Peak-to-average power ratio (PAPR) remains a major limitation of multicarrier modulation schemes such as orthogonal frequency-division multiplexing (OFDM), reducing power amplifier efficiency and limiting practical transmit power. In this work, we propose DeepOFW, a deep learning-driven OFDM-flexible waveform modulation framework that enables data-driven waveform design while preserving the low-complexity hardware structure of conventional transceivers. The proposed architecture is fully differentiable, allowing end-to-end optimization of waveform generation and receiver processing under practical physical constraints. Unlike neural transceiver approaches that require deep learning inference at both ends of the link, DeepOFW confines the learning stage to an offline or centralized unit, enabling deployment on standard transmitter and receiver hardware without additional computational overhead. The framework jointly optimizes waveform representations and detection parameters while explicitly incorporating PAPR constraints during training. Extensive simulations over 3GPP multipath channels demonstrate that the learned waveforms significantly reduce PAPR compared with classical OFDM while simultaneously improving bit error rate (BER) performance relative to state-of-the-art transmission schemes. These results highlight the potential of data-driven waveform design to enhance multicarrier communication systems while maintaining hardware-efficient implementations. An open-source implementation of the proposed framework is released to facilitate reproducible research and practical adoption.

The conformal range (or the real Davis--Wielandt shell), which is a particular planar projection of the Davis--Wielandt shell, can be considered as the hyperbolic version of the numerical range; i. e. it is a ``field of values'' which can be interpreted as a subset of the asymptotically closed hyperbolic plane. Here we explain the analogue of the elliptical range theorem of $2\times2$ complex matrices for the conformal range.

This text highlights issues present in the proof of Lemma 6.10 of the Baumgartner (1943 -- 2011) article "Almost disjoint sets, the dense set problem and the partition calculus" of 1976, and intends to present a correction at the same time it proves a stronger result mentioned in the article to have similar proof.

The sample compression conjecture is: Each concept class of VC dimension d has a compression scheme of size this http URL this paper, for any concept class of finite functions, we present a labeled sample compression scheme of size equals to its VC dimension d. That is, the long standing open sample compression conjecture is resolved.

Hochman asked whether there exists a cellular automaton $F$ such that every cellular automaton is a factor of $F$ in the dynamical sense. In particular, we do not require the factor map to commute with the spatial shifts. We show that no such cellular automaton exists. More generally, if $F$ weakly factors onto the radius-zero $q$-clock automaton $C_q^{(k)}$, then every periodic point of $F$ has period divisible by $q$. For a cellular automaton $F:A^{\mathbb Z^d}\to A^{\mathbb Z^d}$, define $\varphi_F:A\to A$ by $F(\underline a)=\underline{\varphi_F(a)}$, and let $g_F$ be the greatest common divisor of the cycle lengths of $\varphi_F$. We prove that if $C_q^{(k)}$ is a weak factor of $F$, then $q\mid g_F$ holds. It follows that the action of $F$ on constant configurations yields an explicit divisibility obstruction to clock weak factors.

Matrix valued (asymmetric) truncated Toeplitz operators are generally not complex symmetric. In this paper, we define a new conjugation with unique properties and study its relation to matrix valued asymmetric truncated Toeplitz operators. We also explore the connections between matrix valued asymmetric truncated Toeplitz operators and Hankel operators with matrix symbols.

We recently introduced a reverse reconciliation scheme with soft information. In this paper, we assess its performance at ultra-low SNR, thus proving that such scheme is a versatile solution to the reverse reconciliation problem.

Let $ \chi $ be a character of a complex irreducible representation of a finite group $G$. We present a simple formula for the expectation of the random variable $(|\chi|/\chi(1))^{t} $ in terms of character ratios $ (|\chi(g)|/\chi(1))^{t}, \; g \in G, \; t \geq 0 $. As a follow up we briefly discuss asymptotic properties of the formula and its relation to the growth of dimensions of isotypic components in (virtual) tensor powers of irreducible representations

We obtain Hardy inequality for non-local diffusion operator with singular drift, in the case when the strength of attraction to the origin by the drift takes the critical value.

In this short expository note, we give a detailed proof of Markov's theorem on the unsolvability of the homeomorphism problem and of the existence of unrecognizable manifolds in all dimensions larger than 3.

Bilipschitz invariant theory concerns low-distortion embeddings of orbit spaces into Euclidean space. To date, embeddings with the smallest-possible distortion are known for only a few cases, to include: (a) planar rotations, (b) real phase retrieval, and (c) finite reflection groups. Here, we prove that for all three of these cases, the smallest possible distortion is nearly achieved by a composition of a "max filter bank" with a linear transformation. Our proof amounts to a two-step process: first, we show it suffices to demonstrate a certain inclusion of Lipschitz function spaces, and second, we prove that inclusion, using fundamentally different approaches for the three cases. We also show that these cases interact differently with a few related function spaces, which suggests that a unified treatment would be nontrivial.

We study truncated bilinear forms associated with synchronized kernels \[ K(x,y)=k(\phi(x),\psi(y)), \] where the singularity is governed by a one-dimensional kernel $k$, while the geometry is encoded by the phases $\phi$ and $\psi$. The central result of the paper is a framework of exact reduction, analytic transfer, and geometric recomposition for this class of forms. First, we obtain an exact reduction at the level of pushforward measures and data-weighted pushforward measures in the level variable. Under absolute continuity hypotheses, this reduction admits a realization on the Lebesgue layer, where control of the pushforward densities yields an abstract operatorial criterion for reinjecting into the original problem estimates obtained for the reduced model. As a first complete realization of this scheme, we transfer to the synchronized setting a one-dimensional sparse domination for singular truncations with Dini-smooth kernels. The final geometric recomposition then separates two regimes: a uniform regime, in which global consequences are obtained under quantitative control of the pushforward densities, and a critical regime, in which the degeneration of the phases near the critical values forces a localized and pullback-weighted output.

Quantum graphs have been introduced by Duan, Severini, and Winter to describe the zero-error behaviour of quantum channels. Since then, quantum graph theory has become a field of study in its own right. A substantial source of difficulty in working with quantum graphs compared to classical graphs stems from the fact that they are no longer discrete objects. This makes it generally difficult to construct insightful, non-trivial examples. We present a collection of non-trivial quantum graphs that can be thought of in discrete terms, and that can be expressed in the diagrammatic formalism introduced by Musto, Reutter, and Verdon. The examples arise as the quantum graphs acted on by increasingly smaller classical matrix groups, and are parametrised by triples of matrices $(A, B, C)$. The parametrisation reveals a clean decomposition of quantum graph structure into classical and genuinely quantum components: $A$ and $C$ are described by a classical weighted graph called the strange graph, while $B$ provides a purely quantum contribution with no classical analogue. Based on this model, we give exact formulas or establish bounds for quantum graph parameters, such as the number of connected components, the chromatic number, the independence number, and the clique number. Our results provide the first large, parametric families of quantum graphs for which standard graph parameters can be computed analytically.

A variable line through the centroid G of a triangle divides the triangle into two parts each of whose lengths as a fraction of the perimeter fills a closed interval [m,1-m], with m between 0 and 1/2. We show that the range of m taken over all triangles is the interval (3/10,4/9], with 3/10 approached by scales of the triangles approaching the 5-4-1 triangle and their mid-size medians, and 4/9 attained by the equilateral triangles and the lines through G parallel to the sides. This result is the perimeter version of the classical Winternitz theorem for a triangle, asserting that, in the case of area-ratio instead of perimeter-ratio, m=4/9, and this is attained by all triangles and their lines through G and parallel to the sides.

In this paper, we study semiparametric inference for linear multivariate Hawkes processes, a class of point processes widely used to describe self and mutually exciting phenomena. We establish a convolution theorem giving the best limiting distribution for a regular estimator of smooth functional. Then, in the Bayesian setting, we prove a semiparametric Bernstein-von Mises (BvM) theorem for nonparametric random series priors. We apply this result to histogram and wavelet based priors. Taken together, the convolution and BvM theorems show that, from a frequentist point of view, semiparametric Bayesian procedures have asymptotically the optimal behavior. Deriving the BvM property for random series priors led us to prove L2 posterior contraction, complementing for these priors the results of Donnet, Rivoirard and Rousseau (2020).

We establish that the dispersion relations of any physical system composed of two coupled subsystems, governed by a space-time homogeneous Lagrangian, admit a factorized form G_{1}G_{2}=\gamma G_{\mathrm{c}}, where G_{1} and G_{2} are the subsystem dispersion functions, G_{\mathrm{c}} is the coupling function, and \gamma is the coupling parameter. The result follows from a determinant expansion theorem applied to the block structure of the coupled system matrix, and is illustrated through three examples: the traveling wave tube, vibrations of an airplane wing, and the Mindlin-Reissner plate theory. For the Mindlin-Reissner example we carry out a complete asymptotic analysis of the coupled dispersion branches, establishing that the factorized form provides a precise quantitative measure of mode hybridization: all four branches carry the imprint of both subsystem factors for any nonzero coupling, while asymptotically recovering the identity of pure uncoupled modes at large frequencies and wavenumbers. We further analyze the universal local geometry of the coupled dispersion branches near their intersection - the cross-point model - showing it is generically hyperbolic, and present a mechanical analog in which the wavenumber is replaced by a scalar parameter, exhibiting the same factorized structure and avoided crossin

We study compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces with variable exponents on both bounded and unbounded metric measure spaces. We establish sufficient conditions for compactness, and under additional assumptions, we show that they are also necessary. Moreover, we investigate the influence of isometry group actions on the compactness of embeddings. In particular, we answer the open question posed by P. Górka in [P. Górka, Looking For Compactness In Sobolev Spaces On Noncompact etric Spaces, Ann. Acad. Sci. Fenn., Vol 43, 2018, 531-540], proving a Berestycki-Lions type theorem.

The quadratic embedding property of graphs consisting of three paths (theta graphs) is fully characterised. For this aim, a theorem by Winkler (1985) is utilized. An alternative proof of that result using the RKHS technique is presented.

We extend our recently introduced stochastic nonlocal traffic flow model to more general random perturbations, including Markovian noise derived from a discretized Jacobi-type stochastic differential equation. Invoking a deterministic stability estimate, we show that the arising random weak entropy solutions are measurable, ensuring that quantities such as the expectation are well-defined. We show that the proposed Jacobi-type noise is of particular interest as it ensures interpretability, preserves boundedness, and significantly alters the stochastic realizations compared to the previous white noise approach. Moreover, we introduce a local solution operator which provides information on the local effect of the noise and utilize it to derive a mean-value hyperbolic nonlocal PDE, which serves as a proxy for the mean value of the exact solution. The quality of this proxy and the impact of the noise process are analyzed in several simulation studies.

In this work, we extend the concepts of $p$-biharmonic maps and $p$-biharmonic hypersurfaces to provide a broader characterization of $(p,q)$-harmonic hypersurfaces and $(p,q)$-harmonic curves in Riemannian manifolds, including Einstein spaces. Moreover, we present new explicit examples of proper $(p,q)$-harmonic hypersurfaces and $(p,q)$-harmonic curves in space forms.

We show that the classifying space of the flow category of a \emph{tame} Morse function on a smooth, closed manifold $M$ recovers the homotopy type of $M$, thereby addressing a claim in a preprint of Cohen--Jones--Segal. The tameness assumption is that the compactified moduli spaces of broken gradient trajectories are locally contractible, ensuring the flow category is topologically well-behaved. We construct a Morse function and Riemannian metric on $S^2\times S^1$ for which the associated flow category fails to recover the correct homotopy type, showing that the tameness hypothesis is crucial. Together, these results clarify the extent to which transversality assumptions can be relaxed so that the flow category models the homotopy type of the underlying manifold.

We generalize the logarithmic purity theorem of Fujiwara-Kato to torsors which arise in the Kummer log flat topology under finite flat linearly reductive group schemes. As an application, we construct the logarithmic Nori fundamental group of a log regular log scheme classifying those torsors, and compare it to classical Nori fundamental group and tame fundamental group.

We explore the relation between various versions of Ramsey theorem and bounding schemes in model ${N}$ of a fragment of arithmetic $F$. Our goal is to recast, in a different framework, and extend some results of Hirst \cite{Hirst-1987}, see Theorem 1. We will extract Weihrauch reductions from Hirst's and similar proofs. Our results, informally stated in the our terminology, all inside ${N}$, follow: First the following are equivalent: $B\Sigma_2$, the finite union of finite c.e.\ sets is finite, and Infinite Pigeonhole Principle, see Theorem 3. We also discuss the Weihrauch relations between these logically equivalent principles, see Section 4. The Infinite Pigeonhole Principle is Weihrauch reducible to $RT^2_2$, see Theorem 4. There are also another principle logically equivalent to $B\Sigma_2$ which is Weihrauch reducible to $SRT^2_2$, see Theorem 5. We show that there is a principle which is equivalent with $B\Sigma_3$, see Theorem 6, and Weihrauch reducible to $SRT^2_{<\infty}$, Theorem 7. We discuss some equivalencies with $B\Sigma_{n-1}$, see Subsection 6.1, and then end with a problem Weihrauch reducible to $RT^{n+1}_{2}$, Subsection 6.2.

In this paper, we define and study weak expansive and expansive measures for pseudogroups, these two notions appear when analyzing the role of the generating set. We investigate the relations between such properties. We also provide a criterion for a measure to be weak expansive through the positivity of its entropy, generalizing the work of Arbieto and Morales. We also show that in some settings equicontinuous pseudogroups have no expansive measures.

We provide quantitative convergence results for continuous-time dynamical systems in metric spaces that satisfy a continuous-time analog of quasi-Fejér monotonicity. More precisely, we provide a (strong) convergence result for such dynamical systems over compact metric spaces which is quantitatively outfitted with a continuous-time rate of metastability, which moreover can be explicitly and effectively constructed in a very uniform way, only depending on a few moduli representing quantitative witnesses to key properties of the dynamical system and a measure for the compactness of the space. We further show how this convergence result can be extended to non-compact spaces under a regularity assumption of the associated problem, where moreover rates of convergence can then be explicitly constructed which are similarly uniform. In both cases, already the associated ``infinitary'' convergence result is qualitatively novel in its present generality. Beyond this abstract quantitative theory for such dynamical systems, we motivate how the presently studied continuous-time variant of quasi-Fejér monotonicity naturally occurs as a unifying property of many dynamical systems and differential equations and inclusions, and in that way can be used to provide a comprehensive quantitative theory for many such dynamical systems. We illustrate this with three case studies for both classical first- and second-order dynamical systems in Hilbert spaces as well as (generalized) gradient flows and associated semigroups in nonlinear Hadamard spaces.

In this paper, we study the isotropy groups of locally finite derivations of the polynomial ring $\mathbb{K}[X,Y]$, using Van den Essen's classification of locally finite derivations in two variables. We compare the isotropy group of a locally finite derivation with that of its associated exponential automorphism, showing that they coincide in the locally nilpotent case, whereas they may differ when the semisimple part is nontrivial. We also prove that every nonzero locally finite derivation has a nontrivial isotropy group.

In mathematical physics the Galois top, introduced by S. Adlaj, possesses a fixed point on one of two Galois axes through its center of mass. This heavy top has two algebraic motion invariants and an additional transcendental motion-invariant. This third invariant depends on an antiderivative of a variable in the canonical phase space. In this article an abelian semigroup and an abelian group are defined that are related to the application of the Huygens-Steiner theorem to points on the Galois axis of a rigid body.

In this paper, we standardize a homotopy $4$-sphere constructed by Dunfield and Gong. As a corollary, we show that the $18$-crossing knot $18_{\text{nh}00000601}$, which is not known to be ribbon, is slice in the standard $4$-ball. Thus, $18_{\text{nh}00000601}$ serves as a potential counterexample to the Slice-Ribbon Conjecture. In addition, we show that the same knot bounds a fibered handle-ribbon disk in $B^{4}$.

This article provides a synthesis of recent advances in the study of the PI property in various classes of noncommutative algebras of polynomial type.

This paper constructs polynomial bases that capture the structure of the de Rham complex with boundary conditions in disks and cylinders (both periodic and finite) in a way that respects rotational symmetry. The starting point is explicit constructions of vector and matrix orthogonal polynomials on the unit disk that are analogous to the (scalar) generalised Zernike polynomials. We use these to build new orthogonal polynomials with respect to a matrix weight that forces vector polynomials to be normal on the boundary of the disk. The resulting weighted vector orthogonal polynomials have a simple connection to the gradient of weighted generalised Zernike polynomials, and their curl (i.e. vorticity or rot) is a constant multiple of the standard Zernike polynomials which are orthogonal with respect to $L^2$ on the disk. This construction naturally leads to bases in cylinders with simple recurrences relating their gradient, curl and divergence. These bases decouple the de Rham complex into small exact sub-complexes.

We determine the optimal universal growth scale for the integral means of normalized logarithmic derivatives in the Carathéodory class. This resolves a problem of D.~K.~Thomas.

We formulate and solve a discrete-time linear-quadratic regulation (LQR) problem in a finite horizon that penalizes temporal variability and stochastic variability of the state trajectory. Our approach enables the user to strike a balance between regulating the state and reducing temporal variability, with explicit sensitivity to risk. We achieve this by extending a risk measure called predictive variance to a setting with temporally coupled states. Numerical examples demonstrate the effect of temporal coupling in both risk-aware and risk-neutral control settings. Particularly, we observe that explicitly penalizing temporal variability alone can also reduce stochastic variability.

We restate a process presented by Stanley as a technique to prove that there exists exactly one $d$-differential distributive lattice for any positive integer $d$. This process can be trivially extended to apply to distributive finitary lattices that have a variety of differential poset structures. It can be viewed as an algorithm for constructing such lattices. Alternatively, it can be viewed as an algorithm for analyzing and characterizing such lattices. We show that the process can be used to prove properties of all weighted-differential lattices with positive weights. We present this with the hope that this approach can be used as the basis for a complete characterization of distributive lattices with a weighted-differential structure with positive weights.

We characterize the entropy and minimax risk of a broad class of compact pseudodifferential operators. Under suitable decay and regularity conditions on the symbol, we combine a Weyl-type asymptotic relation between the eigenvalue-counting function and the phase-space volume of the symbol with a general correspondence between spectral quantities, entropy, and minimax risk for compact operators. This approach yields explicit asymptotic formulae for both entropy and minimax risk directly in terms of the symbol. As an application, we derive sharp entropy and minimax risk asymptotics for unit balls in Sobolev spaces on unbounded domains, thereby extending Pinsker's theorem for Sobolev classes beyond the bounded-domain setting, and showing that the sharp asymptotic constants are determined by phase-space geometry rather than domain geometry.

We show that if a proper, geodesically complete, CAT(0) homology manifold is quasi-isometric to the Euclidean space R^n then it is homeomorphic to R^n. On the other hand, we show that there exist proper, geodesically complete, CAT(0) spaces quasi-isometric to R^n, which are not homeomorphic to it. We prove that our example is sharp in a suitable sense. Finally, we provide an example of a sequence of proper, geodesically complete, CAT(0) spaces that are not homology manifolds and that converge in the Gromov-Hausdorff sense to a topological manifold: this shows that the set of topological manifolds is not open in the class of proper, geodesically complete, CAT(0) spaces.

Under sharp conditions, we prove the existence and refined asymptotic behaviour near zero (resp., at infinity) for all positive radial solutions to elliptic equations such as \begin{equation}\label{eq11} \tag{*} \mathbb L_{\rho,\lambda}(u)=\Delta u+ (2-N-2\rho)\, \frac{x\cdot \nabla u}{|x|^2}+ \frac{\lambda}{|x|^2}u=|x|^{\theta}\,u^q\, |\nabla u|^m\quad \mbox{in } \Omega\setminus\{0\}, \end{equation} where $\Omega=B_R(0)$ (resp., $\Omega=\mathbb R^{N}\setminus B_{1/R}(0)$) for $R>0$ and $N\geq 2$. The dynamics of such solutions is very rich since $\rho, \lambda,\theta\in \mathbb R$ are arbitrary, $ m>0$, $q\geq 0$ and $\kappa:=m+q-1>0$. To our knowledge, this is the first study of the local properties of the positive solutions of \eqref{eq11} with arbitrary $m>0$ and $\lambda\not=0$. We identify all profiles near zero (and at infinity via a modified Kelvin transform) under optimal conditions, depending on how $\Theta:=(\theta+2-m)/\kappa$ relates to $0$ or the roots $\Theta_\pm$ of $t^2+2\rho t+\lambda$ when $\lambda\leq \rho^2$. For each profile, we advance new methods that unearth the higher order terms in the asymptotic expansion. We highlight two new asymptotic profiles near zero due to the competition between the Hardy potential with $\lambda>0$ and the gradient-dependent absorption: (i) a blow-up profile $\left[ \lambda \left( \frac{\kappa}{m} \right)^m \right]^{\frac{1}{\kappa}} | \log |x||^{\frac{m}{\kappa}} $ if $\Theta=0$ and (ii) a bounded profile if $\Theta<0$. Any radial solution of \eqref{eq11} with $\lim_{r\to 0^+} u(r)=\gamma\in \mathbb R_+$ satisfies $(P_\pm)$ $u(r)=\gamma\pm \lambda^{1/m} \gamma^{1-\kappa/m} (1/\sigma)\, r^\sigma(1+o(1))$ as $r\to 0^+$, where $\sigma=-\kappa \Theta/m$. For any $\gamma\in \mathbb R_+$, there is $R>0$ such that \eqref{eq11} has a radial solution (infinitely many) satisfying $(P_-)$ ($(P_+)$).

We develop a theory of umkehr maps for twisted generalized homology theories. In this theory, interesting umkehr maps, including generalizations of important classical ones, are induced by cartesian morphisms of a certain category opfibred over the category of spaces and continuous maps, making it possible to access them through universal properties.

We prove that all ergodic automorphisms of the $N$-dimensional torus with two dimensional center are stably ergodic. This includes all ergodic automorphisms in dimension $N\leq 5$ or $N=7$. This generalizes a previous result of Rodriguez-Hertz, that required an additional algebraic condition on the carachteristic polynomial of the linear automorphism. The core of the proof is a minimality criterion.

Prediction of wireless channels and their statistics is a fundamental procedure for ensuring performance guarantees in wireless systems. Statistical radio maps powered by Gaussian processes (GPs) offer flexible, non-parametric frameworks, but their performance depends critically on the choice of mean and covariance functions. These are typically learned from dense measurements without exploiting environmental geometry. Digital twins (DTs) of wireless environments leverage computational power to incorporate geometric information; however, they require costly calibration to accurately capture material and propagation characteristics. This work introduces a hybrid channel prediction framework that leverages uncalibrated DTs derived from open-source maps to extract geometry-induced prior information for GP prediction. These structural priors are fused with a small number of channel measurements, enabling data-efficient prediction of channel statistics across the entire environment. By exploiting the uncertainty quantification inherent to GPs, the framework supports principled measurement selection by identifying informative probing locations under resource constraints. Through this integration of imperfect DTs with statistical learning, the proposed method reduces measurement overhead, improves prediction accuracy, and establishes a practical approach for resource-efficient wireless channel prediction.

Many integer sequences arise as numbers of $G$-orbits on $\binom{X}{n}$ as $n$ varies, for a permutation group $G\subseteq \operatorname{Sym}(X)$. For finite $X$, Stanley proved that these finite sequences increase towards the middle using an action of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. For infinite sets $X$, and hence infinite sequences, Cameron provided an argument for monotonicity. He first identifies orbits with a vector space basis of a certain commutative $k$-algebra $\mathsf{H}_{G,X}^{\star}$, called the orbit algebra. He then considers the operator, which forms the product with the constant $1$-function on $X$, and proves its injectivity. In this paper we generalize Stanley's approach to oligomorphic groups, and in particular extend Cameron's operator to a full $\mathfrak{sl}_2(\mathbb{C})$-action on $\mathsf{H}_{G,X}^{\star}$. We define for every oligomorphic permutation group $G\subseteq \operatorname{Sym}(X)$ the $X$-th tensor power $(k^r)^{\otimes X}$, generalizing work of Entova-Aizenbud. We show that this space carries natural commuting actions of $G$ and the Lie algebra $\mathfrak{gl}_r(k)$, the latter depending on a Harman--Snowden measure $\mu$ on $G$. We then show that $\mathsf{H}_{G,X}^{\star}\subseteq (\mathbb{C}^2)^{\otimes X}$ can be decomposed into a direct sum of $\mathfrak{sl}_2(\mathbb{C})$-Verma modules, which gives monotonicity. We explain how our approach applies to Fibonacci numbers, Tribonacci numbers, etc. by constructing measures on products with $(\mathbb{Q},<)$.

A graph $G$ is $D$-distinguishable if there is a labeling of its vertices with $D$ labels such that the only automorphism of $G$ which preserves the labeling is the identity. The distinguishing number of $G$ is the minimum value $D$ for which $G$ is $D$-distinguishable. The fixing number of $G$ is the minimum cardinality of a subset of the vertices of $G$ which is fixed pointwise only by the trivial automorphism. We prove that the fixing number of any $2$-distinguishable tree of order $n \geq 3$ is at most $4n/11$, or at most $(D-1)n / (D+1)$ for a $D$-distinguishable tree ($D \geq 3$). For every $D$ and $r$ at least $2$, we characterize the $D$-distinguishable trees with radius $r$ by constructing a universal tree $T_r^D$ which has the property that a tree $T$ of radius $r$ is $D$-distinguishable if and only if $T$ is a union of branches of $T_r^D$. We obtain a similar collection of universal trees for the property of having a constant paint cost spectrum, i.e., the minimum size of the complement of a color class in a distinguishing $D$-coloring of $T$ is equal to the fixing number. Finally, we prove bounds on the distinguishing and fixing numbers of a tree in terms of the eccentricities of its vertices.

We study characteristic classes for deformations of foliations. Those classes include known classes such as the Godbillon--Vey class and the Fuks--Lodder--Kotschick class. We introduce a certain differential graded algebra (DGA for short) which recovers the Bott vanishing and some formulae by Heitsch. Some basic properties and structures of the cohomology of those DGA's are discussed. In particular, it is shown that at the level of the cohomology of DGA, there are some classes which cannot be described by the Godbillon--Vey class and the Fuks--Lodder--Kotschick class. It is also shown that if a certain type of characteristic classes admit non-trivial deformations in examples, then they yield another kind of classes which admit also non-trivial deformations.

We study the eigenvalues of the localization operator $S_{A, B} = P_A\mathcal{F}^{-1}P_B\mathcal{F} P_A$, where $\mathcal{F}$ is the Fourier transform and $A = cA_0, B = B_0$ for some fixed sets $A_0, B_0\subset \mathbb{R}^d$ and a large parameter $c > 0$. For the counting function of the eigenvalues $|\{n: \varepsilon < \lambda_n(A,B)\le 1-\varepsilon\}|$ we obtain a sharp uniform upper bound if one of the sets is a finite disjoint union of parallelepipeds and a bound which is only a single logarithm off the conjectural optimal bound in the general case. These bounds are applied to the estimation of traces ${\rm{Tr}}\, f(S_{A,B})$ for functions $f$ with a very low regularity, in particular establishing an enhanced area law in the former case.

We study smooth solutions to the three-dimensional stationary Navier--Stokes equations and establish new Liouville-type theorems under refined decay assumptions. Building on the work of Cho et al., we introduce a refinement to previously known integrability criteria and analyze the associated averaged quantities. Our main result shows that if the $L^p$ growth rate of a solution remains bounded for some $3/2 < p < 3$, then the solution must be trivial. The proof combines averaged decay estimates, energy inequalities, and an iteration scheme.

This paper is concerned with the speeds of propagation for the monostable Lotka-Volterra competition-diffusion system in general unbounded domains of $\mathbb{R}^N$. We first establish various definitions of spreading speeds at large time in the situation where one species is an invader and the other is a resident. Then, we study fundamental properties of these new definitions, including their relationships and their dependence on the geometry of the domain and the initial values. Under the conditions that both species possess the same diffusion ability and that the interactions between them are sufficiently weak, we derive an upper bound for the spreading speeds in a large class of domains. Furthermore, we obtain general upper and lower bounds for spreading speeds in exterior domains, as well as a general lower bound in domains containing large half-cylinders. Finally, we construct some particular domains for which the spreading speeds can be zero or infinite.

Terahertz (THz) wireless communication has emerged as a promising solution for future data center interconnects; however, accurate channel characterization and system-level performance evaluation in complex indoor environments remain challenging. In this work, a measurement-calibrated AI-assisted digital twin (DT) framework is developed for THz wireless data centers by tightly integrating channel measurements, ray-tracing (RT), and implicit neural field (INF) modeling. Specifically, channel measurements are first conducted using a vector network analyzer at 300 GHz under both line-of-sight (LoS) and non-line-of-sight (NLoS) scenarios. RT simulations performed on the Sionna platform capture the dominant multipath structures and show good consistency with measured results. Building upon measurement and RT data, an RT-conditioned INF is developed to construct a continuous radio-frequency (RF) field representation, enabling accurate prediction in RT-missing NLoS regions. The comprehensive RF map generated by DT can provide system-level analysis and decisions for wireless data centers.

The tautological rings of strata of differentials are known to be generated by divisor classes. In this paper, we give lower bounds on the degrees of relations among them, depending on the genus $g$ and the number of simple zeros. For strata with more than $4g/3$ simple zeros, our results show that there are no relations in degrees less than $\lfloor g/3 \rfloor + 1$. Moreover, we conjecture that, outside of a few exceptions, there is always a non-trivial relation in degree $\lfloor g/3 \rfloor + 1$, and prove the conjecture for all strata of holomorphic abelian differentials with $g \leq 30$. We also prove that the cohomology rings of strata of holomorphic differentials with sufficiently many simple zeros stabilize to the free algebra on the tautological divisor class. Finally, we show that for a large class of holomorphic abelian strata, containing hyperelliptic differentials, the tautological ring is non-trivial for sufficiently large $g$.

