The paper deals with dead-core solutions to an isothermal reaction-diffusion problem with power-law kinetics for a single reaction that takes place in a chemical reactor represented by a bounded domain in two dimensions. The model boundary value problem for the stationary non-linear diffusion-reaction equation is solved numerically using an appropriate time-marching method. The spatial discretization is based on the lumped finite element method for piecewise linear functions. The effects of the reaction order and Thiele modulus on the concentration profiles and the size of dead zones are investigated numerically. The paper demonstrates that the formation of multiple dead zones is possible for particular reactor geometries.

The paper is a synthesis of several works on the variational principles for application to the mechanics and the physics, inspired from original ideas of Brezis, Ekeland and Nayroles. On this basis, we developed an unified framework for dynamic dissipative systems that leads to a space-time variational principle of minimum constructed with tools of convex analysis and symplectic geometry. We stress the essential ideas and concepts. They are illustrated with various theoretical and numerical examples.

This work investigates a nonlinear two-point boundary value problem arising in diffusion-reaction processes in catalyst slabs with power-law kinetics and fractional reaction order. The model exhibits a free-boundary structure, where an unknown interface separates a dead-core region with vanishing concentration from an active region with positive concentration. We propose a Physics-Informed Neural Network (PINN) framework that incorporates a structured, hard-constrained trial solution embedding the asymptotic behavior near the interface. The dead-core location is treated as a trainable parameter, enabling the simultaneous approximation of the concentration profile and identification of the free boundary without explicit interface tracking. The method is validated against analytical solutions and high-precision numerical shooting. Numerical experiments demonstrate that the approach accurately captures both the solution profile and the free-boundary location while maintaining a computationally manageable training cost.

In this paper, we study the Bohr phenomenon for a generalized subclass of harmonic mappings defined by a second-order differential inequality in the unit disk. Specifically, we consider the class $\mathcal{BH}_0(\gamma, \delta)$, which extends several known subclasses of harmonic and analytic functions. By employing sharp coefficient estimates and growth results, we establish improved versions of Bohr-type inequalities, including refined Bohr radii and Bohr--Rogosinski radii for this class. Furthermore, we derive generalized inequalities involving higher-order coefficient sums and area terms, thereby extending classical Bohr inequalities in a harmonic setting. The sharpness of the obtained results is verified through extremal functions. In addition, we obtain Landau-type theorems for the class $\mathcal{BH}_0(\gamma, \delta)$, providing explicit bounds for the radius of univalence and the size of schlicht disks contained in the image domain. Our results not only unify and extend several earlier works but also provide new insights into the geometric behavior of harmonic mappings under differential constraints.

In this note we explore rational approximations to irrational numbers whose denominators are restricted to a subset of the natural numbers. We introduce a specific notion of density for such subsets and use it to establish a restricted analog of a theorem of Dirichlet's.

Let $T(n)=n/2$ if $n$ is even and $T(n)=(3n+1)/2$ if $n$ is odd. We prove that for each $m\ge1$, exactly $F(m+1)$ odd integers in $\{1,\ldots,2^m\}$ have the property that their orbit under $T$ avoids the residue class $4\pmod6$ during steps $2,\ldots,m$, where $F(m+1)$ is the $(m+1)$-th Fibonacci number; the proportion decays at rate $(\varphi/2)^m$, $\varphi=(1+\sqrt{5})/2$. The proof uses the directed graph $G$ of Collatz transitions modulo $6$ and its unique absorbing strongly connected component $G'=G[\{1,2,4,5\}]$. Removing vertex $4$ from $G'$ yields a subgraph of spectral radius $\varphi$, against $\rho(G')=2$; the Fibonacci count follows from this spectral gap. We construct an explicit bijection $\Psi_m:\{1,\ldots,6\cdot2^m\}\to\mathcal{P}_m(G)$ encoding integers as directed paths in $G$. We further show that no vertex of $G'$ is dispensable: removing any single vertex reduces the spectral radius strictly below $2$, with hierarchy $1<\sqrt{2}<\varphi<2$. In particular, every positive cycle of $T$ must visit residue class $2\pmod6$, and a flow conservation identity forces this class to account for more than $18\%$ of the steps in any such cycle.

Let $\mathcal{A}$ be the mod $2$ Steenrod algebra, and $P(n) = \mathbb{F}_2[x_1, \dots, x_n]$ be the polynomial algebra viewed as an unstable module over $\mathcal{A}$. The symmetric hit conjecture asks whether the symmetrization of a hit monomial in $P(n)$ is always hit in the symmetric invariant subalgebra $B(n) = P(n)^{\Sigma_n}$. While resolved for $n \leq 3$, the case $n=4$ presents significant obstructions due to combinatorial complexity, orbit cancellations intrinsically tied to $\Sigma_4$-stabilizers, and the emergence of strongly spike-free survivor modules. This paper introduces a conditional structural reduction to overcome these obstructions in the domain where the numerical weight satisfies $\mu(d) \leq 4$. By integrating Walker-Wood duality with a new $\Sigma_4$-stabilizer parity analysis, we reduce the global conjecture to localized algebraic conditions: a symmetric lower-spike reduction and a strengthened four-row digital-engineering hypothesis. Assuming these inputs, the conjecture follows by lexicographic induction on the column-sum and row-sum sequences of the binary exponent matrices. Our approach isolates the four-variable repeated-row anomaly into exact local identities, utilizing global Steenrod-kernel functionals lifted from local spike-free quotients to detect potential survivor elements. Finally, we provide explicit monomial-level computations in degrees $8$, $12$, and $14$, explicitly illustrating the stabilizer mechanism in practice and framing the precise algebraic identities required for a future unconditional proof.

Let $r_n^{k}(N)$ be the number of representations of $N$ as the sum of $n$ generalized $k$-gonal numbers and $r_n^{\square}(N)$ be the number of representations of $N$ as the sum of $n$ squares. By modifying the Heath-Brown circle method, we prove a closed-form asymptotic relation between $r_n^{k}(N)$ and $r_n^{\square}(8(k-2)N+n(k-4)^2)$ for any $k\geq3$ and any $n\geq4$. Consequently, we determine the asymptotics of $\sum_{N\leq x}r_4^{k}(N)^2$ and, via a result of Bringmann, Jang, Kane, and Tse, prove a similar closed-form asymptotic relation between the number $r_{4,+}^{k}(N)$ of representations of $N$ as the sum of four ordinary $k$-gonal numbers and $r_4^{\square}(8(k-2)N+n(k-4)^2)$. We also show that if $4\mid k$, any strictly increasing infinite subsequence on which $r_{4,+}^{k}$ is bounded converges $2$-adically to $(k-4)^2/(4-2k)\in\mathbb{Z}_2$, supplementing a result of Meng and Sun, and if $4\nmid k$, there is no strictly increasing infinite subsequence on which $r_{4,+}^{k}$ is bounded.

We investigate the equilibrium (trivial solution) stability, also known as Lyapunov stability, of a certain first-order linear complex constant coefficient dynamic equation on the time scale $\p_{\alpha,\beta}$, which has continuous intervals of length $\alpha>0$ followed by discrete gaps of length $\beta>0$. In particular, we establish results in the case of this specific time scale, for coefficient values in the complex plane, including where the exponential function alternates in sign. In our analysis, we employ the Lambert $W$ function. For increasing gap size $\beta$ relative to $\alpha$, we prove that the complex constant coefficient undergoes a bifurcation in its parameter space. We establish interesting results for both the delta dynamic equation and the nabla dynamic equation. Lastly, we connect these results to those related to Hyers--Ulam stability of the same nabla equations.

A classical result establishes that the area of a minimal graph intersected with the unit ball is at most $2\pi$. A natural question is whether this upper bound is sharp. In this note, we resolve this by constructing a sequence of minimal graphs via solutions to a Dirichlet problem. We show that the areas of these graphs tend to $2\pi$, demonstrating that the bound is sharp.

Let $f(k,s)$ denote the minimum integer $m$ such that any family $\mathcal{F}$ consisting of $k$-sized sets of cardinality at least $m$ always contain a sunflower of size $s$. The Erdős-Rado Sunflower Conjecture states that for every $s >2$, there is an constant $C=C(s)$ such that $f(k,s) \leq C^k$. In this paper, we prove the conjecture.

Devnani and Eyyunni recently studied the maps pre$_k$ on integer partitions, which arise from applying elementary symmetric polynomials to the parts of a partition. They asked whether there exists $n \ge 1$ such that exactly one partition of $n$ lies in the image of pre$_2$. We show that this occurs only for $n$ in {1, 2, 4}, and that for all $n \ge 5$, at least two partitions of n are in the image of pre$_2$.

We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.

A variational analysis of the Bogoliubov--Hartree--Fock (BHF) energy of the translation-invariant, spinless Pauli--Fierz Hamiltonian with massless dispersion relation built up on work of the first author, Breteaux, and Tzaneteas (2013) and of the first author and Hach (2022) is presented. The main results are lower and upper bounds on the BHF energy for fixed total momentum expressed through simpler variational problems defined on the space of positive Hilbert--Schmidt operators and a new variational formulation of the upper bound for zero total momentum. Specifically, we introduce a change of variables which considerably simplifies the energy functional and the derivation of its stationarity condition.

We reinvestigate the theory of spectral sequences by studying the $(\infty,1)$-category of filtered spectra through the cofiber-of-$\tau$ formalism of Burklund-Isaksen-Pstragowski-Wang-Xu. In this framework, we define and analyze hidden extensions along arbitrary maps of filtered spectra, establishing computational principles that extend the generalized Leibniz rule and the generalized Mahowald trick of Lin-Wang-Xu, as well as Burklund's Leibniz rule for total differentials, from the Adams spectral sequence to this broader setup. Our formulation uses a more refined, layered notion of extension, which slightly sharpens these statements even for the Adams spectral sequence. As an application, we study equivariant slice spectral sequences and obtain new families of "exotic transfer" differentials in the $C_4$-slice spectral sequences for the Hill-Hopkins-Ravenel theories $\mathrm{BP}^{((C_4))}\langle m\rangle$ for every $m \ge 1$.

All stabilizing PID controllers form a set in three-dimensional space. A novel viewpoint to its boundary as a ruled surface (or surfaces) being cut with 3D planes is presented. The characterization, being not too new, contributes to an understanding of the stability set as the whole, instead of the classical view as a stack of 2D slices, say, on the P-coefficient. The viewpoint gives clear insight on the structure of the PID stability region, and, in particular, splits its boundary into continuous parts. It is followed by natural 2D unwrapping of the stability set boundary. It also correctly handles pure imaginary zeros in transfer function. A wireframe 3D visualization reveals the structure of the stability set. The presentation is valid both for ideal and filtered PID controllers, as well as for time-delay systems and other linear systems. Finally, based on the viewpoint, a simple formula for stability (fragility) radius is provided.

The paper is concerned with $\alpha$-stable SDEs with singular time-inhomogeneous general drift. Our drifts satisfy a condition that is close to the minimal possible scaling-invariance and can introduce local blow-ups of the type arising in some particle systems with strong attracting interactions.

We propose Adaptive Accelerated Mirror Descent (AAMD), a flow-based method that combines nonlinear preconditioning, acceleration, and adaptivity in mirror geometry. The key ingredient is an accumulated Lyapunov perturbation budget: local descent failures are allowed as long as the total budget remains nonpositive, so line search is used only when stability is at risk. We prove accelerated convergence under dual relative smoothness/convexity and a mirror-geometry compatibility condition, and obtain an $O(1/k^2)$ rate for convex objectives by homotopy under a bounded-sublevel-set assumption. Experiments on relative-smoothness problems show that combining preconditioning, acceleration, and adaptivity gives substantial gains over methods using only part of this structure.

Let $S_{\kappa}$ be the space of cusp forms of weight $\kappa$ and level one, and let $S_{\kappa}^{\ast}$ denote its dual space. In this paper, we find explicit spanning subsets of $S_{\kappa}$ consisting of Rankin-Cohen brackets of Eisenstein series and explicit subsets of periods that span $S_{\kappa}^{\ast}$.

We prove a folklore conjecture identifying two categorical enhancements of the automorphic side of the local Langlands correspondence. Concretely, we construct an equivalence for torsion coefficients between the category considered by Zhu and the one considered by Fargues-Scholze. To achieve this, we revisit Scholze's analytification functor and apply the first author's theory of kimberlites. We discuss unconditional applications to the splitting of the semi-orthogonal decomposition on BunG, and the compatibility with Eisenstein functors. Finally, we formulate a linearity conjecture for our functor with which we can show new vanishing statements for the cohomology of local Shimura varieties, and perverse exactness statements for Hecke operators.

The nullcone membership problem, deciding whether an orbit closure contains the origin, is fundamental in computational invariant theory. For self-adjoint groups, Bürgisser, Franks, Garg, Oliveira, Walter and Wigderson gave a geodesic optimization algorithm whose complexity is controlled by the gap, a condition number of the representation. We study the gap for quiver representations under the action of the special linear group. We prove that the inverse gap is polynomially bounded in the number of vertices and the maximum dimension for type A and $\hat{A}$, as well as tree quivers with uniform dimension vectors. Consequently, the algorithm of Bürgisser et al. solves the nullcone membership problem in polynomial time for these families. In contrast, we construct families of quivers and dimension vectors where the gap is exponentially small in the number of leaves, furthermore, for every connected quiver we exhibit dimension vectors such that the weight margin (a related condition number) is exponentially small in the number of vertices. We also extend our results to $\sigma$-semistability, thereby giving a new proof of a recent result of Iwamasa, Oki, and Soma.

We develop a framework for infinite tensor products of Hilbert spaces, operators, and semigroups tailored to questions of similarity to contraction semigroups. On the operator-theoretic side, we give a systematic treatment of incomplete infinite tensor products, including criteria for existence, non-vanishing, and continuity properties of the associated tensor product semigroups. On the semigroup-theoretic side, we prove a low-regularity similarity theorem showing that global quasi-contractive control together with local contractive information at one positive time implies similarity to a contraction semigroup, with explicit bounds on the similarity constant. These ingredients are then combined to obtain infinite analogues of the finite tensor-product splitting principle for similarity to contraction and quasi-contraction semigroups. We also clarify the role of complete tensor products and show, in particular, that whenever a complete infinite tensor product of semigroups is a \(C_0\)-semigroup, it decomposes along incomplete tensor-product components.

We prove global in time regularity of solutions of the Navier-Stokes equations defined in the complex space.

We show that the Gromov--Hausdorff convergence of non-collapsed polarized constant scalar curvature Kähler (cscK) surfaces can be realized as convergence in a Hilbert scheme. We also derive uniform estimates of Bergman kernels on the effective regular set. As an application, we establish the Zariski openness of cscK metrics for certain smooth polarized families, following the approach of Donaldson.

Given an ordered set $B$ of a finite field, a combinatorial simplex over $B$ is defined as the set of vectors such that the positions of the entries, with respect to $B$, sum up to a fixed integer. CAP codes are Reed-Muller type codes defined over a combinatorial simplex. They were recently introduced by Kopparty et al. as a high-rate alternative to classical Reed-Muller codes, capable of achieving arbitrarily high rates close to one for any fixed minimum distance. In this paper, we use tools from commutative algebra to analyze a combinatorial simplex and its associated CAP code. We give a universal Gröbner basis for the vanishing ideal of a combinatorial simplex. We describe the generalized Hamming weights of a CAP code in terms of the footprint of the vanishing ideal. For the minimum distance case, we proved a closed formula. We give a set of polynomials whose evaluations on the combinatorial simplex generate the dual of the CAP code. We describe the affine permutations that leave invariant a combinatorial simplex and use this information to prove that, in some cases, the permutation group of a CAP code is a symmetric group.

We study the semi-algebraicity of the neuromanifold $\mathcal{M}_\mathbf{d}$ of a feedforward ReLU neural network and its symmetries. We prove that $\mathcal{M}_\mathbf{d}$ is not a semi-algebraic quotient of the space of weights of the network. We introduce and study the notion of \emph{honest} open subset of the space of weights, where the network does not show any hidden symmetries. Finally, we conjecture that the maximal honest open is always semi-algebraic and prove that in the shallow case it is even Zariski.

We prove a sharp Clifford-threshold focal-radius estimate and rigidity for immersed hypersurfaces. Under a $p$-form curvature condition, formulated by the Weitzenböck curvature term together with $\mathrm{Ric}_p\ge p$, any closed two-sided immersion $F:\Sigma^m\to M^{m+1}$ with $b_p(\Sigma;\mathbb R)\neq0$ and $1\le p\le m/2$ satisfies \[ r_f(F,M)\le\frac{\pi}{4}. \] The equality case is rigid: if the ambient manifold is complete, equality forces the hypersurface to be locally the Clifford hypersurface $S^p(1/\sqrt2)\times S^{m-p}(1/\sqrt2)\subset S^{m+1}(1)$; if the ambient manifold is compact and connected, it is a spherical space form. The curvature condition follows from $\sec\ge1$ for $p=1$, from normalized $\mathrm{PIC1}\ge1$ for $p=2$, and from curvature operator bounded below by one in all degrees. By quotient lifting and the Hopf fibrations, we also obtain focal-radius estimates in $\mathbb{CP}^n$ and $\mathbb{HP}^n$, with projective Clifford rigidity, without any Betti-number assumption.

The classical no-$k$-in-line problem asks for the largest number of points that can be placed on an $n \times n$ grid without having $k$ of them collinear. A natural extension, motivated by the analogous question by Erde for $k\in \mathbb{Z}$, is the \emph{extensible no-$(k(n)+1)$-in-line problem}, which seeks a subset of points in $\mathbb{Z}^2$ with maximal possible density such that at most $k(n)$ points are collinear within the subgrid $[1,n]^2$. We construct optimal sets for linear functions and positive-density sets for power functions. We prove that any configuration achieving $\liminf\frac{S_n}{n k(n)} \ge 0.897$ must satisfy $k(n) = \Omega( n^c)$ for some $c>0$ constant; therefore, the extensible no-$k$-in-line problem has no configuration with this property when $k$ is a constant. Finally, we reduce the problem to the extensible no-$k$-in-line problem, showing that if a positive-density point-set exists for a constant limiter function, then one also exists for any sufficiently regular function $k(n)$.

We study Cheeger-type inequalities for persistent Laplacians associated with inclusions of simplicial complexes $\mathcal{K}\hookrightarrow \mathcal{L}$. We introduce a persistent up $p$-Laplacian $\Delta_{q,p,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$ for $p\geq 1$. For $p=2$, this recovers the usual persistent up Laplacian, while for $p=1$ it yields a nonzero persistent Cheeger constant $\varphi_q^{\mathcal{K},\mathcal{L}}$. We prove a Cheeger-type inequality relating $\varphi_q^{\mathcal{K},\mathcal{L}}$ to the smallest nonzero eigenvalue of $\Delta_{q,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$. This gives a persistent extension of recent work by Jost and Zhang (Ann. Sc. Norm. Super. Pisa Cl. Sci., 2024; arXiv:2302.01069). We then study two more structured settings. Under a locally complete $q$-skeleton assumption on $\mathcal{K}$, we extend the complete-skeleton isoperimetric inequality of Parzanchevski--Rosenthal--Tessler (Combinatorica, 2016; arXiv:1207.0638) to the persistent setting. For orientable $(q+1)$-dimensional pseudomanifolds, we prove a Kron-type reduction of the persistent up Laplacian to a vertex- and edge-weighted graph Laplacian, possibly with Dirichlet boundary terms, and obtain two-sided Cheeger inequalities; this is related to the dual-graph perspective in the work of Steenbergen--Klivans--Mukherjee (Adv. Appl. Math., 2014; arXiv:1209.5091). We also describe the nonzero persistent Cheeger constant $\varphi_q^{\mathcal{K},\mathcal{L}}$ explicitly in terms of the dual graph in the non-branching pseudomanifold case. Finally, for graph inclusions $H\hookrightarrow G$, we compare the persistent Cheeger constants introduced here with the Kron-reduction Cheeger constants of Mémoli et al. (SIAM J. Math. Data Sci., 2022; arXiv:2012.02808).

Let $\mathbb Z_n$ be the cyclic group equipped with the uniform probability measure $\pi$, and let $-A_{\psi_n}$ be the Laplacian with respect to the word length $ \psi_n(k) = \min(k,n-k). $ We prove the sharp log-Sobolev inequality $$ \operatorname{Ent}_{\pi}(f^2) \le 2\pi\bigl(f A_{\psi_n} f\bigr), \qquad f:\mathbb Z_n \to \mathbb C, $$ for every $n \ge 4$. The proof is inspired by the recent work of Frank and Ivanisvili~\cite{FrankIvanisvili2026} on a sharp log-Sobolev inequality for nearest-neighbor simple random walk. We use their idea of cubic-majorant reduction, but replace their high-frequency estimate with a Fourier block estimate adapted to the word-length multiplier. The same result was also recently obtained by Yao~\cite{Yao2026} using a completely different approach.

Peskine and Szpiro's Intersection Theorem for finitely generated modules was generalized by Foxby to all modules and bounded complexes of such. We strengthen Foxby's result and prove a dual result, which for complexes with finitely generated homology recovers Iversen's Amplitude Inequality but also applies to derived complete complexes.

The generalized Turán number $ex(n, K_{t, t}, K_{2, t+1})$ is the maximum number of copies of $K_{t, t}$ that a $K_{2, t+1}$-free graph on $n$ vertices can contain. Recently, Pohoata, Tidor, and Yu established that $ex(n, K_{t, t}, K_{2, t+1}) = \Theta_t(n^2)$ for all integers $t \geq 3$. In this short note, we use an explicit construction to establish that when $t$ is a prime power and $n = t^{2e - 1}$, then $$ ex(n, K_{t, t}, K_{2, t+1}) = (1 + o(1))\frac{n^2}{2t(t-1)}. $$

We present the first rigorous analysis of decentralized geometric routing in Geometric Inhomogeneous Random Graphs (GIRGs), a weight-agnostic variant of the greedy routing protocol. While greedy routing in GIRGs is known to explain the algorithmic small-world phenomenon by finding ultra-short paths of length $\Theta (\log \log n)$, it assumes additional knowledge of vertex weights beyond geometry, an assumption that is often restrictive or unavailable. We investigate whether the underlying geometry alone is sufficient for efficient navigation. We prove that for power-law weight exponent $\tau \in (2,3)$ and geometric decay parameter $\alpha > \tau - 1$, geometric routing succeeds with constant probability and finds ultra-short paths of length $\Theta (\log \log n)$, matching the optimal asymptotic guarantees for greedy routing. Our analysis further reveals that, upon success, both protocols follow a similar two-phase trajectory, consisting of a rapid ascent to the heavy vertices, followed by efficient navigation to the target. These results demonstrate that, in the appropriate regime, the network's geometry alone implicitly guides the path to the target through its high-weight core.

We prove that stationary random metric measure spaces have 0,1,2 or a Cantor space of ends. This notion includes stationary random graphs, manifolds and discrete subgroups. In the case of surfaces, we classify all possible homeomorphism types, in analogy with the work of Biringer and Raimbault on unimodular Riemannian manifolds. Our approach relies on a general "no geometric core" principle and an analysis of finite versus infinite expected return times.

Identifying ideal Perron similarities is a problem of central interest in the longstanding nonnegative inverse eigenvalue problem (NIEP). A normalized ideal Perron similarity is called totally extremal if every entry has modulus one. Recently, Gershnik et al. [J. Algebra 694 (2026), 782--800] proved that the character table of a finite Abelian group is totally extremal and conjectured the converse. In this paper, we settle this conjecture in the affirmative by first showing that the rows of a totally extremal normalized ideal Perron similarity form a group under the Hadamard product. Then, it is shown that the rows of a nonsingular matrix form a group under the Hadamard product if and only if it is the character table of a finite Abelian group, and we further show that this group is isomorphic to the underlying group. These results extend the classical theorem due to Romanovsky and Karpelevič on the unimodular eigenvalues of stochastic matrices in the complex unit disk to the setting of spectratopes in the unit ball of complex Euclidean space.

The maximal entries of Dynkin friezes over the positive integers have recently been determined for all finite Dynkin types except $B_n$ and $D_n$. In this note, we explicitly construct large positive integral points on affine cluster varieties of type $B_n$ (resp. $D_n$), giving rise to friezes of types $B_n$ (resp. $D_n$) over the positive integers with largest entries $F_{n+1} F_{n+2} - 1$ (resp. $F_n F_{n+1} - 1$) where $F_k$ is the $k$-th Fibonacci number. We conjecture that these are the maximal possible entries for their respective Dynkin types.

We prove a sparse transference theorem for induced Ramsey graphs. The theorem transfers the weighted random-host proof of Aragão, Campos, Dahia, Filipe, and Marciano to the sparse random setting. It follows that, for every fixed graph $H$ with no isolated vertices and at least two edges, and every $\eta>0$, there is $C>0$ such that, whenever $N\ge r^{Cr}$ and $N^{-1/m_2(H)+\eta}\le p\le \frac12$, with high probability every $r$-colouring of the edges of $G(N,p)$ contains a monochromatic induced copy of $H$. Here, $m_2(H)$ denotes the usual maximum 2-density of $H$.

Pál's isominwidth inequality (1921) answered the Kakeya needle problem (1917) for convex sets. It states that among convex bodies of fixed minimum width $w$ in the Euclidean plane, the regular triangle has minimal area. The isominwidth inequality was generalized to the $2$-dimensional sphere by Bezdek and Blekherman and Freyer and Sagmeister (arXiv:2411.11462). Interestingly, in hyperbolic space, no minimizer exists, as shown by Böröczky, Freyer and Sagmeister (arXiv:2502.04427). The stability of the Euclidean Pál inequality with respect to the Hausdorff metric and the symmetric difference metric was proved by Lucardesi and Zucco (arXiv:2405.18294). Fodor, Robock and Sagmeister (arXiv:2602.19300) proved $r$-ball convex analogs of the isominwidth inequality in all three constant curvature planes connecting Pál's theorem with the Blaschke--Lebesgue inequality. In this paper, we prove optimal stability versions of this statement with respect to the Hausdorff distance and the symmetric difference metric in all three constant curvature planes.

We consider $K3$ surfaces collapsing to a three-dimensional affine base. We show that certain affine lines on the base lift to degenerating sequences of special Lagrangian two-spheres and tori in the collapsing $K3$ surface. In particular, we construct special Lagrangian two-spheres connecting pairs of Taub-NUT bubbles. These examples fit into the broader program of reconstructing special submanifolds from graphs and combinatorial data on a collapsed affine limit.

Despite the prevalence of nonautonomous systems in applications, their statistical properties are much less understood than in the autonomous setting. Building on recent results on response theory for nonautonomous systems, we study the approximation of equivariant families and of their linear response by Ulam-type finite-dimensional reductions. First, we show that coarse-graining procedures associated with the classical Ulam method, and more generally with suitable finite-element projections, provide rigorous approximation of equivariant families for sequential systems with memory loss. Second, for systems whose transfer operators are regularizing, we prove that the linear response of the reduced finite-state Markov model converges to the projected linear response of the original system. To the best of our knowledge, a general approximation result of this type has not previously been established in this form, even in the autonomous case. We complement the analysis with numerical experiments on simple but representative time-dependent diffusive models. These results provide a rigorous foundation for the use of Markov approximations in the study of statistical properties of nonautonomous complex systems which almost invariably relies on finite-scale and finite-precision descriptions of their states and dynamics.

Parametric resonance -- periodic variation of a system parameter driving exponential growth of oscillations -- is among the most fundamental instabilities in physics and engineering. The nondissipative LC circuit with harmonically varying capacitance is one of its simplest realizations: the modulation renders the circuit equation a Hill equation with either bounded or exponentially growing solutions. Identifying the governing equation as a special case of Ince's four-parameter Hill equation yields two main results. First, a sharp structural theorem: instability occurs only at the odd sub-harmonics of the natural frequency, while every even resonance is exactly stable at all modulation amplitudes. This selectivity, invisible to the Mathieu approximation, follows from Krein's collision theory: at odd resonances the colliding Floquet multipliers carry opposite Krein signatures, opening a tongue; at even resonances the signatures agree and the tongue collapses. Second, closed-form formulas for the widths and boundary curves of all surviving tongues, derived by a continued-fraction and Magnus--Winkler method, confirmed against Cambi's 1950 numerics and recovered via the Yakubovich--Starzhinskii series. The continued fraction also gives the Floquet exponent as an exact power series in the modulation amplitude with rational coefficients, and finite-product formulas for all Fourier coefficients of the periodic Floquet factor. The tongue boundaries consist entirely of exceptional points of degeneracy (EPD) of the monodromy matrix, enabling hypersensitive capacitance sensing: at an EPD a small perturbation splits the coincident frequencies by the square root of its size, diverging relative to linear sensing as it shrinks. A closed-form splitting formula is derived, and a work-point strategy shifting slightly into the stable zone keeps the scheme robust while preserving square-root sensitivity.

We introduce disciplined nonlinear programming (DNLP), a syntax for specifying nonlinear programming problems. DNLP is inspired by disciplined convex programming (DCP) and allows smooth functions to be freely mixed with nonsmooth convex and concave functions, with rules governing how the nonsmooth functions can be used. Problems expressed in DNLP form can be automatically canonicalized to a standard nonlinear programming (NLP) form and passed to a suitable NLP solver. As in DCP, the canonicalization relaxes nonsmooth convex and concave functions in a lossless way, allowing them to be handled by NLP solvers that require smooth functions. In addition to extending NLP to include useful nondifferentiable convex and concave functions, transforming the original problem to an equivalent NLP form offers several advantages, including simpler problem initialization. We describe the language and our open-source implementation of DNLP as an extension of CVXPY, a parser for DCP.

We isolate a simple preservation principle governing when it is absolute, between transitive models of set theory, that a given algebraic or topological-algebraic structure has a standard form $F(X)$ indexed by a set $X$. The principle is: if the index $X$ (or a proxy for it) can be recovered from $F(X)$ by a uniform definable construction, then the class of structures isomorphic to some~$F(X)$ is downward absolute from forcing extensions. Answering a question raised by Noah Schweber, we deduce in particular that no group that fails to be a full symmetric group in the ground model can become one after forcing; the result holds already in ZF. The same mechanism applies to full transformation monoids, powerset Boolean algebras, full relation algebras, full clones, full partition lattices, products $R^X$ of finitely generated centrally indecomposable rings, the commutative $C^*$-algebras $\ell_\infty(X)$ and $c_0(X)$, full endomorphism rings, the operator algebras $\mathcal{B}(H)$ and $\mathcal{K}(H)$, and $\ell_1(X)$ as a real Banach lattice. In the motivating symmetric-group case, the same reconstruction gives more than descent: it yields a uniform $\Pi^1_1$ definition of fullness over transitive ZF-models. We then exhibit clean torsor obstructions, in the standard symmetric-model situation: finite covers $Y \times n$ already separate ZF-failure from ZFC-descent without any completeness caveat, and the finite-support normed space $c_{00}(I)$ provides the analogous Banach example. Bare-Banach-space isomorphism with $\ell_1(\Gamma)$ exhibits a genuine ZFC-descent. We conclude with the corresponding, relative, obstructions to $\Pi^1_1$-definability of standardness over transitive ZF-models.

In this work, we introduce a new macroscopic model for crowd motion inspired by the celebrated Hughes' model \cite{Hughes2002, Hughes2003}, which couples a nonlinear conservation law for the pedestrian density with an Eikonal equation describing the shortest path to the target. Our approach can be viewed both as a modification of Hughes' original formulation and as a refinement of the prediction-correction framework proposed in the recent work \cite{ennaji2023prediction}. The resulting model incorporates anticipatory behavior and dynamic route adjustment, offering a more realistic representation of crowd dynamics in complex environments. We present the mathematical formulation of the model, discuss its well-posedness properties, and illustrate its qualitative behavior through numerical simulations. Ultimately, we show, at least from a numerical perspective, that this variant provides a promising avenue towards establishing the well-posedness of the classical Hughes' model, which has remained a challenging open problem for a long time.

We consider supertori which are quotients of affine superspace by translations by algebraically independent odd parameters. Specifically, we describe the ring structure of its space of global sections by generators and relations and completely determine the coherent cohomology groups of its structure sheaf by reducing it to a problem of Lie algebra cohomology. We also show that Poincaré duality on sheaf cohomology is compatible with that of the group cohomology of the translation group and give explicit tables of examples. Finally, we compute the Picard groups of these supertori.

We give a completely explicit Lerch, Chowla--Selberg formula without absolute values of the Dedekind eta function, and give some applications, in particular to the argument of individual CM values of the eta function. We also give precise conjectures on these CM values and a large table supporting them.

In this paper, we study optimal distributional estimates for the multiple Hilbert transform. We obtain pointwise upper and lower distributional estimates of the multiple Hilbert transform in terms of the $d$-fold composition of the Calderón operator with itself. This extends the fundamental results by A. P. Calderón, D. Boyd, and Ch. Fefferman for arbitrary $d\in\mathbb N.$

The Lipschitz-free space $\mathcal{F}(M)$ is a canonical linearization of a complete metric space $M$ whose topological dual is the space of Lipschitz functions on $M$. We review the properties of $\mathcal{F}(M)$ when the underlying space $M$ is purely 1-unrectifiable, that is, it contains no bi-Lipschitz copy of a subset of $\mathbb{R}$ with positive measure. For compact $M$, this is equivalent to several Banach space properties of $\mathcal{F}(M)$, including the Radon-Nikodým and Schur properties or admitting a predual. We shall see how the study of locally flat Lipschitz functions on $M$ reveals these equivalences, and describe a technique that allows most of them to be transferred to the non-compact setting. This manuscript is an expository text based on results by the author in collaboration with C. Gartland, C. Petitjean and A. Procházka, originally published in a Trans. Amer. Math. Soc. paper, and corresponds to a lecture delivered at the Second Winter School in Geometric Measure Theory at Westlake University, Hangzhou, on February 2026.

Let $L$ be a subspace of a complex vector space $V$ and fix $s \leq \dim{L}$. The (type A) Kalman variety consists of all endomorphisms of $V$ that have an $s$-dimensional invariant subspace in $L$. We introduce a generalization where $V$ and $L$ are symplectic vector spaces. We fix an isotropic subspace $W \subseteq V$ satisfying $W^\perp = W \oplus L$. The isotropic (type C) Kalman variety consists of symplectic morphisms of $V$ that have an invariant coisotropic subspace of a prescribed dimension inside $W^\perp$. We are mainly interested in studying the Lagrangian case. In type C, we prove analogues of results known for type A Kalman varieties; in particular, we determine the defining equations, compute geometric invariants, and analyze their singularities. We conjecture the existence of a long exact sequence relating the structure sheaves. Based on the results in the symplectic case, we describe Kalman variety analogues with respect to endomorphisms of odd orthogonal (type B) and even orthogonal (type D) vector spaces.

We prove that if $\theta$ is a continuous unital homomorphism of an operator algebra $A$ into $B(\mathcal{H})$, and $\beta$ is in the dual space of $A$, then the completely bounded norm of $\theta$ is less than or equal to the maximum of $1$ and the completely bounded norm of $\theta + \beta I $. As an application, we give another proof of the Okubo--Ando theorem.

The one-shot approach is a powerful simultaneous optimization framework for design tasks governed by computationally expensive steady-state systems. While previous formulations mainly focused on additional equality constraints, this work extends the one-shot framework to optimization problems with inequality constraints using slack variables embedded into a doubly augmented Lagrangian formulation. After elimination of the slack variables, the resulting formulation contains nonsmooth active-set dependent terms while preserving the characteristic coupled one-shot matrix structure. The resulting generalized gradient system admits a representation of the form $ \nabla \DLag = -Ms, $ analogous to the equality-constrained one-shot framework. To analyze the nonsmooth active-set transitions, generalized second-order subdifferentials in the sense of Clarke and Rockafellar--Wets are employed. We derive explicit positivity conditions ensuring positive definiteness of all admissible generalized Hessians and hence strict local optimality. Furthermore, we give an outlook on the construction of preconditioners based on generalized Hessian approximations.

This paper develops a data-driven realization of the generalized Koopman operator (GeKo), in which states and inputs are lifted independently and the dynamics are expressed as a tensor bilinear system. The first contribution is a time-sequenced multi-step Khatri-Rao kernel regression formulation that exposes the operator to evolved snapshots along trajectories rather than only single one-step pairs, which reduces compounded prediction error. Secondly, we develop a kernel- and input-agnostic structured SVD reduction that compresses the lifted state and input spaces while preserving the Khatri-Rao realization. We instantiate the framework with random Fourier features and describe a complete predictive-control pipeline, including a multi-step roll-out diagnostic that guides the choice of MPC horizon. The framework is validated on the chaotic Lorenz system, where the learned reduced-order GeKo model stabilizes an unstable equilibrium from a range of initial conditions.

We study local finiteness for modal $K4$ algebras via the tunability of their dual general frames. In particular, we provide a sufficient condition for modal $K4$ algebras to be locally finite by identifying a structure which must be present in non-locally finite modal $K4$ algebras. We then show that this condition becomes both necessary and sufficient for complex modal $K4$ algebras. Next, we translate this condition into a pair of order-theoretic conditions on transitive Kripke frames, providing a classification of local finiteness on their dual modal algebras. We further show that the logic of any class of well-founded transitive relations with no infinite antichains has the finite model property, and conclude that the logic of the class of well-quasi orderings has the finite model property.

Mixed structures and their weight filtrations appear in various contexts, prominently Hodge theory and the theory of Galois representations. In the setting of André's formalisation of slopes, we propose an abstract framework explaining why mixed structures have shared characteristics.

We give an alternative formulation of Stein's maximal inequality for generalised spherical averages in terms of derivatives of standard spherical means: if \[ k \ge 0, \qquad d \ge 2 k + 3 , \qquad \frac{d}{d - k - 1} < p < \frac{d - 1}{k} , \] and $\sigma$ is the normalised surface measure on the unit sphere $\mathbb S$, then the maximal operator \[f \mapsto \sup_{r > 0} \, \biggl\lvert r^k (\tfrac{d}{dr})^k \int_{\mathbb S} f(\cdot + r y) \sigma(dy) \biggr\rvert\] is bounded on $L^p$, with a constant that is independent of the dimension $d$.

A basic open question motivated by the study of ribbon knot diagrams asks whether every ribbon knot can be presented as a symmetric union. In this article, we give a negative answer to this question by exhibiting a ribbon Montesinos knot which does not admit a symmetric union presentation.

While the uncrowding algorithm on set-valued tableaux has long been instrumental in proving the Schur positivity of stable symmetric Grothendieck polynomials, lattice models have emerged as a modern framework for investigating symmetric functions, in particular symmetric Grothendieck polynomials. In this work, we synthesize these combinatorial and lattice-theoretic approaches by defining both the Robinson--Schensted--Knuth (RSK) correspondence and the uncrowding operation directly on a 5-vertex model of Motegi and Sakai and its subsequent reinterpretation by Buciumas, Scrimshaw, and Weber. Our lattice-based RSK formulation yields a powerful new result: the direct construction of the associated crystal structure on the states of the 5-vertex model.

Multiplication table of a ring with identity 1 is said to have the diagonal condition if 1s occur only on the main diagonal. In this paper, we study the diagonal condition in the ring of Gaussian integers $\mathbb{Z}[i]$. Furthermore, we also find the Gaussian integers $\alpha$ so that the rings of Gaussian integers modulo $\alpha$ satisfy the diagonal condition.

For a finitely generated group $G$, the Bieri-Neumann-Strebel-Renz (BNSR) invariants are subsets of the character sphere of $G$ that govern the finiteness properties of normal subgroups containing the commutator subgroup. We investigate the BNSR invariants of link groups and $2$-knot groups. In particular, for a link $L$ with at least two components, we prove that the commutator subgroup of the link group is finitely generated if and only if $L$ is a Hopf link. Moreover, we show that there exists a ribbon $2$-knot whose knot group has a non-symmetric BNS invariant.

Motivated by recent developments at the interface of operator algebras and random matrix theory, we propose new conjectures concerning the asymptotic structure of random matrix models of the countable free groups. The first conjecture predicts a random matrix analogue of the Akemann-Ostrand property for free groups, and reveals a succinct approach to recover the Peterson-Thom property for $L(\mathbb{F}_2)$. The second stronger conjecture is motivated by continuous model theory. It predicts that the \emph{random} embedding of the free group factor into a matrix ultraproduct is \emph{existential}. We discuss the interesting relationship between these conjectures.

We obtain the leading term in the solution of the Cauchy problem for the Benjamin-Ono equation in the limit $t\to+\infty$ with $x=O(t^{1/2})$. We show that the rate of decay exceeds that of self-similar solutions and obtain an explicit universal profile for the decaying solution, relating it to the linearization of the profile equation for self-similar solutions. The proof assumes a class of rational initial data $u_0$ in $L^2(\mathbb{R})\cap L^1(\mathbb{R})$ that exhibit generic behavior of the reflection coefficient at the origin.

We study the regularity of spectral measures of self-adjoint operators on infinite weighted graphs in the unimodular setting. This framework encompasses operators in the group algebra of a finitely generated group, random operators whose distribution is quasi-invariant under a group action, and Benjamini--Schramm limits of operators on finite graphs. Under a natural geometric condition on the underlying graph, we prove that the expected spectral measure satisfies a logarithmic Hölder regularity estimate. The proof relies on a strengthened version of the monotone labelling method previously introduced with Sen and Virág to control the pure point part of the spectal measure. Applications include operators in group algebras of indicable groups, Anderson-type models with arbitrary compactly supported potentials on Cayley graphs, anisotropic percolation operators, and operators on quasi-transitive graphs. In particular, our results extend the classical Craig--Simon theorem beyond the Euclidean lattice.

