravitational component normal to the constraint, causing the trajectory to evolve along the constraint surface. The proposed guidance law is evaluated in simulations under various initial conditions, demonstrating accurate landing performance and consistent satisfaction of the path constraints.

Perfect resource placements in dense Eisenstein--Jacobi (EJ) networks partition the network into hexagonal radius-$t$ service cells. This paper studies local repair of such placements after resource failures. For one failed resource, we prove that one replacement cannot cover the failed hexagon and two always suffice, giving $\rho_{\mathrm{EJ}}(t)=2$ for all $t\ge1$. Among minimum-size repairs, the sharp minimum-overlap formula $\Omega_{\mathrm{EJ}}(t)=t^2$ follows from the three-strip geometry of EJ balls. For two failed resources, independent repair gives a four-replacement upper bound, but unlike the Gaussian case EJ repair is not always additive: two infinite neighboring displacement families admit three-replacement repairs, proved optimal by a two-ball impossibility argument. Additive behavior is established algebraically via endpoint-rigidity and diagonal-corridor theorems. For $q$ failed resources, independent canonical repair gives a universal $2q$ upper bound, exact when failed cells are pairwise more than $4t$ apart. Dense cluster subadditivity is proved for infinite four-fault and six-fault families with exact repair numbers four and five, giving savings of four and seven over independent repair. An exact inclusion--exclusion identity governs repeated coverage for arbitrary multi-fault repairs. An audit over 19,400 instances confirms widespread subadditivity. EJ local repair is structurally distinct from the Gaussian case: the one-fault overlap is quadratic, two-fault repair can be non-additive, and clustered repairs reuse replacement balls across multiple failed cells.

In survival analysis, dependent censoring poses significant challenges in accurately estimating model parameters and survival functions. This study introduces a novel framework leveraging Extended Generalized Marshall-Olkin (EGMO) models to address dependent censoring mechanisms. Geometric optimization techniques are employed to develop efficient estimation procedures that capture dependencies between failure and censoring times. We establish their asymptotic properties. Simulation studies and real data applications illustrate the method's robustness and effectiveness.

In this paper we study the class of graphs without cycles of size 4 and independent sets of size 4 as induced subgraphs: $\mathop{Free}\{C_4, 4K_1\}$. This is one of the three minimal minimal open cases for the complexity of the colouring problem when restricted to classes defined by excluding induced subgraphs of order 4. We investigate the clique width of some subclasses of $\mathop{Free}\{C_4, 4K_1\}$. We introduce a new framework: the $(k,l,m)$-decomposition and prove that if all the graphs of a class $\cal G$ are $(k,l,m)$-decomposable, then graphs in $\cal G$ have bounded clique width. We give a few examples of such class, found with the help of a program we designed. We also show, for any graph $G \in \mathop{Free}\{C_4, 4K_1\}$ that is 3 cliques coverable, an infinite family in $\mathop{Free}\{C_4, 4K_1\}$ of supergraphs of $G$ which have unbounded clique width.

Real-time trajectory generation for on-orbit robotic servicing is challenging due to the nonlinear coupling between spacecraft bus motion, manipulator dynamics, visibility cone, and trajectory-level safety constraints. This paper studies learning-based warm-starting for sequential convex programming (SCP) in the terminal approach of a space manipulator toward a tumbling target. The proposed framework decomposes the problem into a system center-of-mass translational planning stage and a coupled attitude--manipulator torque-allocation stage, and applies a causal transformer warm-start to the latter, which constitutes the dominant computational bottleneck. Linear and flow matching action decoders are compared under different action-chunking and training dataset sizes, and the resulting warm-starts are evaluated under both cost-optimal and feasibility projection using SCP. Across 300 held-out scenarios, the learned warm-start reduces the second-stage SCP iteration count by up to 28% and the runtime by 23% while preserving the final control-cost distribution. When the learned warm-starts are used for nonconvex feasibility projection, they nearly halve the runtime relative to cost-optimal SCP, while avoiding the catastrophic high-cost tail behavior observed when initialized heuristically. These results indicate that sequence-model warm-starts can improve both the computational efficiency and trajectory robustness of optimization-based terminal guidance for space manipulation.

Motivated by the optimization of bounded binary black-box functions, we study the problem of learning polynomial surrogates over the Boolean hypercube. To ensure that optimizing the surrogate yields good solutions for the underlying objective, we require uniform $L_\infty$-error guarantees rather than the usual $L_2$-type guarantees. We characterize the minimax sample complexity of uniform estimation under subgaussian noise for two classes of bounded polynomials. First, for polynomials of degree at most $d$ on $n$ variables, the sample complexity scales as $n^{d+1}$. Second, for $s$-sparse Fourier-Walsh polynomials with $s \leq n$, it scales as $ns^2$. These rates differ structurally from the noiseless setting, where uniform exact recovery scales as $n^d$ and $ns$, respectively. Our lower bounds hold even for arbitrary adaptive learners, showing that the additional factors are intrinsic to the noisy cases. Standard Fourier-analysis tools for the $L_2$-norm do not naturally extend to the $L_\infty$-setting in a way that yields uniform guarantees. Our proofs overcome this difficulty by relying on suitably chosen auxiliary norms that serve as proxies for controlling the $L_\infty$-error. Together, our results provide a tight characterization of the sample complexity of learning optimization-safe polynomial surrogates.

The stationary hydrogen atom has Coulomb degeneracy across orbital levels, whereas the Aufbau/Madelung ordering is an empirical, many-electron rule established in atomic physics. We examine the hydrogen atom through a regularized de Broglie--Bohm representation, in which stationary amplitude current constraints generate separable Sturm--Liouville branches. In this formulation, the radial, orbital, and magnetic sectors acquire canonical Langer-like inverse square corrections. The modified boundary value problems allow analytical solutions and produce a hydrogen-like spectrum with regularized radial and angular indices. Consequently, radial Coulomb quantization acquires an orbital dependent shift, lifting the Coulomb degeneracy and producing a spectral ordering that follows the Aufbau/Madelung sequence. On this basis, we construct the ordering of the regularized de Broglie--Bohm states and show that the spectral structure retains the standard degenerate Rydberg sequence in the l=0 sector. The separated amplitudes are represented by generalized special function branches, including the associated Laguerre, Legendre, and Bessel functions with non-integral parameters arising from regularized separation. Therefore, the treatment is intended as an analytical examination of spectral ordering in a regularized one center Coulomb problem rather than as a replacement for the many electron atomic structure theory. Keywords: de Broglie--Bohm representation; Coulomb spectrum; canonical regularization; Langer correction; Sturm--Liouville equations; Aufbau principle; Madelung ordering; associated Legendre functions; associated Laguerre functions; Bessel functions.

Autonomous spacecraft rendezvous and proximity operations (RPO) require controllers that guarantee safety under thrust constraints while minimizing fuel expenditure. Input-constrained control barrier functions (ICCBFs) provide a control method for nonlinear systems with actuation constraints that construct a forward-invariant safe set. Previous work has shown that learning class-$\mathcal{K}$ functions defining the ICCBF recursion via meta reinforcement learning (meta-RL) yields a robust, non-greedy approach to safety-critical control in RPO. This paper extends that framework further by investigating the performance of three recurrent network architectures (Long Short Term Memory (LSTM), Gated Recurrent Unit (GRU), Selective State Space Model (Mamba)) and two training algorithms (Proximal Policy Optimization (PPO) and Soft Actor Critic (SAC)) to identify the best setup for tuning ICCBF class-K functions via meta-RL. In addition to cooperative test cases, performance is evaluated in the presence of adversarial behavior where the target spacecraft behaves in a way that worsens the safety of the chaser spacecraft. Results indicate that state space models such as Mamba when used with PPO achieve superior task completion, safety, and fuel-savings compared to other architectures, across all cooperative and uncooperative scenarios tested.

We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms. In contrast to standard operator-learning settings with a single input function, our framework allows multiple input functions defined on possibly different domains, with different dimensions and Sobolev regularities. The derived rates explicitly quantify the contribution of each input space to the final error bound. In particular, in the balanced regime, the approximation and generalization rates are governed by the interaction between the input dimensions, regularities, and Sobolev orders, while the dependence on the model complexity retains a \(\log\log/\log\)-type structure. Our analysis provides a general theoretical framework for multi-input operator learning, including Sobolev training, and is applicable to operator learning problems arising from partial differential equations and scientific computing.

We consider the concentration properties of functions of infinitely exchangeable random variables. By conditioning on the de Finetti directing measure, we show that the deviation of any function with bounded-difference constants $c_1, \dots, c_n$ decomposes into a conditional sampling fluctuation and a latent mixture fluctuation. When this latent mixture is $\sigma_{\mathrm{mix}}^2$-subgaussian, we establish a concentration inequality with an effective variance proxy of $\frac{1}{4}\sum_i c_i^2 + \sigma_{\mathrm{mix}}^2$. Crucially, we demonstrate that for zero-sum linear contrasts, such as the difference between a subsample mean and a full population mean, the latent mixture term cancels exactly. This cancellation yields a tight, mixture-free Hoeffding-type bound that provides a direct de Finetti mechanism for the infinite-extendibility limit of recent finite-exchangeable concentration results. We apply this framework to quantify uncertainty in composite AI benchmarks, such as MMLU, where question items naturally exhibit exchangeable dependence across domains. Our results provide both a domain-stratified hierarchical model for bounding the uncertainty of accuracy scores, and a distribution-free, cost-saving statistical guarantee for accurately estimating full benchmark scores from random subsets.

Neural operators are widely used as surrogate solution maps for partial differential equations (PDEs), but full-size models can be costly to store, deploy, and evaluate in many-query scientific workflows. This work introduces Operator Boosting, a stagewise residual-learning framework for constructing compact neural-operator surrogates directly, rather than training a large model and compressing it afterward. Starting from the empirical mean predictor in normalized output coordinates, the method trains a sequence of tiny same-family neural operators on residual fields and incorporates each correction through validation-selected shrinkage. We instantiate the framework with Fourier neural operators (FNOs), DeepONets, and convolutional neural operators (CNOs), and compare boosted tiny stacks against full-size monolithic baselines across one-, two-, and three-dimensional PDE benchmarks from PDEBench, APEBench, and The Well. Across 30 dataset-architecture pairs, 21 show positive mean accuracy gains and 17 have positive confidence intervals, while all boosted stacks reduce trainable parameter count by approximately 72-95%. Best-model comparisons show empirical Pareto improvements on 7 of 10 completed PDE benchmarks, including two-dimensional Navier-Stokes, shallow-water dynamics, Darcy flow, one-dimensional transport and reaction systems, and three-dimensional compressible Navier-Stokes. These results show that Operator Boosting often improves the empirical accuracy-parameter Pareto frontier of neural PDE surrogates, while also exposing PDE- and architecture-dependent regimes where residual boosting fails to offset compression.

This paper presents a unified framework for exploiting the boundaries of reconfigurable intelligent surfaces (RIS) joint optimization in multi-user wireless systems, where a single RIS accommodates diverse this http URL first propose an adaptive gradient-scaling mechanism that accelerates the convergence of the underlying optimization algorithm while maintaining stable performance across varying channel and system parameters. The proposed mechanism enables the solver to reach a reasonably good solution rapidly without requiring manual tuning of step sizes or algorithmic hyperparameters when system inputs change. We then propose a low-complexity beamformer recovery method tailored for single-user scenarios, which circumvents the full matrix decomposition required by traditional approaches, thereby significantly reducing computational overhead. Building on these foundations, we develop an element allocation strategy that enables user-specific prioritization through assignment of RIS subsets. This is further extended by a modular add-drop mechanism that supports partial-panel optimization in general multi-user settings. The framework is evaluated across three representative scenarios: (i) signal amplification for all users, (ii) signal suppression for all users, and (iii) selective amplification and suppression. To characterize performance limits, we derive power trade-off boundaries using scalarized joint optimization, which closely align with Monte Carlo simulations. Our unified joint optimization method consistently yield solutions near these boundaries, confirming its near-optimality. Extensive simulations under realistic channel models demonstrate that the proposed approach outperforms conventional semidefinite relaxation techniques, offering a scalable and effective RIS control strategy for cooperative and competitive multi-user environments.

Perfect resource placement in dense Gaussian networks partitions the network into Lee balls centered at resource nodes. The fault-free placement problem is already classified; this paper studies the complementary post-deployment problem of repairing such placements after resource faults. The paper gives exact local repair theorems for the dense Gaussian placement generated by $t+(t+1)i$; by conjugation and rotation symmetry, the same results hold for the companion generator $(t+1)+ti$. For one failed resource, we prove failure-cell locality, derive the exact replacement number $\rho_G(1)=3$ and $\rho_G(t)=2$ for all $t\ge2$, and prove the sharp minimum-overlap formula $\Omega_G(t)=t+1$ among minimum-size repairs. The overlap lower bound is proved from the corner structure of equal-size Lee balls in the rotated coordinates $u=x+y$ and $v=x-y$, where Gaussian Lee balls become parity-constrained squares. For two failed resources, we prove exact additivity: every pair of failed resource cells requires exactly four local replacements for $t\ge2$, and four always suffice. The two-fault lower bound reduces all relevant resource displacements to two canonical neighboring cases and exhibits four mutually incompatible failed-cell corners in each case. For multi-failure repairs, we prove a general inclusion--exclusion identity for overlap inside the failed region; hence the formula remains exact for arbitrary higher-order dense cores. When a canonical repair instance is certified to have maximum multiplicity three, the identity reduces to the compact correction $\Omega_{\rm extra}=P_2-A-C_3$. A ground-truth audit over 7,494 Gaussian cases recomputes coverage from lattice geometry, verifies all exact formulas, and records reproducible multiplicity witnesses.

Dense Gaussian networks are degree-four algebraic interconnection networks with compact diameter and simple modular routing. This paper studies non-redundant one-to-all broadcast repair in the dense Gaussian network generated by $\alpha=k+(k+1)i$. We propose multi-orientation edge-minimum repair (MOEM), which evaluates a constant-size family of Gaussian broadcast-tree orientations, selects a fault-aware orientation, contracts the fault-pruned tree into healthy components, and reconnects those components using external component-crossing repair edges. The resulting structure is a rooted spanning tree of the healthy subgraph, so each healthy node receives the message exactly once and no faulty node is used. We prove that, for a chosen orientation with $c$ fault-pruned components and a connected healthy component graph, the repair step is non-redundant and uses the minimum possible number $c-1$ of external component-repair edges. We also prove that, for every one- or two-fault placement, the MOEM orientation family contains a repair with depth at most $k+2$. The depth proof combines a certificate framework, an explicit four-case off-axis analysis, and a five-component orthogonal-axis certificate. Exhaustive validation for $k=5,\ldots,10$ and large-scale validation through $k=200$ confirm the implementation and show that random two-fault repairs use approximately two external repair edges.

We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

Movable antennas (MAs) are a promising technology to improve wireless data rates by dynamically adjusting their positions to avoid deep fading. However, finding the optimal MA positions requires full channel state information (CSI) for all possible locations within the movement region, creating massive channel estimation overhead. This paper proposes a deep neural network (DNN)-based learning framework to predict the optimal positions of multiple transmit MAs in a multi-user multiple-input single-output (MISO) system, entirely bypassing explicit channel this http URL, we analyze a single-user MISO case, revealing a complex, highly nonlinear mapping between the optimal MA positions and the channel power gains from a specific subset of locations in the transmit region to the user. Because this mapping cannot be mathematically characterized for practical channel models, we train a DNN via supervised learning to capture it. The pre-trained DNN can then determine optimized MA positions in real-time relying only on partial power measurements from the transmit this http URL this to multi-user scenarios is challenging due to complex rate expressions and the lack of globally optimal position solutions to use as training labels. To overcome this, we develop an unsupervised training framework that directly maximizes the multi-user sum-rate. This framework utilizes an attention-based architecture to extract latent features from the partial channel measurements and effectively manage inter-user interference. Simulation results show that our proposed approach achieves near-optimal performance in single-user systems and surpasses conventional CSI-based alternating optimization algorithms in multi-user environments.

Based on the Wigner-Vlasov formalism, this paper investigates the kinematic properties of the Pauli equation. It is shown that the probability current associated with the Pauli equation can be represented as a superposition of two currents with certain expansion coefficients. Each of these currents corresponds to a particular component of the spinor. The expansion coefficients effectively serve as weighting functions that determine the probability contribution of the corresponding spinor component. Therefore, each spin projection corresponds to its own probability flux. A new system of the Hamilton-Jacobi equations and also a system of motion equations in electromagnetic fields are obtained, taking into account the interaction between the spin and the magnetic field. To illustrate how these equations can be applied we have investigated the quantum system kinematics in detail using an exact solution of the Pauli equation in the presence of a uniform magnetic field and an asymmetric quadratic potential.

Prey aggregation is widely regarded as a defense against predation, yet we show that in disease-structured populations subject to predator-induced fear and demographic Allee thresholds, aggregation can paradoxically accelerate ecosystem collapse. We develop and analyze a susceptible-infectious-predator model incorporating dual fear responses -- together with a sublinear aggregation-based predation term and an Allee effect. Critically, we derive an explicit upper bound on the extinction time that decreases as predator pressure increases or aggregation strengthens, quantifying for the first time how behavioral and demographic parameters jointly determine the speed of ecological collapse. This finite-time extinction subsequently triggers a cascade collapse of the infected prey and predator populations, driving the entire ecological community to extinction. Bifurcation analysis reveals transcritical, saddle-node, and Hopf bifurcations as fear intensity, aggregation strength, and Allee threshold vary. Two-parameter continuation further identifies the precise regions of the fear--Allee parameter plane in which stable coexistence, oscillatory coexistence, predator exclusion, and finite-time extinction occur, demonstrating that stronger aggregation monotonically enlarges the finite-time extinction region while weaker aggregation supports a richer landscape of coexistence dynamics. These results demonstrate that behavioral defenses operating at the population level can generate abrupt ecological tipping points when they interact with disease dynamics and demographic vulnerability.

Most quantum gravity theories propose that the fundamental concept of space-time is mostly compatible with quantum theory in noncommutative (NC) space. In the present paper, we revisit the notion of entanglement induced by NC deformations of phase space. The positive partial transpose (PPT) criterion for separability of bipartite Gaussian states is extended to a general class of Bopp's shift. In particular, we have considered both the position-position and momentum-momentum noncommutativity, with deformation parameters $\theta$ and $\eta$, respectively. It turns out that $\theta$ and $\eta$ induce the entanglement. We have directly applied the formalism for an anisotropic two-dimensional harmonic oscillator. Peres-Horodecki separability condition leads to a constraint equation for the parameter values of the oscillator in NC space. It turns out that the bipartite Gaussian state is almost always entangled in deformed space. To implement the theoretical idea, we provide an outline for a gedankenexperiment to identify the signature of phase-space noncommutativity, i.e., quantum gravity. In particular, the gedankenexperiment is devised to test the separability of supposedly separable Gaussian states in the usual commutative space, through the covariance matrix, which is constructed via measured output photocurrents after interaction of input Gaussian states and reference states. If the experiment shows that the supposedly separable states are actually entangled, then the entanglement is created through the intermediate background noncommutative space, which is a signature of the quantum nature of gravity.

This paper considers the problem of a low-thrust spacecraft pursuit-evasion differential game with an arbitrary terminal relative speed constraint. It addresses the terminal phase of the engagement for two relatively close spacecraft near a circular orbit. The problem is formulated as a linear-quadratic zero-sum differential game, with soft constraints on the terminal relative position and velocity, and running costs on the players' control efforts. An analytical, closed-loop, minimum-fuel-consumption optimal guidance law is derived for each player, forming a saddle-point solution. It is proven that any terminal speed can be achieved by properly choosing the weighting parameters of the cost function. To verify the optimality of the solution, a conjugate point analysis is performed when the cost function velocity weighting matrix is either positive or negative definite. The negative-definite case arises at high terminal speeds and is seldom seen in the literature. The performance of the derived guidance law is evaluated in simulations for different target maneuvers and compared to a state-of-the-art optimal-control-based guidance law. The simulations show that the derived guidance law satisfies the constraints and offers a substantial advantage over the optimal-control-based guidance law when the target is optimally evading.

Mediation analysis is essential for decomposing the causal effect of a treatment into direct and indirect pathways. However, many practical settings rely on the stringent assumption that recanting witnesses, defined as treatment-induced mediator-outcome confounders, are either absent or fully known a priori. Such a requirement is often untenable, especially when these variables remain unobservable due to measurement difficulties or privacy constraints. In this paper, we leverage proximal causal inference to develop three novel identification strategies to address the challenge of identifying path-specific effects in the presence of unknown recanting witnesses. Building on this, we develop a semiparametric inference framework that derives the efficient influence function and proposes a proximal multiply robust estimator, which remains consistent if at least one set of nuisance models is correctly specified. When all nuisance models are correctly specified and converge at appropriate rates, the estimator is asymptotically normal and achieves the semiparametric efficiency bound. We provide a minimax optimization-based debiased machine learning procedure for point estimation and constructing valid confidence intervals. The performance of the proposed methods is demonstrated by simulation studies and a real data application.