High-order interpolation on the Grassmann manifold $\Gr(n, p)$ is often hindered by the computational overhead and derivative instability of SVD-based geometric mappings. To solve the challenges, we propose a stabilized framework that combines Maximum-Volume (MV) local coordinates with Arnoldi-orthogonalized polynomial bases. First, manifold data are mapped to a well-conditioned Euclidean domain via MV coordinates. The approach bypasses the costly matrix factorizations inherent to traditional Riemannian normal coordinates. Within the coordinate space, we use the Vandermonde-with-Arnoldi (V+A) method for Lagrange interpolation and its confluent extension (CV+A) for derivative-enriched Hermite interpolation. By constructing discrete orthogonal bases directly from the parameter nodes, the solution of ill-conditioned linear system is avoided. Theoretical bounds are established to verify the stability of the geometric mapping and the polynomial approximation. Extensive numerical experiments demonstrate that the proposed MV-(C)V+A framework can produce highly accurate approximation in high-degree polynomial interpolation.

Paratopism is a well known action of the wreath product $\mathcal{S}_n\wr\mathcal{S}_3$ on Latin squares of order $n$. A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let $\mathrm{Par}(n)$ denote the set of paratopisms that are an autoparatopism of at least one Latin square of order $n$. We prove a number of general properties of autoparatopisms. Applying these results, we determine $\mathrm{Par}(n)$ for $n\le17$. We also study the proportion of all paratopisms that are in $\mathrm{Par}(n)$ as $n\rightarrow\infty$.

We study the Choquard equation involving mixed local and nonlocal operators $$-\Delta u+(-\Delta)^{s}u+V(x)u=(\frac{1}{|x|^{\mu}}* F(u))f(u)\quad\text{in }\R^{2},$$ where $s\in(0,1)$, $\mu\in(0,2)$, $F(t)=\int_{0}^{t} f(\tau)\,d\tau$, and $f$ has subcritical exponential growth of Trudinger--Moser type. Under suitable assumptions on the potential $V$ and the nonlinearity $f$, we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions.

For a class of semilinear elliptic equations, we establish criteria that guarantee that the linearized operator associated with a solution satisfies certain spectral assumptions that are widely used in the analysis of the stability of solitary waves. The criteria only involve the symbol of the linear operator and positivity and symmetry of the solution, and can therefore be verified without an explicit formula for the solution.

In a remarkable paper, Tatsuyuki Hikita settled a longstanding e-positivity conjecture of Stanley and Stembridge. Among many other things, he wrote down a certain formula ${\varphi}_k$, and proved that the ${\varphi}_k$ sum to one, thereby defining a probability distribution. Though Hikita's proof was simple, it remains surprising that the ${\varphi}_k$ sum to one. In this note, we give a combinatorial interpretation of Hikita's probability distribution. The main tool is a certain permutation statistic that we call the watershed. After seeing an early version of our work, Darij Grinberg noticed that the permutation statistic was implicit in a so-called "bulldozer problem" that was on the short list for the 2015 International Mathematics Olympiad. However, our description of the statistic, which makes use of Foata's fundamental bijection, appears to be new.

Mixture-of-Experts (MoE) architectures leverage sparse activation to enhance the scalability of large language models (LLMs), making them suitable for deployment in resource-constrained edge networks. However, the sheer number of experts often exceeds the memory capacity of individual edge nodes, necessitating wireless distributed MoE (WIDE) inference where experts are spread across multiple edge nodes. In this context, expert selection directly affects communication costs. Motivated by the similarity of experts, we propose SiftMoE, which judiciously selects or skips experts to strike a tradeoff between communication costs and inference accuracy. Specifically, we first establish theoretical bounds on the accuracy degradation resulting from expert replacement or skipping. Based on the bounds, we formulate an energy minimization problem for expert selection in WIDE inference subject to latency and accuracy constraints. In particular, for slow-fading channels, we derive optimal expert selection policies for both single-token decoding and multi-token prefilling. For fast-fading channels, we further extend our scheme to cope with rapidly varying channel conditions. Simulation results demonstrate that SiftMoE significantly reduces energy consumption while maintaining inference accuracy compared with conventional Top-K routing in WIDE systems.

We study the semiclassical asymptotics of twisted algebras induced by symbol correspondences for quark systems ($SU(3)$-symmetric mechanical systems) as defined in our previous paper [3]. The linear span of harmonic functions on (co)adjoint orbits is identified with the space of polynomials on $\mathfrak{su}(3)$ restricted to these orbits, and we find two equivalent criteria for the asymptotic emergence of Poisson algebras from sequences of twisted algebras of harmonic functions on (co)adjoint orbits which are induced from sequences of symbol correspondences (the fuzzy orbits). Then, we proceed by ``gluing'' the fuzzy orbits along the unit sphere $\mathcal S^7\subset \mathfrak{su}(3)$, defining Magoo spheres, and studying their asymptotic limits. We end by highlighting the possible generalizations from $SU(3)$ to other compact symmetry groups, specially compact simply connected semisimple Lie groups, commenting on some peculiarities from our treatment for $SU(3)$ deserving further investigations.

A latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots,h_k$ such that $h_1+\dots+h_k = n$ is known as a realization. The existence of realizations is a partially solved problem with a few general results for an arbitrary number of subsquares, $k$. Requiring only that $h_1+\dots+h_k\leq n$ gives a variation of the problem that has few known results. In this paper we prove a general necessary condition for existence and completely determine existence when there are at most three subsquares or the subsquares are all of the same order. Importantly, we prove that if $h_1\geq h_2\geq\dots\geq h_k$ and $n\geq h_1+\sum_{i=1}^kh_i$ then such a latin square always exists.

Recent work of Mao, Wan and Zhang \cite{MWZ} has provided a complete list of strongly tempered hyperspherical varieties and they proposed some new period integrals. In this paper, I will present new period integrals of distinguished polarised strongly tempered hyperspherical varieties and discuss the L-functions these integrals represent, as examples of the Relative Langlands Duality.

We prove that the theory of the Farey graph is pseudofinite by constructing a sequence of finite structures that satisfy increasingly large subsets of its first-order axiomatization. This graph is an important object in the study of curve graphs, and its model-theoretic properties have been explored in the broader context of curve graphs of surfaces in arXiv:2008.10490 The theory of the Farey graph was recently axiomatized by Tent and Mohammadi in arXiv:2503.02121 We show that while no finite planar graph can satisfy these axioms for sufficiently large substructures, they can be satisfied by triangulations densely embedded on orientable surfaces of higher genus. By applying a result of Archdeacon, Hartsfield, and Little on the existence of triangulations with representativity and connectedness, we establish that every finite subset of the theory of the Farey graph has a finite model as desired.

We introduce a deep neural network-based numerical method for solving kinetic Fokker Planck equations, including both linear and nonlinear cases. Building upon the conservative dissipative structure of Vlasov-type equations, we formulate a class of generalized minimizing movement schemes as iterative constrained minimization problems: the conservative part determines the constraint set, while the dissipative part defines the objective functional. This leads to an analog of the classical Jordan-Kinderlehrer-Otto (JKO) scheme for Wasserstein gradient flows, and we refer to it as the kinetic JKO scheme. To compute each step of the kinetic JKO iteration, we introduce a particle-based approximation in which the velocity field is parameterized by deep neural networks. The resulting algorithm can be interpreted as a kinetic-oriented neural differential equation that enables the representation of high-dimensional kinetic dynamics while preserving the essential variational and structural properties of the underlying PDE. We validate the method with extensive numerical experiments and demonstrate that the proposed kinetic JKO-neural ODE framework is effective for high-dimensional numerical simulations.

We study the neutral massive Maxwell (Proca) equation on subextremal Reissner--Nordström exteriors. After spherical-harmonic decomposition, the odd sector is scalar, while the even sector remains a genuinely coupled $2\times2$ system. Our starting point is that this even system admits an exact asymptotic polarization splitting at spatial infinity. The three resulting channels carry effective angular momenta $\ell-1$, $\ell$, and $\ell+1$, and these are precisely the indices that govern the late-time thresholds. % For each fixed angular momentum we develop a threshold spectral theory for the cut-off resolvent. We prove meromorphic continuation across the massive branch cut, rule out upper-half-plane modes and threshold resonances, and obtain explicit small- and large-Coulomb expansions for the branch-cut jump. Inverting this jump yields polarization-resolved intermediate tails together with the universal very-late $t^{-5/6}$ branch-cut law. % At the full-field level, high-order angular regularity allows us to sum the modewise leading terms on compact radial sets and obtain a two-regime asymptotic expansion for the radiative branch-cut component of the Proca field, with explicit coefficient fields and quantitative remainders. We also analyze the quasibound resonance branches created by stable timelike trapping, prove residue and reconstruction bounds, and derive a fully self-contained dyadic packet estimate. As a result, the unsplit full Proca field obeys logarithmic compact-region decay, while the radiative branch-cut contribution retains explicit polynomial asymptotics and explicit leading coefficients.

We provide the first proof of local well-posedness for the two-dimensional gravity water wave equations with spatially quasi-periodic initial conditions. We represent the solution using holomorphic coordinates, which are equivalent to a conformal mapping formulation of the equations of motion. This allows us to compute the Dirichlet-Neumann operator via the Hilbert transform, which has a simple form in the spatially quasiperiodic setting. We use a Littlewood-Paley decomposition adapted to the quasiperiodic setting and establish multiplicative and commutator estimates in this framework. The key step of the proof is the derivation of quasilinear energy estimates for the linearized water wave equations with quasiperiodic initial data.

An $n\times n\times\dots\times n$ hypercube is made from $n^d$ unit hypercubes. Two unit hypercubes are neighbours if they share a $(d-1)$-dimensional face. In each step of a dismantling process, we remove a unit hypercube that has precisely $d$ neighbours. A move is balanced if the neighbours are in $d$ orthogonal directions. In the extremal case, there are $n^{d-1}$ independent unit hypercubes left at the end of the dismantling. We call this set of hypercubes a solution. If a solution is projected in $d$ orthogonal directions and we get the entire $[n]^{d-1}$ hypercube in each direction, then the solution is perfect. We show that it is possible to use a greedy algorithm to test whether a set of hypercubes forms a solution. Perfect solutions turn out to be precisely those which can be reached using only balanced moves. Every perfect solution corresponds naturally to a Latin hypercube. However, we show that almost all Latin hypercubes do not correspond to solutions. In three dimensions, we find at least $n$ perfect solutions for every $n$, and we use our greedy algorithm to count the perfect solutions for $n\le6$. We also construct an infinite family of imperfect solutions and show that the total size of its three orthogonal projections is asymptotic to the minimum possible value. Our results solve several conjectures posed in a proceedings paper by Barát, Korondi and Varga. If our dismantling process is reversed we get a build-up process very closely related to well-studied models of bootstrap percolation. We show that in an important special case our build-up reaches the same maximal position as bootstrap percolation.

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_{\alpha}$-matrix of $G$ is defined as \begin{equation*} A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G), \text{ for any } \alpha \in [0,1]. \end{equation*} The $A_{\alpha}$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_{\alpha}(G)$. A tricyclic graph of order $n$ is a simple connected graph with $n+2$ edges. In this paper, we characterize the unique graph having the largest $A_{\alpha}$-spectral radius for $\alpha \in [\frac{1}{2}, 1)$ among all tricyclic graphs of order $n$ with $k (\geq 1)$ pendant vertices. As an application, we derive a sufficient spectral condition (alternate to the edge condition) to guarantee the absence of the tricyclic structure in a graph with $k$ pendant vertices.

Let $G$ be a graph on $n$ vertices and $m$ edges. For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal matrix of $G$. If $\rho_1 \geq \rho_2 \ldots \geq \rho_n$ are the eigenvalues of $A_{\alpha}(G)$, the $A_{\alpha}$-energy of $G$ is defined as $E_{A_{\alpha}}(G) = \sum_{i=1}^{n} |\rho_i -\frac{2\alpha m}{n}|$. In this paper, we present novel upper and lower bounds for $E_{A_\alpha}(G)$ in terms of standard graph invariants, showing that each bound is sharp and identifying the specific graphs attaining them. For selected bounds, we provide brief comparative analysis with existing results, observing improved estimates. Furthermore, we establish new relations between $E_{A_\alpha}(G)$ and other well known graph energies, including adjacency, Laplacian, as well as the adjacency energy of the line graph.

This paper is concerned with the Riemann problem for the two-dimensional barotropic compressible Euler system with a general strictly increasing pressure law. By means of convex integration, the existence of infinitely many admissible weak solutions is established for certain Riemann initial data for which the corresponding one-dimensional self-similar solution consists solely of a contact discontinuity.

A rational triangle $T$ (one whose angles are rational multiples of $\pi$) unfolds to a translation surface $(X_T,\omega_T)$. The lattice triangle problem asks to classify those $T$ for which $(X_T,\omega_T)$ is a Veech (lattice) surface, which means that the $\operatorname{SL}_2(\mathbb R)$-orbit of $(X_T,\omega_T)$ is closed in its stratum (so its projection to moduli space is a Teichmüller curve). The most mysterious regime is the "hard obtuse window" (largest angle in $(\pi/2,2\pi/3]$), where it is conjectured that no lattice triangles exist. Using an arithmetic reformulation of the Mirzakhani-Wright rank obstruction, we prove a quantitative theorem that rules out all but a density 0 subset of the triangles in this window. The main engine in this paper was autoformalized by AxiomProver in Lean (using mathlib).

A longstanding and important problem in algebraic geometry is the characterization of algebraic function fields. In this paper, we focus on the characterization problem for cyclotomic function field $L(\Lambda_M)$, which is an important class of explicit function fields with applications in number theory and coding theory. Motivated by Arakelian and Quoos' classification of $L(\Lambda_M)$ with an irreducible quadratic modulus, we provide a complete characterization of the cyclotomic function field $L(\Lambda_M)$ with modulus $M = x^2$. More precisely, we prove that a function field $\mathcal{F}$ over $\mathbb{F}_q$ is $\mathbb{F}_q$-isomorphic to $L(\Lambda_{x^2})$ if and only if it satisfies the following three conditions: (i) $\mathcal{F}$ has a subgroup $G$ isomorphic to the direct product $(\mathbb{F}_q,+) \times \mathbb{F}_q^*$; (ii) its genus is $g(\mathcal{F}) = 1 + q(q-3)/2$; and (iii) the cardinality of $\mathbb{F}_q$-rational places is exactly $q+1$.

We study closed manifolds with almost nonnegative curvature operator and address a question of Herrmann--Sebastian--Tuschmann concerning the sign of their Euler characteristic. Our main result shows that if a closed $2n$-dimensional manifold admits an almost nonnegative curvature operator together with a uniform upper bound on the curvature operator, then its Euler characteristic is nonnegative. In addition, under an ANCO-type condition and assuming that the fundamental group is infinite, we prove vanishing results for the Euler characteristic, the signature, and, in the spin case, the $\widehat{A}$-genus, extending recent work of Chen--Ge--Han from almost nonnegative Ricci curvature to the curvature-operator setting.

The notion of degree begins in field theory as the dimension of a field extension. In algebraic geometry, this idea reappears as the degree of a finite morphism, defined using the induced extension of function fields. For proper morphisms that are not necessarily finite, Stein factorization isolates the finite part of the map and leads to the notion of Stein degree. This invariant is especially useful in birational geometry, where it interacts naturally with singularities of pairs and the study of log Calabi-Yau fibrations. In this article we give an expository introduction to these ideas, discuss motivating examples, and explain a boundedness problem for Stein degree arising in recent work of the author and collaborators.

Faber, Muller and Smith used complete sums of conic modules to construct non-commutative crepant resolutions (NCCR) of simplicial toric algebras. We link these conic modules to the Bondal-Thomsen collection of line bundles on smooth toric DM stacks. This viewpoint allows us to establish computational results relating to conic modules, reducing the complexity of the combinatorics involved significantly. We formulate necessary and sufficient conditions for an incomplete sum of conic modules to give an NC(C)R of a toric algebra. Furthermore, we prove that to check if a toric algebra $R$ admits an NCCR in the form of $\End_R(\mathbb{B})$ for an incomplete sum of conic modules, we may reduce to a case where the class group of the affine toric variety $\spec R$ does not have torsion and verify the statement there. Finally, we treat the case of almost simplicial Gorenstein cones, i.e. cones with $|\sigma(1)|=\dim \sigma+1$, classifying when such cones admit NCCRs via endomorphism algebras of conic modules.

We establish three reverse inequalities for strictly convex curves and surfaces. For smooth strictly convex curves in a smooth Minkowski plane we prove an anisotropic reverse isoperimetric inequality controlled by the signed Euclidean area of the Minkowski evolute. For smooth closed strictly convex surfaces in $\mathbb R^{3}$ we prove \[ \left(\int_{M}H\,d\mu\right)^{2}-16\pi |M| \le \frac{8\pi}{3}\int_{M}\frac{|A^{o}|^{2}}{\mathcal K}\,d\mu, \] and relate the right-hand side to the oriented volumes of the focal maps. For smooth simple closed strictly convex curves on \(\mathbb S^{2}\) we prove \[ L^{2}-A(4\pi-A)\le \left(\int_{\gamma}\sqrt{1+k_g^{2}}\,ds\right)^{2}-4\pi^{2}, \] and in fact derive an exact nonnegative remainder formula. Equality in the spherical case holds if and only if $\gamma$ is a geodesic circle.

Reconstruction in limited-angle digital breast tomosynthesis (DBT) suffers from slow convergence of low spatial-frequency components when using weighted data-fidelity terms within primal-dual optimization. We introduce a two-channel fidelity strategy that decomposes the sinogram residual into complementary low-pass and high-pass bands using square-root Hanning (Hann^{1/2}) filter families, each driven by an independent \ell_2-ball constraint and dual update in the PDHG (Chambolle-Pock) algorithm with He-Yuan predictor-corrector relaxation. By assigning a larger dual step size and slightly looser tolerance to the low-frequency channel, the method delivers stronger per-iteration correction to the near-DC band without violating global PDHG stability. Experiments on a 2D digital breast phantom across multiple resolutions demonstrate that the two-channel approach yields 19%--61% RMSE improvement over the single-channel baseline, with larger gains at coarser discretizations where problem conditioning is more favorable, supporting more balanced spectral convergence in clinically realistic limited-angle regimes.

Matérn random fields are one of the most widely used classes of models in spatial statistics. The fixed-domain identifiability of covariance parameters for stationary Matérn Gaussian random fields exhibits a dimension-dependent phase transition. For known smoothness $\nu$, Zhang \cite{Zhang2004} showed that when $d\le3$, two Matérn models with the same microergodic parameter $m=\sigma^2\alpha^{2\nu}$ induce equivalent Gaussian measures on bounded domains, while Anderes \cite{Anderes2010} proved that when $d>4$, the corresponding measures are mutually singular whenever the parameters differ. The critical case $d=4$ for stationary Matérn models has remained open. We resolve this case. Let $d=4$ and consider two stationary Matérn models on $\mathbb R^4$ with parameters $(\sigma_1,\alpha_1)$ and $(\sigma_2,\alpha_2)$ satisfying \[ \sigma_1^2\alpha_1^{2\nu}=\sigma_2^2\alpha_2^{2\nu}, \qquad \alpha_1\neq \alpha_2. \] We prove that the corresponding Gaussian measures on any bounded observation domain are mutually singular on every countable dense observation set, and on the associated path space of continuous functions. Our approach can be viewed as a spectral analogue of the higher-order increment method of Anderes \cite{Anderes2010}. Whereas Anderes isolates the second irregular covariance coefficient through renormalized quadratic variations in physical space, we detect the first nonvanishing high-frequency spectral mismatch via localized Fourier coefficients and use a normalized Whittle score to identify parameters. More broadly, the localized spectral probing framework used here for detecting subtle covariance differences in Gaussian random fields may be useful for studying identifiability and estimation in other spatial models.

Let $L$ be a non-negative self-adjoint operator, we consider some commutators generated by the BMO function $b$ and the area integral operator $S_H$ associated with the heat semigroup $\{e^{-tL}\}_{t>0}$ or the area integral operator $S_P$ associated with the Poisson semigroup $\{e^{-t\sqrt{L}}\}_{t>0}$. The strong-type estimates of these commutators on weighted $L^p$ spaces and weighted Morrey spaces are established. At the same time, we verified that these commutators are compact operators on weighted Morrey spaces.

We investigate linearity of amalgams of subgroups of algebraic groups along intersections with algebraic subgroups. In the process, we establish linearity of certain "doubles" of linear groups, and obtain new examples of finitely generated residually finite groups that fail to be linear.

We develop the theory of Nekovář's Selmer complexes. We prove that, under mild hypotheses, Nekovář's Selmer complexes are canonically quasi-isomorphic to ``Poitou-Tate complexes", which arise from Poitou-Tate global duality exact sequences. We give two applications. Firstly, we prove that the determinant of a Selmer complex is canonically isomorphic to the module of Stark systems and, by using this result, we construct a canonical ``Heegner point Stark system" which controls Selmer groups. Secondly, we prove that the derived $p$-adic height pairing of Bertolini-Darmon concides with that of Nekovář.

Let $S$ be a right group. Then there exist two congruences $\sim$ and $\equiv$ on $S$ such that $S$ is the product of its quotient semigroups $S/{\sim}$ and $S/{\equiv}$, where $S/{\sim}$ is a group and $S/{\equiv}$ is a right zero semigroup. If $E$ is the set of all idempotents of $S$ and we fix an element $e_0\in E$, then the pointed right group $(S,e_0)$ is the coproduct of its pointed subsemigroups $(Se_0,e_0)$ and $(E,e_0)$ in the category of pointed right groups. In general, there is a pretorsion theory in the category of right groups in which the torsion objects are right zero semigroups and the torsion-free objects are groups.

We study stationary scattering for Schrödinger operators in $\R^3$ with finitely many concentric $\delta$--shell interactions of constant real strengths. Starting from the self--adjoint realization and the boundary resolvent formula for this model, we show that, after partial--wave reduction, the same finite-dimensional boundary matrices that arise in the resolvent formula also determine the channel scattering coefficients. More precisely, for each angular momentum $\ell$, the channel coefficient $S_\ell(k)$ satisfies $S_\ell(k)=\det K_\ell(k^2-i0)/\det K_\ell(k^2+i0)$ for almost every $k>0$, where $K_\ell(z)=I_N+m_\ell(z)\Theta$ is the $\ell$--th reduced boundary matrix. Thus, in each channel, the positive--energy scattering problem is reduced to a finite-dimensional matrix problem, and the scattering phase is recovered from $\det K_\ell(k^2+i0)$. We then study the first nontrivial case of two concentric shells in the $s$--wave channel, where the interaction between the shells produces nontrivial threshold effects. We derive an explicit formula for $S_0(k)$ and analyze its behavior as $k\downarrow0$. In the regular threshold regime, we obtain an explicit scattering length. We further identify a threshold--critical configuration characterized by the existence of a nontrivial zero--energy radial solution, regular at the origin, whose exterior constant term vanishes. In the corresponding nondegenerate exceptional case, the usual finite scattering length breaks down, and instead $S_0(k)\to -1$ as $k\downarrow0$.

We prove that for any C*-algebra $A$ and Hilbert $A$-modules $M\subseteq N$ with $M^\perp=\{0\}$, every bounded $A$-linear map $N\to A$ (or $N\to N)$ vanishing on $M$ is the zero map. This verifies the conjectures of the first author and settles the regularity problem for bounded modular functionals and operators on Hilbert C*-modules. As a consequence, kernels of bounded C*-linear operators on Hilbert C*-modules are shown to be biorthogonally complemented, which gives a correct proof of Lemma 2.4 in ``On Hahn-Banach type theorems for Hilbert C*-modules'', Internat. J. Math. 13(2002), 1--19, in full generality.

In this article, we consider the solution to elliptic diffusion problems on a class of random domains obtained by log-Gaussian random homothety of the unit disk respectively an annulus. We model the problem under consideration and verify the existence and uniqueness of the random solution by path-wise pullback to the nominal unit disk respectively annulus. We prove the analytic regularity of the solution with respect to the random input parameter. We consider the numerical approximation of the random diffusion problem by means of continuous, piecewise linear Lagrangian Galerkin Finite Elements with numerical quadrature in the nominal domain, and by sparse grid interpolation and quadrature of Gauss-Hermite Smolyak and Quasi-Monte Carlo type in the parameter domain. The theoretical findings are complemented by numerical results.

An efficient three-color (3C) laser excitation-based Rydberg atomic quantum receiver (RAQR) architecture is investigated for wireless communications, utilizing a five-level (5L) electronic transition mechanism. Specifically, the conventional two-color (2C) RAQR with the four-level (4L) excitation faces three fundamental obstacles: 1) high cost and engineering challenges due to the reliance on unstable blue lasers; 2) a fundamental sensitivity limit in thermal atoms caused by residual Doppler broadening; and 3) the inability to detect low-frequency bands due to the energy-level constraint of two-photon resonance. To address these challenges, this paper analyzes a 3C5L-RAQR architecture with all-red/infrared lasers, which not only solves the engineering cost issues but also enables effective Doppler cancellation and low-frequency detection by exhibiting the three-photon resonance. Bridging atomic physics and communication theory, an end-to-end equivalent baseband signal model is derived. Furthermore, the performance of different RAQR architectures is evaluated in terms of sensitivity, achievable capacity and spectrum access range. Moreover, we provide an exact numerical solution for practical RAQRs by employing the Liouvillian superoperator formalism. Numerical results demonstrate that the exhibited 3C5L-RAQR achieves superior sensitivity compared to the conventional 2C4L-RAQR and the classical receiver based on the conductor antenna. Finally, the inherent sensitivity-capacity trade-off is revealed, showing that the 3C5L-RAQR is more suitable for deployment in power-limited communication scenarios demanding broad spectrum access.

For simultaneous independent events with finitely many outcomes, consider the expected-utility problem with nonnegative wagers and an endogenous cash position. We prove a short support theorem for a broad class of strictly increasing strictly concave utilities. On any fixed support family and at any optimal portfolio with positive cash, summing the active first-order conditions and comparing that sum with cash stationarity yields the exact identity \[ \frac{\lambda}{K_{\ell}^{(U)}}=\frac{1-P_{\ell,A}}{1-Q_{\ell,A}}, \] where $P_{\ell,A}$ and $Q_{\ell,A}$ are the active probability and price masses of event $\ell$, $\lambda$ is the budget multiplier, and $K_{\ell}^{(U)}$ is the continuation factor seen by inactive outcomes of that event. Consequently, after sorting each event by the edge ratio $p_{\ell i}/\pi_{\ell i}$, the exact active support is the eventwise union of the single-event supports, and this support is independent of the utility function. The single-event utility-invariant support theorem is already explicit in the free-exposure pari-mutuel setting in Smoczynski and Miles; the point of the present note is that the simultaneous independent-events analogue follows from the same state-price geometry once the right continuation factor is identified.

This paper establishes quantitative correlation inequalities between monotone events and structured threshold objects in both the discrete cube and Gaussian space. We prove that for any increasing balanced family, there exists a linear threshold function yielding a covariance lower bound of $c \frac{\log n}{\sqrt{n}}$, and extend this principle to halfspaces in Gaussian space. These results verify the conjectures of Kalai, Keller, and Mossel regarding optimal correlation bounds for linear threshold functions and their Gaussian analogues.

Quaternion-valued representations provide a convenient way to model coupled multi-channel signals (e.g., RGB imagery, polarization data, vector fields, and multi-detector time series). Yet practical and numerically reliable software support remains far less mature than those based on the real/complex setting. Here, we present QuatIca, an open-source Python library for quaternion numerical linear algebra and optimization, designed for both research prototyping and reproducible experimentation. QuatIca provides core quaternion matrix operations and norms; dense decompositions and reductions (QR, LU, Q-SVD, eigendecomposition, Hessenberg/tridiagonal reduction, Cholesky decomposition, and Schur helpers); iterative solvers including quaternion GMRES (with preconditioning) and Newton-Schulz pseudoinverse schemes; and domain-focused routines for signal and image processing such as quaternion Tikhonov restoration. The library also includes OptiQ, which solves quaternion Hermitian semidefinite programs using log-det barrier Newton methods with $\mu$-continuation. We highlight design choices that preserve quaternion structure, and we provide end-to-end demonstrations including quaternion image deblurring, Lorenz-attractor filtering, and quaternion image completion. QuatIca is distributed via PyPI and accompanied by open-source development on GitHub and continuously deployed documentation with runnable tutorials.

Near-field beamfocusing with extremely large aperture arrays can effectively enhance physical layer security. Nevertheless, even small estimation errors of the eavesdropper's location may cause a pronounced focal shift, resulting in a severe degradation of the secrecy rate. In this letter, we propose a physics-informed robust beamforming strategy that leverages the electromagnetic (EM) caustic effect for near-field physical layer security provisioning, which can be implemented via phase shifts only. Specifically, we partition the transmit array into caustic and focusing subarrays to simultaneously bypass the potential eavesdropping region and illuminate the legitimate user, thereby significantly improving the robustness against the localization error of eavesdroppers. Moreover, by leveraging the connection between the phase gradient and the EM wave departing angle, we derive the corresponding piece-wise closed-form array phase profile for the subarrays. Simulation results demonstrate that the proposed scheme achieves up to an 80% reduction of the worst-case eavesdropping rate for a localization error of 0.25 m, highlighting its superiority for providing robust and secure communication.