We prove a conjecture of Glasby, Praeger, and Unger concerning the symmetric group $S_{n}$. Let $\pi_{n}$ denote the proportion of elements of $S_{n}$ that are pre-$p$-cycles for some prime $p\in[2, n-3]$. We prove that $\pi_{n} > 1/3$ for all $n\geq 8$.

We study single-error correction for analog codes over $\mathbb{R}$. A key performance measure is the parameter $\Gamma_2(\mathcal{C})$, which quantifies the minimum separation required between large outlying errors that need to be located/corrected and bounded tolerable perturbations. We prove that every real linear $[n,n-2]$ code $\mathcal{C}$ satisfies \[ \Gamma_2(\mathcal{C})\ge \frac{1}{\sin^2(\pi/2n)}. \] This resolves Roth's open problem on the optimality of redundancy-two single-error-correcting analog codes. Our proof combines a zonotope-based geometric characterization of $\Gamma_2(\mathcal{C})$ with a cyclic sine-product inequality. We also construct analog codes with higher fixed redundancy and show that, for every fixed $r\ge 2$, there exists a class of real linear $[n,n-r]$ codes such that \[ \Gamma_2(\mathcal{C})\le O\left(n^{1+\frac{1}{r-1}}\right). \]

We develop a contraction-based framework for proving mixing-time bounds for Markov chain Monte Carlo algorithms. The framework is built around global and local contraction coefficients of Markov kernels under the $\mathsf E_\gamma$-divergence with $\gamma\ge1$. For projected Langevin Monte Carlo on a compact convex domain, we show that Gaussian smoothing yields an explicit global contraction coefficient for the $\mathsf E_\gamma$-divergence. This gives a direct proof of exponential convergence to the discretized stationary distribution for general smooth, possibly non-convex potentials. The rate is explicit, accommodates arbitrary random-batch sampling schemes, and yields convergence guarantees for several divergences, including KL, $\chi^2$, and Rényi divergences. For independent Metropolis--Hastings with target $\pi$, proposal $q$, and unbounded importance weight $w=d\pi/dq$, global contraction coefficients are typically trivial. We therefore introduce a local contraction coefficient on the core $C_R=\{w\le R\}$ and prove that it controls the rejection profile on the core. This yields warm-start convergence bounds governed by the local contraction coefficient and the tail profile $H_R=\pi(w>R)$, recovering sharp existing moment-based convergence rates when $\mathbb E_q[w^p]<\infty$ for some $p>1$, while remaining effective in heavy-tailed regimes where no finite moment of order $p>1$ exists.

Rotatable antenna (RA) technology has emerged as a promising solution to enhance spectrum efficiency by exploiting additional spatial degrees of freedom (DoFs) in multiple access networks. However, the relative performance superiority among different multiple access schemes remains largely unclear due to the unique capability of RA in reconfiguring the directional gain pattern. In this letter, we conduct a theoretical comparison between non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) schemes in RA-assisted communication systems in terms of transmit power minimization, subject to constraints on antenna rotational range and users' target rates. To address the associated non-convex optimization problem, a particle swarm optimization (PSO) algorithm is employed to optimize the rotational angle. Simulation results demonstrate that RA-assisted schemes significantly reduce transmit power compared to fixed-antenna benchmarks. Furthermore, RA-assisted NOMA may perform worse than time-division multiple access (TDMA) for symmetric user deployments, while it exhibits superior robustness and energy efficiency in asymmetric scenarios.

In this article, we study the (full or vertical) linearisation problem under a finite Galois étale cover. As an application, we give sufficient conditions for full linearisation near Hopf manifolds and for vertical or full linearisation near hyperelliptic manifolds.

We study the geometry of feasible value functions in infinite-horizon partially observable Markov decision processes (POMDPs) under memoryless stochastic policies. Our main contribution is a characterization of the feasible set of value functions as a semi-algebraic set, defined by explicit polynomial inequalities determined by the transition dynamics, observation kernel, and reward structure of the POMDP. This result extends prior work for fully observable Markov decision processes, where the feasible set is known to be a polytope, to the substantially more intricate partially observable setting. In contrast to the polyhedral structure arising in MDPs, partial observability induces fundamentally nonlinear constraints, leading to a richer and more complex geometric structure. Our geometric characterization provides new insight into the landscape of policy optimization in both MDPs and POMDPs, and reveals qualitative phenomena unique to partial observability, including the emergence of isolated local maximizers of the long-term reward and their dependence on the initial state distribution.

In this paper, we study the $\mathbb J$-stratification of basic affine Deligne-Lusztig varieties for a minuscule cocharacter $\mu$. This stratification was introduced by Chen-Viehmann and has been expected to serve as an interesting tool for studying basic loci in Shimura varieties. We parametrize the $\mathbb J$-strata whose stabilizers in the Frobenius-twisted centralizer group are parahoric by constructing a natural bijection to combinatorial invariants called small cocharacters. We further prove that the cardinality of these sets is equal to that of a certain subset of the Weyl group orbit of $\mu$. A relationship with the weakly fully Hodge-Newton decomposability of Chen-Tong is also discussed.

In this paper, we study the initial-boundary value problem for the nonlinear Schrödinger equation in $\mathbb{R}^{n}_{+}$ \begin{equation*} i\partial_{t}u+\Delta u+\lambda |u|^pu=0, \qquad (x, t) \in \mathbb{R}_{+}^{n} \times \mathbb{R}_{+},\ \ p\in\mathbb{R}_{+} \end{equation*} with nonhomogeneous Dirichlet boundary conditions. For the corresponding linear problem, endpoint Strichartz estimates are derived. For the nonlinear problem, we prove local well-posedness in $H^{s}(\mathbb{R}^{n}_{+})$ with $s\in[0,\frac{5}{2})$ and $p<\frac{4}{n-2s}$. Moreover, global well-posedness is established in the same regularity range. For $s\in[1,\frac{5}{2})$, the one-dimensional global theory of \cite{figment} in $H^{s}(\mathbb{R}_{+})$ is extended to $H^{s}(\mathbb{R}^{n}_{+})$. Additionally, we obtain global solutions in the lower regularity setting $s\in[0,1)$ for the first time. It is noteworthy that for $s=0$, we overcome the lack of mass conservation resulting from the nonzero boundary data and derive the pivotal $L^{2}(\mathbb{R}^{n}_{+})$ a priori estimates.

In this paper, we establish two loop decomposition theorems for sphere bundles of real vector bundles under appropriate connectivity and characteristic class assumptions. Specifically, we treat two families of sphere bundles: those arising from rank-$(n-1)$ vector bundles and those from rank-$(n-3)$ vector bundles over closed smooth $n$-manifolds.

We study optimal constants and extremizers of smoothing estimates for quantum harmonic oscillators. In particular, we establish harmonic oscillator analogues of free particle results due to Simon (1992), Bez and Sugimoto (2014), and Bez et al. (2015).

We rigorously derive the effective large-population, mean-field dynamics of particle-based reactive Langevin dynamics (PBRLD) models. These models extend particle-based stochastic reaction-diffusion (PBSRD) descriptions by incorporating velocities, inertial effects, and underdamped motion. In Isaacson, Liu, Spiliopoulos, and Yao, SIAP 2026, PBRLD models were formulated and shown to recover Doi volume reactivity PBSRD model in the overdamped limit. In this work we prove convergence of the associated measure-valued stochastic processes, representing species concentration fields on position-velocity phase space, to a deterministic mean-field limit. The limiting equations form a novel system of nonlocal kinetic reaction-diffusion partial integro-differential equations, coupling hypoelliptic transport with reaction terms that retain the spatial and velocity structure of the underlying particle interactions.

We show that higher extension spaces between finite-dimensional nilpotent $\mathbb{F}_1$-representations maybe infinite-dimensional, thereby clarifying a misconception in the literature. Our examples arise from cyclic quivers. In particular, for a cyclic quiver $\Delta_n$, we show that $\operatorname{Ext}^3(-,-)$ vanishes for any pair of finite-dimensional nilpotent $\mathbb{F}_1$-representations of $\Delta_n$, while $\operatorname{Ext}^2(-,-)$ is infinite-dimensional for any pair of simple representations.

Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^{*}$. We introduce a viscosity-Halpern hybrid projection scheme for approximating a common element of the fixed point set of a countable family of generalized nonexpansive-type mappings, the solution sets of countably many variational inequality problems, and the solution sets of countably many equilibrium problems. The method combines a viscosity perturbation generated by a contraction, a Halpern anchor term, equilibrium and variational inequality resolvent steps, and a shrinking generalized projection step. Under monotonicity, continuity, closedness and NST-type assumptions, we prove strong convergence of the generated sequence to the generalized projection of the initial point onto the common solution set. We also give a generalized-projection variational characterization of the selected limit, residual convergence, Hilbert-space specializations, and examples showing that the full countable problem cannot, in general, be recovered from finite truncations.

Let $\Lambda \subset\mathbb{R}^3$ be a region admitting a partition into $n$ compact, connected subregions $\Lambda_1,\dots,\Lambda_n$, each with smooth boundary. Consider a vector field $B$ on $\Lambda$ where $B|_{\Lambda_i}$ is smooth, divergence free, and tangent to $\partial \Lambda_i$ for all $i$. We show that the multi-region relaxed magnetohydrodynamics (MRxMHD) equilibrium equations are necessary and sufficient conditions for $ B $ and a metric to yield a stationary point of the magnetic energy under appropriate constraints. We constrain the pressure, relative helicity, and magnetic flux of $B$ through all smooth surfaces in $\Lambda_i$ whose boundary lies on $\partial \Lambda_i$. We identify a previously overlooked gauge condition. A definition for relative helicity is introduced, its gauge invariance is proved, and the existence of a gauge where relative helicity reduces to conventional helicity is demonstrated. In the case of a single region an additional condition is introduced that is sufficient to ensure a critical point of the magnetic energy is also a minimiser.

A split graph is a graph where the vertices are a disjoint union of a complete part $C=\{x_i,\ldots,x_n\}$ and a stable part $S=\{y_1,\ldots,y_m\}$. We will determine the Betti numbers of the edge ring of all split graphs, in particular show that the only nonzero Betti numbers are $\beta_{0,0}$ and $\beta_{i,i+1}$, $i>0$. The Betti numbers only depend on the multiset of the number of neighbors in $S$ the $x_i$'s have. Singh and Verma have earlier determined the Betti numbers for complete split graphs (where all $y_i$ are neighbors to all $x_j$), and for "nearly complete" split graphs (where all $y_i$ are neighbors to all $x_j$, except that $y_i$ is not a neighbor to $x_i$ for $i=1,\ldots,\min\{m,n\}$). We also determine which split graphs that have Cohen-Macaulay edge ring.

We study the spatial propagation of super-Brownian motion on $\mathbb{R}^d$ with general critical or subcritical branching mechanisms. Under a Keller-Osserman type integrability condition on the spatially homogeneous branching mechanism, we establish a quantitative upper bound for the short-time probability that the support exits a prescribed neighborhood of its initial support. The estimate has a Gaussian-tail form and is obtained through weighted occupation times, Feynman-Kac representations, singular elliptic boundary blow-up estimates, and comparison principles for nonlinear PDEs. As an application, we derive the compact support property for super-Brownian motions started from compactly supported finite measures. We further extend the argument to spatially dependent branching mechanisms by imposing suitable local lower bounds given by spatially homogeneous mechanisms. This yields a sufficient compact-support criterion expressed in terms of the inverse Keller integral. In particular, our results extend those of Sheu for super-Brownian motions with spatially constant branching mechanisms, and those of Engländer-Pinsky and Ren for super-Brownian motions with stable-like spatially dependent branching mechanisms.

Spectral sequences are a key tool for computing the K-theory of a crossed product C$^*$-algebra. However, the impact of a group homomorphism $\Omega\colon G \to H$ on such a spectral sequence was unknown until quite recently, even when $G = \mathbb Z^\ell$, $H = \mathbb Z^{k}.$ Recent work [Mil25] of the fourth-named author in the complex case establishes that ABC spectral sequences are functorial with respect to group homomorphisms. In this paper, we obtain the analogous result for real K-theory and for united K-theory. Specifically, we first show that the ABC spectral sequence approximates KO$_*(G \ltimes_r A)$ with the group homology H$_p(G;KO_q(A))$ when $G$ is a torsion-free discrete group satisfying the Baum--Connes conjecture with coefficients in $A$. Then, for a homomorphism $\Omega \colon G \to H$ of such groups with amenable kernel, and a real $H$-C$^*$-algebra $A$, we show moreover that the map in K-theory induced by the $*$-homomorphism $G \ltimes_r A \to H \ltimes_r A$ is approximated by the natural map in group homology.

An analytic Gentzen-style proof system of first-order Łukasiewicz logic, hypersequent calculus $\text{GŁ}\forall$, and its approximate completeness have been extensively studied. In this paper, we prove the approximate strong completeness of $\text{GŁ}\forall$ by a labelled tableau method. As applications, we prove a variant of Gentzen's midsequent theorem in $\text{GŁ}\forall$ and an approximate Herbrand's theorem. We also introduce a new cut rule $(\text{s-Cut})$ of $\text{GŁ}\forall$ and show the approximate strong completeness of $\text{GŁ}\forall+(\text{s-Cut})$.

We construct $h$- and $p$-robust, degree-preserving space decompositions and additive Schwarz preconditioners for variable-degree $hp$ finite element discretizations of reaction-diffusion and fitted-interface problems. On conforming simplicial meshes in arbitrary dimension, the single-domain result allows an arbitrary elementwise degree distribution subject only to $p_K\ge1$. A minimal-average Falk--Winther bubble transform is introduced by taking each subsimplex average over a fixed adjacent element of minimal polynomial degree. The resulting components remain in the prescribed variable-degree space and satisfy $L^2$- and $H^1$-stable estimates with constants independent of the mesh size, the polynomial degrees, and the way the degrees vary from element to element. Together with a stable continuous piecewise affine component, this yields an $hp$-uniform Schwarz preconditioner for single-domain reaction-diffusion problems with locally comparable coefficients. For three-dimensional fitted-interface problems, we use a symmetric Nitsche discretization on a tetrahedral mesh fitted to a piecewise planar interface. Surface jump components are lifted into the side selected by the penalty scaling, and the conforming remainder is decomposed by a weighted one-sided bubble transform. Grouping the components by vertices gives a practical vertex-patch Schwarz preconditioner. Under a common-degree condition on interface-touching tetrahedra, the condition number is bounded independently of the mesh size, the local polynomial degrees, the diffusion contrast, and the coefficient magnitudes. Numerical experiments for pure diffusion problems support the theory and suggest robustness beyond the common-degree assumption.

In this paper, we establish new connections between lattice graphs and metric grids, providing a unified framework for the study of singular limit problems and Gagliardo--Nirenberg type inequalities on lattice graphs. As applications, we first show that extensions of action ($2<p<2^*$) and energy ($2<p<2+\frac{4}{d}$) ground states of the nonlinear Schrödinger (NLS) equation on $d$-dimensional lattice graphs converge strongly in $H^1(\mathbb{R}^d)$ to the corresponding ground states on $\mathbb{R}^d$ as the edge length of lattice graphs tends to zero. As a by-product of the arguments developed for the singular limit problem of lattice graphs, we obtain multiplicity results for fixed-mass critical points of energy functional on lattice graphs. Furthermore, employing a strategy analogous to the analysis of the singular limit problem of lattice graphs, we investigate the optimal constants of Gagliardo--Nirenberg type inequalities on lattice graphs for $2<p<2^*$. A distinctive feature of this paper is that, going beyond the classical subcritical framework, we also establish novel results on the singular limit problem of action ($d \geq 3$ and $p>2^*$) and energy ($p>2+\frac{4}{d}$) ground states on lattice graphs, and on the optimal constants of Gagliardo--Nirenberg type inequalities for $d \geq 3$ and $p=2^*$ on lattice graphs, thereby substantially extending the existing literature. Notably, we settle an open problem posed by Dovetta [Adv. Math. 444 (2024), 109633] by establishing a new Gagliardo--Nirenberg type inequality.

Let $X \subset \mathbb{R}^2$ be a closed definable set of dimension at most $1$, and let $h : X \to \mathbb{R}^2$ be a definable continuous injective map. In this paper, we establish necessary and sufficient combinatorial conditions, formulated in terms of cyclic orders at topological singular points and orientations of Jordan curves, for $h$ to admit a definable homeomorphic extension to the whole plane.

We investigate the convergence properties of the family of operators $$ \mathcal{T}_{\bf N} f(x)=\sum_{j=N_1}^{N_2} v_j\Bigl( e^{-a_{j+1}(-\Delta_k)^s}f(x) - e^{-a_j(-\Delta_k)^s}f(x) \Bigr), \qquad x\in\mathbb{R}^d, $$ where $\{e^{-t(-\Delta_k)^s}\}_{t>0}$ denotes the fractional heat semigroup generated by the Dunkl Laplacian $\Delta_k$. Here ${\bf N}=(N_1,N_2)\in\mathbb{Z}^2$, $N_1<N_2$, the coefficients $\{v_j\}_{j\in\mathbb{Z}}$ form a bounded sequence of real numbers, and $\{a_j\}_{j\in\mathbb{Z}}$ is a monotone increasing sequence of reals. The primary objective of this work is to establish boundedness results for these differential transform operators on weighted $L^p(\mathbb{R}^d,d\mu_k)$ spaces as well as on Dunkl $\mathrm{BMO}(\mathbb{R}^d)$ spaces. We also establish analogous boundedness properties for the associated maximal operator $$ \mathcal{T}^* f(x)=\sup_{\bf N} |\mathcal{T}_{\bf N} f(x)| $$ and study the pointwise convergence of the corresponding series. In addition, we prove that, for compactly supported functions, the maximal differential transform operator $\mathcal{T}^*$ exhibits local behaviour comparable to that of classical singular integral operators.

The partial Petrial polynomial of a ribbon graph, introduced by Gross, Mansour and Tucker, enumerates partial Petrials by Euler genus. Recently, Deng, Jin and Yan defined an analogue for grafts and showed that it can be expressed as a rank-generating function of an adjacency matrix. In this paper we first prove a recurrence relation that reduces the partial Petrial polynomial of a graph with respect to an arbitrary edge, expressing it as a sum of three terms involving graphs obtained by local complementation and edge pivoting. This recurrence extends the known leaf-reduction formula to vertices of any positive degree. Second, using this recurrence we compare the lowest and highest degree coefficients of the polynomial. We prove that the lowest coefficient is always at most the highest coefficient, and that equality holds if and only if the graph has no edges.

We conjecture that on any compact balanced manifold $(M, \omega_B)$ with $c_{1}(M)=0$, the pluriclosed flow admits long-time solutions $\omega_{t}$ for every initial pluriclosed metric, and that $\omega_{t}$ converges smoothly to a Kähler metric as $t \to \infty$. We verify that this phenomenon occurs when $M$ is a compact quotient of a Lie group by a discrete subgroup, the background metric $\omega_{B}$ is invariant with vanishing Chern--Ricci form, and the initial metric $\omega_{0}$ is invariant. In particular, this provides new evidences for the Fino-Vezzoni conjecture.

The stochastic incompressible Navier-Stokes equations on $\TT^3$, completed by the fluctuation-dissipation noise, have a Fokker-Planck generator that decomposes into a self-adjoint Ornstein-Uhlenbeck (dissipative) part and an antisymmetric (convective) part. We prove two results about this generator. First, the logarithmic Sobolev inequality holds with the same optimal constant as the pure Ornstein-Uhlenbeck operator, $c_\mathrm{LSI} = \nu\lambda_1$ (where $\nu$ is the viscosity and $\lambda_1$ is the smallest nonzero eigenvalue of the Laplacian on $\TT^3$), independent of the number of retained Fourier modes. Second, the full semigroup is hypercontractive with the same rate as the Ornstein-Uhlenbeck semigroup. Both results follow from a single structural property: the convective generator is antisymmetric in $L^2(P_\mathrm{eq})$ (where $P_\mathrm{eq}$ is the Gibbs measure), and therefore contributes nothing to the Dirichlet form or the $L^q$ norm evolution. The antisymmetry is a consequence of two properties of the incompressible Navier-Stokes nonlinearity: energy conservation and phase-space volume preservation (the Liouville property). These are the same properties that underpin the fluctuation-dissipation theorem for the nonlinear Navier-Stokes equations.

In this paper, we study the noncommutative fractional symmetric Sobolev spaces on noncommutative torus. We prove noncommutative distributional fractional Sobolev inequality and as its application, we obtain Sobolev embeddings. In order to obtain these results, we first prove a noncommutative version of the famous O'Neil inequality for the convolution. As a first application of our main results, we obtain a Cwikel-Solomyak-type estimate. As an another application, we show a $L_2$-time decay for the mild solution of the Cauchy problem for the diffusion equation in this noncommutative setting. When $\theta=0,$ our results recover many known results on Sobolev embedding on the torus.

In this paper we prove a conjectured equivalence of categories, showing that the S-dual of $\mathrm{SO}_{2n}\times \mathrm{Sp}_{2n}$ acting on $\mathbb{C}_+^{2n}\otimes \mathbb{C}_-^{2n}$ is equal to $\mathrm{SO}_{2n+1}\times \mathrm{SO}_{2n}\circlearrowright T^*\mathrm{SO}_{2n+1}$. This result is a particular case of a non-polarized version of the (local) relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh. Similar results for the pairs $(\mathrm{SO}_{2n+1}, \mathrm{Sp}_{2n})$ and $(\mathrm{GL}_n, \mathrm{GL}_m)$ were proved by Braverman, Finkelberg, Kazhdan and Travkin and by Fu respectively, whereas the converse result was proved by Braverman, Finkelberg, and Travkin. As a consequence of our main result, we prove that Langlands functoriality of the Derived Satake isomorphism for the pair $\mathrm{Sp}_{2n},\mathrm{SO}_{2n}$ is given by the theta correspondence. Our approach works (with appropriate modifications) in the general even orthosymplectic case of $\mathfrak{osp}(2m|2n)$.

We introduce and study the model category of flexible $2$-categories, which is Quillen equivalent to Lack's model category of $2$-categories, but enjoys several excellent properties not shared by the latter. In particular, every object of this model category is cofibrant, it is a monoidal model category with respect to its cartesian closed structure, and its full subcategory of fibrant objects is equivalent to the category of bicategories and normal pseudofunctors.

This letter studies robust set-point regulation of scalar second-order nonlinear uncertain systems using a classical PID controller with constant gains. The scalar second-order model provides a minimal prototype for nonlinear mechanical and electromechanical dynamics, while its velocity-dependent term captures uncertainties such as physical damping and friction. For a positive velocity-derivative bound, existing Lyapunov sufficient conditions certify fixed-gain PID parameter regions that remain separated from the boundary associated with the necessary condition obtained from the worst-case linear model. To reduce this conservatism, this letter proposes an endpoint-balanced quadratic-plus-integral Lyapunov certificate. The key idea is to choose the quadratic cross-term coefficient so that the mixed-term penalty is balanced at the two endpoints of the admissible effective-damping interval before extracting the scalar PID inequality. The resulting condition guarantees global asymptotic regulation for the full derivative-bounded uncertainty class. When the velocity-derivative bound is positive, the proposed condition certifies a fixed-gain PID region that strictly contains those certified by Zhao--Guo and Zhang--Guo. When this bound is zero, the corresponding boundary coincides with that necessary boundary. At the level of Lyapunov analysis, the construction reduces the uniform mixed-term penalty over the entire effective-damping interval.

Efforts have been made to extend Cassini's identity (also known as Simson's identity) to the k-step or k-bonacci numbers for decades. These efforts have lacked both completeness of result and simplicity of proof, and this question remains open and relevant. In this note, we offer a definitive solution as well as the generalization of both Catalan's and Vajda's identities.

In this paper, a special chain representation of real numbers on a fixed interval is considered, where the elements of the expansion can take only one of two possible values. For this encoding system, a problem in metric number theory and dynamical systems is solved, which is a direct analogue of the classical Gauss problem for simple continued fractions. Specifically, the asymptotic behavior of the Lebesgue measure for a special class of sets is investigated. These sets are formed by those numbers for which the remainder (or infinite tail) of their chain expansion, after discarding the first few elements, is strictly less than a predetermined value.

Pinching antennas are characterized by their capability to create strong line-of-sight connections and realize multi-antenna systems in a flexible manner. Existing works have demonstrated the significant potential of pinching antennas for physical layer design. The aim of this paper is to investigate how pinching antennas can be used to reshape the architecture of future networks. In particular, this paper is motivated by the key advantage of pinching antennas, which is to reconfigure the physical boundaries of wireless cells, and focuses on the impact of pinching antennas on traffic offloading. The models for traffic offloading and pinching antenna transmission are presented first. Then, two traffic offloading strategies are developed based on whether an offloading user releases its bandwidth in its original cell. An overall transmit power minimization problem is formulated, where the optimal solutions for the transmit powers and antenna locations are obtained. The presented simulation results demonstrate that the use of pinching antennas can efficiently support traffic offloading, yield low energy consumption, and achieve balanced cell resource utilization.

In this paper, we consider triangle-free graphs with diameter 2. If a triangle-free graph $G$ with diameter 2 is not isomorphic to a star, then the radius of $G$ is also 2, where such a graph is also called a $2$-self-centered graph. Shekarriz et al. [A characterization for 2-self-centered graphs, Discuss. Math. Graph Theory 38 (2018), 27--37.] gave a characterization of 2-self-centered graphs. However, there is a slight flaw in their characterization. Thus, in this paper, we modify it and prove an accurate characterization of those graphs. Furthermore, by using our characterization, we prove some results concerning the chromatic number of triangle-free graphs with diameter 2.

A Hessian-recovery-based C0 finite element framework is proposed for second-order elliptic equations in non-divergence form. The construction is based on a direct approximation of the strong non-divergence operator: the Hessian D2u is replaced by a recovered Hessian Hhuh, so that A : D2u is approximated by A : Hhuh. The resulting discretizations include a nodal formulation and a Galerkin-type formulation for general Lagrange finite element spaces, as well as a biorthogonal Petrov-Galerkin formulation for linear elements. The analysis focuses on the recovered nodal matrix and identifies two verifiable algebraic solvability mechanisms. The first is a globally monotone regime leading to a discrete maximum principle, and the second is a localized Schur-complement criterion for sign-violating rows. A uniform inverse bound and a condition-number estimate are derived in the globally monotone case. Residual consistency estimates are obtained from the Hessian recovery error. In the globally monotone regime, these estimates combine with the uniform inverse bound to give a nodal L-error estimate for the nodal formulation. Numerical experiments with nonsmooth and discontinuous coefficients support the predicted algebraic diagnostics and show the accuracy of the proposed recovered-residual discretizations. A Monge-Ampere type test further illustrates the use of the recovered Hessian in a Newton iteration for a fully nonlinear problem.

This paper investigates an equality-coupled distributed resource allocation problem with smooth general convex local objective functions. A discrete-time residual-aware dynamic event-triggered algorithm is proposed over time-varying switching undirected graphs. Unlike existing event-triggered resource allocation algorithms that rely on strong convexity, the proposed method establishes convergence for general convex costs without using strong monotonicity or contraction arguments. The key idea is to co-design a resource-allocation search recursion with a dynamic triggering rule that incorporates both local gradient-estimation errors and local gradient-disagreement residuals. The resulting triggering mechanism reduces unnecessary communication and generates a summable error bound, which is embedded into a Mirror-EXTRA-type Lyapunov analysis. Under suitable step-size conditions, the proposed algorithm is proved to converge to an optimal solution. Further, when the local objective functions are strongly convex, a linear convergence result is established. Numerical simulations and comparative tests with related event-triggered methods verify the effectiveness and communication efficiency of the proposed algorithm.

A pair $(x,y)$ of vectors in a Banach space $\mathcal{Y}$ is said to be a triangle equality attaining (or TEA) pair if $\|x+ y\|=\|x\|+\|y\|,$ and a parallel pair if $\|x+\lambda y\|=\|x\|+\|y\|$ holds for some unimodular scalar $\lambda.$ In this article, we explore bounded linear maps $T:\ell_p\to \mathcal{Y}$ preserving parallel and TEA pairs. For $p\in(1,\infty)$ all linear maps trivially preserve parallel pairs. We prove that for $p=\infty,$ if $\ker(T)\neq \{0\},$ then $T$ preserves parallel pairs if and only if $rank(T)\leq 1$. %In particular, $T:\ell_p\to \ell_1$ preserves parallel pairs if and only if $rank(T)\leq 1,$ and TEA pairs if and only if $T=0$. In particular, $T:\ell_\infty\to \ell_1$ preserves parallel (resp. TEA) pairs if and only if $rank(T)\leq 1$ (resp. $T=0$). Analogous characterizations hold if $T$ is defined from $\ell_\infty^n,$ except when $n=2$ and the field is real. In this specific setting, we further characterize such maps $T:\ell_\infty^2\to \ell_1.$ \\ Focusing on $p=1$, we establish a necessary condition for the preservation of parallel pairs. Specifically, we characterize invertible parallel pair preservers $T:\ell_1^n\to\ell_\infty^n$, as well as the general class of such maps $T:\ell_1^2 \to \ell_\infty^m,$ revealing the intricate structure inherent to these mappings. Furthermore, we prove that $(0\neq)~T:\ell_1\to \mathcal{Y}$ preserves TEA pairs if and only if $\Lambda=\{i\in \mathbb{N}:Te_i\neq 0\}$ is singleton, where $\mathcal{Y}$ is either strictly convex or $\ell_\infty^m$ over the complex field. Finally we characterize the TEA pair preservers $T:\ell_1^2\to \ell_\infty^m$ over the real field.

Johnson graph are a family of graphs that play an important role in the theory of constant-weight codes, extremal combinatorics, and combinatorial geometry. We study signed analogues of classical Johnson graphs, denoted by $J_\pm(n,k)$, whose vertices are vectors of the form $\pm e_{i_1}\pm\cdots\pm e_{i_k}$, where two vertices are adjacent whenever their dot product equals $k-1$. We are particularly interested in maximum independent sets in the case $k=4$. An example of such an independent set in $J_\pm(n,4)$, which we call \emph{classical}, is obtained by lifting an arbitrary optimal $(n,4,4)$-code. Such independent sets naturally define kissing arrangements in ${\mathbb R}^n$. We develop an algorithm that is practical for computing all maximum independent sets in $J_\pm(n,4)$ up to signed permutations for $n\le 12$, $n\ne 11$. In addition to obtaining complete lists, we provide structural characterizations of all types of maximum independent sets in these dimensions, excluding $n=5$ and $n=11$. Our most striking results concern the case $n=12$. We identify $1579$ non-isomorphic maximum independent sets in $J_\pm(12,4)$, all corresponding to non-isometric kissing arrangements of size $840$ in ${\mathbb R}^{12}$. Structurally, $1575$ of these independent sets arise from three different constructions, the rest are liftings of one of four $(12,4,4)$-codes. To our knowledge, this is the first dimension in which such a large diversity of potentially optimal kissing arrangements has been observed. Beyond this finite range, we prove that for $n\equiv 2$ or $4 \pmod 6$, every maximum independent set arises from a Steiner quadruple system. We also obtain a characterization of the so-called \emph{nontrivially self-compatible} codes, namely optimal $(n,4,4)$-codes from which non-classical maximum independent sets can be constructed.

In this paper, we study the class \Sigma_{H}^{k}(p) of sense-preserving univalent harmonic mappings in the unit disk \mathbb{D} that possess a simple pole at p\in[0,1) and admit a k-quasiconformal extension to the extended complex plane for k\in[0,1). In 2024, Bhowmik and Satpati established an area theorem and derived a sufficient condition for the k-quasiconformal extension of harmonic mappings belonging to \Sigma_{H}^{k}(p) without logarithmic terms. Motivated by their work, we investigate the corresponding problem when a logarithmic singularity is present. Our main contributions are two-fold: we first prove a generalized area theorem for all mappings in \Sigma_{H}^{k}(p); we then obtain a sufficient condition for sense-preserving univalent harmonic mappings in \mathbb{D} to admit explicit k-quasiconformal extensions. These results extend the aforementioned work to the setting where logarithmic singularities are allowed.

A novel detection method based on maximum-likelihood (ML) detection leveraging Mahalanobis distance is proposed for single-radio-frequency (RF) multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. It can enhance bit error rate (BER) performance and is based on the observation that when using reconfigurable antennas (such as electronically steerable parasitic array radiators (ESPARs) to create a single-RF MIMO system, an additional model error arising from the reconfigurable antennas is introduced. These modeling errors produce an irreducible BER (error floor) at high signal-to-noise ratios (SNRs). Simulation results, using ESPAR as an example, validate our error floor analysis and demonstrate that our proposed enhanced detection method can effectively address the error floor and reduce the BER at high transmit SNRs.

This paper proposes a stochastic control framework for the operation of electric-vehicle (EV) charging stations equipped with on-site photovoltaic (PV) generation and battery storage. To preserve scalability for large fleets, vehicles are aggregated into a finite number of classes according to their residual charging demand, yielding a compact state description and avoiding vehicle-level optimization. The charging-station dynamics are modeled in discrete time and capture stochastic arrivals, charging-induced class transitions, stochastic departures, PV generation, battery operation, and power exchange with the grid. Based on the corresponding expectation model, we formulate a finite-horizon smart-charging problem that jointly optimizes class-wise charging actions and energy-management variables to balance electricity-purchase cost and transitions toward lower residual-demand classes. The resulting stage problem is a linear program, solved in shrinking-horizon form and implemented online after integer discretization of the first charging action. We also derive a robust counterpart that preserves feasibility under interval uncertainty on the first moments of arrivals, departures, and PV generation. To validate scalability and robustness, we conduct an extensive simulation campaign across nine configurations, combining three real-world electricity price patterns with three EV-arrival profiles. Numerical results show that, at comparable but slightly lower service levels, the proposed controller reduces the cost per kWh by up to 17.5% relative to a service-greedy First-In-First-Out baseline; the total daily cost decreases even more substantially because the controller avoids economically unattractive charging. Overall, the results demonstrate a tunable Pareto trade-off between economic efficiency and charging-service quality for large-scale charging hubs.

We prove an upper bound for the number of subgroups of an abelian $p$-group, with constants that are sharp on homocyclic blocks. If $A_\lambda$ has type $\lambda$, the main exponent is $\eta(\lambda)=\sum_i\lfloor(\lambda'_i)^2/4\rfloor$, where the parts of $\lambda'$ are the column heights of the Ferrers diagram of $\lambda$. The leading coefficient is the number of central choices in the blocks of columns of odd height. The proof starts from Delsarte's formula, separates the blocks by column height, and estimates each homocyclic block by successive Durfee squares. We also give three applications: a comparison theorem for $p$-groups of nilpotency class less than $p$, diagonal summatory estimates with the shape fixed, and formulae for leading terms at fixed rank for finite abelian $p$-groups.

We compute an upper bound on the ground state energy of a dilute two-dimensional Fermi gas with repulsive short-range interactions. Our bound can be viewed as the two-dimensional analogue of a formula derived by Huang and Yang in the three-dimensional case. It captures the first three terms in an asymptotic expansion for small $\varrho a^2$, where $\varrho$ denotes the density and $a$ the scattering length of the interaction potential.

Diffusion-based generative models have achieved remarkable success in high-dimensional data generation; however, they fundamentally rely on isotropic diffusion processes that destroy meaningful geometric structures in the forward process. For complex, multimodal, and highly correlated distributions such as biologically constrained genetic data, isotropic noise merges distinct modes and distorts intrinsic dependencies. This forces the reverse process to recover structure from heavily degraded signals, leading to slow convergence, mode averaging, and biologically implausible samples. To address this, we introduce the Geometry-aware Ornstein-Uhlenbeck (GOU) process, a structured drift design that embeds data geometry into forward and backward dynamics. By employing a variance-aware anisotropic drift, GOU contracts low-variance directions rapidly while preserving high-variance directions longer, maintaining key multimodal structures as stable channels over time. Crucially, we show that GOU's backward initialization error is governed by local rather than global variance. This geometry-adaptive initialization improves convergence rates by reducing initial mismatch and preserving cluster-level structures. Synthetic and real-world genetic experiments demonstrate that GOU significantly improves mode separation, correlation preservation, and statistical validity over standard isotropic models.

Let $\mathcal{I}$ be a $\sigma$-ideal on the Cantor space $2^{\omega}$. A set $X\subseteq 2^\omega$ is called everywhere $\mathcal{I}$ if for any $S\in [\omega]^\omega$ the set $X\!\restriction\! S\in\mathcal{I}(2^S).$ We will discuss relations between such families in the class of perfect and Borel sets.

We propose a shifted-symplectic formulation of a classical continuous analogue of the symmetry TFT paradigm. Let $G$ be an algebraic or Lie group acting by topological defects on an $n$-dimensional classical topological sigma model with target an $(n-1)$-shifted symplectic derived stack $(X,\omega)$ via the AKSZ construction. We argue that the corresponding $(n+1)$-dimensional bulk theory should be the AKSZ theory with target the shifted cotangent stack $T^*[n] (\mathrm B G)$, equivalently the $(n+1)$-dimensional BF theory for $G$. We characterize the Dirichlet and Neumann boundary conditions, and more general topological boundaries, in terms of shifted Lagrangians in $T^*[n] (\mathrm B G)$. We realize the gauging of the $G$-symmetry in the original theory as inserting a topological domain wall between the corresponding topological boundaries in the BF bulk, and introduce the notion of Hamiltonian reduction, syplectic reduction, and Lagrangian reduction in the shifted symplectic setting. We also discuss prequantum refinements of continuous SymTFTs. In this refinement, higher gerbes on $\mathrm B G$ encode classical analogues of 't Hooft anomaly data by decorating the shifted cotangent bulk and its Lagrangian boundary conditions. Finally, in dimension three we compare the infinitesimal BF model $\mathrm B(\mathfrak g\ltimes\mathfrak g^\vee)$ with the factorizable double $\mathrm B(\mathfrak g\oplus \mathfrak g)$. The resulting topological boundaries are described by Lagrangian Lie subalgebras, and the factorizable case relates the SymTFT dictionary to $r$-matrices and Belavin--Drinfeld data.

We construct a natural Legendrian cycle $\mathcal{N}_\mathcal{S}$ associated with any $F_nW^{2,n}$-set $\mathcal{S}$, that is, a closed set locally described as a finite union of graphs of $(C^0\cap W^{2,n})$-regular functions with integer multiplicity. The construction relies on the fact that, at $\mathcal{H}^n$-almost every point where two or more graphs overlap, their normals coincide up to sign. As a consequence, we prove Reilly-type variational formulae for the higher-order mean curvature integrals of $\mathcal{S}$, extending the classical results of Reilly to this non-smooth setting.

We study integrability of generalized almost complex structures on the six-dimensional sphere $\mathbb{S}^6$. Two notions of integrability are considered: integrability with respect to brackets determined by an affine connection $[,]_\nabla$ (in particular the Levi-Civita connection), and the Courant integrability for strong generalized almost complex structures. After recalling the necessary background on the generalized tangent bundle and on spherical combinations of the canonical generalized structures determined by an almost Hermitian triple $(J,g,\omega)$, we derive local coordinate criteria for $[,]_\nabla$-integrability of weak generalized structures. Applying these formulae to the nearly Kähler structure on S^6 induced by the octonionic product, we prove that no nontrivial spherical combinations $J=aJ_{1,J} + bJ_g + cJ_\omega$ with smooth coefficients such that $a^2+b^2+c^2=1$ (except $J_g$) is integrable with respect to $[,]_{\nabla^{LC}}$. We then turn to Courant integrability: we give sufficient local conditions for Courant integrability of strong generalized almost complex structures, prove a gluing result for local Courant algebroids and b-field transforms, and use it to exhibit obstruction results characterizing the impossibility of constructing, via certain gluing procedures, a Courant integrable strong generalized almost complex structures on $\mathbb{S}^6$.

We investigate a recursive variant of the classical Frank-Wolfe algorithm for minimizing a convex differentiable function over a convex compact domain. Unlike the traditional setting, we assume that both the problem domain and the objective function are initially unknown and must be learned from data. To address this, we integrate estimators into the optimization process, allowing the algorithm to iteratively refine approximations of the problem domain and the objective function. Our approach maintains the projection-free nature of the classical Frank-Wolfe algorithm while adapting to the uncertainty inherent in data-driven settings. We establish convergence guarantees for the recursive method, showing that the optimization error scales with the accuracy of the learned estimators. Two experiments support our theoretical findings, demonstrating that the proposed method achieves convergence behavior comparable to that of the classical Frank-Wolfe algorithm under exact knowledge of the problem domain and objective function, while offering significant computational savings.

Stone demonstrated that the category $\mathbf{DLATT_{0,1}}$ of bounded distributive lattices is dually equivalent to the category $\mathbf{Spec}$ of spectral spaces and Priestley showed that $\mathbf{DLatt_{0,1}}$ is dually equivalent to the category $\mathbf{Priest}$ of Priestley spaces so that $\mathbf{Spec}$ is equivalent $\mathbf{Priest}$. Cornish strengthened this by showing that $\mathbf{Spec}$ and $\mathbf{Priest}$ are in fact isomorphic. In this study, we investigate the duality theory of various lattice expansions of certain bounded distributive lattice-ordered algebras, known as De Morgan algebras. In particular we obtain spectral duality results for the category $\mathbf{S4DM}$ of De Morgan algebras equipped with a closure operator, which we call S4 De Morgan algebras, as well as for the category $\mathbf{DMGrp}$ of De Morgan groupoids. This is achieved by an appropriate adaptation of Bimbó's Priestley-style duality for general De Morgan algebras together with Urquhart's Priestley-style duality for relevance algebras under the isomorphism between $\mathbf{Priest}$ and $\mathbf{Spec}$.