We study the one-loop torus vacuum of Type IIB Superstring theory through sector-resolved modular integrals. Building on the i\varepsilon-prescription and the E_s-regularized modular-integral framework of Manschot and Wang [1], we construct regularized sector functionals for the closed oriented torus before the final GSO projection. The construction keeps the unprojected spin-sector data explicit and fixes the compact-domain and cusp contributions within a single modular prescription. We also independently cross-check the result with the Lorentzian-inversion reconstruction of modular integrals by Baccianti et al. [2] This provides a first direct regularized construction of the unprojected sectors of the Type IIB Superstring torus vacuum.

Given a simple graph on $n$ vertices and a parameter $k$, the triangle-densest-$k$-subgraph problem is known to be computationally hard in the worst case. To circumvent the computational hardness, we study an average-case model where a triangle-dense subgraph on $k$ vertices is planted in an Erdős-Rényi random graph on $n$ vertices. For the recovery of the planted subgraph, we propose a simple spectral algorithm and a semidefinite program, both of which use a graph matrix whose entries are local signed triangle counts. Theoretical guarantees for these algorithms are established through spectral analysis of the graph matrix. Finally, we provide evidence showing a statistical-to-computational gap analogous to that for the planted clique problem. The computational threshold in terms of the subgraph size $k$ is at least $\sqrt{n}$ in the framework of low-degree polynomial algorithms, while the information-theoretic threshold is at most logarithmic in $n$.

We present a real geometric algebra framework designed for the direct translation of the Dirac formalism into geometric algebra representations. Unlike previous approaches based on positive-definite signatures, QCA employs a split-signature construction that enables a natural realization of quantum states and operators while simplifying computational implementation. We further present an implementation of QCA using the \textit{GAALOP} software and show how quantum gates and multi-qubit systems can be efficiently represented and generated computationally. As an application, we demonstrate the use of QCA in quantum game theory, where the real-algebraic formulation provides computational advantages for modeling entangled strategies and quantum interactions. The proposed framework establishes a practical bridge between the abstract formalism of quantum computation and efficient geometric algebra implementations.

We prove almost tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of reversible quantum Markov semigroups satisfying the positive off-diagonal scaling (PODS) property. This class includes qubit examples and generalized depolarizing semigroups with respect to full-rank states in arbitrary finite dimensions. For any such semigroup $(\Phi_t)_{t\ge 0}$ and every tensor power $n$, we show that the log-Sobolev constant of the product semigroup $\Phi_t^{\otimes n}$ is at least $2/(3\ln 2)$, approximately 0.96, times the log-Sobolev constant of the single-site semigroup $\Phi_t$, independently of $n$ and the local dimension $d$. The proof first establishes exact tensorization of the $(q,2)$-hypercontractive inequality for integer $q$, in particular $q=3$, and then extends the estimate to all real $q>2$ by complex interpolation; the standard implication from hypercontractivity to logarithmic-Sobolev inequalities yields the stated almost tensorization result. As an application of the same method, we also obtain sharp $(q,2)$-hypercontractivity estimates for qubit depolarizing channels.

We study the large--\(d\) dynamics of the mass--deformed bosonic \(\mathrm{BFSS}_{d+1}\) matrix quantum mechanics using a Hubbard--Stratonovich localization of the Yang--Mills interaction. After integrating out the matrix coordinates, the theory reduces to a holonomy--dependent effective action for an auxiliary adjoint kernel. We introduce a commuting--symmetric saddle and its maximally symmetric specialization, in which the interaction is encoded in a single dynamically generated mass shift \(k_0\). The resulting large--\(d\) description is a gauged matrix harmonic oscillator with self--consistent frequency \(s^2=m+k_0\), fixed by a gap equation. We analyze the low--temperature \(X\)-space physics, the holonomy effective action, the Yang--Mills observable, and the associated phase structure. We then identify a correlated double--scaling limit in which \(d\to\infty\), \(m\to\infty\), and \(\kappa=m^{3/2}/d\) is held fixed. In this limit the Yang--Mills interaction and the explicit mass deformation remain parametrically balanced: the theory interpolates between the commutator--dominated BFSS regime and the mass--dominated Gaussian regime. The double--scaled theory exhibits two complementary large--\(d\) regimes. At low temperature, the enhanced gap pushes the deconfinement scale upward and opens a parametrically large uniform--holonomy region, where the bulk dynamics behaves as weakly coupled \(\mathrm{BFSS}_2\)--type gauged harmonic--oscillator sectors. At the same time, the high--temperature branch reveals an overlap window in which the Gaussian description remains self--consistent while the commutator contribution per matrix pair is parametrically suppressed. The resulting dynamics is therefore \(\mathrm{BFSS}_2\)--like in its enlarged uniform--holonomy sector and IKKT--like in its almost--commuting matrix behavior.

Response-adaptive randomization (RAR) in clinical trials aims to improve ethical and statistical efficiency by dynamically allocating patients to treatments based on observed outcomes. While RAR based on a target optimal allocation have been extensively studied for two-arms settings, their extension to multi-treatment experiments ($K \geq 2$) remains theoretically fragmented, with most existing methods focusing on specific algorithms or restricted target allocations. In this paper, we introduce a unified framework for response-adaptive targeting, the $\alpha$-Rebalancing Targeting Strategies ($\alpha$RTS), which generalize the ERADE two-armed strategy of Hu et al. [2009]. We prove that all designs in this family share fundamental asymptotic properties: strong consistency, asymptotic normality of allocation proportions and treatment effect estimators, and asymptotic efficiency. To address sparse target regimes (where some treatments are asymptotically eliminated), we further propose $\alpha$RTS with Forced Exploration, a variant that guarantees infinite sampling for all treatments while preserving the asymptotic guarantees. Extensive simulations illustrate the finite-sample behavior of $\alpha$RTS variants in a 3-armed context, highlighting in particular the critical role of forced exploration in sparse settings.

We develop a decision-theoretic framework for distributed Bayesian experimental design in which local agents evaluate candidate experiments using expected information gain and transmit their local design decisions to a fusion center. Unlike centralized Bayesian design, where all likelihood components and information-gain values are available to a single planner, the fusion center in the distributed setting chooses a global experiment from compressed local recommendations. We derive the Bayes-optimal fusion rule, which selects the experiment with largest conditional expected centralized information gain given the observed local design decisions. This rule is analogous in spirit to optimal fusion rules in distributed detection, but differs fundamentally because the underlying utility is expected information gain and the resulting loss is information-gain regret rather than classification error. We also establish information-loss bounds and identify conditions under which the decision-only fusion rule is asymptotically equivalent to the centralized design. Numerical experiments show that Bayes-optimal fusion closely approximates the centralized oracle, whereas majority voting can be highly suboptimal when a minority of sites carry disproportionate information.

Heat conduction in semiconductor crystals is fundamentally governed by the linearized Peierls-Boltzmann equation (LPBE) for phonon transport, that arises out of a kinetic theory for phonon quasiparticles. Yet, continuum theories such as the Fourier's heat diffusion, weakly quasiballistic and hydrodynamic heat equations are often used to explain the experimental observations of heat flow in these materials. Here, we show that a systematic reduction of the LPBE into such equivalent continuum descriptions are possible only for the limiting values of a set of generalized Knudsen numbers. We further show that all of these continuum heat flow regimes, along with the ballistic heat flow, can be described by a single continuum equation for the temperature field that originates from the eigenmode analysis of the LPBE, thus offering a unified picture of all possible heat flow regimes in semiconducting crystals. Using quantitative examples on twenty three technologically important semiconductors, we show that several previously-unidentified features of the non-Fourier heat flow regimes emerge from this generalized Knudsen number framework such as (1) the mutual exclusivity of the weakly quasiballistic and the hydrodynamic heat flow regimes, (2) length-dependent velocity of the hydrodynamic second sound temperature wave and a characteristic heating length for the strongest hydrodynamic second sound, (3) characteristic frequency-domain temperature response distinguishing the hydrodynamic second sound from the ballistic heat flow regime and, (4) a new non-oscillatory signature of transient hydrodynamic heat flow. Our work formally bridges the continuum and the particulate descriptions of heat flow, and provides insights into the important signatures of temperature dynamics in each of these heat flow regimes, that will aid in their unambiguous experimental observations in the future.

In 1965, Wegner conjectured that every finite family \(\mathcal R\) of axis-parallel rectangles in the plane satisfies \(\tau(\mathcal R) \le 2\nu(\mathcal R)-1\), where \(\tau(\mathcal R)\) denotes the minimum number of points needed to pierce all rectangles in \(\mathcal R\), and \(\nu(\mathcal R)\) denotes the maximum size of a pairwise disjoint subfamily. Over the last six decades, the conjecture has motivated a long line of work: it has been verified for several special classes of rectangle families, and the best known general upper bounds have been progressively improved, but the conjecture itself had remained open. We give an explicit counterexample. More precisely, we construct a triangle-free rectangle-intersection graph on \(n\) vertices whose independence number is at most \(n/4\). Since the graph is triangle-free, no point of the plane can lie in three rectangles; hence every piercing point hits at most two rectangles. Consequently, \(\tau(\mathcal R) \ge n/2 \ge 2\nu(\mathcal R)\), contradicting Wegner's conjectured bound. We also give a slightly more general construction for which \(\tau(\mathcal R) \ge 2.21\nu(\mathcal R)\). This shows that the standard point relaxation, equivalently the clique relaxation, for the Maximum Independent Set of Rectangles problem has integrality gap at least \(2.21\).

Equalization-enhanced phase noise is avoided by applying phase noise compensation (PNC) before chromatic dispersion compensation. Feedforward and iterative PNC algorithms based on expectation propagation are proposed. Both achieve information rates close to channels without phase noise for 100 GBaud 64-QAM and 10,000 km of fiber.

Collective migration of heterogeneous cell populations is central to many biological and physiological processes, including development and immune response. Recent experimental and theoretical advances have shown how asymmetric interactions with self-generated chemical gradients shape the spatial distribution of distinct cell types within migrating collectives. However, the principles governing robust spatial organisation of heterogeneous cell populations remain poorly understood. Here, we use asymptotic analysis to systematically derive a nonlinear analytical theory for heterogeneous cell collectives guided by self-generated chemotaxis. Our theory disentangles how heterogeneity in cell diffusivity, chemoattractant consumption, and chemotactic sensitivity shape the density profiles of migrating heterogeneous collectives, revealing four distinct dynamical behaviours that together capture all possible regimes. We calibrate our framework to experimental data on the co-migration of dendritic and T cells. We predict that this system operates in a parameter regime that balances intercellular mixing with T-cell localisation at the leading front of the migrating collective. Our theory reveals that this behaviour is enabled by intermediate long-range chemoattractant signalling generated through strong chemoattractant consumption by dendritic cells. Overall, our framework provides general principles for understanding how non-reciprocal chemical interactions shape robust collective migration in heterogeneous cell populations.

We extend the construction of single-valued polylogarithms at genus one from arXiv:2511.15240 to once-punctured Riemann surfaces of higher genera. The resulting functions have a trivial monodromy representation with respect to the fundamental group, hence they descend to well-defined functions on the surface. Our construction of single-valued polylogarithms is based on Enriquez' connection and relates them to the polylogarithms from D'Hoker-Hidding-Schlotterer. Finally, we identify the Arakelov Green's function within our framework.

Parametric PDE-constrained optimal control with pointwise state constraints requires repeated solution of restricted Schur-complement systems on parameter-dependent inactive sets. In a primal active-set method, each inactive-set system is symmetric positive definite, but the active set can change nonsmoothly with the parameter. The resulting operator may vary in dimension, sparsity pattern, and spectrum, limiting reuse of sparse factorizations, multigrid hierarchies, and Krylov information. We propose a reusable spectral-deflation strategy anchored to one full-domain reference Schur complement. Low reference eigenmodes are computed once, restricted online to each inactive set, and used as an A-DEF2 deflation basis for Jacobi-preconditioned CG. The framework also supports POD enrichment, Rayleigh-Ritz reselection, coarse-grid or analytical reference modes, and conditioning safeguards. Given the active set, the method preserves the high-fidelity inactive-set system and solves it to the prescribed CG tolerance; it accelerates the linear algebra rather than replacing the optimal-control solve with a surrogate. We explain the method through a spectral-coherence view, motivated by interlacing and perturbation arguments and assessed with principal-angle diagnostics. Across diffusion, convection-diffusion, nonlinear thermal, and conjugate-heat-transfer benchmarks, deflation reduces CG iterations by about 55 to 98 percent. GPU deployments also show wall-time gains over CPU sparse-direct and algebraic-multigrid baselines, because the reference basis is built once whereas competing solver structures are rebuilt per instance. Coarse-grid or analytical modes amortize the offline cost within a single parameter sweep; fine-grid eigensolves remain more precompute-limited. Timings isolate the inactive-set linear-solve kernel; reducing the active-set outer loop is outside the present scope.

In this paper, we propose a general-purpose multi-dimensional symbol construction for computing an arbitrary symmetric function with digital over-the-air computation (OAC) and discuss the practical aspects of coherent aggregation. For our first contribution, we discuss the categorical representation of a symmetric function. By using this representation and leveraging the sufficiency of the histogram to evaluate a symmetric function, i.e., inspired by type-based multiple access (TBMA), we introduce a general approach to design a single set of OAC symbols to compute any digital function. For our second contribution, we use a comprehensive platform based on low-cost nodes that maintain synchronization in time, frequency, phase, and amplitude via a trigger mechanism, enabling coherent OAC experiments without Global Positioning System (GPS) or cable-based synchronization. Using measurements from the platform, we characterize the phase and amplitude statistics of the composite channel to derive a realistic impairment model for coherent OAC. Through a comprehensive analysis, we demonstrate the effectiveness of the proposed scheme under impairments captured by the proposed model

Modern AI systems increasingly solve a task not with a single model call but with several imperfect agents working together: some propose pieces of a solution, others verify them, and the results are combined. These systems often outperform any single model, yet it is rarely clear why they succeed or when they will fail. We model such a system as message passing on a sparse graph, the structure that underlies low-density parity-check (LDPC) codes, and extend the density-evolution machinery of coding theory to this richer setting. In our model a task is a set of coupled binary subclaims, and an agent architecture is a sparse, role-typed factor graph whose check nodes are noisy Boolean verifier nodes, each computing a local Boolean function of the subclaims it touches. Three distinct failure modes, all modeled as erasures (an agent abstaining, a verifier returning no usable output, and a message lost between two agents), propagate as the agents exchange set-valued messages. The check agents combine these messages by a single logical-forcing rule that specializes to XOR, AND, OR, implication, and Horn constraints. This is more than a relabeling of LDPC theory: the verifier functions are nonlinear and value-asymmetric, and the three failure modes do not reduce to a single effective channel, so they require new threshold, finite-length, and converse results rather than a direct reuse of parity-check density evolution. We prove a density-evolution theorem that predicts the asymptotic fraction of unresolved subclaims on random role-typed architectures, with an extension to deterministic, locally tree-like graph sequences. The XOR case recovers the classical LDPC recursion on the binary erasure channel (BEC); the AND case exposes an asymmetry between positive and negative verifier certificates.

We study the action principle for space-times whose boundary is singular. We suggest that it is natural to treat the singularity as a {\it free} boundary, where the variation is unconstrained. Demanding that the action is stationary under such free variations then implies certain (on-shell) boundary conditions at the singularity. We derive these boundary conditions for the case of Einstein gravity coupled to matter and show that, when applied to an initial spacelike singularity, they exclude Kasner-like or BKL space-times, but admit conformally regular space-times (including FLRW models) sourced by fluids satisfying $0 \leq P < \rho$. For standard hot big bang FLRW cosmologies, the admissible linear (scalar, vector, tensor) perturbations satisfy reflecting boundary conditions at the bang, in agreement with large-scale cosmological observations.

Channel state information (CSI)-based neural positioning learns a mapping from CSI measurements to user equipment (UE) positions using neural networks. However, most existing performance evaluations utilize randomly partitioned train/test CSI-dataset splits, which fail to reflect the generalization requirements of practical deployments and present optimistic results. In this paper, we study the spatial and temporal generalization of neural positioning with standard-compliant Wi-Fi and 5G NR systems for three real-world CSI datasets acquired in indoor and outdoor environments. We assess generalization with two different architectures, a conventional multilayer perceptron (MLP) and a novel transformer architecture, to unseen spatial regions, unseen UE trajectories, and CSI measurement campaigns separated by one week. Our experiments show that both architectures generalize well in space and time, and the proposed transformer consistently outperforms the MLP in positioning accuracy while requiring fewer model parameters.

Channel charting (CC) in real-world coordinates is a recently proposed self-supervised machine learning method that maps high-dimensional channel state information (CSI) to user equipment (UE) position. In this paper, we extend CC to also estimate UE orientation, which can further assist tasks such as beamfinding, precoding, and beam- and cell-assignment. To this end, we propose a novel orientation triplet loss that accounts for angle periodicity and an alignment loss that embeds estimated orientations in real-world coordinates in a self-supervised fashion. Using real-world CSI measurements from a standard-compliant 5G NR system, we demonstrate that the proposed method achieves position and orientation estimation accuracy close to that of supervised approaches trained with ground-truth labels.

Temporal difference (TD) learning with linear function approximation is a core method for policy evaluation. Its classical continuous-time description is an ordinary differential equation (ODE), which captures the asymptotic mean dynamics but neglects stochastic fluctuations determining the error floor. We introduce a stochastic differential equation (SDE) approximation for linear TD(0) under Markovian noise. The resulting model distinguishes the contraction dynamics governed by the projected Bellman operator from the influence of Markovian sampling. As a consequence, the model explains the constant-stepsize error floor through the interaction between Markovian long-run covariance and the contraction geometry of the projected Bellman operator.

We introduce Ergodic Deviation-Robust Equilibrium (EDRE), a dynamics-relative equilibrium concept for repeated finite games in which agents learn via entropic mirror descent (EMD). EDRE requires three properties to hold simultaneously for the same profile and learning run: (E1) the limit profile is an $\varepsilon$-Nash equilibrium at a product distribution; (E2) along the entire learning trajectory, every fixed coalition's cumulative aggregate (summed-unilateral) deviation gain is $\tilde{\mathcal{O}}(\sqrt{T})$ with high probability; and (E3) the limit profile is a fixed point of the EMD map, so that it is selected by the dynamics rather than merely certified as an equilibrium. We prove that the $\sqrt{T}$ deviation-regret rate is order-tight, establish existence in exact-potential games (via Nash's theorem, with a constructive proximal route under concavity) together with Lyapunov monotonicity of EMD (and pointwise convergence when the fixed-point set is a singleton), and extend the selection property to monotone polymatrix games through variational inequalities. Although a static EDRE coincides with an $\varepsilon$-Nash equilibrium, its content is dynamic: robust (positive-measure) selection under EMD excludes linearly unstable equilibria, so EDRE acts as a Nash equilibrium equipped with a dynamic certificate rather than a static refinement. On the complexity side, we show that computing EDRE is PPAD-hard in general polymatrix games and belongs to promise-PPAD for potential games. A worked $2\times 2$ coordination-game example illustrates all components of the framework. Additional results, including a bandit-feedback extension, a period-doubling route to Li-Yorke chaos for the two-strategy EMD map at large step size, a linear-program formulation for minimum-cost steering, and supporting simulations, appear in the appendices.

Learned PDE simulators are increasingly used as low-cost replacements for expensive numerical solvers, but standard relative $L^2$ error does not determine whether a learned model behaves as a coherent numerical time propagator. This paper presents a diagnostic software suite for auditing learned PDE simulators as approximate evolution operators. The suite provides architecture-independent, post hoc diagnostics for relative state error, semigroup consistency, finite-difference generator discrepancy, energy behavior, integral balance, admissibility constraints, perturbation response, and scaling-law consistency. The software is designed around a minimal contract: reference trajectories, a learned propagator or saved predictions, equation metadata, and a diagnostic configuration specifying which structures are meaningful for the problem under study. We validate the suite on five benchmark PDE tasks: two-dimensional incompressible Navier-Stokes, shallow-water dynamics, active matter, three-dimensional compressible Navier-Stokes, and three-dimensional magnetohydrodynamics, using FNO, DeepONet, U-Net, and ResNet-style surrogate models together with controlled underfit and oversmoothed variants. The validation study shows that relative $L^2$ error can remain moderate, or even improve, while structural diagnostics deteriorate substantially. The package therefore supports software-level auditing of learned PDE simulators by reporting an interpretable diagnostic panel rather than collapsing model behavior into a single state-error score.

In learning theory, the sign rank of a binary concept class captures the smallest dimension in which it can be represented by points and halfspaces. Despite tremendous interest, lower bounds on sign rank are notoriously difficult to come by. Two recent approaches to the problem establish lower bounds on sign rank by measures that are easier to analyze: the $\mathbb{Z}_2$-index and the list replicability number. We order these measures, showing that the $\mathbb{Z}_2$-index is upper-bounded by a linear function of the list replicability number. As a main consequence, we obtain a strong separation between sign rank and $\mathbb{Z}_2$-index, thereby resolving a question of Frick, Hosseini, and Vasileuski. This motivates a thorough study of list replicability, the stronger of the two lower-bounding measures. We establish upper bounds on the list replicability number by two combinatorial measures: height and minimum star number. We also prove a fundamental composition result, showing that the product of two concept classes has list replicability number bounded by the sum of the list replicability numbers of the two classes.