Semantic communication, enabled by deep joint source-channel coding (DeepJSCC), is widely expected to inherit the vulnerability of deep learning to adversarial perturbations. This paper challenges this prevailing belief and reveals a counterintuitive finding: semantic communication systems exhibit unanticipated adversarial robustness that can exceed that of classical separate source-channel coding systems. On the theoretical front, we establish fundamental bounds on the minimum attack power required to induce a target distortion, overcoming the analytical intractability of highly nonlinear DeepJSCC models by leveraging Lipschitz smoothness. We prove that the implicit regularization from noisy training forces decoder smoothness, a property that inherently provides built-in protection against adversarial attacks. To enable rigorous and fair comparison, we develop two novel attack methodologies that address previously unexplored vulnerabilities: a structure-aware vulnerable set attack that, for the first time, exploits graph-theoretic vulnerabilities in LDPC codes to induce decoding failure with minimal energy, and a progressive gradient ascent attack that leverages the differentiability of DeepJSCC to efficiently find minimum-power perturbations. Designing such attacks is challenging, as classical systems lack gradient information while semantic systems require navigating high-dimensional, non-convex spaces; our methods fill these critical gaps in the literature. Extensive experiments demonstrate that semantic communication requires up to $14$-$16\times$ more attack power to achieve the same distortion as classical systems, empirically substantiating its superior robustness.

Despite the growing interest in fractional generalizations of classical fluid dynamics equations, the fractional Rayleigh--Stokes problem has previously been studied almost exclusively using the Riemann--Liouville fractional derivative. To the authors' knowledge, an explicit analytical form of the solution for the Caputo derivative case has not been established in the literature, and before this work, no systematic study of the existence, uniqueness, or regularity properties of this formulation has been conducted. In this paper, we fill this gap by considering the Rayleigh--Stokes equation with the Caputo fractional time derivative of order $\rho \in (0, \, 1)$. Using the Laplace transform and Fourier methods, as well as special functions, we perform a rigorous well-posedness analysis of the corresponding initial boundary-value, non-local, and backward problems.

We study the quasilinear elliptic system \[ -\textbf{div}(A(x,\boldsymbol u)|D\boldsymbol u|^{p-2}D\boldsymbol u) +\frac{1}{p}\nabla_{\boldsymbol s}A(x,\boldsymbol u)|D\boldsymbol u|^p = \boldsymbol g(x,\boldsymbol u) \quad \text{in } \Omega, \qquad \boldsymbol u = 0 \text{ on } \partial\Omega, \] where $p>1$, $\Omega\subset\mathbb R^N$ is a bounded domain with $N>1$, and $\boldsymbol g$ satisfies a subcritical growth condition. In this setting, the associated energy functional is, in general, neither differentiable nor locally Lipschitz in the natural Sobolev space. By exploiting a nonsmooth critical point theory, we prove the existence of infinitely many weak solutions by means of an Equivariant Mountain Pass Theorem. In addition, we establish $L^\infty$-bounds for weak solutions by adapting a Moser-type iteration.

In this paper, we are interested in the existence and asymptotic behavior of least energy solutions to the upper critical Choquard equation \begin{equation*} \begin{cases} -\Delta u+au=\displaystyle\left(\int_{\Omega}\frac{u^{6-\alpha}(y)}{|x-y|^\alpha}dy\right)u^{5-\alpha}&\mbox{in}\ \Omega, u>0 \ \ &\mbox{in}\ \Omega, u=0 \ \ &\mbox{on}\ \partial \Omega, \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^{3}$ is a bounded domain with a $C^{2}$ boundary, $\alpha \in (0,3)$, $a \in C(\overline{\Omega}) \cap C^{1}(\Omega)$, and the operator $-\Delta + a$ is coercive. We first establish that the following three properties are equivalent: the existence of least energy solutions, the validity of a strict inequality in the associated minimization problem, and the positivity of the Robin function somewhere in the domain. This leads naturally to the definition of a critical function $a$. Under the perturbation $a \mapsto a + \varepsilon V$ with $a$ critical and $V \in L^{\infty}(\Omega)$, we prove that least energy solutions exist. Furthermore, we establish a refined energy estimate and describe their asymptotic profile.

The recent work of Morini-Oronzio-Spadaro and the third author shows that, in three dimensions, a flat-flow solution of the volume-preserving mean curvature flow that converges to a single ball, which is the case for instance when the initial perimeter is less than that of two disjoint balls, converges exponentially fast in Hausdorff distance. In this paper we strengthen this result by proving that after a finite time the flow becomes smooth, satisfies the equation in the classical sense and converges exponentially fast to the limiting ball in every C^k-norm. In the proof we develop a version of Brakke's epsilon regularity theorem adapted to our setting and derive the necessary nonlinear PDE estimates directly at the level of the discrete minimizing-movement scheme. The same result holds in the planar case.

In this paper, we consider the existence of positive solutions to the following slightly supercritical Choquard equation \begin{equation*} \begin{cases} -\Delta u=\displaystyle\Big(\int\limits_{\Omega}\frac{u^{2^*_{\alpha}+\varepsilon}(y)}{|x-y|^\alpha}dy\Big)u^{2^*_{\alpha}-1+\varepsilon},\quad u>0\ \ &\mbox{in}\ \Omega, \quad \ \ u=0 \ \ &\mbox{on}\ \partial \Omega, \end{cases} \end{equation*} where $N\geq 3$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $\alpha\in (0,N)$, $2^*_{\alpha}:=\frac{2N-\alpha}{N-2}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and $\varepsilon>0$ is a small parameter. In contrast with the slightly subcritical Choquard equation studied by Chen and Wang (Calculus of Variations and Partial Differential Equations, 63:235, 2024), we find that there is no chance to construct a family of single-bubble solutions as $\varepsilon\to 0^{+}$.

This research paper introduces an SQIR-V epidemic model to investigate the transmission of infectious diseases. Particular attention is paid to the roles of vaccination and quarantine (incorporating physical distancing interventions) in protecting susceptible individuals. The model features nonlinear transition rates that depend on the history of infection, allowing the emergence of periodic solutions. We calculate the basic reproduction number, R 0 , and analyze the local asymptotic stability of the equilibrium points. Additionally, we demonstrate that the diseasefree equilibrium is globally asymptotically stable when R 0 $\le$ 1. The study further explores the existence of periodic solutions through a Hopf bifurcation, showing the occurrence of epidemic waves. A condition was derived to determine the direction of the crossing of the imaginary axis. We finish by presenting some numerical simulations to illustrate how vaccination and isolation delays influence disease dynamics. Those findings highlight potential areas for further research and validation.

We investigate finite-time blow-up of solutions to the Cauchy problem for a semilinear heat equation posed on infinite graphs. Assuming that the initial datum is sufficiently large, we establish a general blow-up criterion valid on arbitrary infinite graphs. We then apply this result to specific classes of graphs, including trees and the integer lattice. The approach developed in the paper can be regarded as a discrete counterpart of Kaplan's method, suitably adapted to the graph setting. In a companion paper, which is the second part of this work, we also complement the blow-up analysis by addressing arbitrary initial data and proving global existence for sufficiently small data.

This paper is the second part of the study initiated in a companion work and is devoted to finite-time blow-up and global existence for a semilinear heat equation on infinite weighted graphs. We first establish basic results on mild and classical solutions (which, to the best of our knowledge, were not previously available in the setting of graphs) proving their equivalence under suitable assumptions and showing the existence of a solution between a given sub- and supersolution. We then analyze blow-up and global existence on $\mathbb Z^N$, providing proofs based on methods different from those used on $\mathbb Z^N$ in the existing literature. Moreover, for graphs with positive spectral gap, we prove global existence for small initial data. In contrast with previous functional analytic approaches yielding mild solutions, our method relies on the construction of global-in-time supersolutions and leads to the existence of classical solutions.

We extend a result by Ikeda and Suriajaya (2025) to find the asymptotic behaviour of the average number of representations of an integer $n$, over multiples of a fixed $q\ge 2$, as a sum of two prime $k$-th powers, for $k\ge 2$.

In this article, we study a model of random permutations, which we call random standardized permutations, based on a sequence of i.i.d. random variables. This model generalizes others, such as the riffle-shuffle and the major-index-biased permutations. We first establish an exact result on the joint distribution of the number of cycles of given lengths, involving the notion of primitive words. From this result, we obtain various convergence results, most of which are proved using the method of moments. First we prove that the number of small cycles may have either a Poisson limit distribution, or a limit distribution given by a countable sum of independent geometric distributions. Then we establish a limit distribution for large cycles, which is the Poisson-Dirichlet process. Finally we prove a central limit theorem for the total number of cycles.

We study the degree landscape of the partition graph $G_n$, whose vertices are the integer partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts, followed by reordering. Using the previously established local degree formula, we introduce the degree layers $D_d(n)$, the degree spectrum $Spec_D(n)$, and the numerical invariants $\Delta_n$, $m_\Delta(n)$, and $s(n)$. The main theorem provides an exact formula for the maximal degree. If $$ \rho(n):=\max\{r:T_r\le n\},\qquad T_r=\frac{r(r+1)}{2}, $$ and $$ \nu:=n-T_{\rho(n)}, $$ then $$ \Delta_n=\rho(n)\bigl(\rho(n)-1\bigr)+\beta_{\rho(n)}(\nu), $$ where $\beta_r$ is an explicit budget function governed by a square--pronic threshold rule. We also prove that every maximal-degree vertex lies on the maximal-support stratum, and we obtain exact extremal classifications at the levels $n=T_t$, $n=T_t+1$, and $n=T_t+2$. The paper also includes a finite computation on the range $1\le n\le 60$, recording extremal multiplicities, representative extremal shapes, spectrum sizes, selected degree histograms, and first data on contact between the extremal layer and the self-conjugate axis. This computational part is deliberately limited in scope. It is descriptive rather than exhaustive, and is included only as a first numerical profile of the degree landscape.

In this paper, given a certain regularity of a function $v$, we derive an explicit formula relating the order $\nu_0\in(0,1)$ of the leading fractional derivative in a fractional differential operator $\mathbf{D_t}$ with the variable coefficients $r_i=r_i(x,t)$ and the function $v$ on which this operator acts. Moreover, we discuss application of this result in the reconstruction of the memory order of semilinear subdiffusion with memory terms. To achieve this aim, we analyze some inverse problems to multi-term fractional in time ordinary and partial differential equations with smooth local or nonlocal additional measurements for small time. In conclusion, we discuss how this formula may be exploited to numerical computation of $\nu_0$ in the case of discrete noisy observation in the corresponding inverse problems. Our theoretical results along with the computational algorithm are supplemented by numerical tests.

Quantile regression (QR) relies on the estimation of conditional quantiles and explores the relationships between independent and dependent variables. At high probability levels, classical QR methods face extrapolation difficulties due to the scarcity of data in the tail of the distribution. Another challenge arises when the number of predictors is large and the quantile function exhibits a complex structure. In this work, we propose an estimation method designed to overcome these challenges. To enhance extrapolation in the tail of the conditional response distribution, we model block maxima using the generalized extreme value (GEV) distribution, where the parameters depend on covariates. To address the second challenge, we adopt an approach based on generalized random forests (grf) to estimate these parameters. Specifically, we maximize a penalized likelihood, weighted by the weights obtained through the grf method. This penalization helps overcome the limitations of the maximum likelihood estimator (MLE) in small samples, while preserving its optimality in large samples. The effectiveness of our method is validated through comparisons with other approaches in simulation studies and an application to U.S. wage data.

Stochastic rounding (SR) is a probabilistic rounding mode that mitigates errors in large-scale numerical computations, especially when prone to stagnation effects. Beyond numerical analysis, SR has shown significant benefits in practical applications such as deep learning and climate modelling. The definition of classical SR requires that results of arithmetic operations are known with infinite precision. This is often not possible, and when it is, the resulting hardware implementation can become prohibitively expensive in terms of energy, area, and latency. A more practical alternative is limited-precision SR, which only requires that the outputs of arithmetic operations are available in higher, finite, precision. We extend previous work on limited-precision SR presented in [El Arar et al., SIAM J. Sci. Comput. 47(5) (2025), B1227-B1249], which developed a framework to evaluate the trade-off between accuracy and hardware resource cost in SR implementations. Within this framework, we study the Horner algorithm and pairwise summation, providing both theoretical insights and practical experiments in these settings when using limited-precision SR.

The paper treats density measures as typical examples of finitely additive measures in $\mathbb{R}^n$. We study their structure and derive basic properties. In addition, estimates for related integrals are provided. The results are applied to the precise representative of general integrable functions and then they are specialized to functions of bounded variation. Moreover, a new representation of the generalized gradients in the sense of Clarke is given for the finite dimensional case.

We prove that, given a wavelet $\psi$, it is possible to choose some multi-integers $(p_j=(p_{j,1},...,p_{j,d}))_{j \in \mathbb{Z}} \in \mathbb{Z}^d$ such that, for every $x=(x_1,...,x_d) \in \mathbb{R}^d$, for infinitely many integers $j$, the tensorized wavelet $\prod_{i=1}^d \psi(2^j x_i-p_{j,i})$ does not vanish at $x$. This non-vanishing property is essential for analyzing some generic regularity properties in certain Sobolev and Besov spaces. The proof relies on an assumption regarding the zeros of $\psi$, which we numerically verify for the first Daubechies wavelets.

We establish a couple of dynamical properties of surjective rational maps $f: X \dashrightarrow X$ for smooth projective surfaces $X$. We also give a numerical characterization of regular $f$ in the case when $X$ is a del Pezzo surface. Some explicit constructions and calculations, related to the topological entropy of $f$, are provided.

We verify Tutte's $3$-flow conjecture in the class of Cayley graphs on solvable groups of order $2n$, where $n$ is square-free. The proof relies on a new necessary and sufficient condition for a simple $5$-valent graph to admit a nowhere-zero $3$-flow in terms of a pseudoforest decomposition.

We propose an alternating optimization framework for maximizing energy efficiency (EE) in reconfigurable intelligent surface (RIS) assisted distributed MIMO (D-MIMO) systems under both coherent and non-coherent reception modes. The framework jointly optimizes access point (AP) power allocation and RIS phase configurations to improve EE under per-AP power and signal-to-interference-plus-noise ratio (SINR) constraints. Using majorization-minimization for power allocation together with per-element RIS adaptation, the framework achieves tractable optimization of this non-convex problem. Simulation results for indoor deployments with realistic power-consumption models show that the proposed scheme outperforms equal-power and random-scatterer baselines, with clear EE gains. We evaluate the performance of both reception modes and quantify the impact of RIS phase-shift optimization, RIS controller architectures (centralized vs. per-RIS control), and RIS size, providing design insights for practical RIS-assisted D-MIMO deployments in future 6G networks.

We characterize in certain basic cases when a module over a ring is either {\it hereditarily Bassian} or {\it super Bassian} in the sense that either each its proper submodule is Bassian or, respectively, each its proper epimorphic image is Bassian. We prove several structural criteria for both hereditarily Bassian and super Bassian modules over non-primitive Dedekind prime rings, and in particular Dedekind domains. Over these rings, we establish that a singular module is super Bassian exactly when it is Bassian, which is true if and only if it is Bassian. In addition, for an arbitrary (not necessarily singular) module over a non-primitive Dedekind prime ring, the property of being super Bassian curiously implies the property of being hereditary Bassian always. Our results somewhat continue and supply recent results due to Tuganbaev in Mathematics (2026) and Blacher in J. Algebra (2026).

We introduce the concept of volume term adaptivity for high-order discontinuous Galerkin (DG) schemes solving time-dependent partial differential equations. Termed v-adaptivity, we present a novel general approach that exchanges the discretization of the volume contribution of the DG scheme at every Runge-Kutta stage based on suitable indicators. Depending on whether robustness or efficiency is the main concern, different adaptation strategies can be chosen. Precisely, the weak form volume term discretization is used instead of the entropy-conserving flux-differencing volume integral whenever the former produces more entropy than the latter, resulting in an entropy-stable scheme. Conversely, if increasing the efficiency is the main objective, the weak form volume integral may be employed as long as it does not increase entropy beyond a certain threshold or cause instabilities. Thus, depending on the choice of the indicator, the v-adaptive DG scheme improves robustness, efficiency and approximation quality compared to schemes with a uniform volume term discretization. We thoroughly verify the accuracy, linear stability, and entropy-admissibility of the v-adaptive DG scheme before applying it to various compressible flow problems in two and three dimensions.

We prove compactness with respect to $\Gamma$-convergence for a general class of non-local energies modelled after the ones considered in [Gobbino, CPAM (1998)]. We give an integral representation result for the limits, which are free discontinuity functionals defined on the space of generalised special functions of bounded variation. We then characterise the bulk and surface energy densities of the obtained limits by means of minimisation problems on small cubes for the approximating energies.

Let $B$ be a compact Riemann surface and $B_0\subset B$ a bordered hyperbolic subsurface obtained by removing finitely many disjoint closed disks. Fix a nontrivial loop $\alpha$ in $B_0$. For $s\ge 0$, let $L(\alpha,s)$ denote the supremum, over all finite subsets $S\subset B_0$ with $\#S\le s$, of the minimal Kobayashi length of a loop in $B_0\smallsetminus S$ that is freely homotopic to $\alpha$ in $B_0$. Phung in [7] proved that $L(\alpha,s)$ grows at most linearly and at least as $\sqrt{s}/\log s$. We sharpen the upper bound to $O\left(\sqrt{s\log s}\right)$, which determines $\lim_{s\to\infty}\frac{\log L(\alpha,s)}{\log s}=\frac{1}{2}$, answering a question raised in [7, Question 1.4]. As an application, we improve the counting bound for generalized integral points on abelian varieties over complex function fields: for an abelian variety of dimension $n$ over $\mathbb C(B)$, Phung proved that the number of $(s, B_0)$-generalized integral points modulo the constant trace grows at most as $s^{2nk}$, where $k=\operatorname{rk}(\pi_1(B_0))$. We sharpen this to $s^{nk+\varepsilon}$ for every $\varepsilon>0$, halving the exponent.

We prove a globally hyperbolic spacetime with locally Lipschitz continuous metric and timelike distributional Ricci curvature bounded from below obeys the timelike measure contraction property. The remarkable class of examples of spacetimes that are covered by this result includes impulsive gravity waves, thin shells, and matched spacetimes. As applications, we get new comparison theorems for Lipschitz spacetimes in sharp form: d'Alembert, timelike Brunn-Minkowski, and timelike Bishop-Gromov. Under appropriate nonbranching assumptions (conjectured to hold in even lower regularity), our results also yield the timelike curvature-dimension condition, a volume incompleteness theorem, as well as exact representation formulas and sharp comparison estimates for d'Alembertians of Lorentz distance functions from general spacelike submanifolds. Moreover, we establish the sharp timelike Bonnet--Myers inequality ad hoc using the localization technique from convex geometry. Alongside, we prove a timelike diameter estimate for spacetimes whose timelike Ricci curvature is positive up to a "small" deviation (in an $L^p$-sense). This adapts prior theorems for Riemannian manifolds by Petersen-Sprouse and Aubry to Lorentzian geometry, a transition the former two anticipated almost 30 years ago.

We examine the existence of thick bubble rings within the framework of the free-boundary capillary Euler equations, focusing on the regime of low Weber numbers. Although spheroidal bubbles are known to approach a spherical shape in this limit, the possibility of thick bubble rings persisting at low Weber numbers has remained uncertain. In contrast to the ordinary Euler equations, which admit thick vortex ring solutions, our analysis reveals that the free-boundary capillary Euler equations do not support thick bubble rings at low Weber numbers. This distinction highlights the significant impact of surface tension on the behavior of vortex rings in the capillary regime.

We establish monotonicity formulas for a parabolic frequency function associated with sign-changing solutions to a class of doubly nonlinear parabolic equations of the form $\partial_t u = \mathcal{L}_{p,\varphi} u^q$ on weighted complete Riemannian manifolds without any curvature assumption, where $\mathcal{L}_{p,\varphi}$ denotes the weighted $p$-Laplacian and $p>1$, $q>0$. As a consequence, we obtain results on backward uniqueness for $q(p-1)\geq 1$ and unique continuation at infinity for $q(p-1) > 1$. We further consider equations with a controlled nonlinear perturbation term and derive an almost-monotonicity formula for the parabolic frequency. By employing the parabolic frequency, we also establish some Liouville-type results for ancient solutions in the case $q(p-1)\geq 1$.

Let $s\in(\frac{n}{n+1},1)$, $\Omega\subset\mathbb{R}^n$ be an open set and let $f\in W^{s,n/s}(\Omega,\mathbb{R}^n)$ be mapping with positive distributional Jacobian $\mathcal{J}_f>0$ which models some deformation in fractional Nonlinear Elasticity. We show change of variables formula in this class and as a consequence we show that the analogue of Ciarlet-Nečas condition $\mathcal{J}_f(\Omega)=|f(\Omega)|$ implies that our mapping is one-to-one a.e.

In this work, the first initial-boundary value problem for a sub-diffusion equation involving the regularized Prabhakar fractional derivative is studied. The problem is solved by reducing it to two initial-boundary value problems using the superposition method. An explicit representation of the solution and the corresponding Green's function is obtained. The explicit form of the Green's function is expressed in terms of a bivariate Mittag-Leffler type function. Then, it is proved that the obtained solution indeed constitutes the solution of the considered problem.

The design of numerical integrators for solving stochastic dynamics with high weak order relies on tedious calculations and is subject to a high number of order conditions. The original approaches from the literature consider strong approximations and adapt them for the weak approximation by replacing the iterated stochastic integrals by appropriate random variables. The methods obtained this way are sub-optimal in their number of function evaluations and the analysis of order conditions is unnecessarily complicated. We provide in this paper a novel approach, relying on well-chosen sets of random Runge-Kutta coefficients, that greatly reduce the number of order conditions. The approach is successfully applied to the creation of a collection of new stochastic Runge-Kutta methods of second weak order with an optimal number of function evaluations and a smaller number of random variables. The efficiency of the new methods is confirmed with numerical experiments and a modern algebraic approach using Hopf algebras is provided for the derivation and the study of the order conditions.

We conjecture the existence of almost integer invariants governing the all-genus equivariant Gromov-Witten theory of Calabi-Yau fivefolds with a torus action. We prove the conjecture for skeletal, locally anti-diagonal torus actions by establishing a vertex formalism evaluating the Gromov-Witten invariants via the topological vertex of Aganagic, Klemm, Marino and Vafa. We apply the formalism in several examples.

The additional information carried by an enlarged filtration and its measurement was studied by several authors. Already Meyer (Sur un theoreme de J. Jacod, 1978) and Yor (Entropie d'une partition, et grossissement initial d'une filtration, 1985), investigated stability of martingale spaces with respect to initial enlargement with atomic sigma-field. We extend these considerations to the case where information is disclosed progressively in time. We define the entropy of such information and we prove that its finiteness is enough for stability of some martingale spaces in progressive setting. Finally we calculate additional logarithmic utility of a discrete information disclosed progressively in time.

A classical proposal to derive weights from a pairwise comparison matrix is the right eigenvector. The literature has identified some potential weaknesses of this method in previous decades. This chapter discusses five of these issues. First, right-left asymmetry emerges because of the difference between the right and inverse left eigenvectors. Second, group incoherence for choice means that, in group decision-making problems, the ranking given by the aggregated individual weight vectors is not guaranteed to coincide with the ranking derived from the aggregated pairwise comparison matrix. Third, the ranking based on the right eigenvector may depend on the intensity of the preferences, represented by taking a positive power of all comparisons. Fourth, both the ranking position and the normalised weight of an object might change counter-intuitively after modifying a particular comparison. Fifth, the right eigenvector is not necessarily Pareto efficient: a dominating weight vector that approximates each pairwise comparison at least as well, with an improvement in at least one position, could exist. All violations of the theoretical properties are highlighted by illustrative examples. We also present several open questions in order to inspire future research.

The Miquel-Steiner theorem for a quadrilateral in the Euclidean plane states that the circumcircles of the four component triangles intersect at a single point, which now is called the Miquel-Steiner point of the quadrilateral. In elliptic and in hyperbolic planes, the Miquel-Steiner theorem does not hold in this form. Instead, a weaker version applies: The circumcircles of the four component triangles of a quadrilateral have a common radical center, which we will also call the Miquel-Steiner point. The Miquel-Steiner theorem for Euclidean planes also needs to be modified for Minkowski and Galilean planes: Either the circumcircles of the four component triangles touch each other at a point on the line at infinity, or they intersect transversely at an anisotropic point. For specific quadrilaterals (such as cyclic quadrilaterals), the location of the Miquel-Steiner point can be determined more precisely.

We define a partition of a reductive group into finitely many subsets, refining the partition of the group into strata. We state some conjectural properties of these subsets (called substrata) and verify them in some examples.

A graph is called strongly $\Z_{2k+1}$-connected if for each boundary function $\beta: V(G)\mapsto \Z_{2k+1}$ with $\sum_{v\in V(G)}\beta(v)\equiv 0\pmod{2k+1}$, there exists an orientation $D$ of $G$ such that $d_D^+(v) - d_D^-(v) \equiv \beta(v) \pmod{2k+1}$ for each $v \in V(G)$. We show that every planar multigraph with $5$ edge-disjoint spanning trees is strongly $\Z_{5}$-connected. This verifies a special case of the Additive Base Conjecture when restricted to planar graphs. Hence, every $10$-edge-connected directed planar graph admits an antisymmetric $\Z_5$-flow. So, by duality, every orientation of a planar graph of girth at least $10$ admits a homomorphism to a $5$-vertex tournament. Our result also gives a new proof of the known result that every planar graph of girth at least $10$ has a homomorphism to the $5$-cycle.

In this short note, we study dg categories with homotopy kernels, whose homotopy categories are known to admit a natural left triangulated structure. Prototypical examples of such dg categories arise as dg quotients of exact dg categories. We demonstrate that the stablization of the homotopy category of such a dg category admits a canonical dg enhancement via its bounded derived dg category.

In this work we introduce a new method for the construction of minimal submanifolds of codimension two in even dimensional spheres and hyperbolic spaces. This is based on the theory of complex-valued harmonic morphisms. This gives the first explicit examples of such maps defined on the sphere $S^4$ and the hyperbolic space $H^4$.

A graph is chordal if it contains no induced cycle of length four or more. While finite chordal graphs are precisely those admitting tree-decompositions into cliques, this fails for infinite graphs. We establish two results extending the known theory to the infinite setting. Our first result strengthens sufficient conditions of Halin, Kříž-Thomas, and Chudnovsky-Nguyen-Scott-Seymour: We show that every chordal graph without a strict comb of cliques admits a tree-decomposition into maximal cliques. Our second result characterises the chordal graphs admitting tree-decompositions into finite cliques: a connected graph admits such a decomposition if and only if it is chordal, admits a normal spanning tree, and does not contain $\mathcal{H}$ $\unicode{x2013}$ an infinite clique with two non-adjacent dominating vertices $\unicode{x2013}$ as an induced minor. Combined with the characterisation of graphs with normal spanning trees, this yields a description by three types of forbidden minors. Both proofs proceed via greedy constructions of length $\omega$, with the key new ingredient for the second result being an Extension Lemma that uses a finiteness theorem of Halin on minimal separators to produce suitable finite clique extensions at each step.

When dealing with stiff conservation laws, explicit time integration forces to employ very small time steps, due to the restrictive CFL stability condition. Implicit methods offer an alternative, yielding the possibility to choose the time step according to accuracy constraints. However, the construction of high-order implicit methods is difficult, mainly because of the non-linearity of the space and time limiting procedures required to control spurious oscillations. The Quinpi approach addresses this problem by introducing a first-order implicit predictor, which is employed in both space and time limiting. The scheme has been proposed in (Puppo et al., Comm. Comput. Phys., 2024) for systems of conservation laws in one dimension. In this work the multi-dimensional extension is presented. Similarly to the one-dimensional case, the scheme combines a third-order Central WENO-Z reconstruction in space with a third-order Diagonally Implicit Runge-Kutta (DIRK) method for time integration, and a low order predictor to ease the computation of the Runge-Kutta stages. Even applying space-limiting, spurious oscillations may still appear in implicit integration, especially for large time steps. For this reason, a time-limiting procedure inspired by the MOOD technique and based on numerical entropy production together with a cascade of schemes of decreasing order is applied. The scheme is tested on the Euler equations of gasdynamics also in low Mach regimes. The numerical tests are performed on both structured and unstructured meshes.

One approach to studying the dynamics of a singular Lagrangian system is to attempt to regularize it, that is, to find an equivalent and regular system. In the case of time-independent singular Lagrangians, an approach due to \textit{A. Ibort} and \textit{J. Marín-Solano} is to use the coisotropic embedding theorem proved by \textit{M.J. Gotay} which states that any pre-symplectic manifold can be coisotropically embedded in a symplectic manifold. In this paper, we revisit these results and provide an alternative approach, also based on the coisotropic embedding theorem, that employs the Tulczyjew isomorphism and almost product structures, and allows for a slight generalization of the construction. In this revision, we also prove uniqueness of the Lagrangian regularization to first order. Furthermore, we extend our methodology to the case of time-dependent singular Lagrangians.

Landing methods have recently emerged in Riemannian matrix optimization as efficient schemes for handling nonlinear equality constraints without resorting to costly retractions. These methods decompose the search direction into tangent and normal components, enabling asymptotic feasibility while maintaining inexpensive updates. In this work, we provide a unifying geometric framework which reveals that, under suitable choices of Riemannian metric, the landing algorithm encompasses several classical optimization methods such as projected and null-space gradient flows, Sequential Quadratic Programming (SQP), and a certain form of the augmented Lagrangian method. In particular, we show that a quadratically convergent landing method essentially reproduces the quadratically convergent SQP method. These connections also allow us to propose a globally convergent landing method using adaptive step sizes. The backtracking line search satisfies an Armijo condition on a merit function, and does not require prior knowledge of Lipschitz constants. Our second key contribution is to analyze landing methods through a geometric parameterization of the metric in terms of fields of oblique projectors and associated metric restrictions. This viewpoint disentangles the roles of orthogonality, tangent and normal metrics, and elucidates how to design the metric to obtain explicit tangent and normal updates. For matrix optimization, this framework not only recovers recent constructions in the literature for problems with orthogonality constraints, but also provides systematic guidelines for designing new metrics that admit closed-form search directions.