Implication algebras were introduced by Abbott as algebraic models of the operation of Boolean implication in the classical propositional calculus. In this work, we study additional operators and constants on bounded implication algebras by introducing monadic and cylindric implication algebras. It is demonstrated that the category $\mathbf{MIA}$ of monadic implication algebras is isomorphic to the category $\mathbf{MBA}$ of monadic Boolean algebras and moreover, that the category $\mathbf{CIA}$ of $I$-dimensional cylindric implication algebras is isomorphic to the category $\mathbf{CBA}$ of $I$-dimensional cylindric Boolean algebras. As an application of the obtained categorical isomorphisms, we provide spectral duality results for $I$-dimensional cylindric implication algebras along the lines of Bezhanishvili and Holliday's spectral duality for Boolean algebras combined with McDonald's extension of their duality to monadic and $I$-dimensional cylindric Boolean algebras.

The 2026 disproof of Erdős's unit-distance conjecture and Sawin's subsequent explicit quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes finite parameters whose choice is not fully optimized. This report formulates the finite parameter-selection task as a variant of a nonlinear integer programming problem and proposes an open-source Python verification pipeline, first validated by reproducing Sawin's published parameter choice and then applied to computationally improved certificates. The main computational contribution is an integer optimization and checking procedure for the sets of primes $T$ and $S_Q$, the integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. The optimization pipelines are intentionally lightweight and replicable on standard hardware: we propose a deterministic greedy construction heuristic, a Tailored Integer Evolution Strategy with repair operators for number-theoretic feasibility, and a two-parent discrete-recombination variant. Four certificate levels are compared: Sawin's published example with $\delta=0.0141144286784982\ldots$, a greedy optimization certificate with $\delta=0.0151718056372133\ldots$, a Tailored Integer Evolution Strategy certificate with rational $R=6672416/100000$ and $\delta=0.0152616610684193\ldots$, and a Tailored Integer Evolution Strategy with discrete recombination, again with $R=6672416/100000$, giving $\delta=0.0152628688170072\ldots$. Consequently, subject to Sawin's explicit criterion being applied exactly as cited, the best current certificate supports the cautious clean statement $u(n)>n^{1.0152}$ for arbitrarily large $n$.

We prove that the Poisson semigroup on free group von Neumann algebras satisfies hypercontractivity with optimal time.

Bondal's conjecture in Poisson geometry gives lower bounds on the degeneracy loci of Poisson Fano manifolds, where the rank of the Poisson structure drops. By work of several authors, it was previously known to hold for Fano manifolds of dimension at most four. We give the first proof of this conjecture for Fano manifolds of dimension five, and partial results for Fano manifolds of all odd dimensions. The proof uses: (i) an algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, extending a previous result of the first author; (ii) the "modular residues" of Poisson structures introduced by Gualtieri and the third author; and (iii) a cohomological constraint on invariant subvarieties for Pfaff fields, extending earlier results of Esteves--Kleiman to the case in which the Pfaff distribution on the subvariety admits a closed strongly directed positive current.

In this article, we analyze how (projective and injective) complexity, curvature, and complete intersection dimension behave under linkage of modules and ideals. Let $R$ be a Gorenstein local ring. Consider a Gorenstein perfect ideal $\mathfrak{a}$ (e.g., $\mathfrak{a}$ is generated by an $R$-regular sequence). Let $M$ and $N$ be two Cohen-Macaulay $R$-modules linked by $\mathfrak{a}$. We prove that $\mathrm{cx}_R(M)= \mathrm{inj\,cx}_R(N)$ and $\mathrm{curv}_R(M)= \mathrm{inj\,curv}_R(N)$. In particular, when $R$ is complete intersection, $\mathrm{cx}_R(M)= \mathrm{cx}_R(N)$ and $\mathrm{curv}_R(M)= \mathrm{curv}_R(N)$. Furthermore, we show that $\mathrm{pd}_R(M)= \mathrm{pd}_R(N)$ and $\operatorname{CI-dim}_R(M)= \operatorname{CI-dim}_R(N)$. If any of these dimensions is finite, it is equal to $\mathrm{ht}(\mathfrak{a})$. Similar results are obtained for linkage of ideals. All these results highly extend a classical result of Peskine and Szpiro in many directions. We construct several examples that complement our results. These also show how properties like `integrally closed', `$\mathfrak{m}$-full' and `Burch' behave under linkage of ideals.

In this paper we give estimate of $|S'_{f}(0)|$, where $S_{f}(z)= \left(\frac{f''(z)}{f'(z)}\right)'-\frac{1}{2}\left(\frac{f''(z)}{f'(z)}\right)^{2}$ is the Schwartian derivative, and $f$ belongs to different classes of functions univalent in the open unit disc $\ID$.

A long-standing question is whether chromatic symmetric functions can distinguish non-isomorphic trees. Campbell introduced chromatic noncommutative symmetric functions for digraphs, which lift chromatic symmetric functions to NSym, and asked to what extent they can distinguish non-isomorphic oriented trees. In this article, we prove that chromatic noncommutative symmetric functions can reconstruct oriented stars, oriented double stars, some oriented caterpillars, and some oriented paths.

The paper consists of a brief survey of the use of Stein's method in network analysis. Topics covered include normal and Poisson approximation of graph--based statistics, approximating an exponential random graph by a Bernoulli model, and comparison of different random geometric graph models.

A primary method for constructing orthogonal polynomials on the unit ball consists of combining a Jacobi-type radial component with a spherical harmonic angular part. Building upon this framework and using Jacobi-Piñeiro multiple orthogonal polynomials, this paper introduces Type I and Type II multiple orthogonal polynomials on the multidimensional ball. To demonstrate the practical utility of these definitions, we establish multivariate extensions of several fundamental results from univariate multiple orthogonality. Finally, we extend the construction to more general domains by introducing multiple orthogonality with respect to radial weights.

We employ the recently developed Kac-Rice formalism for non-Hermitian random matrices to derive the joint distribution of an eigenvalue and its associated normalised right eigenvector in a Gaussian ensemble that interpolates between complex Ginibre (Class A) and complex symmetric matrices (Class AI$^\dagger$). This distribution is valid at finite matrix size, $N$, for any value of the interpolation parameter $\sigma \in [0,1]$, with $0$ and $1$ corresponding to classes A and AI$^\dagger$ respectively. The marginal distribution for the density of the eigenvalues is derived at finite $N$ and then considered asymptotically as $N \to \infty$. When considering bulk eigenvalues, we recover the standard circular law for all $\sigma$. Furthermore, for edge eigenvalues we find that for fixed $\sigma$, the eigenvalues follow the edge density associated with matrices in Class A. However, a transitional regime is discovered for the interpolation parameter being scaled as $\sigma = 1 - \kappa N^{-1/2}$, where new edge behaviour is observed for the density of eigenvalues - smoothly interpolating two previously known results. This transitional regime and the associated density of eigenvalues is conjectured to be universal for non-Gaussian matrices and we provide numerical evidence in support of this.

In this paper, we present a complete characterization of solvable symplectic Lie algebras via a symplectic double extension process. We demonstrate that any such algebra is either symplectically irreducible or can be constructed through a finite sequence of symplectic double extensions by a line or a plane, starting from symplectically irreducible Lie algebras. Furthermore, we show that if a symplectic Lie algebra has a nondegenerate derived ideal, then it is necessarily unimodular and, in particular, solvable. Finally, we present a novel algebraic proof of a classical structural theorem on symplectically irreducible symplectic Lie algebras and classify all Lie algebras of dimension up to $6$ that admit such structures.

Let $\mathbb D$ be a division ring, and let $\mathcal{R}\subseteq M_n(\mathbb{D})$ be a structural matrix ring over $\mathbb{D}$, that is, the subring of $M_n(\mathbb{D})$ supported on the ordered pairs of a preorder on $\{1,\ldots,n\}$. We study injective Jordan semi-triple maps $\phi:\mathcal{R}\to M_n(\mathbb{D})$, namely injective maps satisfying \[ \phi(XYX)=\phi(X)\phi(Y)\phi(X), \qquad \text{for all } X,Y\in\mathcal{R}. \] Assuming that the centre of $\mathbb{D}$ has more than two elements, we give a criterion for automatic additivity and show that there are exactly two obstructions. The first one is scalar: it occurs precisely when $\mathcal{R}$ has a direct ring summand isomorphic to $\mathbb{D}$ and $\mathbb{D}$ is isomorphic to neither $\mathbb{F}_3$ nor $\mathbb{F}_4$. The second one is order-theoretic: it occurs when a nonsymmetric comparable pair $i\preceq j$, $j\not\preceq i$, admits no third index $k\notin\{i,j\}$ comparable with both $i$ and $j$. If neither obstruction occurs, all injective Jordan semi-triple maps are additive. The centre-size hypothesis is sharp: for $n\ge3$, the upper-triangular ring $T_n(\mathbb{F}_2)$ has neither obstruction but nevertheless admits nonadditive injective Jordan semi-triple maps. Finally, in the additive case, we describe the maps componentwise, in terms of endomorphisms, anti-endomorphisms, and transitive multipliers.

Physics-informed operator learning provides an efficient framework for approximating solution operators of partial differential equations by combining observational data with governing physical laws. However, most existing methods implicitly assume that the prescribed governing equation is accurate. This assumption may fail in practical applications, where model simplifications, missing physical effects, parameter drift, or incomplete constitutive relations can lead to model misspecification. In this work, we propose a physics-guided operator correction framework for learning solution operators under misspecified governing equations. At the operator level, the target mapping is decomposed into a prior operator induced by an approximate physical model and a learnable correction operator that accounts for the remaining discrepancy. Although the formulation is architecture independent, we realize it using a serial DeepONet architecture, where the first DeepONet provides a prior prediction and the second DeepONet learns an additive correction conditioned on both the input function and the prior prediction. The learned correction is incorporated into the physics residual and trained together with data-consistency constraints, allowing the model to retain useful physical structure while adapting to inaccurate governing equations. Numerical experiments on diffusion-reaction, Burgers, cavity flow, and hyperelastic problems show that the proposed method substantially reduces errors induced by misspecified physics. Additional tests under sparse and noisy observations further demonstrate the robustness of the framework and its ability to provide informative uncertainty estimates through deep ensembles.

Let $Q=-x_1^1-x_2^2-x_3^2+x_4^2$ be the standard signature $(1,3)$ quadratic form. To each non-degenerate rational plane $L$ in the four-dimensional quadratic space $(\mathbb{Q}^4,Q)$ we can naturally attach a periodic geodesic on the Bianchi orbifold $\mathrm{SL}_2(\mathbb{Z}[i])\backslash \mathbb{H}^3$ which records the position of $L$ in the Grassmannian up to integer rotations. Moreover, each such plane $L$ defines a CM point and a periodic geodesic on the modular curve through restriction of $Q$ to $L$ and its orthogonal complement. Lastly, the local isomorphism between $\mathrm{SO}_{1,3}(\mathbb{R})$ and $\mathrm{SL}_2(\mathbb{C})$ gives rise to a further periodic geodesic on the Bianchi orbifold. In this article, we exhibit a natural coupling of all the above objects and prove simultaneous equidistribution under a Linnik-type splitting condition. The main ingredient is the classification of joinings of higher-rank diagonalizable actions on homogeneous spaces due to Einsiedler and Lindenstrauss.

Regev and Stephens-Davidowitz conjectured that the integer lattice maximizes Gaussian mass among integral lattices of a given rank. We prove this, including the equality case, for all unimodular Construction-A lattices arising from binary self-dual codes. The proof reduces the theta-series inequality to a sharp majorization statement for codes: if $C$ is a binary self-dual $[2k,k]$ code, then the half-weight distribution of $C$ is dominated in convex order by $\operatorname{Bin}(k,1/2)$, which is the corresponding distribution for the repetition-code model of $\mathbb{Z}^{2k}$. Indeed, after putting $C$ in systematic form $[I\mid A]$, self-duality gives $AA^T=I$ over $\mathbb{F}_2$, so for a uniformly random message $a$ the two weights $\wt(a)$ and $\wt(aA)$ have the same binomial law. The half-weight of the resulting codeword is their average, and Jensen's inequality then gives convex-order domination. Applied to the convex test functions that build the theta series, this yields a sum-of-squares formula for the Gaussian-mass gap; applied to hinge functions, it gives coefficientwise nonnegativity of the reduced gap polynomial.

Recall that a module is called weakly uniserial if its submodules are comparable regarding embedding. Weakly uniserial modules are a nontrivial generalization of uniserial modules. In this paper we define and study a new dimension, which measure how far a module deviates from being weakly uniserial. We call this dimension, weakly uniserial dimension. Also, we define and study monoartinian (mononoetherian) modules. We say that an $R$-module $M$ is monoartinian (mononoetherian) if in every descending (ascending) chain of submodules of $M$, except probably a finite number, each module in chain embedded in the next (previous) one. We show that a module has weakly uniserial dimension if and only if it is monoartonian.

This paper investigates the secrecy sum-rate (SSR) performance of optical intelligent reflecting surface (OIRS)-assisted multi-user visible light communication (VLC) systems under line-of-sight (LoS) blockages. To mitigate physical obstructions and internal eavesdropping, a joint optimization problem is formulated to maximize the SSR through the co-design of the transmission precoder and OIRS units assignment. Due to the binary constraints and coupled variables, the problem is highly non-convex. To solve it efficiently, an alternating optimization (AO) framework integrating the concave-convex procedure (CCCP) and first-order Taylor approximations is developed. Simulation results demonstrate the convergence of the proposed algorithm and show that increasing the number of OIRS reflecting units yields significant SSR gains.

Long-time simulations of evolution equations present substantial computational challenges due to the inherently sequential nature of conventional time-stepping schemes. The parareal method, a leading parallel-in-time (PinT) algorithm, offers a promising approach to overcome the challenge by introducing concurrency in the time domain. While its convergence theory is well-established for linear problems, extending the theory to nonlinear problems, particularly when the problem data have only limited regularity, remains a significant challenge. In this work, we provide the convergence analysis of the parareal algorithm for solving semilinear parabolic equations with an $H^2$ initial data. We employ stable rational approximations and first-order linearization as coarse propagators, establish the linear convergence of the parareal algorithm and provide a sharp estimate for the convergence factor. The analysis combines the error-splitting technique from the superlinear convergence analysis of the parareal method, a refined linear convergence theory for linear parabolic equations, and \textsl{a priori} error estimates that are optimal with respect to the regularity of the problem data. The analysis shows the close connection between the convergence behavior of nonlinear models and their linear counterparts. Numerical experiments fully support the theoretical findings.

We study finite extremal problems for palettes, which arise from the palette framework for the uniform Turán densities of $3$-uniform hypergraphs. Recent work has developed reductions from palette colorability questions to extremal problems for digraphs. In this paper we prove an exact degree-square refinement of these reductions for a natural family of left and right tournament palettes. For a tournament $T$, let $P_T^L$ and $P_T^R$ denote the left and right palettes generated by $T$. We prove that if $T$ is self-converse and has at least two vertices, then for every $m\ge 1$ the maximum number of admissible triples in an $m$-color palette avoiding both $P_T^L$ and $P_T^R$ is \[ \operatorname{ex}_2^+(m,T) = \max\left\{ \sum_{v\in V(D)} d_D^+(v)^2: |V(D)|=m,\; D\text{ is }T\text{-free} \right\}. \] The proof attaches two auxiliary digraphs to each palette and converts the palette optimization into a degree-square Turán problem. We also prove a general majorization principle for convex out-degree moments in $F$-free digraphs. Whenever an ordinary Turán extremal construction has extremal initial segments, the same construction maximizes every nondecreasing convex function of the out-degree sequence. Applying this to the Brown--Harary and Zhou--Li extremal digraphs for directed cycles gives exact formulas for all convex out-degree moments in $\overrightarrow{C}_{\ell}$-free digraphs. In particular, $\operatorname{ex}_2^+(m,\overrightarrow{C}_{3}) =\frac{m(m^2-1)}{3}.$ Consequently, for $m$ color the sharp density avoiding the two cyclic-triangle palettes is $\frac13-\frac1{3m^2}.$ Combining this exact finite endpoint with the palette classification theorem, we obtain finite $3$-graphs $H_m$ satisfying \[ \frac13-\frac1{3m^2} \le \pi_{\mathrm u}(H_m) \le \frac13. \] Thus the densities of these finite hypergraphs converge to $\frac13$.

In this paper, we introduce a new type of ring, called quasi-nil clean ring, where each element of the ring is the sum of a quasi-idempotent and a nilpotent element. We also investigate a particular class of ring, called strongly quasi-nil clean ring whose each element is the sum of a quasi-idempotent and a nilpotent element, where they commute. Our primary objective is to study the structural properties of these new class of rings, explore their relationships with existing classes of rings and establish some key characterizations. In the commutative setting, we provide a complete characterization in terms of quasi-Boolean quotients. Moreover we discuss about quasi-cleanness of amalgamated algebra and group ring.

Neurons receive information through their dendrites. During development, when synaptic connections are forming, dendrites grow, retract, and branch. The resulting dendritic tree shapes the structure of the broader neural network. Crucially, retraction and branching make it necessary to track whole dendritic paths rather than only their endpoints. While this is handled implicitly in some existing simulations, here we construct an explicitly path-valued stochastic process for dendrite growth. Combining this with a branching process, using ideas from measure-valued branching particle systems, we show that the model produces the typical tree structures of real dendrites. To complement this analytical work, we also outline several methods for numerical simulation, including time discretisations at different temporal scales and an approximation using a dynamic graph. This provides both a more rigorous mathematical framework and more structured simulation methods for modelling dendrite development.

We study a general class of non-selfadjoint indefinite Sturm--Liouville problems of the form $$ -(p\,y')' + q\,y = \lambda\, p\, y, $$ on a finite interval with complex-valued coefficients, where $p$ is piecewise in $W^{2,\infty}$, non-vanishing, and satisfies a non-degenerate interface condition, and $q$ is bounded. We prove that all eigenvalues in the open left half-plane are contained in a bounded set, which, by classical Sturm--Liouville theory, implies their finiteness. A prominent instance of this class arises in the lamellar grating diffraction problem with transverse-magnetic (TM) polarization, where $p=\epsilon(x)^{-1}$ is the inverse of a spatially varying permittivity profile. Our result provides a simple and rigorous criterion for identifying non-physical spurious modes in low-loss metallic gratings -- a notorious source of instability in Fourier modal methods. Numerical examples illustrate the practical utility of the criterion.

In this paper, we study the incompressible, viscous and resistive Hall-magnetohydrodynamic (Hall-MHD) system. We first prove that every two-dimensional Leray-Hopf weak solution becomes smooth after a finite time. In three dimensions, where eventual smoothness for arbitrary Leray-Hopf weak solutions is not known, we construct Leray-Hopf weak solutions for which the magneto-vorticity field $B+\nabla\times u$ eventually gains additional regularity. Finally, under suitable low-frequency pseudomeasure assumptions on initial data, we establish decoupled algebraic decay rates for the velocity and magnetic fields by combining a generalized Fourier splitting method with the eventual smoothness in two dimensions and strong regularity in three dimensions.

Burrows-Wheeler-transform-based compressors rely on local context regularity, but structured text also contains dates, counters, identifiers, coordinates, and other digit runs whose values vary differently from their surrounding tokens. STC is a practical BWT-family compressor that separates this source of variation before the component BWT stage. It replaces digit runs in the main stream with an unambiguous placeholder and stores the removed digits in length- and context-conditioned side streams. The side streams use stable bucket ordering and compact digit packing, so the decoder can reconstruct the original run order from the normalized main stream without storing a separate permutation. The resulting components are encoded by a fixed internal BWT/M03-style component coder. On enwik9, STC produces a 157,388,188-byte archive with a 183,174-byte decoder source package, giving a local LTCB-style total of 157,571,362 bytes. A full-enwik9 same-coder ablation shows that the digit-context decomposition reduces the archive by 2,629,561 bytes relative to the no-split control. The result is locally verified by full decode and SHA-256 matching; official benchmark status requires independent maintainer-side verification.

The neighborhood complex $N(G)$ and the dominance complex $D(G)$ are fundamental simplicial complexes associated with a graph $G$. We characterize precisely when the Stanley-Reisner ring $k[N(G)]$ admits a $2$-linear resolution, thereby answering an open question posed by Fröberg. We prove that this occurs if and only if $G$ is neighborhood conformal and its common neighbor graph is chordal. Equivalently, $G$ is a bipartite graph whose only induced cycles are $4$-cycles. As a consequence, we show that Katzman's lower bound becomes an equality for this class, yielding $\operatorname{reg}(S/I(G))=\operatorname{im}(G)$. Using recent results on glued clique complexes, we derive explicit combinatorial formulas for the exact graded Betti numbers of these neighborhood complexes. Finally, utilizing combinatorial Alexander duality, we obtain a corresponding classification of Cohen-Macaulay dominance complexes, and prove that the dominance complex of a graph without isolated vertices admits a $2$-linear resolution if and only if the graph is a star graph.

For a possibly ramified $\mathbb{Z}_p^d$-covering of connected graphs, we establish an Iwasawa-type asymptotic formula for the growth of the $p$-adic valuations of the complexities. The formula is expressed as a polynomial in $n$ and $p^n$ with explicit leading coefficients $\lambda$ and $\mu$; in particular, we eliminate the error term of the form $O(p^{(d-1)n})$ appearing in earlier work. We then establish a Kida-type formula describing the behavior of $\lambda$ under a $p$-covering between $\mathbb{Z}_p^d$-coverings, assuming $\mu = 0$. Finally, for any fixed $p$ and integer $d \geq 2$, we construct an unramified $\mathbb{Z}_p^d$-covering of a bouquet with prescribed $\lambda$- and $\mu$-invariants.

We study selection problems for the vanishing discount limit of first-order stationary mean field games with local coupling. The associated ergodic problem may admit several value functions for the same density and ergodic constant. We decompose the state space into the positive-density region, the essential zero-density interior, and a residual set, and show that possible non-uniqueness of selected value functions can occur only on the gap part of the residual set. If the gradients of selected value functions coincide on this gap residual set, then the selected value function is unique up to additive constants; under compactness and stability assumptions, this yields convergence of the whole normalized discounted family. We show that the gap residual set is null for one-dimensional problems and for a specific Hamiltonian in the multidimensional setting, and hence obtain convergence results in these cases.

In this paper we prove that an $(n+1)$-manifold, compactly $n$-enlargeable, where $3\leq (n+1)\leq 7$, has connected, immersed Morse index one, closed minimal hypersurfaces with unbounded volumes for bumpy metrics. We prove that in the three-dimensional case the hypersurfaces are geometrically distinct using cyclic coverings of manifolds with boundary. The proof extends to $(n+1)$-fiberings. We prove a scalar curvature rigidity theorem for area-nonincreasing maps of three-dimensional manifolds. The case of stable surfaces is also discussed by using cohomology classes and incompressible surfaces.

In this technical note, we show that the relativized bounded arithmetic $T^{1}_{2}(R)$ equipped with $UCP(\Delta^{b}_{1}(R))$ cannot prove $injPHP^{n+1}_{n}(R)$. The result partially supports [K, 2024] Conjecture 1.

A class of Kuramoto models with a general coupling function that can be expressed in terms of a finite number of harmonics, each comprising sinusoidal terms, is studied. We propose a novel approach for certifying local phase synchronization in this class for all initial conditions lying on an arc. The trace parametrization property and Gram matrix representation of a trigonometric polynomial are utilized along with Putinar's Positivstellensatz to obtain semidefinite programming certificates for the stability of the phase-difference system, which in turn implies synchronization of the original system. The results can be extended to any system of coupled oscillators where the forward-invariance on arcs can be established.

This paper compares three different multirate splitting approaches for the application on field-circuit coupled magnetoquasistatic simulations. For these methods, again three different variants for exchanging values between the field and circuit are tested, namely voltages, currents and flux correction terms. All scenarios are applied on two different benchmark problems, i.e. a coil inductor and transformer model coupled to different circuits. The convergence behavior of different time steppers (Implicit Euler and Trapezoidal Rule) is determined for all possible settings, and guidelines for practical applications are derived.

Band-structure calculations for three-dimensional anisotropic phononic crystals require the repeated solution of large elastic generalized eigenvalue problems along Bloch paths. In standard staggered-grid discretizations, anisotropic coupling may involve derivative components located at incompatible grid positions, so additional interpolation or averaging closures are often introduced. This paper proposes a parity-blocked rotated staggered discretization based on four Bloch-periodic body-diagonal differences. The directional derivatives are reconstructed from these diagonal differences, leading to a Hermitian $B_hC_hB_h^H$ generalized eigenvalue formulation that incorporates anisotropic derivative coupling without separate interpolation closures. On even grids, when the stiffness and mass matrices are nodewise local multiplication matrices, the body-diagonal shifts preserve two independent parity invariants. The discrete velocity space is then decomposed exactly into four mutually independent block subspaces, and the full discrete spectrum can be recovered by solving the four smaller eigenvalue problems and merging their spectra. The full and block formulations are further organized in a unified Fourier SVD framework, which supports $\Gamma$-point zero-mode treatment, shift-invert Krylov iteration, inner PCG solves, and GPU matrix-vector products. Numerical experiments for a three-dimensional two-phase anisotropic phononic crystal show that the block implementation preserves the full-space spectrum while substantially reducing the wall-clock time. The results demonstrate that the proposed method provides a structured and efficient solver for large-scale band-structure computations of three-dimensional anisotropic phononic crystals.

We develop a hierarchical multi-fidelity (MF) framework for efficient uncertainty quantification of porous-medium equation (PME) tumor growth models with moving free boundaries. The proposed approach combines coarse-grid PME solvers, level-set approximations of the Hele--Shaw limit, and fine-grid asymptotic-preserving PME discretizations, thereby integrating both discretization-based and asymptotic-model-based fidelity reduction. To guide the selection of high-fidelity samples, we introduce a residual-based farthest-point sampling (RFPS) criterion that combines projection residual information with a distance-based separation term in the low-fidelity snapshot space. Based on this criterion, we construct both bi-fidelity and tri-fidelity approximations, together with empirical error indicators for adaptive refinement. Numerical experiments are conducted in both bi-fidelity and tri-fidelity settings under several uncertainty scenarios, showing that the proposed multi-fidelity approximations achieve accurate results with reduced high-fidelity sampling cost in the reported tests.

For integers $n \geq k \geq 2$ and $0 \leq a,b \leq k-1$, let $m_{k,n}(a,b)$ denote the maximum size of an $(a,b)$-town (mod $k$) family of an $n$-element set, a collection of subsets of whose cardinalities are congruent to $a$ modulo $k$ and whose pairwise intersections are congruent to $b$ modulo $k$. This notion generalizes the classical Oddtown and Eventown problems. We prove that $m_{k,n}(a,b)\leq n$ whenever $a\not\equiv b\pmod{k}$, thereby resolving a conjecture of Veselinov and Marinov. We also disprove another conjecture of theirs by showing that $m_{3,11}(2,2)>m_{3,11}(1,1)$. For the diagonal case $a\equiv b\pmod{k}$, we establish the general bound $m_{k,n}(a,a)\leq 2^{\lfloor n/2\rfloor}$ and completely determine when equality holds. We further obtain improved bounds and exact values in several special cases. The proofs combine characteristic-zero linear algebra with methods from coding theory and finite geometry.

Let $\Gamma$ be a group acting properly and cocompactly on the product of two trees $T_1$ and $T_2$. An anti-torus is a non-periodic flat plane in $T_1 \times T_2$ that is the convex hull of two secant periodic lines. That notion was introduced by Dani Wise as a tool to show that $\Gamma$ is irreducible. We establish a new criterion ensuring the existence of anti-tori, and use it to prove that if $\Gamma$ is an $S$-arithmetic lattice in a product of simple algebraic groups of rank one, then $T_1\times T_2$ contains anti-tori. We also introduce a new class of irreducible lattices acting regularly on the vertex set of a product of trees, containing anti-tori that are fractal aperiodic tilings of the plane. This establishes a connection between lattices in products of trees and substitution tilings.

This paper extends the short-time Fourier transform (STFT), a fundamental tool in time-frequency analysis, to the quantum group setting of unimodular Kac algebras. For a unimodular Kac algebra \mathbb{G}, we introduce a time-frequency shift operator that combines left translation and modulation operators. Using a window vector in the Hilbert space L^2(\mathbb{G}), we define the corresponding STFT and establish its essential analytic properties, including a Plancherel theorem, the Moyal identity, an inversion formula, and a fundamental identity. Furthermore, we explore the projective corepresentation structure of the time-frequency shift operator, and prove that its reflected version induces a continuous projective left representation of the dual quantum group of the quantum double. Finally, we derive the covariance principle and several uncertainty principles.

Let $\mathcal{A}$ be a (not necessarily rational) conformal net with an action of a discrete group $G$. We show that the category $\text{Rep}^G(\mathcal{A})$ of $G$-twisted representations of $\mathcal{A}$ is canonically a $G$-crossed balanced $\mathrm{W}^*$-tensor category. This extends the results of Müger arXiv:math/0403322, in the language of localized endomorphisms, that $\text{Rep}^G(\mathcal{A})$ is a $G$-crossed braided tensor category.

We introduce a method based on horizontal slicing and a plane-then-line stopping-time decomposition for the prime field spherical restriction problem in four dimensions. The method is designed to overcome the Kloosterman obstruction in the spherical Bochner--Riesz kernel by decomposing each critical horizontal slice into rich-plane, rich-line-and-poor-plane, and poor-line-and-poor-plane components, which are then treated by distinct affine-geometric mechanisms. As a quantitative consequence of this structural method, we prove that \[ R_{S_j}^*(2\to r)\lesssim 1 \] for every nonzero sphere $S_j\subset\mathbb{F}^4$ and every $r>23/7$. As an application, we obtain the first improvement over the twenty-year-old $(d+1)/2$ threshold in the four-dimensional prime field Erdős-Falconer distance problem.

This article presents a finite-horizon linear quadratic regulator for the control of the first-order Lighthill-Whitham-Richards traffic model with a triangular fundamental diagram. The in-domain control action is realized through variable speed limits implemented as a source term in the governing hyperbolic partial differential equation. Unlike prior studies on infinite-horizon formulations, this article develops a finite-horizon LQR framework, deriving a space and time varying state feedback function for hyperbolic PDEs. The solution to the finite time optimal control problem relies on the solution of another PDE, called the Riccati PDE. The resulting nonlinear Riccati PDE is solved analytically via the parametric method of characteristics. The Riccati PDE solution is a function of both time and space, as well as the traffic regime. A sensitivity analysis demonstrates the effects of the LQR parameters for both the infinite and finite time horizon problem in different traffic situations, while siulations validate the finite-horizon LQR's ability to guarentee finite-time convergence. Comapred to the infinite-horizon LQR, the proposed approach achieves significantly improved control performance across various scenarios, making it particularly suitable for time-sensitive traffic management applications.

Cyclic codes with dimensions exceeding half of the code length and minimum distance greater than the square root of the code length are of significant interest due to their high transmission efficiency and strong error-correcting capability. Such codes are well suited for demanding applications, including communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. Motivated by the work of Ding \cite{P3}, this paper extends the binary framework of Ding and Zhou \cite{P2} to a non-binary setting. By employing power functions with known differential uniformity over finite fields of odd characteristic, we present several infinite families of $q$-ary cyclic codes of length $q^m-1$ with dimensions exceeding $(q^m-1)/2$ and the lower bounds on the minimum distances greater than the square root of the code length, thereby achieving a favorable balance between code rate and error-correcting capability. We also determine the exact minimum distance of some of these codes. Furthermore, we partially resolve Open Problem $5.31$ posed by Ding in \cite{P3}.

Let $P(G)$ be the Grothendieck ring of the semiring of endomorphisms of the point in the $1$-category of bispans of finite $G$-sets for a finite group $G$. This is the bispan analogue of the Burnside ring of $G$. The ring $P(G)$ admits a third operation from composition of bispans. We produce a character map for $P(G)$ landing in a plethory built out of polynomial rings and the poset of conjugacy classes of subgroups of $G$. We prove that the character map sends composition of bispans to the plethysm operation -- which is a generalization of composition of polynomials.

This paper presents a novel application of Jet Transport, a high-order automatic differentiation technique, to enhance classical numerical methods, with a focus on Newton's method. We prove a central theorem establishing that, under appropriate conditions, applying Jet Transport within a Newton iteration doubles the number of correct coefficients in the Taylor series approximation of the solution. This theoretical result is then extended to the practical case where the exact solution is unknown, demonstrating the expected quadratic convergence (error reduction from \( \varepsilon \) to \( \varepsilon^2 \)) while simultaneously doubling the order of accuracy in the series expansion. The efficacy of the resulting Jet-Newton method is demonstrated through three illustrative examples: an academic problem validating the theoretical convergence rates, the solution of Kepler's equation, and a new continuation algorithm for computing zero-velocity curves in the circular restricted three-body problem. These examples showcase the method's capability to provide high-order semi-analytical approximations.

We study the $b$-$6j$ symbols -- an analytic extension of the $6j$-symbols for the principal series of the modular double of $\mathrm U_q\mathfrak{sl}(2;\mathbb R)$ -- with complex index $b,$ refereed to as the \emph{complex $b$-$6j$ symbols}. Then we relate their asymptotics, when the six parameters scale according to the dihedral angles of a hyperideal hyperbolic tetrahedron, to the volume and the determinant of the Gram matrix of the tetrahedron. In the case $\arg b=\pm \frac{\pi}{4},$ we believe that this work is closely related to the complex Liouville string\,\cite{CEMR}.

We study Gromov's problem on the minimal normal curvature of immersions. Our main result is a lower bound for the average normal curvature of a closed submanifold immersed in a Riemannian domain. The bound is expressed in terms of an invariant measuring the optimal $n$-trace convexity of the domain under a unit-gradient normalization. As applications, we recover and extend Petrunin's lower bound for closed submanifolds immersed in Euclidean balls to geodesic balls in Cartan-Hadamard manifolds and, more generally, to Riemannian domains satisfying suitable convexity conditions. In the Cartan-Hadamard setting, under a natural assumption on the average scalar curvature, we show that equality forces the submanifold to lie minimally in the boundary sphere and that the radial sectional curvature vanishes along it. We also obtain sharper estimates for immersions into hyperbolic balls and Euclidean tubes.

In this work we introduce a class of non-local differential operators defined through a beta-weighted averaging of the ordinary derivative. We investigate their analytical properties and establish connections with the Caputo and Erdélyi-Kober operators. Differential equations involving the beta-weighted derivative are studied by Mellin transform methods, leading to solutions represented in terms of Barnes G-functions and a new class of G-hypergeometric functions. We also analyze asymptotic properties, Laplace transforms, and the second-order equation involving the sequential beta-weighted derivative. Finally, we present stochastic applications of these results, showing that continuous-time random walks, with waiting times characterized by the beta-weighted derivative, converge to Brownian motions time-changed by a scaled inverse stable subordinator. We compare this anomalous-diffusion model with a time-changed Brownian motion whose one-dimensional distribution solve a heat-type equation with beta-weighted derivative.

This paper studies the global regularity problem for the two-dimensional incompressible Boussinesq equations with fractional dissipation given by $(-\Delta)^{\frac\alpha2}u$ and $(-\Delta)^{\frac\beta2} \theta$. Attention is focused on the subcritical regime where $\alpha+ \beta>1$. The case $\alpha >\frac23$ was recently settled in a joint work of the authors [Math. Ann., \textbf{391} (2025), 5965-6012], which established global regularity under this condition. This paper addresses the remaining case $\alpha \leq \frac23$. We obtain the sharpest regularity result by minimizing assumptions on $\alpha$ and $\beta$. We derive nonlinear lower bounds for the fractional Laplacian operator and implement an iterative procedure.

A $\mathbf{GL}$-variety is a typically infinite dimensional variety equipped with a suitable action of the infinite general linear group $\mathbf{GL}$. In earlier work, we established the unirationality theorem: an irreducible $\mathbf{GL}$-variety admits a dominant map from a particularly simple $\mathbf{GL}$-variety, namely, the product of an irreducible finite-dimensional variety with trivial $\mathbf{GL}$-action and an infinite-dimensional affine space on which $\mathbf{GL}$ acts linearly. The main result of this paper states that this map can in fact be constructed to be surjective rather than merely dominant. An immediate application is that secant varieties to varieties of tensors, which are typically constructed as image closures of certain $\mathbf{GL}$-equivariant maps, are in fact also images of (more complicated) $\mathbf{GL}$-equivariant maps. We derive several consequences of this improved unirationality theorem.

We prove that, on a bounded open convex domain $\Omega\subset\mathbb{R}^n$, the first Dirichlet eigenfunction of the Laplacian or the Ornstein--Uhlenbeck operator is $\alpha$-logconcave for every $\alpha\in(0,1/2]$. This extends the recent $1/2$-logconcavity theorem of Crasta--Fragalà for the Laplacian to the weighted Gaussian setting and, simultaneously, to a broader range of exponents. More precisely, if $u$ denotes the first eigenfunction normalized by $\|u\|_\infty=1$, then for every $\alpha\in(0,1/2]$, the function $-\bigl(-\log(\kappa u(x))\bigr)^{\alpha}$ is concave in $\Omega$ provided the scaling parameter $\kappa$ lies below an explicit threshold $\kappa_\alpha(\Omega)\in(0,1)$, which depends on the first Dirichlet eigenvalue and on the diameter of~$\Omega$. For the Ornstein--Uhlenbeck operator, $\kappa_\alpha(\Omega)$ also depends on the distance between $\Omega$ and the origin. Moreover, we establish a local counterpart: for every $\kappa\in(0,1)$, the function $\bigl(-\log(\kappa u)\bigr)^{\alpha}$ is convex on a convex neighborhood $\Omega_\kappa$ of the unique maximum point of~$u$. We also provide counterexamples showing that unscaled $1/2$-logconcavity may fail for the first Dirichlet eigenfunction of a Schrödinger operator with a smooth convex potential, and for the first Dirichlet eigenfunction of a weighted Laplacian associated with an affine log-concave weight.

The $r$-neighbor bootstrap percolation process on a graph is a vertex-activation process that begins with a set of initially active vertices. In each subsequent round, every inactive vertex having at least $r$ active neighbors becomes active. A set of initially active vertices whose activation eventually spreads to all vertices of the graph is called a percolating set. Let $m(G,r)$ denote the minimum size of a percolating set in the $r$-neighbor bootstrap percolation process on a graph $G$. In this paper, among other results, we prove that \[ \frac{k^2}{4}+\Omega(k)\leqslant m(\mathbbmsl{O}_k,2)\leqslant \frac{k^2}{3}+O(k), \] where $\mathbbmsl{O}_k$ denotes the odd graph on a ground set of cardinality $2k+1$. As a consequence, we confirm a conjecture posed in 2021 by Grippo, Pastine, Torres, Valencia-Pabon, and Vera.

Integrated sensing and communication (ISAC) has emerged as a promising key technology for future wireless networks, enabling the efficient coordination of sensing and communication functions within limited resources. This work investigates a secure ISAC system assisted by an uncrewed aerial vehicle (UAV). By incorporating the extended Kalman filter (EKF), the proposed system is capable of delivering communication services to legitimate users while simultaneously jamming eavesdroppers and performing joint prediction and tracking of the trajectories of both legitimate and illegitimate users. Considering practical constraints such as {sensing beamwidth}, transmit power, and UAV's propulsion energy consumption, the secrecy rate is maximized through the joint design of transmit beamforming and UAV trajectory. To tackle the resulting highly non-convex optimization problem, an efficient iterative algorithm is developed by integrating block coordinate descent, successive convex approximation, and EKF, thereby yielding a high-quality suboptimal solution. Extensive simulation results validate the superior performance of the proposed scheme compared to benchmarks.

A classical theorem of Nash-Williams states that if $G$ is a $2$-connected graph on $n$ vertices with minimum degree at least $(n+2)/3$, then for every longest cycle $C$ of $G$, the graph $G-V(C)$ is edgeless. Motivated by a higher-connectivity analogue, Bondy conjectured in 1980 that if $G$ is a $k$-connected graph on $n$ vertices with minimum degree at least $(n+k(k-1))/(k+1)$, then for every longest cycle $C$ of $G$, every path in $G-V(C)$ has at most $k-1$ vertices. This conjecture is known for $k\le 3$ and remains open for all $k\ge 4$. In this paper, we prove Bondy's conjecture for all sufficiently large graphs. The key ingredient is a new Dirac-type theorem that gives a lower bound on the length of a longest cycle in a $k$-connected graph, which also yields a partial solution to a conjecture of Jung from 1990. Along the way, we develop several new tools, including a DFS lemma and an average-degree analogue of the Bondy--Jackson theorem. We conclude with a discussion of related problems and a counterexample to a conjecture of Voss from 1991.

We find a necessary and sufficient condition for a smooth functional on a subspace to admit a norm-preserving smooth extension to the entire space in polyhedral norms. The characterization is geometric: such an extension exists if and only if the unique absolute norm-attaining point of the smooth functional is an extreme point of both the unit ball of the subspace and that of the ambient space. We show by example that such a result is not true in non-polyhedral norms, even under sufficiently strong hypothesis. Extremity of the norm preserving restrictions of extreme functionals are also discussed.