If H is a commutative connected graded Hopf algebra over a commutative ring k, then a certain canonical k-algebra homomorphism H -> H (x) QSym is defined, where QSym denotes the Hopf algebra of quasisymmetric functions over k. This homomorphism generalizes the "internal comultiplication" on QSym, and extends what Hazewinkel (in Section 18.24 of his "Witt vectors") calls the Bernstein homomorphism. We construct this homomorphism with the help of the universal property of QSym as a combinatorial Hopf algebra (a well-known result by Aguiar, Bergeron and Sottile) and extension of scalars (the commutativity of H allows us to consider, for example, H (x) QSym as an H-Hopf algebra, and this change of viewpoint significantly extends the reach of the universal property).

We explore several generalizations of Whitney's theorem -- a classical formula for the chromatic polynomial of a graph. Following Stanley, we replace the chromatic polynomial by the chromatic symmetric function. Following Dohmen and Trinks, we exclude not all but only an (arbitrarily selected) set of broken circuits, or even weigh these broken circuits with weight monomials instead of excluding them. Following Crew and Spirkl, we put weights on the vertices of the graph. Following Gebhard and Sagan, we lift the chromatic symmetric function to noncommuting variables. In addition, we replace the graph by an "ambigraph", an apparently new concept that includes both hypergraphs and multigraphs as particular cases. We show that Whitney's formula endures all these generalizations, and a fairly simple sign-reversing involution can be used to prove it in each setting. Furthermore, if we restrict ourselves to the chromatic polynomial, then the graph can be replaced by a matroid. We discuss an application to transitive digraphs (i.e., posets), and reprove an alternating-sum identity by Dahlberg and van Willigenburg.

We show a determinant identity which generalizes both the Chio pivotal condensation theorem and the Matrix-Tree theorem.

We will invest quite some computer power to find double octic threefolds that are connected to weight four modular forms.

We study Manneville-Pomeau maps on the unit interval and prove that the set of points whose forward orbits miss an interval with left endpoint 0 is strong winning for Schmidt's game. Strong winning sets are dense, have full Hausdorff dimension, and satisfy a countable intersection property. Similar results were known for certain expanding maps, but these did not address the nonuniformly expanding case. Our analysis is complicated by the presence of infinite distortion and unbounded geometry.

The concept of a classical player, corresponding to a classical random variable, is extended to include quantum random variables in the form of self adjoint operators on infinite dimensional Hilbert space. A quantum version of Von Neumann's Minimax theorem for infinite dimensional (or continuous) games is proved.

We give a characterization of the largest $2$-intersecting families of permutations of $\{1,2,\ldots,n\}$ and of perfect matchings of the complete graph $K_{2n}$ for all $n \geq 2$.

A generic uniformly distributed random sequence on the unit interval has Poissonian pair correlations. Usually, the pair correlations statistic is therefore studied for equidistributed sequences. At the same time, there are only very few explicitly known examples of sequences with this property and many types of deterministic sequences have been proven to fail having the Poissonian pair correlation property. In this paper we study the pair correlation statistic in the non-uniform case and analyze the first elementary example of such a sequence, namely $x_n := \left\{ \frac{\log(2n-1)}{\log(2)} \right\}$, which is a standard low-dispersion sequence. The proof heavily relies on a full understanding of the gap structure of $(x_n)_{n=1}^N$. Furthermore, we discuss differences to the weak pair correlation function which turns out to be linear.

Let $f$ be a primitive form of weight $2k+j-2$ for $\SL_2(\ZZ)$, and let $\frkp$ be a prime ideal of the Hecke field of $f$. We denote by $\SP_m(\ZZ)$ the Siegel modular group of degree $m$. Suppose that $k \equiv 0 \mod 2, \ j \equiv 0 \mod 4$ and that $\frkp$ divides the algebraic part of $L(k+j,f)$. Put ${\bf k}=(k+j/2,k+j/2,j/2+4,j/2+4)$. Then under certain easily checkable conditions, we prove that there exists a Hecke eigenform $F$ in the space of modular forms of weight $(k+j,k)$ for $\SP_2(\ZZ)$ such that $[\scri_2(f)]^{\bf k}$ is congruent to $\scra^{(I)}_4(F)$ modulo $\frkp$. Here, $[\scri_2(f)]^{\bf k}$ is the Klingen-Eisenstein lift of the Saito-Kurokawa lift $\scri_2(f)$ of $f$ to the space of modular forms of weight ${\bf k}$ for $\SP_4(\ZZ)$, and $\scra^{(I)}_4(F)$ is a certain lift of $F$ to the space of cusp forms of weight ${\bf k}$ for $\SP_4(\ZZ)$. As an application, we prove Harder's conjecture on the congruence between the Hecke eigenvalues of $F$ and some quantities related to the Hecke eigenvalues of $f$. This version gives proofs of Lemmas 7.2 and 7.3 and Corollaries 7.4 and 7.5 of the paper arXiv:2306.07582v2.

Markov categories have recently turned out to be a powerful high-level framework for probability and statistics. They accommodate purely categorical definitions of notions like conditional probability and almost sure equality, as well as proofs of fundamental results such as the Hewitt--Savage 0/1 Law, the de Finetti Theorem and the Ergodic Decomposition Theorem. In this work, we develop additional relevant notions from probability theory in the setting of Markov categories. This comprises improved versions of previously introduced definitions of absolute continuity and supports, as well as a detailed study of idempotents and idempotent splitting in Markov categories. Our main result on idempotent splitting is that every idempotent measurable Markov kernel between standard Borel spaces splits across another standard Borel space, and we derive this as an instance of a general categorical criterion for idempotent splitting in Markov categories.

In a previous paper we constructed higher theta series for unitary groups over function fields, and conjectured their modularity properties. Here we prove the generic modularity of the $\ell$-adic realization of higher theta series in cohomology. The proof debuts a new type of Fourier transform, occurring on the Borel-Moore homology of moduli spaces for shtuka-type objects, that we call the arithmetic Fourier transform. Another novelty in the argument is a sheaf-cycle correspondence extending the classical sheaf-function correspondence, which facilitates the deployment of sheaf-theoretic methods to analyze algebraic cycles. Although the modularity property is a statement within classical algebraic geometry, the proof relies on derived algebraic geometry, especially a nascent theory of derived Fourier analysis on derived vector bundles, which we develop.

Given a $d$-dimensional vector space $V \subset \mathbb{C}[u]$ of polynomials, its Wronskian is the polynomial $(u + z_1) \cdots (u + z_n)$ whose zeros $-z_i$ are the points of $\mathbb{C}$ such that $V$ contains a nonzero polynomial with a zero of order at least $d$ at $-z_i$. Equivalently, $V$ is a solution to the Schubert problem defined by osculating planes to the moment curve at $z_1, \dots, z_n$. The inverse Wronski problem involves finding all $V$ with a given Wronskian $(u + z_1) \cdots (u + z_n)$. We solve this problem by providing explicit formulas for the Grassmann-Plücker coordinates of the general solution $V$, as commuting operators in the group algebra $\mathbb{C}[\mathfrak{S}_n]$ of the symmetric group. The Plücker coordinates of individual solutions over $\mathbb{C}$ are obtained by restricting to an eigenspace and replacing each operator by its eigenvalue. This generalizes work of Mukhin, Tarasov, and Varchenko (2013) and of Purbhoo (2022), which give formulas in $\mathbb{C}[\mathfrak{S}_n]$ for the differential equation satisfied by $V$. Moreover, if $z_1, \dots, z_n$ are real and nonnegative, then our operators are positive semidefinite, implying that the Plücker coordinates of $V$ are all real and nonnegative. This verifies several outstanding conjectures in real Schubert calculus, including the positivity conjectures of Mukhin and Tarasov (2017) and of Karp (2021), the disconjugacy conjecture of Eremenko (2015), and the divisor form of the secant conjecture of Sottile (2003). The proofs involve the representation theory of $\mathfrak{S}_n$, symmetric functions, and $\tau$-functions of the KP hierarchy.

In this paper, we introduce novel concepts and establish a formal framework for twisted differential operators in the context of several variables. The focus is on twisted coordinates within Huber rings, which facilitate the construction of diverse rings of twisted differential operators. We establish an equivalence between modules equipped with twisted connections and those endowed with actions of twisted derivatives. Furthermore, we examine the convergence properties of twisted differential operators under specific conditions. As one of the main results, we extend the confluence theorem of Le Stum and Quirós to several variables. This work aligns with the ongoing advancements in $p$-adic Hodge cohomology and prismatic cohomology.

We show that every two-parameter period map admits a Kato--Nakayama--Usui completion to a morphism of log manifolds, and the map onto the image is a morphism of compact algebraic spaces. This result also applies to the case of mixed period maps and we use it to give a construction of generalized Nèron models.

We exhibit closed hyperbolic surfaces of genus $10$, $17$, and $37$ such that the multiplicity of the first nonzero eigenvalue of their Laplacian is larger than the maximum conjectured by Yves Colin de Verdière in 1986. In order to determine these multiplicities, we apply the twisted Selberg trace formula to the representations induced by the isometry groups of these surfaces on corresponding triangle groups.

We study general $p$-means of convex bodies, extending the classical definitions by W. J. Firey via support and gauge functions to two families ranging over all $p \in [-\infty,\infty]$. For values of $p$ beyond the classical ranges, we show that $p$-means of polytopes are again polytopes, yielding simpler structural descriptions. Using a natural characterization of dilates of convex bodies based on their boundary structure, we characterize the equality cases between the two types of $p$-means for the same $p$-value. Extending recent results on standard mean-symmetrizations of convex bodies, we further establish (in almost all instances tight) inequalities quantifying how well arbitrary $p$-means of convex bodies approximate each other. These bounds lead to characterizations and sharp stability results for the equality cases between $p$-means for different $p$-values. As a corollary, every Minkowski centered convex body is equidistant from all its $p$-symmetrizations with respect to the Banach-Mazur distance.

An odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, there is a color appearing an odd number of times in $N_G(v)$. Odd coloring of graphs was studied intensively in recent few years. In this paper, we introduce the notion of a strong odd coloring, as not only a strengthened version of odd coloring, but also a relaxation of square coloring. A strong odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, if a color appears in $N_G(v)$, then it appears an odd number of times in $N_G(v)$. We denote by $\chi_{so}(G)$ the smallest integer $k$ such that $G$ admits a strong odd coloring with $k$ colors. We prove that if $G$ is a graph with $mad(G)\le\frac{20}{7}$, then $\chi_{so}(G)\le \Delta(G)+4$, and the bound is tight. We also prove that if $G$ is a $C_4$-free graph with $mad(G)\le\frac{30}{11}$, then $\chi_{so}(G)\le \Delta(G)+3$.

In the present paper, we provide cohomology groups for matroids, as a categorification of the characteristic polynomial of matroids. The construction depends on the ``quasi-representation'' of a matroid. For a certain choice of the quasi-representation, we show that our cohomology theory gives a generalization of the chromatic cohomology introduced by L. Helme--Guizon and Y. Rong, and also the characteristic cohomology introduced by Z. Dancso and A. Licata.

Given a compact, oriented surface $S$ of finite genus and finitely many boundary components, we provide examples of finite covers $\tilde{S}$ of $S$ and non-simple closed curves $\gamma$ on $S$ which lifts to simple closed curves on $\tilde{S}$. In particular, given any positive integer $n\geq 2$, we construct explicit non-simple closed curves on $S$ which has a simple lift to a degree $n$ cover of $S$.

We prove that, for an equivalence class of irreducible smooth representations of the symplectic group Sp(2n,F) over a non-Archimedean local field F, the Jacquet functor with respect to the maximal Levi subgroup GL(l,F)\times Sp(2n-2l,F) is multiplicity-free. The proof is based on an explicit computation of Jacquet modules for a broader family of Sp(2n,F)-representations induced from segments, yielding a detailed structural description that may be of independent interest.

This work studies the statistical implications of using features comprised of general linear combinations of covariates to partition the data in randomized decision tree and forest regression algorithms. Using random tessellation theory in stochastic geometry, we provide a theoretical analysis of a class of efficiently generated random tree and forest estimators that allow for oblique splits along such features. We call these estimators oblique Mondrian trees and forests, as the trees are generated by first selecting a set of features from linear combinations of the covariates and then running a Mondrian process that hierarchically partitions the data along these features. Quadratic risk bounds and convergence rates are obtained for the flexible function class of multi-index models for dimension reduction, where the output is assumed to depend on a low-dimensional relevant feature subspace of the input domain. The results highlight how the risk of these estimators depends on the choice of features and quantify how robust the risk is with respect to error between the selected features along which the data is split and the true relevant feature subspace. The asymptotic analysis also provides conditions on the convergence rate a set of estimated relevant features must satisfy for oblique Mondrian estimators to obtain minimax optimal rates of convergence with respect to the dimension of the relevant feature subspace. Additionally, a lower bound on the risk of axis-aligned Mondrian trees (where features are restricted to the set of covariates) is obtained, proving that these estimators are suboptimal for general ridge functions, no matter how the distribution over the covariates used to divide the data at each tree node is weighted.

We establish a sharp upper bound for the bottom spectrum of the Beltrami Laplacian on universal covers of closed Riemannian manifolds with a scalar curvature lower bound. Moreover, we prove a scalar curvature rigidity theorem when this bound is achieved. Additionally, we prove a net characterization of scalar curvature for general complete noncompact Riemannian manifolds.

We analyze Allen-Cahn functionals with stationary ergodic coefficients in the regime where the length scale $\delta$ of the heterogeneities is much smaller (microscopic) than the interface width $\epsilon$ (mesoscopic). In the main result of this paper, we prove that if the ratio $\delta \epsilon^{-1}$ decays fast enough compared to $\epsilon$, then homogenization effects dominate, and the $\Gamma$-limit of the energy is the same as if the coefficients had been replaced by their homogenized values. As a byproduct of the proof, this implies that homogenization holds in the periodic setting whenever $\delta \epsilon^{-1}$ vanishes with $\epsilon$, no matter how slowly. In a companion paper, we prove this is sharp: if $\delta \epsilon^{-1}$ decays too slowly, then improbable or atypical local configurations of the medium begin to play a role, and the $\Gamma$-limit may be smaller than the one predicted by homogenization theory. We refer to this as the rare events regime, and we prove that it can occur in both random and almost periodic media.

A derived algebraic geometric study of classical $\mathrm{GL}_n$-Yang-Mills theory on the $2$-dimensional square lattice $\mathbb{Z}^2$ is presented. The derived critical locus of the Wilson action is described and its local data supported in rectangular subsets $V =[a,b]\times [c,d]\subseteq \mathbb{Z}^2$ with both sides of length $\geq 2$ is extracted. A locally constant dg-category-valued prefactorization algebra on $\mathbb{Z}^2$ is constructed from the dg-categories of quasi-coherent complexes on the derived stacks of local data.

We develop a general framework of rank-preserving, element-wise matrix transformations for engineering fermion mass hierarchies in theory-space constructions. We prove that preservation of massless modes requires the transformation function to be separable, $g_f(i,j)=g^{(L)}_f(i)g^{(R)}_f(j)$, which in turn enables independent control of left- and right-chiral zero-mode profiles directly at the level of the theory-space mass matrix. This formalism unifies and extends the clockwork mechanism, permits controlled deformation of Kaluza--Klein spectra, and enhances hierarchy generation in GIM-like fine-cancellation scenarios. As a concrete application, we show that in theory-space models for neutrino masses, suitable transformations allow sub-eV light neutrinos to arise from TeV-scale new physics with only $\mathcal{O}(40)$ additional fermionic sites, while remaining consistent with charged-lepton flavor-violation bounds. In contrast, the corresponding untransformed models asymptote at the MeV scale and cannot access the phenomenologically required regime without extreme field multiplicities or hierarchical parameters.

We investigate the convergence rates of variational posterior distributions for statistical inverse problems involving nonlinear partial differential equations (PDEs). Departing from exact Bayesian inference, variational inference transforms the inference problem into an optimization problem by introducing variational sets. Based on a modified ``prior mass and testing'' framework, we propose general conditions for three categories of inverse problems: mildly ill-posed, severely ill-posed, and those with unknown model parameters. Concentrating on the variational sets comprising the restricted Gaussian or widely utilized Gaussian mean-field families, we demonstrate that for all three categories, the convergence rate can be decomposed into a true distribution term and a variational approximation term. Moreover, we illustrate that the true distribution term dominates the convergence rates, thereby substantiating the effectiveness of the variational inference method for inverse problems of PDEs. As specific examples, we examine a collection of non-linear inverse problems, including the Darcy flow problem, the inverse potential problem for a subdiffusion equation, and the inverse medium scattering problem. Besides, we show that our convergence rates are minimax optimal for these inverse problems.

We explore an optimal impulse control problem wherein an electronic device owner strategically calibrates protection levels against cyber attacks. Utilizing epidemiological compartment models, we qualitatively characterize the dynamics of cyber attacks within the network. We determine the optimal protective measures against effective hacking by formulating and solving a stochastic control problem with optimal switching. We demonstrate that the value function for the cluster owner constitutes a viscosity solution to a system of coupled variational inequalities associated with a fully coupled reflected backward stochastic differential equation (BSDE). Furthermore, we devise a comprehensive algorithm alongside a verification procedure to ascertain the optimal timing for network protection across various cyber attack scenarios. Our findings are illustrated through numerical approximations employing deep Galerkin methods for partial differential equations (PDEs). We visualize the optimal protection strategies in the context of two distinct attack scenarios: (1) a constant cyber attack, (2) an exogenous cyber attack strategy modeled with a Poisson process.

This article describes certain ratios that attend pairs of complementary Gauss-Jordan pivotings transforming skew-symmetric matrices. Our interest in those ratios was motivated by a need to prove a crucial Claim stated in a recently proposed strongly polynomial-time algorithm for the general LP problem. That Claim is proved in this article and, as a consequence of this proof, a compact implementation of the strongly polynomial-time algorithm is suggested.

The 0/1 D-optimality problem and the Maximum-Entropy Sampling problem are two well-known NP-hard discrete maximization problems in experimental design. Algorithms for exact optimization (of moderate-sized instances) are based on branch-and-bound. The best upper-bounding methods are based on convex relaxation. We present ADMM (Alternating Direction Method of Multipliers) algorithms for solving these relaxations and experimentally demonstrate their practical value.

Motivated by recent developments in mean field games in ecology, in this paper we introduce a connection between the best response dynamics in evolutionary game theory, the minimization of the highest income of a game, and minimizing movement schemes. The aim of this work is to develop a variational approach to compute solutions of first order ergodic mean field games that may not possess a priori a variational structure. The study is complemented by a discussion and successful implementation of numerical algorithms, and comparisons between them in a variety of cases.

We study the time-dependent spatial averages of a critical stochastic partial differential equation, namely the stochastic heat equation in dimension $d\geq 3$ with noise white in time and colored in space with covariance kernel $\|\cdot\|^{-2}$. The solution to this SPDE is a singular measure and was constructed by Mueller and Tribe in [MT04]. We show that the time-dependent spatial averages of this SPDE over a ball of radius $R$ at time $t$ have different limits under different space-time scales. In particular, when $t\ll R^2$, the central limit theorem holds; when $t=R^2$, the spatial average is a non-Gaussian random variable; when $t\gg R^2$, the spatial average becomes extinct.

Measurement-constrained problems frequently arise in modern applications such as electronic health record studies. In such problems, despite the availability of large datasets, collecting labeled data can be highly costly or time-consuming, allowing only a small portion of the data to be labeled within a given budget. This raises a critical question: which data points are most beneficial to label given the budget constraint? We study this question in the context of estimating an optimal individualized threshold under a measurement-constrained M-estimation framework. In particular, our goal is to estimate a high-dimensional parameter $\theta$ in a linear threshold $\theta^TZ$ for a continuous variable $X$ such that the discrepancy between whether $X$ exceeds the threshold $\theta^TZ$ and a binary outcome $Y$ is minimized. In the measurement-constrained setting, we propose a novel $K$-step active subsampling algorithm to estimate $\theta$, which iteratively samples the most informative observations in the dataset and solves a regularized M-estimator. Our theoretical analysis reveals a sharp phase transition phenomenon with respect to $\beta$, the smoothness of the conditional density of $X$ given $Y$ and $Z$. Please see the paper for the full abstract.

Let Y be a smooth K3 surface of Picard rank 1. We prove that the subgroup G of Aut D^b(Y) generated by spherical twists with respect to all spherical objects is free. Moreover, we provide a precise recipe to find free generators of G and determine the cases when G is finitely generated, depending on the degree of Y. This description in particular yields a precise classification of spherical objects in Aut D^b(Y). We apply these results to verify the first three-dimensional case of a conjecture due to Bondal and Polishchuck, namely, we establish the transitivity of the braid group action on full exceptional collections for Fano threefolds of Picard rank 1.

In 2009, R. Siefring introduced a homotopy-invariant generalized intersection number and singularity index for punctured pseudoholomorphic curves, by adding contributions from curve's asymptotic behavior to the standard intersection number and singularity index. In this article, we provide a stratification of the moduli space that describes the rate of asymptotic convergence of the pseudoholomorphic curves. Using this stratification, we provide a more intricate characterization of the curves for which these added contribution to the intersection number and singularity index vanishes. In doing so, we prove that the asymptotic contribution to intersection number and singularity index vanishes under generic perturbations. This means that in generic situations we only need to consider the usual intersections of the curves.