We prove that the geodesic flow on any hyperbolic surface $S$ with at least one cusp is not expansive. The proof is based on the study of strong-stable sets.

We introduce a zeta function counting imaginary quadratic number fields by their class numbers. It is proved that such a function is rational depending only on the eight roots of unity of degrees $1$ and $2$. As a corollary, one gets a lower bound $2p$ for the number of imaginary quadratic fields of the prime class number $p$. Our method is based on the study of periodic points of a dynamical system arising in the representation theory of the Drinfeld modules by the bounded linear operators on a Hilbert space.

This study presents a high-order finite volume scheme capable of large time-step integration for three-temperature radiation diffusion (3TRD) equations, where conservation is naturally achieved through energy update. To handle local large gradients and discontinuities in temperature, a central generalized ENO (GENO) reconstruction is developed for diffusion systems, which achieves essentially non-oscillatory reconstruction for discontinuous solutions. Compared to conventional nonlinear reconstruction methods, its most distinctive feature is the central-type symmetric sub-stencils, which ensure consistency between the numerics and the isotropic nature of thermal diffusion. Additionally, the central GENO method provides smooth states of temperature and temperature gradient at interfaces, facilitating the evaluation of numerical fluxes. Furthermore, interface flux evaluation for cases with discontinuous physical property parameters is modeled. To address the extremely small time-step issue caused by stiff diffusion and source terms, a dual-time-stepping method based on implicit time discretization is developed for the first time in 3TRD systems, with the advantage of decoupling temporal discretization from complex nonlinear spatial discretization. A series of numerical examples validates the high accuracy, physical property preservation, strong robustness, and large time-step integration capability of the present high-order central GENO scheme.

We first fully implement, in Maple, the ingenious method of Robert Stoyan and Volker Strehl from 1995 to automatically derive generating functions for the number of Hamiltonian cycles in an m by n grid graph ,for a fixed width m, but general length n, and actually compute these generating functions for all m up to ten. We also show how to generate a uniformly-at-random such Hamiltonian cycle, and also derive more informative generating functions for other parameters besides the length of the grid graph.

The cyclic hoist scheduling problem originates in electroplating lines, where a single or multiple hoists transport parts between processing tanks subject to technological constraints. The objective is typically to determine a cyclic sequence of hoist movements that minimizes the cycle time while satisfying travel and processing constraints. Although the problem has been widely studied for several decades, the literature contains a puzzling phenomenon: different studies often report different optimal cycle times for the same benchmark instances, which limits the comparability and reproducibility of computational results. In this paper, we revisit the modeling of cyclic hoist scheduling problems from a unified perspective. We introduce a consistent modeling approach for single-hoist problems and analyze several mixed-integer linear programming (MIP) formulations proposed in the literature. Our analysis identifies modeling inconsistencies and clarifies the relationships between existing formulations. Based on these observations, we propose straightforward constraint programming (CP) models that can serve as baseline approaches, and we also derive improved MIP~formulations. Extensive computational experiments compare the strength and performance of the investigated formulations. To support reproducible research, we also provide a publicly available library containing benchmark instances and implementations of several CP and MIP~formulations for single-hoist cyclic hoist scheduling.

We determine the squarefree part of the scalar factor that arises when the quartic invariant of the generic binary form $F$ of odd degree $2n+1$ is expressed as the discriminant of the unique quadratic covariant $(F,F)_{2n}$. This squarefree part is exactly $p$ when $n+2$ is a power of an odd prime $p$, and $1$ otherwise. Equivalently, for each prime $p$: $v_2(S(n))$ is always even, and for odd $p$, $v_p(S(n))$ is odd if and only if $n+2$ is a power of $p$. This generalizes the classical identity $\operatorname{disc}(H(F))=-3\cdot\operatorname{disc}(F)$ for binary cubics, which dates back to the work of Cayley and Sylvester in the 1850s. The proof, which involves substantial explicit coefficient analysis and $p$-adic deformation arguments, was developed using an AI-assisted research workflow: the author's earlier partial attempts were completed through systematic collaboration with Claude Code (Anthropic) and Codex (OpenAI), and key arithmetic lemmas were formally verified in Lean~4 using Aristotle (Harmonic). We describe this workflow in detail as a case study in AI-assisted mathematical research. We also discuss representation-theoretic, geometric, and arithmetic interpretations of the quadratic covariant.

Recently, Forré (arXiv:2104.11547, 2021) introduced transitional conditional independence, a notion of conditional independence that provides a unified framework for both random and non-stochastic variables. The original paper establishes a strong global Markov property connecting transitional conditional independencies with suitable graphical separation criteria for directed mixed graphs with input nodes (iDMGs), together with a version of causal calculus for iDMGs in a general measure-theoretic setting. These notes aim to further illustrate the motivations behind this framework and its connections to the literature, highlight certain subtlies in the general measure-theoretic causal calculus, and extend the "one-line" formulation of the ID algorithm of Richardson et al. (Ann. Statist. 51(1):334--361, 2023) to the general measure-theoretic setting.

We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to projective space. Let $A$ be a simple abelian variety, $f:A \rightarrow \mathbb{P}^n$ be a morphism which is finite onto its image, and $\Gamma \subseteq A(\overline{K})$ be a finite-rank subgroup. We show that for any affine chart $\mathbb{A}^n \subseteq \mathbb{P}^n$ and any finite subset $X \subseteq f(\Gamma) \cap \mathbb{A}^n$, the energy and the sumset of $X$ satisfy quadratic bounds in $\lvert X \rvert$. The proof uses the uniform Mordell-Lang conjecture.

Tashiro and Tachibana proved that there exist no totally umbilical hypersurfaces in complex space forms with nonzero constant holomorphic sectional curvature, and it is also known that the shape operator of such hypersurfaces cannot be parallel. Motivated by these results, we study real hypersurfaces in products of complex space forms. We establish rigidity and nonexistence results for totally umbilical real hypersurfaces in this setting. In particular, we show that if a real hypersurface in a product of complex space forms does not admit a local product structure, then its shape operator cannot be parallel. Moreover, we provide a classification of totally umbilical real hypersurfaces, showing that those admitting a local almost product structure are necessarily totally geodesic or extrinsic hyperspheres.

We investigate shape optimization for the principal eigenvalue of the Pucci extremal operator \[ \left\{ \begin{aligned} -\mathcal{M}^+_{\lambda,\Lambda}(D^{2}u)&=\mu^{+}_{1}(\Omega)u &&\text{in }\Omega,\\ u &=0 &&\text{on }\partial\Omega, \end{aligned} \right. \] in dimension three. Since $\mathcal{M}^+_{\lambda,\Lambda}$ is fully nonlinear, in non-divergence form, and non-variational, classical symmetrization and rearrangement methods are not available. We introduce a three-dimensional family of double--pyramidal domains $\{\Omega^\omega_{\gamma,a}\}$ parametrized by an anisotropy factor $\gamma \in \left[\frac{1}{\sqrt{\omega}},\sqrt{\omega}\right]$ and an affine shear parameter $a\in(-\pi,\pi)$, under fixed ellipticity ratio $\omega=\Lambda/\lambda \ge 1$. Within this family and under a fixed-volume constraint, we prove that the volume-normalized principal eigenvalue is uniquely minimized at the symmetric unsheared configuration $(\gamma,a)=(1,0)$ among domains in the family $\{\Omega^\omega_{\gamma,a}\}$. The proof combines an explicit construction of positive eigenfunctions on seven patches with a lower bound under affine shear deformations. Using the homogeneity and orthogonal invariance of the Pucci operator, we identify an involutive symmetry $\gamma\mapsto \gamma^{-1}$ in the associated volume functional and establish strict monotonicity away from the self-dual point $\gamma=1$. In particular, for $\omega>1$, any nontrivial anisotropy or shear strictly increases the normalized principal eigenvalue. This reveals a genuinely three-dimensional rigidity mechanism for a fully nonlinear spectral problem and extends to dimension three the symmetry-minimization phenomenon previously known in the planar case.

We study a nonlocal SIS epidemic model with free boundaries, advection, and spatial heterogeneity, where the dispersal kernels are not assumed to be symmetric. The model describes the evolution of susceptible and infected populations in a bounded infected habitat whose endpoints move according to nonlocal boundary fluxes. Our goal is to determine the sharp threshold between disease spreading and vanishing, and to characterize the long-time behavior of solutions. The analysis faces several essential difficulties. The linearization around the disease-free equilibrium gives rise to a genuinely coupled nonlocal system with drift, so the relevant spectral quantity cannot be reduced directly to a standard scalar eigenvalue problem. In addition, the presence of advection terms and possibly non-symmetric kernels destroys self-adjointness, so no useful variational characterization is available; in particular, classical Rayleigh quotient and minimax arguments cannot be applied. To overcome these difficulties, we employ the generalized principal eigenvalue theory for nonlocal operators developed by Coville and Hamel, together with the Harnack inequality for non-symmetric nonlocal operators established therein. This non-variational framework is particularly well suited to our setting. Combined with comparison principles, sub- and supersolution constructions, and uniform estimates on time-dependent spatial intervals, it allows us to derive the precise asymptotic behavior of the generalized principal eigenvalue with respect to the spatial domain and the diffusion rate, identify the sharp threshold and the critical habitat size, and determine the long-time dynamics of $S$ and $I$ via an $\omega$-limit set approach. To the best of our knowledge, this is the first work on a free-boundary SIS epidemic model with non-symmetric nonlocal dispersal kernels, advection, and spatial periodicity.

This paper addresses the Service Network Design (SND) problem for a logistics service provider (LSP) operating in a multimodal freight transport network, considering uncertain travel times and limited truck fleet availability. A two-stage optimization approach is proposed, which combines metaheuristics, simulation and machine learning components. This solution framework integrates tactical decisions, such as transport request acceptance and capacity booking for scheduled services, with operational decisions, including dynamic truck allocation, routing, and re-planning in response to disruptions. A simulated annealing (SA) metaheuristic is employed to solve the tactical problem, supported by an adaptive surrogate model trained using a discrete-event simulation model that captures operational complexities and cascading effects of uncertain travel times. The performance of the proposed method is evaluated using benchmark instances. First, the SA is tested on a deterministic version of the problem and compared to state-of-the-art results, demonstrating it can improve the solution quality and significantly reduce the computational time. Then, the proposed SA is applied to the more complex stochastic problem. Compared to a benchmark algorithm that executes a full simulation for each solution evaluation, the learning-based SA generates high quality solutions while significantly reducing computational effort, achieving only a 5% difference in objective function value while cutting computation time by up to 20 times. These results demonstrate the strong performance of the proposed algorithm in solving complex versions of the SND. Moreover, they highlight the effectiveness of integrating diverse modeling and optimization techniques, and the potential of such approaches to efficiently address freight transport planning challenges.

We study higher-order curvature estimates along Kähler-Ricci flows on compact Kähler manifolds of intermediate Kodaira dimension. We prove that away from singular fibers, the Ricci curvature is uniformly bounded in $C^1$, the Laplacian of the Ricci curvature in $C^0$, and the scalar curvature in $C^2$. We identify a geometric obstruction to higher-order curvature bounds, whose non-vanishing causes a specific third-order derivative of the Ricci curvature to blow up at rate $e^{t/2}$. Uniform $C^k$ bounds for every $k$ hold for the Ricci curvature in the isotrivial case, and for the full Riemann curvature in the torus-fibered case.

In recent years it has been shown how approximate bosonization can be used to justify the random phase approximation for the correlation energy of interacting fermions in a mean-field scaling limit. At the core is the interpretation of particle-hole excitations close to the Fermi surface at bosons. The main two approaches however differ in emphasizing collective degrees of freedom (particle-hole pairs delocalized over patches on the Fermi surface) or particle-hole pairs exactly localized in momentum space. Both methods lead to equal precision for the correlation energy with regular interaction potentials. This poses the question how big the influence of delocalizing particle-hole pairs really is. In the present note we show that a description with few, completely collective bosonic degrees of freedom only yields an upper bound of about 92% of the optimal value. Nevertheless it is remarkable that such a simple approach comes that close to the optimal bound.

We provide a comprehensive overview of the fundamental structural properties of weighted projective Reed-Muller codes. We give a recursive construction for these codes, under some conditions for the weights, and we use it to derive bounds on the generalized Hamming weights and to obtain a recursive construction for their subfield subcodes and their dual codes. The dual codes are further studied in more generality, where the recursive constructions may not apply, obtaining a description as an evaluation code when the degree is low. We also provide insights into the Schur products of these codes when they are not degenerate.

We prove the global well-posedness of the one-dimensional Navier-Stokes-Korteweg equations driven by a stochastic multiplicative noise. The analysis is performed for the general case of capillarity and viscosity coefficients $k(\rho)= \rho^\beta, \, \beta \in \mathbb{R},\, \mu(\rho)=\rho^\alpha, \, \alpha \ge 0,$ which are not coupled through a BD relation. Global existence and uniqueness of solutions is obtained in the regularity class of strong pathwise solutions, which are strong solutions in PDEs and also in the sense of probability. We first make use of a multi-layer approximation scheme and a stochastic compactness argument to establish the local well-posedness result for any $\alpha$ and $\beta.$ Then, we apply a BD entropy method which provides control of the vacuum states of the density and allows to perform an extension argument. Global well-posedness is thus obtained in the range where there is no vacuum and the strong coercivity condition $2\alpha-4 \le \beta \le 2 \alpha-1,$ introduced in [49], holds. As a byproduct, we also cover the deterministic setting $\mathbb{F}=0,$ which to the best of our knowledge is likewise an open problem in the fluid dynamics literature.

Let $\mathbb{K}$ be an algebraically closed field, and $A \subset \mathbb{K}[x_{1}, \ldots, x_n]$ be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of $\mathbb{K}$-algebras \[ A = A_{0} \subset A_{1} \subset \ldots \subset A_m = \mathbb{K}[x_{1}, \ldots, x_n], \] where each $A_i$ can be written as the kernel of some linear functional $L_{i + 1} : A_{i + 1} \to \mathbb{K}$, and each $L_i$ is either a derivation or of the form $L_i : f \to c(f(\mathbf{\alpha}) - f(\mathbf{\beta}))$ for some $\mathbf{\alpha}, \mathbf{\beta} \in \mathbb{K}^{n}$ and $c \in \mathbb{K}$. We investigate the structure of these filtrations and linear functionals. Our main result shows that each such $L_i$ which is a derivation may be written as a linear combination of partial derivatives evaluated at points of $\mathbb{K}^{n}$.

Given a polynomial $\sum_\nu a_\nu X^\nu$ of degree $<d$, bounded by one on the unit disk, how large can $\lvert a_0+a_1+\ldots+a_n \rvert$ ($n<d$) get? This question dates back at least to the 1952 thesis work of H. S. Shapiro. In 1978, D. J. Newman gave an exact answer for $d=2(n+1)$, but there does not seem to have been further progress on the question since. We study variations on this theme, obtaining exact answers for some related coefficient sums, and answer the original question in an asymptotic sense, provided that $n$ is not too large in terms of $d$. The latter is achieved via a quantitative Eneström--Kakeya theorem, while the former is based on certain identities for carefully selected Lagrange interpolators. From the interpolation approach we also obtain a general inequality for coefficient sums $\lvert t_0 a_0 + \ldots + t_{d-1} a_{d-1} \rvert$ for arbitrary complex numbers $t_0,\ldots,t_{d-1}$. This inequality fails to be sharp in general, yet it is in some cases and also yields non-trivial bounds for Shapiro's problem for some choices of $n$ and $d$.

We investigate a class of locally complicated self-affine functions defined via the $Q_s$-representation of real numbers. In particular, we compute local Hölder exponents at points with given asymptotic frequencies of digits in their $Q_s$-representation. Furthermore, we establish conditions under which these functions possess continuum level sets. Finally, for self-affine functions satisfying additional conditions, we describe the geometric structure of the set of maximum points and show that this set can be fractal.

We propose an extension of the plane-wave representation for wave fields defined on the real sphere $§^2$. This representation is well-known in the planar setting but has never been developed for curved surfaces. To achieve this, we need to carefully study the geometry of the complexification of $§^2$ and the properties of the Laplace--Beltrami operator, while using concepts of multidimensional complex analysis. We extend the region of validity of such plane-wave representation by developing a sliding-contours method. Our methodology is illustrated through the study of the Green's function on the real sphere.

We identify a geometric principle governing the location of Hopf and Bogdanov--Takens bifurcations in planar predator--prey systems. The prey coordinate of any coexistence equilibrium undergoing such a bifurcation lies between consecutive critical points of the prey nullcline. The mechanism is algebraic. At critical points of the nullcline, the vanishing of its derivative induces constraints on the Jacobian that prevent the spectral conditions required for bifurcation from being satisfied. We refer to this phenomenon as \emph{spectral rigidity}. The principle is established for three model families and one discrete counterpart with qualitatively different nullcline geometries: a quadratic case (Bazykin model), a cubic case (Holling type~IV with harvesting), and a rational case (Crowley--Martin functional response). In each case, the localization follows from explicit parametric characterizations and symbolic reduction. The analysis extends to discrete-time systems. For a map obtained by forward Euler discretization of the Crowley--Martin model, the Neimark--Sacker bifurcation occurs on the descending branch of the nullcline, providing a continuous--discrete duality governed by the same mechanism. We conjecture that this localization holds for general smooth prey nullclines, with critical points acting as spectral barriers that organise the bifurcation structure.

Sublinearly Morse directions in proper geodesic spaces are defined by sublinearly Morse stability. In this paper we offer an alternative characterization for sublinearly Morse geodesic lines via middle recurrence. We then study first passage percolation (FPP) on proper geodesic graphs of bounded degree. We associate an i.i.d. collection of random passage times to each edge. Under suitable conditions on the passage time distribution, we prove that sublinearly Morse boundaries are invariant under first passage percolation.

Folded sampling replaces clipping in analog-to-digital converters by reducing samples modulo a threshold, thereby avoiding saturation artifacts. We study the reconstruction of bandlimited functions from folded samples and show that, for equispaced sampling patterns, the recovery problem is inherently unstable. We then prove that imposing any a priori energy bound restores stability, and that this regularization effect extends to non-uniform sampling geometries. Our analysis recasts folded-sampling stability as an infinite-dimensional lattice shortest-vector problem, which we resolve via harmonic-analytic tools (the spectral profile of Fourier concentration matrices) and, alternatively, via bounds for integer Tschebyschev polynomials. Our work brings context to recent results on injectivity and encoding guarantees for folded sampling and further supports the empirical success of folded sampling under natural energy constraints.

We consider a far generalization of the well-known stable roommates and non-bipartite stable allocation problems. In its setting, one is given a finite non-bipartite graph $G=(V,E)$ with nonnegative integer edge capacities $b(e)\in{\mathbb Z}_+$, $e\in E$, in which for each vertex (``agent'') $v\in V$, the preferences on the set $E_v$ of its incident edges are given via a choice function $C_v$ acting on the vectors in ${\mathbb Z}_+^{E_v}$ bounded by the capacities and obeying the standard axioms of substitutability and size monotonicity. We refer to the related stability problem as the stable partnership problem with integer choice functions, or SPPIC for short. Extending well-known results for particular cases, we give a solvability criterion for SPPIC and develop an algorithm of finding a stable solution, called a stable partnership, or establishing that there is none. Moreover, in general the algorithm constructs a pair $(x,{\cal K})$ such that $x\in {\mathbb Z}_+^E$ and ${\cal K}$ is a set of pairwise edge-disjoint odd cycles in $G$ satisfying the following properties: if ${\cal K}=\emptyset$, then $x$ is a stable partnership, whereas if ${\cal K}$ is nonempty, then a stable partnership does not exist, and in this case, the set ${\cal K}$ is determined canonically. Our constructions essentially use earlier author's results on the corresponding bipartite counterpart of SPPIC. Keywords: stable marriage problem, stable roommates problem, stable partition, stable allocation, choice function

In this paper we give a complete description of all possible automorphism groups of real $\mathbb{R}$-rational del Pezzo surfaces $X$ of degree $4$, using the description of $X$ as the blow-up of some smooth real quadric surface $Q$ in $\mathbb{P}^{3}_{\mathbb{R}}$. We examine all possible ways to blow up $4$ geometric points on $Q$, illustrate in each case the $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$-action on the conic bundle structures on $X_{\mathbb{C}}$, and use it to give a geometric description of the real automorphism group $\operatorname{Aut}_{\mathbb{R}}(X)$ by generators in terms of automorphisms and birational automorphisms of $Q$. As a consequence, we get which finite subgroups of $\operatorname{Bir}_{\mathbb{C}}(\mathbb{P}^{2})$ can act faithfully by automorphisms on real $\mathbb{R}$-rational del Pezzo surfaces of degree $4$.

In this paper, we investigate the existence and limit behaviours of travelling solitary waves of the form $\psi(t,x)=e^{i\lambda t}\varphi\left(x-vt\right)$ to the nonlinear pseudo-relativistic Schrödinger equation \[ i\partial_t \psi=(\sqrt{-\Delta+m^2})\psi - |\psi|^{\frac{2}{N}}\psi-\mu|\psi|^{q}\psi~~\text{ on }\mathbb{R}^N, \] for $m\ge 0$ and $|v|<1$. To this end, we introduce and analyse an associated constrained variational problem, whose minimizers are termed boosted ground states and the parameter $\lambda$ is obtained as a Lagrangian multiplier. We first provide a complete classification for the existence and nonexistence of such boosted ground states. Based on this classification, we then study several limiting profiles, for which the exact blow-up rate is also established.

A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive low accuracy approaches such as lower precision computation of the implicit solve, under-resolved iterative solvers, or simpler, less accurate models for the implicit stages. Understanding the effect of the perturbation errors introduced by the low accuracy computations enables the design of stable and accurate mixed accuracy DIRK methods where the errors from the low-accuracy computation are damped out by multiplication by \dt at multiple points in the simulation, resulting in a more accurate simulation than if low-accuracy was used for all computation. To improve upon this, explicit corrections were previously proposed and analyzed for accuracy, and their performance was tested in related work. Explicit corrections work well when the time-step is sufficiently small, but may introduce instabilities when the time-step is larger. In this work, the stability of the mixed accuracy approach is carefully studied, and used to design novel stabilized correction approaches.

This article is concerned with the parabolic Monge-Ampère equation $-u_t\det D_x^2u=f$, where $f=f_1(x)f_2(t)$ and $f_1,f_2$ are positive periodic functions. We prove that any classical parabolically convex ancient solution $u$ must be of the form $-\tau t+p(x)+v(x,t)$, where $\tau$ is a positive constant, $p(x)$ is a convex quadratic polynomial, and $v$ inherits both the spatial and temporal periodicity from $f$. This work extends previous contributions by Caffarelli-Li \cite{cl04} on periodic frameworks for the elliptic Monge-Ampère equations, and generalizes Zhang-Bao \cite{zb18}'s Liouville theorem for $f_2\equiv1$ in parabolic case.

In this paper, we obtain optimal asymptotic behavior of parabolically convex $C^{2,1}$ solution to the parabolic Monge-Ampère equation $-u_t\det D_x^2u=f$, where $f$ converges to $1$ at infinity with a slow rate. This result extends the elliptic estimate in \cite{lb5} to the parabolic setting.

The Abelian sandpile model serves as a canonical example of self-organized criticality. This critical behavior manifests itself through large cascading events triggered by small perturbations. Such large-scale events, known as avalanches, are often regarded as stylized representations of catastrophic phenomena, such as earthquakes or forest fires. Motivated by this perspective, we study strategies to reduce avalanche sizes. We provide a first rigorous analysis of the impact of interventions in the Abelian sandpile model, considering a setting in which an external controller can perturb a configuration by removing sand grains at selected locations. We first develop and formalize an extended method to compute the expected size of an avalanche originating from a connected component of critical vertices, i.e., vertices at maximum height. Using this method, we characterize the structure of avalanches starting from square components and explicitly analyze the effect of interventions in such components. Our results show that the optimal intervention locations strike an interesting balance between reduction of largest avalanche sizes and increasing the number of mitigated avalanches.

Sterile Insect Technique (SIT) is widely regarded as a promising, environmentally friendly and chemical-free strategy for the prevention and control of dengue and other vector-borne diseases. In this paper, we develop and analyze a spatio-temporal reaction-diffusion model describing the dynamics of three mosquito subpopulations involved in SIT-based biological control of Aedes aegypti mosquitoes. Our sex-structured model explicitly incorporates fertile females together with fertile and sterile males that compete for mating. Its key features include spatial mosquito dispersal and the incorporation of spatially heterogeneous external releases of sterile individuals. We establish the existence and uniqueness of global, non negative, and bounded solutions, guaranteeing the mathematical well-posedness and biological consistency of the system. A fully discrete numerical scheme based on the finite element method and an implicit-explicit time-stepping scheme is proposed and analyzed. Numerical simulations confirm the presence of a critical release-size threshold governing eradication versus persistence at a stable equilibrium with reduced total population size, in agreement with the underlying ODE dynamics. Moreover, the spatial structure of the model allows us to analyze the impact of spatial distributions, heterogeneous releases, and periodic impulsive control strategies, providing insight into the optimal spatial and temporal deployment of SIT-based interventions.

We study Schrödinger operators on the real line whose potentials are generated by the Fibonacci substitution sequence and a rule that replaces symbols by compactly supported potential pieces. We consider the case in which one of those pieces is identically zero, and study the dimension of the spectrum in the large-coupling regime. Our results include a generalization of theorems regarding explicit examples that were studied previously and a counterexample that shows that the naïve generalization of previously established statements is false. In particular, in the aperiodic case, the local Hausdorff dimension of the spectrum does not necessarily converge to zero uniformly on compact subsets as the coupling constant is sent to infinity.

We prove that the intersection cohomology of the Baily-Borel compactification of a complex Shimura variety is identified with the top weight quotient of the mixed Hodge structure on the reductive Borel-Serre compactification. This yields canonical cup products and functorial pullbacks on the intersection cohomology. As an application, we introduce canonical cycle classes associated to special cycles, relating analytic geometric volumes of non-compact Shimura varieties to topological terms.

The conformal dimension of a metric space $(X, d)$ is equal to the infimum of the Hausdorff dimensions among all metric spaces quasisymmetric to $(X, d)$. It is an important quasisymmetric invariant which lies non-strictly between the topological and Hausdorff dimensions of $(X, d)$. We consider the conformal dimension of the Brownian sphere (a.k.a. the Brownian map), whose law can be thought of as the uniform measure on metric measure spaces homeomorphic to the standard sphere $\mathbf S^2$ with unit area. Since the Hausdorff dimension of the Brownian sphere is $4$, its conformal dimension lies in $[2, 4]$. Our main result is that its conformal dimension is equal to $2$, its topological dimension.

We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Ampère equation with smooth periodic data, generalizing Caffarelli-Li's result \cite{cl04} in 2004 to the parabolic background. To achieve this, we obtain a necessary and sufficient condition for the existence of the smooth periodic solution of the equation $\left(1-u_t\right)\det \left(D_x^2u+I\right)=f$ in $\mathbb{R}^{n+1}$, where $f$ is smooth and periodic in both spatial and temporal variables. This parabolic existence theorem parallels the elliptic counterpart established by Li \cite{l90} in 1990.

We study a variational problem modeling equilibrium configurations of charged liquid droplets resting on a surface under a convexity constraint. In the two-dimensional case with Coulomb interactions, we establish the validity of Young's law for the contact angle for small enough charges.

Model Predictive Path Integral (MPPI) control is a popular sampling-based method for trajectory optimization in nonlinear and nonconvex settings, yet its optimization structure remains only partially understood. We develop a variational, optimization-theoretic interpretation of MPPI by lifting constrained trajectory optimization to a KL-regularized problem over distributions and reducing it to a negative log-partition (free-energy) objective over a tractable sampling family. For a general parametric family, this yields a preconditioned gradient method on the distribution parameters and a natural multi-step extension of MPPI. For the fixed-covariance Gaussian family, we show that classical MPPI is recovered exactly as a preconditioned gradient descent step with unit step size. This interpretation enables a direct convergence analysis: under bounded feasible sets, we derive an explicit upper bound on the smoothness constant and a simple sufficient condition guaranteeing descent of exact MPPI. Numerical experiments support the theory and illustrate the effect of key hyperparameters on performance.

We study cyclic adjoint modules arising from the relative locally finite part of the adjoint action of a quantum Levi subalgebra on a quantized enveloping algebra. We analyze the realization of irreducible modules inside the quantized enveloping algebra via cyclic generators and describe embeddings of a fixed type. This leads to a natural map to isomorphism classes, whose fibers reflect the non-uniqueness of such realizations. We further introduce a partial order on cyclic adjoint modules and relate its minimal elements to irreducible submodules. In addition, we show that every cyclic adjoint module is generated by finitely many irreducible submodules.

While the mathematical foundations of score-based generative models are increasingly well understood for unconstrained Euclidean spaces, many practical applications involve data restricted to bounded domains. This paper provides a statistical analysis of reflected diffusion models on the hypercube $[0,1]^D$ for target distributions supported on $d$-dimensional linear subspaces. A primary challenge in this setting is the absence of Gaussian transition kernels, which play a central role in standard theory in $\mathbb{R}^D$. By employing an easily implementable infinite series expansion of the transition densities, we develop analytic tools to bound the score function and its approximation by sparse ReLU networks. For target densities with Sobolev smoothness $\alpha$, we establish a convergence rate in the $1$-Wasserstein distance of order $n^{-\frac{\alpha+1-\delta}{2\alpha+d}}$ for arbitrarily small $\delta > 0$, demonstrating that the generative algorithm fully adapts to the intrinsic dimension $d$. These results confirm that the presence of reflecting boundaries does not degrade the fundamental statistical efficiency of the diffusion paradigm, matching the almost optimal rates known for unconstrained settings.