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.

Spectral characterizations of algebraic structure have a long history in the theory of Banach algebras. It is known that weak spectral information may force strong algebraic consequences, such as commutativity throughout the algebra or centrality of a single element. In this paper we study when spectral invariance, applied to the Jordan product $x \circ y = (xy + yx)/2$, forces commutativity or centrality in a semisimple Banach algebra $A$. We first investigate permutations of three elements, proving that equality of the spectral radii $\rho(xyz)=\rho(xzy)$ for all $x,y,z\in A$ implies $A$ is commutative, thereby complementing earlier cardinality and diameter results of Braatvedt et al. We next consider Jordan products and show that if the spectrum (or spectral radius) cannot distinguish the Jordan product from the ordinary product then $A$ must be commutative. By using representation-theoretic methods, we also obtain local spectral characterizations of central elements, showing that boundedness or omission properties of the spectrum (or spectral argument) of the Jordan product $x\circ(ax^{-1})$, as $x$ runs through the exponential group of $A$, imply $a$ belongs to the center of $A$. These results show that coarse spectral data associated with the Jordan product can determine commutativity and centrality.

We consider the compressible time-dependent Navier-Stokes equations in a bounded perforated domain in dimensions two and three. Provided the perforations are small enough, we show that the limiting equations do not change their form when the perforation size goes to zero while their number increases to infinity. The novelty of this result is the form of the pressure: we consider a hard-sphere pressure law, giving an \emph{a priori} bound for the density while, compared to the barotropic case, having worse regularity for the pressure, therefore causing significant problems in the homogenization procedure. To the best of our knowledge, the homogenization for this kind of pressures has not been addressed in the literature yet.

The Borel lifting problem asks when a nonsingular near action of a Polish group can be represented by a genuine Borel action. For locally compact Polish groups, a classical theorem of Mackey and Ramsay gives an affirmative answer. At the opposite extreme, Glasner-Tsirelson-Weiss proved that every probability preserving Borel action of a Lévy group is trivial, and asked whether a Lévy group can admit a nontrivial nonsingular Borel action. We prove a fixed-point theorem which gives a negative answer for most of the standard Lévy groups in the literature: if $G$ is a locally Wasserstein group, then every quasi-invariant $\sigma$-finite Borel measure for a Borel action of $G$ is supported on the fixed points. The proof introduces Wasserstein stability, a local-to-global principle for compact measure metric groups that refines the Gromov-Milman framework for concentration of measure in topological groups. It asserts for a compact measure metric group that the local total variation of the Haar measure tilted by density kernels, controls global Wasserstein distance from the Haar measure. We show that Wasserstein stability implies concentration, and prove a functional inequality using martingales in the spirit of Milman-Schechtman, which is governed by $\ell^{1}$-sums of martingale increments. This gives a geometric criterion by which most of the standard Lévy groups in the literature are Wasserstein or locally Wasserstein, including $L^{0}$-groups with compact targets, measure preserving and nonsingular automorphism groups, full groups of amenable equivalence relations, isometry groups of $L^{p}$-spaces for $p\neq2$, the unitary group of the hyperfinite $\mathrm{II}_{1}$-factor, the infinite dimensional unitary and orthogonal groups, the Cameron-Martin affine group, and the isometry group of the Urysohn space.

Carlen, Frank and Lieb studied stability estimates for the lowest eigenvalue of a Schrödinger operator by decomposing the problem into a stability estimate for Hölder's inequality and a stability estimate for a Gagliardo--Nirenberg--Sobolev inequality. In this note we point out that, if the Hölder step is replaced by the optimal $L^1$-stability theorem of Leng and Lu in probabilistic form, then one obtains a density-distance version of the Carlen--Frank--Lieb stability theorem. The new formulation measures the $L^1$ distance between the normalized density $V_-^s/\int V_-^s$ induced by the negative part of the potential and the corresponding density induced by an optimal potential, where $s=\gamma+d/2$. As a geometric application of the same idea, we also derive a density-stability version of the $L_p$ mixed volume inequality. In the case where one of the two convex bodies is centrally symmetric and both bodies are trapped between two concentric Euclidean balls, this gives an averaged stability estimate for the non-evenness of the support function.

We study Ulam stability for approximate *-homomorphisms of C*-algebras. We prove stability results for several classes of nuclear C*-algebras with respect to von Neumann algebra targets, including abelian C*-algebras and large classes arising in the Elliott classification program. We also discuss permanence properties, counterexamples, and related stability phenomena. As applications, we obtain rigidity and independence results for corona algebras.

The (generalized) Hu algebra is a nontrivial quantization of the wreath product $\Sigma_m \wr \Sigma_d$ between symmetric groups, whose representation theory controls the Hecke algebra of the complex reflection group $G(d,d,md)$. In this paper, we construct a unified basis for this algebra and establish its cellular algebra structure in the case $d = 2$. As an application, our construction provides an elementary realization of the simple modules for the Hecke algebra of type $D_{2m}$ that are parameterized by bipartitions of size $(m,m)$.

For an integer $k\ge 0$ and a graph $G$, the \emph{token-sliding reconfiguration graph $\mathsf{TS}_k(G)$} has the independent $k$-sets of $G$ as vertices. Two vertices are adjacent if one token can slide along an edge of $G$ and the resulting $k$-set is still independent. We study the following realizability problem: for fixed $k\ge 2$, which graphs are isomorphic to $\mathsf{TS}_k(G)$ for some graph $G$? This inverse viewpoint asks which abstract state spaces can occur exactly under a local token rule. We give positive realizability results for the complement targets $\overline{K_n}$, $\overline{K_{m,n}}$, and $\overline{K_n-e}$, and we determine sharp cutoffs for complements of paths and cycles. We also prove a product formula for token-sliding graphs of disjoint unions and apply it to Cartesian products of complete graphs, paths, and cycles. For every grid $\Gamma_{m,n}=P_m\square P_n$ with $2\le m\le n$, we realize $\Gamma_{m,n}$ at token value $m+n-2$ and at every token value $k\ge 4$. At small token values, we prove that $C_4\square C_n$ is not a $\mathsf{TS}_2$-graph for $n\ge 4$, classify ladders $\Gamma_{2,n}$, and settle the first non-ladder grid: for $k\ge 2$, $\Gamma_{3,3}$ is realizable if and only if $k\ge 4$.

We propose a novel method for computing invariant densities of certain multidimensional continued fraction algorithms. Inspired by Rauzy induction, our approach builds on the formalism of simplicial systems developed by Fougeron. We introduce a win-lose induction on a graph that is conjugate to the original algorithm, and construct its natural extension by introducing the notion of a dual graph. This method explicitly reconstructs the complete dynamics of the algorithm, yielding a partition of the invariant domain of the natural extension into pieces that map to one another. We further study the ergodic properties of the algorithms within this framework; in particular, we prove that the Modified Triangle algorithm in any dimension admits a unique ergodic measure equivalent to the Lebesgue measure.

We study the robustness of stochastic mirror descent (SMD) under heavy-tailed noise, focusing on whether the method retains its convergence guarantees when run with infinite-variance stochastic gradient input. To address this question in a principled manner, we begin by introducing a continuous-time model of SMD as a stochastic differential equation (SDE) driven by a centered Lévy noise process with finite $p$-th order moments, $1 < p \leq 2$. This scheme -- which we call the Lévy mirror flow (LMF) -- arises naturally as the scaling limit of SMD in the presence of heavy-tailed noise. In particular, when $p < 2$ -- the heavy noise regime -- the trajectories of LMF generically exhibit jump discontinuities of arbitrary magnitude which, if frequent enough, lead to infinite variance. Nonetheless, despite this highly singular behavior, we show that LMF attains $\epsilon$-optimality within $\mathcal{O}(\epsilon^{-p/(p-1)})$ time in the convex case, and within $\mathcal{\tilde O}(\epsilon^{-1/(p-1)})$ time for (relatively) strongly convex objectives. These guarantees provide a transparent characterization of the impact of frequent long jumps on the convergence of the process, and percolate to a series of matching discrete-time guarantees for several variants of SMD under heavy-tailed noise.

We prove results on moduli spaces of slope stable bundles of projective spaces on a hyperkähler manifold of Type $K3^{[2]}$. Let $X$ be projective of Type $K3^{[2]}$ and $h$ be a (generic) ample class. We prove that the moduli space $M_{\overline{\bf w}_a}(X,h)$ parametrizing $h$ slope stable bundles with a suitable mock Mukai vector ${\overline{\bf w}}_a$ contains an irreducible component $M_{\overline{\bf w}_a}(X,h)^{*}$ whose normalization $\widetilde{M}_{\overline{\bf w}_a}(X,h)^{*}$ is a (projective) HK manifold of Type $K3^{[a^2+1]}$, and that conversely every projective HK manifold $W$ of Type $K3^{[a^2+1]}$ is isomorphic to $\widetilde{M}_{\overline{\bf w}_a}(X,h)^{*}$ for a suitable $(X,h)$ as above. Moreover the universal bundle of projective spaces on $X\times \widetilde{M}_{\overline{\bf w}_a}(X,h)^{*}$ defines a vector bundle whose $2nd$ Chern class defines a rational Hodge isometry $H^2(X)\to H^2(\widetilde{M}_{\overline{\bf w}_a}(X,h)^{*})$. From this and a result of Markman one gets that the analogue of the Shafarevich conjecture (a special case of the Hodge conjecture) holds for rational Hodge isometries $H^2(W_1) \to H^2(W_2)$ between projective hyperkähler manifolds $W_1,W_2$ of Types $K3^{[a_1^2+1]}$ and $K3^{[a_2^2+1]}$ respectively. We prove results also for $(X,\omega)$ a general HK manifold of Type $K3^{[2]}$. In fact one ingredient in our proof is Verbistsky's theory of projectively hyperhomolorphic vector bundles.

We explicitly calculate Pan's geometric intertwining operator and the Fontaine operator on modular curves at infinite level via $q$-expansions, using Heuer's theory of cusps at infinite level. We prove that these two operators coincide on such expansions up to an explicit constant. As an application, we combine this result with $q$-expansion principles to provide a new proof of Pan's theorem that these operators are equal on the locally analytic vectors of completed cohomology of modular curves.

Let $\mathcal{M}_g$ be the moduli space of smooth curves of genus $g$. The image of a non-constant morphism from a curve $T$ to $\mathcal{M}_g$ is a curve in $\mathcal{M}_g$. By work of González Díez and Harvey, for every integer $g \geq 3$, there exists a complete curve in $\mathcal{M}_g$. Here we generalize the construction to produce new complete curves in $\mathcal{M}_g$. We also find a formula for the genus of each curve $T$ using Galois theory for function fields.

For fixed $\alpha>0$, we show that the sequence $\{\alpha n^{\theta}\}$ has Poissonian pair correlation for Lebesgue-almost all $\theta \in (0,\frac{3}{5})\cup(3,\infty)$. This improves a result of Technau and Yesha, who proved the same for almost all $\theta>7$. The approach of Technau and Yesha was based on a repulsion principle, which roughly allows one to estimate the variance of the pair correlation function using the fourth derivative of the phase. In our approach, we split the $\theta$-integration in the variance into many short intervals and show that most of the integrals can be estimated using the first derivative. The problem is then reduced to several counting estimates, which we prove using moments of the Riemann zeta function and exponent pairs.

A group is called realizable if it is the group of units in a ring with identity. The classification of realizable groups is a difficult open problem -- originally posed by László Fuchs -- and is an active area of research. Realizable groups seem rare, but their proportion within a fixed class of groups (cyclic, dihedral, finite abelian, etc.) varies. To quantify this proportion, we introduce the realizable density of a class of finite groups as an analog of natural density for subsets of the natural numbers. The realizable finite cyclic groups and the realizable finite abelian $p$-groups for $p$ odd have been classified; we prove that their realizable densities are 1/4 and 0, respectively. The realizable finite abelian 2-groups -- and more generally the realizable finite abelian groups -- have not been fully classified, and these special cases appear quite difficult. Nonetheless, we prove that the realizable density of finite abelian 2-groups is 1 and the realizable density of finite abelian groups is 1/2. Our work combines existing classification theorems for realizable groups with tools from analytic number theory.

Switched systems are a class of dynamical systems where trajectories switch between different systems based on a switching rule. This rule can depend on time and/or the position of the trajectory in the state space. The existence of a Lyapunov function implies the existence of a uniformly asymptotically stable equilibrium point at the origin, which we prove in this paper. A method to construct Lyapunov functions for switched systems using meshfree collocation and quadratic programming was described in a related article. We prove that, under suitable assumptions, the algorithm described in this previous work converges as the fill distance between the collocation points tends to zero.

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.

A finite group $G$ is $\varepsilon$-quasirandom if all its nontrivial irreducible complex representations have degree at least $|G|^\varepsilon$. Building on recent work of Golsefidy-Srinivas, we prove that expansion in a quasirandom group is controlled by expansion in its simple quotients. As a consequence, we remove the product theorem from the hypotheses of the Bourgain-Gamburd expansion machine. Moreover, we combine this result with crown theory to deduce that $1 + \lfloor \varepsilon^{-1} \rfloor$ random elements give an expander Cayley graph with high probability. Finally, generalizing results of Breuillard-Green-Tao and Pyber-Szabó, we prove that the diameter of any connected Cayley graph of a quasirandom group is polylogarithmic.

We uncover a finite-temperature extension of de Bruijn's identity -- the classical relation $\frac{d}{dt}h(X+\sqrt{t}Z)=\frac{1}{2}J(X)$ connecting differential entropy and Fisher information. Our framework is the spectral theory of Blahut--Arimoto (BA) dynamics, recently developed by Wang~\cite{Wang2026} for the analysis of rate-distortion optimization. The central observation is elementary yet profound: for Gaussian sources, the spectral gap $\lam$ of the BA relaxation kernel $\G$ satisfies $\lam = 1/(2\beta\sigma^2)$~\cite{Wang2026}, while the Fisher information of the source is $J = 1/\sigma^2$. Hence \[ {\lam = \frac{J}{2\beta}} \] for all inverse temperatures $\beta > 1/(2\sigma^2)$. This identifies the BA spectral gap as a \emph{finite-temperature regularization of Fisher information}. From this observation we derive an exact finite-temperature de Bruijn identity: \[ \frac{\partial F_\beta}{\partial \sigma^2} = \frac{1}{2\beta\sigma^2} = \lam, \] where $F_\beta$ is the BA free energy. This identity holds for all finite $\beta$ without any limit procedure. The classical de Bruijn identity follows as the exact consequence $\beta\,\partial F_\beta/\partial\sigma^2 = J/2$. The significance is structural: classical de Bruijn is not an isolated fact about Gaussian convolutions, but the $\beta\to\infty$ shadow of a one-parameter family of exact identities living in the spectral geometry of rate-distortion optimization. We discuss implications for the entropy power inequality, the $\chi^2$-dissipation structure of BA dynamics, and the geometric unification of information inequalities.

It is well-known that the midpoint rule preserves the dissipation inequality if applied to a certain class of energy-based models. We introduce an appropriate scaling of the state variables such that the symmetric part of the resulting iteration matrix is guaranteed to be positive definite. This allows the application of three-term iteration schemes such as the methods of Widlund and Rapoport. Special emphasis is put on examples where the symmetric part is block diagonal such that the computations decouple. This then leads to efficient dissipation-preserving numerical schemes as illustrated in two numerical examples, namely the biharmonic heat equation and linear poroelasticity.

In this paper, we study the non-symmetric system of semilinear damped $\sigma$-evolution equations with different damping types, where two power exponents of nonlinearities belong to the critical curve, by using moduli of continuity in nonlinear terms. Our goal is to determine the sharp conditions on these moduli of continuity that guarantee the global (in time) existence of Sobolev solutions or, conversely, lead to finite-time blow-up. Furthermore, by employing the analysis introduced in the proof of our blow-up result, we provide a positive answer to an open problem for the symmetric models posed in the literature.

We present and analyze an embedded Trefftz discontinuous Galerkin method for reaction-diffusion problems on anisotropic meshes. The method is constructed by imposing a relaxed local Trefftz condition via an embedding into a tensor-product DG space, yielding a reduced global system while preserving the approximation properties of the underlying high-order discretization. We prove stability and quasi-optimality on anisotropic, possibly curved, quadrilateral elements, and derive anisotropic a priori error estimates. Numerical experiments for $h$- and $hp$-refinement, including curved-domain examples, validate the theoretical results.

We present new short exponential sum approximations of length $N$ for $f_1(x)=\frac{1}{a+x}$ with $a>0$ on $[0, \infty)$ and for $f_2(x)= {\mathrm e}^{-x^2/2\sigma}$ with $\sigma>0$ on ${\mathbb R}$ with geometric error decay ${\rho}^{-2N}$ for user-defined $N \ge 2$ and $\rho >1$. The approximations are built over consecutive intervals $[b_j, \, b_{j+1}) \subset [0, \infty)$, $j \in {\mathbb N}_{0}$, with interval lengths that depend on $\rho$ and grow exponentially for $f_1$ and are equidistant for $f_2$. All parameters determining the exponential sum approximations on $[b_j, \, b_{j+1})$ are easily computed from the initial parameters on $[b_0, \, b_{1})$, ensuring numerical stability. Our method is based on Gauss-Laguerre and Gauss-Hermite quadrature, respectively, applied to suitable parametric integral representations of $f_1$ and $f_2$. This technique ensures consistent relative errors across all intervals. Using the obtained exponential sum approximations, we achieve highly accurate approximations of $\log(x)$ on $[1,\infty)$ and of the error function $\mathrm{erf}(x)$ with predictable geometric error decay. Numerical examples for $N=8$ and $N=10$ clearly illustrate the theoretical error estimates.

Extending recent advances on constructive quantum field theory, we study cumulants of one of the simplest matrix models in the regime of bounded rank. We analyze both ordinary cumulants and scalar cumulants, which arise from the Weingarten calculus and are essential to the topological expansion in quantum field theory. Our results are valid for arbitrarily large positive coupling and provide new instances where techniques such as the variational approach are applied to cu- mulants.

A longstanding open problem posed by Andrew Pitts is whether every Heyting algebra is the lattice of truth values (i.e., of subterminal objects) of some elementary topos. A positive answer is known for complete Heyting algebras (i.e., locales) via sheaves, and for Boolean algebras via a construction due to Peter Freyd. We extend Freyd's construction to all étale-finite Heyting algebras, in the sense of Evgeny Kuznetsov. These are the Heyting algebras satisfying a generalisation of the law of excluded middle relative to some finite Heyting subalgebra. For every étale-finite Heyting algebra $H$, we use Esakia duality to construct an elementary topos whose lattice of truth values is isomorphic to $H$, thereby extending the class of Heyting algebras for which a positive answer to Pitts' question is known. The toposes we construct are categories of certain compact étale spaces. As a consequence, they are finitely propositional: every object has a finite cover by subterminal objects. We show that a Heyting algebra occurs as the lattice of truth values of some finitely propositional topos if and only if it is étale-finite. This exhibits an obstruction to extending our use of compact étale spaces beyond the étale-finite case.

We give a proof of the cosmological splitting conjecture of Robert Bartnik from 1988, which expresses the rigidity of the cosmological Hawking--Penrose singularity theorem. It states that a timelike geodesically complete, globally hyperbolic spacetime which has compact Cauchy surfaces and satisfies the strong energy condition must split isometrically as a Lorentzian product. Our methods combine the construction of global viscosity solutions to the Lorentzian eikonal equation by Zhu--Wu--Cui with our recently developed elliptic approach to the proof of Lorentzian splitting theorems in joint work with Braun, Gigli and Sämann, where we make use of the $p$-d'Alembertian operator for $p < 1$.

Motivated by the recent work of Li and Wang on parametric generalizations of Van Hamme's $(C.2)$ supercongruence in the $q$-setting, we establish $q$-analogues of a supercongruence related to Van Hamme's $(B.2)$ supercongruence, recently obtained by the authors. In particular, we derive parametric extensions of these $q$-supercongruences by constructing suitable pairs of hypergeometric functions through the $q$-WZ method.

We study the Abel--Jacobi map over the twistor-$\mathbb{P}^1$ in the context of Scholze's geometrisation of the real local Langlands correspondence. In a similar spirit to a result of Fargues over the Fargues--Fontaine curve, we prove that pullback along the Abel--Jacobi map induces an equivalence on Picard groupoids and use this to recover local class field theory for archimedean local fields.

Motivated by the vision of making sixth-generation (6G) networks sustainable, we study the sparse antenna/array activation problems in uplink cell-free massive multiple-input multiple-output (CF mMIMO) networks. We first develop an antenna-level optimal bilinear equalizer (OBE) weighting framework, in which each access point-user equipment (AP-UE) pair is assigned a matrix-valued long-term weight to shape the contribution of individual antenna elements, thereby generalizing the conventional large-scale fading decoding (LSFD) strategy from scalar coefficients to antenna-element-aware weighting. Building on this structure, we formulate sparse antenna activation as structured sparsity-inducing mean square error (MSE) minimization problems, and design four activation schemes at two granularities: antenna-level and array-level, each with UE-specific and network-wide (all-UEs) variants. The resulting convex problems are solved efficiently via the proximal method with closed-form group-wise updates, while the network-wide schemes are modeled through hierarchical sparsity and handled by a tree-structured proximal operator. Numerical results under correlated Rician channels and a detailed power consumption model demonstrate that the OBE weighting scheme consistently improves spectral efficiency over the LSFD, with gains increasing with the number of antennas. Meanwhile, the studied sparse activation schemes can achieve substantial energy efficiency improvement and power reduction with controllable spectral efficiency loss.

Primitive positive constructions of first order structures have been shown to be a very useful tool in universal algebra for the study of constraint satisfaction problems. However, they seemed to be very rarely studied in classical algebra such as group theory. This paper fills in this gaps by looking at structures and obstructions based on permutation groups and giving a full classification in this sub-area. This special case is also very important for the generalization to all first order structures as every permutation group describes an easy-to-check necessary condition for the existence of primitive positive constructions, also between structures that are not at all linked to permutation groups.

The study on shift spaces in ergodic theory has been beyond the classical setting, but there is a room to discuss an extension of the Kolmogorov-Sinai entropy in the ergodic theoretical point of view. On the other hand, the rate distortion dimension recently attracted attention in mean dimension theory because it behaves like the Kolmogorov-Sinai entropy on dynamical systems in the ``large" spaces in which the usual entropies is in general infinite. According to these background, we investigate the connection between the Gibbs measure on the product spaces and the variational principle based on the rate distortion dimension: we concretely calculate the rate distortion dimension of the Gibbs measure on the concrete setting and it satisfies the simplest case of thermodynamical formalism based on the rate distortion dimension: the extension of the maximal measure of topological entropy. Remark that the result shows a new phenomenon which does not hold in the classical setting. We also discuss another variational principle under more concrete settings.

These lecture notes provide an introduction to first-order optimization methods with a particular emphasis on stochastic gradient methods. We begin with deterministic gradient based methods for unconstrained optimization and study their convergence under standard assumptions such as smoothness, convexity, strong convexity, and the Polyak-Lojasiewicz condition. We then turn to stochastic approximation and stochastic gradient descent, motivated by empirical and expected risk minimization in machine learning. The main focus is on convergence theory: we discuss almost sure convergence and convergence in expectation, derive classical convergence rates, and present selected advanced topics, including almost sure convergence rates and variance reduction methods.

In this paper we confirm several conjectures on determinants and permanents. For example, we prove that for any prime $p\equiv3\pmod 4$ the number $2\det[a_{jk}]_{0\le j,k\le (p-1)/2}$ is congruent to a square modulo $p$, where $a_{jk}=(\frac{j+k}{p})+(\frac{j^2+k^2}{p})$ with $(\frac{\cdot}{p})$ the Legendre symbol. We also prove that ${\rm per}[j^{k-1}]_{1\leq j,k\leq n-1}\equiv0\pmod n$ for any integer $n>1$ with $n\not\equiv2\pmod 4$.

We show that the lower bound in the majorizing measures theorem holds for a large class of random vectors. Specifically, suppose $X \sim \mu$ is a centered random vector in $\mathbf{R}^n$ with \[ C_{\mathrm{KL}}(\mu) = \sup_{\substack{\theta \neq \eta \\ \theta, \eta \in \mathbf{R}^n}} \frac{\mathrm{KL}(\mu_\theta \| \mu_\eta)}{\|\theta - \eta\|_2^2} < \infty, \] where $\mu_\theta$ denotes the law of the translate $\theta + X$. Then, for every nonempty, bounded $T \subset \mathbf{R}^n$, \[ \sqrt{C_{\mathrm{KL}}(\mu)}\, \mathbf{E}_\mu \Big[\sup_{t \in T} \, \langle X, t \rangle \Big] \gtrsim \gamma_2(T), \] where the righthand side denotes Talagrand's generic chaining functional. This result recovers, as a special case, the lower bound in the majorizing measures theorem for centered Gaussian processes. Our argument critically relies on the rate-distortion integral, recently introduced by J. Liu

We construct an explicit two-parameter family of complete, non-compact, three-dimensional, smooth steady gradient generalized Ricci solitons with $\mathrm{SO}(2)\times\mathbb{R}$ symmetry, providing a cylindrical counterpart to the spherically symmetric solitons recently found by Podestà and Raffero. The family is parametrized by a flux constant $k>0$ and a conserved quantity $\mathcal{C}\ge 0$. For $\mathcal{C}=0$, the asymptotic geometry exhibits power-law decay; for $\mathcal{C}>0$, the metric converges exponentially fast to a flat cylinder of finite radius.

In 1983, Kusner conjectured that the largest equilateral set in $\mathbb{R}^{n}$ with metric $\ell_{p}$ has cardinality $n+1$ when $1<p<\infty$ and $2n$ when $p=1.$ This conjecture was proved only in the isolated cases $p=2$ and $p=4$, and was disproved when $1<p<2$. The best general upper bound $O_p(n^{\frac{2p+2}{2p-1}})$ is due to the celebrated work of Alon and Pudlák~[GAFA, 2003]. Our main contributions include: (1) We prove Kusner's conjecture for every dimension $n\ge 1$ when $2\le p\le 4$. More generally, for every integer $k\ge 0$ and every $p\in[4k+2,4k+4]$, every equilateral set in \(\mathbb{R}^{n}\) with metric $\ell_p$ has cardinality at most $(2k+1)n+1$. On the complementary intervals $p\in(4k,4k+2)$ with $p\geq 1$, we obtain the almost linear bound $O_p(n\log n)$. (2) We also consider the analogous problem on the torus $\mathbb{T}^n$, recently initiated by Alon, where the cyclic distance makes the problem substantially more delicate than in $\mathbb R^n$. We prove the almost linear bound $O_p(n\log n)$ for $1\le p\le 2$ and $O_p(n^{\frac{3}{2}-\frac{1}{p}})$ for every fixed real $p>2$, improving Alon's bounds $O_p(n^{2+\frac{2}{\lfloor p\rfloor}})$ for all finite $p\ge 1$.

We consider the problem of generating a large collection of initial guesses for local minima of multimodal non-convex continuous optimization problems. The goal is for these initial guesses to be high-quality (i.e., a numerical solver converges quickly) and diverse (i.e., represent many different local minima). Identifying multiple locally optimal solutions enables flexible downstream decision-making, but typically requires expensive global search. Existing data-driven methods predict initial guesses using only the final converged optima from offline solver runs, which discards information about the local neighborhoods of solutions and limits the available training data. We propose GLENS (Global Search via Learning from Solver Iterates), a data-efficient global search method that leverages intermediate solver iterates as free data augmentation. GLENS consists of two components: a neighborhood structure model that uses diffusion models to learn the local geometry around optima conditioned on problem parameters, and a solver behavior model that learns refinement directions to further guide samples towards nearby optima during diffusion sampling. Experiments on modified non-convex benchmark problems and a two-robot obstacle-avoidance navigation problem show that GLENS generates high-quality initial guesses while preserving the multimodal distribution of diverse local optima. The resulting initial guesses lead to faster solver convergence across different problem settings and solvers. We also analyze how key hyperparameter choices affect the performance.

In this paper, we study the efficient numerical solution of Darcy equations in strongly heterogeneous media with high-contrast permeability and propose a hybrid framework that combines learning with multiscale numerical methods. The learning component is used for the prediction of multiscale basis functions in the mixed generalized multiscale finite element method (mixed GMsFEM), with the goal of reducing the repeated local computations required in the offline stage. Once these basis functions are predicted, the global system is assembled and the pressure field is computed by a two-grid preconditioned solver. The resulting method accelerates the costly local basis-construction stage while retaining the multiscale discretization and preconditioned iterative structure of the underlying solver. Numerical experiments on two-dimensional heterogeneous Darcy problems show that the proposed framework yields more accurate final pressure reconstruction than several representative learning-based methods and remains stable under strong heterogeneity and high-contrast coefficients. In comparison with the traditional mixed GMsFEM, its main advantage lies in the efficiency of the basis-generation stage, while the quality of the global solve is still ensured by the two-grid preconditioner. These results indicate that accelerating multiscale basis construction through learning, while preserving a mature numerical solver for the global problem, provides a viable approach for high-resolution Darcy-type simulations.

We describe a multi-neck spacetime wormhole with a simple metric tensor and a simple injective map without coordinate patching. An intra-universe, non-thin-shell, non-spherically-symmetric 3-neck spacetime wormhole is geometrically constructed by spherically inverting a 3-torus. We place the resulting Dupin hypercyclide in a synchronous reference frame. The three necks are arranged around a central point and satisfy topological and geometric spacetime wormhole definitions. Asserting this metric tensor as an exact solution of Einstein's field equations in global coordinates generates diagonal Ricci and stress-energy tensors, and a Riemann curvature tensor with only six nonzero entries. The local inertial frame at every point of the coordinate system is comoving with the triple wormhole. This non-vacuum solution answers affirmatively the question posed by Einstein and Rosen (1935) of whether or not multi-neck solutions exist. The wormhole solution contains negative energy density as is expected to hold the necks open; however, geodesic paths through each neck exist which encounter only positive energy density. The spatial manifold is a trivariate Dupin hypercyclide. The spherically inverted equal-radii 3-torus is unbounded, asymptotically flat and admits a global isothermal coordinate system that further simplifies the curvature tensors.

We derive and analyze the exact solutions of the inverted Dirac-Moshinsky oscillator (IDMO) in $(1+1)$ dimensions, obtained from the standard model via the substitution $p \to p + im\omega\beta x$. The upper spinor component satisfies a Weber equation with complex spectral parameter $\lambda = (E^2-m^2)/(2m\omega)+i/2$, whose solutions are parabolic cylinder functions $D_\nu(\xi)$ with complex order $\nu = \lambda - 1/2$. The physical spectrum is purely continuous ($|E|>m$), with no discrete bound states. Three normalization schemes are developed, and the discrete Gamow resonances at $E_n^\pm = \pm\sqrt{m^2+(2n+1)m\omega-im\omega}$ are identified as poles of the resolvent. The negative-energy sector describes antiparticle anti-resonances whose positive imaginary part signals vacuum instability and spontaneous pair production, analogous to the Schwinger effect. The algebraic structure is governed by the principal series of $SU(1,1)$, and the Hamiltonian is $\mathcal{PT}$-symmetric with unbroken symmetry for $|E|>m$.

We introduce temporal matrix scale invariance (tMSI), a mathematical structure for the two-time correlation kernel of a multivariate observable. A kernel $C(t,t')$ satisfies tMSI of order $\alpha$ if $C(kt, kt') = k^{-\alpha}C(t,t')$ for all $k>0$; this condition holds near a tipping point, where the divergence of the coherence time produces temporal scale freedom. By a kernel factorization theorem, every tMSI kernel separates into a power-law envelope $(tt')^{-\alpha/2}$ and a shape function $F(t/t')$ diagonalized by the Mellin transform. This reveals a decoupling of two independent exponents: the dynamical exponent $\alpha$, carried by the envelope, and the spectral relaxation exponent $\beta$, determined by the eigenvalue decay of the finite-dimensional truncation. Their equality $\alpha = \beta$ characterizes a simple critical point; their inequality $\alpha \neq \beta$ is the signature of temporal multicriticality. We provide a classification of tipping points. The Landau quartic coefficient $a_4$ is given exactly by $a_4 = p^2 + q^2 - 2\lambda pq - g^2_{\alpha\alpha\beta}\Gamma(\sigma_\alpha, \sigma_\beta)$, where $\lambda = 2\sqrt{\sigma_\alpha\sigma_\beta}/(\sigma_\alpha+\sigma_\beta) \in (0, 1]$, $g_{\alpha\alpha\beta}$ is the three-point structure constant, and $\Gamma > 0$ is in explicit closed form. The transition is continuous for $a_4 > 0$, tricritical for $a_4 = 0$, and discontinuous for $a_4 < 0$. The simple critical point $\alpha = \beta$ is maximally fragile: any nonzero operator mixing drives $a_4 < 0$, placing the synchronized state generically at the edge of catastrophe. The framework yields a matrix-valued early warning diagnostic, computable from a multivariate time series without knowledge of the underlying equations, that classifies an approaching tipping point as recoverable or catastrophic. Applications to epilepsy and acute myocardial infarction are discussed.

External regret certifies stability only against replacing one's behavior by a fixed alternative. In a quantum game, this misses a natural physical move: a player can apply a local completely positive trace-preserving (CPTP) map to the state it actually received or prepared. We introduce coherent swap regret as the regret benchmark against all such local CPTP deviations, and give an algorithm achieving $O(\sqrt{dT\log d})$ coherent swap regret via entropic mirror ascent on the CPTP Choi slice with a fixed-point play rule. The main result is a three-level deviation-class landscape. Replacement channels recover ordinary external regret at rate $\Theta(\sqrt{T\log d})$. Unital channels, including unitary deviations and mixtures of unitaries, have zero minimax regret. Deterministic measurement-and-preparation channels already force $\Omega(\sqrt{dT\log d})$ regret in the moderate-horizon regime, and this rate is also sufficient for all CPTP deviations. Thus the hardness comes from non-unital use of the recommendation register, not from quantum coherence alone. As an application, decentralized full-information learning in finite quantum games reaches an $\varepsilon$-approximate separable quantum correlated equilibrium after $T=O(\max_i d_i\log d_i/\varepsilon^2)$ rounds. We identify these equilibria with channel-proofness of mediated quantum recommendation protocols, give an SDP audit for local CPTP exploitability applicable to arbitrary finite-dimensional states, and include a probing-bandit extension with pseudo-regret $O(d^{4/3}T^{2/3}(\log d)^{1/3})$ under Haar-random pure-state probes.

In this manuscript, we explore the application of neural networks to predict the natural parameter $\kappa \geq 0$ of Schramm-Loewner Evolution (SLE$_\kappa$) theory. SLE$_\kappa$ is a family of random fractal curves that has significant implications in Statistical Mechanics and Conformal Field Theory. This parameter $\kappa \geq 0$ plays an important role in the theory as there are models of Planar Statistical Physics that are proven to have SLE as scaling limits as well as others that are conjectured to have this limit for various choices of the parameter $\kappa \geq 0$. In addition, there are three different statistical behaviors of the SLE curves as the parameter $\kappa$ changes in $[0, \infty).$ Leveraging the powerful pattern recognition capabilities of neural networks, this study aims to develop a predictive model that can estimate the $\kappa$ parameter with good accuracy.

Quantum transit times have a long history of inequivalent definitions, including phase times, dwell times, and quantum-clock constructions. In this context we revisit the Salecker--Wigner--Peres stationary quantum clock as a phase-sensitive scattering observable, with clock time defined by the energy derivative of the transmission phase shift across the interaction region. For real compactly supported one-dimensional potentials, we show that the raw stationary Peres clock generically contains a universal \(1/\sqrt{E}\) continuum-edge term whose coefficient is fixed by low-energy scattering data. For the attractive square well, this threshold singularity is inherited from the vanishing exterior momentum and the associated scattering matching, rather than from resonant delay itself. We derive the exact stationary clock time for the square well and introduce a new threshold-subtracted clock observable. Away from exceptional zero-energy tuning, the subtraction removes the universal low-energy term and isolates the resonant contribution. Comparison with the dwell time and the transmission Wigner phase delay shows that the threshold-subtracted clock acquires the expected local Lorentzian form near isolated transmission resonances. Near the continuum edge, if \(\varepsilon\) denotes the detuning from threshold, the resonant peak grows only as \(\varepsilon^{-1/2}\), whereas the unsubtracted threshold background grows as \(\varepsilon^{-3/2}\). A symmetric barrier--well--barrier cavity and a numerical asymmetric two-step attractive well provide complementary controls. The result is a new threshold-subtracted stationary-clock candidate that separates universal threshold kinematics from pole-sensitive resonant delay.

Real-world datasets often contain spurious correlations that are not causally related to the target label. When such correlations dominate the majority of training samples, models tend to rely on them, leading to misclassification of minority samples that do not exhibit the same spurious patterns. While a potential approach is to select subsets of data to better represent the minority samples, this may require access to group labels, which are typically unknown. Furthermore, as we demonstrate, widely used sample scoring functions in the invariant subset or coreset selection literature largely depend on spurious features and therefore fail to accurately capture the importance or difficulty of core, causally relevant features. Accordingly, we propose to mitigate spurious correlations by developing a two-stage sample scoring function that disentangles the learning dynamics of core and spurious features and evaluates their difficulty separately. Based on our proposed metric, we introduce a new algorithm to find and prioritize informative samples both with and without spurious correlations. Extensive experiments demonstrate that a standard ERM model trained on our selected samples achieves superior performance compared to state-of-the-art debiasing techniques, while requiring as little as 10\% of the original training data.

Deep neural networks learn representations where individual features often lack interpretable meaning; a single neuron may activate for scattered, unrelated inputs. We introduce coherence, a geometric property inspired by neural coding in the brain, where neurons like grid cells and head direction cells respond to contiguous regions of state space. A non-negative matrix is coherent if each row (sample) attends to geometrically clustered columns (features) and vice versa, and in addition every sample is well described by some feature and every feature is needed by some sample. We prove that coherent matrices induce a bounded interleaving between the Vietoris-Rips filtrations of samples and features, guaranteeing that both spaces share compatible topological structure. This geometric constraint facilitates interpretability. For example, if data lies on a circle, coherent features must tile that circle into contiguous arcs. We introduce Coh, a differentiable objective function based on Fréchet variance that enforces coherence during training. Unlike sparsity, which bounds how many samples a feature activates on, coherence bounds which samples, requiring geometric connectivity rather than only rarity. This yields not just interpretable features but an interpretable feature space. We validate Coh in an auto-encoder using synthetic and rotated MNIST datasets and in a token embedding of BERT using language data.

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.

Deep learning weather models now match numerical weather prediction accuracy while running orders of magnitude faster, but produce deterministic forecasts without uncertainty estimates, a critical gap for high-stakes decisions during extreme weather events. This paper proposes Neural Tangent Kernel-based uncertainty quantification (NTK-UQ) using last-layer empirical features. Theoretical analysis predicts that UQ quality is architecture-dependent through two mechanisms. First, a variance collapse mechanism explains when UQ fails: when the eigenvalue truncation rank approaches the effective rank of the feature space, the GP correction term consumes nearly all prior variance, destroying discrimination between tropical cyclones and routine conditions; architectures with concentrated spectra (spectral operators) require aggressive truncation ($k \leq 10$), while attention-based models tolerate full-rank computation. Second, decomposition performance depends on the non-Gaussian, heavy-tailed structure of extreme weather: Independent Component Analysis exploits higher-order statistics (kurtosis, negentropy) to isolate heavy-tailed extreme-event features, achieving higher discrimination than singular value decomposition, which captures only second-order variance. A data-driven selection rule chooses ICA or SVD from the feature eigenspectrum concentration ratio, correctly prescribing the superior decomposition for all four evaluated architectures. Compared to split conformal prediction (the natural post-hoc baseline), NTK-UQ achieves 31--37\% sharper prediction intervals at 90\% coverage, and uniquely produces \emph{adaptive} intervals that scale with extreme event severity, which conformal prediction cannot achieve by construction. The framework requires no retraining; inference-time uncertainty requires only a single matrix-vector product per sample.

Symmetric nonnegative matrix factorization (Symmetric NMF) approximates a matrix as $WW^T$ with nonnegative rectangular factor $W$. It has broad applications in graph clustering and machine learning. In contrast to the NMF, projected gradient methods for the symmetric problem had been associated with slow convergence. To address this, we introduce SNMPBB, the first adaptation of nonmonotone projected Barzilai-Borwein methods to Symmetric NMF, demonstrating that gradient algorithms are significantly more effective than previously understood. We further extend SNMPBB to graph clustering using the graph Laplacian regularization (Graph-SNMPBB) and to large problems with low-rank approximations (LAI-SNMPBB). For all variants we prove global convergence to first-order stationary points and also that Barzilai-Borwein curvature information is preserved with randomized approximations. On synthetic data, SNMPBB achieves 6 times speedup over the alternative SymANLS for similar residuals, with advantages growing at higher ranks. Across six real-world clustering benchmarks, Graph-SNMPBB matches or exceeds SymANLS accuracy. Lastly, LAI-SNMPBB outperforms state-of-the-art LAI-SymPGNCG on 34 SuiteSparse matrices in both runtime and residual quality.