We construct non-negative weak solutions of fast diffusion equations with a divergence type of drift term satisfying the $L^q$-energy inequality and speed estimate in Wasserstein spaces under some integrability conditions on the drift term. Furthermore, in the case that the drift term has a divergence-free structure, it turns out that its integrability conditions can be relaxed, which is also applicable to porous medium equations, thereby improving previous results. As an application, the existence of weak solutions is also discussed for a viscous Boussinesq system of the fast diffusion type.

Generative modeling aims to generate new data samples that resemble a given dataset. When using diffusion models for this task, one of the main challenges is solving the problem in the input space, which tends to be very high-dimensional. To address this, recent approaches solve diffusion models in the latent space through an encoder that maps from the data space to a lower-dimensional latent space, improving training efficiency and achieving state-of-the-art results. The variational autoencoder (VAE) is the most commonly used encoder/decoder framework in this domain, known for its ability to learn latent representations and generate data samples. In this paper, we introduce a novel encoder/decoder framework with theoretical properties distinct from those of the VAE, specifically designed to preserve the geometric structure of the data distribution. We demonstrate the significant advantages of this geometry-preserving encoder in the training process of both the encoder and decoder. Additionally, we provide theoretical results proving convergence of the training process, including convergence guarantees for encoder training, and results showing faster convergence of decoder training when using the geometry-preserving encoder.

We find explicit formulae for poset probabilities \(\mathbf{Prob}(P_\lambda; \alpha < \beta)\) in partition posets (cell posets) \(P_{\lambda}\) when \(\lambda=(\lambda_{1},\lambda_{2})\) is a two-row partition. These probabilities are given as rational expressions in \(f^{\sigma / \tau}\), where \(\tau \subseteq \sigma \subseteq \lambda\). We then use well-known formulae, such as the hook-length formula for \(f^\lambda\), the number of standard Young tableaux on a partition \(\lambda\), and the corresponding determinantal formula by Jacobi-Trudi-Aitken for \(f^{\lambda / \mu}\), the number of standard Young tableaux on a skew partition \(\lambda / \mu\), to make the aforementioned expressions explicit. We also calculate the limit probabilities of \(\mathbf{Prob}(P_\lambda; \alpha < \beta)\) when the elements \(\alpha,\beta\) are fixed cells, but the arm-lengths of \(\lambda=(\lambda_{1},\lambda_{2})\) tend to infinity with bounded difference \(\lambda_{1} - \lambda_{2}\).

In this note, we consider a Stochastic Differential Equation under a strong confluence and Lipschitz continuity assumption of the coefficients. For the unique stationary solution, we study the rate of convergence of its empirical measure toward the invariant probability measure. We provide rate for the Wasserstein distance in the mean quadratic and almost sure sense.

Feature allocation models are Bayesian nonparametric tools tailored to data in which each observation can simultaneously exhibit multiple characteristics, or features. A fundamental limitation of standard formulations is that feature labels are assumed to be independent and identically distributed, and therefore play no role in posterior inference. The present paper introduces a unified Bayesian framework for extended feature allocation models, in which feature labels and proportions are modeled jointly, thereby enabling the simultaneous discovery of features and learning of dependencies among their labels. Building on point process theory, we develop a full Bayesian analysis of these models. Within this general setting, we also characterize previously proposed priors as those leading to poor predictive distributions, which cannot capture label dependencies and are insensitive to the observed frequency spectrum. Our methodology is designed to move beyond such standard formulations by leveraging the information carried by feature labels. We demonstrate the usefulness of our approach by introducing: (i) a Cox process prior that clusters genomic variant embeddings while predicting new variants and new variant clusters; (ii) a determinantal point process prior for repeated forest surveys, where prediction concerns both the number and the locations of unobserved trees.

We construct and study a bicategory of super 2-line bundles over graded Lie groupoids, providing a unified framework for geometric models of twistings of (Real) K-theory. The core of our work is to exhibit a wide range of models from the literature as special cases, among them several variants of bundle gerbes (Real/equivariant/Jandl), Freed-Moore's twisted groupoid extensions, Freed-Hopkins-Teleman's K-theory twistings, Moutuou's Real twistings, Freed's invertible algebra bundles, and Distler-Freed-Moore's orientifold twistings.

This paper is devoted to parameter estimation for partially observed polynomial state space models. This class includes discretely observed affine or more generally polynomial Markov processes. The polynomial structure allows for the explicit computation of a Gaussian quasi-likelihood estimator and its asymptotic covariance matrix. We show consistency and asymptotic normality of the estimating sequence and provide explicitly computable expressions for the corresponding asymptotic covariance matrix.

We obtain rates of convergence in the weak invariance principle (functional central limit theorem) for $\mathbb{R}^d$-valued Hölder observables of nonuniformly hyperbolic maps. In particular, for maps modelled by a Young tower with superpolynomial tails (e.g. the Sinai billiard map, and Axiom A diffeomorphisms) we obtain a rate of $O(n^{-\kappa})$ in the Wasserstein $p$-metric for all $\kappa<1/4$ and $p<\infty$. Additionally, this is the first result on rates that covers certain invertible, slowly mixing maps, such as Bunimovich flowers.

We introduce a notion of weak definability of first order structures, show that various classification-theoretic properties are or are not preserved under it, and that the properties which are preserved can also be characterized in terms of it.

Recent work of Trusiani implies that the existence of a constant scalar curvature Kähler metric on a smooth polarised variety with discrete automorphism group is equivalent to uniform arc K-stability. We prove that uniform arc K-stability is essentially algebraic in flat families of polarised varieties. When the polarised varieties are further smooth and have discrete automorphism group, combining these two results implies that the constant scalar curvature Kähler locus is very general. We use this result to give the first examples of constant scalar curvature Kähler metrics whose existence only follows from the recent solution of the Yau--Tian--Donaldson conjecture. Our technique is to prove a general result stating that stability of a pair in the sense of Paul is essentially an algebraic property in families, and to employ prior work with Reboulet relating uniform arc K-stability to stability of an associated pair.

In this paper, we propose two second-order methods for solving the \(\ell_1\)-regularized composite optimization problem, which are developed based on two distinct definitions of approximate second-order stationary points. We introduce a hybrid proximal gradient and negative curvature method, as well as an adaptive hybrid proximal gradient-Newton-CG method with negative curvature directions, to find a strong* approximate second-order stationary point and a weak approximate second-order stationary point for \(\ell_1\)-regularized optimization problems, respectively. Comprehensive analyses are provided regarding the iteration complexity, operation complexity (including gradient evaluations and Hessian-vector products), and the local superlinear convergence rates of the first phases of these two methods under specific error bound conditions. We demonstrate that the proximal gradient-Newton-CG method achieves the best-known iteration complexity for attaining the proposed weak approximate second-order stationary point, which is consistent with results for finding an approximate second-order stationary point in unconstrained optimization. Through a toy example, we show that our proposed methods can effectively escape first-order approximate solution. Numerical experiments implemented on the \(\ell_1\)-regularized Student's t-regression problem validate the effectiveness of both methods.

We introduce the concept of a restriction semigroupoid S, which unifies the notion of restriction semigroups and restriction categories within a single structure. We prove a representation theorem, showing that every restriction semigroupoid can be embedded into a determined category of partial maps. Furthermore, we construct the Szendrei expansion Sz(S) of S and establish that each premorphism between two restriction semigroupoids S and T is uniquely factorized by a morphism between the Szendrei expansion Sz(S) and T.

We prove existence of weak solutions of the 3D compressible semi-geostrophic (SG) equations with compactly supported measure-valued initial data. These equations model large-scale atmospheric flows. Our proof uses a particle discretisation and semi-discrete optimal transport techniques. We show that, if the initial data is a discrete measure, then the compressible SG equations admit a unique, twice continuously differentiable, energy-conserving and global-in-time solution. In general, by discretising the initial measure by particles and sending the number of particles to infinity, we show that for any compactly supported initial measure there exists a global-in-time solution of the compressible SG equations that is Lipschitz in time. This significantly generalises the original results due to Cullen and Maroofi (2003), and it provides the theoretical foundation for the design of numerical schemes using semi-discrete optimal transport to solve the 3D compressible SG equations.

In this paper we develop the Direct Method in the Calculus of Variations for free-discontinuity energies whose bulk and surface densities exhibit superlinear growth, respectively for large gradients and small jump amplitudes. A distinctive feature of this kind of models is that the functionals are defined on $SBV$ functions whose jump sets may have infinite measure. Establishing general lower semicontinuity and relaxation results in this setting requires new analytical techniques. In addition, we propose a variational approximation of certain superlinear energies via phase-field models.

We show that the measure-valued solutions of the Euler system of gas dynamics generated by oscillatory sequences of consistent approximations violate the principle of maximal entropy production formulated by Dafermos. Numerical results illustrate that solutions obtained by standard numerical methods may be oscillatory and thus do not comply with the Dafermos criterion.

This article shows the geometric decay rate of Euler-Maruyama scheme for one-dimensional stochastic differential equation towards its invariant probability measure under total variation distance. Firstly, the existence and uniqueness of invariant probability measure and the uniform geometric ergodicity of the chain are studied through introduction of non-atomic Markov chains. Secondly, the equivalent conditions for uniform geometric ergodicity of the chain are discovered, by constructing a split Markov chain based on the original Euler-Maruyama scheme.

Let $\Omega$ be an open subset of $\mathbb R^n$, and let $f: \Omega \to \mathbb R$ be differentiable $\mathcal H^k$-almost everywhere, for some nonnegative integer $k < n$, where $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure. We show that $\|\nabla f\|_{L^\infty (\mathcal H^k)} = \|\nabla f\|_{L^\infty(\mathcal H^n)}.$ We deduce that convergence in the Sobolev space $W^{1, \infty}$ preserves everywhere differentiability.

We establish an Esakia duality for the categories of temporal Heyting algebras and temporal Esakia spaces. This includes a proof of contravariant equivalence and a congruence/filter/closed-upset correspondence. We then study two notions of « reachability » on the relevant spaces/frames and show their equivalence in the finite case. We use these notions of reachability to give both lattice-theoretic and dual order-topological characterisations of simple and subdirectly-irreducible temporal Heyting algebras. Finally, we apply our duality results to prove the relational and algebraic finite model property for the temporal Heyting calculus. This, in conjunction with the proven characterisations, allows us to prove a relational completeness result that combines finiteness and the frame property dual to subdirect-irreducibility, giving us a class of finite, well-understood frames for the logic.

We investigate the limiting distribution of geometric Brownian motion conditional on its running maximum taking large values. The Freidlin-Wentzell large deviations theory predicts that the conditional distribution of the sample paths converge weakly to a deterministic exponential curve. We complement this result by showing that the conditional sample paths in fact converge in strong sense, and obtain quantitative bounds on the rate of convergence. As an application of our results to financial mathematics, we obtain new closed form asymptotic formulae for the fair price of barrier options with general path dependent payoff in the short maturity limit, with quantitative error estimates. We provide exact formulae for Asian and lookback style payoffs.

Motivated by the geometry of Levi degenerate CR hypersurfaces, we define a \emph{pre-Kähler structure} on a complex manifold as a pre-symplectic structure compatible with the almost complex structure, i.e. a closed (1,1)-form. Extending \emph{Freeman filtration} to the pre-Kähler setting, we define holomorphic degeneration and finite-nondegeneracy and show that the symmetry algebra of a real analytic pre-Kähler structure is finite-dimensional if and only if it is finitely nondegenerate. Concurrently, we extend the classical correspondence between Kähler and Sasakian structures to the pre-Kähler setting, i.e. a one-to-one (local) correspondence between $k$-nondegenerate CR hypersurfaces equipped with a transverse infinitesimal symmetry and $k$-nondegenerate pre-Kähler structures. Focusing on the lowest dimensional case, we solve the equivalence problem of non-Kähler pre-Kähler complex surfaces that are $2$-nondegenerate by associating a Cartan geometry to them and explicitly express their local invariants in terms of the fifth jet of a potential function. We describe the vanishing of their basic invariants in terms of a double fibration, which gives a pre-Kähler characterization of the twistor bundle of symplectic connections on surfaces. Lastly, we study the pre-Kähler complex surfaces arising as symmetry reductions of homogeneous $2$-nondegenerate CR 5-manifolds, which leads to a characterization of certain \emph{critical} symplectic connections on surfaces. For such pre-Kähler manifolds, their moduli space of geometrically distinct structures contain $2$-dimensional open dense subsets, and they all have nontrivial infinitesimal symmetries. Finally, we show that all locally homogeneous pre-Kähler complex surfaces are locally flat.

We are concerned with discretisations of the classical Kolmogorov equation by a standard space-time discontinuous Galerkin method. {The} Kolmogorov equation serves as simple, yet rich enough in the present context, model problem for a wide range of kinetic-type equations: although it involves diffusion in one of the two spatial dimensions only, the combined nature of the first order transport/drift term and the degenerate diffusion are sufficient to `propagate dissipation' across the spatial domain in its entirety. This is a manifestation of the celebrated concept of hypocoercivity, a term coined and studied extensively by Villani in \cite{villani}. We show that the {classical} space-time discontinuous Galerkin method {admits} a corresponding hypocoercivity property at the discrete level, asymptotically for large times. To the best of our knowledge, this is the first result of this kind for any standard Galerkin scheme. This property is shown by proving one part of a discrete inf-sup-type stability result for the method in a family of norms dictated by a modified scalar product motivated by the theory in \cite{villani}. This family of norms contains the full gradient of the numerical solution, thereby allowing for a full spectral gap/Poincaré-type inequality at the discrete level, thus, showcasing a subtle, discretisation-parameter-dependent, numerical hypocoercivity property. Further, we show that the space-time discontinuous Galerkin method is inf-sup stable in the family of norms containing the full gradient of the numerical solution, which may be a result of independent interest.

We design strategies in nonlinear geometric analysis to temper the effects of adversarial learning for sufficiently smooth data of numerical method-type dynamics in encoder-decoder methods, variational and deterministic, through the use of geometric flow regularization. We augment latent spaces with geometric flows to control structure, relying on adaptations of curvature and Ricci flow. All of our flows are solved using physics-informed learning. Traditional geometric meaning is traded for computing ability, but we maintain key geometric invariants, the primary of which are maintained, intrinsically-low structure, nontriviality due to sufficient lower bounds on curvature, distortion of volume element, that develop quality in the inference stage. We instill representations that are canonical, smooth, curvature-aware, geodesic-aware, and non-topologically void or sparse. The primary bottleneck of a Ricci curvature flow is that Ricci curvature is high order, thus expensive to compute, so we will attempt to overcome this with properly justified proxies. Our primary contributions are fourfold. We develop a loss based on Gaussian curvature using closed path circulation integration for surfaces, bypassing automatic differentiation of the Christoffel symbols through use of Stokes' theorem. We invent a new parametric flow valid under a Taylor expansion derived from the Gauss equation. We develop two strategies based on time differentiation of functionals, one with a special case of scalar curvature for conformally-changed metrics, and another with harmonic maps, their energy, and induced metrics. Our losses are diminished analytically and mostly heuristic but maintain overall integral latent structure. We showcase that curvature flows and the formulation of geometric structure in intermediary encoded settings enhance learning and overall zero-shot and adversarial fidelity.

We demonstrate the generic invariance of the Fano type property in cases where the volumes of anti-canonical divisors of Fano type fibers are a constant over a Zariski-dense subset, or the Fano type fibers are dimension $2$. Additionally, paralleling this theorem, we establish a conjecture by Schwede and Smith under the condition that the volumes of anti-canonical divisors remain constant in the reduction mod $p$.

In this paper, we study the almost everywhere convergence of sequences of two-parameter ergodic averages over rectangles in the plane. On the one hand, we show that if the rectangles we consider have their sides with slopes in a finitely lacunary set, then the averages converge almost everywhere in all $L^p$ spaces, $1 < p < \infty$. On the other hand, given some non-lacunary sets of directions, we construct sequences of rectangles oriented along these directions for which the associated ergodic averages fail to converge almost everywhere in any $L^p$ space, $1 < p < \infty$.

We present a construction of non-rectifiable, repetitive Delone sets in every Euclidean space $\mathbb{R}^d$ with $d \geq 2$. We further obtain a close to optimal repetitivity function for such sets. The proof is based on the process of encoding a non-realisable density in a Delone set, due to Burago and Kleiner.

We investigate the role of the conductor in analytic rank bounds for elliptic curves over \(\mathbb{Q}\). Let \(E/\mathbb{Q}\) be an elliptic curve with conductor \(N_E\). We consider hypothetical degree-two \(L\)-functions associated to (E) that satisfy analytic continuation, a functional equation involving an arithmetic invariant \(\Phi(E)\), and yield rank bounds of the form \[ \operatorname{rank}(E)\ll \log \Phi(E). \] Using the Modularity Theorem, we show that any such invariant must satisfy \[ \Phi(E)\ge N_E. \] Thus the conductor is minimal among arithmetic invariants that can appear in this analytic framework. In particular, the standard logarithmic rank bounds arising from the conductor cannot be improved by replacing \(N_E\) with a strictly smaller invariant while preserving the same degree-two functional equation structure. These results provide a structural explanation for the distinguished role of the conductor in analytic approaches to the rank problem.

We introduce a hierarchical clustering framework for weighted projective spaces $\mathbb{P}_{\mathbf{q}}$ built on Finsler geometry. From an optimization-based Finsler norm that quotients out the weighted scaling action, we construct a scaling-invariant distance $d_F([z], [w])$ and a rational analogue $d_{F,\mathbb{Q}}([z], [w])$ for points of $\mathbb{P}_{\mathbf{q}}(\mathbb{Q})$. The norm carries a shape parameter $p$: the case $p=2$ is Riemannian and admits a closed-form distance, while $p\neq 2$ is genuinely Finsler, and the metric and clustering guarantees below hold for every $p\in[1,\infty)$. Whereas earlier work measured proximity in these spaces through non-metric dissimilarities, we prove that $d_F$ satisfies the triangle inequality and is therefore a genuine metric; this is what equips the induced clustering with its theoretical guarantees, including monotone dendrograms and Gromov--Hausdorff stability under perturbation of the data. The metric respects the intrinsic scaling symmetry and weighted topology of $\mathbb{P}_{\mathbf{q}}$, avoiding the distortions of a flat-space embedding. We develop the framework's arithmetic applications -- clustering rational points in the moduli space of genus two curves and analyzing rational functions in arithmetic dynamics -- and indicate prospective extensions to quantum state spaces, where the weights $\mathbf{q}$ model anisotropic noise. More broadly, the construction offers a rigorous metric foundation for graded neural networks and related machine-learning techniques on graded algebraic varieties.

In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements. We also tackle three concrete challenges. First, we consider the process of information propagation through a neural network, i.e., we study the input-output map for different architectures. We explain the universal embedding property for augmented neural ODEs representing arbitrary functions of given regularity, the classification of multilayer perceptrons and neural ODEs in terms of suitable function classes, and the memory-dependence in neural delay equations. Second, we consider the training aspect of neural networks dynamically. We describe a dynamical systems perspective on gradient descent and study stability for overdetermined problems. We then extend this analysis to the overparameterized setting and describe the edge of stability phenomenon, also in the context of possible explanations for implicit bias. For stochastic gradient descent, we present stability results for the overparameterized setting via Lyapunov exponents of interpolation solutions. Third, we explain several results regarding mean-field limits of neural networks. We describe a result that extends existing techniques to heterogeneous neural networks involving graph limits via digraph measures. This shows how large classes of neural networks naturally fall within the framework of Kuramoto-type models on graphs and their large-graph limits. Finally, we point out that similar strategies to use dynamics to study explainable and reliable AI can also be applied to settings such as generative models or fundamental issues in gradient training methods, such as backpropagation or vanishing/exploding gradients.

We study K-stability of smooth Fano threefolds of Picard rank $2$ and degree $22$ which can be obtained by blowing up a smooth complete intersection of two quadrics in $\mathbb{P}^5$ along a conic. We also describe the automorphism groups of these threefolds.

Let G be a simple, finite, connected, and undirected graph. The middle graph M(G) of G is obtained from the subdivision graph S(G) after joining pairs of subdivided vertices that lie on adjacent edges of G and the central graph C(G) of G is obtained from S(G) after joining all non-adjacent vertices of G. We show that if the order of G is at least 4, then Aut(G), Aut(C(G)), and Aut(M(G)) are isomorphic (as abstract groups) and apply this result to obtain new upper bounds of the distinguishing number and the distinguishing index of C(G) and M(G) and provide examples showing that these bounds cannot be improved in general. Moreover, we use idempotent commutative Latin squares and a theorem of Galvin on list edge colorings of bipartite graphs to study the total distinguishing chromatic number of central graphs.

This survey deals with two closely connected topics: first, the stability of Einstein metrics under the Einstein-Hilbert functional, and second, their deformation theory and the study of the moduli space of Einstein metrics on a compact manifold. To first order, both problems reduce to studying the spectrum and eigentensors of the Lichnerowicz Laplacian. We give an introduction to the classical theory and survey recent results and advances.

We introduce and study the Rhaly operator on Köthe spaces, with a primary focus on understanding its well-definedness, continuity, and compactness. We especially examine operators acting on power series spaces of both infinite and finite type. In the sequel, we provide integral representations for the Rhaly operator on the space of entire functions $H(\mathbb{C})$ and the space of holomorphic functions on the unit disc $H(\mathbb{D})$. We also investigate the topologizability and power boundedness of the Rhaly operators, which leads to findings about their mean ergodicity, uniform mean ergodicity, and Cesàro boundedness.