We study an inverse problem for the viscoacoustic wave equation, an integro-differential model describing wave propagation in viscoacoustic media with memory in the leading order term. The medium is characterized by a spatially varying sound speed and a space-time dependent memory kernel. Assuming that waves are generated by sources supported outside the region of interest, we consider exterior measurements encoded by the source-to-solution map. To study this inverse problem, we construct solutions concentrating near fixed geodesics and establish a corresponding propagation of singularities result for the semiclassical wave front set. These results are valid without any restriction on the underlying sound speed. Then, under certain geometric conditions, we prove that the exterior data uniquely determine not just the sound speed inside the domain but also all time derivatives at zero of the memory kernel. This involves a reduction to the lens rigidity and geodesic ray transform inverse problems. As an application, we establish uniqueness for the recovery of variable parameters in the extended Maxwell model.

There is a well understood way of generating random coverings of a fixed manifold by sampling homomorphisms from the fundamental group of this manifold into the symmetric group. We prove a central limit theorem for the number of connected components of these random coverings when the fundamental group is nilpotent. This provides a nonabelian generalization of an earlier result by the author and Shannon Starr in the case of the torus where the fundamental group is a free abelian group of rank at least two. Our result relies on the work of du Sautoy and Grunewald on the subgroup growth zeta functions of nilpotent groups, and on Delange's generalization of the Wiener-Ikehara Tauberian theorem.

A countable group $G$ is said to be matricial field (MF) if it admits a strongly converging sequence of approximate homomorphisms into matrices; i.e, the norms of polynomials converge to those in the left regular representation. $G$ is then said to be purely MF (PMF) if this sequence of maps into matrices can be chosen as actual homomorphisms. $G$ is further said to be purely finite field (PFF) if the image of each homomorphism is finite. By developing a new operator algebraic approach to these problems, we are able to prove the following result bringing several new examples into the fold. Suppose $G$ is a MF (resp., PMF, PFF) group and $H<G$ is separable (i.e., $H=\cap_{i\in \mathbb{N}}H_i$ where $H_i<G$ are finite index subgroups) and $K$ is a residually finite MF (resp., PMF, PFF) group. If either $G$ or $K$ is exact, then the amalgamated free product $G*_{H}(H\times K)$ is MF (resp., PMF, PFF). Our work has several applications. Firstly, as a consequence of MF, the Brown--Douglas--Fillmore semigroups of many new reduced $C^*$-algebras are not groups. Secondly, we obtain that arbitrary graph products of residually finite exact MF (resp., PMF, PFF) groups are MF (resp., PMF, PFF), yielding a significant generalization of the breakthrough work of Magee--Thomas. Thirdly, our work resolves the open problem of proving PFF for 3-manifold groups. More generally all groups that virtually embed into RAAGs are PFF. Prior to our work, PFF was not known even in the case of free products. Our results are of geometric significance since PFF is the property that is used in Antoine Song's approach in the theory of minimal surfaces.

We prove Conjecture~2 of Bondarenko, Ortega-Cerdà, Radchenko, and Seip for the three-term recurrence attached to the Hörmander--Bernhardsson extremal function $\varphi$. More precisely, define \[ \widetilde u_{-1}=0,\qquad \widetilde u_0=1, \] and \[ \widetilde u_{n+1} = \frac{4n+2}{n+1}\bigl(n(n+1)-\lambda\bigr)\widetilde u_n + \frac{4n}{n+1}x\,\widetilde u_{n-1}. \] Then \[ \widetilde u_n(x,\lambda)\in\mathbb Z[x,\lambda] \qquad(n\ge0). \] The proof is a determinant comparison in the scaled Legendre basis. After sign reversal and central-binomial normalization, the recurrence becomes exactly the continuant recurrence of a finite tridiagonal compression. In particular, if $T_n(a,\lambda)$ denotes the $n$th BOCRS tridiagonal truncation, then \[ \widetilde u_{n+1}(a^2,\lambda)=\binom{2n+2}{n+1}\det T_n(a,\lambda). \] As consequences, we derive that \[ \left(\frac{\pi}{4C}\right)^2 \quad\text{and}\quad -\frac{L_\tau(1)}{2C} \] are not simultaneously rational, where \(C\) is the sharp point-evaluation constant for $PW^1$, $\pm\tau_n$ are the nonzero zeros of $\varphi$, and $ L_\tau(1)=\sum_{n\ge1}\frac{(-1)^n}{\tau_n}.$ Finally, if we write $\varphi(z)=\sum_{n\ge0}c_n z^{2n},$ then \[ c_n\in C^n\,\mathbb Z[\pi^2,C,L_\tau(1)] \qquad(n\ge0). \]

We study the linear-quadratic optimal control problem for infinite-dimensional dissipative systems with possibly indefinite cost functional. Under the assumption that a storage function exists, we show that this indefinite optimal control problem is equivalent to a linear-quadratic optimal control problem with a nonnegative cost functional. We establish the relationship between the corresponding value functions and present the associated operator Lur'e equation. Finally, we illustrate our results with several examples.

A conjecture of Verstraëte states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this conjecture was verified by Kühn and Osthus in 2004. We identify two counterexamples to this conjecture for $\ell = 4$ and $k=5$: the first comes from a recent construction of a dense $C_{10}$-free subgraph of the hypercube and the second from Wenger's construction for extremal $C_{10}$-free graphs.

Convolutional codes are a class of error-correcting codes that performs very well over erasure channels with low delay requirements. In particular, Maximum Distance Profile (MDP) convolutional codes, which are defined to have optimal column distances, are able to correct a maximal number of erasures in decoding windows of fixed sizes. However, the required field size in the known constructions for MDP convolutional codes increases rapidly with the code parameters. On the other hand, if the code parameters are small, larger bursts of erasures cannot be corrected. In this paper, we present a new class of convolutional codes, which we call Pseudo-MDP convolutional codes. By definition these codes can correct large bursts of erasures within a prescribed time-delay and still keep part of the advantageous properties of MDP convolutional codes, in the sense that we require some but not all column distances to be optimal. This release in the condition on the column distances allows us to construct Pseudo-MDP convolutional codes over fields of smaller size than those required for MDP convolutional codes with the same code parameters.

Given a $\vartheta$-Anosov representation into a real reductive group $G$, we construct a natural resonance spectrum associated with the representation. This spectrum is a complex analytic variety of codimension $1$ in $(\mathfrak{a}_\vartheta^*)_{\mathbb{C}}$, the complexified dual of the split component of the associated Levi group $L_\vartheta < G$. We reinterpret several objects from the theory of Anosov representations within this spectral framework and investigate, in higher rank, questions that are classically related to Ruelle-Pollicott theory in the rank-one setting. In particular, the ``leading resonance'' -- which is now a hypersurface -- is identified with the critical hypersurface of the representation. As a corollary of our work, we prove that the zeta functions and Poincaré series associated with Anosov representations admit a meromorphic extension to $(\mathfrak{a}_\vartheta^*)_{\mathbb{C}}$. We also establish sharp mixing estimates for the refraction flow under a Diophantine condition on the representation. Most of our results concerning Anosov representations are obtained as a byproduct of a general theory of free Abelian cocycles over hyperbolic flows. This article is intended as a foundational work toward more advanced results such as higher-rank quantum/classical correspondence, the detection of topological invariants of representations via the value at zero of Poincaré series or the order of vanishing of zeta functions, sharp counting results for the Lyapunov spectrum, etc.

Barren plateaus present a major challenge in the training of variational quantum algorithms (VQAs), particularly for large-scale discretizations of nonlinear partial differential equations. In this work, we introduce a domain decomposition framework to mitigate barren plateaus by localizing the cost functional. Our strategy is based on partitioning the spatial domain into overlapping subdomains, each associated with a localized parameterized quantum circuit and measurement operator. Numerical results for the time-independent Gross-Pitaevskii equation show that the domain-decomposed formulation, allowing subdomain iterations to be interleaved with optimization iterations, exhibits improved solution accuracy and stable optimization compared to the global VQA formulation.

Nonlinear Schwarz methods are a type of nonlinear domain decomposition method used as an alternative to Newton's method for solving discretized nonlinear partial differential equations. In this article, the first parallel implementation of a two-level nonlinear Schwarz method leveraging the GDSW-type coarse spaces from the Fast and Robust Overlapping Schwarz (FROSch) framework in Trilinos is presented. This framework supports both additive and hybrid two-level nonlinear Schwarz methods and makes use of modifications to the coarse spaces constructed by FROSch to further enhance the robustness and convergence speed of the methods. Efficiency and excellent parallel performance of the software framework are demonstrated by applying it to two challenging nonlinear problems: the two-dimensional lid-driven cavity problem at high Reynolds numbers, and a Neo-Hookean beam deformation problem. The results show that two-level nonlinear Schwarz methods scale exceptionally well up to 9\,000 subdomains and are more robust than standard Newton-Krylov-Schwarz solvers for the considered Navier-Stokes problems with high Reynolds numbers or, respectively, for the nonlinear elasticity problems and large deformations. The new parallel implementation provides a foundation for future research in scalable nonlinear domain decomposition methods and demonstrates the practical viability of nonlinear Schwarz techniques for large-scale simulations.

We study the problem of detecting local geometry in random graphs. We introduce a model $\mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $\mathbb{S}^{d-1}$, while all remaining edges follow the Erdős--Rényi model $\mathcal{G}(n, p)$. The random geometric graph is generated by thresholding inner products of latent vectors on $\mathbb{S}^{d-1}$, with each edge having marginal probability equal to $p$. This implies that $\mathcal{G}(n, p, d, k)$ and $\mathcal{G}(n, p)$ are indistinguishable at the level of the marginals, and the signal lies entirely in the edge dependencies induced by the local geometry. We investigate both the information-theoretic and computational limits of detection. On the information-theoretic side, our upper bounds follow from three tests based on signed triangle counts: a global test, a scan test, and a constrained scan test; our lower bounds follow from two complementary methods: truncated second moment via Wishart--GOE comparison, and tensorization of KL divergence. These results together settle the detection threshold at $d = \widetilde{\Theta}(k^2 \vee k^6/n^3)$ for fixed $p$, and extend the state-of-the-art bounds from the full model (i.e., $k = n$) for vanishing $p$. On the computational side, we identify a computational--statistical gap and provide evidence via the low-degree polynomial framework, as well as the suboptimality of signed cycle counts of length $\ell \geq 4$.

In this paper, we analyze $m$-dimensional ($m$D) convolutional codes with finite support, viewed as a natural generalization of one-dimensional (1D) convolutional codes to higher dimensions. An $m$D convolutional code with finite support consists of codewords with compact support indexed in $\mathbb{N}^m$ and taking values in $\mathbb{F}_{q}[z_1,\ldots,z_m]^n$, where $\mathbb{F}_{q}$ is a finite field with $q$ elements. We recall a natural upper bound on the free distance of an $m$D convolutional code with rate $k/n$ and degree~$\delta$, called $m$D generalized Singleton bound. Codes that attain this bound are called maximum distance separable (MDS) $m$D convolutional codes. As our main result, we develop new constructions of MDS $m$D convolutional codes based on superregularity of certain matrices. Our results include the construction of new families of MDS $mD$ convolutional codes of rate $1/n$, relying on generator matrices with specific row degree conditions. These constructions significantly expand the set of known constructions of MDS $m$D convolutional codes.

The unconditional, i.e. without assuming validity of RH, sharp limit relationship (as p tends to infinity) is found between the remainder in the modified Mertens asymptotic formula for the sums of primes' reciprocals and maximal values of Gronwall numbers G(N) among all integers whose greatest prime factor is p and which are divided by any prime q<p. The proofs are based on the properties of G(N) studied in previous author's preprints.

Given any compact connected matrix Lie group $G$ and any lattice dimension $d\ge 2$, we construct a massive Gaussian scaling limit for the $G$-valued lattice Yang-Mills-Higgs theory in the "complete breakdown of symmetry" regime. This limit arises as the lattice spacing tends to zero and the (inverse) gauge coupling constant tends to infinity sufficiently fast, causing the theory to "abelianize" and yield a Gaussian limit. This complements a recent work by Chatterjee (arXiv:2401.10507), which obtained a similar scaling limit in the special case $G= SU(2)$.

We show that the homology of strata of abelian differentials stabilizes in a range where the number of simple zeros is large relative to the homological degree. In this range, we show that the rational cohomology agrees with the restriction of the tautological classes to the stratum, and that the rational Picard group is trivial for unprojectivized strata. Our proof method is to develop an $h$-principle for these strata, valid in a range of homological degrees that increases with the number of simple zeros. The same approach also applies to higher order differentials.

We develop the first algorithms for computing the Skyscraper Invariant [FJNT24]. This is a filtration of the classical rank invariant for multiparameter persistence modules defined by the Harder-Narasimhan filtrations along every central charge supported at a single parameter value. Cheng's algorithm [Cheng24] can be used to compute HN filtrations of arbitrary acyclic quiver representations in polynomial time in the total dimension, but in practice, the large dimension of persistence modules makes this direct approach infeasible. We show that by exploiting the additivity of the HN filtration and the special central charges, one can get away with a brute-force approach. For $d$-parameter modules, this produces an FPT $\varepsilon$-approximate algorithm with runtime dominated by $O( 1/\varepsilon^d \cdot T_{\mathsc{dec}})$, where $T_{\mathsc{dec}}$ is the time for decomposition, which we compute with \textsc{aida} [DJK25]. We show that the wall-and-chamber structure of the module can be computed via lower envelopes of degree $d - 1$ polynomials. This allows for an exact computation of the Skyscraper Invariant whose runtime is roughly $O(n^d \cdot T_{\mathsc{dec}})$ for $n$ the size of the presentation of the modules and enables a faster hybrid algorithm to compute an approximation. For 2-parameter modules, we have implemented not only our algorithms but also, for the first time, Cheng's algorithm. We compare all algorithms and, as a proof of concept for data analysis, compute a filtered version of the Multiparameter Landscape for biomedical data.

The central limit theorem provides the theoretical foundation for the universality of the normal distribution: under broad conditions, the asymptotic distribution of a sum of independent random variables approaches a Gaussian. Yet, physical systems described by positive random variable -- from earthquakes to microbial growth to epidemic spreading -- consistently exhibit gamma rather than Gaussian statistics -- what leads to field-specific mechanistic explanations that are non robust to small changes in the model details. We show that gamma distributions emerge naturally from large deviation theory when Padé approximants replace polynomial expansions of the derivative of the scaled cumulant generating function, respecting positivity constraints that the central limit theorem violates. Gamma universality thus emerges as the constrained analog of Gaussian universality, providing a mechanism-free explanation for its pervasive appearance across different disciplines.

Gradient boosting, a method of building additive ensembles from weak learners, has established itself as a practical and theoretically-motivated approach to approximate functions, especially using decision tree weak learners. Comparable methods for smooth parametric learners, such as neural networks, remain less developed in both training methodology and theory. To this end, we introduce \texttt{VPBoost} ({\bf V}ariable {\bf P}rojection {\bf Boost}ing), a gradient boosting algorithm for separable smooth approximators, i.e., models with a smooth nonlinear featurizer followed by a final linear mapping. \texttt{VPBoost} fuses variable projection, a training paradigm for separable models that enforces optimality of the linear weights, with a second-order weak learning strategy. The combination of second-order boosting, separable models, and variable projection give rise to a closed-form solution for the optimal linear weights and a natural interpretation of \VPBoost as a functional trust-region method. We thereby leverage trust-region theory to prove \VPBoost converges to a stationary point under mild geometric conditions and, under stronger assumptions, achieves a superlinear convergence rate. Comprehensive numerical experiments on synthetic data, image recognition, and scientific machine learning benchmarks demonstrate that \VPBoost learns an ensemble with improved evaluation metrics in comparison to gradient-descent-based boosting and attains competitive performance relative to an industry-standard decision tree boosting algorithm.

Positive and negative flows of the Chen-Lee-Liu model and its various reductions, including Burgers hierarchy, are formulated within the framework of Riemann-Hilbert-Birkhoff decomposition with the constant grade two generator. Two classes of vacua, namely zero vacuum and constant non-zero vacuum can be realized within a centerless Heisenberg algebra. The tau functions for soliton solutions are obtained by a dressing method and vertex operators are constructed for both types of vacua. We are able to select and classify the soliton solutions in terms of the type of vertices involved. A judicious choice of vertices yields in a closed form a particular set of multi soliton solutions for the Burgers hierarchy. We develop and analyze a class of gauge-Bäcklund transformations that generate further multi soliton solutions from those obtained by dressing method by letting them interact with various integrable defects.

The main purpose of this paper is to study the Dynamical behaviors of a stochastic SIS epidemic model using mean-reverting inhomogeneous geometric brownian motion process. First we demonstrate the existence of a global-in-time solution and establish that is unique and remains positive. Then we derive a sufficient condition for exponential extinction of infectious diseases and we show that our extinction threshold in the stochastic case coincides with that of the deterministic case. Finaly, we define an appropriate theoretical framework to guarantee the existence of an ergodic stationary distribution.

Many scientific systems, such as cellular populations or economic cohorts, are naturally described by probability distributions that evolve over time. Predicting how such a system would have evolved under different forces or initial conditions is fundamental to causal inference, domain adaptation, and counterfactual prediction. However, the space of distributions often lacks the vector space structure on which classical methods rely. To address this, we introduce a general notion of parallel dynamics at a distributional level. We base this principle on parallel transport of tangent dynamics along optimal transport geodesics and call it ``Wasserstein Parallel Trends''. By replacing the vector subtraction of classic methods with geodesic parallel transport, we can provide counterfactual comparisons of distributional dynamics in applications such as causal inference, domain adaptation, and batch-effect correction in experimental settings. The main mathematical contribution is a novel notion of fanning scheme on the Wasserstein manifold that allows us to efficiently approximate parallel transport along geodesics while also providing the first theoretical guarantees for parallel transport in the Wasserstein space. We also show that Wasserstein Parallel Trends recovers the classic parallel trends assumption for averages as a special case and derive closed-form parallel transport for Gaussian measures. We deploy the method on synthetic data and two single-cell RNA sequencing datasets to impute gene-expression dynamics across biological systems.

Adapting large-scale foundation models to new domains with limited supervision remains a fundamental challenge due to latent distribution mismatch, unstable optimization dynamics, and miscalibrated uncertainty propagation. This paper introduces an uncertainty-aware probabilistic latent transport framework that formulates domain adaptation as a stochastic geometric alignment problem in representation space. A Bayesian transport operator is proposed to redistribute latent probability mass along Wasserstein-type geodesic trajectories, while a PAC-Bayesian regularization mechanism constrains posterior model complexity to mitigate catastrophic overfitting. The proposed formulation yields theoretical guarantees on convergence stability, loss landscape smoothness, and sample efficiency under distributional shift. Empirical analyses demonstrate substantial reduction in latent manifold discrepancy, accelerated transport energy decay, and improved covariance calibration compared with deterministic fine-tuning and adversarial domain adaptation baselines. Furthermore, bounded posterior uncertainty evolution indicates enhanced probabilistic reliability during cross-domain transfer. By establishing a principled connection between stochastic optimal transport geometry and statistical generalization theory, the proposed framework provides new insights into robust adaptation of modern foundation architectures operating in heterogeneous environments. These findings suggest that uncertainty-aware probabilistic alignment constitutes a promising paradigm for reliable transfer learning in next-generation deep representation systems.

Physics-informed neural networks (PINNs) are increasingly used in mathematical epidemiology to bridge the gap between noisy clinical data and compartmental models, such as the susceptible-exposed-infected-removed (SEIR) model. However, training these hybrid networks is often unstable due to competing optimization objectives. As established in recent literature on ``gradient pathology," the gradient vectors derived from the data loss and the physical residual often point in conflicting directions, leading to slow convergence or optimization deadlock. While existing methods attempt to resolve this by balancing gradient magnitudes or projecting conflicting vectors, we propose a novel method, conflict-gated gradient scaling (CGGS), to address gradient conflicts in physics-informed neural networks for epidemiological modelling, ensuring stable and efficient training and a computationally efficient alternative. This method utilizes the cosine similarity between the data and physics gradients to dynamically modulate the penalty weight. Unlike standard annealing schemes that only normalize scales, CGGS acts as a geometric gate: it suppresses the physical constraint when directional conflict is high, allowing the optimizer to prioritize data fidelity, and restores the constraint when gradients align. We prove that this gating mechanism preserves the standard $O(1/T)$ convergence rate for smooth non-convex objectives, a guarantee that fails under fixed-weight or magnitude-balanced training when gradients conflict. We demonstrate that this mechanism autonomously induces a curriculum learning effect, improving parameter estimation in stiff epidemiological systems compared to magnitude-based baselines. Our empirical results show improved peak recovery and convergence over magnitude-based methods.

There remain theoretical gaps in deep neural network estimators for the nonparametric Cox proportional hazards model. In particular, it is unclear how gradient-based optimization error propagates to population risk under partial likelihood, how pointwise bias can be controlled to permit valid inference, and how ensemble-based uncertainty quantification behaves under realistic variance decay regimes. We develop an asymptotic distribution theory for deep Cox estimators that addresses these issues. First, we establish nonasymptotic oracle inequalities for general trained networks that link in-sample optimization error to population risk without requiring the exact empirical risk optimizer. We then construct a structured neural parameterization that achieves infinity-norm approximation rates compatible with the oracle bound, yielding control of the pointwise bias. Under these conditions and using the Hajek--Hoeffding projection, we prove pointwise and multivariate asymptotic normality for subsampled ensemble estimators. We derive a range of subsample sizes that balances bias correction with the requirement that the Hajek--Hoeffding projection remain dominant. This range accommodates decay conditions on the single-overlap covariance, which measures how strongly a single shared observation influences the estimator, and is weaker than those imposed in the subsampling literature. An infinitesimal jackknife representation provides analytic covariance estimation and valid Wald-type inference for relative risk contrasts such as log-hazard ratios. Finally, we illustrate the finite-sample implications of the theory through simulations and a real data application.

Symbolic discovery of governing equations is a long-standing goal in scientific machine learning, yet a fundamental trade-off persists between interpretability and scalable learning. Classical symbolic regression methods yield explicit analytic expressions but rely on combinatorial search, whereas neural networks scale efficiently with data and dimensionality but produce opaque representations. In this work, we introduce Symbolic Kolmogorov-Arnold Networks (Symbolic-KANs), a neural architecture that bridges this gap by embedding discrete symbolic structure directly within a trainable deep network. Symbolic-KANs represent multivariate functions as compositions of learned univariate primitives applied to learned scalar projections, guided by a library of analytic primitives, hierarchical gating, and symbolic regularization that progressively sharpens continuous mixtures into one-hot selections. After gated training and discretization, each active unit selects a single primitive and projection direction, yielding compact closed-form expressions without post-hoc symbolic fitting. Symbolic-KANs further act as scalable primitive discovery mechanisms, identifying the most relevant analytic components that can subsequently inform candidate libraries for sparse equation-learning methods. We demonstrate that Symbolic-KAN reliably recovers correct primitive terms and governing structures in data-driven regression and inverse dynamical systems. Moreover, the framework extends to forward and inverse physics-informed learning of partial differential equations, producing accurate solutions directly from governing constraints while constructing compact symbolic representations whose selected primitives reflect the true analytical structure of the underlying equations. These results position Symbolic-KAN as a step toward scalable, interpretable, and mechanistically grounded learning of governing laws.

Semantic communication has emerged as a promising paradigm for improving transmission efficiency and task-level reliability, yet most existing reliability-enhancement approaches rely on retransmission strategies driven by semantic fidelity checking that require additional check codewords solely for retransmission triggering, thereby incurring substantial communication overhead. In this paper, we propose S3CHARQ, a Joint Source-Channel-Check Coding framework with hybrid automatic repeat request that fundamentally rethinks the role of check codewords in semantic communications. By integrating the check codeword into the JSCC process, S3CHARQ enables JS3C, allowing the check codeword to simultaneously support semantic fidelity verification and reconstruction enhancement. At the transmitter, a semantic fidelity-aware check encoder embeds auxiliary reconstruction information into the check codeword. At the receiver, the JSCC and check codewords are jointly decoded by a JS3C decoder, while the check codeword is additionally exploited for perceptual quality estimation. Moreover, because retransmission decisions are necessarily based on imperfect semantic quality estimation in the absence of ground-truth reconstruction, estimation errors are unavoidable and fundamentally limit the effectiveness of rule-based decision schemes. To overcome this limitation, we develop a reinforcement learning-based retransmission decision module that enables adaptive, sample-level retransmission decisions, effectively balancing recovery and refinement information under dynamic channel conditions. Experimental results demonstrate that compared with existing HARQ-based semantic communication systems, the proposed S3CHARQ framework achieves a 2.36 dB improvement in the 97th percentile PSNR, as well as a 37.45% reduction in outage probability.

Galaxy surveys provide finite catalogs of objects observed within bounded volumes, yet clustering statistics are often interpreted using theoretical frameworks developed for infinite point processes. In this work, we formulate key statistical quantities directly for finite point processes and examine the structural consequences of finite-number and finite-window constraints. We show that several well-known features of galaxy survey analysis arise naturally from finiteness alone. In particular, non-vanishing higher-order connected correlations can occur even in statistically independent samples when the total number of points is fixed, and the integral constraint in two-point statistics appears as an exact identity implied by the finite-number condition rather than as an estimator artifact. We further demonstrate that counts-in-cells and point-centered environmental measures correspond to distinct statistical ensembles. Using Palm conditioning, we derive an exact relation between random-cell and point-centered statistics, showing that the latter probe a tilted version of the underlying distribution. These results provide a probabilistic framework for separating structural effects imposed by finite sampling from correlations reflecting genuine astrophysical processes. The formulation presented here remains valid for realistic survey geometries and finite data sets and clarifies the interpretation of commonly used clustering statistics in galaxy surveys.

Graph states are a key resource for measurement-based quantum computation and quantum networking, but state-preparation costs limit their practical use. Graph states related by local complement (LC) operations are equivalent up to single-qubit Clifford gates; one may reduce entangling resources by preparing a favorable LC-equivalent representative. However, exhaustive optimization over the LC orbit is not scalable. We address this problem using the split decomposition and its quotient-augmented strong split tree (QASST). For several families of distance-hereditary (DH) graphs, we use the QASST to characterize LC orbits and identify representatives with reduced controlled-Z count or preparation circuit depth. We also introduce a split-fuse construction for arbitrary DH graph states, achieving linear scaling with respect to entangling gates, time steps, and auxiliary qubits. Beyond the DH setting, we discuss a generalized divide-and-conquer split-fuse strategy and a simple greedy heuristic for generic graphs based on triangle enumeration. Together, these methods outperform direct implementations on sufficiently large graphs, providing a scalable alternative to brute-force optimization.

Online reinforcement learning in infinite-horizon Markov decision processes (MDPs) remains less theoretically and algorithmically developed than its episodic counterpart, with many algorithms suffering from high ``burn-in'' costs and failing to adapt to benign instance-specific complexity. In this work, we address these shortcomings for two infinite-horizon objectives: the classical average-reward regret and the $\gamma$-regret. We develop a single tractable UCB-style algorithm applicable to both settings, which achieves the first optimal variance-dependent regret guarantees. Our regret bounds in both settings take the form $\tilde{O}( \sqrt{SA\,\text{Var}} + \text{lower-order terms})$, where $S,A$ are the state and action space sizes, and $\text{Var}$ captures cumulative transition variance. This implies minimax-optimal average-reward and $\gamma$-regret bounds in the worst case but also adapts to easier problem instances, for example yielding nearly constant regret in deterministic MDPs. Furthermore, our algorithm enjoys significantly improved lower-order terms for the average-reward setting. With prior knowledge of the optimal bias span $\Vert h^\star\Vert_\text{sp}$, our algorithm obtains lower-order terms scaling as $\Vert h^\star\Vert_\text{sp} S^2 A$, which we prove is optimal in both $\Vert h^\star\Vert_\text{sp}$ and $A$. Without prior knowledge, we prove that no algorithm can have lower-order terms smaller than $\Vert h^\star \Vert_\text{sp}^2 S A$, and we provide a prior-free algorithm whose lower-order terms scale as $\Vert h^\star\Vert_\text{sp}^2 S^3 A$, nearly matching this lower bound. Taken together, these results completely characterize the optimal dependence on $\Vert h^\star\Vert_\text{sp}$ in both leading and lower-order terms, and reveal a fundamental gap in what is achievable with and without prior knowledge.

We suggest a tight inequality for norms in $d$-dimensional space $l_p $ which has simple formulation but appears hard to prove. We give a proof for $d=3$ and provide a detailed numerical check for $d\leq 200$ confirming the conjecture. We conclude with a brief survey of solutions for kin problems which anyhow concern minimization of the output entropy of certain quantum channel and rely upon the symmetry properties of the problem. Key words and phrases: $l_p $-norm, Rényi entropy, tight inequality, maximization of a convex function.

We investigate the extension of pure-state symmetry protected topological phases to mixed-state regime with a strong U(1) and a weak $\mathbb{Z}_2$ symmetries in one-dimensional spin systems by the concept of quantum channels. We propose a corresponding topological phase order parameter for short-range entangled mixed states by showing that it is quantized and its distinct values can be realized by concrete spin systems with disorders, sharply signaling phase transitions among them. We also give a model-independent way to generate two distinct phases by various types of translation and reflection transformations. These results on the short-range entangled mixed states further enable us to generalize the conventional Lieb-Schultz-Mattis theorem to mixed states, even without the concept of spectral gaps and lattice Hamiltonians.