We present a new method for the efficient imaginary time evolution of quantum many-body wavefunctions represented by matrix product states (MPS). We first show that logarithmic time grids are sufficient to resolve long imaginary time dynamics, yielding an exponential reduction in the number of time steps compared with standard approaches. We then show that A-stable implicit time-stepping methods for ordinary differential equations allow stable propagation for any time step size. The resulting scheme requires only matrix-vector products and linear solves, standard operations in the MPS toolbox. We validate our approach with two examples: a Heisenberg spin chain, which we use to demonstrate a speedup of several orders of magnitude over the standard time-dependent variational principle method with uniform time steps, and a single-site Anderson impurity model with a metallic bath, for which propagation to large imaginary times allows one to observe the exponential dependence of the Kondo temperature on the interaction strength.

Public scientific and metrology releases can leak the hidden settings that produced them. We formalize and quantify this risk as a profiled statistical side-channel audit: a release map exposes finite-band statistics of a power spectral density (PSD), a profiled observer trains labeled template spectra under an explicit budget, and a challenge release is drawn from one of two utility-equivalent recipes separated by a protected coordinate. Averaged PSD bins follow a gamma channel, replaced by a covariance-weighted log-spectrum channel when the bins are correlated; this yields exact Kullback-Leibler divergences, Chernoff exponents, protected-bit advantage bounds, and finite-training, finite-library, finite-compute, and model-mismatch corrections. Our headline result is a finite-band transport-leakage law: after amplitude and blur are eliminated, the protected acid-transport information obeys $I_{\lambda|\alpha,\beta}(K) = (64/1225)\, w \lambda^{6} K^{9} + O(w \lambda^{8} K^{11})$ for $K\lambda \ll 1$, a ninth-order exponent with a closed-form safe band. A step-by-step protocol turns a measured release into these numbers, and a fixed-seed reproducibility package regenerates every table and figure. We instantiate the audit on screened extreme-ultraviolet (EUV) roughness spectra as a model-conditioned case study, with deployment on measured releases the next step.

We study an infinite-horizon optimal consumption-investment problem for an investor with Epstein-Zin stochastic differential utility with stochastic investment opportunities in an incomplete market. Risk aversion and intertemporal substitution are separated, and we work in the regime $\theta\in(0,1)$, where there exists a unique generalised utility process for arbitrary non-negative progressively measurable consumption streams. Our main contribution is a variational characterisation of the value function. We show that the value function is the unique minimiser of a functional whose Euler-Lagrange equation coincides with the Hamilton-Jacobi-Bellman equation. Although the functional may be non-convex, the direct method yields existence, and we prove every minimiser is strictly positive, bounded, and classical. A verification theorem identifies any minimiser with the value function and gives feedback representations for optimal consumption and investment policies. The proof combines a change of measure to the myopic probability with uniqueness results for Epstein-Zin BSDEs and a perturbation argument for optimality. Examples with stochastic volatility, Gaussian excess returns, and fat-tailed excess returns illustrate the scope of the framework and its implications for intertemporal hedging.

This paper examines three approaches for modeling the dynamics of a flexible-link 2-DoF robotic arm to address unmodeled dynamics not captured by rigid-body models. Two physics informed models combine rigid-body dynamics (RBD) formulations with a Gaussian Mixture Model (GMM) to capture residual model errors and linkage flexibility. A kinematics-based regression model serves as a purely data-driven baseline. Using an open-source dataset, torque predictions are first estimated using Ridge regression on kinematic features, while the physicsbased baseline is constructed from published specifications, and ordinary least-squares regression is subsequently used to estimate the same parameter set directly from data. Results show that the physics-based parameters yield the poorest accuracy, while regularized and least-squares estimators align more closely with measured torques. Residual analysis and error metrics highlight the limitations of purely parametric models for flexible-link systems and underscore the value of regularization and data-driven identification, supporting developments of semi-parametric residual learning methods.

Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict $g_1 \star g_2$ for elements of a finite group $G$. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.

We investigate the energetics, interactions, and ordering tendencies of bimerons (cholesteric fingers of the second type, CF--2) and hopfions in thin films of chiral magnets and chiral liquid crystals hosting a conical background state. Although isolated bimerons possess positive eigen-energy with respect to the conical phase, they develop an attractive interaction mediated by the restructuring and partial overlap of their positive-energy shells, i.e., intermediate regions formed relative to the conical state. This attraction promotes the formation of bound pairs and extended bimeron chains, even in parameter regimes where a periodic bimeron lattice is no longer thermodynamically stable. Extending the analysis to three dimensions, we show that circularization of bimerons into hopfions renders their energy finite and gives rise to a well-defined metastability window closely linked to the stability range of cholesteric fingers. Isolated hopfions likewise exhibit an attractive interaction within the conical phase, leading to the formation of hexagonally ordered clusters. The attraction originates from the competition between favorable and unfavorable twist regions and from the energetic cost of the shell structures imposed by the conical background. Despite the presence of attractive pair potentials and cluster formation, we demonstrate that hexagonal hopfion lattices do not exhibit an equilibrium lattice period. Instead, the system evolves toward states in which the conical spiral or the CF--1 phase (cholesteric fingers of the first type) progressively invade the inter-soliton regions, thereby preventing crystallization. Our results reveal a regime of attraction without stable long-range order and clarify the interplay between topology, confinement, and conical-phase frustration in chiral magnet and liquid-crystal thin films.

Modeling interactions among multimodal, high-dimensional data is intrinsically challenging due to ultra-high dimensionality and complex dependence structure with high level noise. Screening methods are effective for reducing dimensionality, but most existing approaches shrink only the predictor space while retaining all outcomes. In cross-modal analyses, different outcomes often select different predictor subsets, so the union remains large and the response dimension is unchanged, limiting the practical benefit of screening. This gives rise to heavy computational burdens and poor interpretability. To address these limitations, we propose a new screening framework, Graph Independence Dual Screening (GIDS), which simultaneously reduces the dimensionality of response variables and predictors. We design computationally efficient algorithms that facilitate downstream selection procedures, improving accuracy and scalability, and establish supporting theoretical results. Extensive simulation studies demonstrate that GIDS outperforms existing methods that screen only predictors. To illustrate its utility, we applied GIDS to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset, analyzing interactions between genome-wide 865,353 DNA methylation and 49,386 transcriptomic variables. GIDS reduced the feature space to approximately 9,000 CpGs and 2,000 transcripts, uncovering blockwise interaction structures: clusters of CpG sites and gene transcripts with strong associations. These findings not only improve computational tractability but also yield interpretable biological insights, highlighting coordinated regulatory mechanisms underlying Alzheimer's disease.

The Membership Problem for Pedigree Polytope (M3P) asks, given $X\in\mathbb{Q}^{\binom{n}{3}}$, whether $X\in\mathrm{conv}(P_n)$, where $P_n$ is the set of all pedigrees. A pedigree is a structured encoding of a Hamiltonian cycle construction in $K_n$. We establish that M3P is solvable in strongly polynomial time via a recursively constructed layered network $(N_k, R_k, \mu)$ and a multicommodity flow problem MCF$(k)$. The necessary and sufficient condition for membership established is that the optimal total flow in MCF$(n-1)$ equals the maximum possible flow $z_{\max}$. The complexity analysis, grounded in Tardos's strongly polynomial algorithm for combinatorial linear programs (1986), shows that this condition can be checked in strongly polynomial time in the dimension of the matrix involved. By sufficiency, this implies M3P~$\in$~P. Since the Symmetric Travelling Salesman Problem (STSP) reduces to M3P via the Multistage Insertion (MI) formulation (Arthanari 1983), STSP is solvable in polynomial time, and the P this http URL question is resolved. The proofs leading to this result are fully machine-verified in Lean~4/Mathlib4, with zero unresolved \texttt{sorry}s in the main proof chain. The main contribution is the Lean~4 machine verification of all proofs in the main chain, resulting in \texttt{theorem p\_equals\_np}: P = NP. The Lean~4 formal verification covers the sufficiency of MCF(n-1) for membership in $\mathrm{conv}(P_n)$, and the P = NP chain via Maurras (2002), Grötschel--Lovász--Schrijver (1988), Cook (1971), and Karp (1972). The complete lean project (36 Lean~4 files, 2968/2968 build targets clean) is available at this https URL.

Automatic-differentiation-based inverse analysis methods, including physics-informed neural networks (PINNs) and differentiable programming, have recently shown great promise due to their ability to compute accurate gradients and convergence efficiency. However, their applicability to falling weight deflectometer (FWD) backcalculation remains unexplored. This study critically evaluates PINN-based inverse analysis for a multilayer pavement system and investigates differentiable finite element method (DiffFEM) as an alternative based on a synthetic benchmark. The standard PINN does not recover layer moduli because of the sharp domain discontinuities inherent to layered pavement systems. Although we use an extended PINN with domain decomposition (XPINN), which shows better performance on discontinuous domains, its performance remains highly sensitive to loss weighting and network architecture, and degrades under measurement noise. By contrast, DiffFEM consistently achieves more accurate, stable, and computationally efficient inversion results. These results indicate that DiffFEM, which enforces the governing physics as a hard constraint, yields better accuracy, robustness, and computational efficiency than PINN-based approaches, in which the governing physics is imposed as a soft constraint through the loss function. More broadly, the findings suggest that the choice between PINN- and DiffFEM-based inverse analysis needs careful consideration, with DiffFEM offering practical advantages when an efficient and robust differentiable forward solver is available.

Neural operators learn mappings between infinite-dimensional function spaces and provide a data-driven surrogate modeling paradigm for parametric partial differential equations (PDEs). Existing architectures typically obtain expressivity by parameterizing integral kernels in prescribed transform domains or by applying attention-like interactions over discretized spatial points. While these approaches have achieved substantial progress, they often face a persistent trade-off among physical interpretability, nonlocal spatial communication, mesh scalability, and computational cost. We propose a Light-inspired neural operator(LiNO), an operator-learning architecture whose latent evolution is decomposed into three mechanisms motivated by elementary light transport: reflection, refraction, and scattering. Reflection and refraction act as adaptive pointwise transformations in latent feature space, enabling local feature reorientation and anisotropic modulation, whereas scattering performs input-dependent nonlocal propagation over the physical domain. We first formulate scattering as a normalized pairwise kernel with relative positional bias, and then develop an efficient scattering variant that replaces explicit pairwise interactions with positive-feature global propagation and a local diffusion branch, reducing the dominant spatial complexity from quadratic to linear. This yields a structured neural operator that separates local feature modulation from global spatial communication while retaining a modular and interpretable latent evolution.

We present a framework to generalize the standard b-ary Gray code to get the k-bonacci ones obtained in [5] as well as many others by using theoretical tools that allow to make calculations on lists. We introduce the notion of Z-Gray product, from which we deduce sequences of lists of finite words avoiding a predefinite list Z of factors and which satisfy a power-associativity property as well a generalizations of the classical flipping digit property.

We identify two constructions from different mathematical traditions. In linear logic and realisability, logical types are generated rather than fixed in advance: one begins with a universe of realisers equipped with execution, uses orthogonality to test their interactions, and takes types to be the biorthogonally closed subsets. In enriched Isbell duality, a quantitative relation induces an adjunction whose fixed points form a category, its nucleus. These constructions proceed by different means; we show that, in the present setting, they produce the same objects. The shared datum is minimal: an associative product, called execution, and a real-valued measurement, with no compatibility assumed between them. The failure of the measurement to be additive is at once the relation defining orthogonality and the quantitative relation whose Isbell nucleus we form, and the types cut out by orthogonality are exactly the fixed points of the associated adjunction. The identification pays off in both directions. The most natural product of types fails to be associative; repairing this failure forces a different notion of type, sensitive to both sides of a composite, on which the induced product is associative and, when execution has units, carries two residuals. What emerges is a noncommutative Lambek calculus, derived directly from execution and orthogonality rather than imposed. In the reverse direction, each such type, read on the categorical side, generates a quantitative relation of its own, and with it a derived adjunction and a further generation of types; these derived types are again types of the original situation, computed by the residuals of the Lambek calculus. We also prove a coherence theorem for the threefold arrangements of this construction and, in the finite-dimensional case, give explicit formulas for the product.

Recently, the mapping of the replicator equation onto Bayes' theorem has been recognised, leading to an analogy between evolutionary dynamics and Bayesian learning. However, this analogy holds only for pure selection in infinite populations and breaks down when mutations -- a central mechanism of evolution -- are introduced. Here I propose that evolution by natural selection, at least for populations of haploid replicators in static environments, is best understood not as a learning process but as a process of causal inference. Each mutation event constitutes a natural experiment in which the parent serves as the control and the mutant offspring as the treated unit. Natural selection screens the causal effect of the mutation on fitness, retaining mutations with non-negative effects. I formalise this view within the Neyman-Rubin potential-outcomes framework. I first develop the general theory using a generic fitness outcome and show how the core identification assumptions in causal inference (Stable Unit Treatment Value Assumption, Consistency, Unconfoundedness, Positivity) map onto evolutionary biology. Using the unnormalised quasispecies equation, I prove that the intergenerational change in mean fitness decomposes exactly into a selection term -- recovering Fisher's Fundamental Theorem -- plus a mutation term that corresponds to a fitness-weighted average of the cumulated effect of all mutations over all parental genotypes. I show that this decomposition extends, under suitable assumptions, to the generalised replicator-mutator equation and that the frequencies of populations of matched parents-offspring update in proportion to the average causal effect of mutations on fitness.

We study a heat transfer equation in spatial dimensions $n = 1$, $2$, and $3$. A group classification with respect to the functional parameter $k = k(T)$ is done and symmetry algebras are presented. Recursion operators are found in the case $n = 1$ and infinite hierarchies of symmetries are constructed. We also find a number of exact solution in all the three cases.

We give detailed analysis and circuit design of structure-preserving quantum algorithms for second-order linear evolutionary PDEs, including parabolic equations and hyperbolic equations with mixed Dirichlet, Neumann, and periodic boundary conditions and source terms. While prior quantum algorithms usually neglect the stability problem from the PDE-to-ODE reduction, our method-of-lines approach investigates the boundary lifting via Coons interpolation and boundary-aware discretization, so that the resulting semi-discrete systems are stable and compatible with efficient quantum ODE primitives. For the parabolic problem, we use a diagonal similarity transform to ensure the semi-discrete generator must have a positive semi-definite Hermitian part, and then solve the resulting ODE system by the optimal linear combination of Hamiltonian simulation (LCHS). For the hyperbolic problem, we rewrite the semi-discrete equation as an equivalent first-order system and solve it by Hamiltonian simulation. We implement our quantum algorithms with explicit block-encoding constructions and circuit implementations, as well as demonstrating the end-to-end complexity bounds together with spatial and quadrature error estimates. We conduct classical numerical experiments on the convection-diffusion equation, inhomogeneous heat equation, and Klein-Gordon equation to validate our structure-preserving analysis and algorithmic constructions.

We introduce and investigate non-wellfounded and cyclic linear nested sequent calculi, and, as a case study, develop such systems for linear temporal logic (LTL). The paper addresses two central problems, which we call 'cycle recognition' and 'unraveling.' Cycle recognition concerns identifying cycles in non-wellfounded proofs in order to extract corresponding cyclic proofs, while unraveling studies the converse transformation, from cyclic proofs to non-wellfounded ones. Although these processes are well understood for Gentzen sequents, they have received little attention for more expressive sequent formalisms and become more challenging in the linear nested sequent setting. To address cycle recognition, we show the completeness of non-wellfounded proofs relative to a particular normal form exhibiting a property we call 'saturation recurrence,' which enables the systematic extraction of cyclic proofs. To address unraveling, we introduce a specialized procedure that shifts rule applications forward along linear nested sequents, allowing non-wellfounded proofs to be reconstructed from cyclic ones. Overall, our work provides new proof-theoretic techniques for cycle recognition and unraveling in expressive multisequent formalisms.

Modeling high-dimensional data is challenging, yet essential to understanding many complex systems. Maximum entropy models such as Ising and Potts models have been used extensively to capture pairwise interactions from correlation patterns in data, allowing to infer graphical representations of complex systems from observations (e.g., from protein sequences or neural population activity). Recently, there has been growing interest in modeling higher-order correlation patterns involving simultaneously three or more variables. While progress has been made in binary data with high-order Ising models, we extend this framework to the more general case of discrete data. We introduce q-state spin models, a complete family of maximum entropy models that generalize the vector Potts model to include long-range and arbitrary high-order interactions. In the pairwise case, our models allow for more diverse interaction types compared to the standard vector Potts model. We discuss their statistical interpretation with examples and relate them to discrete Fourier analysis. Using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions. We define gauge transformations under which this structure, and thus the partition function, remains invariant. Models equivalent under gauge transformations can be seen as different representations of the same abstract statistical model, despite generally having interactions of different orders, extending results from the binary case. For practical application to data analysis, we focus on a subset of models known in the binary case as Minimally Complex Models, generalizing them to discrete data. We obtain a closed-form expression for the marginal likelihood of these models, enabling fast model selection. We illustrate their use with simple real-world examples.

Building on the recently discovered origami-amplituhedron correspondence, we prove that the BCFW (Britto-Cachazo-Feng-Witten) cells triangulate the $m=4$ amplituhedron in full generality at all loop orders, both in momentum and momentum-twistor space. Along the way, we develop two natural "$L$-punctured" extensions of the positive Grassmannian and relate them via T-duality.

We answer a question posed by Poggiolesi concerning a syntactic decidability proof for GL in the tree-hypersequent system CSGL, and resolve a challenge identified by Maggesi and Perini Brogi, who sought a PSPACE proof-search algorithm for GL in expressive sequent-based formalisms. We work with a notational variant of CSGL formulated in terms of (labeled) tree sequents. Our answer is complexity-optimal: we present a proof-search algorithm that decides the (in)validity of formulae and runs in PSPACE, matching the known PSPACE-completeness of GL. To achieve this, we introduce a "linearization method," which constructs only a single branch of a derivation and of a tree sequent at a time, avoiding the exponential blowup typical of naive proof-search in sequent formalisms. We show how to systematically combine fragments of tree sequents generated during proof-search to extract finite counter-models, which serves as a theoretical device for establishing the correctness of the algorithm when proof-search fails. Finally, we show that every valid formula admits a proof consisting solely of line sequents, which correspond to linear nested sequents. This establishes a connection between depth-first proof-search and linear nested sequent calculi. Our results not only answer the aforementioned questions, but also provide new insights into proof-search and correctness arguments in tree sequent systems for modal logics.

The paper discusses renormalization aspects of the quantum four-dimensional Yang-Mills theory with a cutoff regularization in the coordinate representation. The background field method is used to formulate a generating functional, and the regularization is introduced through quasi-local probabilistic averaging. Two main types of regularization are proposed: strong deformation, which consists in averaging fluctuation fields, and weak deformation, which is a covariant generalization of the first case with respect to gauge transformations of the background field. We study singular contributions for the first two quantum corrections in this paper and compare them in detail with the case of dimensional regularization. The consistency of the action and the equation of motion after introducing the regularization and making a renormalization procedure is analyzed. New counter-vertices are studied, in particular their locality properties and dependence on the regularization parameter.

Quantum computing presents a promising method to overcome the efficiency and memory constraints in large-scale mechanical problems, with numerous successful applications demonstrated in fluid mechanics. However, solid mechanics problems usually require irregular grids for spatial discretization, due to the Lagrange formulations and complex boundaries, which makes the quantum simulation of the system matrix, e.g., the mass or stiffness matrix which is often referred to as the Hamiltonian in quantum computing, difficult to be effectively conducted. This study proposes a voxel-based quantum computing method (VBQC) for the quantum simulation of Hamiltonians in solid mechanics. VBQC applies voxel grids to discretize the spatial domain, thereby enabling the system matrix to exhibit the tridiagonal fractal property. Based on this property, the system matrix can be decomposed into three groups of fundamental matrices, $\mathbf{k}_{n}$, $\mathbf{c}_{n}$, and $\mathbf{q}_{n}$. This decomposition process is referred to as the KCQ decomposition. By integrating the KCQ decomposition with the quantum Fourier transform and the quantum multiplexer, VBQC enables efficient quantum simulation of Hamiltonians in solid mechanics. Three specific solid problems with different dimensions and numbers of variables are applied to preliminarily verify the correctness of the proposed VBQC for solid mechanics problems.

Hyperparameter optimization (HPO) for Random Forest faces a specific difficulty in tuning the number of trees: the predictive score typically improves monotonically with ensemble size, so standard methods such as Tree-structured Parzen Estimator (TPE) and Hyperband require a predefined search range and often drive the estimate toward its right boundary. Early-stopping strategies avoid fixing such a range, but can be sensitive to score noise and prone to premature stopping. To address this, we propose an integrated triplet-based plateau-search algorithm that removes the number of trees from the direct TPE search space and still exploits information accumulated across HPO trials. The method adaptively tracks a near-minimal sufficient ensemble size by monitoring relative changes in the out-of-bag (OOB) score across a triplet of forest sizes and shifting this triplet accordingly. This yields an automated and user-interpretable procedure based on a tolerance parameter. We also provide a theoretical analysis: we relate the proposed relative OOB-score criterion to the gap between the current and limiting scores, and derive an asymptotic variance estimate for the corresponding OOB-based absolute relative difference. Experiments show that the selected number of trees can differ substantially from the common heuristic: for most classical benchmark datasets it is smaller, whereas for some high-dimensional bioinformatics datasets, such as Arcene and Dorothea, it is larger. The source code and reproducible experiments are available at this https URL.

While principal component analysis (PCA) is a fundamental tool for dimensionality reduction, its dense representations make it ill-suited for high-dimensional data. Existing methods address this by promoting sparsity through explicit $\ell_1$-penalties, but these are not obvious to tune due to the unsupervised nature of the task. In contrast, we propose Adversarial PCA (AdvPCA), which leverages robust optimization to achieve sparsity by optimizing the reconstruction objective against bounded, worst-case latent space perturbations. We show that this formulation admits a closed-form reduction, leading to a practical iterative algorithm that alternates between adversarial linear regression-style updates for the sparse encoder and orthogonal updates for the decoder. By theoretically characterizing the solution, we derive a data-adaptive parameterization that allows the algorithm to perform effectively out of the box. We validate these claims through numerical experiments on synthetic and real-world genomics data.

For nonconvex optimization problems whose objective is the prediction function of a trained Support Vector Regression (SVR) model with the Gaussian radial basis function (RBF) kernel (RBF-SVR), we present a framework that applies the difference of convex functions (DC) algorithm (DCA) by exploiting the analytical structure of the RBF kernel to construct an explicit DC decomposition. Specifically, we derive in closed form both the lower bound $\mu$ of the strong convexity parameter of the DC components and the upper bound $L$ of the gradient Lipschitz constant of the subproblem. Both $\mu$ and $L$ are determined solely by the post-training dual-coefficient sum $C_{\alpha}$ and the RBF kernel parameter $\gamma$, together with the DC decomposition parameter $\rho$, and they share a common leading term $C_{\alpha}\rho$. Through numerical experiments on six benchmark functions, we show that $C_{\alpha}\rho$ is the primary single quantity characterizing both the convergence properties and the initial-point dependence of DCA, and further demonstrate that it decomposes into two independent pathways, $C \to C_{\alpha}$ and $\gamma \to \rho$, with its primary variation governed by the SVR hyperparameters $(C, \gamma)$. Together, these results allow the convergence properties of DCA on RBF-SVR to be assessed in advance through the single scalar quantity $C_{\alpha}\rho$: approximately from $(C, \gamma)$ before training, and exactly in closed form after training.

Channel charting enables location-based services (LBSs) without requiring explicit position information by using pseudo-locations from the channel chart. While this property implies inherent privacy advantages, it does not provide formal privacy guarantees. In this work, we address location privacy in channel charting referred to as chart location indistinguishability (CLI), which extends geo-indistinguishability (GI) to channel charting representations. In order to achieve CLI, a standard planar Laplace mechanism is investigated and a geometry-aware Mahalanobis norm planar Laplace (MNPL) mechanism is devised. The proposed MNPL mechanism perturbs the channel chart by injecting noise aligned with the local structure of the chart. In the CLI framework with MNPL, privacy is defined in latent channel chart manifolds using locally adaptive covariance derived from chart neighborhoods, while preserving manifold topology under privacy constraints. In addition, differential privacy is considered as a privacy baseline. The proposed approach is evaluated across multiple channel charting schemes. The performance is assessed using utility metrics such as quality loss (QL) and range query error (RQE), as well as geometry-aware metrics including trustworthiness (TW) and continuity (CT). Numerical results demonstrate that the proposed privacy mechanism provides strong privacy guarantees while preserving the channel chart for LBSs tasks.

Non-negative reduced biquaternion matrix factorization (NRBMF) uses the product of reduced biquaternion (RB) matrices to incorporate the non-negativity constraints of color image pixels into the factorization process. However, NRBMF mainly focuses on reconstruction accuracy and does not exploit the local geometric structure of image data, which may limit the discriminative ability of the learned low-dimensional features. To address this issue, we propose a graph regularized non-negative reduced biquaternion matrix factorization (GNRBMF) model for color image recognition. The proposed model incorporates a graph Laplacian regularizer into the reduced biquaternion coefficient matrix, encouraging nearby samples in the original space to have similar representations in the learned feature space. Meanwhile, GNRBMF retains the non-negativity-preserving property of NRBMF in the reduced biquaternion domain. To solve the optimization problem, a component-wise alternating projected gradient algorithm is derived, and its convergence properties are analyzed. Experimental results demonstrate that the proposed GNRBMF model achieves competitive or superior recognition performance in some tested settings.

We study projection-based diagnostics for distinguishing directional asymmetry from tail-ratio departure in multivariate data. The procedure reduces the problem to one-dimensional projections and computes two quantile-based summaries: a directional skewness measure evaluated over several quantile levels, and an interquantile tail-ratio evaluated relative to a chosen benchmark. The two summaries lead to a four-regime classification: symmetric benchmark-tail, symmetric tail-departed, skewed benchmark-tail, and skewed tail-departed. The quantile formulation avoids relying on third and fourth moments, which can be unstable in heavy-tailed settings. We establish population properties under central symmetry and ellipticity, uniform finite-sample bounds over the searched directions, and consistency of the threshold classifier under separated regimes. A sparse rank-one calculation is also used to show why coordinate directions can complement random directions in high dimensions. The resulting diagnostic is meant to guide subsequent modelling choices, for example whether a symmetric, skewed, tail-departed, or combined multivariate model is appropriate.

LLM pipelines waste substantial token budgets on low-information content: repeated context, verbose responses, and redundant boilerplate. We introduce Entropy Gate, a token compression framework applying entropy quenching $-$ a thermodynamic process that progressively freezes out low-energy tokens while preserving semantic fidelity. Each token receives a multi-factor information energy $E(t)$ combining statistical, structural, and positional components. An adaptive quenching schedule $T(\tau) = T_0 / (1 + \alpha \tau)$ removes tokens whose Boltzmann survival probability $p_i = \exp(-E_i / kT)$ falls below threshold, with a fidelity gate halting compression when energy-weighted similarity drops below $\theta$. We prove token selection by descending $E(t)$ maximizes expected semantic preservation, that quenching produces nested survival sets, and that achievable compression approaches the information-theoretic limit $\text{CR} \to 1 - I(P; T)/H(P)$. A Phase 1 heuristic achieves 40-60% compression across five prompt categories while maintaining $S_E > 0.80$, with energy-squared amplification $E \to E^2$ adding 10-25 percentage points. Context deduplication adds 50-70% savings on repeated blocks. Output-side quenching, motivated by findings that brevity improves accuracy, further reduces response overhead. Combined with external memory, reduction composes multiplicatively to 88-96% for agentic workloads. The framework is stateless, model-agnostic, and deploys as an OpenAI-compatible HTTP proxy.

A fermionic three-site generalization of the popular bosonic Swanson model is studied as providing an exactly solvable five-parametric example of the quantum-mechanical unitary-evolution process leading to an ultimate loss of the observability and fall in an exceptional-point singularity (EP3). The instant of degeneracy is found to have an explicit one-parametric form. Its unitarity-compatible vicinity (i.e., the corridor of access to EP3) is also specified in closed form. The exact, numerical-error-independent solvability is found essential due to another, avoided, false energy-level crossing which is found to occur not too far from the true EP3 singularity.

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.

Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. Existing rigorous methods are largely limited to special settings, while variational numerical approaches typically provide estimates rather than certified bounds. Here we introduce a complete family of certified upper bounds on the bulk spectral gap of quantum many-body systems. These upper bounds are obtained by solving a series of semidefinite programs and they become arbitrarily tight at the cost of more computational resources. This shows that the bulk spectral gap is semi-decidable, in contrast to undecidability results for alternative notions of spectral gap based on sequences of finite systems with prescribed boundary conditions. As a proof of principle, we apply our algorithm to the spin-$\frac{1}{2}$ kagome lattice Heisenberg antiferromagnet and obtain, to our knowledge, the first nontrivial certified upper bounds on its bulk spectral gap.

Computing the Lipschitz constant of the solution map of a multi-parametric quadratic program is important for the analysis of optimization-based control. This problem is governed by three factors: the parameter dimension, the number of decision variables, and the number of constraints. While empirical evidence has long suggested exponential complexity, a rigorous complexity-theoretic proof has been lacking. In this paper, we fill this gap by proving that this problem is not only NP-hard but also APX-hard. Furthermore, we reveal that: (a) the problem becomes polynomial-time solvable when the number of constraints or decision variables is fixed; and (b) both NP-hardness and APX-hardness persist even in the scalar parameter case. These results confirm that the complexity stems from the number of constraints and variables, rather than the parameter dimension. Numerical experiments further validate these theoretical findings.

Motivated by recent developments in stochastic modeling of clock jitter in Analog-to-Digital Converters (ADCs) as autoregressive processes of order one (AR(1)), we study the density and stability properties of AR(1)-jittered sampling sets for Paley-Wiener signals. We show that, despite having the correct asymptotic density both on average and almost surely, such sets almost surely fail to be stable sampling sets. We complement this negative result with a finite-dimensional analysis, showing that the corresponding jittered sinc matrices are nonetheless well-conditioned with high probability.

In this paper, we review hypergeometric functions $\mathscr{F}^{\rm Dw}_{\underline{a}}(t),$ $\mathscr{F}^{(\sigma)}_{\underline{a}}(t)$ and $\widehat{\mathscr{F}}^{(\sigma)} _{\mathbf{a}}(t)$ together with their conjectured transformation formulas, and show that one transformation formula implies the other.

We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $\gamma$ fraction of the vertices classified correctly, where $\gamma\in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati (2017) for non-uniform HSBMs. Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularized adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularization of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

We develop a theory of Burnside rings in the context of birational equivalences of algebraic varieties equipped with logarithmic volume forms. We introduce a residue homomorphism and construct an additive invariant of birational morphisms. We also define a specialization homomorphism. -- Nous proposons une théorie d'anneaux de Burnside dans le contexte de la géométrie birationnelle des variétés algébriques munies d'une forme volume à pôles logarithmiques. Nous introduisons un homomorphisme « résidu », construisons un invariant additif des morphismes birationnels. Nous définissons aussi un homomorphisme de spécialisation.

In directed acyclic graph (DAG)-based distributed ledgers, unreferenced blocks (tips) form the backlog of a distributed queueing system. Each new block creates one tip and attempts to remove up to $k$ existing tips by referencing them. With heterogeneous propagation delays, these service decisions are made from delayed local information, so nodes may disagree on the backlog and some reference attempts are wasted. We study a continuous-time Poisson model with bounded heterogeneous delays and uniform tip selection. We prove that the embedded tip-configuration chain is irreducible, aperiodic, and positive Harris recurrent, and hence admits a unique stationary regime. The observer and local tip-pool sizes have stationary exponential moments, converge to their stationary limits, and satisfy almost-sure ergodic averages. We also derive a Little-type identity relating the stationary mean observer tip count to the mean time until a typical block is first referenced. Simulations are included as qualitative illustrations of the effects of delay variability and issuance heterogeneity.

The symmetric case of the Szemerédi-Trotter theorem says that any configuration of $N$ lines and $N$ points in the plane has at most $O(N^{4/3})$ incidences. We describe a recipe involving just $O(N^{1/3})$ parameters which sometimes (that is, for some choices of the parameters) produces a configuration of N point and N lines. (Otherwise, we say the recipe fails.) We show that any near-extremal example for Szemerédi Trotter is densely related to a successful instance of the recipe. We obtain this result by getting structural information on cell decompositions for extremal Szemerédi-Trotter examples. We obtain analogous results for unit circles.

Consider the community detection problem in random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), where each hyperedge appears independently with some given probability depending only on the labels of its vertices. We establish, for the first time in the literature, a sharp threshold for exact recovery under this non-uniform case, subject to minor constraints; in particular, we consider the model with multiple communities. One crucial point here is that by aggregating information from all the uniform layers, we may obtain exact recovery even in cases when this may appear impossible if each layer were considered alone. Besides that, we prove a wide-ranging, information-theoretic lower bound on the number of misclassified vertices \emph{for any algorithm}, depending on a \emph{generalized Chernoff-Hellinger} divergence involving model parameters. We provide two efficient algorithms which successfully achieve exact recovery when above the threshold, and attain the lowest possible mismatch ratio when the exact recovery is impossible, proved to be optimal. The theoretical analysis of our algorithms relies on the concentration and regularization of the adjacency matrix for non-uniform random hypergraphs, which could be of independent interest. We also address some open problems regarding parameter knowledge and estimation.

In this paper, we propose a novel, computationally efficient reduced order method to solve linear parabolic inverse source problems. Our approach provides accurate numerical solutions without relying on specific training data. The forward solution is constructed using a Krylov sequence, while the source term is recovered via the conjugate gradient (CG) method. Under a weak regularity assumption on the solution of the parabolic partial differential equations (PDEs), we establish convergence of the forward solution and provide a rigorous error estimate for our method. Numerical results demonstrate that our approach offers substantial computational savings compared to the traditional finite element method (FEM) and retains equivalent accuracy.

The entropic doubling $\sigma_{\operatorname{ent}}[X]$ of a random variable $X$ taking values in an abelian group $G$ is a variant of the notion of the doubling constant $\sigma[A]$ of a finite subset $A$ of $G$, but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of Pálvölgyi and Zhelezov on the ``skew dimension'' of subsets of $\mathbf{Z}^D$ with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of $\mathbf{Z}^D$ with small doubling; (3) A proof that the Polynomial Freiman--Ruzsa conjecture over $\mathbf{F}_2$ implies the (weak) Polynomial Freiman--Ruzsa conjecture over $\mathbf{Z}$.

This paper is concerned with normalized solutions of the magnetic focusing Gross-Pitaevskii equations with anharmonic potentials in $\mathbb{R}^N$, where $N=2$ or $3$. We construct axially symmetric normalized concentrating solutions as the parameter $a>0$ approaches $a_*(N)$, where $a_*(N)\geq0$ is a critical constant depending only on $N$. We further prove that up to a constant phase (and a rotational transformation for $N=2$), normalized concentrating solutions are unique and axially symmetric as $a\to a_*(N)$. When $N=3$, we also prove that the corresponding unique normalized concentrating solution is free of vortices as $a\to a_*(3)$, even if the anharmonic potential is non-radially symmetric.

In mirror symmetry, symplectic Landau-Ginzburg models are mirror to a large class of examples, in particular to Fano varieties and hypersurfaces of many Calabi-Yau and Fano varieties. When studying their Fukaya categories on the A-model in homological mirror symmetry, one needs to calculate the weights of pseudo-holomorphic discs bounded by Lagrangian branes. While these calculations simplify for exact and Lefschetz fibrations, we generalize the machinery for computing these weights by dropping the exact and Lefschetz assumptions. For a general symplectic Landau-Ginzburg model, a singular symplectic fibration, we prove that the weights and Lagrangian gradings split into base and fiber components. This is used in many calculations of Fukaya-Seidel categories to provide evidence of Kontsevich's homological mirror symmetry conjecture.

Let $p\geq 3$ be a prime. The hyper-algebraic elements in the $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ form an algebraically closed subfield $\mathbb{L}_p^{\operatorname{ha}}$. In this article, we clarify the relations among the fields $\mathbb{L}_p^{\operatorname{ha}}$, $\overline{\mathbb{Q}}_p$ and $\mathbb{C}_p$. We introduce two arithmetic invariants (hyper-tame index and hyper-inertia index) of hyper-algebraic elements and study the relation between these invariants and classical arithmetic invariants of $p$-adic algebraic numbers. Finally, we give a criterion for hyper-algebraic elements to be tamely ramified over $\mathbb{Q}_p$.

String attractors are a combinatorial tool coming from the field of data compression. It is a set of positions within a word which captures an occurrence of every factor. While one-sided infinite words admitting a finite string attractor are eventually periodic, the situation is different for two-sided infinite words. In this article, we characterise the bi-infinite words admitting a finite string attractor as the characteristic Sturmian words and their morphic images. For words that do not admit finite string attractors, we study the structure and properties of their infinite string attractors.

The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems and to some Higman-Thompson groups.

Let $R$ be a commutative noetherian local ring, and let $M$ be a finitely generated $R$-module. Inspired by works of Vasconcelos and Briggs on characterization of complete intersection local rings through the homological properties of the conormal module, in this paper, we define the characteristic module T$_M$ and the cocharacteristic module E$_M$ of $M$, and investigate their properties. Our main results include characterizations of Cohen--Macaulay and Gorenstein local rings. Also, we show that if the injective dimension of the conormal module over an almost complete intersection ring is finite, then $R$ is a complete intersection.

In this work, we consider the $H_2$ optimal model reduction of dynamical systems that are linear in the state equation and up to quadratic nonlinearity in the output equation. As our primary theoretical contributions, we derive gradients of the squared $H_2$ system error with respect to the reduced model quantities and, from the stationary points of these gradients, introduce Gramian-based first-order necessary conditions for the $H_2$ optimal approximation of a linear quadratic output (LQO) system. The resulting $H_2$ optimality framework neatly generalizes the analogous Gramian-based optimality framework for purely linear systems. Computationally, we show how to enforce the necessary optimality conditions using Petrov-Galerkin projection; the corresponding projection matrices are obtained from a pair of Sylvester equations. Based on this result, we propose an iteratively corrected algorithm for the $H_2$ model reduction of LQO systems, which we refer to as LQO-TSIA (linear quadratic output two-sided iteration algorithm). Numerical examples are included to illustrate the effectiveness of the proposed computational method against other existing approaches.

For a polarized toric variety, we provide a sufficient criterion ensuring that a uniformly K-stable polarized toric variety $(X,L)$ is asymptotically Chow polystable, under the assumption that the obstruction to asymptotic Chow semistability (the Futaki-Ono invariant) vanishes. Our approach is based on a detailed study of triangulations of neighborhoods of the vertices of the associated moment polytope $\Delta$. As an application, we prove that every uniformly K-stable polarized smooth toric variety $(X,L)$ with vanishing Futaki-Ono invariant is asymptotically Chow polystable.

The patch dynamics scheme in equation-free multiscale modelling has the potential to efficiently predict the macroscopic behaviours by simulating the microscale problem in a fraction of the space-time domain. The patch dynamics schemes developed so far are mainly on rectangular domains with uniform grids and uniform rectangular patches. In real-life problems, the geometry of the domain is not regular or simple, where rectangular and uniform grids or patches may not be useful. To address this kind of complexity, for the first time, a generalised orthogonal curvilinear coordinate system is employed in the patch dynamics scheme, applicable to both rectangular domains with non-uniform grids and non-rectangular domains; while applying this, the concept of non-uniform and non-rectangular patch configurations in the physical domain is also adopted for the first time. An explicit representation of a patch dynamics scheme on a generalised curvilinear coordinate system in a two-dimensional domain is proposed for unsteady, linear, heterogeneous convection-diffusion-reaction (CDR) problems. The proposed scheme is validated through heterogeneous convection-diffusion-reaction and non-axisymmetric diffusion problems on generalised curvilinear coordinate systems. The results demonstrate excellent accuracy and show that the method significantly outperforms full-domain simulations in terms of computational efficiency, memory usage and overall performance.

We consider the problem of minimizing a polynomial $f$ over the (binary) hypercube. We show that, for a specific set of polynomials, their binary non-negativity (i.e. on the hypercube) can be checked in polynomial time via minimum cut algorithms, from which we construct a linear programming representation for this set of polynomials. We categorize binary polynomials according to their signed support patterns and develop parameterized linear programming representations for binary non-negative polynomials. This allows the construction of signed certificates of binary non-negativity with adjustable signed support patterns and representation complexities; and we propose a method for minimizing $f$ by decomposing it as a sum of signed certificates. This method yields new hierarchies of linear programming relaxations for binary polynomial optimization. Moreover, since our decomposition depends only on the support of $f$, the new hierarchies are sparsity-preserving.

In this paper, we begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solution to the Pythagorean equation $a^2 + b^2 = c^2$, these closed forms become even more intriguing. This approach allows us to provide new integral representations for the error function. Additionally, by leveraging an identity we derived for the inverse Laplace transform and applying a result based on Srivastava and Yürekli's identity, we provide a closed form for a nontrivial generalized integral.