In this article we study the algebraic structure of fine Mordell--Weil groups, plus/minus Mordell--Weil groups, Selmer groups, and plus/minus Selmer groups in the cyclotomic $\mathbb{Z}_p$-extensions of abelian number fields. As a first, we prove theorems on the equivariant structure of fine Mordell--Weil groups and plus/minus Mordell--Weil groups. In other words, we study the explicit shape of the fine, plus/minus objects as a $\Lambda(\mathcal{G})$-module with $\mathcal{G} \simeq \mathbb{Z}_p \times G$ and $G$ a finite abelian group. We prove refinements of previously known results over $\mathbb{Q}$ for the classical Selmer group and the plus/minus Selmer group, and subsequently also the Shafarevich--Tate group, and the plus/minus Shafarevich--Tate group. This gives new evidence towards an affirmative answer for the Kurihara--Pollack problem.

The purpose of this note is to provide a family of explicit examples of $4$-dimensional operator systems contained in the Calkin algebra $\mathcal{Q}(\mathcal{H})$ on a separable infinite-dimensional Hilbert space $\mathcal{H}$ for which the identity map has no unital completely positive (ucp) lift to $\mathcal{B}(\mathcal{H})$ with respect to the canonical quotient map $\pi:\mathcal{B}(\mathcal{H}) \to \mathcal{Q}(\mathcal{H})$. More specifically, to each unital $C^*$-algebra $\mathcal{A}$ generated by $n$ unitaries and unital $*$-homomorphism $\rho:\mathcal{A} \to \mathcal{Q}(\mathcal{H})$ with no ucp lift, we construct a four-dimensional operator subsystem $\mathcal{S}$ of $M_{n+1}(\mathcal{A})$ without the lifting property. As a result, for each $n \geq 2$ we exhibit a four-dimensional operator system $\mathcal{S}$ in $M_{n+1}(C_r^*(\mathbb{F}_n))$ without the lifting property. We also obtain explicit examples where the generalized Smith-Ward problem for liftings of joint matrix ranges for three self-adjoint operators has a negative answer.

It is shown that two conjectures put forward in the recent article Iksanov and Kostohryz (2025) are true. Namely, we prove a functional central limit theorem (FCLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series $\sum_p \frac{\eta_p}{p^{1/2+s}}$ as $s\to 0+$, where $\eta_1$, $\eta_2,\ldots$ are independent identically distributed random variables with zero mean and finite variance, and $\sum_p$ denotes the summation over the prime numbers. As a consequence, an FCLT and an LIL are obtained for $\log \sum_{n\geq 1} \frac{f(n)}{n^{1/2+s}}$ as $s\to 0+$, where $f$ is a Rademacher random multiplicative function.

Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the coefficients of the system. In this paper we assume that the symbolic parameters appear polynomially in the coefficients and that the only variables to be solved for are those of the linear system. The consistency of the system and expression of the solutions may vary depending on the values of the parameters. It is well-known that it is possible to specify a covering set of regimes, each of which is a Zariski-constructible condition on the parameters together with a solution description valid under that condition. We provide a method of solution that requires time polynomial in the matrix dimension and the degrees of the polynomials when there are up to three parameters. We also discuss examples suggesting how the method may be useful beyond the formal three-parameter setting. In previous methods the number of regimes needed is exponential in the system dimension and polynomial degree of the parameters. Our approach exploits the Hermite and Smith normal forms that may be computed when the system coefficient domain is mapped to the univariate polynomial domain over suitably constructed fields. Our method identifies {intrinsic singularities} and {ramification points} where the algebraic and geometric structure of the matrix changes. Parametric eigenvalue problems are addressed as well.

We give sparsity results and present algorithms for calculating minimum (vector) 1-norm universal solvers connected to least-squares problems. In particular, besides universal least-squares solvers, we consider minimum-rank universal least-squares solvers, and simultaneous universal minimum-norm/least-squares solvers. For all of these, we present and compare several new alternative linear-programming formulations and very effective proximal-point algorithms. Overall, we found that our new Douglas-Rachford splitting algorithms for these problems performed best.

We study how conditioning on the first $k$ steps, which we think of as training, affects the long-term behavior of the Elephant Random Walk. When the elephant is conditioned to be at position $k$ at time $k$, the first return time to the origin scales as $k^{(4-4p)/(3-4p)}$ in the diffusive regime, and grows exponentially in the critical regime. We loosely interpret this as a measurement of the rate at which the elephant forgets its training.

This paper studies the global existence of component-wise nonnegative solutions of the Gray-Scott model in $\Omega \subset \mathbb{R}^n$, $n \ge 1$, with a mixture of both local and nonlocal diffusion operators. We use semigroup theory with duality arguments to establish the global existence and boundedness of solutions of our model. We also present the patterns of both local and mixed Gray-Scott models.

We analyze the Gray-Scott reaction--diffusion system on $\Omega\subset\mathbb{R}^n$ ($n\ge 1$) with mixed diffusion combining local and nonlocal operators. Using semigroup methods and duality estimates, we prove global existence of component-wise nonnegative solutions and establish uniform-in-time bounds. Numerical simulations illustrate pattern formation and highlight qualitative differences between the purely local and mixed-diffusion models.

This is the third and the last of a series of three papers, in which we prove a formula expressing the modular height of a unitary Shimura variety over a CM number field in terms of the logarithmic derivative of the Hecke L-function associated with the CM extension. The main idea of our proof is to compare the holomorphic projection of the derivative of a certain mixed Eisenstein-theta series and the arithmetic degree of a generating series of divisors on unitary Shimura varieties. In this paper, we compute the arithmetic degree of the arithmetic generating series of divisors on unitary Shimura varieties, and then, combining with the results from the first two papers in this series, derive the modular height formula for unitary Shimura varieties.

We address the efficient computation of power-law-based interaction potentials of homogeneous $d$-dimensional bodies with an infinite $n$-dimensional array of copies, including their higher-order derivatives. This problem forms a serious challenge in micromagnetics with periodic boundary conditions and related fields. Nowadays, it is common practice to truncate the associated infinite lattice sum to a finite number of images, introducing uncontrolled errors. We show that, for general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law potentials can be obtained by complementing a small direct sum by a correction term that involves efficiently computable derivatives of generalized zeta functions. We show that the resulting representation converges exponentially in the derivative order, reaching machine precision at a computational cost no greater than that of truncated summation schemes. In order to compute the generalized zeta functions efficiently, we provide a superexponentially convergent algorithm for their evaluation, as well as for all required special functions, such as incomplete Bessel functions. Magnetic fields and related quantities can thus be evaluated to machine precision in arbitrary cuboidal domains periodically extended along one or two dimensions. We benchmark our method against known formulas for magnetic interactions and against direct summation for Riesz potentials with sufficiently large exponents, consistently achieving full precision. In addition, we identify new corrections to the asymptotic limit of the demagnetization field and tabulate high-precision benchmark values that can be used as a reliable reference for micromagnetic solvers. The techniques developed are broadly applicable, with direct impact in other areas such as molecular dynamics.

We prove central limit theorems for the number of descents and inversions of permutations produced by shelf-shuffles. These are a model for casino card shuffling machines. We show the asymptotic normality of the number of descents in two limiting regimes depending on the ratio of cards to shelves. On the other hand, we study the inversions by employing a modification of the techniques from Islak's analysis of the statistics of riffle shuffles. In particular, we obtain a bound for the rate of convergence for inversions that is independent of the number of shelves.

We present an enhanced version of the row-based randomized block-Kaczmarz method to solve a linear system of equations. This improvement makes use of a regularization during block updates in the solution, and a dynamic proposal distribution based on the current residual vector and effective mutual orthogonality between all blocks. The improved method provides significant gains in solving highly ill-conditioned linear systems that are either sparse, or dense least-squares problems that are significantly over/under determined. Considering the poor guarantees in effectively preconditioning iterative solutions for such ill-conditioned problems, it may also serve as a pre-solver for accelerating other iterative numerical methods, and as an inner iteration in certain types of GMRES solvers for linear systems.

In this work, we propose two step-size strategies for the Golden ratio proximal ADMM (GrpADMM) to solve linearly constrained separable convex optimization problems. Both strategies eliminate explicit operator norm estimates by relying on inexpensive local information computed at the current iterate and requiring no backtracking. However, the key difference is that the second step-size strategy allows recovery from poor initial steps and can increase from iteration to iteration. Under standard assumptions, we establish global convergence of the generated iterates and derive sublinear convergence rates for both algorithms. We also obtain pointwise convergence rate results for the iterates of the algorithms. In addition, we show that the first proposed step-size rule for GrpADMM reduces to the fixed-step-size counterpart when the initial step-size is chosen below a certain threshold. Preliminary numerical experiments demonstrate the practical adaptability and effectiveness of the proposed approaches.

By introducing an intrinsic perimeter measure for intrinsic countably rectifiable sets, we prove a compactness result and a Poincaré inequality for special functions with bounded variation in equiregular Carnot-Carathéodory spaces which satisfy an additional natural assumption, called property $\mathcal R$.

Let $\mathbb{G}$ be a Lie group with a normal abelian subgroup $\mathbb{A}$, and let $(M,\omega)$ be a symplectic manifold endowed with a Hamiltonian $\mathbb{G}$-action. We investigate conditions under which symplectic reduction by $\mathbb{G}$ coincides with the symplectic reduction by the abelian subgroup $\mathbb{A}$. Using the reduction-by-stages framework (Marsden et al Springer Notes in Math., 1913, (2007)), we prove that, under a mild assumption, the corresponding reduced spaces are symplectomorphic if and only if they have the same dimension. Both this assumption and the dimension condition depend only on the groups $\mathbb{G}$ and $\mathbb{A}$, and on the momentum value $\mu\in \mathfrak{g}^*$ at which the symplectic reduction by $\mathbb{G}$ is performed; in particular, they are independent of the symplectic manifold $(M,\omega)$. We then provide a broad class of examples by identifying a large family of nilpotent Lie groups, including classical Carnot groups such as the Heisenberg group and jet-space $\mathcal{J}^k(\mathbb{R}^n,\mathbb{R}^m)$, for which the two reduced spaces are symplectomorphic for generic momentum values.

In this report, we present a proof of Levinson's theorem, following the ideas of Matthew P. Young in 2010, which states that one-third of the non-trivial zeros of the Riemann zeta function lie on the critical line, i.e. the line Re(s) = 1/2, using a mollified second moment of the zeta-function. Later, we present a generalized result for Dirichlet L-functions by Xiaosheng Wu in 2018, using Levinson's method, showing that more than two-fifths of the non-trivial zeros of Dirichlet L-functions are on the critical line. Moreover, more than two-fifths of the non-trivial zeros are simple and on the critical line, using a longer mollifier than in Levinson's original proof. This generalizes a result by Conrey from 1989 that the Riemann zeta-function has at least two-fifths of its zeros on the critical line.

We prove formulas for generalized rank deviations for overpartitions. These formulas are in terms of Appell-Lerch series and sums of quotients of theta functions and extend work of Lovejoy and the second author. As an application, we compute a dissection.

We consider the hyperbolic three-space in the half-space model endowed with a metric Lie group structure. In this setting, translation surfaces are defined as products of two curves $\alpha$ and $\beta$ with respect to the Lie group operation. We investigate minimal surfaces and solitons to the mean curvature flow arising from specific types of products of these curves. In particular, we provide classification results for minimal surfaces, hyperbolic translators, and conformal solitons to the mean curvature flow.

In this article, we study a Hénon-type equation in $\mathbb{R}^N$ driven by a nonlinear operator given by the combination of a local and a nonlocal term. This equation was originally proposed to model spherically symmetric stellar clusters. Here, we prove that, under a suitable relation among the parameters, there exists a threshold separating the existence and non-existence of solutions. Moreover, we establish regularity properties of the solutions.

In 1977, Chung, Chung and Liu generalized the definition of the Ramsey number. They introduced the $s$-chromatic Ramsey number as follows. Let $1\leq s< t$ be integers and let $A_{1}, A_{2}, \dots, A_{c}$ be subsets with size $s$ of $[t]$, where $c= {t\choose s}$. For given graphs $G_{1}, G_{2}, \dots, G_{c}$, the {\it $s$-chromatic Ramsey number} $r^{s, t}(G_{1}, G_{2}, \dots, G_{c})$, is the minimum positive integer $N$ such that every $t$-coloring of $E(K_{N})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [c]$. The {\it star-critical $s$-chromatic Ramsey number} $r_{*}^{s, t}(G_{1}, G_{2}, \dots, G_{c})$, is the minimum integer $\ell$ such that every $t$-coloring of the edges in $K_{N}- E(K_{1, N- 1- \ell})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [c]$, where $N= r^{s, t}(G_{1}, G_{2}, \dots, G_{c})$. If $G_{1}= G_{2}= \dots= G_{c}= G$, then we simplify them to $r^{s, t}(G)$ (also called the {\it weakened Ramsey number}) and $r^{s, t}_{*}(G)$, respectively. In this paper, we determine all the values of $r^{s, t}(K_{1, m})$ and $r_{*}^{s, t}(K_{1, m})$, and part of the value of $r^{s, t}(K_{1, m_{1}}, K_{1, m_{2}}, \dots, K_{1, m_{c}})$.

In this paper we initiate the study of quivers carrying quantum Yang--Baxter and Hecke structure, and we apply this framework to study path algebras over quivers whose loop spaces carry RTT relations determined by Hecke $R$-matrices. We show that the quantum matrix algebra $\mathcal{O}_q(M_n)$ is isomorphic as a bialgebra to the face algebra over a rose quiver deformed by RTT relations of the $GL_q(n)$ Hecke $R$-matrix.

We prove cyclic sieving phenomena satisfied by corner-rooted plane trees (alias ordered trees). The sets of rooted plane trees that we consider are: (1) all trees with $n$ nodes; (2) all trees with $n$ nodes and $k$ leaves; (3) all trees with a given degree distribution of the nodes. Moreover, we consider four different cyclic group actions: (1) the root is moved to the next corner along a tour of the tree; (2) only trees in which the root is at a leaf are considered, and the action moves the root to the next leaf; (3) only trees in which the root is at a non-leaf are considered, and the action moves the root to the next non-leaf corner; (4) only trees in which the root is at a node of degree $\delta$ are considered, for a fixed $\delta$, and the action moves the root to the next corner of this type. We prove a cyclic sieving phenomenon for each meaningful combination of these sets and actions. As a bonus, we also establish corresponding cyclic sieving phenomena for tree-rooted planar maps.

Inspired by the pullback method in the recent work of Sangiovanni-Vincentelli--Skinner, we reconstruct the diagonal class of Darmon--Rotger. Moreover, we reinterpret the computation of the $p$-adic regulator formula.

We study clinically motivated capacity-constrained treatment-adherence outreach through a belief-state restless multi-armed bandit model, in which each patient is a partially observed two-state Markov decision process and interventions induce reset-type belief dynamics. For the discounted criterion, partial conservation law (PCL)-based conditions are used to establish single-patient threshold-policy optimality and indexability (threshold-indexability) and yield a closed-form Whittle index, threshold performance metrics, and an explicit optimal threshold map. We also prove a single-patient long-run average analogue on the invariant belief core and obtain an explicit average-criterion Whittle index. For the multi-patient model, the PCL-derived formulas give an analytic Lagrangian relaxation, efficient dual bounds, and computable Lagrangian index benchmark policies, including a forced-capacity variant. We analyze how the Whittle index depends on lapse and spontaneous-recovery probabilities. In large-scale experiments with two-type, three-type, and jittered finite-mixture populations, the Whittle and forced-capacity Lagrangian index policies are the strongest performers, while myopic prioritization can be substantially worse under tight capacity.

Preference cycles are prevalent in problems of decision-making, and are contradictory when preferences are assumed to be transitive. This contradiction underlies Condorcet's Paradox, a pioneering result of social choice theory, wherein intuitive and seemingly desirable constraints on decision-making necessarily lead to contradictory preference cycles. Topological methods have since broadened social choice theory and elucidated existing results. However, characterisations of preference cycles in topological social choice theory are lacking. In this paper, we address this gap by introducing a framework for topologically modelling preference cycles that generalises Baryshnikov's existing topological model of strict, ordinal preferences on 3 alternatives. In our framework, the contradiction underlying Condorcet's Paradox topologically corresponds to the non-orientability of a surface homeomorphic to either the Klein bottle or real projective plane, depending on how preference cycles are represented. These findings allow us to reformulate Arrow's Impossibility Theorem in terms of the orientability of a surface as well.

The Compact Discontinuous Galerkin method was introduced by Peraire and Persson in (SIAM J. Sci. Comput., 30, 1806-1824, 2008). In this work, we present the stability and convergence analysis for the $hp$-version of this method applied to elliptic problems on polytopal meshes. Moreover, we introduce fast and practical algorithms that allow the CDG, LDG, and BR2 methods to be implemented within a unified framework. Our numerical experiments show that the CDG method yields a compact stencil for the stiffness matrix, with faster assembly and solving times compared to the LDG and BR2 methods. We numerically study how coercivity depends on the method parameters for various mesh types, with particular focus on the number of facets per mesh element. Finally, we demonstrate the importance of choosing the correct directions for the numerical fluxes when using variable polynomial degrees.

Local Polynomial Regression (LPR) and Moving Least Squares (MLS) are closely related nonparametric estimation methods, developed independently in statistics and approximation theory. While statistical LPR analysis focuses on overcoming sampling noise under probabilistic assumptions, the deterministic MLS theory studies smoothness properties and convergence rates with respect to the \textit{fill distance} (a resolution parameter). Despite this similarity, the deterministic assumptions underlying MLS fail to hold under random sampling. We begin by quantifying the probabilistic behavior of the fill distance $h_n$ and \textit{separation} $\delta_n$ of an i.i.d. random sample. That is, for a distribution satisfying a mild regularity condition, $h_n\propto n^{-1/d}\log^{1/d} (n)$ and $\delta_n \propto n^{-2/d}$ in probability. We then prove that, for MLS of degree $k\!-\!1$, the approximation error associated with a differential operator $Q$ of order $m\leq k-1$ decays as $h_n^{\,k-m}$, establishing stochastic analogues of the classical MLS estimates. Additionally, we show that the MLS approximant is locally smooth with high probability. This work provides the first unified stochastic analysis of MLS, demonstrating that - despite the failure of deterministic sampling assumptions - the classical convergence and smoothness properties persist under natural probabilistic models.

The action of the finite Hilbert transform defined on $L^\infty(-1,1)$ and taking its values in the Zygmund space $L_{\textnormal{exp}}(-1,1)$ is studied in detail. This is a reciprocal situation to the investigation recently undertaken in [11] of the finite Hilbert transform defined on the Zygumd space $L\textnormal{log} L(-1,1)$ and taking its values in $L^1(-1,1)$. The fact that both $L^\infty(-1,1)$ and $L_{\textnormal{exp}}(-1,1)$ fail to be separable generates new features not present in[11].

We present an algorithm for determining the set of $S$-integral points on an affine curve based on the Affine Chabauty method developed in the first part of this series. We achieve this by constructing explicit logarithmic differentials whose integrals take on prescribed values on $S$-integral points. Along the way, we prove a $p$-adic residue theorem for Coleman integrals of log differentials.

Fast computation of a matrix product $W^\top X$ is a workhorse of modern LLMs. To make their deployment more efficient, a popular approach is that of using a low-precision approximation $\widehat W$ in place of true $W$ (``weight-only quantization''). Information theory demonstrates that an optimal algorithm for reducing precision of $W$ depends on the (second order) statistics of $X$ and requires a careful alignment of vector quantization codebook with PCA directions of $X$ (a process known as ``waterfilling allocation''). Dependence of the codebook on statistics of $X$, however, is highly impractical. This paper proves that there exist a universal codebook that is simultaneously near-optimal for all possible statistics of $X$, in the sense of being at least as good as an $X$-adapted waterfilling codebook with rate reduced by 0.11 bit per dimension in the case when $W$ is Gaussian. Such universal codebook would be an ideal candidate for the low-precision storage format, a topic of active modern research, but alas the existence proof is non-constructive. Equivalently, our result shows existence of a net in $\mathbb{R}^n$ that is a nearly-optimal covering of a sphere simultaneously with respect to all Hilbert norms.

We introduce a refinement of Boolean-Catalan numbers and call them Boolean-Narayana numbers. We provide an explicit formula for these numbers, and prove unimodality, log-concavity, and real-roots-only results for their sequences. We also prove a three-term recurrence relation for their generating polynomials.

This work presents a novel agglomeration-based multilevel preconditioner designed to accelerate the convergence of iterative solvers for linear systems arising from the discontinuous Galerkin discretization of the monodomain model in cardiac electrophysiology. The proposed approach exploits general polytopic grids at coarser levels, obtained through the agglomeration of elements from an initial, potentially fine, mesh. By leveraging a robust and efficient agglomeration strategy, we construct a nested hierarchy of grids suitable for multilevel solver frameworks. The effectiveness and performance of the methodology are assessed through a series of numerical experiments on two- and three-dimensional domains, involving different ionic models and realistic unstructured geometries. The results demonstrate strong solver effectiveness and favorable scalability with respect to both the polynomial degree of the discretization and the number of levels selected in the multigrid preconditioner.