Neural operator learning directly constructs the mapping relationship from the equation parameter space to the solution space, enabling efficient direct inference in practical applications without the need for repeated solution of partial differential equations (PDEs) - an advantage that is difficult to achieve with traditional numerical methods. In this work, we find that explicitly decoupling linear and nonlinear effects within such operator mappings leads to markedly improved learning efficiency. This yields a novel network structure, namely the Linear-Nonlinear Fusion Neural Operator (LNF-NO), which models operator mappings via the multiplicative fusion of a linear component and a nonlinear component, thus achieving a lightweight and interpretable representation. This linear-nonlinear decoupling enables efficient capture of complex solution features at the operator level while maintaining stability and generality. LNF-NO naturally supports multiple functional inputs and is applicable to both regular grids and irregular geometries. Across a diverse suite of PDE operator-learning benchmarks, including nonlinear Poisson-Boltzmann equations and multi-physics coupled systems, LNF-NO is typically substantially faster to train than Deep Operator Networks (DeepONet) and Fourier Neural Operators (FNO), while achieving comparable or better accuracy in most cases. On the tested 3D Poisson-Boltzmann case, LNF-NO attains the best accuracy among the compared models and trains approximately 2.7x faster than a 3D FNO baseline.

At the end, the house always wins! This simple truth holds for all public games of chance. Nevertheless, since lotteries have existed, people have tried everything to give luck a helping hand. This article compares objective scientific approaches to tackle the 6/49 lottery: probabilistic methods and combinatorial designs. The mathematical models developed herein can be modified and applied to other lotteries. Additionally, this work introduces the newly constructed (49, 6, 5) covering design, which meets the Schönheim bound. For lottery designs and for covering designs, a benchmark based on probabilistic methods is presented. It is demonstrated that common attempts to outwit the odds correspond to limitations of numbers to subsets, which disproportionately reduce the chances of winning.

In this paper, we introduce a novel model for the meta-analysis of proportions that integrates the standard random-effects model (REM) with an extreme value theory (EVT)-based component. The proposed model, named XT-REM (Extreme-Tail Random Effects Model), extends the classical REM framework by explicitly accounting for extreme proportions through a partial segmentation of the study set based on a predefined threshold. While the majority of proportions are modeled using REM, proportions exceeding the threshold are analyzed using the Generalized Pareto Distribution (GPD). This formulation enables a dual interpretation of meta-analytic results, providing both an aggregate estimate for the central bulk of studies and a separate characterization of tail behavior. The XT-REM framework accommodates heteroskedastic variance structures inherent to proportion data, while preserving identifiability and consistency. Using real-world data on immunotherapy-related adverse events, together with simulation studies calibrated to empirical settings, we demonstrate that XT-REM yields a comparable central estimate while enabling a more explicit assessment of tail behavior, including high-percentile extreme proportions. Compared with the classical REM, XT-REM achieves higher log-likelihood values and lower AIC, in the considered scenarios, indicating a better fit within this modeling framework. In summary, XT-REM offers a theoretically grounded and practically useful extension of random-effects meta-analysis, with potential relevance to clinical contexts in which extreme event rates carry important implications for risk assessment.

Survival analysis provides a well-established framework for modeling time-to-event data, with hazard and survival functions formally defined as population-level quantities. In applied work, however, these quantities are often interpreted as representing individual-level risk, despite the absence of a clear generative account linking individual risk mechanisms to observed survival data. This paper develops a latent hazard framework that makes this relationship explicit by modeling event times as arising from unobserved, individual-specific hazard mechanisms and viewing population-level survival quantities as aggregates over heterogeneous mechanisms. Within this framework, we show that individual hazard trajectories are not identifiable from survival data under partial information. More generally, the conditional distribution of latent hazard mechanisms given covariates is structurally non-identifiable, even when population-level survival functions are fully known. This non-identifiability arises from the aggregation inherent in survival data and persists independently of model flexibility or estimation strategy. Finally, we show that classical survival models can be systematically reinterpreted according to how they handle this unresolved conditional mechanism distribution. This paper provides a unified framework for understanding heterogeneity, identifiability, and interpretation in survival analysis, and clarifies how population-level survival models should be interpreted when individual risk mechanisms are only partially observed, thereby establishing explicit information constraints for principled modeling and inference.

We classify $1+1$d bosonic SPT phases with non-invertible symmetry $\mathrm{Rep}(G)\times G$, equivalently the fusion-category symmetry $\mathcal{H}=\mathrm{Rep}(G)\times\mathrm{Vec}_G$. Focusing on \emph{intrinsically mixed} phases (trivial under either factor alone), we use the correspondence between $\mathcal{H}$-SPTs, $\mathcal{H}$-modules over $\mathrm{Vec}$, and fiber functors $\mathcal{H}\to\mathrm{Vec}$ to obtain a complete classification: such phases are parametrized by $\phi\in\operatorname{End}(G)$. For each $\phi$ we identify the associated condensable (Lagrangian) algebra $\mathcal{A}_\phi$ in the bulk $\mathcal{Z}(\mathcal{H})\simeq\mathcal{D}_G^2$. We further provide an explicit lattice realization by modifying Kitaev's quantum double model with a domain wall $\mathcal{B}_\phi$ and smooth/rough boundaries, and then contracting to a 1D chain, yielding a (possibly twisted) group-based cluster state whose ribbon-generated symmetry operators encode the same $\phi$.

We study two-dimensional periodic metamaterials in which idealised cylindrical inclusions are modelled by boundary conditions. In the scalar time-harmonic setting, the background field satisfies the Helmholtz equation, and high-contrast inclusion limits reduce to Dirichlet or Neumann conditions, with direct analogues in dielectric and acoustic media. By switching the condition assigned to selected inclusions, we break point-group symmetries of the primitive cell and thereby lift symmetry-induced degeneracies in the Floquet--Bloch spectrum of hexagonal and square lattices, opening valley-type band gaps with Berry curvature localised near opposite valleys. To analyse infinite and finite structures within a unified framework, we derive matched-asymptotic point-scatterer approximations for mixed Dirichlet--Neumann arrays. For doubly periodic systems, this yields a finite-dimensional generalised eigenvalue problem for the Floquet--Bloch spectrum; for finite arrays, it yields a generalised Foldy multiple-scattering system. In both hexagonal and square lattices, geometrically identical crystals can realise distinct valley-Hall phases solely through boundary-condition assignment while retaining an overlapping bulk gap. Spatially varying this assignment therefore creates and relocates internal interfaces without altering the underlying geometry, enabling the associated valley-Hall interfacial modes to be repositioned within the same crystal.

Anisotropic two-dimensional diffraction signals contain more information than the conventional isotropic signals for both gas phase ultrafast electron and X-ray diffraction experiments and are common in typical time-resolved diffraction experiments due to the use of linearly polarized lasers to excite the sample that imprints spatial anisotropy on the molecules. We report an iterative algorithm to restore the missing data at low scattering angles in a two-dimensional diffraction signal, which is essential to obtain real-space representation. The iterative algorithm transforms two-dimensional signals back and forth between the momentum transfer domain and the real space domain through Fourier and Abel transforms and apply real space constraints to retrieve missing signal at low scattering angles. The algorithm only requires an approximate a-priori knowledge of the shortest and longest internuclear distances in the molecule. We demonstrated successful retrieval of the missing signal in simulated patterns and in experimentally measured diffraction patterns from laser-induced alignment of trifluoroiodomethane molecules.

In this work, we continue the investigation of correlation numbers in $\mathcal{N}=1$ super Minimal Liouville Gravity (SMLG), with physical fields in the Ramond sector. Building upon our previous construction of physical operators and the evaluation of three-point correlation functions involving Ramond and Neveu-Schwarz (NS) insertions, we now turn to the analytic computation of four-point correlation numbers. This development is motivated by the framework established for the bosonic Minimal Liouville Gravity and its supersymmetric NS analog, where the integration over moduli space in correlation functions can be performed explicitly using the higher equations of motion (HEM) in Liouville theory. In particular, if one of the insertions corresponds to a degenerate field, the four-point amplitude can be expressed in terms of boundary contributions obtained from the OPE structure of logarithmic counterparts of ground ring elements. We aim to adapt and generalize this approach to the Ramond this http URL result is a closed-form analytic expression for four-point correlation numbers involving Ramond fields.

Accurate modeling of ultrasound wave propagation is essential for high-fidelity simulation and imaging in ultrasonic testing. A primary challenge lies in characterizing the excitation source, particularly for transducers with large apertures relative to the acoustic wavelengths. In such cases, non-uniform excitation and spatial interference significantly affect the resulting radiation patterns. This paper proposes a distributed source inversion strategy to reconstruct an effective spatio-temporal transducer model that reproduces experimentally measured wavefields. The reconstructed source model captures aperture-dependent phase and amplitude variations without the need for detailed knowledge of the transducer structure. The approach is validated using directivity measurements on an aluminum half-cylinder, where simulations incorporating the reconstructed source model show close agreement with experimental directivity patterns and waveform shapes. Finally, synthetic studies on reverse time migration and full-waveform inversion demonstrate that accurate transducer modeling is critical for the success of simulation-based imaging and inversion workflows and significantly improves reconstruction quality.

A recurring debate in the philosophy of statistics concerns what, exactly, should count as a measure of evidence for or against a given hypothesis. P-values, likelihood ratios, and Bayes factors all have their defenders. In this paper we add two additional candidates to this list: the e-value and its sequential analogue, the e-process. E-values enjoy several desirable properties as measures of evidence: they combine naturally across studies, handle composite hypotheses, provide long-run error rates, and admit a useful interpretation as the wealth accrued by a bettor in a game against the null distribution. E-processes additionally handle optional stopping and optional continuation. This work examines the extent to which e-values and e-processes satisfy the evidential desiderata of different statistical traditions, concluding that they combine attractive features of p-values, likelihood ratios, and Bayes factors, and merit serious consideration as interpretable and intuitive measures of statistical evidence.

Following the Bethe ansatz we determine the dynamical spectra of the one-dimensional supersymmetric t-J model. A series of fractionalized excitations are identified through two sets of Bethe numbers. Typical patterns in each set are found to yield wavefunctions containing elementary spin and charge carriers, manifested as distinct boundaries of the collective excitations in the spectra of single electron Green functions. In spin channels, gapless excitations fractionalized into two spin and a pair of postive and negative charge carriers, extending to finite energy as multiple continua. These patterns connect to the half-filling limit where only fractionalized spinons survive. In particle density channel, apart from spin-charge fractionalization, excitations involving only charge fluctuations are observed. Furthermore, nontrivial Bethe strings encoding bound state structure appear in channels of reducing or conserving magnetization, where spin and charge constituents can also be identified. These string states contribute significantly even to the low-energy sector in the limit of vanishing magnetization.

Uniform laws of large numbers form a cornerstone of Vapnik--Chervonenkis theory, where they are characterized by the finiteness of the VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces, under assumptions on the underlying distribution that are compatible with the product structure. Specifically, we assume that the distribution is absolutely continuous with respect to the product of its marginals, a condition that captures many natural settings, including product distributions, sparse mixtures of product distributions, distributions with low mutual information, and more. We show that, under this assumption, a uniform law of large numbers holds for a family of events if and only if the linear VC dimension of the family is finite. The linear VC dimension is defined as the maximum size of a shattered set that lies on an axis-parallel line, namely, a set of vectors that agree on all but at most one coordinate. This dimension is always at most the classical VC dimension, yet it can be arbitrarily smaller. For instance, the family of convex sets in $\mathbb{R}^d$ has linear VC dimension $2$, while its VC dimension is infinite already for $d\ge 2$. Our proofs rely on estimator that departs substantially from the standard empirical mean estimator and exhibits more intricate structure. We show that such deviations from the standard empirical mean estimator are unavoidable in this setting. Throughout the paper, we propose several open questions, with a particular focus on quantitative sample complexity bounds.

The freight industry is undergoing a digital revolution, with an ever-growing volume of transactions being facilitated by digital marketplaces. A core capability of these marketplaces is the fulfillment of demand for truckload movements (loads) by procuring the services of carriers who execute them. Notably, these services are procured both through long-term contracts, where carriers commit capacity to execute loads (e.g., contracted fleet of drivers or lane-level commitments), and through short-term spot marketplaces, where carriers can agree to move individual loads for the offered price. This naturally couples two canonical problems of the transportation industry: contract assignment and spot pricing. In this work, we model and analyze the problem of coordinating long-term contract supply and short-term spot supply to minimize total procurement costs. We develop a Dual Frank Wolfe algorithm to compute shadow prices which allow the spot pricing policy to account for the committed contract capacity. We show that our algorithm achieves small relative regret against the optimal -- but intractable -- dynamic programming benchmark when the size of the market is large. Importantly, our Dual Frank Wolfe algorithm is computationally efficient, modular, and only requires oracle access to spot-pricing protocols, making it ideal for large-scale markets. Finally, we evaluate our algorithm on semi-synthetic data from a major Digital Freight Marketplace, and find that it yields significant savings ($\approx 10\%$) compared to a popular status-quo method.

We construct nested Calderbank-Shor-Steane code pairs with non-vanishing coding rate from Hsu-Anastasopoulos codes and MacKay-Neal codes. In the fixed-degree regime, we prove relative linear distance with high probability. Moreover, for several finite degree settings, we prove Gilbert-Varshamov distance by a rigorous computer-assisted proof.

We introduce the Multilevel Euler-Maruyama (ML-EM) method compute solutions of SDEs and ODEs using a range of approximators $f^1,\dots,f^k$ to the drift $f$ with increasing accuracy and computational cost, only requiring a few evaluations of the most accurate $f^k$ and many evaluations of the less costly $f^1,\dots,f^{k-1}$. If the drift lies in the so-called Harder than Monte Carlo (HTMC) regime, i.e. it requires $\epsilon^{-\gamma}$ compute to be $\epsilon$-approximated for some $\gamma>2$, then ML-EM $\epsilon$-approximates the solution of the SDE with $\epsilon^{-\gamma}$ compute, improving over the traditional EM rate of $\epsilon^{-\gamma-1}$. In other terms it allows us to solve the SDE at the same cost as a single evaluation of the drift. In the context of diffusion models, the different levels $f^{1},\dots,f^{k}$ are obtained by training UNets of increasing sizes, and ML-EM allows us to perform sampling with the equivalent of a single evaluation of the largest UNet. Our numerical experiments confirm our theory: we obtain up to fourfold speedups for image generation on the CelebA dataset downscaled to 64x64, where we measure a $\gamma\approx2.5$. Given that this is a polynomial speedup, we expect even stronger speedups in practical applications which involve orders of magnitude larger networks.

In this article, we prove commutativity principal for linear, symplectic and transvection groups. This principle is a consequence of Quillen-Suslin local global principle and using a non-symmetric application of it as done by A. Bak. The existence of a Local-Global Principle enables us to prove similar results in various groups. We restrict ourselves to the classical symplectic, orthogonal groups (and their relative versions); and to the automorphism groups of a projective module (with a unimodular element), a symplectic module (with ahyperbolic summand), and an orthogonal module (with a hyperbolic symmand). We could show that the symplectic quotients were abelian, but we could only establish that the orthogonal quotients are solvable of length atmost two. We do believe that the orthogonal quotient groups are also abelian; and prove this when the base ring is a regular local ring containing a field.

We show that the mod $\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\ell$ admits the structure of a module over the mod $\ell$ cohomology of the free loop space of the classifying space $BG$ of the corresponding compact Lie group $G$, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain class in the homology of the finite group of Lie type, arising from the fundamental class of $G$, is nontrivial, then this module structure is free of rank one, providing a highly structured isomorphism between the two cohomologies. We verify the nontriviality of the class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $\ell$. We also show how to deal with twistings and avoid the congruence condition by replacing $BG$ by a certain $\ell$-compact fixed point group depending on the order of $q$ mod $\ell$, without changing the finite group. With this modification, we know of no examples where the class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings.

In this paper, we study limiting laws and consistent estimation criteria for the extreme eigenvalues in a spiked covariance model of dimension $p$. Firstly, for fixed $p$, we propose a generalized estimation criterion that can consistently estimate, $k$, the number of spiked eigenvalues. Compared with the existing literature, we show that consistency can be achieved under weaker conditions on the penalty term. Next, allowing both $p$ and $k$ to diverge, we derive limiting distributions of the spiked sample eigenvalues using random matrix theory techniques. Notably, our results do not require the spiked eigenvalues to be uniformly bounded from above or tending to infinity, as have been assumed in the existing literature. Based on the above derived results, we formulate a generalized estimation criterion and show that it can consistently estimate $k$, while $k$ can be fixed or grow at an order of $k=o(n^{1/3})$. We further show that the results in our work continue to hold under a general population distribution without assuming normality. The efficacy of the proposed estimation criteria is illustrated through comparative simulation studies.

In this work, we consider the Cauchy problem for a diffusive Oldroyd-B model in three dimensions. Some optimal time-decay rates of the solutions are derived via analysis of upper and lower time-decay estimates provided that the initial data are small and that the absolute value of Fourier transform of the initial velocity is bounded below away from zero in a low-frequency region. It is worth noticing that the optimal rates are independent of the fluid viscosity or the diffusive coefficient, which is a different phenomenon from that for incompressible Navier-Stokes equations.

In this note we show how to construct a number of nonconvex quadratic inequalities for a variety of physics equations appearing in physical design problems. These nonconvex quadratic inequalities can then be used to construct bounds on physical design problems where the objective is a quadratic or a ratio of quadratics. We show that the quadratic inequalities and the original physics equations are equivalent under a technical condition that holds in many practical cases which is easy to computationally (and, in some cases, manually) verify.

In this paper, we will provide the the finite element method for the electro-osmotic flow in micro-channels, in which a convection-diffusion type equation is given for the charge density $\rho^e$. A time-discrete method based on the backward Euler method is designed. The theoretical analysis shows that the numerical algorithm is unconditionally stable and has optimal convergence rates. To show the effectiveness of the proposed model, some numerical results for the electro-osmotic flow in the T-junction micro-channels and in rough micro-channels are provided. Numerical results indicate that the proposed numerical method is suitable for simulating electro-osmotic flows.

Given a formally integrable almost complex structure $X$ defined on the closure of a bounded domain $D \subset \mathbb C^n$, and provided that $X$ is sufficiently close to the standard complex structure, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism defined on $\overline D$ that transforms $X$ into the standard complex structure, under certain geometric and regularity assumptions on $D$. In this paper we prove a quantitative result of this problem. Assuming $D$ is a strictly pseudoconvex domain in $\mathbb C^n$ with $C^2$ boundary, and that the almost structure $X$ is of the Hölder-Zygmund class $\Lambda^r(\overline D)$ for $r>\frac{3}{2}$, we prove the existence of a global diffeomorphism (independent of $r$) in the class $\Lambda^{r+\frac12-\varepsilon}(\overline D)$, for any $\varepsilon>0$.

This paper concerns the lattice counting problem for the mapping class group of a surface $S$ acting on Teichmüller space with the Teichmüller metric. In that problem the goal is to count the number of mapping classes that send a given point $x$ into the ball of radius $R$ centered about another point $y$. For the action of the entire group, Athreya, Bufetov, Eskin and Mirzakhani have shown this quantity is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichmüller space. We refine the problem by considering the action various distinguished subsets of elements and counting these separately. For the set of finite-order elements, we show the associated count grows coarsely at the rate of $e^{hR/2}$, that is, with half the exponent. For the reducible elements, the associated count grows coarsely at the rate of $e^{(h-1)R}$. Finally, for the set of all multitwists, the coarse growth rate is also $e^{hR/2}$. To obtain these quantitative estimates, we introduce a new notion in Teichmüller geometry, called complexity length, which reflects some aspects of the negative curvature of curve complexes and also has applications to counting problems.

We study the geometric particle-in-cell methods for an electrostatic hybrid plasma model. In this model, ions are described by the fully kinetic equations, electron density is determined by the Boltzmann relation, and space-charge effects are incorporated through the Poisson equation. By discretizing the action integral or the Poisson bracket of the hybrid model, we obtain a finite dimensional Hamiltonian system, for which the Hamiltonian splitting methods or the discrete gradient methods can be used to preserve the geometric structure or energy. The global neutrality condition is conserved under suitable boundary conditions. Moreover, the results are further developed for an electromagnetic hybrid model proposed in [Vu H X. J Comput Phys, 124(2):417-430]. Numerical experiments of finite grid instability, Landau damping, and resonantly excited nonlinear ion waves illustrate the behaviour of the proposed numerical methods.

We prove several rigidity results on multiplier spectrum and length spectrum. For example, we show that for every non-exceptional rational map $f:\mathbb{P}^1(\mathbb{C})\to\mathbb{P}^1(\mathbb{C})$ of degree $d\geq2$, the $\mathbb{Q}$-vector space generated by all the (finite) characteristic exponents of periodic points of $f$ has infinite dimension. This answers a stronger version of a question of Levy and Tucker. Our result can also be seen as a generalization of recent results of Ji-Xie and of Huguin which proved Milnor's conjecture about rational maps having integer multipliers. We also get a characterization of postcritically finite maps by using its length spectra. Finally as an application of our result, we get a new proof of the Zariski-dense orbit conjecture for endomorphisms on $(\mathbb{P}^1)^N, N\geq 1$.

We establish sharp nonlinear stability results for fronts that describe the creation of a periodic pattern through the invasion of an unstable state. The fronts we consider are critical, in the sense that they are expected to mediate pattern selection from compactly supported or steep initial data. We focus on pulled fronts, that is, on fronts whose propagation speed is determined by the linearization about the unstable state in the leading edge, only. We present our analysis in the specific setting of the FitzHugh-Nagumo system, where pattern-forming uniformly translating fronts have recently been constructed rigorously, but our methods can be used to establish nonlinear stability of pulled pattern-forming fronts in general reaction-diffusion systems. This is the first stability result of critical pattern-selecting fronts and provides a rigorous foundation for heuristic, universal wave number selection laws in growth processes based on a marginal stability conjecture. The main technical challenge is to describe the interaction between two separate modes of marginal stability, one associated with the spreading process in the leading edge, and one associated with the pattern in the wake. We develop tools based on far-field/core decompositions to characterize, and eventually control, the interaction between these two different types of diffusive modes. Linear decay rates are insufficient to close a nonlinear stability argument and we therefore need a sharper description of the relaxation in the wake of the front using a phase modulation ansatz. We control regularity in the resulting quasilinear equation for the modulated perturbation using nonlinear damping estimates.

We show that every element of $\mathrm{SL}_{n}(\mathbb{Z}/q\mathbb{Z})$ can be lifted to an element of $\mathrm{SL}_{n}(\mathbb{Z})$ of norm at most $Cq^2\log q$, while there exists an element such that every lift of it is of norm at least $q^{2+o(1)}$. This should be compared to the recent result that almost every element has a lift of norm bounded by $q^{1+1/n+o(1)}$. The main step in the proof is showing that for every $q$, there is a small element in $(\mathbb{Z}/q\mathbb{Z})^\times$ with a large $n$-th root, which is a result of independent interest.

The simplex algorithm is one of the most popular algorithms to solve linear programs (LPs). Starting at an extreme point solution of an LP, it performs a sequence of basis exchanges (called pivots) that allows one to move to a better extreme point along an improving edge-direction of the underlying polyhedron. A key issue in the simplex algorithm's performance is degeneracy, which may lead to a (potentially long) sequence of basis exchanges which do not change the current extreme point solution. In this paper, we prove that one can employ any improving feasible direction at an extreme point to limit the number of consecutive degenerate pivots that the simplex algorithm performs to $n-m-1$, where $n$ is the number of variables and $m$ is the number of equality constraints of a given LP in standard equality form.

We present a new analytical framework on the uplink data detection for massive multiple-input multiple-output systems with 1-bit analog-to-digital converters (ADCs). We first characterize the expected values of the soft-estimated symbols (after the linear receiver and prior to the data detection), which are affected by the 1-bit quantization during both the channel estimation and the uplink data transmission. In our analysis, we consider conventional receivers such as maximum ratio combining (MRC), zero forcing, and minimum mean squared error (MMSE), with multiple user equipments (UEs) and correlated Rayleigh fading. Additionally, we design a linear minimum mean dispersion (LMMD) receiver tailored for the data detection with 1-bit ADCs, which exploits the expected values of the soft-estimated symbols previously derived. Then, we propose a joint data detection (JD) strategy that exploits the interdependence among the soft-estimated symbols of the interfering UEs, along with its low-complexity variant. These strategies are compared with the robust maximum likelihood data detection with 1-bit ADCs. Numerical results examining the symbol error rate show that MMSE exhibits a considerable performance gain over MRC, whereas the proposed LMMD receiver significantly outperforms all the conventional receivers. Lastly, the proposed JD and its low-complexity variant provide a significant boost in comparison with the single-UE data detection.

We consider the set $\mathcal M_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper bounds on the number of matrices from $\mathcal M_n(\mathbb Z; H)$, for which the characteristic polynomial has a fixed discriminant $d$. When $d=0$, this corresponds to counting matrices with a repeated eigenvalue, and thus is related to counting non-diagonalisable matrices. For $d\ne 0$, this problem seems not to have been studied previously, while for $d=0$, both our approach and the final result improve on those of A. J. Hetzel, J. S. Liew and K. Morrison (2007).

We introduce the concept of a viable generically Gorenstein toroidal crossing (ggtc) space $Y$. This generalizes the concept of Gorenstein toroidal crossing scheme, which in turn generalizes that of a simple normal crossing scheme. On such a space $Y$, we define a sheaf $\mathcal{LS}_Y$, intrinsic to $Y$, by means of an explicit construction. Our main theorem establishes a bijection between the set $\operatorname{LS}(Y)$ of isomorphism classes of log structures on $Y$ over the log point $\operatorname{Spec} k^\dagger$ that are compatible with the ggtc structure and the set $\Gamma(Y,\mathcal{LS}_Y^\times)$ of nowhere vanishing global sections of $\mathcal{LS}_Y$. The definition of $\mathcal{LS}_Y$ by explicit construction permits the effective construction of log structures on $Y$; it also enables logarithmic birational geometry, in particular the construction - in some cases - of resolutions of singular log structures. Our work generalizes [GS06], Theorem 3.22, adapting the original proof with techniques from the theory of $2$-groups and local line bundle systems.

Products in double categories, as found in cartesian double categories, are an elegant concept with numerous applications, yet also have a few puzzling aspects. In this paper, we revisit double-categorical products from an unbiased perspective, following up an original idea by Paré to employ a double-categorical analogue of the family construction, or free product completion. Defined in this way, double categories with finite products are strictly more expressive than cartesian double categories, while being governed by a single universal property that is no more difficult to work with. We develop the basic theory and examples of such products and, by duality, of coproducts in double categories. As an application, we introduce finite-product double theories, a categorification of finite-product theories that extends recent work by Lambert and the author on cartesian double theories, and we construct the virtual double category of models of a finite-product double theory.

Lipolysis is a life-essential metabolic process, which supplies fatty acids stored in lipid droplets to the body in order to match the demands of building new cells and providing cellular energy. In this paper, we present a first mathematical modelling approach for lipolysis, which takes into account that the involved enzymes act on the surface of lipid droplets. We postulate an active region near the surface where the substrates are within reach of the surface-bound enzymes and formulate a system of reaction-diffusion PDEs, which connect the active region to the inner core of lipid droplets via interface conditions. We establish two numerical discretisations based on finite element method and isogeometric analysis, and validate them to perform reliably. Since numerical tests are best performed on non-zero explicit stationary state solutions, we introduce and analyse a model, which describes besides lipolysis also a reverse process (yet in a physiologically much oversimplified way). The system is not coercive such that establishing well-posedness is a non-standard task. We prove the unique existence of global and equilibrium solutions. We establish exponential convergence to the equilibrium solutions using the entropy method. We then study the stationary state model and compute explicitly for radially symmetric solutions. Concerning the finite element methods, we show numerically the linear and quadratic convergence of the errors with respect to the $H^{1}$- and $L^{2}$-norms, respectively. Finally, we present numerical simulations of a prototypical PDE model of lipolysis and illustrate that ATGL clustering on lipid droplets can significantly slow down lipolysis.

In this note we generalize the construction, due to Ghilardi, of the free Heyting algebra generated by a finite distributive lattice, to the case of arbitrary distributive lattices. Categorically, this provides an explicit construction of a left adjoint to the inclusion of Heyting algebras in the category of distributive lattices This is shown to have several applications, both old and new, in the study of Heyting algebras: (1) it allows a more concrete description of colimits of Heyting algebras, as well as, via duality theory, limits of Esakia spaces, by knowing their description over distributive lattices and Priestley spaces; (2) It allows a direct proof of the amalgamation property for Heyting algebras, and of related facts; (3) it allows a proof of the fact that the category of Heyting algebras is co-distributive. We also study some generalizations and variations of this construction to different settings. First, we analyse some subvarieties of Heyting algebras -- such as Boolean algebras, $\mathsf{KC}$ and $\mathsf{LC}$ algebras, and show how the construction can be adapted to this setting. Second, we study the relationship between the category of image-finite posets with p-morphisms and the category of posets with monotone maps, showing that a variation of the above ideas provides us with an appropriate general idea.

We develop a notion of sampling, called \emph{generic sampling}, for the context of global Keisler measures where the standard product is replaced by the Morley product. Choosing a point randomly in this space with respect to our distribution yields a \emph{random generic type} in infinitely many variables. We investigate several natural model-theoretic events and provide conditions under which they occur for almost all random generic types.

Very differently from those perturbative techniques of Deng-Musso in [26], we use the assumption of a $C^1$-stable critical point to construct positive or sign-changing solutions with arbitrary $m$ isolated bubbles to the boundary value problem $-\Delta u=\lambda u|u|^{p-2}e^{|u|^p}$ under homogeneous Dirichlet boundary condition in a bounded, smooth planar domain $\Omega$, when $0<p<2$ and $\lambda>0$ is a small but free parameter. We build a vanishing identity of first order and an identity of second order to prove that for any $0<p<1$ the delicate energy expansion of these bubbling solutions always converges to $4\pi m$ from below, but for any $1<p<2$ the energy always converges to $4\pi m$ from above, where the latter case sharply recurs a result of De Marchis-Malchiodi-Martinazzi-Thizy in [32] involving concentration and compactness properties at any critical energy level $4\pi m$ only for positive bubbling solutions. A sufficient condition on the intersection between the nodal line of these sign-changing solutions and the boundary of the domain is founded. Moreover, for $\lambda$ small enough, we prove that when $\Omega$ is an arbitrary bounded domain, this problem has not only at least two pairs of bubbling solutions which change sign exactly once and whose nodal lines intersect the boundary, but also a bubbling solution which changes sign exactly twice or three times; when $\Omega$ has an axial symmetry, this problem has a bubbling solution which alternately changes sign arbitrarily many times along the axis of symmetry through the domain.