Semidefinite programs (SDPs) and their solvers are powerful tools with many applications in machine learning and data science. Designing scalable SDP solvers is challenging because by standard the positive semidefinite decision variable is an $n \times n$ dense matrix, even though the input is often an $n \times n$ sparse matrix. However, the solution may not require a full-rank matrix, as shown by Barvinok and Pataki. Two decades ago, Burer and Monteiro developed an SDP solver \texttt{SDPLR} that optimizes over a low-rank factorization instead of the full matrix. This greatly decreases the storage cost and works well for many problems. The original solver \texttt{SDPLR} tracks only the primal infeasibility of the solution, preventing early termination at moderate accuracy. We use a suboptimality bound for trace-bounded SDP problems that enables us to track the progress better and perform early termination. We then develop \texttt{SDPLR+}, which starts the optimization with an extremely low-rank factorization and dynamically updates the rank based on the primal infeasibility and suboptimality. This further speeds up the computation and saves storage. Numerical comparisons on Max Cut, Minimum Bisection, Cut Norm, and Lovász Theta problems with many recent memory-efficient scalable SDP solvers demonstrate the scalability of \texttt{SDPLR+} up to problems with million-by-million decision variables. It is often the fastest solver to a moderate accuracy of $10^{-2}$. Further experiments on $\mu$-conductance, matrix completion, and $k$-means clustering show the potential of \texttt{SDPLR+} on a broader range of data science applications.

In this paper a proof is given of Sugawara's conjecture from 1936, that the ray class field of conductor $\mathfrak{f}$ over an imaginary quadratic field $K$ is generated over $K$ by a single primitive $\mathfrak{f}$-division value of the $\tau$-function, first defined by Weber and then modified by Hasse in his 1927 paper giving a new foundation of complex multiplication.

Kronecker observed that either all roots or only one root of a solvable irreducible equation of odd prime degree with integer coefficients are real. This gives a possibility to construct specific examples of equations not solvable by radicals. A relatively elementary proof without using the full power of Galois theory is due to Weber. We give a rather short proof of Kronecker's theorem with a slightly different argument from Weber's. Several modern presentations of Weber's proof contain inaccuracies, which can be traced back to an error in the original proof. We discuss this error and how it can be corrected.

Let $X$ be a smooth projective toric variety, and let $\widetilde{X}$ denote the blow-up of $X$ at finitely many distinct torus-invariant points. In this paper, we derive an explicit combinatorial formula for the Chow weight of $\widetilde{X}$ in terms of the underlying toric manifold $X$ and the symplectic cuts of its associated Delzant polytope. As an application, we study toric blow-ups of the projective plane and compare their Chow stability with that of blow-ups at general points.

We consider a general Kinetic Fokker-Planck (KFP) equation in a domain with Maxwell reflection condition on the boundary, not necessarily with conservation of mass. We establish the wellposedness in many spaces including Radon measures spaces, and in particular the existence and uniqueness of fundamental solutions. We also establish a Krein-Rutman theorem with constructive rate of convergence in an abstract setting that we use for proving that the solutions to the KFP equation converge toward the conveniently normalized first eigenfunction. Both results use the ultracontractivity of the associated semigroup in a fundamental way.

We introduce and document a class of probability distributions, called bilateral generalized inverse Gaussian (BGIG) distributions, that are obtained by convolution of two generalized inverse Gaussian distributions supported by the positive and negative semi-axis. We prove several results regarding their analyticity, shapes and asymptotics, and we introduce the associated Lévy processes as well as their main properties. We study the behaviour of these processes under change of measure, their simulations and the structure of their sample paths, and we introduce a stock market model constructed by means of exponential BGIG processes. Based on real market data, we show that this model is easy to calibrate thanks notably to idiosyncratic properties of BGIG distributions, and that it is well suited to Monte Carlo and Fourier option pricing.

In 1939 Pál Turán and János Erőd initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a matter of normalization they considered the family $\mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given convex, compact subset $K\Subset {\mathbb C}$. While Turán obtained the first results for the interval $I:=[-1,1]$ and the disk $D:=\{ z\in {\mathbb C}~:~ |z|\le 1\}$, Erőd extended investigations to other compact convex domains, too. The order of the optimal constant was found to be $\sqrt{n}$ for $I$ and $n$ for $D$. It took until 2006 to clarify that all compact convex \emph{domains} (with nonempty interior), follow the pattern of the disk, and admit an order $n$ inequality. For $L^q(\partial K)$ norms with any $1\le q <\infty$ we obtained order $n$ results for various classes of domains. Further, in the generality of all convex, compact domains we could show a $c n/\log n$ lower bound together with an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all compact convex domains admit an order $n$ Turán type inequality. Here we prove this for all \emph{polygonal} convex domains and any $0< q <\infty$.

Let $\Bbbk$ be an algebraically closed field of characteristic $0$ and let $H$ be a finite-dimensional Hopf algebra over $\Bbbk$ with the dual Chevalley property. In this paper, we give a description of the link quiver of $H$ for different corepresentation types. Moreover, we show that $\operatorname{gr}^c(H)$ is of tame corepresentation type if and only if $\operatorname{gr}^c(H)\cong (\k\langle x,y\rangle/I)^* \times H_0$ for some special ideals $I$. Using the methods of link quivers and bosonization, we then discuss which of the above ideals occur when $(\Bbbk\langle x,y\rangle/I)^* \times H_0$ is a Hopf algebra of tame corepresentation type under certain assumptions.

We give several formulas for how Iwasawa $\mu$-invariants of abelian varieties over unramified $\mathbb{Z}_{p}$-extensions of function fields change under isogeny. These are analogues of Schneider's formula in the number field setting. We also prove that the validity of the Birch--Swinnerton-Dyer conjecture (including the leading coefficient formula) over function fields is invariant under isogeny, without using the result of Kato--Trihan.

The motivic version of the $c_1$-spherical cobordism spectrum is constructed. A connection of this spectrum with other motivic Thom spectra is established. Using this connection, we compute the $\mathbb{P}^1$-diagonal of the homotopy groups of the special linear algebraic cobordism $\pi_{2*,*}(\mathrm{MSL})$ over a local Dedekind domain $k$ with $1/2\in k$ after inverting the exponential characteristic of the residue field of $k$. We discuss the action of the motivic Hopf element $\eta$ on this ring, obtain a description of the localization away from $2$ and compute the $2$-primary torsion subgroup. The complete answer is given in terms of the special unitary cobordism ring. An important component of the computation is the construction of Pontryagin characteristic numbers with values in the hermitian K-theory. We also construct Chern numbers in this setting, prove the motivic version of the Anderson-Brown-Peterson theorem and briefly discuss classes of Calabi-Yau varieties in the $\mathrm{SL}$-cobordism ring.

In this contribution, kernel approximations are applied as ansatz functions within the Deep Ritz method. This allows to approximate weak solutions of elliptic partial differential equations with weak enforcement of boundary conditions using Nitsche's method. A priori error estimates are proven in different norms leveraging both standard results for weak solutions of elliptic equations and well-established convergence results for kernel methods. This availability of a priori error estimates renders the method useful for practical purposes. The procedure is described in detail, meanwhile providing practical hints and implementation details. By means of numerical examples, the performance of the proposed approach is evaluated numerically and the results agree with the theoretical findings.

In his book "Metric structures for Riemannian and non-Riemannian spaces", Gromov defined two properties of Riemannian manifolds, ellipticity and quasiregular ellipticity, and suggested that there may be a connection between the two. Since then, groups of researchers working independently have proved strikingly similar results about these two concepts. We obtain new topological obstructions to the two properties: most notably, we show that closed manifolds of both types must have virtually abelian fundamental group. We also give the first examples of open manifolds which are elliptic but not quasireguarly elliptic and vice versa. Whether there is a direct connection between these properties -- and, in particular, whether they are equivalent for closed manifolds -- remains elusive.

We present a comprehensive study of two new Poisson-type algebras. Namely, we are working with $\delta$-Poisson and transposed $\delta$-Poisson algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to shift associative algebras, $F$-manifold algebras, algebras of Jordan brackets, etc. We classify simple $\delta$-Poisson and transposed $\delta$-Poisson algebras and found their depolarizations. We study $\delta$-Poisson and mixed-Poisson algebras to be Koszul and self-dual. Bases of the free $\delta$-Poisson and mixed-Poisson algebras generated by a countable set $X$ are constructed.

Bounded minimizers of double phase problems at nearly linear growth have locally Hölder continuous gradient within the sharp maximal nonuniformity range $q<1+\alpha$.

We prove that triangle Artin groups of the type $A_{2,3,2n}$ are residually finite for all $n\geq4$. This requires splitting these triangle Artin groups as graphs of groups and then proving that each of these graphs of groups has finite stature with respect to its vertex groups.

Nested integration problems arise in various scientific and engineering applications, including Bayesian experimental design, financial risk assessment, and uncertainty quantification. These nested integrals take the form $\int f\left(\int g(\boldsymbol{y},\boldsymbol{x})\mathrm{d}\boldsymbol{x}\right)\mathrm{d}\boldsymbol{y}$, for nonlinear $f$, making them computationally challenging, particularly in high-dimensional settings. Although widely used for single integrals, traditional Monte Carlo (MC) methods can be inefficient when encountering complexities of nested integration. This work introduces a novel multilevel estimator, combining deterministic and randomized quasi-MC (rQMC) methods to handle nested integration problems efficiently. In this context, the inner number of samples and the discretization accuracy of the inner integrand evaluation constitute the level. We provide a comprehensive theoretical analysis of the estimator, deriving error bounds demonstrating significant reductions in bias and variance compared with standard methods. The proposed estimator is particularly effective in scenarios where the integrand is evaluated approximately, as it adapts to different levels of resolution without compromising precision. We verify the performance of our method via numerical experiments, focusing on estimating the expected information gain of experiments. When applied to Gaussian noise in the experiment, a truncation scheme ensures finite error bounds. The results reveal that the proposed multilevel rQMC estimator outperforms existing MC and rQMC approaches, offering a substantial reduction in computational costs and offering a powerful tool for practitioners dealing with complex, nested integration problems across various domains.

We introduce a new analytic method for studying Witten zeta function of a root system $\Phi$, based on a refined manipulation of an integral representation involving the Hurwitz zeta function. As an application, we prove high-order vanishing at negative even integers. This technique also describes non-trivially, the arithmetic nature of the leading term, in which the highest root of $\Phi$ makes a surprising appearance.

We study analytic properties of graph product of finite groups with a hyperbolic defining graph. This is done by studying dynamics on the Bowditch compactification of the extension graph, or the crossing graph, of graph product. In particular, we provide a new class of convergence groups and identify the if and only if condition for this convergence action to be geometrically finite. We also provide a new class of properly proximal groups, relatively bi-exact groups, and groups with strongly solid group von Neumann algebras.

Let $\Psi(\textbf{z},\textbf{a},q)$ a the fundamental solution matrix of the quantum difference equation of a Nakajima variety $X$. In this work, we prove that the operator $$ \Psi(\textbf{z},\textbf{a},q) \Psi\left(\textbf{z}^p,\textbf{a}^p,q^{p^2}\right)^{-1} $$ has no poles at the primitive complex $p$-th roots of unity $q=\zeta_p$. As a byproduct, we show that the iterated product of the operators ${\bf M}_{\mathcal{L}}(\textbf{z},\textbf{a},q )$ from the $q$-difference equation on $X$: $$ {\bf M}_{\mathcal{L}} (\textbf{z} q^{(p-1)\mathcal{L}},\textbf{a},q) \cdots {\bf M}_{\mathcal{L}} (\textbf{z} q^{\mathcal{L}},\textbf{a},q) {\bf M}_{\mathcal{L}} (\textbf{z} ,\textbf{a},q) $$ evaluated at $q=\zeta_p$ has the same eigenvalues as ${\bf M}_{\mathcal{L}} (\textbf{z}^p,\textbf{a}^p,q^p)$. Upon a reduction of the quantum difference equation of $X$ to the quantum differential equation over the field of finite characteristic, the above iterated product transforms into a Grothendiek-Katz $p$-curvature of the corresponding quantum connection whreas ${\bf M}_{\mathcal{L}} (\textbf{z}^p,\textbf{a}^p,q^p)$ becomes a certain Frobenius twist of that connection. In this way, we give an explicit description of the spectrum of the $p$-curvature of quantum connection for Nakajima varieties.

Jacobi diffusion is a representative diffusion process whose solution is bounded in a domain under certain drift and diffusion coefficient conditions. However, the process without such conditions has not been thoroughly investigated. We explore a Jacobi diffusion whose drift coefficient is affected by another deterministic process, causing the process to hit the boundary of a domain in finite time. The Kolmogorov equation (a degenerate elliptic partial differential equation) for evaluating the boundary hitting of the proposed Jacobi diffusion is then presented and analyzed, with several conditional arguments, some of which are addressed computationally. We also investigate a related mean-field-type (McKean-Vlasov) self-consistent model arising in tourism management, where the drift depends on the index for sensor boundary hitting, thereby confining the process to a domain with higher probability. We propose a finite difference method for the linear and nonlinear Kolmogorov equations, which yields a unique numerical solution because of discrete ellipticity if the discount is positive. The accuracy of the finite difference method critically depends on the regularity of the boundary condition, and the use of high-order discretization is not always effective. Finally, we computationally investigate the mean field effect.

We construct an action of the positive part of the Witt algebra on Khovanov--Rozansky $\mathfrak{gl}_N$-link homology and show that link cobordisms induce equivariant maps between twists of the homology. Moreover, the state spaces of simple webs are identified with standard representations of the Witt algebra on polynomials. Some simple relations to Lee homology and genus bounds are derived from the analysis of this presentation.

We construct bulk-deformed orbifold Hamiltonian Floer theory for a global quotient orbifold, that is the quotient of a smooth closed symplectic manifold by a finite group acting faithfully via symplectomorphisms. The moduli spaces define an `ordered marked Flow category', which we equip with a coherent presentation via derived orbifolds. The global charts for orbifold Floer cylinders are built from moduli spaces of holomorphic curves in a quotient of projective space by a free action of the given finite group.

The global well-posedness of the multi-dimensional compressible Euler equations with damping remains a longstanding open problem. This problem has been partially resolved in the isentropic regime ({\it i.e.}, the adiabatic exponent \(\gamma>1\)) for small smooth initial data (see \cite{WY, STW}). In this paper, we establish the global well-posedness and asymptotic decay of smooth solutions of the Cauchy problem of the three-dimensional compressible Euler equations with damping for the isentropic regime \(\gamma>1\) and the isothermal regime \(\gamma=1\), allowing for partially large initial data. More precisely, the \(L^2\)-norm of the initial data is allowed to be large, while the third-order Sobolev norm of the initial data is assumed to be small. For the isentropic case, we develop a new analytical framework in which all required {\it a priori} estimates of solution $(\rho,u)$ can be derived under the condition that $\int_0^T \big( \|\nabla\rho\|_{L^\infty} + \|\nabla u\|_{L^\infty} \big) \, \mathrm{d}t$ remains sufficiently small. Moreover, we obtain the optimal algebraic decay rates of global solutions. Furthermore, we study the isothermal limit of solutions of the isentropic regime as $\gamma \to 1$, and establish the global well-posedness and asymptotic decay of solutions to the isothermal Euler equations with damping.

The classical and extended deficiency one theorems by Feinberg apply to reaction networks with mass-action kinetics that have independent linkage classes or subnetworks, each with a deficiency of at most one and exactly one absorbing strong component. The theorems assume the existence of a positive equilibrium and guarantee the existence of a unique positive equilibrium in every stoichiometric compatibility class. In our work, we use the $\textit{monomial dependency}$ which extends the concept of deficiency. First, we provide a dependency one theorem for parametrized systems of polynomial equations that are essentially univariate and decomposable. As our main result, we present a corresponding theorem for mass-action systems, which permits subnetworks with arbitrary deficiency and arbitrary number of absorbing strong components. Finally, to complete the picture, we derive the extended deficiency one theorem as a special case of our more general dependency one theorem.

In this paper, we present an independent proof of the Cauchy identities for staircase matrices, originally discovered in arXiv:2411.03117, using the combinatorics of the Bruhat poset and the bubble-sort procedure. Additionally, we derive new insights into certain coefficients appearing in one of these identities. The first part of the paper focuses on combinatorial aspects. It is self-contained, of independent interest, and introduces a generalization of parabolic Bruhat graphs for monotone functions on an arborescent poset. The second part examines the intersections of Demazure modules within a given integrable representation. Finally, we propose a generalization of the classical Howe duality for staircase matrices in terms of the corresponding distributive lattice of Demazure submodules. Computing the associated character yields the desired Cauchy identities for staircase matrices.

The field of proof-theoretic semantics (P-tS) offers an alternative approach to meaning in logic that is based on inference and argument (rather than truth in a model). It has been successfully developed for various logics; in particular, Sandqvist has developed such semantics for both classical and intuitionistic logic. In the case of classical logic, P-tS provides a conception of consequence that avoids an a priori commitment to the principle of bivalence, addressing what Dummett identified as a significant foundational challenge in logic. In this paper, we propose an alternative P-tS for classical logic, which essentially extends the P-tS for intuitionistic logic by operating over literals rather than atomic propositions. Importantly, literals are atomic and not defined by negation but are related by a primitive duality encoded inferentially at the atomic level. This semantics illustrates the perspective that classical logic can be understood as intuitionistic logic supplemented by a principle of duality, offering fresh insights into the relationship between these two systems.

We prove that any complex local representation of the flat virtual braid group, $FVB_2$, into $GL_2(\mathbb{C})$, has one of the types $\lambda_i: FVB_2 \rightarrow GL_2(\mathbb{C})$, $1\leq i\leq 12$. We find necessary and sufficient conditions that guarantee the irreducibility of representations of type $\lambda_i$, $1\leq i\leq 5$, and we prove that representations of type $\lambda_i$, $6\leq i\leq 12$, are reducible. Regarding faithfulness, we find necessary and sufficient conditions for representations of type $\lambda_6$ or $\lambda_7$ to be faithful. Moreover, we give sufficient conditions for representations of type $\lambda_1$, $\lambda_2$, or $\lambda_4$ to be unfaithful, and we show that representations of type $\lambda_i$, $i=3, 5, 8, 9, 10, 11, 12$ are unfaithful. We prove that any complex homogeneous local representations of the flat virtual braid group, $FVB_n$, into $GL_{n}(\mathbb{C})$, for $n\geq 3$, has one of the types $\gamma_i: FVB_n \rightarrow GL_n(\mathbb{C})$, $i=1, 2$. We then prove that representations of type $\gamma_1: FVB_n \rightarrow GL_n(\mathbb{C})$ are reducible for $n\geq 6$, while representations of type $\gamma_2: FVB_n \rightarrow GL_n(\mathbb{C})$ are irreducible if and only if $b\neq y$, for $n\geq 3$. Then, we show that representations of type $\gamma_1$ are unfaithful for $n\geq 3$ and that representations of type $\gamma_2$ are unfaithful if $y=b$. Furthermore, we prove that any complex homogeneous local representation of the flat virtual braid group, $FVB_n$, into $GL_{n+1}(\mathbb{C})$, for all $n\geq 4$, has one of the types $\delta_i: FVB_n \rightarrow GL_{n+1}(\mathbb{C})$, $1\leq i\leq 8$. We prove that these representations are reducible for $n\geq 10$. Then, we show that representations of types $\delta_i$, $i\neq 5, 6$, are unfaithful, while representations of types $\delta_5$ or $\delta_6$ are unfaithful if $x=y$.

We prove that cw-hyperbolic homeomorphisms with jointly continuous stable/unstable holonomies satisfy the periodic shadowing property and, if they are topologically mixing, the periodic specification property. We discuss difficulties to adapt Bowen's techniques to obtain a measure of maximal entropy for cw-hyperbolic homeomorphisms, exhibit the unique measure of maximal entropy for Walter's pseudo-Anosov diffeomorphism of $\mathbb{S}^2$, and prove it can be obtained, as in the expansive case, as the weak* limit of an average of Dirac measures on periodic orbits. As an application, we exhibit the unique measure of maximal entropy for the homeomorphism on the Sierpiński Carpet defined in [12], which does not satisfy the specification property.

We address the enumeration of Eulerian orientations of 4-valent planar maps according to three parameters: the number of vertices, the number of alternating vertices (having in/out/in/out incident edges), and the number of clockwise oriented faces. This is a refinement of the six vertex model studied by Kostov, then Zinn-Justin and Elvey Price, where one only considers the first two parameters. Via a bijection of Ambjorn and Budd, our problem is equivalent to the enumeration of Eulerian partial orientations of general planar maps, counted by the number of edges, the number of undirected edges, and the number of vertices. We first derive from combinatorial arguments a system of functional equations characterising the associated trivariate series $Q(t,\omega,v)$. We then derive from this system a compact characterisation of this series. We use it to determine $Q(t,\omega,v)$ in three two-parameter cases. The first two cases correspond to setting the variable $\omega$ counting alternating vertices (or undirected edges after the AB bijection) to $0$ or $1$: when $\omega=0$ we count Eulerian orientations of general planar maps by edges and vertices, and when $\omega=1$ we count Eulerian orientations of quartic maps by vertices and clockwise faces. The final forms of these two series, namely $Q(t,0, v)$ and $Q(t,1,v)$, refine those obtained by the authors in an earlier paper for $v=1$. The third case that we solve, namely $v=1$ (but $\omega$ arbitrary), is the standard six-vertex model, for which we provide a new proof of the formula of Elvey Price and Zinn-Justin involving Jacobi theta functions. This new derivation remains purely in the world of formal power series, not relying on complex analysis. Our results also use a more direct approach to solving the functional equations, in contrast to the guess and check approaches used in previous work.

We study feedback control of the Kuramoto model with uniformly spaced natural frequencies defined on uniform graphs which may be complete, random dense or random sparse. The control objective is to drive all nodes to the same constant rotational motion. For the case of node number $n\ge 3$, we establish the existence of exactly $2^n$ synchronized solutions in the controlled Kuramoto model (CKM) and their saddle-node and pitchfork bifurcations, and determine their stability. In particular, we show that only a solution converging to the desired motion in the limit of infinite feedback gain is stable and the others are unstable. Based on the previous results, it is shown that (i) the solution to which the stable synchronized solution in the CKM converge as $n\to\infty$ is always asymptotically stable in the continuous limit (CL) if it exists, and (ii) the asymptotically stable solution of the CL captures the asymptotic behavior of the CKM when the node number is sufficiently large, even if the graphs are random dense or sparse. We demonstrate the theoretical results by numerical simulations for the CKM on complete simple, and uniform random dense and sparse graphs.

For a generalized step reinforced random walk, starting from the origin, the first step is taken according to the first element of an innovation sequence. Then in subsequent epochs, it recalls a past epoch with probability proportional to a regularly varying sequence $\{\mu_n\}$ of index $\gamma>-1$; recalls and repeats the step taken with probability $p$, or with probability $1-p$ takes a fresh step from the innovation sequence. The innovation sequence is assumed to be i.i.d.\ with mean zero. We study the corresponding step reinforced random walk process with linearly scaled time as an r.c.l.l.\ function on $[0, \infty)$. We prove law of large numbers for the linearly scaled process almost surely and in $L^1$ for all possible values of $p$ and $\gamma$. Assuming finite second moments for the innovation sequence, we obtain interesting phase transitions based on the boundedness of a sequence associated with $\{\mu_n\}$. The random walk suitably scaled converges almost surely to a process, which may not be Gaussian, when the sequence is bounded and the convergence is in distribution to a Gaussian process otherwise. This phase transition introduces the point of criticality at $p_c=\frac{\gamma+1/2}{\gamma+1}$ for $\gamma>-\frac12$. For the subcritical regime, the process is diffusive, while it is superdiffusive otherwise. However, for the critical regime, the scaled process can converge almost surely or in distribution depending on the choice of sequence $\{\mu_n\}$. Almost sure convergence in the critical regime is new. In the critical regime, the scaling can include many more novel choices in addition to $\sqrt{n \log n}$. Further, we use linear time scale and time independent scales even for the critical regime. We argue the exponential time scale for the critical regime is not natural. All the convergences in all the regimes are obtained for the process as an r.c.l.l.\ function.

In this paper, we prove that the Hilbert scheme $\mathbf{H}_4(X_5)$ of rational quartic curves on the quintic del Pezzo threefold $X_5$ is isomorphic to a Grassmannian bundle over the Hilbert scheme of lines on $X_5$. In particular, $\mathbf{H}_4(X_5)$ is smooth and irreducible. Our approach builds upon the geometry of rational quartic curves on $X_5$ studied by Fanelli-Gruson-Perrin in their work on the moduli space of stable maps to $X_5$.

We investigate the convergence to (quasi--)equilibrium of a density dependent Markov chain in~${\mathbb Z}^d$, whose drift satisfies a system of ordinary differential equations having an attractive fixed point. For a sequence of such processes~${\mathbb X}^N$, indexed by a size parameter~$N$, the time taken until the distribution of~${\mathbb X}^N$, started in some given state, approaches its (quasi--)equilibrium distribution~$\pi^N$ typically increases with~$N$. To first order, it corresponds to the time~$t_N$ at which the solution to the drift equations reaches a distance of~$\sqrt N$ from their fixed point. However, the length of the time interval over which the total variation distance between ${\mathcal L} ({\mathbb X}^N(t))$ and its (quasi--)equilibrium distribution~$\pi^N$ changes from being close to~$1$ to being close to zero is asymptotically of smaller order than~$t_N$. In this sense, the chains exhibit `cut--off', and we are able to prove that the cut-off window is of (optimal) constant size.

We consider the eigenvalue problem for certain classes of elliptic operators, namely inhomogeneous membrane operators $ L = \tfrac{1}{ \rho } ( -\Delta + V ) $ and divergence form operators $ L = -\operatorname{div} A \nabla $, on bounded domains. For these operators, we prove ordering inequalities between the Dirichlet and the Neumann eigenvalues, generalizing results of Levine--Weinberger and Friedlander--Filonov for the Laplacian. We take inspiration from their proofs and derive sufficient conditions on the coefficients of the operator that ensure that the inequalities remain valid.

We propose transformed Diffsuion-Wave fractional Physics-Informed Neural Networks (tDWfPINNs) for efficiently solving time-fractional diffusion-wave equations with fractional order $\alpha\in(1,2)$. Conventional numerical methods for these equations often compromise the mesh-free advantage of Physics-Informed Neural Networks (PINNs) or impose high computational costs when computing fractional derivatives. The proposed method avoids first-order derivative calculations at quadrature points by introducing an integrand transformation technique, significantly reducing computational costs associated with fractional derivative evaluation while preserving accuracy. We conduct a comprehensive comparative analysis applying this integrand transformation in conjunction with both Monte Carlo integration and Gauss-Jacobi quadrature schemes across various time-fractional PDEs. Our results demonstrate that tDWfPINNs achieve superior computational efficiency without sacrificing accuracy. Furthermore, we incorporate the proposed approach into adaptive sampling approaches such as the residual-based adaptive distribution (RAD) for the time-fractional Burgers equation with order $\alpha\in(1,2)$, which exhibits complex solution dynamics. The experiments show that the Gauss-Jacobi method typically outperforms the Monte Carlo approach; however, careful consideration is required when selecting the number of quadrature points. Overall, the proposed tDWfPINNs offer a significant advancement in the numerical solution of time-fractional diffusion-wave equations, providing an accurate and scalable mesh-free alternative for challenging fractional models.

Let $X$ be a singular, projective variety. For every $p>0$, $H^{2p}(X,\mathbb{Q})$ is equipped with a mixed Hodge structure. The elements of $\mathrm{Gr}^W_{2p}H^{2p}(X,\mathbb{Q}) \cap H^{p,p} \mathrm{Gr}^W_{2p}H^{2p}(X,\mathbb{C})$ will be called Hodge (p,p)-classes. The purpose of this article, is to study the Bloch-Gillet-Soulé (BGS) cycle class map from the $p$-th operational Chow group $A^p(X)$ to the space of $(p,p)$-Hodge classes. We show that if $p=1$ and $X$ is a normal surface with at worst rational singularities, then the BGS cycle class map is surjective. This extends the Lefschetz $(1,1)$-theorem to the setup of rational surface singularities. However, the BGS map is not always surjective. For this reason we introduce extended operational Chow group $A^p_{\mathrm{ext}}(X)$ which contains the operational Chow group. We show that the BGS cycle class map extends to $A^p_{\mathrm{ext}}(X)$. Moreover, if $p=1$ and $X$ has at worst isolated singularity (not necessarily a surface), then the extended BGS map is surjective. This further extends the Lefschetz $(1,1)$-theorem to the case of isolated singularities.

We investigate the existence and uniqueness of non-Markovian second-order backward stochastic differential equations with an uncertain terminal horizon and establish comparison principles under the assumption that the driver is Lipschitz continuous. The terminal time is both random and exogenous, and it may not be adapted to the Brownian filtration, leading to a singular jump in the 2BSDE decomposition. We also provide a connection between this new class of 2BSDE and a fully nonlinear PDE in a Markovian setting. Our theoretical results are applied to non-Markovian stochastic control problems in two settings: (1) when an agent seeks to maximize utility from a payoff received at an uncertain terminal time by controlling both the drift and volatility of a diffusion process; and (2) when the agent contends with volatility uncertainty stemming from an external source, referred to as Nature, and optimizes the drift in a worst-case scenario for the ambiguous volatility. We term this class of problems erratic stochastic control, reflecting the dual uncertainty in both model parameters and the timing of the terminal horizon.

We present a computational framework for simulating finite-dimensional ordinary differential equations by combining classical Koopman-Lie product formulas with coordinate-wise frozen subflows. The setting is model-known, since the vector field is assumed to be available, and no data-driven approximation of the Koopman operator is attempted. Under standard assumptions, the Koopman-Lie generator associated with the flow admits a coordinate decomposition into partial generators. This decomposition leads to elementary updates in which all but one state variable are frozen, and the resulting frozen scalar subproblems are evaluated either in closed form or by one-dimensional solves. Lie-Trotter, Strang, and higher-order exponential compositions are then converted into state-update algorithms for two- and three-dimensional systems, with the semigroup and product-formula theory used as background justification for the constructions. We also record the exponential-term counts produced by the recursive constructions used in the implementation. These counts are presented as implementation costs. Numerical experiments on the Lotka-Volterra, Van der Pol, and Lorenz systems compare the coordinate-wise splitting algorithms with high-accuracy RK45 reference solutions using root-mean-square errors and work-precision curves. The results illustrate the practical trade-off between splitting order, number of time steps, number of exponential factors, and runtime.

We prove the existence of a localization spectral sequence for the hat variant of Guth and Manolescu's recent construction of real Heegaard Floer homology, and apply it to branched double covers and strongly invertible knots. Our construction applies to real Lagrangian Floer homology in exact symplectic manifolds equipped with anti-symplectic involutions more generally, and may be of independent interest to symplectic geometers.

We study the vortex formation in extreme type-II superconductors immersed in strong magnetic fields in the framework of the the Ginzburg-Landau theory. We focus on the regime where superconductivity survives in the bulk of the material but the magnetic field penetrates the sample, i.e., for applied field much larger than the first critical one, but below the transition to surface superconductivity. Through a two-scale vortex construction, we obtain precise estimates for the vortex distribution and prove the existence of isolated defects with non-trivial winding numbers. In this respect, our work provides the first rigorous mathematical proof of the existence of isolated vortices for fields comparable to the second critical one.

Given any generically étale morphism of varieties $f \colon X \to Y$, we define the relative Mather discrepancy function on the arc space $X_\infty$ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of $\widehat K$-equivalence, which agrees with $K$-equivalence for smooth varieties, and prove that $\widehat K$-equivalent varieties of arbitrary characteristic define the same class in the motivic ring.

A graph $G(V, E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that for distinct letters $x,y\in V$, $x$ and $y$ alternate in $w$ if and only if they are adjacent in $G$. In general, determining whether a graph is word-representable is an NP-complete problem. A graph is co-bipartite if its complement is bipartite. Therefore, the vertex set of a co-bipartite graph can be partitioned into two disjoint subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are cliques. In this paper, we obtain necessary and sufficient conditions for a co-bipartite graph to be word-representable in terms of a vertex ordering. Based on this ordering, we study the representation number of word-representable co-bipartite graphs and analyse the speed and entropy of this graph class. We show that the representation number of any word-representable co-bipartite graph is at most $3$, and that permutation graphs are the only co-bipartite graphs with representation number $2$. We prove that the speed is $2^{O(n \log n)}$ and the entropy is $0$. This provides an asymptotic bound on the number of labelled graphs in this class, which is significantly smaller than the known bound for the class of all co-bipartite graphs. These results provide a better understanding of the structure and enumeration of word-representable co-bipartite graphs and show that vertex ordering is an effective tool for studying this class.

We prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite. This can be considered as a generalization of hyperfiniteness of the boundary action of cubulated hyperbolic groups by Huang-Sabok-Shinko.

We study actions of (infinitely presented) graphical small cancellation groups on the Gromov boundaries of their coned-off Cayley graphs. We show that a class of graphical small cancellation groups, including (infinitely presented) classical small cancellation groups, admit hyperfinite boundary actions, more precisely, the orbit equivalence relation that they induce on the boundaries of the coned-off Cayley graphs is hyperfinite.

We propose a novel feedback controller for a class of uncertain higher relative degree nonlinear systems, subject to delays in both state measurement and control input signals. Building on the prescribed performance control framework, a delay-dependent performance correction mechanism is introduced to ensure the boundedness of all signals in the closed-loop and to keep the output tracking error strictly within a dynamically adjusted performance envelope. This mechanism adapts in response to large delays that may cause performance degradation. In the absence of delays, the correction term vanishes, and the controller recovers the nominal (user-defined) performance envelope. The effectiveness of the proposed approach is validated through simulation studies.

This paper focuses on the decentralized stochastic optimization problem $f(\mathbf{x})=\frac{1}{m}\sum_{i=1}^m f_i(\mathbf{x})$ over a connected network of $n$ agents, where each local function has the form of $f_i(\mathbf{x}) = {\mathbb E}\left[F(\mathbf{x};{\boldsymbol \xi}_i)\right]$ which satisfies the $(L_0,L_1)$-smooth condition but possibly nonconvex and each random variable ${\boldsymbol \xi}_i$ follows distribution ${\mathcal D}_i$. We propose a novel algorithm called decentralized normalized stochastic gradient descent (DNSGD), which can achieve an $\epsilon$-stationary point at each local agent. We present a new framework for analyzing decentralized first-order methods in the $(L_0,L_1)$-smooth setting, based on the Lyapunov function related to the product of the gradient norm and the consensus error. We show that the proposed algorithm attains the upper bounds on the sample complexity of ${\mathcal O}(m^{-1}(L_f\sigma^2\Delta_f\epsilon^{-4} + \sigma^2\epsilon^{-2} + L_f^{-2}L_1^3\sigma^2\Delta_f\epsilon^{-1} + L_f^{-2}L_1^2\sigma^2))$ per agent and the communication complexity of $\tilde{\mathcal O}((L_f\epsilon^{-2} + L_1\epsilon^{-1})\gamma^{-1/2}\Delta_f)$, where $L_f=L_0 +L_1\zeta$, $\sigma^2$ is the variance of the stochastic gradient, $\Delta_f$ is the initial optimal function value gap, $\gamma$ is the spectral gap of the network, and $\zeta$ is the degree of the gradient dissimilarity. In the special case of $L_1=0$, the above results (nearly) match the lower bounds of decentralized stochastic nonconvex optimization under the standard smoothness. We also conduct numerical experiments to show the empirical superiority of our method.

Menger's theorem tells us that if $S,T$ are sets of vertices in a graph $G$, then (for $k\ge0$) either there are $k+1$ vertex-disjoint paths between $S$ and $T$, or there is a set of $k$ vertices separating $S$ and $T$. But what if we want the paths to be far apart, say at distance at least $c$? One might hope that we can find either $k+1$ paths pairwise far apart, or $k$ sets of bounded radius that separate $S$ and $T$, where the bound on the radius is some $\ell$ that depends only on $k,c$ (the ``coarse Menger conjecture''). The last three authors showed in an earlier paper that this is false for all $k\ge 2$ and $c\ge3$, by constructing a sequence of finite graphs giving counterexamples for larger and larger values of $\ell$ with $k=2$ and $c=3$. These counterexamples contained subdivisions of uniform binary trees with arbitrarily large depth as subgraphs, and so had unbounded path-width. Here we show that, if $H$ is a graph that can be drawn in the plane such that each region shares a vertex with the infinite region, then the coarse Menger conjecture is true for all graphs not containing $H$ as a minor. Consequently, the conjecture is true for all graphs with bounded path-width (by taking $H$ to be a sufficiently large tree), and it is true for series-parallel graphs (by taking $H=K_4$). The first is somewhat surprising, since the conjecture is false for bounded tree-width.

We introduce a two-level hybrid restricted additive Schwarz (RAS) preconditioner for heterogeneous steady-state convection-diffusion equations at high Péclet numbers. Our construction builds on the multiscale spectral generalized finite element method (MS-GFEM), wherein the coarse space is spanned by locally optimal basis functions obtained from local generalized eigenproblems on operator-harmonic spaces. Extending the theory of Ma (2025) to convection-diffusion problems in conservation form, we establish exponential convergence of the MS-GFEM approximation with respect to the dimension of the local approximation space. Rewriting MS-GFEM as a RAS-type iteration, we show for coercive problems that this exponential convergence property is inherited by the RAS-type iterative method (at least in the continuous setting). Employed as a preconditioner within the generalized minimal residual method (GMRES), the resulting method requires only a few iterations for high accuracy even with low-dimensional coarse spaces. Through extensive numerical experiments on problems with high-contrast diffusion and non-divergence-free, rotating velocity fields, we demonstrate robustness with respect to the grid Péclet number and the number of subdomains (tested up to $10^5$ subdomains), while coarse-space dimensions remain small as grid Péclet numbers increase. By adapting the coarse space and oversampling size, we are able to achieve arbitrarily fast convergence of preconditioned GMRES. As an extension, for which we do not have theory yet, we show effectiveness of the method even for indefinite problems and in the vanishing-diffusion limit.

The problems that we consider in this paper are as follows. Let $A_1, \ldots, A_k$ be square matrices (over reals). Let $W=w(A_1, \ldots, A_k)$ be a random product of $n$ matrices. What is the expected absolute value of the largest (in the absolute value) entry in such a random product? What is the (maximal) Lyapunov exponent for a random matrix product like that? We give an answer to the first question under some mild restrictions on the entries of $A_i$. For the second question, we offer a very simple and efficient method to produce an upper bound on the Lyapunov exponent.

In this paper, we propose DeMuon, a method for decentralized matrix optimization over a given communication topology. DeMuon incorporates matrix orthogonalization via Newton-Schulz iterations-a technique inherited from its centralized predecessor, Muon-and employs gradient tracking to mitigate heterogeneity among local functions. Under heavy-tailed noise conditions and additional mild assumptions, we establish the iteration complexity of DeMuon for reaching an approximate stochastic stationary point. This complexity result matches the best-known complexity bounds of centralized algorithms in terms of dependence on the target tolerance. To the best of our knowledge, DeMuon is the first direct extension of Muon to decentralized optimization over graphs with provable complexity guarantees. We conduct preliminary numerical experiments on decentralized transformer pretraining over graphs with varying degrees of connectivity. Our numerical results demonstrate a clear margin of improvement of DeMuon over other popular decentralized algorithms across different network topologies.

This paper investigates a cooperative jamming (CJ)-aided secure wireless communication system. Conventional CJ schemes transmit Gaussian noise (GN) to improve security, which inherently degrades the legitimate receiver's performance. While encoded jamming (EJ) mitigates this interference, its superiority over GN is highly channel-dependent. To overcome this limitation, we introduce a joint optimization framework integrating a reconfigurable intelligent surface (RIS) with EJ to maximize the secrecy rate. \RED{We first establish the information-theoretic relationship between the EJ and GN schemes, identifying the spatial channel conditions that limit EJ performance. For the multiple-input single-output (MISO) scenario, we analytically derive the ergodic secrecy gap as the number of RIS elements grows large and obtain a positive EJ-over-GN gap under explicit power and channel conditions.} Furthermore, for the general multiple-input multiple-output (MIMO) setup, we develop a low-complexity algorithm based on the weighted minimum mean-square-error (WMMSE) framework to handle the resulting non-smooth max-min structure through a WMMSE-based mode-selection framework. By introducing a parameterized function abstraction, the transmit precoding matrices and the RIS phase shift matrix are jointly optimized via block coordinate descent (BCD). Simulation results support the analysis and show that, under the evaluated settings, RIS-assisted EJ can overcome the identified spatial bottlenecks and outperform the optimized GN baseline.

For a Galois number field $K$, the Galois group $\text{Gal}(K/\mathbb{Q})$ acts on the class group $\text{Cl}_K$ in a very natural way: $\sigma\cdot[I]=[\sigma(I)]$ for any $\sigma \in \text{Gal}(K/\mathbb{Q})$, $[I]\in \text{Cl}_K$. In this paper, we will explore how the unique properties of this group action work together to elucidate the relationship between these two groups -- developing and expanding upon some known results from a new perspective. To this end, we explore the class groups of localizations of the ring of integers $\mathcal{O}_K$. These turn out to be powerful tools for understanding $\text{Cl}_K$ and overrings of $\mathcal{O}_K$. The paper concludes with some interesting observations about normset arithmetic and complexity -- topics intimately related to this action.

We prove the stability and global convergence of a coupled actor-critic gradient flow for infinite-horizon and entropy-regularised Markov decision processes (MDPs) in continuous state and action space with linear function approximation under Q-function realisability. We consider a version of the actor critic gradient flow where the critic is updated using temporal difference (TD) learning while the policy is updated using a policy mirror descent method on a separate timescale. For general action spaces, the relative entropy regularizer is unbounded and thus it is not clear a priori that the actor-critc flow does not suffer from finite-time blow-up. Therefore we first demonstrate stability which in turn enables us obtain a convergence rate of the actor critic flow to the optimal regularised value function. The arguments presented show that timescale separation is crucial for stability and convergence in this setting.