In this paper, based mainly on the method of Iwasawa and Kida, by studying in detail the Hasse's unit index and the ramifications of prime ideals, we obtain explicit results of Iwasawa invariants $ \lambda_{2} $ of the cyclotomic $ \Z_{2}$-extensions of number fields. In particular, under Greenberg's conjecture, we obtain an explicit formula of $ \lambda_{2} $ for imaginary multi-quadratic number fields. As an application, we give a criteria of determining class number parity of multi-quadratic number fields.

We study local generalized additive decompositions (GADs) of homogeneous polynomials and their associated points schemes through their local inverse systems. We verify that their construction and algebraic properties are independent of the chosen apolarity action. We propose a determinantal method for computing minimal local GADs by minimizing the rank of a symbolic inverse system. When the locus of minimal supports is finite, this procedure provides a practical tool to determine all minimal local decompositions without tensor extensions. We prove that this finiteness is guaranteed whenever the local GAD-rank of the form does not exceed its degree. We analyze both generic and special cases, provide computational evidence assessing the impact of different choices for minors in the determinantal algorithm, and compare our approach with existing algorithms for local apolar schemes.

We investigate which invariants of groups are powerful in distinguishing non-isomorphic p-groups. Based on this, we devise an effective algorithm to solve the group identification problem for the $10,494,213$ groups of order $2^9$. We exhibit a list of groups of order $2^9$ which are difficult to distinguish y invariants and thus are the most difficult cases for our algorithm.

In a recent paper the last author proved that absence of complex zeros of the partition function of the hard-core model near a parameter $\lambda>0$ implies a form of correlation decay called strong spacial mixing. In this paper we investigate the reverse implication. We introduce a strengthening of strong spatial mixing that we call very strong spatial mixing (VSSM). Our main result is that if VSSM holds at a parameter $\lambda>0$ for a family of graphs, this implies that the partition function has no zeros near that parameter for each graph in the family. We also demonstrate that a closely related variant of very strong spatial mixing does not imply zero-freeness. As a consequence of our main result, we moreover obtain that VSSM implies spectral independence. Our proof relies on transforming the problem to the analysis of an induced non-autonomous dynamical system given by Möbius transformations.

We propose a generalized version of the bisection method where the cutting point between the two subintervals is chosen at random following an arbitrary distribution. We compute expected convergence rates with respect to any arbitrary a priori distribution for the position of the root in the initial interval and proved that it depends only on the the expectation $\mathbb{E}[c(1-c)]$ of the cut $c$. We also provide a generalization of the method for $K$ random cuts and study its convergence properties. Most probabilistic derivations are kept fairly simple for the ease of understanding of a larger audience. Our theoretical results are then validated numerically using statistical simulation.

We investigate novel scenarios of self-similar finite-time blowups of the generalized Constantin-Lax-Majda model with a parameter $a$, which are induced by a new setting where the smooth initial data satisfy certain derivative degeneracy condition. In this setting, our numerical study reveals distinct self-similar blowup behaviors depending on the sign of $a$. For $a>0$, we observe one-scale self-similar blowups with regular profiles that have not been found in previous studies. In contrast, for $a\le 0$, we discover a novel two-scale self-similar blowup scenario where the outer profile converges to a singular function at the blowup time while the inner profile remains regular on a much smaller scale. Correspondingly, an $a$-parameterized family of singular self-similar profiles with explicit expressions are constructed for $a<0$ and shown to match nicely with the limiting profiles obtained in numerical simulation. In particular, for the specific case of $a=0$, we rigorously prove the convergence of the outer profile to an explicit singular function in self-similar coordinates. Furthermore, we demonstrate the two-scale nature of the blowup in this scenario by showing that the local inner profile behavior around the singularity point of the outer profile is governed by a traveling wave on a smaller scale. To support this observation, we rigorously establish the existence of such traveling wave solutions via a fixed-point method.

We characterize Hilbert-Schmidt Hankel operators on the Bergman spaces of smooth bounded strongly pseudoconvex domains in $\mathbb{C}^n$ for $n \geq 2$. We consider harmonic symbols of class $C^3$ up to the closure of the domain and show $H_{\phi}$ is Hilbert-Schmidt if and only if $\phi$ is holomorphic on the domain.

We consider scheduling in a quantum switch with stochastic entanglement generation, finite quantum memories, and decoherence. The objective is to design a scheduling algorithm with polynomial-time computational complexity that stabilizes a nontrivial fraction of the capacity region. Scheduling in such a switch corresponds to finding a matching in a graph subject to additional constraints. We propose an LP-based policy, which finds a point in the matching polytope, which is further implemented using a randomized decomposition into matchings. The main challenge is that service over an edge is feasible only when entanglement is simultaneously available at both endpoint memories, so the effective service rates depend on the steady-state availability induced by the scheduling rule. To address this, we introduce a single-node reference Markov chain and derive lower bounds on achievable service rates in terms of the steady-state nonemptiness probabilities. We then use a Lyapunov drift argument to show that, whenever the request arrival rates lie within the resulting throughput region, the proposed algorithm stabilizes the request queues. We further analyze how the achievable throughput depends on entanglement generation rates, decoherence probabilities, and buffer sizes, and show that the throughput lower bound converges exponentially fast to its infinite-buffer limit as the memory size increases. Numerical results illustrate that the guaranteed throughput fraction is substantial for parameter regimes relevant to near-term quantum networking systems.

In this short note we present new results on a higher-dimensional generalization of De~Giorgi's conjecture for Allen--Cahn type equations, focusing on dimensions $n \ge 9$. Although counterexamples are known in this regime, our goal is to identify assumptions on solutions that still enforce one-dimensional symmetry. We prove an analogue of Savin's theorem in arbitrary dimension: for energy-minimizing solutions whose level sets enjoy n-7 directions of monotonicity, we deduce one-dimensional symmetry. In the same spirit, we extend these ideas to nonlocal phase transitions, and we discuss an application to free boundary problems Finally, we establish a counterpart of the Ambrosio--Cabré theorem for solutions that are not necessarily energy minimizers and may lack bounded energy density, assuming instead n-2 directions of monotonicity everywhere. Overall, this note aims to further strengthen the connection between phase transition models and minimal surface theory.

In 1998, E. Couselo, S. González, V. T. Markov, and A. A. Nechaev introduced the notions of recursive codes and recursively differentiable quasigroups. They conjectured that recursive MDS codes of dimension $2$ and length $4$ exist over every finite alphabet of size $q \not\in \{2, 6\}$, and verified this conjecture in all cases except $q \in \{14, 18, 26, 42\}$. In 2008, V. T. Markov, A. A. Nechaev, S. S. Skazhenik, and E. O. Tveritinov resolved the case $q=42$ by providing an explicit construction. The present paper settles the outstanding case $q=26$. The construction rests upon methods for producing recursively differentiable quasigroups and recursive MDS codes via perfect cyclic Mendelsohn designs. Moreover, we sharpen several known bounds concerning the existence of recursively $n$-differentiable quasigroups of small orders.

Let $X$ be a projective complex 3-manifold. An effective curve class $\beta\in H_2(X,\mathbb Z)$ is called positive if $c_1(X)\cdot\beta>0$, and superpositive if all the effective summands of $\beta$ are positive. If $X$ is Fano then all curve classes are superpositive. In arXiv:2111.04694 the second author developed a theory of enumerative invariants in abelian categories and wall-crossing formulae. We use this theory to prove conjectures by Pandharipande and Thomas on the rationality and poles of generating functions of Pandharipande-Thomas invariants of $X$ with descendent insertions, for superpositive curve classes.

The mean-field Schrödinger bridge (MFSB) problem concerns designing a minimum-effort controller that guides a diffusion process with nonlocal interaction to reach a given distribution from another by a fixed deadline. Unlike the standard Schrödinger bridge, the dynamical constraint for MFSB is the mean-field limit of a population of interacting agents with controls. It serves as a natural model for large-scale multi-agent systems. The MFSB is computationally challenging because the nonlocal interaction makes the problem nonconvex. We propose a generalization of the Hopf-Cole transform for MFSB and, building on it, design a Sinkhorn-type recursive algorithm to solve the associated system of integro-PDEs. Under mild assumptions on the interaction potential, we discuss convergence guarantees for the proposed algorithm. We present numerical examples with repulsive and attractive interactions to illustrate the theoretical contributions.

We develop a semigroup-theoretic analogue of liaison for relative ideals of a numerical semigroup. Two parallel linkage notions are proposed: a theory based on translates of the semigroup and a theory based on translates of the canonical ideal.

We determine the limiting density of the largest sum-free subset of the lattice cube $\{1,2,\ldots,n\}^d$ for all $d$, thus resolving the natural conjecture that it is constructed by two appropriate hyperplane slices. Equivalently, we show that the largest measure of a sum-free subset of the hypercube $(0,1)^d\subset \mathbb{R}^d$ is attained by $\setcond{x\in (0,1)^d}{1\leq L(x)<2}$ for some linear map $L:\mathbb{R}^d\to \mathbb{R}$. It is natural to conjecture that the same phenomenon might hold if one replaces the hypercube by any convex set not containing the origin, but we give an example to show that for sufficiently large $d$ this is not the case.

It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$ goes to infinity and $np$ converges to some $\lambda$. In this note we use instead the coupling technique to show a much more general result. Moreover, we only use elementary results from probability.

This paper considers functional series whose terms are higher-order derivatives of Chebyshev polynomials of the second kind, where the degree of the polynomial is related to the order of the derivative. Analytic summation is used to determine the rational functions to which these series converge. These functions are expressed in terms of Chebyshev polynomials evaluated at a specific argument. Connections are established between derivatives of Chebyshev polynomials of the second kind and special numerical sequences generated by linear recurrence relations. New closed-form formulas are obtained for the sums of the series at various values of the argument. As consequences, combinatorial identities are derived for the Fibonacci, Lucas, and Pell numbers, for sections of the Fibonacci sequence, and for their convolutions. By means of analytic continuation, sums of formally divergent series are obtained, which in special cases correspond to the classical Euler formulas.

Calibration of an honest confidence interval means choosing, for each $\alpha\in(0,1)$, how the corresponding $\alpha$-critical value is converted into a radius yielding coverage probability at least $1-\alpha$. Standard-normal critical-value calibration (SNC) is the default route for many confidence intervals based on nonparametric smoothers in nonparametric econometrics. However, this calibration method creates a structural difficulty: the normalization yielding a limiting distribution also makes a small estimation bias become a non-negligible inferential bias. We take a different calibration route by combining the tail control of empirical Bernstein inequalities with a fixed-length-radius optimization from bias-aware inference. We establish the formal theory in canonical scalar-covariate regression and density settings, with the regression theory ranging from local-polynomial to weighted-average estimators. The resulting empirical Bernstein confidence intervals (EBCIs) are "safe" and "sharp". Safety means that, uniformly over functions with some $S$-th order local smoothness, both one-sided and two-sided intervals attain the nominal coverage level up to a remainder $o(n^{-\frac{2S}{2S+1}})$, or an exponential remainder in bounded or sub-Gaussian settings. Sharpness means that interval widths shrink at the minimax rate $n^{-\frac{S}{2S+1}}$. Moreover, in the small-$\alpha$ regime, the EBCI radius is first-order aligned with the radii of bias-aware fixed-length confidence intervals. Thus, EBCI safely converts correctly specified smoothness into both coverage accuracy and interval-length efficiency. The contribution is not a new bias-control approach, but a new calibration principle for the radius of a confidence interval. The method can be combined with existing ideas such as bias-aware inference (BA) and robust bias correction (RBC), while avoiding the bias inflation induced by SNC.

We prove that the KSGNS construction can be viewed as an endofunctor on a category whose objects are positive $C^*$-correspondences from a fixed $C^*$-algebra and morphisms are given by intertwiners which account for automorphisms of the fixed $C^*$-algebra. Using this perspective, we provide a functorial perspective for strict positive equivariant $C^*$-correspondences of $C^*$-dynamical systems and show every strict positive equivariant $C^*$-correspondence of $C^*$-dynamical systems unitarily uniquely dilates under the KSGNS construction to an equivariant $C^*$-correspondence of the dynamical systems.

We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining isometries and linear order embeddings. We show that the class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé, and that the limit is the countable rational Urysohn ultrametric space $\mathbb{U}$. The space $\mathbb{U}$ is dc-universal for all countable ultrametric spaces, and its Cauchy completion $\overline{\mathbb{U}}$ is dc-universal for all separable ultrametric spaces, which is in contrast with the situation of classical ultrametric spaces and isometric embeddings, where no such universal space can exist. We study further properties of $\mathbb{U}$, of its variants, and of its automorphism group, which is richer than its group of isometries. In particular, we provide two types of tree representations of the two-sorted ultrametric spaces, discuss connections to valued fields, and characterize the automorphism group of $\mathbb{U}$ as the semidirect product of a group of order preserving bijections and a group of isometries. Furthermore, we show universality of $\operatorname{Aut}(\mathbb{U})$ and identify its universal minimal flow.

A global packet may simultaneously contain an automorphic representation and a non-automorphic representation. The global $\mathcal S$-group is expected, and known in some cases, to specify the automorphic representations in each global packet. For a covering group of an algebraic torus, it is not obvious from the definition whether the analogue of a global $\mathcal S$-group has finite order. In this paper, we verify this finiteness for a Brylinski-Deligne covering group of a torus.

In this article, we study the zero-divisor graph of the commutative ring with identity $R= \mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p,$ where $u^2 = 0, v^2 = 0, uv = vu$ and $p$ is an odd prime. We determine several graph-theoretic properties associated with the zero-divisor graph $\Gamma(R),$ including the clique number, chromatic number, vertex connectivity, edge connectivity, diameter and girth. In addition, we compute certain topological indices of the graph $\Gamma(R).$ Furthermore, we find the eigenvalues, energy and spectral radius of the adjacency matrix, the Laplacian matrix and the Eccentricity matrix of the zero-divisor graph $(\Gamma(R).$

We denote by $\mathcal{R}_{g;m}$ the moduli space of $m$--pointed Prym curves of genus $g$, that is, tuples $[\widetilde C / C; x_1, \dots, x_m]$ where $[C, x_1, \dots, x_m]$ is an $m$--pointed curve of genus $g$ and $\widetilde C/ C$ is an étale double cover of $C$. In this paper, we address the problem of the non--tautology of the Chow ring of $\mathcal{R}_{g;m}$. The locus which allows us to achieve earlier bounds for the non--tautology of $\mathrm{CH}^\bullet(\mathcal{R}_{g})$ compared to $\mathcal{M}_g$ is the component $\mathcal{R}\mathcal{B}_g^0$ of the locus of bi--elliptic Prym curves. This parametrises covers $[\widetilde C/ C]$ such that, if $C \rightarrow E$ is the bi--elliptic structure, the composition $\widetilde C \rightarrow E$ factors through an elliptic cover of $E$. Our main contribution is thus the non--tautology of the class $[\mathcal{R}\mathcal{B}_8^0] \in \mathrm{CH}^*(\mathcal{R}_8)$. In the course of establishing this theorem, a similar result for the compact moduli spaces $\overline{\mathcal{R}}_{g; 2m}$ for $g + m \geq 8$ is proven.

We investigate the category of ``matricial order operator spaces,'' which generalize operator systems, being equipped with both matricial norms and matricial order. For these objects, we develop duality theory. Taking a cue from the theory of ordered normed spaces, we introduce two important properties describing the interplay between order and norm -- ``normality'' and ``generation,'' and show that they are dual to each other. As examples, we consider operator systems (in particular, C*-algebras), and Schatten spaces. We also describe the minimal and maximal matricial order structures (which, again, turn out to be in duality), and show how Banach lattices can be equipped with such structures.

We present a graph-theoretic reformulation of pitch-class set theory in which each set in $\mathbb{Z}_n$ is represented as a complete weighted graph whose edge weights are interval classes. We show that this construction is invariant under the dihedral group $D_n$, and that the full interval structure is encoded by a cyclic step composition, from which all interval data are recovered via an additivity principle. This framework yields a direct correspondence between T/I equivalence and graph isomorphism, and reinterprets Z-relation as non-isomorphic graphs with identical edge-weight multisets. We extend the model to weighted clique complexes, linking higher-order homometry to simplex-weight structure, and introduce a cent-weighted formulation enabling comparisons across different equal temperaments. Finally, we define a polynomial invariant derived from antipodal step pairings for algebraic analysis of pitch class space.

We show that the geometry of Hertling-Manin F-manifolds $(M,\circ,e)$ provide the appropriate theoretical framework for studying the integrability of quasilinear systems of first-order evolutionary partial differential equations of the form ${\bf u}_t=X\circ {\bf u}_x$ (F-systems) under the mild assumption that the unit vector field is cyclic with respect to the operator of multiplication by the vector field $X$. This approach is very general and allows us to treat even non-regular systems that were previously beyond the scope of existing techniques. Like in the regular case the information about integrability is contained in a torsionless connection associated with the system and the integrability condition reduces to a geometric condition involving the Riemann tensor of the connection and the structure functions of the product. We prove that a locally conservative F-system is integrable and, in the analytic setting, also the converse statement, thereby providing a full characterization of integrability. Moreover, in the analytic case, we prove the existence of a family of analytic symmetries providing, in principle, the unique local analytic solution of the Cauchy problem through the generalised hodograph method.

We show that if $G$ is a $d$-regular Vizing-class-1 graph, then the proper additive chromatic index of $G$, denoted $\eta'_p(G)$, is equal to its chromatic index. This verifies that a strengthening of the Additive Coloring Conjecture of Czerwiński et al. holds for line graphs of $d$-regular Vizing-class-1 graphs. We show that if $G$ is a $d$-regular Vizing-class-2 graph, $\eta'_{p}(G)\leq \frac{(2^{\lceil \log_2 (d+1)\rceil})^2+2}{3}$, and if $G$ is a $d$-regular Vizing-class-2 graph that admits a proper edge-coloring with a smallest color class of size $r$ and $\text{girth}(G)\geq 6r-5$, then $\eta_p'(G)\leq 2d$, among other results.

For positive integers $s$ and $k$, the Ramsey number $r(s,k)$ is the minimum integer $n$ such that any graph on $n$ vertices contains a clique of size $s$ or an independent set of size $k$. We prove that for any fixed $s \ge 3$ and $k$ tending to infinity, the off-diagonal Ramsey numbers satisfy \[ r(s, k) \ge \Omega \left(\frac{k^{s-1}}{(\log k)^{2s-4}} \right), \] which matches, up to polylogarithmic factors, the upper bound established over 90 years ago by Erdős and Szekeres. For $s \ge 5,$ this improves the best known lower bound of the form $r(s, k) \ge k^{\frac{s+1}{2} + o(1)}$ which was first established by Spencer in 1977 and has since only seen polylogarithmic improvements.

Let $G$ be an arbitrary group and let $\mathbb{F}$ be a finite field. In this paper, we determine bases for the $T_G$-ideals of graded polynomial identities of the algebra $M_2(\mathbb{F})$ for all possible $G$-gradings. The bases obtained consist of finitely many non-trivial graded identities, and are finite whenever $G$ is finite.

Ballantine, Beck, and Merca defined the elementary symmetric partition map pre$_j$ that sends a partition $\lambda$ to a larger partition whose parts are the summands appearing in the evaluation of the $j$-th elementary symmetric polynomial on $\lambda$. They conjectured that pre$_j$ is injective on the set of partitions of $n$ with length $\ell \geq j$. The $\ell = j$ case was disproved by Devnani and Eyyunni; they instead conjectured the statement to be true for $\ell > j$. In this article, we answer this refined conjecture in the negative by proving that pre$_j$ is not injective on partitions of $n$ with length $2j$ for $j \geq 3$. We also prove that the analogous map prh$_j$ defined via the complete homogenous symmetric polynomial is injective on the set of all partitions.

Motivated by their applications in DNA-based storage systems, codes capable of correcting consecutive deletions have attracted significant attention. An important class of such codes consists of those that can correct multiple consecutive deletion errors, commonly referred to as multiple $b$-burst deletion-correcting codes. In this paper, we investigate the fundamental limits of multiple $b$-burst deletion-correcting codes. Specifically, we first characterize several structural properties of the associated deletion balls. Then, leveraging these properties, we derive several upper bounds and a combinatorial lower bound on the maximum size of such codes. As a consequence, our bounds improve upon the previously known results for general parameter regimes and are shown to be asymptotically optimal for certain cases.

We define the analogue of Lie-Rinehart algebras over $C^\infty$-rings. We show that given a Poisson $C^\infty$-ring $\mathcal{A}$ its module $\Omega_{\mathcal{A}}^{1}$ of $C^\infty$-Kähler differentials is (part of) a Lie-Rinehart algebra. Conversely, given a Lie-Rinehart algebra $\mathcal{M} \xrightarrow{\rho} C^\infty\mathrm{Der}(\mathcal{A})$ over a $C^\infty$-ring $\mathcal{A}$, there is a natural Poisson bracket on the $C^\infty$-ring $\mathcal{F}(\mathcal{M})$ associated with the $\mathcal{A}$-module $\mathcal{M}$ (the $C^\infty$-ring analogue of an $\mathcal{A}$-algebra freely generated by the module $\mathcal{M}$). In the case where $\mathcal{A}$ is the $C^\infty$-ring of smooth functions on a manifold $M$ and $\mathcal{M}$ is the module $\Gamma(E)$ of sections of a Lie algebroid $E \to M$, the $C^\infty$-ring $\mathcal{F}(\Gamma(E))$ is the ring of functions $C^\infty(E^\vee)$ on the total space of the vector bundle $E^\vee \to M$ dual to the vector bundle $E$.