In this paper we focus on various aspects of singular complex plane curves, mostly in the context of their homological properties and the associated combinatorial structures. We formulate some challenging open problems that can point to new directions in research, for example by introducing weak Ziegler pairs of curve arrangements. Moreover, we construct new examples of different Ziegler pairs, in both the classical and the weak sense, and present new geometric approaches to construction problems of singular plane curves.

We derive an uncertainty principle for Lipschitz maps acting on subsets of Banach spaces. We show that this nonlinear uncertainty principle reduces to the Heisenberg-Robertson-Schrodinger uncertainty principle for linear operators acting on Hilbert spaces.

We introduce a local non-determinism condition for Volterra Itô processes that captures smoothing properties of possibly degenerate noise. By combining the stochastic sewing lemma with one-step Euler approximations, we first prove the joint space-time regularity for their occupation measure, self-intersection measure, and time marginals for such Volterra Itô processes. As an application, we obtain the space-time regularity of local times and self-intersection times for rough perturbations of Gaussian Volterra processes, and construct a class of non-Gaussian Volterra Iô processes that are $C^{\infty}$-regularising. Secondly, for the particular class of stochastic Volterra equations with Hölder continuous coefficients, using disintegration of measures for their Markovian lifts, we further establish the absolute continuity of finite-dimensional distributions. Finally, we prove the existence, uniqueness, and stability for self-interacting stochastic equations with distributional drifts.

Our main goal is to prove that every invertible generalized hyperbolic operator on a Banach space has a stability property, known as time-dependent stability, which was introduced by J. M. Franks (Invent. Math. 24 (1974), 163--172) and is stronger than structural stability.

We construct the geometric Satake equivalence for quasi-split reductive groups over nonarchimedean local fields, using étale Artin-Tate motives with $\mathbb{Z}[\frac{1}{p}]$-coefficients. We consider local fields of both equal and mixed characteristic. Along the way, we extend the work of Gaussent--Littelmann on the connection between LS galleries and MV cycles to the case of residually split reductive groups. As an application, we generalize Zhu's integral Satake isomorphism for spherical Hecke algebras to ramified groups. Moreover, for residually split groups, we define generic spherical Hecke algebras, and construct generic Satake and Bernstein isomorphisms.

Let $G$ be a plane elementary bipartite graph whose infinite face is forcing. We provide a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal resonant sets of $G$, which generalizes a main result in [MATCH Commun. Math. Comput. Chem. 68 (2012) 65-77], where $G$ was only considered as an elementary benzenoid graph without nice coronenes. For a special case when $G$ is a peripherally 2-colorable graph, it follows that there is a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal independent sets of a tree that is the inner dual of $G$. We then show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if it is the simplex graph of the complement of a forest. Finally, we characterize trees with at most 5 maximal independent sets to determine daisy cubes that are simplex graphs of the complements of trees and having at most five maximal vertices.

Multiscale differentials arise as limits of holomorphic differentials with prescribed zero orders on nodal curves. In this paper, we address the conjecture concerning Gorenstein contractions of multiscale differentials, originally proposed by Ranganathan and Wise and further developed by Battistella and Bozlee. Specifically, in the case of a one-parameter degeneration, we show that multiscale differentials can be contracted to Gorenstein singularities, level by level, from the top down. At each level, these differentials descend to generators of the dualizing bundle at the resulting singularities. Moreover, the global residue condition, which governs the smoothability of multiscale differentials, appears as a special case of the residue condition for descent differentials.

In this paper, we address the distributed prescribed-time convex optimization (DPTCO) problem for a class of nonlinear multi-agent systems (MASs) under undirected connected graph. A cascade design framework is proposed such that the DPTCO implementation is divided into two parts: distributed optimal trajectory generator design and local reference trajectory tracking controller design. The DPTCO problem is then transformed into the prescribed-time stabilization problem of a cascaded system. Changing Lyapunov function method and time-varying state transformation method together with the sufficient conditions are proposed to prove the prescribed-time stabilization of the cascaded system as well as the uniform boundedness of internal signals in the closed-loop systems. The proposed framework is then utilized to solve robust DPTCO problem for a class of chain-integrator MASs with external disturbances by constructing a novel variables and exploiting the property of time-varying gains. The proposed framework is further utilized to solve the adaptive DPTCO problem for a class of strict-feedback MASs with parameter uncertainty, in which backstepping method with prescribed-time dynamic filter is adopted. The descending power state transformation is introduced to compensate the growth of increasing rate induced by the derivative of time-varying gains in recursive steps and the high-order derivative of local reference trajectory is not required. Finally, theoretical results are verified by two numerical examples.

This paper investigates the distribution of non-rigid families in a moduli space $\mathcal{M}$ of polarized projective manifolds for which the infinitesimal Torelli theorem holds. Guided by the analogy with unlikely intersection in Shimura varieties, we show that the image of any non-rigid classifying morphisms into $\mathcal{M}$ is contained in the Hodge locus as long as the derived Mumford-Tate group is $\mathbb{Q}$-simple and the period map is generically finite. If moreover the period domain is not Hermitian of rank at least 2, then the Hodge locus can be replaced by a closed subscheme, which yields a finiteness theorem of geometric Bombieri-Lang type. Inspired by the Zilber-Pink conjecture, we also characterize the geometry of non-rigid locus by the specialness of bi-Hom schemes and the finiteness of "structurally-atypical" intersections. Finally, we specialize to the moduli spaces of polarized Calabi-Yau manifolds, formulate an unobstructedness conjecture for non-rigid maps which implies the specialness of bi-Hom schemes, prove a geometric André-Oort theorem describing the Zariski closure of non-rigid locus, and test the theory and the conjecture for the explicit Viehweg-Zuo family of Calabi--Yau quintics in $\mathbb{P}^4$.

In high-dimensional principal component analysis, important inferential targets include both leading spikes and the associated principal eigenspaces. Such problems arise naturally in high-dimensional factor models, where leading principal directions are interpreted as dominant loading directions and spike magnitudes reflect the strength of the corresponding common factors. We study inference based on the sample covariance matrix $\bS$ and the sample correlation matrix $\widehat{\bR}$ under generalized spiked models with arbitrary bulk spectrum. We establish almost sure limits and central limit theorems for spiked sample eigenvalues, and derive asymptotic distributions for functionals of sample spiked eigenspaces. Building on this theory, we develop procedures for one-sample inference for benchmark principal directions and for two-sample comparison of leading spike strengths across populations. Even in the covariance setting, our results substantially extend the existing literature by allowing a non-identity bulk structure. A real-data analysis on stock returns further illustrates the practical relevance of the proposed procedures, showing that covariance-based and correlation-based PCA can lead to markedly different conclusions.

We consider a null-recurrent randomly biased walk $\mathbb{X}$ on a Galton-Watson tree in the (sub)-diffusive regime and we prove that properly renormalized, the local time in a critical generation converges in law towards some function of a stable continuous-state branching process. We also provide an explicit equivalent of the probability that critical generations are reached by the random walk $\mathbb{X}$.

Let $R$ be a ring. In \cite{MD4} Mao and Ding defined an special class of $R$-modules that they called \( FP_n \)-projective $R$-modules. In this paper, we give some new characterizations of \( FP_n \)-projective $R$-modules and strong $n$-coherent rings. Some known results are extended and some new characterizations of the \( FP_n \)-injective global dimension in terms of \( FP_n \)-projective $R$-modules are obtained. Using the \( FP_n \)-projective dimension of an $R$-module defined by Ouyang, Duan and Li in \cite{Ouy} we introduce a slightly different \( FP_n \)-projective global dimension over the ring $R$ which measures how far away the ring is from being Noetherian. This dimension agrees with the $(n,0)$-projective global dimension of \cite{Ouy} when the ring in question is strong $n$-coherent.

We study combinatorial properties of plateaued functions $F \colon \mathbb{F}_p^n \rightarrow \mathbb{F}_p^m$. All quadratic functions, bent functions and most known APN functions are plateaued, so many cryptographic primitives rely on plateaued functions as building blocks. The main focus of our study is the interplay of the Walsh transform and linearity of a plateaued function, its differential properties, and their value distributions, i.e., the sizes of image and preimage sets. In particular, we study the special case of ''almost balanced'' plateaued functions, which only have two nonzero preimage set sizes, generalizing for instance all monomial functions. We achieve several direct connections and (non)existence conditions for these functions, showing for instance that plateaued $d$-to-$1$ functions (and thus plateaued monomials) only exist for a very select choice of $d$, and we derive for all these functions their linearity as well as bounds on their differential uniformity. We also specifically study the Walsh transform of plateaued APN functions and their relation to their value distribution.

In this paper, we prove energy and Morawetz estimates for solutions to the scalar wave equation in spacetimes with metrics that are perturbations, compatible with nonlinear applications, of Kerr metrics in the full subextremal range. Central to our approach is the proof of a global in time energy-Morawetz estimate conditional on a low frequency control of the solution using microlocal multipliers adapted to the $r$-foliation of the spacetime. This result constitutes a first step towards extending the current proof of Kerr stability in \cite{GCM1} \cite{GCM2} \cite{KS:Kerr} \cite{GKS} \cite{Shen}, valid in the slowly rotating case, to a complete resolution of the black hole stability conjecture, i.e., the statement that the Kerr family of spacetimes is nonlinearly stable for all subextremal angular momenta.

We consider the problem of shared randomness-assisted multiple access channel (MAC) simulation for product inputs and characterize the one-shot communication cost region via almost-matching inner and outer bounds in terms of the smooth max-information of the channel, featuring auxiliary random variables of bounded size. The achievability relies on a rejection-sampling algorithm to simulate an auxiliary channel between each sender and the decoder, and producing the final output based on the output of these intermediate channels. The converse follows via information-spectrum based arguments. To bound the cardinality of the auxiliary random variables, we employ the perturbation method from [Anantharam et al., IEEE Trans. Inf. Theory (2019)] in the one-shot setting. For the asymptotic setting and vanishing errors, our result expands to a tight single-letter rate characterization and consequently extends a special case of the simulation results of [Kurri et al., IEEE Trans. Inf. Theory (2022)] for fixed, independent and identically distributed (iid) product inputs to universal simulation for any product inputs. We broaden our discussion into the quantum realm by studying feedback simulation of quantum-to-classical (QC) MACs with product measurements [Atif et al., IEEE Trans. Inf. Theory (2022)]. For fixed product inputs and with shared randomness assistance, we give a quasi tight one-shot communication cost region with corresponding single-letter asymptotic iid expansion.

We study a class of dynamical systems generated by random substitutions, which contains both intrinsically ergodic systems and instances with several measures of maximal entropy. In this class, we show that the measures of maximal entropy are classified by invariance under an appropriate symmetry relation. All measures of maximal entropy are fully supported and they are generally not Gibbs measures. We prove that there is a unique measure of maximal entropy if and only if an associated Markov chain is ergodic in inverse time. This Markov chain has finitely many states and all transition matrices are explicitly computable. Thereby, we obtain several sufficient conditions for intrinsic ergodicity that are easy to verify. A practical way to compute the topological entropy in terms of inflation words is extended from previous work to a more general geometric setting.

The third author introduced the $g$-polynomial $g_M(t)$ of a matroid, a covaluative matroid statistic which is unchanged under series and parallel extension. The $g$-polynomial of a rank $r$ matroid $M$ has the form $g_1 t + g_2 t^2 + \cdots + g_r t^r$. The coefficient $g_1$ is Crapo's classical $\beta$-invariant. In this paper, we study the coefficient $g_r$, which we term the $\omega$-invariant of $M$. We show that, if $M/F$ is connected for every proper flat $F$ of $M$, and $\omega(N)$ is nonnegative for every minor $N$ of $M$, then all the coefficients of $g_M(t)$ are nonnegative. We give several simplified versions of Ferroni's formula for $\omega(M)$, and compute $\omega(M)$ when $r$ or $|E(M)|-2r$ is small.

We introduce geometric invariants for $p$-groups of class $2$ and exponent $p$. We report on their effectiveness in distinguishing among 5-generator $p$-groups of this type.

Inequalities between Dirichlet and Neumann eigenvalues of the Laplacian and of other differential operators have been intensively studied in the past decades. The aim of this paper is to introduce differential forms and the de Rham complex in the study of such inequalities. We show how differential forms lie hidden at the heart of the work of Rohleder on inequalities between Dirichlet and Neumann eigenvalues for the Laplacian on planar domains.

The variance of the number of particles in a set is an important quantity in understanding the statistics of non-interacting fermionic systems in low dimensions. An exact map of their ground state in a harmonic trap in one and two dimensions to the classical Gaussian unitary and complex Ginibre ensemble, respectively, allows to determine the counting statistics at finite and infinite system size. We will establish two new results in this setup. First, we uncover an interpolating central limit theorem between known results in one and two dimensions, for linear statistics of the elliptic Ginibre ensemble. We find an entire range of interpolating weak non-Hermiticity limits, given by a two-parameter family for the mesoscopic scaling regime. Second, we considerably generalize the proportionality between the number variance and the entanglement entropy between Fermions in a set $A$ and its complement in two dimensions. Previously known only for rotationally invariant sets and external potentials, we prove a holographic principle for general non-rotationally invariant sets and random normal matrices. It states that both number variance and entanglement entropy are proportional to the circumference of $A$.

We consider the $r$-neighbor bootstrap percolation process on the graph with vertex set $V=\{0,1\}^n$ and edges connecting the pairs at Hamming distance $1,2,\dots,k$, where $k\ge 2$. We find asymptotics of the critical probability of percolation for $r=2,3$. In the deterministic setting, we obtain several results for the size of the smallest percolating set for $k\ge 2$.

Closure operations such as tight and integral closure and test ideals have appeared frequently in the study of commutative algebra. This articles serves as a survey of the authors' prior results connecting closure operations, test ideals, and interior operations via the more general structure of pair operations. Specifically, we describe a duality between closure and interior operations generalizing the duality between tight closure and its test ideal, provide methods for creating pair operations that are compatible with taking quotient modules or submodules, and describe a generalization of core and its dual. Throughout, we discuss how these ideas connect to common constructions in commutative algebra.

Introduced by Takagi and Watanabe, the F-pure threshold is an invariant defined in terms of the Frobenius homomorphism. While it finds applications in various settings, it is primarily used as a local invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the defect of the F-pure threshold of a local ring $(R,\mathfrak{m})$ by setting ${\rm dfpt}(R)=\dim (R) - {\rm fpt}(\mathfrak{m})$. It turns out that this invariant defines an upper semi-continuous function on a scheme and satisfies Bertini-type theorems. We also study the behavior of the defect of the F-pure threshold under flat extensions and after blowing up the maximal ideal of a local ring.

Spectral Barron spaces, which quantify the absolute value of weighted Fourier coefficients of a function, have gained considerable attention due to their capability for universal approximation across certain function classes. By establishing a connection between these spaces and a specific positive linear operator, we investigate the interpolation and scaling relationships among diverse spectral Barron spaces. Furthermore, we introduce a link condition by relating the spectral Barron space to inverse problems, illustrating this with three exemplary cases. We revisit the notion of universal approximation within the context of spectral Barron spaces and validate an error bound for Tikhonov regularization, penalized by the spectral Barron norm.

Given a connected graph, the principal eigenvector of the adjacency matrix (often called the Perron vector) can be used to assign positive weights to the vertices. A natural way to measure the homogeneousness of this vector is by considering the ratio of its $\ell^1$ and $\ell^2$ norms. It is easy to see that the most balanced graphs in this sense (i.e., the ones with the largest ratio) are the regular graphs. What can we say about the least balanced (or most centralized) graphs with the smallest ratio? It was conjectured by Rücker, Rücker and Gutman that, for any given $n \geq 6$, among $n$-vertex connected graphs the smallest ratio is achieved by the complete graph $K_4$ with a single path $P_{n-4}$ attached to one of its vertices. In this paper we confirm this conjecture. We also verify the analogous conjecture for trees: for any given $n \geq 8$, among $n$-vertex trees the smallest ratio is achieved by the star graph $S_5$ with a path $P_{n-5}$ attached to its central vertex.

In this long note, we investigate various purely topological aspects of non-Hausdorff manifolds (NH-manifolds for short). Our emphasis is on manifolds which exhibit homogeneity or weakenings thereof, in particular being everywhere non-Hausdorff. Homogeneous NH-manifolds and everywhere non-Hausdorff manifolds are respectively called HNH- and ENH-manifolds. We write $NH_X(x)$ for the subset of points of a space $X$ which cannot be separated of $x$ by open sets. The topics covered in this note are the following. -- General (basic) properties of manifolds and their quasi-compact or quasi-countably compact subspaces. -- Covering properties implying the Hausdorffness of (weakly) homogeneous manifolds. -- (Non-)existence of hereditarily separable ENH-manifolds (under set theoretic hypotheses). -- Non-existence of a quasi-countably compact ENH-manifold. -- Properties of NH-manifolds which imply that $NH_M(x)$ is discrete, or at least ``simple''. -- Constructions of HNH-manifolds such that $NH(x)$ is non-homogeneous, for instance a countable union of closed intervals and $n$-torii. -- Constructions of NH-manifolds $M$ with a point $x$ such that $NH_M(x)$ is homeomorphic to various ``complicated'' spaces, in particular in dimension $1$ and $2$. We use elementary (or at least well known) methods of general or set theoretic topology, with a little bit of conformal theory and dynamical systems (flows) in some constructions. Many pictures are given to illustrate the constructions, and the proofs are rather detailed, which is the main reason for the length of this note.

In this paper, we consider the Cauchy problem for the 3D Euler equations with the Coriolis force in the whole space. We first establish the local-in-time existence and uniqueness of solution to this system in $B^s_{p,r}(\R^3)$. Then we prove that the Cauchy problem is ill-posed in two different sense: (1) the solution of this system is not uniformly continuous dependence on the initial data in the same Besov spaces, which extends the recent work of Himonas-Misiołek \cite[Comm. Math. Phys., 296, 2010]{HM1} to the more general framework of Besov spaces; (2) the solution of this system cannot be Hölder continuous in time variable in the same Besov spaces. In particular, the solution of the system is discontinuous in the weaker Besov spaces at time zero. To the best of our knowledge, our work is the first one addressing the issue on the failure of Hölder continuous in time of solution to the classical Euler equations with(out) the Coriolis force.

We analyze a continuous-time optimal trade execution problem in multiple assets where the price impact and the resilience can be matrix-valued stochastic processes that incorporate cross-impact effects. In addition, we allow for stochastic terminal and running targets. Initially, we formulate the optimal trade execution task as a stochastic control problem with a finite-variation control process that acts as an integrator both in the state dynamics and in the cost functional. We then extend this problem continuously to a stochastic control problem with progressively measurable controls. By identifying this extended problem as equivalent to a certain linear-quadratic stochastic control problem, we can use established results in linear-quadratic stochastic control to solve the extended problem. This work generalizes [Ackermann, Kruse, Urusov; FinancStoch'24] from the single-asset setting to the multi-asset case. In particular, we reveal cross-hedging effects, showing that it can be optimal to trade in an asset despite having no initial position. Moreover, as a subsetting we discuss a multi-asset variant of the model in [Obizhaeva, Wang; JFinancMark'13].

Let $\Gamma$ be a countable discrete group. We say that $\Gamma$ has $C^*$-invariant subalgebra rigidity (ISR) property if every $\Gamma$-invariant $C^*$-subalgebra $\mathcal{A}\le C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for some normal subgroup $N\triangleleft\Gamma$. We show that all torsion-free, non-amenable (cylindrically) hyperbolic groups with property-AP and a finite direct product of such groups have this property. We also prove that an infinite group $\Gamma$ has the C$^*$-ISR property only if $\Gamma$ is simple amenable or $C^*$-simple.

Providing theoretical guarantees for parameter estimation in exponential random graph models is a largely open problem. While maximum likelihood estimation has theoretical guarantees in principle, verifying the assumptions for these guarantees to hold can be very difficult. Moreover, in complex networks, numerical maximum likelihood estimation is computer-intensive and may not converge in reasonable time. To ameliorate this issue, local dependency exponential random graph models have been introduced, which assume that the network consists of many independent exponential random graphs. In this setting, progress towards maximum likelihood estimation has been made. However the estimation is still computer-intensive. Instead, we propose to use so-called Stein estimators: we use the Stein characterizations to obtain new estimators for local dependency exponential random graph models.

We present a rigorous mathematical framework establishing the equivalence of four classical notions of synchronization full phase-locking, phase-locking, frequency synchronization, and order parameter synchronization in generalized Kuramoto models, via a non-perturbative, finite-dimensional analysis. Our approach avoids linearization, mean-field limits, and restrictions on initial conditions, relying instead on global phase-space geometry, periodic vector field structure, and compactness arguments based on contradiction. These results clarify the foundational role of the order parameter and provide a unified understanding of synchronization across a broad class of heterogeneous oscillator networks.

We investigate a globalized inexact semismooth Newton method applied to strongly convex optimization problems in Hilbert spaces. Here, the semismooth Newton method is appplied to the dual problem, which has a continuously differentiable objective. We prove global strong convergence of iterates as well as transition to local superlinear convergence. The latter needs a second-order Taylor expansion involving semismooth derivative concepts. The convergence of the globalized method is demonstrated in numerical examples, for which the local unglobalized method diverges.

We study the problem of optimally projecting the transition matrix of a finite ergodic multivariate Markov chain onto a lower-dimensional state space, as well as the problem of finding an optimal partition of coordinates such that the factorized Markov chain gives minimal information loss compared to the original multivariate chain. Specifically, we seek to construct a Markov chain that optimizes various information-theoretic criteria under cardinality constraints. These criteria include entropy rate, information-theoretic distance to factorizability, independence, and stationarity. We formulate these tasks as best subset or partition selection problems over multivariate Markov chains and leverage the (k-)submodular (or (k-)supermodular) structures of the objective functions to develop efficient greedy-based algorithms with theoretical guarantees. Along the way, we introduce a generalized version of the distorted greedy algorithm, which may be of independent interest. Finally, we illustrate the theory and algorithms through extensive numerical experiments with publicly available code on multivariate Markov chains associated with the Bernoulli--Laplace and Curie--Weiss models.

Determinantal processes provide mathematical modeling of repulsion among points. In quantum mechanics, Slater determinant states generate such processes, reflecting Fermionic behavior. This note exploits the connections between the former and the latter structures by establishing quantitative bounds in terms of trace/total variation and Wasserstein distances.

In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemarié wavelet system of nonnegative integer order, $n$. We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to $s < n+1$, which includes values of $s$ where the Battle-Lemarié system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case $s=n+1$. This builds off of earlier work by G. Garrigós, A. Seeger, and T. Ullrich, where they proved the case $n=0$, i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemarié system to give an unconditional basis of the Triebel-Lizorkin spaces.

We investigate the automorphism groups of the algebraic curves \[ \mathcal{C}_d : y^d = \varphi_d(x), \] where $\varphi_d(x)$ denotes the Chebyshev polynomial of degree $d$, defined over a field $k$ with $p:=\operatorname{char}(k) \nmid 2d$. We determine the full automorphism group of $\mathcal{C}_d$ in all the cases considered in this paper, namely for $d=4$, and more generally when $2d = p^r+1$ or $4d = p^r+1$. For all other $d>4$, Expectation~\ref{3.19} predicts what the automorphism group should be. As an application, we show that certain maximal curves of the same genus are not isomorphic.

Let $\A$ be a free arrangement of $d$ lines in the complex projective plane, with exponents $d_1\leq d_2$. Let $m$ be the maximal multiplicity of points in $\A$. In this note, we describe first the simple cases $d_1 \leq m$. Then we study the case $d_1=m+1$, and describe which line arrangements can occur by deleting or adding a line to $\A$. When $d \leq 14$, there are only two free arrangements with $d_1=m+2$, namely one with degree $13$ and the other with degree $14$. We study their geometries in order to deepen our understanding of the structure of free line arrangements in general.

The apportionment problem asks how to assign representation to states based on their populations. That is, given census data and a fixed number of seats, how many seats should each state be assigned? Various algorithms exist to solve the apportionment problem, but by the Balinski-Young Impossibility Theorem, every such algorithm will be flawed in some way. This paper focuses on divisor methods of apportionment, where the possible flaws are known as quota violations. This paper presents a detailed analysis of quota violations that can arise under divisor methods for three states, By focusing on the three-state case, the paper makes the consequences of the Balinski-Young theorem particularly transparent and allows for a precise classification of quota violations that is difficult to obtain in more general formulations. The study focuses on quota violations in the Adams, Jefferson, Dean, and the Huntington-Hill methods when allocating M seats, but is expandable to a wider class of divisor functions. Theoretical results are proved about the behavior of these methods, particularly focusing on the types of quota violations that may occur, their frequency, and their structure and geometry. The key results of the paper are tests to detect quota violations which are employed to understand the geometry of violations and construct a probability function which calculates the likelihood of such violations occurring given an initial three state population vector whose components follow varying distributions.

We prove that a log Fano cone $(X,\Delta,\xi_0)$ satisfying $\delta_\mathbb{T}(X,\Delta,\xi_0)\ge 1$ is K-polystable for normal test configurations if and only if it is K-polystable for special test configurations. We also establish the reduced uniform K-stability of $(X,\Delta,\xi_0)$ and show that it is equivalent to K-polystability.

Given an $\infty$-category $C$ equipped with suitable wide subcategories $I, P \subset E\subset C$, we show that the $(\infty,2)$-category $\text{S}{\scriptstyle\text{PAN}}_2(C,E)_{P,I}$ of higher (or iterated) spans defined by Haugseng has the universal property that 2-functors $\text{S}{\scriptstyle\text{PAN}}_2(C,E)_{P,I} \to \mathbb D$ correspond precisely to $(I, P)$-biadjointable functors $C^\text{op} \to \mathbb D$, i.e. functors $F$ where $F(i)$ for $i \in I$ admits a left adjoint and $F(p)$ for $p \in P$ admits a right adjoint satisfying various Beck-Chevalley conditions. We also extend this universality to the symmetric monoidal and lax symmetric monoidal settings. This provides a conceptual explanation for - and an independent proof of - the Mann-Liu-Zheng construction of 6-functor formalisms from suitable functors $C^\text{op}\to\text{CAlg}(\text{Cat})$.

In this paper, we consider in $\mathbb{R}^3$ the following zero mass Schrödinger-Bopp-Podolsky system \[ \begin{cases} -\Delta u +q^2\phi u=|u|^{p-2}u\\ -\Delta \phi+a^2\Delta^2\phi=4\pi u^2 \end{cases} \] where $a>0$, $q\ne 0$ and $p\in (3,6)$. Inspired by [Ruiz, Arch. Ration. Mech. Anal. 198 (2010)], we introduce a Sobolev space $\mathcal{E}$ endowed with a norm containing a nonlocal term. Firstly, we provide some fundamental properties for the space $\mathcal{E}$ including embeddings into Lebesgue spaces. Moreover a general lower bound for the Bopp-Podolsky energy is obtained. Based on these facts, by applying a perturbation argument, we finally prove the existence of a weak solution to the above system.

We derive a formula for the Nielsen number $N(f)$ for every $n$-valued self-map $f$ of an infra-solvmanifold. To do this, we express $N(f)$ in terms of Nielsen coincidence numbers of single-valued maps on solvmanifolds, and derive a formula for Nielsen coincidence numbers in that setting.

The rank correlation \xi(X,Y), recently established by Sourav Chatterjee and already popular in the statistics literature, takes values in [0,1], where 0 characterizes independence of X and Y, and 1 characterizes perfect dependence of Y on X. Unlike concordance measures such as Spearman's \rho, which capture the degree of positive or negative dependence, \xi quantifies the strength of functional dependence. In this paper, we study the attainable set of pairs (\xi(X,Y),\rho(X,Y)). The resulting {\xi}-\r{ho}-region is a convex set whose boundary is characterized by a novel family of absolutely continuous, asymmetric copulas having a diagonal band structure. Moreover, we prove that \xi(X,Y)\leq|\rho}(X,Y)| whenever Y is stochastically increasing or decreasing in X, and we identify the maximal difference \rho(X,Y)-\xi(X,Y) as exactly 0.4. Our proofs rely on a convex optimization problem under various equality and inequality constraints, as well as on ordering properties for \xi and \rho. Our results contribute to a better understanding of Chatterjee's rank correlation, which typically yields substantially smaller values than Spearman's \rho when quantifying positive dependencies. In particular, when interpreting the values of Chatterjee's rank correlation on the scale of \rho, the quantity \sqrt{\xi} appears to be more appropriate.

While initial applications of artificial intelligence (AI) in wireless communications over the past decade have demonstrated considerable potential using specialized models for targeted communication tasks, the revolutionary demands of sixth-generation (6G) networks for holographic communications, ubiquitous sensing, and native intelligence are propelling a necessary evolution towards AI-native wireless networks. The arrival of large AI models paves the way for the next phase of Wireless AI, driven by wireless foundation models (WFMs). In particular, pre-training on universal electromagnetic (EM) principles equips WFMs with the essential adaptability for a multitude of demanding 6G applications. However, existing large AI models face critical limitations, including pre-training strategies disconnected from EM-compliant constraints leading to physically inconsistent predictions, a lack of embedded understanding of wave propagation physics, and the inaccessibility of massive labeled datasets for comprehensive EM-aware training. To address these challenges, this article presents an electromagnetic information theory-guided self-supervised pre-training (EIT-SPT) framework designed to systematically inject EM physics into WFMs. The EIT-SPT framework aims to infuse WFMs with intrinsic EM knowledge, thereby enhancing their physical consistency, generalization capabilities across varied EM landscapes, and overall data efficiency. Building upon the proposed EIT-SPT framework, this article first elaborates on diverse potential applications in 6G scenarios of WFMs, then validates the efficacy of the proposed framework through illustrative case studies, and finally summarizes critical open research challenges and future directions for WFMs.