Girko matrices have independent and identically distributed entries of mean zero and unit variance. In this note, we consider the random matrix model formed by the ratio of two independent Girko matrices, its entries are dependent and heavy-tailed. Our main message is that divided by the square root of the dimension, the spectral radius of the ratio converges in distribution, when the dimension tends to infinity, to a universal heavy-tailed distribution. We provide a mathematical proof of this high-dimensional phenomenon, under a fourth moment matching with a Gaussian case known as the complex Ginibre ensemble. In this Gaussian case, the model is known as the spherical ensemble, and its spectrum is a determinantal planar Coulomb gas. Its image by the inverse stereographic projection is a rotationally invariant gas on the two-sphere. A crucial observation is the invariance in law of the model under inversion, related to its spherical symmetry, and that makes, in a sense, edge and bulk equivalent. Our approach involves Girko Hermitization, local law estimates for Wigner matrices, lower bound estimates on the smallest singular value, and convergence of kernels of determinantal point processes. The universality of the high-dimensional fluctuation of the spectral radius of the ratio of Girko matrices turns out to be remarkably more accessible mathematically than for a single Girko matrix!

In this paper, we derive new results on the asymptotic behavior of eigenvalues of perturbed one-dimensional massive Dirac operators in the weak coupling limit. Two classes of potentials are considered. For bounded Hermitian potentials $V$ satisfying $|V(x)| \lesssim |x|^{-1}$ for large $|x|$, we recover the leading term, which may include a logarithmic correction if $V(x) \sim |x|^{-1}$ at infinity. For possibly non-Hermitian $L^1$ potentials satisfying a suitable moment condition, we obtain the second term in the asymptotic expansion. The first result is based on a min-max principle adapted to the non-relativistic limit, while the second result is obtained via the Birman-Schwinger principle and resolvent expansions.

We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities whose divisor class group has rank one. More precisely, such toric NCCRs are in bijection with non-trivial upper sets in a certain quotient of the divisor class group equipped with a natural partial order. This classification allows us to prove that all toric NCCRs of such toric singularities are connected by iterated Iyama--Wemyss mutations, and hence are derived equivalent to one another. We further give a dimer-model realization of this classification in the non-pyramidal case. More precisely, we construct periodic quivers with cuts on a $d$-dimensional torus, establish a cut-upper set correspondence, and prove that the resulting cut quiver with relations presents the corresponding toric NCCR. For $d=2$, this recovers the quiver-theoretic part of the usual dimer-model construction. In the appendix, we give an explicit formula for the volume of $d$-dimensional lattice polytopes with $d+2$ vertices. As an application, we verify Van den Bergh's conjectural equality, for Gorenstein toric singularities with divisor class group of rank one, between the number of indecomposable direct summands of a toric NCCR and the normalized volume of the corresponding lattice polytope.

We appeal to the theory of Jouanolou torsors to model the coherent cohomology of configuration spaces of points in d-dimensional affine space. Using this model, we develop the operadic notion of chiral operations, thus generalizing the notion of chiral algebras of Beilinson and Drinfeld to higher dimensions. To produce examples, we use a higher-dimensional conceptualization of the residue which is inspired by Feynman graph integrals. One of our main results is the realization, using higher chiral operations, of the higher-dimensional Kac--Moody and Virasoro algebras.

We consider an integral transform given by $T_{\nu} f(s) := \pi \int_0^\infty rs J_{\nu}(r s)^2 f(r) \, dr$, where $J_{\nu}$ denotes the Bessel function of the first kind of order $\nu$. As shown by Walther (2002, doi:https://doi.org/10.1006/jfan.2001.3863), this transform plays an essential role in the study of optimal constants of smoothing estimates for the free Schrödinger equations on $\mathbb{R}^d$. On the other hand, Bez et al. (2015, doi:https://doi.org/10.1016/j.aim.2015.08.025) studied these optimal constants using a different method, and obtained a certain alternative expression for $T_{\nu} f$ involving the $d$-dimensional Fourier transform of $x \mapsto f(\lvert x \rvert)$ when $\nu = k + d/2 - 1$ for $k \in \mathbb{N}$. The aims of this paper are to extend their identity for non-integer indices and to derive several inequalities from it.

The fluctuations of the passage time in first passage percolation are of great interest. We show that the non-random fluctuations in planar FPP are at least of order $\sqrt{\log n}$ under some conditions that are known to be met for a large class of absolutely continuous edge weight distributions. This improves the ${\log(\log(n))}$ bound proven by Nakajima and is the first result showing divergence of the fluctuations for arbitrary directions. Our proof is an application of recent work by Dembin, Elboim and Peled on the BKS midpoint problem and the development of Mermin-Wagner type estimates.

We explore a method, initiated by Guichard in 1890, which allows to generate sequences of Voss surfaces, starting from an arbitrarily chosen pseudospherical surface and a seed solution of the Moutard equation, by means of two simple transformations. In this paper we 1) identify the Guichard transformations with the well-known recursion operator for symmetries of the sine-Gordon equation and its inverse; 2) prove a lemma which allows us to derive the length of Guichard's sequences from the invariance properties of the initial sine-Gordon solution; 3) introduce an extended class of inverted operators, expanding the class of Voss surfaces obtainable by quadratures. A number of Voss nets are presented explicitly. As the main instrument, we reinterpret Guichard's transformations in terms of the recursion operators for sine-Gordon symmetries. Simultaneously, we clarify relevant aspects of Guthrie's formalism, paving the way for the future employment of the entire division algebra of recursion operators.

Let $K$ be a discretely valued Henselian field. Creutz and Viray show that the degree set of a curve $C$ over a $p$-adic field can miss infinitely many multiples of the index of $C$, a phenomenon that cannot occur over finitely generated fields. For curves $C/K$ with a cyclic cover of $\mathbb{P}^1$ of prime degree, under mild assumptions, we completely characterize how and when this behavior can occur, and give a method for computing degree sets of curves of this type.

For a locally defined real analytic function $f$, we study the relation between the oscillation index of oscillatory integrals and the real log canonical threshold. The former is always negative, and its absolute value is greater than or equal to the latter. They coincide very often, but there are certain exceptional cases, and it is not very clear when the equality holds. In this note we give some sufficient conditions for the coincidence to hold or to fail. In the Newton-nondegenerate convenient homogeneous case, we show that the strict inequality holds if the number of variables $n$ is even and smaller than the degree $d$ of $f$ (or $f^{-1}(0)=\{0\}$), and the equality holds if $n$ is odd and $f^{-1}(0)=\{0\}$ (in particular, $d$ is even). The first assertion does not seem to be compatible with some standard formula in the literature, and there must be some error somewhere, although it does not seem easy to detect it inside this paper.

In this paper, we study an optimal dividend and capital-injection problem in a Cramér--Lundberg model where claim arrivals follow a Hawkes process, capturing clustering effects often observed in insurance portfolios. We establish key analytical properties of the value function and characterise the optimal capital-injection strategy through an explicit threshold. We also show that the value function is the unique viscosity solution of the associated HJB variational inequality. For numerical purposes, we first compute a benchmark solution via a monotone finite-difference scheme with Howard's policy iteration. We then develop a reinforcement learning approach based on policy-gradient and actor-critic methods. The learned strategies closely match the PDE benchmark and remain stable across initial conditions. The results highlight the relevance of policy-gradient techniques for dividend optimisation under self-exciting claim dynamics and point toward scalable methods for higher-dimensional extensions.

We explicitly compute Fourier coefficients of Eisenstein series on the special orthogonal group $G={\rm SO}(3,n+1)$ over $\mathbb{Q}$ with $n\ge 2$ which splits everywhere at finite places. We show that it has a bounded denominator.

High-fidelity spectrum cartography is important for spectrum monitoring and wireless situational awareness, especially in satellite-based wide-area sensing scenarios where measurements are sparse, noisy, and often low-bit quantized. In such settings, two coupled challenges arise: accurate reconstruction from severely incomplete measurements and efficient allocation of additional sensing resources under a limited sensing budget. Existing methods usually address these problems separately, and, for reconstruction, they often rely on priors that are insufficiently expressive under sparse and quantized measurements. This paper proposes Generative Spectrum Cartography (GSC), a diffusion-based posterior inference framework for spectrum cartography with uncertainty-aware active sensing. Specifically, spectrum map recovery is formulated as a Bayesian inverse problem under a learned diffusion model prior, and closed-form posterior mean updates are derived for both linear and quantized measurement models. By embedding these updates into the reverse diffusion process, GSC enables gradient-free and measurement-consistent posterior sampling without relying on computationally costly likelihood-gradient guidance. The resulting posterior samples are further used to estimate spatial uncertainty and to guide diversity-aware selection of additional measurement locations for active sensing. Experiments on simulated electromagnetic maps and a high-fidelity simulated satellite monitoring scenario show that GSC achieves higher PSNR, lower LPIPS, and more efficient sensing than representative baseline methods under sparse, noisy, and low-bit quantized measurements.

We study the structure of the fibre operators corresponding to periodic magnetic pseudo-differential operators having periodic magnetic potentials. We obtain explicit formulas for their distribution kernel, both when these fibres are seen as operators on the $d$-dimensional torus, and also when they are seen as infinite matrices acting on a discrete $\ell^2$ space via a discrete Fourier transform. Moreover, using these distribution kernels we prove that the fibre operators are toroidal pseudo-differential operators.

We study the crossing matrix of a braid and introduce a polynomial invariant for braid systems that is invariant under Hurwitz equivalence. As an application to the study of surface braids and surface links, we also define an invariant that can be used as an indicator of the necessity of Euler fusion or fission between braid systems.

We introduce a new two-parameter fractional time operator with Volterra structure, denoted by ${}^{W}D_{t}^{\alpha,\beta}$, defined through the Laplace symbol \[ \Phi_{\alpha,\beta}(s) = \frac{s^\alpha}{\bigl(1+(1-\alpha)s^{\alpha-1}\bigr)^\beta}, \qquad 0<\alpha<1, \ \beta\ge0. \] The operator preserves the Caputo-type high-frequency behavior while allowing a controlled modification of the low-frequency regime via $\beta$. We develop an explicit symbolic/Volterra theory: Prabhakar-type kernels, a left-inverse Volterra integral, and a fractional fundamental theorem of calculus. A central contribution is a sharp clarification of the Bernstein structure of the symbol. We show that the natural factorization $\Phi_{\alpha,\beta}(s)=s^\alpha h_\alpha(s)^\beta$ does not fit the classical Bernstein product mechanism for any $\beta>0$. Nevertheless, by a direct complete-monotonicity argument on $\Phi'_{\alpha,\beta}$, we prove the exact Bernstein threshold \[ \Phi_{\alpha,\beta}\in\mathcal{BF} \quad\Longleftrightarrow\quad 0\le\beta\le1. \] where $\mathcal{BF}$ denotes the class of Bernstein functions \noindent For $\beta>1$, the Bernstein property fails by a low-frequency asymptotic convexity obstruction. This shows that the Bernstein nature of the natural range $0\le\beta\le1$ is genuine but is not produced by the standard product mechanism. We then establish well-posedness of abstract W-fractional Cauchy problems with sectorial generators by resolvent estimates and Laplace inversion, yielding a W-resolvent family with temporal regularity and smoothing properties. As an illustration, we apply the theory to a W-fractional diffusion model and discuss the effect of $\beta$ on the relaxation of spectral modes.

Optimal control of obstacle problems arises in a wide range of applications and is computationally challenging due to its nonsmoothness, nonlinearity, and bilevel structure. Classical numerical approaches rely on mesh-based discretization and typically require solving a sequence of costly subproblems. In this work, we propose a single-loop bilevel deep learning method, which is mesh-free, scalable to high-dimensional and complex domains, and avoids repeated solution of discretized subproblems. The method employs constraint-embedding neural networks to approximate the state and control and preserves the bilevel structure. To train the neural networks efficiently, we propose a Single-Loop Stochastic First-Order Bilevel Algorithm (S2-FOBA), which eliminates nested optimization and does not rely on restrictive lower-level uniqueness assumptions. We analyze the convergence behavior of S2-FOBA under mild assumptions. Numerical experiments on benchmark examples, including distributed and obstacle control problems with regular and irregular obstacles on complex domains, demonstrate that the proposed method achieves satisfactory accuracy while reducing computational cost compared to classical numerical methods.

The existence of a finite global attractor for polynomial curve system has been known since the work of Belk et al. [4]. However, except in the hyperbolic case, the rate at which the pullback of a curve under a polynomial converges to the attractor remained unclear. In this paper, we introduce the notions of $\textit{quick returns}$ and $\textit{barrier lakes}$ to analyze the combinatorial models of curves. These concepts allow us to show that if a certain number of successive pullbacks do not decrease the complexity of the curve by a definite proportion, then the curve admits a $\textit{thick}$-$\textit{thin}$ $\textit{decomposition}$: most of the curve is organized into finitely many disjoint annuli whose core curves have bounded homotopy type. In this case, we can show that some number of successive pullbacks must decrease the complexity of the curve by a definite factor. This implies that the complexity of a curve $C$ decreases exponentially under iteration of the pullback by a polynomial $f$: \[ N_{\mathcal{F}}(\eta) \le A \, N_{\mathcal{F}}(C) \, e^{-n \delta} + D, \qquad \forall n\ge 1, \] where $\mathcal{F}$ is an admissible family of separation arcs, $N_{\mathcal{F}}(\cdot)$ denotes the minimal intersection number of the curves in its homotopy class and the arcs in $\mathcal{F}$, $\delta> 0$ is a constant depending only on $f$, $A, D > 0$ are constants depending only on $\mathcal{F}$ and $f$, and $\eta$ is any component of $f^{-n}(C)$. Consequently, the pullback of a curve contracts exponentially to the attractor. In particular, this provides a quantitative proof of the finite global attractor conjecture for the polynomial case.

An example originally given by F.~Delon shows the existence of an algebraically maximal discretely valued field of characteristic $p>0$ which admits purely inseparable extensions of degree $p^2$ with defect $p$. These extensions are not generated by a single element. Using a trick introduced in an earlier paper of the author, we construct algebraically maximal valued fields, of characteristic $p$ as well as of characteristic $0$, which admit separable extensions of degree $p^2$ with defect $p$. They are of rank 2 and it is an open question whether such examples having rank 1 exist.

In this paper, we consider an age-structured jump model that arises as a description of continuous time random walks with infinite mean waiting time between jumps. We prove that under a suitable rescaling, this equation converges in the long time large scale limit to a time fractional subdiffusion equation.

Optimization is central to both modern machine learning (ML) and scientific machine learning (SciML), yet the structure of the underlying optimization problems differs substantially across these domains. Classical ML typically relies on stochastic, sample-separable objectives that favor first-order and adaptive gradient methods. In contrast, SciML often involves physics-informed or operator-constrained formulations in which differential operators induce global coupling, stiffness, and strong anisotropy in the loss landscape. As a result, optimization behavior in SciML is governed by the spectral properties of the underlying physical models rather than by data statistics, frequently limiting the effectiveness of standard stochastic methods and motivating deterministic or curvature-aware approaches. This document provides a unified introduction to optimization methods in ML and SciML, emphasizing how problem structure shapes algorithmic choices. We review first- and second-order optimization techniques in both deterministic and stochastic settings, discuss their adaptation to physics-constrained and data-driven SciML models, and illustrate practical strategies through tutorial examples, while highlighting open research directions at the interface of scientific computing and scientific machine learning.

We introduce a nearly parameter-free algorithm for minimizing piecewise smooth (PWS) convex functions under the quadratic-growth (QG) condition, where the locations and structure of the smooth regions are entirely unknown. Our algorithm, APEX (Accelerated Prox-Level method for Exploring Piecewise Smoothness), is an accelerated bundle-level method designed to adaptively exploit the underlying PWS structure. For this setting, APEX achieves the best-known oracle-complexity result among existing first-order methods, improving the dependence on the condition number relative to prior bundle-level guarantees. Furthermore, APEX generates a verifiable and accurate termination certificate, enabling a robust, nearly parameter-free implementation. To the best of our knowledge, APEX is the first algorithm to simultaneously achieve the best-known first-order oracle complexity for PWS optimization and provide certificate guarantees.

We study the well-posedness of a non-linear heat equation with power nonlinearity with positive initial data on quantum Euclidean spaces. We prove a noncommutative analogue of the classical Fujita theorem by identifying the critical exponent separating finite-time blow-up from global existence for small initial data. Moreover, we establish a fundamental inequality in general semifinite von Neumann algebras that is of independent interest and plays a crucial role in the study of global existence and local well-posedness of solutions of nonlinear equations in noncommutative setting.

We prove a new rigidity criterion for families of polarized Calabi--Yau manifolds. Motivated by known non-rigid examples, we conjecture that a family over a quasi-projective curve is rigid if, near a boundary point, the total space is smooth, the relative canonical bundle is trivial, and the boundary fiber contains an isolated singular point. We verify this conjecture when one such isolated singularity has a concentrated mixed Hodge spectrum, a class including ordinary double points and cusps. The proof combines a local vanishing-cycle analysis with a global tensor-product decomposition of the associated variation of Hodge structures.

We prove new arithmetic results for parametric linear recurrence sequences specialized at roots of unity, denoted by $(U_n(\zeta))_{n\geq 0}$. In particular, we obtain effective lower bounds for the largest prime ideal divisor and norm of the radical of the principal ideal generated by $U_n(\zeta)$. We further derive an effective upper bound for the $S$-part of $U_n(\zeta)$, showing that it is strictly smaller than a fixed power of its absolute norm for sufficiently large $n$.

Deep learning-based hybrid iterative methods (DL-HIMs) integrate classical numerical solvers with neural operators, utilizing their complementary spectral biases to accelerate convergence. Despite this promise, many DL-HIMs stagnate at false fixed points where neural updates vanish while the physical residual remains large, raising questions about reliability in scientific computing. In this paper, we provide evidence that performance is highly sensitive to training paradigms and update strategies, even when the neural architecture is fixed. Through a detailed study of a DeepONet-based hybrid iterative numerical transferable solver (HINTS) and an FFT-based Fourier neural solver (FNS), we show that significant physical residuals can persist when training objectives are not aligned with solver dynamics and problem physics. We further examine Anderson acceleration (AA) and demonstrate that its classical form is ill-suited for nonlinear neural operators. To overcome this, we introduce physics-aware Anderson acceleration (PA-AA), which minimizes the physical residual rather than the fixed-point update. Numerical experiments confirm that PA-AA restores reliable convergence in substantially fewer iterations. These findings provide a concrete answer to ongoing controversies surrounding AI-based PDE solvers: reliability hinges not only on architectures but on physically informed training and iteration design.

We establish the existence of many holomorphic Hecke eigenforms $f$ of large weight $k$ for the full modular group, for which the least positive integer $n_f$ such that $\lambda_f(n_f)<0$ satisfies $n_f \ge (\log k)^{1-o(1)}.$ This is believed to be best possible up to the $o(1)$ term in the exponent, and improves on a result of Kowalski, Lau, Soundararajan and Wu, who showed that, when restricted to primes, the least prime $p$ such that $\lambda_f(p)<0$ can be as large as $(\log k)^{1/2+o(1)}$. We also discuss an extension of our result to primitive holomorphic cusp forms of weight $k$ and squarefree level $N\geq 1$.

In this paper we show that the volume of a maximal globally hyperbolic Cauchy-compact anti-de Sitter $3$-manifold $M$ is at least $\pi^2|\chi(M)|$, and that this minimum value is attained if and only if $M$ is Fuchsian.

We first present a reduction of properties of compact sets of real numbers to properties of countable orders. Then discuss a variant of homogeneity of compact subsets of real numbers, focusing on the family of $t$-sets. Finally, we prove that there are exactly $\omega_1$ many non-homeomorphic $t$-sets.

We answer a question of Biran and Cornea about the density of iterated cones of fibers in the Fukaya category of a cotangent bundle. We prove that indeed if we take a dense set of basepoints, the iterated cones of the cotangent fibres are dense in the Filtered Fukaya category. In an appendix we prove that the space of exact Lagrangians in a symplectic manifold is never totally bounded for the spectral distance (unless it is empty). This was implicit in \cite{MCA-VH-CV} for $n=1$ and proved for cotangent bundles of negatively curved manifolds in \cite{A-B-C}.

We propose an approach to construct three-dimensional lattice models using line defects in state integral models on shaped triangulations of 3-manifolds. The Boltzmann weights for these models satisfy a variant of the tetrahedron equation, which implies integrability under suitable assumptions on R-matrices and transfer matrices. As an explicit example, we present a solution produced by Teichmüller TQFT.

Let $M$ be a compact oriented $3$-manifold with boundary consisting of tori, and let $G$ be a semisimple algebraic group. We define the adjoint torsion function on the moduli stack of $G$-local systems on $M$ satisfying a certain regularity condition, extending the construction by Porti for $G = \mathrm{SL}_2$. When $M$ is a cusped hyperbolic manifold, we prove that the local system associated with the image of the complete hyperbolic structure via a principal embedding $\mathrm{PGL}_2 \to G$ satisfies the regularity condition. Moreover, we provide a formula expressing its adjoint torsion as a product of $\mathrm{PGL}_2$-torsions associated with the simple $\mathrm{PGL}_2$-modules with multiplicity given by the exponents of the Lie algebra of $G$. We compute the adjoint $\mathrm{PGSp}_4$-torsions of the figure-eight knot complement for two boundary-unipotent local systems, one is arising from the complete hyperbolic structure via a principal embedding, and the other is defined over a number field of degree $6$ and not arising from any $\mathrm{PGL}_2$-local system via principal embeddings.

We study whether a logic based on team semantics can be enriched with a conditional satisfying minimal requirements, such as preservation of the closure property of the logic, Modus Ponens, and the Deduction Theorem. We show that such well-behaved conditionals exist for downward or upward closed logics, but do not typically exist for union closed, convex or intersection closed logics. We also briefly investigate conditionals satisfying weaker requirements.

We resolve two problems pertaining to indefinite integral ternary quadratic forms, one highlighted by Margulis and the other initiated by Serre, both from 1990. To do so we develop tools for dealing with high ramification in problems involving sums over classes of such forms weighted by their diophantine invariants.

According to the classical Wagstaff heuristic, the probability that a Mersenne number $M_p=2^p-1$ is prime depends primarily on the size of the exponent $p$. We investigate whether the divisor structure of $p-1$ produces detectable secondary variations within this asymptotic framework. We introduce the normalized divisor parameter $S(p)=\log \tau(p-1)/\log\log p$, which measures the divisor complexity of p-1, including prime multiplicities. Using the currently known Mersenne prime exponents (excluding small cases), we compare $S(p)$ against nearby prime controls of comparable size. Across several complementary distribution-free methods, including percentile analysis, conditional likelihood estimation, and permutation tests, Mersenne prime exponents consistently exhibit elevated values of $S(p)$. To interpret this effect, we develop a heuristic framework based on the cyclotomic decomposition $2^{p-1}-1=\prod_{d|(p-1)}\Phi_d(2)$, in which divisors of $p-1$ generate effective modular constraint layers. This motivates a heuristic refinement of the Wagstaff model of the form $\Pr(M_p\ \text{prime}) \approx C(\log p)^{S(p)}/p$. The proposed refinement preserves the classical Wagstaff scale in the typical regime $S(p)\approx 1$, while suggesting that the finite-scale distribution of Mersenne prime exponents exhibits a weak arithmetic bias linked to the divisor structure of $p-1$.

A field in which the (logarithmic) Weil height is bounded from below by a strictly positive constant is said to have the Bogomolov property (property (B)). Given a normalized eigenform $f\in S_k(\Gamma_0(N))$ Amoroso and Terracini proved (B) for the field "cut out" by the adelic representation associated to $f$ under some assumptions on $f$, generalizing the earlier work of Habegger on elliptic curves. In this paper we extend this result to the case of normalized eigenforms with nontrivial nebentypus character. We also introduce the notion of ADZ field, inspired by earlier work of Amoroso, David and Zannier, exhibiting a class of fields in which property (B) is preserved under (arbitrary) composition.

Saddle point problems with smooth convex-concave objective functions are often used to model min-max problems arising in machine learning. First-order methods are the standard paradigm for solving such problems. Therefore, it is important to know how those methods behave in the worst-case scenarios. In order to derive the guarantees, one would require the inequalities that appropriately constrain the iterates, gradients and function values. In this paper, we present stronger constraints for smooth convex-concave functions and show that they could allow tighter upper bounds for first-order methods.

The Debreu Koopmans theorem restricts separable aggregation to at most one nonconvex component. We solve this by proving that a separable, additive or multiplicative, function is star quasiconvex, those with star shaped sublevel sets about minimizers, if and only if each component is star quasiconvex. This immediately yields star quasiconvexity of separable sums of quasiconvex functions, formally bridging diversification theory with the S shaped value functions of Prospect Theory. Furthermore, we develop a complete calculus, monotonic composition, pointwise minima, quasi arithmetic means, and we apply it to Cobb-Douglas functions, multifactor risk models, and constant function market makers in decentralized finance. Star quasiconvexity thus provides a unified framework for applications in optimization and economic modeling beyond the classical Debreu Koopmans constraint. The introduction discuss economic motivations.

In this note, we study the equivalence of Morse and stable subgroups in the framework of the coset intersection complex. Under certain conditions on a coset intersection complex of a group, we prove that infinite-index Morse subgroups are stable.

This paper provides necessary and sufficient conditions for the existence of free boundaries in overdetermined problems for the Laplacian, and sufficient conditions for the bi-Laplacian, when the overdetermined boundary condition is non-constant. Using classical integral inequalities (Cauchy-Schwarz, Hölder, Hardy, eigenvalue bounds, Pohozaev and Reilly identities), we derive existence results for a broad class of free boundary problems arising in potential theory, plate theory, electromagnetism, and shape optimization. A regularity result for minimizers in the $C$-GNP class is established using the thickness function and the Wiener criterion, based on the geometric description of cusp points given in \cite{Barkatou2002}. We provide a new, self-contained geometric result: for almost every $t$, the level sets of the solution inherit the $C$-GNP property. This inheritance theorem justifies the variational framework and guarantees that the entire foliation generated by the state function remains within the admissible class. New results include refined estimates via interpolation inequalities, stability under perturbations, and connections with isoperimetric inequalities. The physical interpretation of the bi-Laplacian problem $\mathcal{B}(f,g)$ in the Kirchhoff-Love theory of thin plates is emphasized.

This work uses mostly model-theoretic methods to establish new proof-theoretic theorems about several axiomatic theories of truth over KP (Kripke-Platek set theory) and stronger theories, especially ZF (Zermelo-Fraenkel set theory).

We revisit the Gaussian broadcast channel (GBC) and explore the rate region achieved by purely discrete inputs with treating interference as noise (TIN) decoding. Specifically, we introduce a simple scheme based on superposition coding with identically and independently distributed (i.i.d.) inputs drawn from discrete constellations, e.g., pulse amplitude modulations (PAM). Most importantly, we prove that the resulting achievable rate region under TIN decoding is within a constant gap to the capacity region of the GBC, where the gap is independent of all channel parameters. In addition, we show via simulation that the weak user can achieve a higher rate with PAM than with Gaussian signaling in some cases.

We establish the existence of a positive bounded weak solution for a class of Kirchhoff-type $p(\cdot)$-Laplacian problems involving an arbitrary growth and a sandwich-type growth $s(\cdot)\in (\inf p,\sup p)$. This setting leads to substantial analytical difficulties in the variational analysis of the associated energy functional. By combining truncation arguments with a priori estimates, we prove the existence result under suitable assumptions on the data.

In this paper, we study a system of bosons trapped in a confining potential, interacting with a quantized field of coherent photons in the mean field description of non-relativistic Quantum Electrodynamics (QED) obtained by [N. Leopold and P. Pickl , 2017]. We derive the effective nonlinear cascade equations governing the emission and absorption of coherent photons by the boson subsystem in a combined weak-coupling and macroscopic time limit. We demonstrate that solutions to this nonlinear cascade describe a monotone decreasing energy flow in the boson subsystem. Thereby, we prove that a Bose-Einstein condensate (BEC) forms dynamically, under conservation of the total boson $L^2$ mass. We note that this process is crucially different from thermal relaxation to the ground state, and fundamentally depends on the nonlinear nature of the cascade dynamics.

Let $\Theta=\{\theta_n\}, \Lambda=\{\lambda_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by $$ f_{\Theta,\Lambda}(x)= \left( \sum_{n=0}^{\infty} a^n\cos\bigl(2\pi (b^n x+\theta_n)\bigr),\ \sum_{n=0}^{\infty} a^n\sin\bigl(2\pi (b^n x+\lambda_n)\bigr) \right), \; x\in[0,1], $$ where $0<a<1<b,\ ab> 1$. The Hausdorff dimension of the graph of this function is proved to be $$\dim_H G(f_{\Theta,\Lambda}) = \min\left\{-\frac{\log b}{\log a}, \, 3 +2\frac{\log a}{\log b}\right\} \quad \text{a.s.}$$

Let \(\Lambda\) be a finite-dimensional quadratic monomial algebra and let \(\Lambda^{!}\) be its Koszul dual. We investigate the structure of graded modules over \(\Lambda^{!}\) and derive several consequences for Koszul duality. We prove that \(\Lambda^{!}\) is both graded coherent and graded co-coherent. Moreover, finitely presented and finitely copresented graded \(\Lambda^{!}\)-modules coincide with perfect and coperfect modules, respectively. As a consequence, the associated tails and cotails categories are hereditary abelian categories admitting explicit descriptions in terms of linear and colinear modules. We further show that every finite-dimensional quadratic monomial algebra is absolutely Koszul and has global linearity defect at most one. In particular, finitely presented graded modules have rational Poincaré and Hilbert series. Using these structural results, we refine graded and ungraded derived Koszul dualities, singular Koszul dualities, and the Bernstein--Gelfand--Gelfand correspondence. We obtain explicit descriptions of the associated triangulated categories and of the induced nonstandard \(t\)-structures. As an application, we derive new bounds on the finitistic dimension of quadratic monomial algebras in terms of finite paths in the Koszul dual algebra.

We study the generalized Lam'e equation (GLE) on an elliptic curve $E$ with multiple regular singularities $\mathbf{p} = (p_i)_{i = 1}^r$ of weights $\mathbf{n} = (n_i)_{i = 1}^r$. By analyzing the locus admitting quasi-periodic solutions, we construct two fundamental algebraic curves: (i) The generalized Lam'e curve (GLC), $\mathcal{Y}_{\mathbf{n}, \mathbf{p}}$, which lies in an affine bundle over $\operatorname{Sym}^n E$ for total weight $n:=\sum n_i \in \mathbb{Z}_{\geq 0}$ and parametrizes generalized Hermite--Halphen ansatz solutions. (ii) The log-free curve, $V_{\mathbf{n}, \mathbf{p}}$, a non-complete intersection variety arising when all $n_i \in \frac{1}{2}\mathbb{N}$, which we prove is a reduced curve, confirming a conjecture of Wang. We analyze the GLC as an algebraic family over the pole configuration space. By studying the addition map$$\sigma \colon \operatorname{Sym}^n E \longrightarrow E,$$where we establish a generically finite, universal degree formula, we show that the geometry of boundary degenerations under pole collisions perfectly mirrors the tensor algebra of $\mathfrak{sl}_2(\mathbb{C})$-modules within the BGG category $\mathcal{O}$. This provides the local structural limits needed to establish the global flatness of the GLC. Furthermore, we develop a framework of twisted isomonodromic deformations and construct $(\mathbf{n}, \mathbf{p})$-deformed pre-modular forms parameterized by twisted monodromy data $(t,s)$. Their vanishing solves the underlying monodromy problem and factorizes along boundary strata, allowing an arbitrary configuration to be continuously deformed down to the classical Lam'e equation. Finally, using an asymptotic scaling technique, we completely solve the Treibich conjecture for $r=2$ symmetric pairs, extend it to $r \leq 4$, and propose a general formula enumerating symmetric finite-gap KdV potentials for all $r$.

In this paper we study continuum-marginal optimal transport. Given a time-continuous family of probability marginals, the problem is to recover the minimum-energy velocity field whose flow reproduces every marginal. This problem is the continuum limit of the classical two-marginal Benamou--Brenier formulation, and also the deterministic limit of the Nelson problem of stochastic optimal transport. We propose a practical mesh-free solver for this problem. The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial discretization. The velocity is parametrized by any linear-in-parameters dictionary or neural network, and is optimized by mini-batch stochastic methods. Synthetic experiments confirm that the method achieves accurate drift recovery and marginal consistency. The same computational framework also applies to the stochastic Nelson problem.

We present a novel and unifying framework for constructing spectral approximations to fractional integral operators. These spectral approximations are based on transplanted Chebyshev polynomials, which are obtained by composing Chebyshev polynomials with a variable transform. When an algebraic transform is used, the framework produces spectral approximations based on Jacobi fractional polynomials. When an exponential transform is used, it yields a versatile spectral approximation that is applicable to a much broader class of fractional calculus problems. The construction of such spectral approximations is both numerically stable and optimal in terms of complexity. These spectral approximations lead to stable and fast spectral methods for fractional calculus. The spectral approximation based on the double-exponential transform is demonstrated through extensive numerical examples that are intractable for existing spectral methods.

Let $\zeta$ be Euclidean norm of the degree sequence of a graph normalized by the graph size. We prove that when the vertices of a graph are randomly colored with $s$ colors such that the fraction of vertices in each color class is bounded away from zero, only two asymptotic regimes emerge. If $\zeta=o(1)$, then the sizes of the subgraphs induced by the color classes concentrate around their expected values. If $\zeta=\Theta(1)$, then concentration depends on the color balance: for colorings with persisting imbalance, the total number $M$ of monochromatic edges stays bounded away from its mean with positive probability; otherwise, for vanishing imbalance, $M$ still concentrates. The same dichotomy holds for a broad class of randomly colored random graphs.

Classroom dynamics depend on various elements that influence teaching performance and learning activities. A key challenge is to determine the most effective seating plan, where students will seat in a specific classroom setting to achieve the best learning environment. This paper introduces the Student Seat Allocation Problem (SSAP) for strategically organizing student seating in traditional classrooms to minimize interpersonal conflicts. We propose a mathematical model and an Iterated Local Search (ILS) heuristic to solve the SSAP. Computational experiments demonstrated that ILS outperformed in more complex scenarios when compared to the results obtained by a commercial solver on the introduced mathematical model. ILS was particularly efficient in real and artificial instances that exhibited a higher number of conflicts.

We show that information on the first server influences the expected total number of games and margin in a tennis match under the standard assumption that each player's serve point win probability remains constant, and identify the exact regions, in terms of these probabilities, in which this effect is non-negligible. We confirm numerically that this effect is bounded by at most $0.4$ games at both the set and match level. This translates, for example, to roughly a $9$ percent shift in the probability that a match exceeds $19.5$ games when the players' serve point win probabilities differ by $10$ percent. We complement the analysis with an empirical comparison on professional match data, illustrating the adequacy of the constant-probability assumption for modelling the total number of games.

Well-known chaotic maps, such as the logistic and tent maps, have been used to generate cryptographically secure pseudorandomness, yet we know of no efforts which attempt to utilize the Gauss continued-fraction map, a known chaotic map, as a starting point for producing quality pseudorandom output. In this paper, we consider the family of $r$-continued-fraction maps, which generalize the Gauss map, and use them to generate pseudorandom output which outperforms many standard generators, such as the Mersenne Twister, in statistical quality, as ascertained by the use of the Dieharder, PractRand, and TestU01 suites. In this way, we demonstrate the potential viability of these maps as a starting point for novel generators, and provide practical motivation for further study of the properties of both the exact and finite-precision $r$-continued fraction maps.

In this paper, we study generalized $m$-quasi-Einstein $(M^n,g,X,\lambda)$ under natural conditions on the potential vector field. We show that, under suitable integral assumptions, the potential vector field is Killing, extending earlier results of Sharma to the generalized setting. Moreover, we show that divergence-free vector fields are Killing in this context, and we derive consequences under sign conditions on $m$ and $\lambda$, including triviality results. We also revisit a recent theorem of Ghosh \cite{ghosh}, discuss a subtle issue in the argument, and provide a new formulation and proof. Finally, we establish rigidity results for manifolds with geodesic potential vector fields.

We study the rational coefficients that arise when the Eisenstein series $G_k$ is expressed as a polynomial in $G_4$ and $G_6$, proving a conjecture that gives an exact formula for their minimal 2-adic valuation in terms of the binary expansion of the weight. The proof uses lacunary recurrences for Eisenstein series and yields refined information about the first valuation levels. As an application, we prove irreducibility results for Faber polynomials associated to dyadic linear combinations of powers of Eisenstein series.

We develop a Kubo--Ando theory on order intervals of completely positive maps. Using Arveson's Radon--Nikodym theorem as a structural tool, we define relative Kubo--Ando means \(\Phi\sigma_\Omega\Psi\) for completely positive maps dominated by a common ambient map \(\Omega\). The special choice \(\Omega=\Phi+\Psi\) yields an intrinsic mean of two completely positive maps. We prove that these means are independent of the chosen Stinespring representation and satisfy the expected order-theoretic properties, including monotonicity, transformer inequalities, Jensen-type inequalities, data processing, and monotonicity with respect to the ambient map. For the geometric mean, we obtain a block-positivity characterization and show that the intrinsic geometric mean vanishes exactly when the two maps have no nonzero common completely positive submap. Finally, we compare the construction with existing finite-dimensional and form-theoretic approaches: for maps between matrix algebras it agrees with the Choi-matrix mean, and in the geometric case it agrees with Okayasu's Pusz--Woronowicz mean on their common domain.

We develop an adaptive-metric framework for norm-minimization-based outer approximation algorithms in bounded convex vector optimization. The key idea is to let the scalarization metric vary across iterations while measuring approximation error in a fixed Euclidean norm. This enables the algorithm to exploit problem geometry dynamically. Our approach rests on two theoretical foundations. First, we prove that the improved Euclidean convergence rate $O(k^{2/(1-q)})$ -- previously known only for the standard $\ell_2$-norm -- extends to all fixed inner-product norms. Second, we establish a dispersion theorem showing that the cut-normals generated by the algorithm naturally spread across all directions when the upper image has a strictly convex boundary with bounded curvature. This geometric condition guarantees that the adaptive metric remains well-conditioned throughout execution. Building on these results, we derive explicit convergence bounds that quantify how metric conditioning influences the Hausdorff error estimates. Numerical experiments on three test problems validate the theoretical convergence rate; on the problems whose Pareto fronts have sufficient curvature, the adaptive metric additionally reduces the iteration count relative to the fixed Euclidean norm. Our results provide a rigorous foundation for adaptive metric selection in convex vector optimization.

A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. Following this principle, we first provide a unified perspective on bi-orthogonally equivariant updates for general matrix layers, as employed by stochastic spectral descent, Muon, Scion, and polar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symmetries differ from those of general matrix layers: embedding and LM head matrices, SwiGLU MLP projections, and MoE router matrices. These constructions include one-sided spectral, row-norm, hybrid row-norm/spectral, row-aware, column-aware, centered row-norm, and left-spectral updates. They yield an end-to-end layerwise optimizer stack in which each major matrix-valued parameter class is assigned an update whose equivariance matches its symmetry group. We corroborate this principle through pre-training experiments on dense and sparse MoE language models, including Qwen3-0.6B-style, Gemma 3 1B-style, OLMoE-1B-7B-style, and downsized gpt-oss architectures. Across these experiments, symmetry-compatible update rules consistently improve final validation loss, reduce load imbalance in sparse MoE models, and in several cases improve training stability over the corresponding AdamW updates.

For a smooth projective curve of genus $g$, we study some positivity properties of (twisted) rank-$g$ Picard bundles on the $g$-fold symmetric product. As an application, we prove that the degree of irrationality of any genus $g$ Jacobian is bounded from above by $2^g$, and the degree of irrationality of any $(g-1)$-dimensional Prym variety is bounded from above by $2^{2g-3}$.

Let $G$ be a group and let $R$ be a $G$-graded ring. We show that every nonzero central idempotent in $R$ has finite support group in two broad settings: when $G$ is abelian, and when $G$ is arbitrary but the grading satisfies a certain one-sided non-annihilation condition on nonzero homogeneous elements. In particular, under the respective hypotheses, if $G$ is torsion-free, then every central idempotent lies in the principal component of the grading. Our results generalize earlier results by H. Bass, R. G. Burns, and A. A. Bovdi--S. V. Mihovski, from group rings and crossed products, to non-commutative, possibly non-unital, group-graded rings. We demonstrate the utility of our results by applying them to semigroup-graded rings, Leavitt path rings, fractional skew monoid rings, partial skew group rings, and algebraic Cuntz-Pimsner rings.

We prove that there are globally hyperbolic spacetimes which are refocusing but not strongly refocusing. In fact, every globally hyperbolic strongly refocusing spacetime of dimension at least $3$ admits globally hyperbolic metrics which are refocusing but not strongly refocusing. This answers a question by Chernov, Kinlaw, and Sadykov. We then prove that globally hyperbolic spacetimes which are Legendrian refocusing (a notion introduced in this paper) admit globally hyperbolic strongly refocusing metrics.

We develop a first and second fundamental theorem for $n$--tuples of bosonic and fermionic matrices, by developing graded analogues of the classical case.