We use recent results, among which modular inequalities for curves, to determine the Alexander polynomials for some classes of pencil-type conic-line arrangements. For these classes of curves we prove that the Alexander polynomial is (at least partially) combinatorial. To this end, we exemplify new techniques that are suitable for broader use, lending themselves to more general classes of curves.

We study the tri-plane crossing number, that is, the minimal number of crossings in a tri-plane diagram for a bridge trisection of a knotted sphere in $S^4$. We show that every 2-knot in $S^4$ that admits a bridge trisection with at most five crossings is ribbon. As a consequence, we show that the 2-twist spin of the trefoil has crossing number 6. This is the first such computation for a non-trivial knotted surface.

We show that every derivation of a simple and rational vertex operator algebra of CFT type is an inner derivation.

We address two open questions in the rough volatility literature. First, we prove finite positive moments for the rough Bergomi price process, and for a wider class of Gaussian Volterra Bergomi models, in the whole subcritical range under negative correlation. More precisely, if \(\rho\in[-1,0)\), then \(\E[S_T^p]<\infty\) for every \(0<p<p_\rho\), where \(p_{-1}=\infty\) and \(p_\rho=(1-\rho^2)^{-1}\) for \(-1<\rho<0\). For the fractional rough Bergomi kernel, this gives the finite side of the sharp critical moment threshold, complementing the known explosion result above the threshold. Second, we prove that the rough Heston variance process, equivalently the scalar Volterra square-root process with fractional kernel \(K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)\) and \(\alpha\in(1/2,1)\), has a positive atom at zero at every positive time. Consequently, zero is hit with positive probability before every positive time horizon. This rules out any Feller-type condition making the zero boundary inaccessible in the fractional rough Heston model.

Operations research and management science models support decisions that affect patients, workers, citizens, and public institutions. Decision-makers, such as clinicians approving surgical schedules, planners allocating disaster relief resources, or managers designing workforce rotations, increasingly require clear and actionable justifications that bridge the gap between mathematical optimization outputs and the intuitive reasoning stakeholders need to trust, contest, and implement recommended decisions. Yet the field has traditionally evaluated optimization models through computational criteria such as feasibility, optimality, scalability, and solution time, while treating explanation as a secondary concern. Mathematical transparency, provided through access to objectives, constraints, shadow prices, or sensitivity reports, does not automatically offer the forms of justification that stakeholders need to understand, trust, contest, or implement optimization-based decisions. This paper calls for the development of explainable optimization (XOpt) as a distinct interdisciplinary area that moves beyond algorithmic efficiency and incorporates behavioral, cognitive, and pragmatic perspectives to address this explanatory deficit.

In this paper we develop a combinatorial algorithm to compute the Gelfand--Kirillov (GK) dimension of simple highest weight modules for basic classical Lie superalgebras. Building upon the results for classical Lie algebras via Lusztig's {\bf a}-function and the Robinson--Schensted (RS) insertion algorithm, we extend these techniques to the super setting, providing explicit formulas for types $\mathfrak{sl}(m|n)$ and $\mathfrak{osp}(2|2n)$. Our results show that the GK dimension of a simple highest weight module is determined entirely by the even part of the Lie superalgebras.

Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, \[ (\kappa^2 - \Delta)^{\alpha/2} u = \mathcal{W}, \;\; \kappa \in \mathbb{R}, \; \alpha \in \mathbb{N}. \] of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to $\alpha, \kappa$ and thus allows a universal approximation scheme for the precision and covariance matrices of the entire $(\alpha, \kappa)$-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.

We present a discrete parametric characterization of the Mahler functional $\mathcal{V}_M$ for centrally symmetric polytopes in $\mathbb{R}^n$. By formulating the first variation of the volume with respect to the radial immersions of the vertices, we derive an exact KKT stationarity condition. Through this formalism, we demonstrate that the Hanner orbit constitutes a strict local minimum in the stratum of radial dilations modulo $GL(n, \mathbb{R})$, for all dimensions $n \ge 2$. Spectral analysis of the second variation reveals that the radial Hessian matrix is analytically equivalent to a positive semi-definite discrete Graph Laplacian. Coupling this isochoric result with a simplicial truncation analysis in the Hausdorff topology, we establish a quadratic quantitative stability bound against local polyhedral perturbations. This discrete framework eludes the degenerations of traditional smooth continuous analysis, providing an explicit algebraic resolution for the strict local minimality of this geometric configuration and its topological isolation in the subspace of polytopes.

We develop a general framework for analyzing representation costs of parametric data-fitting methods through their parameter-space regularizers. From this abstract perspective, we define representation costs for arbitrary parametric models and reveal their induced (native) function spaces. This unifies recent function-space views of data-fitting methods. We also prove that many natural results hold in this abstract setting, including representer theorems for parametric methods on their native spaces. The framework also rigorously connects parametric methods with their equivalent nonparametric descriptions under sufficient overparameterization. Classical methods and their native spaces, such as kernel methods / reproducing kernel Hilbert spaces, wavelets / Besov spaces, and shallow neural networks / variation spaces emerge as special cases of our abstract framework. A byproduct of "axiomatizing" the study of representation costs is that we also immediately obtain new results for deep neural networks: For depth-$L$ feedforward ReLU networks, their induced native spaces are $p$-normable quasi-Banach spaces with $p = 2/L$. This reveals that the inductive bias of deep neural networks (as given by the representation cost) cannot be captured by norms for depths $L > 2$.

This paper studies local controllability of nonlinear control-affine systems subject to state-dependent box constraints that strictly exclude the zero input. Such constraints arise naturally in cable-driven robots and other systems with strictly positive actuation, but fall outside classical small-time local controllability theory and existing frameworks for positive or cone-constrained controls. We introduce the admissible balancing set, an input-space object that classifies reference states without requiring the control distribution to have full rank. When an admissible balancing input lies in the interior of the input set, a locally uniform input shift recovers a symmetric-control system, allowing classical accessibility and small-time local controllability criteria to be applied. When no admissible balancing input exists, the feasible-velocity set is strictly separated from the origin. We show that the resulting separating covector defines a local barrier functional that increases at a uniform positive rate along every admissible trajectory, thereby providing a quantitative obstruction to small-time local controllability. This obstruction does not exclude finite-time reachability through trajectories leaving the barrier neighborhood, which motivates the notion of admissible excursions. The framework is illustrated on an underactuated planar cable-driven parallel robot, for which the barrier is certified numerically over a prescribed state neighborhood.

We show that every optimal transport plan for the $1$-Wasserstein distance in $\mathbb R^n$ is supported on a closed $n+1$ rectifiable set. In particular, the support of any optimal plan has Hausdorff dimension at most $n+1$.

We prove Gromov's Euclidean endpoint $C^0$ rigidity conjecture. Let $g$ be a smooth complete metric on $\R^3$ with non-negative scalar curvature. If $$ |g-g_{\Euc}|=o(r^{-1}),\qquad r=|x|\to\infty, $$ then $(\R^3,g)$ is isometric to Euclidean space.

We study families of smooth, embedded, regular planar curves $ \alpha : \left [-1,1 \right ]\times \left [0,T \right )\to \mathbb{R}^{2}$ with generalised Neumann boundary conditions inside cones, satisfying three variants of the fourth-order nonlinear curve diffusion flow: (1) curve diffusion flow with a length penalisation, (2, 3) two forms of constrained curve diffusion flow with fixed length. We prove in cases (2) and (3) for cone angle less than $\pi$, if the initial curve has small oscillation of curvature and the initial curve is sufficiently far from the cone tip, then the solution exists for all time and converges exponentially in the $C^\infty$-topology to a circular arc with the same length as the initial curve. In case (1), a similar result holds under suitable rescaling. In all cases, the limiting arc is centred at the cone tip.

The perfectly matched layer-based boundary integral equation (PML-BIE) method (Lu et al., \emph{SIAM J. Appl. Math.} 78 (2018)) has become an effective tool for wave scattering problems in unbounded domains. Despite its successful applications, a rigorous convergence theory has remained incomplete in many physically relevant settings. In this paper, we present a general framework for establishing convergence of PML-BIE methods, using acoustic scattering in an impedance half-space as the first illustrative example. The framework separates the analysis into three components: convergence of the PML truncated partial differential equation, equivalence between the PML problem and an exact PML-BIE, and convergence of a computable PML-BIE obtained by replacing the exact kernel with a stretched free-space kernel. For the impedance half-space problem, we prove convergence of both the exact and computable PML-BIE formulations. The final error bound decays exponentially as the PML absorption power increases. The resulting theory provides a rigorous foundation for the PML-BIE method and gives a reference model for more general scattering problems in layered and inhomogeneous media.

We introduce the notion of local pseudofunction spectrum $\sigma_{PF}(A)$ for the infinitesimal generator $A$ of a bounded $C_0$-semigroup $\mathcal{T} = (T(t))_{t \geq 0}$ on a Banach space $X$ and show it is the right spectral concept to deliver a full characterization of the strong stability of $\mathcal{T}$: $$ \forall x \in X : ~ \lim_{t \to \infty} \| T(t) x \|_X = 0 \quad \Longleftrightarrow \quad \sigma_{PF}(A) = \varnothing. $$ We demonstrate how this yields a quick proof of the well-known Arendt-Batty-Lyubich-Vũ theorem and establish novel stability results through local range density conditions for semigroups whose local pseudofunction spectra are a null subset of the imaginary axis. We also obtain similar stability characterization theorems for individual orbits and for semi-uniform stability. As an application of our results, we provide spectral characterizations of almost periodic $C_0$-semigroups with countable spectrum. In addition, we prove optimal Tauberian theorems of Katznelson-Tzafriri type and discuss connections with Wiener kernels.

In this paper, we introduce a new idea to study modules of logarithmic differential forms of hyperplane arrangements, which we call residue ideals. We first establish basic properties of these ideals, including their radicals and primary decompositions, and obtain applications for freeness of restrictions of arrangements. Then we apply these ideals to the study of modules of logarithmic differential $1$-forms for graphic arrangements. We give an explicit generating set for these modules and find a new connection to cover ideals of graphs studied in combinatorial commutative algebra. As a consequence we establish several new connections between arrangement theory and Stanley--Reisner theory.

We study balanced subfamilies of the middle layer $\binom{[2n]}{n}$ of the Boolean lattice $2^{[2n]}$. A family $\mathcal{F}\subseteq\binom{[2n]}{n}$ is said to be balanced if every element in $[2n]$ appears in the same number of members of $\mathcal{F}$. A balanced subfamily of size 2 is exactly a complementary pair $\{A,[2n]\setminus A\}$, and therefore a family with no balanced subfamily of size $2$ has at most $\frac{1}{2}\binom{2n}{n}$ members. We show that for every $k\geq 1$ and all sufficiently large $n$, this maximum size is compatible with delaying the smallest size of a balanced subfamily until $2k+2$. More precisely, there exists a family $\mathcal{F}\subseteq\binom{[2n]}{n}$ of size $\frac{1}{2}\binom{2n}{n}$ with no balanced subfamilies of sizes $2,4,\ldots,2k$, but with a balanced subfamily of size $2k+2$. The proof is constructive and is obtained by lifting Taylor-Zwicker trade-robust magic-square games to self-dual selectors in the middle layer. This proves a recent conjecture of Moss and Pedersen.

For systems of non-relativistic charged particles minimally coupled to the soft modes of the quantized radiation field, we prove self-adjointness on the domain of the free Hamiltonian. This result is not new, but the proof we give shortens previous arguments. It is based on a comparison of graph norms and Nelson's commutator theorem.

Model risk in credit portfolio models is a serious issue for banks but has so far not been tackled comprehensively. We will demonstrate how to deal with uncertainty in all model parameters in an all-embracing, yet easy-to-implement way.

We extend the study of force-free magnetospheres from non-rotating to rotating black holes \cite{Sheikhahmadi} and investigate the influence of a force-free magnetic field on the geometry around a Kerr black hole. Using the Newman-Penrose formalism, we explicitly construct the electromagnetic field strength tensor in the Kerr background and compute the corresponding stress-energy tensor. The resulting metric perturbation is then obtained by solving the linearised Einstein equations. In this modified geometry, we analyse key observables of thin accretion disks, including the innermost stable circular orbit (ISCO), effective potential, energy flux, temperature, and efficiency parameter. Our results demonstrate that magnetic backreaction significantly alters the spacetime near the black hole, with important consequences for accretion physics and jet-launching mechanisms such as the Blandford-Znajek process. This work underlines the essential role of magnetic fields in shaping relativistic astrophysical environments.

We propose two graph neural network layers for graphs with features in a Riemannian manifold. First, based on a manifold-valued graph diffusion equation, we construct a diffusion layer that can be applied to an arbitrary number of nodes and graph connectivity patterns. Second, we model a tangent multilayer perceptron by transferring ideas from the vector neuron framework to our general setting. Both layers are equivariant under node permutations and the feature manifold's isometries. These properties have led to a beneficial inductive bias in many deep-learning tasks. Furthermore, they enable novel, more flexible feature designs. Numerical examples on synthetic data and an Alzheimer's classification application on triangle meshes of the right hippocampus demonstrate the usefulness of our new layers: While they apply to a much broader class of problems, they outperform task-specific state-of-the-art networks.

Categorical variables are of uttermost importance in biomedical research. When two of them are considered, it is often the case that one wants to test whether or not they are statistically dependent. We show weaknesses of classical methods -- such as Pearson's and the G-test -- and we propose testing strategies based on distances that lack those drawbacks. We first develop this theory for classical two-dimensional contingency tables, within the context of distance covariance, an association measure that characterizes general statistical independence of two variables. We then apply the same fundamental ideas to one-dimensional tables, namely to the testing for goodness of fit to a discrete distribution, for which we resort to an analogous statistic called energy distance. We prove that our methodology has desirable theoretical properties, and we show that we can calibrate the null distribution of our test statistics without resampling. We illustrate all this in simulations, as well as with some real data examples, demonstrating the adequate performance of our approach for biostatistical practice.

In this paper, we study the problem of sampling from log-concave distributions supported on convex and compact sets, with a particular focus on the randomized midpoint discretization of both overdamped and kinetic Langevin diffusions in constrained domains. We revisit the proximal framework for handling constraints through projection operators and develop a more general formulation that encompasses Euclidean, Bregman, and Gauge projections. The resulting smooth approximation allows a unified and tractable analysis of Langevin algorithms and their variants under constraints. Within this framework, we establish convergence guarantees in Wasserstein-$q$ $(q\geqslant 1)$ distances between the smooth surrogate and the target distribution. We further derive complementary lower bounds, showing that the results are near-optimal in order. Building upon this tight approximation analysis, we obtain new convergence guarantees for the randomized midpoint Langevin algorithms and refined bounds for both vanilla and kinetic Langevin Monte Carlo methods under constraints, thereby advancing the theoretical understanding of constrained diffusion-based sampling.

In airport operations, optimally using dedicated personnel for baggage handling tasks plays a crucial role in the design of resource-efficient processes. Teams of workers with different qualifications must be formed, and loading or unloading tasks must be assigned to them. Each task has a time window within which it can be started and should be finished. Violating these temporal restrictions incurs severe financial penalties for the operator. In practice, various components of this process are subject to uncertainties. We consider the aforementioned problem under the assumption of time-dependent stochastic travel times across the apron. We present two binary program formulations to model the problem at hand and propose a novel solution approach that we call Branch-Price-Cut-and-Switch, in which we dynamically switch between two master problem formulations. Furthermore, we use an exact separation method to identify violated rank-1 Chvátal-Gomory cuts and utilize an efficient branching rule relying on task finish times. We test the algorithm on instances generated based on real-world data from a major European hub airport with a planning horizon of up to two hours, 30 flights per hour, and three available task execution modes to choose from. Our results indicate that our algorithm is able to significantly outperform existing solution approaches. Moreover, an explicit consideration of stochastic travel times allows for solutions that utilize the available workforce more efficiently, while simultaneously guaranteeing a stable service level for the baggage handling operator.

We develop Riemannian approaches to variational autoencoders (VAEs) for PDE-type ambient data with regularizing geometric latent dynamics, which we refer to as VAE-DLM, or VAEs with dynamical latent manifolds. We redevelop the VAE framework such that manifold geometries, subject to our geometric flow, embedded in Euclidean space are learned in the intermediary latent space developed by encoders and decoders. By tailoring the geometric flow in which the latent space evolves, we induce latent geometric properties of our choosing, which are reflected in empirical performance. We reformulate the traditional evidence lower bound (ELBO) loss with a considerate choice of prior. We develop a linear geometric flow with a steady-state regularizing term. This flow requires only automatic differentiation of one time derivative, and can be solved in moderately high dimensions in a physics-informed approach, allowing more expressive latent representations. We discuss how this flow can be formulated as a gradient flow, and maintains entropy away from metric singularity. This, along with an eigenvalue penalization condition, helps ensure the manifold is sufficiently large in measure, nondegenerate, and a canonical geometry, which contribute to a robust representation. Our methods focus on the modified multi-layer perceptron architecture with tanh activations for the manifold encoder-decoder. We demonstrate, on our datasets of interest, our methods perform at least as well as the traditional VAE, and oftentimes better. Our methods can outperform this and a VAE endowed with our proposed architecture, frequently reducing out-of-distribution (OOD) error between 15% to 35% on select datasets. We highlight our method on ambient PDEs whose solutions maintain minimal variation in late times. We provide empirical justification towards how we can improve robust learning for external dynamics with VAEs.

Temporal graphs are graphs whose edges are labelled with times at which they are active. Their time-sensitivity provides a useful model of real networks, but renders many problems studied on temporal graphs more computationally complex than their static counterparts. To contend with this, there has been recent work devising parameters for which temporal problems become tractable. One such parameter is vertex-interval-membership (VIM) width. Broadly, this gives a bound on the number of vertices we need to keep track of at any given time to solve many problems. Our contributions are two-fold. Firstly, we introduce a new parameter, tree-interval-membership (TIM) width, that generalises both VIM width and several existing generalisations. Secondly, we provide meta-algorithms for both VIM and TIM width which can be used to prove fixed-parameter-tractability for large families of problems, bypassing the need to give involved dynamic programming arguments for every problem. In doing this, we provide a characterisation of problems in FPT with respect to both parameters. We apply these algorithms to temporal versions of Hamiltonian path, dominating set, matching, and edge deletion to limit maximum reachability.

The rapid deployment of AI systems in high-stakes domains, including those classified as high-risk under the The EU AI Act (Regulation (EU) 2024/1689), has intensified the need for reliable compliance auditing. For binary classifiers, regulatory risk assessment often relies on global fairness metrics such as the Disparate Impact ratio, widely used to evaluate potential discrimination. In typical auditing settings, the auditee provides a subset of its dataset to an auditor, while a supervisory authority may verify whether this subset is representative of the full underlying distribution. In this work, we investigate to what extent a malicious auditee can construct a fairness-compliant yet representative-looking sample from a non-compliant original distribution, thereby creating an illusion of fairness. We formalize this problem as a constrained distributional projection task and introduce mathematically grounded manipulation strategies based on entropic and optimal transport projections. These constructions characterize the minimal distributional shift required to satisfy fairness constraints. To counter such attacks, we formalize representativeness through distributional distance based statistical tests and systematically evaluate their ability to detect manipulated samples. Our analysis highlights the conditions under which fairness manipulation can remain statistically undetected and provides practical guidelines for strengthening supervisory verification. We validate our theoretical findings through experiments on standard tabular datasets for bias detection. Code is publicly available at this https URL.

We are surrounded by spatio-temporal patterns resulting from the interaction of the numerous basic units constituting natural or human-made systems. In presence of diffusive-like coupling, Turing theory has been largely applied to explain the formation of such self-organized motifs both on continuous domains or networked systems, where reactions occur in the nodes and the available links are used for species to diffuse. In many relevant applications, those links are not static, as very often assumed, but evolve in time and more importantly they adapt their weights to the states of the nodes. In this work, we make one step forward and we provide a general theory to prove the validity of Turing idea in the case of adaptive symmetric networks with positive weights. The conditions for the emergence of Turing instability rely on the spectral property of the Laplace matrix and the model parameters, thus strengthening the interplay between dynamics and network topology. A rich variety of patterns are presented by using two prototype models of nonlinear dynamical systems, the Brusselator and the FitzHugh-Nagumo model. Because many empirical networks adapt to changes in the system states, our results pave the way for a thorough understanding of self-organization in real-world systems.

Survival analysis provides statistical methods to model the time until an event occurs. Reporting delays arise when event times are not observed at their occurrence but are only revealed upon reporting. This issue is particularly critical for timely risk evaluation when the observation window is short due to administrative censoring. In this study, we incorporate right-censored reporting delays by jointly modeling parametric hazards for the event and reporting processes. We then construct a consistent estimator for the model parameters and develop a Monte Carlo expectation-maximization algorithm to compute it. To address the challenges posed by administrative censoring, we leverage these findings and propose a transfer-learning procedure. Experimental results demonstrate that our method improves the accuracy of timely risk evaluation under administrative censoring.