The prime number graph is the set of points $(n,p_n)$ where $p_n$ denotes the $n^{\rm th}$ prime. Let $L(n)$ be the minimum number of straight line segments needed to cover the first $n$ points in this set. Let $B(n)$ be the largest number of points $(k,p_k)$ with $k\le n$ covered by a single line. Recently Sloane conjectured that $L(n) = O(n/\log n)$. We show that $L(n)=O(n \log \log n / \log n)$ and $B(n)\ge c\log n$ for a constant $c>0$ and all large $n$. Under RH we show that for large $n$ we have $B(n)=O(n^{3/4}(\log n)^{1/2})$ and $ L(n)\ge c' n^{1/4} (\log n) ^{-1/2}$ for some constant $c'>0.$

We consider the ODE/IM correspondence for the value $c=-2$ of the Virasoro central charge (free-fermion point) and the associated quantum KdV model $-$ the quantization of the second hamiltonian structure of the classical periodic KdV model. We prove that the ODE/IM correspondence is complete (in the sense of V. Bazhanov, S. Lukyanov, and A. Zamolodchikov), namely that any solution of the Bethe equations coincides with the spectrum of a rational extension of the (quantum) harmonic oscillator. To this end, on the ODE side we consider Crum$-$Darboux transformations of the harmonic oscillator and the associated Laguerre Wronskians, which are remarkable special functions parametrized by pairs of partitions which we study in depth. As a further result, on the IM side, we diagonalize explicitly the first three hamiltonian operators of quantum KdV (in the free field representation): the eigenstates are Schur functions and the eigenvalues are shifted symmetric functions on partitions. We give two applications of this result: i) we prove that the eigenvalues are given by the evaluation of the Newton symmetric polynomials at the poles of the associated monster potentials, as further conjectured by V. Bazhanov, S. Lukyanov, and A. Zamolodchikov; ii) we show that these hamiltonian operators also belong to the algebra of hamiltonian operators obtained by quantizing the first hamiltonian structure of the classical periodic dispersionless KdV model.

We investigate Fourier multipliers associated with the Strichartz Fourier transform on the Heisenberg group. In particular, we establish Hörmander-type $L^{p}-L^{q}$ boundedness results for the range $1<p\leq 2\leq q<\infty$. The analysis is based on deriving suitable analogues of the Hausdorff-Young and Paley inequalities for the Strichartz Fourier transform, followed by interpolation arguments to obtain the desired multiplier estimates. As an application, we study the local well-posedness of certain nonlinear partial differential equations. Furthermore, we establish an $L^{p}$-boundedness theorem for Fourier multipliers associated with the Strichartz Fourier transform for the full range $1<p<\infty$.

Itenberg-Katzarkov-Mikhalkin-Zharkov gave an isomorphism of tropical cohomology and cohomology of some maximally degenerate algebraic varieties. Their proof was based on tropical analogs of Steenbrink's geometric monodromy-weight spectral sequences. These were generalized to the non-realizable case by Amini-Piquerez. In this paper, we give a new construction of these tropical spectral sequences in the same way as Steenbrink's ones. For this purpose, we introduce reductions of tropical varieties. We also show that eigenwave actions are given by tropical Gauss-Manin connections.

Recently, [25] observed that, among the currently unresolved cases of the axially symmetric Navier-Stokes equations (ASNS), the most relatively tractable one is where the fluid passes the exterior of a cone. In this paper, we investigate this case with the classical Navier total-slip boundary condition. We show that there exists an absolute constant $C_* > 0$ such that if \[ \sup_{x\in D}r|v_{0,\theta}|\leq C_* \quad\text{and}\quad \int_{D} r v_{0,\theta}(x) \mathrm{d} x = 0, \] then there exists a unique global bounded strong solution with finite energy. We point out that there is no size restriction on other components of the initial velocity. Compared with [25], no parity symmetry assumption on $\boldsymbol{v}_0$ is required. There are four key ingredients in the proof. (1) In spherical coordinates, we introduce three new quantities \[ \mathcal{K}\buildrel\hbox{def}\over =\frac{\sin\phi}{\rho^2}\partial_\phi\Big(\frac{v_\theta}{\sin\phi}\Big) \,,\quad\quad\mathcal{F}\buildrel\hbox{def}\over =-\partial_\rho\Big(\frac{v_\theta}{\rho}\Big) \,,\quad\quad \mathcal{O}\buildrel\hbox{def}\over =\frac{1}{\rho\sin\phi}\Big(\omega_\theta-\frac{2v_\phi\eta(\rho)}{\rho}\Big) \,, \] and derive a self-closed energy estimate for them, where $\eta$ is a cut-off function which vanishes near the origin and equals $1$ away from the origin. (2) A boundary value problem of the pressure $P$ is proposed and an elliptic estimate for $P$ is established in order to control boundary terms arising from the Navier total-slip boundary condition. (3) A De Giorgi iteration scheme is applied to establish the boundedness of the quantity $rv_\theta$ whose integral on $D$ vanishes for all the time. (4) A new anisotropic Hardy's inequality is derived for functions whose integral on $D$ vanish to overcome the lack of parity symmetry of $\boldsymbol{v}$.

We study a symmetry problem for the $h$-polynomials of edge rings of bipartite graphs. Let $G$ be a bipartite graph and write $h(\mathbb{k}[G];t)=h_0+h_1t+\cdots+h_st^s$. We prove that if $\Bbbk[G]$ is pseudo-Gorenstein and $h_1=h_{s-1}$, then $\Bbbk[G]$ is Gorenstein. Equivalently, under these assumptions the $h$-polynomial of $\Bbbk[G]$ is palindromic. The proof treats the $2$-connected case first by translating the numerical condition $h_1=h_{s-1}$ into a tight-separation condition for non-edges, and then passes to arbitrary bipartite graphs using the block decomposition. We also construct a blockwise minimal Gorenstein closure, obtained by adjoining all non-edges not separated by tight acceptable sets, and show that this construction preserves the next-to-leading coefficient of the $h$-polynomial.

We prove that the topological flow category $\mathcal{M}$ arising from a Morse-Smale pair $(f,\xi)$ on a smooth closed manifold $X$ is equivalent, as an $\infty$-category, to Lurie's $\infty$-category $\mathrm{Sing}_A(X)$ of exit paths in $X$ with respect to the stratification by the stable manifolds of $\xi$. The objects of $\mathcal{M}$ are the critical points of $f$, and for every pair of critical points, the space of morphisms of $\mathcal{M}$ between these is the space of possibly broken trajectories of $\xi$ connecting them; it can be identified up to homotopy with the space of unbroken ones. The latter maps naturally to the space of exit paths connecting these critical points; we prove this map to be a weak homotopy equivalence. Then, we combine these ingredients with several others to construct a zigzag of equivalences between the homotopy coherent nerve of $\mathcal{M}$, denoted $\mathcal{N}(\mathcal{M})$, and $\mathrm{Sing}_A(X)$. The $n$-simplices of $\mathcal{N}(\mathcal{M})$ are homotopy coherent diagrams of $n$ composable morphisms of $\mathcal{M}$; we introduce the notion of unbroken diagram, yielding an $\infty$-subcategory of $\mathcal{N}(\mathcal{M})$, which we refer to as the flow coherent nerve of $\mathcal{M}$. The simplices of the latter give rise to stratified maps out of a family of stratified cubes, into $X$. We organize this family into a functor from the category of finite ordered sequences of critical points, to the category of $A$-stratified topological spaces, and we prove a comparison result with the usual stratified geometric realization functor. We finally use a theorem of Tanaka that associates a functor of $\infty$-categories to a map a semi-simplicial sets satisfying some conditions. Our theorem has implications regarding constructible sheaves and the description of homotopy types in terms of flow categories.

We prove that infinite Galois extensions of number fields with Galois group of finite exponent have the Northcott property. The main novelty of our approach lies in the application of a theorem of Segal on profinite groups.

In chapter 9 of his book "The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal", Woodin shows how to force the Strong Chang Conjecture over models of determinacy using $\mathbb{P}_{\mathrm{max}}$. We show here how a modification of the proof implies that such extensions actually verify the stronger cofinal version of the conjecture. This stronger version has important consequences on the semi-properness of small forcing, allowing us to prove the consistency of the theory "ZFC + Namba forcing is semiproper + $\Theta^{UB}=\omega_3$". We then use the constructions of this proof to also show that Woodin $(\ast)_{UB}$ axiom implies the conjecture.

We give examples of proper minimal immersions in Euclidean space with very rapid area growth. The first is a proper embedding into $\bf{R}^4$ that yields a stable minimal surface, while the second is a proper immersion into $\bf{R}^3$. These results are motivated by [CM1] that shows that proper minimal submanifolds confined in space satisfy strong structural constraints.

We propose Acc-Sinkhorn, a simple accelerated variant of Sinkhorn for entropy-regularized optimal transport (EOT). The method is derived from a bilevel optimization view: Sinkhorn row scaling solves the inner variable $u$ exactly and defines the reduced dual objective $f(v)=\min_u F(u,v)$, while the remaining column scaling is a unit-step dual mirror descent step in $v$. This structure yields a Hessian-driven Nesterov acceleration that keeps Sinkhorn's scaling form and per-iteration cost, using only extrapolated combinations of Sinkhorn iterates. We prove an $\mathcal{O}(1/k^2)$ rate under a verifiable stability condition. For an $\varepsilon$-approximation of unregularized OT, the resulting complexity is $\widetilde{\mathcal{O}}(n^2/\varepsilon)$, improved from $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$ for Sinkhorn. On synthetic problems, color transfer, and word alignment, Acc-Sinkhorn gives a $10\times$--$30\times$ speedup over Sinkhorn at small regularization.

We define a quantum graph to be vertex-transitive if the join of its automorphism group is the maximum quantum relation on its quantum vertex set, in direct analogy with the classical case. All simple quantum graphs in $M_2(\mathbb C)$ are vertex-transitive, but many simple quantum graphs in $M_3(\mathbb C)$ are not vertex-transitive. We provide a complete classification of vertex-transitive quantum graphs in $M_3(\mathbb C)$ up to isomorphism. To do this, we introduce a polynomial invariant for quantum graphs in $M_n(\mathbb C)$, which we call the panoramic polynomial.

Underdetermined quadratic interpolation is a central model-construction tool in model-based derivative-free trust-region methods: it limits sampling costs but leaves an affine family of interpolating quadratics. Classical solvers select one element of this family by prescribing a fixed norm or model-change measure, such as the least-Frobenius-change Hessian update in Powell-type methods. We introduce BUP-TR (Bayesian Underdetermined Projection Trust-Region), which instead completes the model by projecting a prior quadratic onto the affine interpolation set in the precision norm supplied by the prior. The same precision matrix defines a spectral geometry certificate, MAP-poisedness, and a repair mechanism for interpolation sets. Under standard smoothness assumptions, uniform precision bounds, MAP-poisedness, and a trust-region-scale prior-accuracy condition, the hard-MAP models are fully linear. Consequently, BUP-TR attains global first-order convergence and O(epsilon^{-2}) evaluation complexity, with geometry-repair evaluations included. A NEWUOA-style implementation, BUP-NEWUOA, improves fixed-budget performance on the reported benchmark suite at moderate and stringent accuracy targets while retaining the computational structure of a Powell-type trust-region method.

Can gradient descent be accelerated by just choosing better stepsizes? Surprisingly, the answer is yes. This short expository article provides an accessible introduction to this phenomenon of stepsize hedging.

We show that axisymmetric Willmore surfaces admit a first-order formulation obtained by combining two independent first integrals. If $\rho$ denotes the distance from the axis of revolution and $\Psi=\sin\psi$, where $\psi$ is the tangent angle of the generating curve, then the profile satisfies \begin{equation*} \left[\frac{\Psi(\rho\Psi'-\Psi)^2+2(\rho\Psi'-\Psi)+2C_1\rho}{\sqrt{1-\Psi^2}}\right]^2 +\left[(\rho\Psi'-\Psi)^2-2\right]^2=C_2, \end{equation*} where $C_1$ and $C_2$ are constants of integration and the prime denotes differentiation with respect to $\rho$. This equation reduces the axisymmetric Willmore equation to a first-order ordinary differential equation and provides a convenient classification scheme for Willmore surfaces of revolution. The sphere and the Clifford torus are discussed as elementary checks of the formulation.

We present a non-convergence theory for probabilistic direct search, a randomized derivative-free optimization method, where non-convergence means the failure to produce iterates that achieve stationarity asymptotically. The motivation is to understand whether the submartingale-like assumption in the existing convergence theory is essential or merely an artifact of the analysis techniques. For convex objectives, we prove that the probability of non-convergence is positive, provided that the polling directions satisfy a probabilistic ascent condition that is essentially the opposite of the submartingale-like convergence condition. Furthermore, we establish a lower bound for the non-convergence probability. For the typical implementation of this method, where each iteration draws a fixed number of random polling directions independently and uniformly from the unit sphere, our theory implies that the method is not globally convergent if the number of directions is below the threshold specified in the convergence theory, and the submartingale-like assumption is confirmed to be essential for convergence. Our theory is obtained by examining two random series that control the distance from any iterate to the starting point and estimating the probability for these series to stay below certain bounds.

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 $E$.

Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. For any $a,b\in\mathbb{F}_p$, it is known that the Kloosterman sum $$K_p(a,b)=\sum_{x\in\mathbb{F}_p\setminus\{0\}}e^{\frac{2\pi i}{p}(ax+\frac{b}{x})}$$ can be viewed as a finite field analogue of certain Bessel function. In this paper, using the arithmetic properties of character sums over $\mathbb{F}_p$, we study some cyclotomic matrices involving Kloosterman sums. For example, we prove that the matrix $[K_p(1,i^2+j^2)]_{1\le i,j\le (p-1)/2}$ is singular if and only if $p\ge11$.

In our previous work, we generalized the concept of Galois points for irreducible plane curves to the case of reduced plane curves. We also introduced the concept of simultaneous Galois points, which is an equivalent concept to Galois points, and studied their number when the irreducible components are nonsingular. In this paper, we consider the remaining cases where the irreducible components are of degree $d=2$ or $3$. For the case of $d=2$, we establish a generalized version of the theorem in our previous paper. For the case of $d=3$, we classify simultaneous Galois points into the first and second kinds. We give a necessary condition for the former and provide examples for the latter.

Let $n$ be an integer not equal to $-2$, $0$ or $2$. We consider the unit equation $\varepsilon + \delta = n$ in units $\varepsilon, \delta$ of cubic fields. We show that this equation has no solutions for 100% of cubic fields, when ordered by discriminant. This is consistent with a recent conjecture of the authors.

Continuous-variable quantum key distribution (CV-QKD) requires highly efficient reconciliation techniques to operate at low signal-to-noise ratios and long distances. Multidimensional reconciliation addresses this challenge by transforming the physical Gaussian quantum channel into a virtual binary-input additive white Gaussian noise (BIAWGN) channel, enabling the use of modern errorcorrecting codes. In this work, we review the principles of multidimensional reconciliation, with a particular focus on high-dimensional constructions beyond the algebraic dimensions 1, 2, 4, 8. We describe the construction of the virtual channel, discuss practical coding schemes for reverse reconciliation, and analyse their integration with linear error-correcting codes. We also present an opensource simulation framework, HDirac, implementing multidimensional reconciliation for arbitrary dimensions, and use it to evaluate state-of-the-art LDPC codes. The results highlight key trade-offs between dimension, reconciliation efficiency, and frame error rate, providing practical guidance for CV-QKD system design.

We derive a novel canonical decomposition of factor models encompassing both the static factor model - where factors are loaded only contemporaneously - and the Generalised Dynamic Factor Model - where factors are loaded with lags. This decomposition features a new term: the weak common component, defined as the difference between the dynamic and static common components. It is driven by (possibly infinitely many) non-pervasive weak factors which belong to the dynamically common space. Through theoretical and empirical examples - both on U.S. macroeconomic indicators and global financial volatilities - we show that, in general, the weak common component shall not be neglected. Furthermore, we show that, by accounting for the presence of weak common components, we are likely to obtain Impulse Response Functions with more plausible shapes than those obtained from purely static approaches. In addition, we provide consistent estimators for all terms of the canonical decomposition and for the weak factors.

Constrained maximization of submodular functions is a central problem in combinatorial optimization. In many realistic scenarios, multiple agents each need to maximize their own submodular objective over a common ground set, subject to individual constraints, with the requirement that their solutions be disjoint. We study this setting through the lens of algorithmic fairness and constrained fair division. Inspired by the fair division literature, we propose and analyze a simple Round-Robin protocol in which agents take turns building their solutions one item at a time; each agent is free to use any internal algorithm, and the protocol itself performs no computation. We show that agents following simple greedy policies enjoy solid guarantees for both monotone and non-monotone objectives subject to constraints as general as $p$-systems. For monotone objectives, a greedy agent $i$ with a $p_i$-system constraint achieves a $1/(n+p_i)$ fraction of the best value available when they first get to choose. On instances that are robust to competition -- where no agent's optimal value is greatly affected by losing some items to others -- these guarantees improve to a $1/\Theta(p_i)$ approximation of the unconstrained optimum, which is asymptotically best-possible in polynomial time. We further establish novel fairness guarantees: greedy agents produce approximately feasible-envy-free-up-to-one-item (FEF1) and approximately feasible-envy-free-towards-unallocated-items (FEFu) allocations for monotone and non-monotone objectives. Via a simple augmented protocol and a self-contained polynomial-time proxy algorithm, we also obtain the first $\Theta(1/p_i)$-approximate feasible maximin share (FMMS) guarantees for submodular agents with combinatorial constraints. Finally, although greedy policies may not be individually optimal, consistently improving upon them is NP-hard even in the simplest settings.

The G-Wishart distribution is a core component for the Bayesian analysis of Gaussian graphical models as the conjugate prior for the precision matrix. Evaluating the marginal likelihood of such models usually requires computing high-dimensional integrals to determine the G-Wishart normalising constant. Closed-form results are known for decomposable or chordal graphs, while an explicit representation as a formal series expansion has been derived recently for general graphs. The nested infinite sums, however, do not lend themselves to computation, remaining of limited practical value. Borrowing techniques from random matrix theory and Fourier analysis, we provide novel exact results well suited to the numerical evaluation of the normalising constant for classes of graphs beyond chordal graphs. We additionally develop a Monte Carlo scheme for general graphs, which can be orders of magnitude more efficient than current approaches.

We are considering the problem of optimal portfolio delegation between an investor and a portfolio manager under a random default time. We focus on a novel variation of the Principal-Agent problem adapted to this framework. We address the challenge of an uncertain investment horizon caused by an exogenous random default time, after which neither the agent nor the principal can access the market. This uncertainty introduces significant complexities in analyzing the problem, requiring distinct mathematical approaches for two cases: when the random default time falls within the initial time frame [0,T] and when it extends beyond this period. We develop a theoretical framework to model the stochastic dynamics of the investment process, incorporating the random default time. We then analyze the portfolio manager's investment decisions and compensation mechanisms for both scenarios. In the first case, where the default time could be unbounded, we apply traditional results from Backward Stochastic Differential Equations (BSDEs) and control theory to address the agent problem. In the second case, where the default time is within the interval [0,T], the problem becomes more intricate due to the degeneracy of the BSDE's driver. For both scenarios, we demonstrate that the contracting problem can be resolved by examining the existence of solutions to integro-partial Hamilton-Jacobi-Bellman (HJB) equations in both situations. We develop a deep-learning algorithm to solve the problem in high-dimension with no access to the optimizer of the Hamiltonian function.

We study the problem of determining the minimal genus of a simple finite connected graph. We present an algorithm which, for an arbitrary graph $G$ with $n$ vertices and $m$ edges, determines the orientable genus of $G$ in $O(n(4^m/n)^{n/t})$ steps where $t$ is the girth of $G$. This algorithm avoids difficulties that many other genus algorithms have with handling bridge placements which is a well-known issue. The algorithm has a number of useful properties for practical use: it is simple to implement, it outputs the faces of an optimal embedding, and it iteratively narrows both upper and lower bounds. We illustrate the algorithm by determining the genus of the $(3,12)$ cage (which is 17); other graphs are also considered.

Semi-supervised inference assumes access to a labeled dataset together with a large unlabeled dataset in which the outcome variable is missing, and it is widely used to improve statistical efficiency and support generalizability across populations. In many modern applications, however, individual-level unlabeled data may not be directly accessible due to privacy restrictions, data-sharing limits, or storage constraints, while summary statistics such as sample means and covariances from the unlabeled population are often available. In this work, we study this constrained semi-supervised setting where, in addition to labeled data with individualized observations, auxiliary information from the unlabeled population is available only through summary statistics. We propose new semi-supervised inference methods for mean estimation under both covariate-independent and covariate-dependent labeling and show that unlabeled summaries can still improve efficiency and help correct selection bias. The proposed methods apply in high dimensions and are robust to model misspecification. Valid inference is obtained under sparsity conditions comparable to those required by semi-supervised methods that assume access to individual-level unlabeled samples. Our approach relies on a specialized cross-fitting procedure, where sample splitting is applied only to the labeled data, which removes the need for individualized unlabeled covariates. We further extend this framework to average treatment effect estimation, enabling generalizability and transportability of causal conclusions in this constrained semi-supervised setting.

We derive universal approximation results for the class of (countably) $m$-rectifiable measures. Specifically, we prove that $m$-rectifiable measures can be approximated as push-forwards of the one-dimensional Lebesgue measure on $[0,1]$ using ReLU neural networks with arbitrarily small approximation error in terms of Wasserstein distance. What is more, the weights in the networks under consideration are quantized and bounded and the number of ReLU neural networks required to achieve an approximation error of $\varepsilon$ is no larger than $2^{b(\varepsilon)}$ with $b(\varepsilon)=\mathcal{O}(\varepsilon^{-m}\log^2(\varepsilon))$. This result improves Lemma IX.4 in Perekrestenko et al. as it shows that the rate at which $b(\varepsilon)$ tends to infinity as $\varepsilon$ tends to zero equals the rectifiability parameter $m$, which can be much smaller than the ambient dimension. We extend this result to countably $m$-rectifiable measures and show that this rate still equals the rectifiability parameter $m$ provided that, among other technical assumptions, the measure decays exponentially on the individual components of the countably $m$-rectifiable support set.

In this work, we develop a scalable approach for a flexible latent factor model for high-dimensional dynamical systems. Each latent factor process has its own correlation and variance parameters, and the orthogonal factor loading matrix can be either fixed or estimated. We utilize an orthogonal factor loading matrix that avoids computing the inversion of the posterior covariance matrix at each time of the Kalman filter, and derive closed-form expressions in an expectation-maximization algorithm for parameter estimation, which substantially reduces the computational complexity without approximation. Our approach has several applications, including noise filtering for high-dimensional time series, estimating nonseparable covariance structure between different time series, and estimating latent physical processes from real-world measurements. Extensive simulated studies illustrate higher accuracy and scalability of our approach compared to alternatives. Furthermore, by applying our method to geodetic measurements to estimate slow slip events from geodetic data in the Cascadia region, our estimated slip better agrees with independently measured seismic data of tremor events. The substantial acceleration from our method enables the use of massive noisy data for geological hazard quantification and other applications.

In this paper, we investigate the data-driven identification of asymmetric interaction kernels in the Motsch-Tadmor model based on observed trajectory data. The model under consideration is governed by a class of semilinear evolution equations, where the interaction kernel defines a normalized, state-dependent Laplacian operator that governs collective dynamics. To address the resulting nonlinear inverse problem, we propose a variational framework that reformulates kernel identification using the implicit form of the governing equations, reducing it to a subspace identification problem. We establish an identifiability result that characterizes conditions under which the interaction kernel can be uniquely recovered up to scale. To solve the inverse problem robustly, we develop a sparse Bayesian learning algorithm that incorporates informative priors for regularization, quantifies uncertainty, and enables principled model selection. Extensive numerical experiments on representative interacting particle systems demonstrate the accuracy, robustness, and interpretability of the proposed framework across a range of noise levels and data regimes.

The Boolean Fourier representation has been widely used in learning theory, particularly for learning Disjunctive Normal Form (DNF) under uniform and product distributions. Extending these results to non-product distributions has remained a longstanding open problem. We address this challenge by introducing a generalized Fourier representation that enables learning under a broad class of non-product distributions. Our approach represents any distribution $D$ as a Bayesian network (BN) and derives a corresponding Fourier expansion. We show that standard Fourier-based learning techniques using membership queries to identify heavy coefficients can be adapted to this generalized representation with minor modifications. We prove that the $L_1$ spectral norm of conjunctions remains bounded under this expansion for difference-bounded tree BNs, significantly generalizing the known result for uniform distributions; matching lower bounds demonstrate the necessity of these constraints. Using these results, we establish the learnability of DNF and the agnostic learnability of decision trees under such distributions. Finally, we present an algorithm for learning difference-bounded tree BN distributions, extending our results to settings where the distribution is unknown.

We solve the one-dimensional Helmholtz equation in several scenarios using the quantum annealer provided by D-Wave Systems within a pseudospectral framework, where the solution is encoded in an appropriate set of basis functions. We evaluate the performance of different encoding strategies based on algebraic conditioning and adiabatic considerations, and benchmark their performance against the classical simulated annealing algorithm. In particular, we analyze the minimum energy gap, the so-called dynamic range, and the mean squared error to assess the numerical stability, consistency, and accuracy of the solutions returned by each strategy. Our work highlights the importance of developing custom embedding techniques ensuring well-conditioned algebraic systems. In particular, we show that encoding strategies producing full-rank matrices with reduced dynamic ranges enhance the performance of the quantum annealer even under polychromatic driving and nontrivial initial conditions. We further discuss the potential of developing hybrid quantum-classical schemes designed to simultaneously satisfy algebraic conditioning and adiabatic requirements.

Supersymmetry algebras can be used to obtain algebraic expressions for constant Yang-Baxter solutions, also known as braid group generators. This was done for non-invertible braid operators in \cite{maity2025non}. In this work we extend this construction for the invertible ones. The resulting expressions are then shown to obey relations analogous to those satisfied by quotients of braid groups. Examples of the latter include the Iwahori-Hecke algebra and the Birman-Murakami-Wenzl (BMW) algebra. As a result, we can Baxterize the constant Yang-Baxter solutions to yield spectral parameter dependent $R$-matrices. The regularity of these $R$-matrices depends on the representation of SUSY generators. In some cases they are regular in the usual sense and in the remaining they are `almost' regular. In the latter case they are also non-invertible. Nevertheless, we show that they can still help us construct integrable models in all dimensions of the local Hilbert space. These models can be described by Hamiltonian densities that are either local or non-local, depending on the representation chosen for the SUSY generators. We demonstrate this for all constant $4\times 4$ invertible Yang-Baxter solutions. Apart from finding new nearest-neighbor interaction spin $\frac{1}{2}$ systems, we also find their higher spin analogs due to the algebraic [representation independent] approach.

The default Gaussian latent in flow-based generative models poses challenges when learning certain distributions such as heavy-tailed ones. We introduce a general framework for learning data-adaptive parametric prior distributions (latent noise) using one-dimensional quantile functions, optimized via the Wasserstein distance between noise and data. The quantile-based prior parameterization naturally adapts to both heavy-tailed and compactly supported distributions and shortens transport paths. Numerical results on heavy-tailed weather and image datasets confirm the method's flexibility and effectiveness achieved with negligible computational overhead.

Small influential data subsets can dramatically impact model conclusions, with a few data points overturning key findings. While recent work identifies these most influential sets, there is no formal way to tell when maximum influence is excessive rather than expected under natural random sampling variation. We address this gap by developing a principled framework for most influential sets. Focusing on linear least-squares, we derive a convenient exact influence formula and identify the extreme value distributions of maximal influence - the heavy-tailed Fréchet for constant-size sets and heavy-tailed data, and the well-behaved Gumbel for growing sets or light tails. This allows us to conduct rigorous hypothesis tests for excessive influence. We demonstrate through applications across economics, biology, and machine learning benchmarks, resolving contested findings and replacing ad-hoc heuristics with rigorous inference.

We study the typical structure of games in terms of their connectivity properties. A game is `connected' if it has a pure Nash equilibrium and there is a best-response path from every action profile which is not a pure Nash equilibrium to every pure Nash equilibrium; a game is generic if it has no indifferences. In previous work we showed that, among all $n$-player $k$-action generic games that admit a pure Nash equilibrium, the fraction that are connected tends to $1$ as $n$ gets sufficiently large relative to $k$. Here, we consider the large-$k$ regime, which behaves differently: we show that the connected fraction tends to $1-\zeta_n$ as $k$ gets large, where $\zeta_n>0$ is an explicit constant. Thus, a constant fraction of many-action games are \emph{not} connected. However, for $n\geq3$, $\zeta_n$ is small and tends to $0$ rapidly with $n$, so as $n$ increases all but a vanishingly small fraction of many-player-many-action games are connected. Since connectedness is conducive to equilibrium convergence, we find a simple adaptive dynamic that is guaranteed to converge to a pure Nash equilibrium in all but a vanishingly small fraction of generic games that have one. We rely on new probabilistic and combinatorial arguments to tackle the large-$k$ regime.

A single geometric invariant fixes the relative normalization and structure of gravity, Yang--Mills theory, and fermion kinetic terms. The restricted invariant basis contains known linearized ghost-free quadratic gravitational sectors with propagating torsion. Our results establish a minimal geometric route to unification that does not rely on extra dimensions or symmetry breaking by hand. Unlike previous gauge--gravity constructions, the relative normalizations and ghost freedom emerge from a single Clifford--algebraic invariant.

We propose physics-informed spectral diffusion (PISD), a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems. We learn the joint distribution of PDE parameters and solutions via a diffusion process in a latent space of scaled spectral representations, where Gaussian noise corresponds to functions with controlled regularity. This spectral formulation enables significant dimensionality reduction compared to grid-based diffusion models and ensures that the induced process in function space remains within a class of functions for which the PDE operators are well defined. Building on diffusion posterior sampling, we enforce physics-informed constraints and measurement conditions during inference, applying Adam-based updates at each diffusion step. We evaluate the proposed approach on Poisson, Helmholtz, and incompressible Navier-Stokes equations, demonstrating improved accuracy and computational efficiency compared with existing diffusion-based PDE solvers, which are state of the art for sparse observations. Code is available at this https URL.

Effective initialization in deep networks requires an understanding of random neural networks. In this work, a rigorous probabilistic analysis of deep bias-free random Leaky ReLU networks is provided. We prove a Law of Large Numbers and a Central Limit Theorem for the logarithm of the norm of network activations, establishing that, as the number of layers increases, their growth is governed by a parameter called the Lyapunov exponent. This parameter characterizes a sharp phase transition between vanishing and exploding activations, and we calculate the Lyapunov exponent explicitly for Gaussian or orthogonal weight matrices. Our results reveal that standard methods, such as He initialization or orthogonal initialization, do not guarantee activation stability for deep networks of low width. Based on these theoretical insights, we propose a novel initialization method, referred to as Lyapunov initialization, which sets the Lyapunov exponent to zero and thereby ensures that the neural network is as stable as possible, leading empirically to improved learning.

The Nekrasov instanton partition function of the 4d $\mathcal{N}=2^*$ $U(N)$ gauge theory (a mass deformation of 4d $\mathcal{N}=4$ super-Yang-Mills theory), which is a generating series of equivariant integrals over instanton moduli spaces, is given by a sum over colored partitions weighted by a counting parameter $\mathfrak{q}$. This note proves convergence of the series in the unit disk $|\mathfrak{q}|<1$ for generic parameters. Specifically, the absolute convergence radius of this sum is determined, assuming that mass and Coulomb branch parameters avoid some lattice. If the ratio $b^2=\epsilon_1/\epsilon_2$ of equivariant parameters is in $\mathbb{C}\setminus[0,+\infty)$, the radius is $1$, as expected. If $b^2$ is non-negative, three cases arise: the radius is finite if $b^2$ has finite exponential type (a generalization of Brjuno numbers), namely there exists $C>0$ such that $|b^2-p/q|>\exp(-Cq)$ for all integers $p,q\neq 0$; the series diverges if $b^2$ is super-exponentially well approximable by rationals; and if $b^2$ is rational some terms are singular. The AGT correspondence translates these results to convergence of torus one-point conformal blocks of the Virasoro and $W_N$ algebras with non-real $b$, within the unit disk. For the Virasoro algebra this corresponds to a central charge in $\mathbb{C}\setminus[25,+\infty)$.

We introduce a diagonalization-based optimization for Linear Echo State Networks (ESNs) that reduces the per-step computational complexity of reservoir state updates from quadratic to linear. By reformulating reservoir dynamics in the eigenbasis of the recurrent matrix, the recurrent update becomes a set of independent element-wise operations, eliminating the matrix multiplication. We further propose three methods to use our optimization depending on the situation: (i) Eigenbasis Weight Transformation (EWT), which preserves the dynamics of standard and trained Linear ESNs, (ii) End-to-End Eigenbasis Training (EET), which directly optimizes readout weights in the transformed space and (iii) Direct Parameter Generation (DPG), that bypasses matrix diagonalization by directly sampling eigenvalues and eigenvectors, achieving comparable performance to standard Linear ESNs. Across all experiments, both our methods preserve predictive accuracy while offering significant computational speedups, making them a replacement for standard Linear ESNs computations and training, and suggesting a shift of paradigm in linear ESN towards the direct selection of eigenvalues.

We investigate within the framework of linear theory the behaviour of the total (hydrodynamic) pressure and of the dynamic pressure in a regular wave train which propagates at the surface of water with a flat bed in a flow with constant vorticity. We show that nonzero vorticity, the hallmark of a non-uniform underlying current, may strongly alter the behaviour with respect to the case of irrotational flows, for which the maximum and minimum of the dynamic pressure always occur at the wave crest and at the wave trough, respectively (the extrema of the dynamic pressure may occur along the flat bed or along the critical level, depending on the vorticity strength). While vorticity does not modify the increase of the hydrodynamic pressure with depth, it can significantly alter the location of the extrema of the hydrodynamic pressure at a fixed depth level.

This paper considers the problem of converting a given dense linear layer to low precision. The tradeoff between compressed length and output discrepancy is analyzed information theoretically (IT). It is shown that a popular GPTQ algorithm may have an arbitrarily large gap to the IT limit. To alleviate this problem, a novel algorithm, termed ``WaterSIC'', is proposed and is shown to be within a rate gap of 0.255 bits to the IT limit, uniformly over all possible covariance matrices of input activations. The key innovation of WaterSIC's is to allocate different quantization rates to different columns (in-features) of the weight matrix, mimicking the classical IT solution known as "waterfilling". Applying WaterSIC to the Llama and Qwen family of LLMs establishes new state-of-the-art performance for all quantization rates from 1 to 4 bits. Our code is available at this https URL.

The Optimal Morse Matching (OMM) problem asks for a discrete gradient vector field on a simplicial complex that minimizes the number of critical simplices. It is NP-hard and has been studied extensively in heuristic, approximation, and parameterized complexity settings. Parameterized by treewidth $k$, OMM has long been known to be solvable on triangulations of $3$-manifolds in $2^{O(k^2)} n^{O(1)}$ time and in FPT time for triangulations of arbitrary manifolds, but the exact dependence on $k$ has remained an open question. We resolve this by giving a new $2^{O(k \log k)} n$-time algorithm for any finite regular CW complex, and show that no $2^{o(k \log k)} n^{O(1)}$-time algorithm exists unless the Exponential Time Hypothesis (ETH) fails.

We study one dimensional binary Probabilistic Cellular Automaton (PCA) that interpolate between Wolfram's classical rules 23, 77, 178 and 232. These rules are the only ones that satisfy two criteria: (i) in the case of a majority in the neighborhood states, the central site takes either the majority state or the opposite and (ii) if the neighborhood states are tied, the central site either changes its current state or keeps it. The PCA is defined by two Bernoulli random variables with parameters $p,r \in [0,1]$, and we analytically solve small size cases by using a Markov process formulation. We derive analytical expressions for the probability of asymptotically reaching each possible global configuration as a function of $p$ and $r$, for all initial states. We show that for $0 < p,r < 1$, the asymptotic probability distributions of achieving any of the states for the PCA are independent of the initial conditions. This contrasts with the behavior of the deterministic Wolfram's rules 23 ($p=0,r=0$), 77 ($p=1,r=0$), 178 ($p=0,r=1$) and 232 ($p=1,r=1$), for which additional asymptotic states can occur, in particular periodic configurations Finally, we discuss applying this kind of PCA to describe opinion dynamics involving hesitant agents.

Using the short-wavelength instability method, we investigate the linear instability of an exact solution describing upward-propagating mountain waves, derived in A. Constantin, \emph{J. Phys. A: Math. Theor.} (2023), under the assumption of a dry adiabatic flow. Within this approach, the stability problem reduces to analysing a system of ordinary differential equations along fluid trajectories. Our results show that the flow becomes unstable when the wave steepness exceeds the critical threshold of $\frac{1}{3}$. Given the representation of the solution in Lagrangian coordinates, the instability analysis will show the existence of an unstable layer beneath the tropopause, where instability may occur, finally leading to a chaotic 3-dimensional fluid motion.

We extend the CDPR's quantum attack from ideal lattices to module lattices over $2^k$-th cyclotomic rings. Using trace orthogonality of the power basis, we decompose a rank-$d$ module into mutually orthogonal rank-$1$ submodules, and apply CDPR's analysis to each independently and return the shortest candidate. The Hermite factor $\exp(\tilde{O}(\sqrt{n}))$ matches the ideal case, with a module reduction factor $\alpha_d=O(1)$ independent of the rank, under a balance hypothesis (proved for Gaussian distribution) automatic for MLWE-distributed bases. To enable a bounded-precision implementation, we replace coordinate-wise rounding with Chinese Remainder Theorem-scaled rounding at totally split primes, reducing the Gram-Schmidt rounding radius from $n/2$ to $\le 1$ at cost $O(d^2 r n \log n)$. Finally, we reformulate the CDPR's sign-selection step as a mixed-integer linear program and prove its optimum is no more than 1/2 for all $k$ ($\approx 0.4407$ for all tested $k\le 12$, conjecturally universal). This replaces the previous heuristic discrepancy $\Theta(\sqrt{nk})$. All results build on the class number condition $h_k^+=1$ established in Part I of this series.

We study the classification of the gapped phases or massive renormalization group (RG) flows dual to the massless RG flows under changing the sign of the coupling constants. Whereas our formalism is based on combining Higgs- or Nambu-Goldstone-type arguments with Cardy's smeared boundary conformal field theories (SBCFTs), several puzzling structures arise. More specifically, the established Higgs or Nambu-Goldstone type arguments on the duality imply that the natural basis for the gapped states should be constructed from a set of smeared Ishibashi states, which are unphysical in boundary critical phenomena. Hence, the module of the gapped phases can be outside of that of boundary critical phenomena, whereas one can still calculate characterizing quantities by applying SBCFTs to the models. For example, we demonstrate that the massive RG flow dual to the massless RG flow from the tricritical Ising model to the Ising model, one of the simplest massless RG flows, has this unusual structure. This can be regarded as a quantum field-theoretic analogue of order-disorder coexistence in lattice models. More generally, the resultant gapped phases usually spontaneously break non-group-like symmetry (or noninvertible symmetry). Our work provides systematic quantum field theoretic descriptions of such unusual phases with spontaneous symmetry breaking of non-group-like (or noninvertible) symmetries.

We develop a rigorous evaluation of Dirac brackets for classical observables on the phase space of radiative gravitational modes at null infinity that naturally incorporates memory effects. Considering the Ashtekar-Streubel phase space, with boundary conditions in time given by vanishing {\it news} and purely electric {\it shear}, and taking into account the infinite dimensionality of the phase space, we identify the algebra of proper observables (understood as functions on phase space that can be associated with smooth symplectic flows). We show that the action of supertranslation charges generate the correct transformations on the shear. We also show that the conventional definition of the ``Goldstone mode'' adopted in the literature cannot be associated with a proper observable, but nevertheless there exists an infinite family of proper observables, which we call {\it Goldstone probes}, that are capable of measuring the Goldstone mode. We notice that there are no Goldstone probes constructed only out of the shear {\it or} the news, providing a possible explanation for why attempts to construct a (separable) Hilbert space with different memory states have failed so far. Finally, we derive formulas for distributional Dirac brackets between local shear and news, and show that they contain non-local corrections.

Some time ago, we published the full count of degrees of freedom in the linearised weak gravity limit of arbitrary New GR models. We did it by considering the linear weak gravity equations and presented a thorough analysis with no ambiguity left. A bit later, we generalised it to linear cosmological perturbations and discussed the strong coupling issues that appear already at this level. Recently, there were claims that some dynamical modes had been missed in our work. However, the authors of the new claims did not look at the equations of motion and analysed the quadratic Lagrangian densities instead. In this paper, I take one of the most elementary cases, namely the vector modes in New GR of Type 3, and show what was their mistake that had led them to claiming that those were dynamical. The main message: Do not substitute constraint equations into a Lagrangian.

We study the contraction in Wasserstein distance of the coordinate ascent variational inference algorithm. This is shown to hold under a transport-information inequality at the fixed points and a functional smoothness condition. The results are general and sharp, allow for local convergence guarantees, hold for general smooth manifolds, and also in some non-smooth spaces. We consider applications to Bayesian Gaussian Mixture Models, and high-dimensional Bayesian Probit Regression, and Logistic Regression with Pólya-Gamma random variables (i.e. Jaakkola-Jordan's algorithm).