We investigate the fundamental problem of leveraging offline data to accelerate online reinforcement learning - a direction with strong potential but limited theoretical grounding. Our study centers on how to \emph{learn} and \emph{apply} value envelopes within this context. To this end, we introduce a principled two-stage framework: the first stage uses offline data to derive upper and lower bounds on value functions, while the second incorporates these learned bounds into online algorithms. Our method extends prior work by decoupling the upper and lower bounds, enabling more flexible and tighter approximations. In contrast to approaches that rely on fixed shaping functions, our envelopes are data-driven and explicitly modeled as random variables, with a filtration argument ensuring independence across phases. The analysis establishes high-probability regret bounds determined by two interpretable quantities, thereby providing a formal bridge between offline pre-training and online fine-tuning. Empirical results on tabular MDPs demonstrate substantial regret reductions compared with both UCBVI and prior methods while remaining competitive with related approaches.

We prove the dynamical formation of a marginally outer trapped surface in pure vacuum spacetime from smooth asymptotically flat Cauchy data which initially contain no MOTS. The mechanism is a boundary effect rather than a collapse mechanism. We work in a Cauchy--double-null framework and use Yau's boundary criterion \cite{yau}, which gives the existence of an interior MOTS from a lower bound for the generalized boundary mean curvature relative to the Schoen--Yau radius of the domain. We construct an explicit class of vacuum initial data for which this criterion is strictly subcritical on the initial hypersurface, while the Einstein evolution drives the same domain into the supercritical regime. More precisely, a mild incoming gravitational radiation field increases the generalized boundary mean curvature of an isotropically large interior region sufficiently to force the formation of a MOTS in its future development. A characteristic feature of the initial data is a large interior anisotropic curvature component: the trace-free Ricci curvature is of larger order than the scalar curvature, which is balanced at the vacuum constraint scale. Thus the MOTS forms not from matter concentration or standard gravitational collapse, but from the interaction between boundary geometry, large-scale interior geometry, and the vacuum Einstein dynamics. This gives a rigorous realization of a long-suspected physical idea that apparent horizons may form from global geometric effects in vacuum general relativity.

In this work, we investigate the existence and effect of percolation in training deep Neural Networks (NNs) with dropout. Dropout methods are regularisation techniques for training NNs, first introduced by G. Hinton et al. (2012). These methods temporarily remove connections in the NN, randomly at each stage of training, and update the remaining subnetwork with Stochastic Gradient Descent (SGD). The process of removing connections from a network at random is similar to percolation, a paradigm model of statistical physics. If dropout were to remove enough connections such that there is no path between the input and output of the NN, then the NN could not make predictions informed by the data. We study new percolation models that mimic dropout in NNs and characterise the relationship between network topology and this path problem. The theory shows the existence of a percolative effect in dropout. We also show that this percolative effect can cause a breakdown when training NNs without biases with dropout; and we argue heuristically that this breakdown extends to NNs with biases.

Statistical functions such as the moment-generating, characteristic, cumulant-generating, and second characteristic functions are standard tools in classical statistics and probability theory. They provide a systematic means to analyze the statistical properties of a system and find applications in diverse fields. While these functions are ubiquitous in classical theory, a quantum counterpart has remained underdeveloped because of the noncommutativity of operators. The absence of such a framework has obscured the connections between statistical quantities and the nonclassical features of quantum mechanics. Here, we construct a framework for quantum statistical functions that addresses these limitations and unifies the languages of quantum statistics. We show that the functions reproduce standard statistical quantities such as expectation values, variance, and covariance upon differentiation. By extending the framework to include pre- and post-selection, we define conditional functions that generate conditional statistical quantities, including the weak value and the weak variance. We further show that multivariable functions, defined with specific operator orderings, correspond to the Kirkwood--Dirac, Margenau--Hill, and Wigner distributions. By generalizing Bochner's theorem within the theory of compactly supported distributions, we obtain a criterion that separates classical statistics from quantum statistics, linking the failure of positive definiteness of the multivariable function to the emergence of quasiprobability. As an application, we import the classical method of moments and generalized method of moments into quantum estimation, introducing quantum estimators that exploit the proposed functions. Our framework reproduces quantum statistical quantities and incorporates the nonclassical features of quasiprobability, providing a basis for further study of quantum statistics.

Thompson sampling (TS) is widely used for stochastic multi-armed bandits, yet its inferential properties under adaptive data collection are subtle. Classical asymptotic theory for sample means can fail because arm-specific sample sizes are random and coupled with the rewards through the action-selection rule. We study adaptive inference for Thompson sampling with Gaussian randomized indices in $K$-armed stochastic bandits with independent sub-Gaussian reward noises, and identify \emph{optimism} as a key mechanism for restoring \emph{stability}, meaning that each arm's pull count concentrates around a deterministic scale. This stability yields asymptotically valid Wald inference despite adaptive sampling. First, we prove that variance-inflated TS is stable for any $K \ge 2$, including the challenging regime where multiple arms are optimal, with asymptotically uniform allocation over optimal arms and sharp logarithmic pull-count asymptotics for suboptimal arms. This resolves the $K$-armed extension question raised by \citet{halder2025stable}, using new winner-map and Lyapunov-drift techniques to control allocation among multiple optimal arms. Second, we analyze an alternative optimistic modification that keeps the Gaussian index variance unchanged but adds an explicit mean bonus to the index center, and establish a similar stability conclusion. In summary, suitably implemented optimism stabilizes Thompson sampling and enables asymptotically valid Wald inference in multi-armed bandits, while incurring only a mild additional regret cost.

Synchronization of coupled oscillators is observed in many natural and engineered systems and emerges due to the interactions within the system. It can be both beneficial, e.g., in power grids, and harmful, e.g., in epileptic seizures. In the latter case, efficient control methods to desynchronize the systems are crucial. Recent studies have shown that interactions are not always pairwise, but higher-order, i.e., many-body, and this greatly affects the dynamics. For instance, higher-order interactions increase the linear stability of synchronized states but simultaneously shrink their attraction basin, with potentially opposite effects on control methods. Here, we use a minimally invasive pairwise control based on Hamiltonian control theory, and investigate its efficiency on phase oscillators with higher-order interactions. We show that, if the initial phases are close to the synchronized state, higher-order interactions make desynchronization more difficult to achieve. Otherwise, a non-monotonic effect appears: intermediate strengths of higher-order interactions impede desynchronization while larger ones facilitate it. In all cases, the control can desynchronize the system with a sufficient number of controlled nodes and intensity.

Data collection is a critical component of modern statistical and machine learning pipelines, particularly when data must be gathered from multiple heterogeneous sources to study a target population of interest. In many use cases, such as medical studies or political polling, different sources incur different sampling costs. Observations often have associated group identities - for example, health markers, demographics, or political affiliations - and the relative composition of these groups may differ substantially, both among the source populations and between sources and target population. In this work, we study multi-source data collection under a fixed budget, focusing on the estimation of population means and group-conditional means. We show that naive data collection strategies (e.g. attempting to "match" the target distribution) or relying on standard estimators (e.g. sample mean) can be highly suboptimal. Instead, we develop a sampling plan which maximizes the effective sample size - the total sample size divided by $D_{\chi^2}(q\mid\mid\overline{p}) + 1$, where $q$ is the target distribution, $\overline{p}$ is the aggregated source distribution, and $D_{\chi^2}$ is the $\chi^2$-divergence. We pair this sampling plan with a classical post-stratification estimator and upper bound its risk. We provide matching lower bounds, establishing that our approach achieves the budgeted minimax optimal risk. Our techniques also extend to prediction problems when minimizing the excess risk, providing a principled approach to multi-source learning with costly and heterogeneous data sources.

Smooth words over an alphabet of non-negative integers $\{a,b\}$ are infinite words that are infinitely derivable, the emblematic example being the Oldenburger-Kolakoski word over $\{1,2\}$. The main way to study their language is to consider a finite version of smooth words that we call f-smooth words. In this paper we prove that the f-smooth words are exactly the factors of smooth words, and we make progress towards the conjecture of Sing that the complexity of f-smooth words over $\{a,b\}$ grows like $\Theta\left(n^{\log(a+b)/\log((a+b)/2)}\right)$: we prove it over even alphabets, we prove the lower bound over any binary alphabet and we improve the known upper bound over odd alphabets.

Chimera states are one of the most intriguing phenomena in nonlinear dynamics, characterized by the coexistence of coherent and incoherent behavior in systems of coupled identical oscillators. Despite extensive studies and numerous observations in different settings, the development of reliable and systematic methods to classify chimera states and distinguish them from other dynamical patterns remains a challenging task. Existing approaches are often limited in scope and lack robustness. In this work, we propose a method based on Fourier analysis combined with statistical classification to identify chimera behavior. The method is applied to a system of topological signals coupled via the Dirac operator, where it successfully captures the rich dynamical regimes exhibited by the model. We demonstrate that the proposed approach is robust with respect to variations in network topology and system parameters. Beyond the specific model considered, the framework provides a general and automated tool for distinguishing different dynamical regimes in complex systems.

Using loop equations, we derive an exact solution for the statistical distribution of freely decaying incompressible turbulence in arbitrary spatial dimension $d>1$. By applying the Mandelstam identity to the loop dynamics, we prove that the nonlinear advection term reduces to a pure derivative and drops out of the momentum-loop equation. As a result, the momentum-loop equation becomes purely diffusive, admitting an exact geometric solution as a random walk on a circle. Despite this distinct local loop algebra, the dimension-independent Euler ensemble dictates macroscopic observables via the Mellin transform. This Mellin transform $M(p)$ for the energy scaling function $H(k\sqrt{\nu t})$ emerges as completely universal, independent of $d$. The applications for $d=3$ were studied previously; here we extend the theory to $d=2$. Our analytical solution extends the empirically observed $k^{-3.5}$ spectrum to a continuous effective index, providing an exact analytic alternative to classical Kraichnan--Batchelor--Leith phenomenology. We prove that previously reported ``multifractal'' transient exponents are merely local tangents of a single universal scaling function. We find an infinite cascade of finite-time transitions (a Stokes staircase associated with complex zeros $z = 1/2 + i\rho_n$ of the Riemann zeta function), mimicking finite-time discontinuities with Berry smoothing by the error function. Thus, there are no true finite-time singularities; instead, as a consequence of the Riemann hypothesis, an essential singularity emerges at infinite time, manifesting as rapid transitions at $t_n \propto \rho_n^3$, sharpening as $1/\log t_n$. We compare the predicted energy spectrum with recent 3D DNS in two independent ways, each yielding a close match within statistical errors.

Kunchenko's method of polynomial maximization provides a semiparametric apparatus for parameter estimation under non-Gaussian errors, but its classical power basis relies on finite higher-order integer moments. This paper introduces the Parametrically Adaptive Transition Polynomial (PATP), a signed-parity fractional-power family controlled by a continuous parameter alpha in [0,1]. The quadratic exponent map p_i(alpha) connects the fractal regime p_i(0)=1/i, the degenerate linear point p_i(1/2)=1, and the signed-parity integer-power regime p_i(1)=i. For the degree-S=2 case we derive a closed-form variance-reduction coefficient g_2(alpha) in terms of signed and absolute fractional moments, identify the singular behavior at alpha=1/2, and state the moment and regularity conditions under which the formula is meaningful. The construction should be read as a Form-B PATP analogue within Kunchenko's generalized apparatus, not as an exact recovery of the canonical even-power PMM basis at alpha=1. Numerical illustrations on canonical distributions are used to examine the finite-sample behavior of the signed-parity estimator and to mark the boundary of applicability for extremely heavy-tailed cases such as Cauchy.

We propose a betweenness centrality measure and algorithms for stochastic networks, where edges can fail and weights vary across realizations, making the most central node random. Our approach models the sequence of reported central nodes as an absorbing Markov chain and measures node importance by the share of pre-absorption time spent at each node. This produces a way to study centrality under uncertainty, which can then be estimated with Monte Carlo simulation. We also analyze robustness when the transition kernel is only approximately known, using row-wise perturbations to assess sensitivity and potential ranking changes. The framework further admits extensions to weighted rewards and restricted candidate sets without altering the Markov chain formulation. Experiments on Erdős-Rényi, Watts-Strogatz, and Les Misérables networks with stochastic edges show that the method identifies a small set of dominant nodes, reveals stable versus sensitive rankings under perturbations, and supports reward-based and structure-constrained variants.

We study parameter estimation for univariate stochastic differential equations with locally Lipschitz drift and Hölder continuous multiplicative diffusion, a class commonly arising in several applications. Existing inference methods typically rely on either the Euler-Maruyama discretisation, despite its lack of strong convergence and failure to preserve the state space, or on approximations, e.g. Gaussian approximation or truncation of Hermite's expansions, impacting on their stability and computational efficiency. We introduce the first explicit pseudo-likelihood estimators based on numerical splitting schemes that are both strong mean-square convergent and state space preserving for this class of SDEs. Our approach is based on a novel decomposition of the SDE that exploits reducibility and the Lamperti transform, leading to Lie-Trotter (LT) and Strang splitting schemes yielding explicit pseudo-likelihoods and maximum likelihood estimators based on them. We prove strong mean-square convergence, state space preservation, and improved robustness with respect to the discretisation step compared to Euler-Maruyama-based methods. We further establish consistency and asymptotic normality of the LT estimator. Because the proposed numerical scheme couples drift and diffusion parameters in the pseudo-likelihood, the asymptotic analysis requires new proof techniques. Extensive simulations demonstrate that the proposed estimators outperform existing methods in both accuracy and computational efficiency.

Recent work in random matrix theory (RMT) has developed the notion of deterministic equivalents: typically linear surrogate models that approximate the spectral behavior of large nonlinear random matrices, such as nonlinear feature maps in neural networks (NNs). Such equivalents make theoretical predictions tractable by reducing a complex model to a simpler one with properties that fall under the umbrella of classical RMT tools. However, this leaves open the question of whether this idealized linear equivalence remains meaningful for classification of high-dimensional nonlinearly separable data. Motivated by this, we consider the conjugate kernel (CK), which is the nonlinear feature map of a one-layer feedforward NN, under a canonical nonlinearly separable dataset for the XOR problem; and we use the study of informative outlier eigenvalues in the CK and whether their corresponding eigenvectors asymptotically align with XOR labels as a proxy for nonlinear learnability. We develop a robust quadratic equivalent of the CK matrix that enables a precise analysis of emergent informative spikes, as one modifies various knobs common in ML practice: sample complexity, signal-to-noise ratio (SNR), nonlinear activation choice, and pretrained features. We identify regimes in which these knobs move the CK beyond the linear equivalent and produce BBP-type transitions to label-aligned outlier eigenspaces. Our analysis helps bring deterministic-equivalence tools from RMT to bear on problems of practical relevance in ML.

For fans of Gabriel's "Worse is Better" it may be ironic that C++, by way of MLIR, serves as the scaffold for compiling an ML-family language whose correctness properties are structural. A crucial intersection in our Composer compiler initiates its lowering with a fixed-point combinator that preserves the dimensional, grade, escape, and numeric-representation structure from the Program Semantic Graph. And the MLIR that's witnessed from the PSG is no passive host. Its use of static single assignment, attribute system and dialects carry that structure materially. We show that our compiler middle end uses categorical construction for lowering code with companion verification to that strata: a functor from the compilation poset to a target category, subject to the compositionality equation. The grounding of our approach comes from three sources, each on its own algebraic object: Ohori's machine-code proof theory grounds the compilation axis, parametricity grounds the content at the base, and adjoint mode logic grounds the traversal between our verification tiers. To extend the thesis we introduce compact-closed negative and fractional types, and show the type machinery can be carried with preserved structure and realized through tooling MLIR provides. More broadly, the same fixed-point primitive that preserves types through compilation also supplies proof terms that can continue to be exercised in MLIR to verify its integrity as lowering proceeds through the pipeline. We argue that this foundation is a unique additional point anticipated by our framework that includes dimensional types, Tarau's groupoid, and cellular sheaves. Throughout, the formalism is instrumented as an internal scaffold: the abstractions support the compiler's mechanics, where a developer is never required to reach for category theory in order to rely on the guarantees the compiler provides.

Formal rigor distinguishes mathematics from other disciplines, in the sense that mathematical statements are derived from explicit axioms by logically verifiable steps. Interactive theorem provers support this by expressing definitions, theorems, and proofs in a fully formal language and verifying them mechanically. We consider the benchmark problem of formalizing all published mathematics as a machine verifiable and continuously updated corpus of mathematical knowledge. This viewpoint treats mathematics as a structured database of interdependent results and raises questions about scalability and organization of large formal libraries. As a case study, we present an ongoing formalization in categorical algebra, namely dilatations of categories, extending classical localizations and illustrating what such an implementation looks like in practice.

We propose the data augmented bootstrap (DAB), a framework for constructing confidence intervals from approximately invariant transformations of the data. As special cases, DAB recovers popular methods that rely on exact group symmetries, such as conformal prediction, wild bootstrap for Maximum Mean Discrepancy U-statistics and the recently proposed SymmPI. Meanwhile, DAB also recovers the classical bootstrap method, which exploits the dataset's approximate invariance under uniform sampling of data indices as the dataset size grows. For all DAB methods, we establish theoretical coverage results that interpolate between finite-sample and asymptotic guarantees according to the strength of the invariance, and without assuming a group structure. The approximate invariance is measured in the Kolmogorov distance and, for statistics that satisfy Gaussian universality, reduces to conditional mean and variance matching. This allows us to incorporate data augmentation (DA), a widely used machine learning heuristic based on approximate invariances, into known statistical methods. We empirically test the performance of incorporating DA into bootstrap, wild bootstrap and conformal prediction for simulated settings as well as for image, language and scientific data.

Cancer treatment is at the core a sequential decision-making problem with partial observability, latent patient heterogeneity, and explicit constraints on the budget for medical measurements. Unlike standard Reinforcement Learning (RL) approaches that control state trajectories, cancer treatments permanently modify patients' transition dynamics, changing how states evolve over time. We model cancer treatment as a belief-space planning problem using active inference, deriving an expected free-energy objective that unifies goal-directed control and information acquisition under measurement budgets without. We implement this framework using real clinical cancer data from the AACR Project GENIE Biopharma Collaborative dataset. Results on clinical data demonstrate a simultaneous patient categorization and high treatment efficacy, under real measurement and treatment constraints.

This note is a tutorial on the deterministic version of the Kalman filter (state estimator), which is formulated as finding the state trajectory consistent with the system's equations with the minimal amount of $L^2$ process and measurement uncertainty. As stated, this is an input signal design problem with linear dynamics and an objective that is affine-quadratic in the state and inputs. The first step is to convert this problem to one with a purely quadratic objective by embedding in a larger system using ``homogeneous coordinates''. This converts the problem to a purely quadratic (i.e. an LQR) problem, but with non-standard initial or final state constraints. This latter problem can then be solved using a version of the matrix Differential Riccati Equation (DRE) for the larger LQR problem. The second step is a partitioning of this larger problem, which then yields the optimal dynamic observer and the DRE of the traditional Kalman filter. For comparison, the solution of the traditional LQ-tracking (Servomechanism) problem is also treated using a similar construction.

Periodically driven dissipative systems can settle into steady orbits - fixed loops on their dynamical manifolds. In quantum mechanics, steady orbits occur in cooling engines (used to initialise quantum devices), coherent oscillators (such as lasers and masers), precision metrology devices (atomic clocks, optical and spin magnetometers), and magnetic resonance (steady state free precession, dynamic nuclear polarisation). Steady orbits and stroboscopic steady states are a promising target for quantum optimal control, but the numerical complexity is prohibitive: the infinite loop defeats gradient ascent pulse engineering (GRAPE) which relies on explicit numerical propagation in the time domain. Here we propose an efficient quantum control strategy for stroboscopic steady states and limit cycles that are approached asymptotically when a control sequence is repeated infinitely many times. The formalism is different from Floquet-Lindblad state engineering and effective Hamiltonian theories: it finds control sequences that drive a dissipative quantum system towards a steady orbit passing through user-specified waypoints. The software implementation (same numerical complexity scaling as GRAPE) is done for the Spinach library.

Predicting the initiation and propagation of cracks in heterogeneous wellbore systems under complex in-situ conditions remains challenging. We present a hybrid phase-field fracture framework to model crack growth in heterogeneous wellbore systems with weak interfaces. The framework is first validated against benchmark problems with available analytical and numerical solutions. Subsequently, numerical experiments are conducted to isolate the effects of interface strength and casing eccentricity on crack growth. The results show that casing eccentricity strongly influences both the pressure at crack initiation and the resulting crack paths, reducing the crack initiation pressure by up to 30% relative to the concentric configuration. Beyond a critical eccentricity threshold of 50%, localized shear stresses drive the nucleation of inclined cracks in the formation in addition to radial cracking -- a failure mode absent in concentric configurations. For sufficiently weak interfaces (i.e., interfaces with 30% of the strength of the surrounding bulk material), radially propagating cracks in the cement sheath are deflected along the interface rather than penetrating into the formation. This deflection delays stress relaxation within the sheath, promotes the nucleation of additional radial cracks, and increases the risk of sustained casing pressure and wellbore failure. Finally, a three-dimensional simulation reveals depth-dependent crack nucleation, stress-shadow effects that suppress full-depth crack growth and crack coalescence along the cement-formation interface -- phenomena that are fundamentally inaccessible under plane-strain assumptions - demonstrating the applicability of the framework to realistic heterogeneous wellbore systems.