We describe and compute various families of commuting elements of the matrix shuffle algebra of type $\mathfrak{gl}_{n|m}$, which is expected to be isomorphic to quantum toroidal $\mathfrak{gl}_{n|m}$. Our formulas are given in terms of partial traces of products of $R$-matrices of the quantum affine algebra $U_t(\dot{\mathfrak{gl}}_{n|m})$, and have a lattice path interpretation. Our calculations are based on the machinery of the quantum toroidal algebras and a new anti-homomorphism between matrix shuffle algebras.

Beltran \& Cladek~\cite{BC} use $L^r$ to $L^s$ bounds to prove sparse form bounds for pseudodifferential operators with Hörmander symbols in $S^m_{\rho,\delta}$ up to, but not including, the sharp end-point in decay $m$. We further develop their technique, obtaining pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables and an alternative proof of their results which avoids proving geometrically decaying sparse bounds. We also provide sufficient conditions for sparse form bounds to hold and use these to reprove know sparse bounds for pseudodifferential operators with symbols in $S^0_{1,\delta}$ for $\delta < 1$.

Given a holomorphic differential on a smooth complex algebraic curve, we associate to it a Gorenstein curve singularity with $\mathbb G_m$-action via a test configuration. This construction decomposes the strata of holomorphic differentials with prescribed orders of zeros into negatively graded miniversal deformation spaces of such singularities. Additionally, it provides a natural description for the singular curves that appear in the boundary of the miniversal deformation spaces. Our approach leads to a number of applications. We classify the unique Gorenstein singularity with $\mathbb G_m$-action for each nonvarying stratum of holomorphic differentials and study when these nonvarying strata can be compactified by weighted projective spaces. Moreover, extending the classical results about $ADE$ singularities, we establish the $K(\pi,1)$-property for non-hypersurface complete intersection singularities of type $U_7$, $U_8$, $U_9$, and $S_{k}$. We also study singularities with bounded $\alpha$-invariants in the log minimal model program for $\overline{\mathcal M}_g$ and utilize them to bound the slopes of effective divisors in $\overline{\mathcal M}_g$. Finally, we show that the loci of subcanonical points with fixed semigroups have trivial tautological rings and provide a criterion to determine whether they are affine varieties.

We study the De Giorgi-Moser-Nash estimates of higher-order parabolic equations in divergence form with complex-valued, measurable, bounded, uniformly elliptic (in the sense of G$\mathring{a}$rding inequality) and time-independent coefficients. We also obtain Gaussian upper bounds and Hölder regularity estimates for the fundamental solutions of this class of parabolic equations.

We establish some conditions under which $\text{GL}(d,\mathbb{R})$-valued cocycles over a subshift of finite type, equipped with an equilibrium state, exhibit exponential asymptotics for the spectral radius. Specifically, we show that the exponential growth rate of the spectral radius converges to the top Lyapunov exponent of the cocycle. This result provides a partial answer to a question posed by Aoun and Sert in their paper "Law of large numbers for the spectral radius of random matrix products" (2021). Our approach relies on large deviation estimates for linear cocycles, which may be of independent interest.

We develop a central limit theorem (CLT) for a non-parametric estimator of the transition matrices in controlled Markov chains (CMCs) with finite state-action spaces. Our results establish precise conditions on the logging policy under which the estimator is asymptotically normal, and reveal settings in which no CLT can exist. We then build on it to derive CLTs for the value, Q-, and advantage functions of any stationary stochastic policy, including the optimal policy recovered from the estimated model. Goodness-of-fit tests are derived as a corollary, which enable to test whether the logged data is stochastic. These results provide new statistical tools for offline policy evaluation and optimal policy recovery, and enable hypothesis tests for transition probabilities.

Motivated by some known problems concerning combinatorial structures associated with finite one-dimensional affine permutation groups, we study subgroups which are closed in $\operatorname{\Gamma{L}}_1(q)$. This brings us to a description of the relatively closed subgroups of permutation groups with a cyclic regular normal subgroup. Our results, in particular, provide a classification of the minimal nontrivial one-dimensional affine association schemes which generalizes the recent Muzychuk classification of the one-dimensional affine rank 3 graphs.

In this paper, we study the Prym map associated to degree 4 étale cyclic covers of genus $g$ hyperelliptic curves restricted to the irreducible component $\mathcal{RH}_g[4]^{hyp}$ of the moduli space of such covers where an intermediate cover is hyperelliptic. We show that for $g \geq 3$ the Prym map is injective on $\mathcal{RH}_g[4]^{hyp}$. In the case $g=2$ (where $\mathcal{RH}_2[4]^{hyp} = \mathcal{RH}_2[4]$) we prove that non-empty fibers of the Prym map, apart from two exceptional fibers, are isomorphic to the projective line without 8 points. Moreover, we obtain a new description of the space $\mathcal{RH}_g[4]^{hyp}$ in terms of tuples of complex numbers and find equations of hyperelliptic curves arising from such covers.

We extend the microlocal Kakeya--Nikodym bounds for eigenfunctions of Blair--Sogge to a larger range of exponents, which is optimal in all dimensions $n\ge3$ on general manifolds. On manifolds of constant sectional curvature, we introduce a new anisotropic variant of the microlocal Kakeya--Nikodym norm that further enlarges the admissible $p$-range. As a corollary, by combining our results with a recent theorem of Hou, we obtain improved $L^p$ bounds for Hecke--Maass forms on compact hyperbolic $3$-manifolds. In particular, our method applies to general Hörmander operators, and we characterize the $L^q \to L^p$ boundedness of Hörmander operators with positive-definite phase in all dimensions $n\ge3$, thereby fully resolving a question going back to Hörmander. Further applications include improved $L^q \to L^p$ Fourier extension bounds, and improved bounds related to the Bochner--Riesz conjecture in $\mathbb R^3$.

We consider the limiting fluctuations of the geodesic in the directed landscape, conditioning on its length going to infinity. It was shown in \cite{Liu22b,Ganguly-Hegde-Zhang23} that when the directed landscape $\mathcal{L}(0,0;0,1) = L$ becomes large, the geodesic from $(0,0)$ to $(0,1)$ lies in a strip of size $O(L^{-1/4})$ and behaves like a Brownian bridge if we zoom in the strip by a factor of $L^{1/4}$. Moreover, the length along the geodesic with respect to the directed landscape fluctuates of order $O(L^{1/4})$ and its limiting one-point distribution is Gaussian \cite{Liu22b}. In this paper, we further zoom in a smaller neighborhood of the endpoints when $\mathcal{L}(0,0;0,1) = L$ or $\mathcal{L}(0,0;0,1) \ge L$, and show that there is a critical scaling window $L^{-3/2}:L^{-1}:L^{-1/2}$ for the time, geodesic location, and geodesic length, respectively. Within this scaling window, we find a nontrivial limit of the one-point joint distribution of the geodesic location and length as $L\to\infty$. This limiting distribution, if we tune the time parameter to infinity, converges to the joint distribution of two independent Gaussian random variables, which is consistent with the results in \cite{Liu22b}. We also find a surprising connection between this limiting distribution and the one-point distribution of the upper tail field of the KPZ fixed point recently obtained in \cite{Liu-Zhang25}.

For $p\in (1,\infty)$ and $\alpha\in\mathbb{R}$, we consider measurable functions $g$ on $\mathbb{S}^{N-1}$ that satisfy the following weighted Hardy inequality: \begin{equation}\label{abs} \int_{\mathbb{R}^N}\frac{ g (x/|x|)}{|x|^{p+\alpha}}|u(x)|^p dx \leq C\int_{\mathbb{R}^N}\frac{|\nabla u(x)|^p}{|x|^\alpha} dx, \quad\forall\,u\in \mathcal{C}_c^\infty(\mathbb{R}^N), \end{equation} for some constant $C>0$. Depending on $N$, $p$, and $\alpha$, we identify suitable function spaces for $g$ so that \eqref{abs} holds. The constant obtained is sharp, in the sense that it is sharp when $g \equiv 1$. Furthermore, we establish the sharp fractional Hardy inequality with homogeneous weights.

We develop an index theory for variational problems on noncompact quantum graphs. The main results are a spectral flow formula, relating the net change of eigenvalues to the Maslov index of boundary data, and a Morse index theorem, equating the negative directions of the Lagrangian action with the total multiplicity of conjugate instants along the edges. These results extend classical tools in global analysis and symplectic geometry to graph based models, with applications to nonlinear wave equations such as the nonlinear Schroedinger equation. The spectral flow formula is proved by constructing a Lagrangian intersection theory in the Gelfand-Robbin quotients of the second variation of the action. This approach also recovers, in a unified way, the known formulas for heteroclinic, halfclinic, homoclinic, and bounded orbits of (non)autonomous Lagrangian systems.

Using orbifold Hilbert schemes, we compactify all two-dimensional Hitchin systems corresponding to types A0-tilde, D4-tilde, E6-tilde, E7-tilde, and E8-tilde, thereby obtaining four rational elliptic surfaces with C*-actions. Their singular fibers and relative minimal models are listed in the main table. A particularly interesting point is that we found they can all be obtained by performing a finite number of blow-ups on the second Hirzebruch surface. To this end, we prove that Hilbert schemes of orbifold surfaces are connected smooth projective schemes under suitable conditions, and we use the Hilbert-Chow morphism to construct the minimal resolutions of the coarse moduli spaces.

This paper studies kernel Radon-Nikodym derivatives for the one-step shift of time-indexed positive definite kernels associated with random matrix products. The problem is to determine when the shifted kernel is dominated by the original kernel and to identify the corresponding Radon-Nikodym derivative. We treat two concrete classes of multiplicative walks: ensembles with inhomogeneous variances and Gaussian Kraus products. In both settings, the shifted kernel inequality reduces to a one-step condition on the diagonal moments, and the Radon-Nikodym derivative is described explicitly by a fiberwise sequence in the time variable. In the inhomogeneous variance model, the diagonal compression is governed by a nonnegative matrix $S$, which yields an explicit coordinate formula for the fibers. In the Gaussian Kraus model, the diagonal moments are generated by a completely positive map $\Psi$, and the shifted kernel inequality is equivalent to the condition ${\Psi\left(I\right)\le I}$.

We consider a parabolic stochastic partial differential equation (SPDE) on $[0\,,1]$ that is forced with multiplicative space-time white noise with a bounded and Lipschitz diffusion coefficient and a drift coefficient that is locally Lipschitz and satisfies an $L\log L$ growth condition. We prove that the SPDE is well posed when the initial data is in $L^2[0\,,1]$. This solves a strong form of an open problem.

This work proposes a novel Alternating Direction Method of Multipliers (ADMM)-based Ensemble Kalman Inversion (EKI) algorithm for solving constrained nonlinear model predictive control (NMPC) problems. First, stage-wise nonlinear inequality constraints in the NMPC problem are embedded via an augmented Lagrangian with nonnegative slack variables. We then show that the resulting unconstrained augmented-Lagrangian primal subproblem admits a Bayesian interpretation: under independent Gaussian virtual observations, its minimizers coincide with MAP estimators, enabling solution via EKI. However, since the nonnegativity constraint on the slacks is a hard constraint not naturally encoded by a Gaussian model, our proposed algorithm yields a two-block ADMM scheme that alternates between (i) an inexact primal step that minimizes the augmented-Lagrangian objective (implemented via EKI rollouts), (ii) a nonnegativity projection for the slacks, and (iii) a dual ascent step. To balance exploration and convergence, an annealing schedule tempers sampling covariances while a penalty schedule increases constraint enforcement over outer iterations, encouraging global search early and precise constraint satisfaction later. We evaluate the proposed controller on a 6-DOF UR5e manipulation benchmark in MuJoCo, comparing it against DIAL-MPC (an iterative MPPI variant) as the arm traverses a cluttered tabletop environment.

Coxeter pointed out that a number of polytopes can be projected orthogonally into two dimensions in such a way that their vertices lie on a number of concentric regular triacontagons (or 30-gons). Among them are the 600-cell and 120-cell in four dimensions and Gosset's polytope 4_21 in eight dimensions. We show how these projections can be modified into Kochen-Specker diagrams from which parity proofs of the Bell-Kochen-Specker theorem are easily extracted. Our construction trivially yields parity proofs of fifteen bases for all three polytopes and also allows many other proofs of the same type to be constructed for two of them. The defining feature of these proofs is that they have a fifteen-fold symmetry about the center of the Kochen-Specker diagram and thus involve both rays and bases that are multiples of fifteen. Any proof of this type can be written as a word made up of an odd number of distinct letters, each representing an orbit of fifteen bases. Knowing the word representing a proof makes it possible to infer all its characteristics without first having to recover its bases. A comparison is made with earlier approaches that have been used to obtain parity proofs in these polytopes, and some directions in which this work can be extended are discussed.

In the framework of quantum mechanics over a quadratic extension of the ultrametric field of p-adic numbers, we introduce a notion of tensor product of p-adic Hilbert spaces. To this end, following a standard approach, we first consider the algebraic tensor product of p-adic Hilbert spaces. We next define a suitable norm on this linear space. It turns out that, in the p-adic framework, this norm is the analogue of the projective norm associated with the tensor product of real or complex normed spaces. Eventually, by metrically completing the resulting p-adic normed space, and equipping it with a suitable inner product, we obtain the tensor product of p-adic Hilbert spaces. That this is indeed the correct p-adic counterpart of the tensor product of complex Hilbert spaces is also certified by establishing a natural isomorphism between this p-adic Hilbert space and the corresponding Hilbert-Schmidt class. Since the notion of subspace of a p-adic Hilbert space is highly nontrivial, we finally study the tensor product of subspaces, stressing both the analogies and the significant differences with respect to the standard complex case. These findings should provide us with the mathematical foundations necessary to explore quantum entanglement in the p-adic setting, with potential applications in the emerging field of p-adic quantum information theory.

We study a generalized version of Zermelo's navigation problem where the set of admissible velocities is a general compact convex set, replacing the classical Euclidean ball. After establishing existence results under the natural assumption of weak currents, we derive necessary optimality conditions via Pontryagin's maximum principle and convex analysis. Consequently, in the planar case, the domain of any optimal control is shown to be partitioned into regular and singular regimes. In the former, the optimal control is regular and satisfies a Zermelo-like navigation equation while in the latter it is largely undetermined. A necessary condition that can exclude singular regimes is stated and proved, providing a useful tool in applications. In regular regimes our results extend the classical Zermelo navigation equation to general convex control sets within a non-parametric setting. Furthermore, we discuss direct applications to the case of a non-convex control set. As an application, we develop the relevant case of an affine current. The results are illustrated with examples relevant to sailing and ship routing with asymmetric or sail-assisted propulsion, including the presence of waves.

In this paper, we obtain the optimal rigidity of dimension estimate for holomorphic functions with polynomial growth on Kähler manifolds with non-negative holomorphic bisectional curvature. There is a specific gap between the largest and the second largest dimension. We also determine the optimal dimension that ensures the maximal volume growth which implies the manifold is biholomorphic to the complex Euclidean space.

In the March 2025 issue of Pour la Science, Jean-Paul Delahaye described a wonderful solution to the following problem: How many ways can you divide a 3 by 2n rectangle into two connected, congruent pieces? We show that this problem can be solved by the transfer matrix method, and demonstrate this by computing the generating function for the number of ways to divide a 4 by n rectangle into two connected, congruent parts.

We prove that the linearised operator around any sufficiently small solitary wave of the one-dimensional Zakharov system has no internal mode. This spectral result, along with its proof, is expected to play a role in the study of the asymptotic stability of solitary waves.

We propose a level-set-based semi-Lagrangian method on graded adaptive Cartesian grids to address the problem of surface reconstruction from point clouds. The goal is to obtain an implicit, high-quality representation of real shapes that can subsequently serve as computational domain for partial differential equation models. The mathematical formulation is variational, incorporating a curvature constraint that minimizes the surface area while being weighted by the distance of the reconstructed surface from the input point cloud. Within the level set framework, this problem is reformulated as an advection-diffusion equation, which we solve using a semi-Lagrangian scheme coupled with a local high-order interpolator. Building on the features of the level set and semi-Lagrangian method, we use quadtree and octree data structures to represent the grid and generate a mesh with the finest resolution near the zero level set, i.e., the reconstructed surface interface. The complete surface reconstruction workflow is described, including localization and reinitialization techniques, as well as strategies to handle complex and evolving topologies. A broad set of numerical tests in two and three dimensions is presented to assess the effectiveness of the method.

Histopolation is the approximation procedure that associates a degree $ d-1 $ polynomial $ p_{d-1} \in \mathscr{P}_{d-1} (I) $ with a locally integrable function $ f $ imposing that the integral (or, equivalently, the average) of $p$ coincides with that of $f$ on a collection of $ d $ distinct segments $s_i$. In this work we discuss unisolvence and conditioning of the associated matrices, in an asymptotic linear algebra perspective, i.e., when the matrix-size $d$ tends to infinity. While the unisolvence is a rather sparse topic, the conditioning in the unisolvent setting has a uniform behavior: as for the case of standard Vandermonde matrix-sequences with real nodes, the conditioning is inherently exponential as a function of $d$ when the monomial basis is chosen. In contrast, for an appropriate selection of supports, the Chebyshev polynomials of second kind exhibit a bounded conditioning. A linear behavior is also observed in the Frobenius norm.

The numerical solution of partial differential equations (PDEs) is fundamental to scientific and engineering computing. In the presence of strong anisotropy, material heterogeneity, and complex geometries, however, classical iterative solvers often suffer from reduced efficiency and require substantial problem-dependent tuning. The Fourier neural solver (FNS) is a learning-based hybrid iterative solver for such problems without extensive manual parameter tuning, but its original design is primarily effective for scalar PDEs on structured meshes and is difficult to extend directly to unstructured meshes or strongly coupled PDE systems. Building on the FNS framework, we introduce block smoothing operators and graph neural networks to construct a solver for unstructured systems, termed the graph Fourier neural solver (G-FNS). We further incorporate a coordinate transformation network to develop the adaptive graph Fourier neural solver (AG-FNS), and then extend this formulation to a frequency-domain multilevel variant, ML-AG-FNS. Rigorous analysis shows that, under suitable mathematical assumptions, the proposed method achieves mesh-independent convergence rate. Error-spectrum visualizations further indicate that AG-FNS can capture complex multiscale error modes. Extensive experiments on two-dimensional anisotropic diffusion and on two- and three-dimensional isotropic/anisotropic linear elasticity problems over unstructured meshes demonstrate strong robustness and efficiency. The proposed framework can be used either as a solver or as a preconditioner for Krylov subspace methods. Overall, it substantially extends the original FNS methodology and broadens the applicability of this class of neural solvers.

In this work, we prove the irrationality of $\pi$ based on the nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $\pi$ with increasing the integer $k$ are presented.

Based on entropy and symmetrical uncertainty (SU), we define a metric for categorical random variables and show that this metric can be promoted into an appropriate quotient space of categorical random variables. Moreover, we also show that there is a natural commutative monoid structure in the same quotient space, which is compatible with the topology induced by the metric, in the sense that the monoid operation is continuous.

Orthogonal polynomials of a continuous variable in the Askey scheme satisfying second order difference equations, such as the Askey-Wilson polynomial, can be studied by the quantum mechanical formulation, idQM (discrete quantum mechanics with pure imaginary shifts). These idQM systems have the shape invariance property, which relates the Hilbert space $\mathsf{H}_{\lambda}$ ($\lambda$ : a set of parameters) and that with shifted parameters $\mathsf{H}_{\lambda+\delta}$ ($\delta$ : shift of $\lambda$), and gives the forward and backward shift relations for the orthogonal polynomials. Based on the forward shift relation and the Christoffel's theorem with some polynomial $\check{\Phi}(x)$, which is expressed in terms of the quantities appeared in the forward and backward shift relations, we obtain some difference relations for the orthogonal polynomials. The multiplication of $\sqrt{\check{\Phi}(x)}$ gives a surjective map from $\mathsf{H}_{\lambda+2\delta}$ to $\mathsf{H}_{\lambda}$. Similarly, for the orthogonal polynomials in the Askey scheme satisfying second order differential equations, such as the Jacobi polynomial, we obtain some differential relations, and the multiplication of $\sqrt{\check{\Phi}(x)}$ in this case gives a surjective map from $\mathsf{H}_{\lambda+\delta}$ to $\mathsf{H}_{\lambda}$.

This work introduces security for unsourced random access (URA) by employing physical layer security techniques. To achieve confidentiality, the proposed system opportunistically exploits intrinsic features of feedback-aided URA without adding any overhead or altering its original structure or operational characteristics. As a result, the proposed system preserves the low-cost advantages of URA, including low delay and minimal signaling overhead, while providing secure communication. To secure transmission, each user generates a secret key from a feedback signal broadcast by the BS in a previous transmission round. This feedback depends on the BS-user channel, making it a private signal for each user. Secure transmission is achieved not only through encryption using the secret key, but also by transmitting only the parity bits of the LDPC-encoded key, thereby enabling its recovery at the legitimate receiver via Slepian-Wolf decoding with side information. For reception, a receiver algorithm is designed for the legitimate receiver, and a leakage analysis is provided to quantify the information available to the eavesdropper. The simulation results show that meaningful secrecy is achieved in URA without modifying its structure.

Consider the additive Gaussian model $Y = X + \sigma Z$, where $X \sim P$ is an unknown signal, $Z \sim N(0,1)$ is independent of $X$, and $\sigma > 0$ is known. Let $Q$ denote the law of $Y$. We construct a hierarchy of denoisers $T_0, T_1, \ldots, T_\infty \colon \mathbb{R} \to \mathbb{R}$ that depend only on higher-order score functions $q^{(m)}/q$, $m \geq 1$, of $Q$ and require no knowledge of the law $P$. The $K$-th order denoiser $T_K$ involves scores up to order $2K{-}1$ and satisfies $W_r(T_K \sharp Q, P) = O(\sigma^{2(K+1)})$ for every $r \geq 1$; in the limit, $T_\infty$ recovers the monotone optimal transport map (Brenier map) pushing $Q$ onto $P$. We provide a complete characterization of the combinatorial structure governing this hierarchy through partial Bell polynomial recursions, making precise how higher-order score functions encode the Brenier map. We further establish rates of convergence for estimating these scores from $n$ i.i.d.\ draws from $Q$ under two complementary strategies: (i) plug-in kernel density estimation, and (ii) higher-order score matching. The construction reveals a precise interplay among higher-order Fisher-type information, optimal transport, and the combinatorics of integer partitions.

The classification of homomorphisms between certain unital simple nuclear C*-algebras lead to the discovery of a natural transformation as part of the classifying invariant. We develop a uniqueness result and an abstract characterization of said transformation.

The residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ of a finitely generated group $G$ is a function that gives the smallest value of the index $[G:N]$ with $N$ a normal subgroup not containing a non-trivial element $g$, in function of the word norm of that element $g$. It has been studied for several classes of finitely generated groups, including free groups, linear groups and virtually abelian groups. This paper shows that if $G$ is virtually nilpotent, then $\text{RF}_G = \log^\delta$ for some $\delta\in \mathbb{N}\cup\{0\}$, with moreover an explicit formula for $\delta$ in terms of Lie algebras. This implies in particular that it is an invariant of the complex Mal'cev completion, leading to the application that residual finiteness growth is a profinite invariant for virtually nilpotent groups.

The use of machine learning models in system identification has increased due to their ability to approximate complex nonlinear dynamics with high accuracy. However, often it is not clear how the performance of trained models scales with given resources such as data, compute, and model size. To allow for a better understanding of the scalability of the performance of machine learning models, we verify neural scaling laws (NSLs) in the context of system identification from input-state-output data using different evaluation metrics for accuracy and different system architectures, including input-affine and physics-informed port-Hamiltonian representations. Our verified NSLs can help to forecast performance improvements and guide model design or data acquisition.

We study the amplitude-constrained additive white Gaussian noise channel. It is well known that the capacity-achieving input distribution for this channel is discrete and supported on finitely many points. The best known bounds show that the support size of the capacity-achieving distribution is lower-bounded by a term of order $A$ and upper-bounded by a term of order $A^2$, where $A$ denotes the amplitude constraint. It was conjectured in [1] that the linear scaling is optimal. In this work, we establish a new lower bound of order $A\sqrt{\log A}$, improving the known bound and ruling out the conjectured linear scaling. To obtain this result, we quantify the fact that the capacity-achieving output distribution is close to the uniform distribution in the interior of the amplitude constraint. Next, we introduce a wrapping operation that maps the problem to a compact domain and develop a theory of best approximation of the uniform distribution by finite Gaussian mixtures. These approximation bounds are then combined with stability properties of capacity-achieving distributions to yield the final support-size lower bound.

The problem of enumerating meanders -- pairs of simple plane curves with transverse intersections -- was formulated about forty years ago and is still far from solved. Recently, it was discovered that meanders admit a factorization into prime components. This factorization naturally leads to a broader class of objects, which we call singular meanders, in which tangential intersections between the curves are also allowed. In the present paper we initiate a systematic study of singular meanders: we develop a basic combinatorial framework, point out connections with other combinatorial objects and known integer sequences, and completely enumerate several natural families of singular meanders.

We establish two correspondences between reverse-mode automatic differentiation (backpropagation at a given forward-pass point) and compositions of projection maps in Kullback--Leibler (KL) geometry. In both settings, message passing enforces agreement and factorization constraints through KL projections. In the first setting, backpropagation arises as the differential of a KL projection map on a lifted deterministic computation graph. In the second setting, on complete and decomposable sum--product networks, the same reverse-mode quantities coincide with exact probabilistic marginals and are realized by a KL I-projection. The distinction is that, in the first setting, projection induces structure, whereas, in the second, structure makes the projection exact. This pedagogical note highlights the relation among backpropagation, belief propagation, and KL projection algorithms and provides a perspective that unifies learning, sampling, and inference under a common geometric operator.

Fix a prime $p > 3$. Working over $\mathbb{Z}_p$, we show that the quantum connection of any closed Calabi-Yau threefold gives rise to a Fontaine-Laffaile module when restricted to the even degree and torsion-free part of $p$-adic quantum cohomology, whose associated Frobenius endomorphism has leading order term prescribed by the $p$-adic Gamma class. After reducing mod $p$, the divided Frobenius endomorphism defines an analogue of the inverse Cartier operator on mod $p$ quantum cohomology. We establish an $A$-model analogue of a classical result due to Katz: the conjugation of the $p$-curvature of the mod $p$ quantum connection by the inverse Cartier operator is equal to the Frobenius pullback of the quantum product, the $A$-model counterpart of the Kodaira-Spencer class. Moreover, we identify the quantum Steenrod operation with the $p$-curvature of the mod $p$ quantum connection in this setting for any prime $p$. We propose several conjectures concerning how these arithmetic structures may extend to quantum connections on more general semi-positive symplectic manifolds.

We present a nesting-free normal form for the formalism of nested conditions and constraints in the context of finite lattices of subgraphs.

We consider a Markovian growth process on a partially ordered set $\Lambda$, equivalent to last passage percolation (LPP) with independent (not necessarily identical) exponentially distributed weights on the elements of $\Lambda$. Such a process includes inhomogeneous exponential LPP on the Euclidean lattice $\mathbb{N}_0^d$. We give non-asymptotic bounds on the mean and variance, as well as higher, central, and exponential moments of the passage time $\tau_A$ to grow any set $A \subseteq \Lambda$ in terms of characteristics of $A$. We also give a limit shape theorem when $\Lambda$ is equipped with a monoid structure. Methods involve making use of the backward equation associated to the Markovian evolution and comparison inequalities with respect to the time-reversed generator.

We calculate the mechanical response $r(x,t$) of an initially quiescent semi-infinite homogeneous medium to a pulse applied at the origin, and this is achieved within the framework of the Kelvin-Voigt model. Although this problem has been extensively studied in the literature because of its wide range of applications -- particularly in seismology -- here, we present a solution in a novel integral form. This integral solution avoids the numerical computation of the solution in terms of the inverse Laplace transform; that is, numerical integration in the complex plane. In particular, we derive integral form expressions for both delta-pulse and step-pulse excitations which are simpler and more computationally efficient than those previously reported in the literature. Furthermore, the obtained expressions allow us to obtain simple asymptotic formulas for $r(x,t$ as $x,t \to 0,\infty$ for both step- and delta-type pulses.

We propose a novel computational method for unit commitment UC, which does not require linearized approximation and provides several orders of magnitude performance improvement over current state-of-the-art. The performance improvement is achieved by introducing a heuristic tailored for UC problems. The method can be implemented using existing continuous optimization solvers and adapted for different applications. We demonstrate value of the new method in examples of advanced UC analyses at the scale where use of current state-of-the-art tools is infeasible. We expect that the capability demonstrated in this paper will be critical to address emerging power systems challenges with more volatile large loads, such as data centers, and generation that is composed of larger number of smaller units, including significant behind-the-meter generation.

Let $X$ be a compact arithmetic congruence hyperbolic surface, and let $\psi$ be an $L^2$-normalized Hecke-Maass form on $X$ with sufficiently large spectral parameter $\lambda$. We give a new proof to obtain an improved power saving for the global $L^6$-norm bound of $\psi$ over the local bound of Sogge. Our method uses a microlocal decomposition for $\psi$ and reduces the $L^6$-norm problem to microlocal Kakeya-Nikodym estimates for $\psi$, and we establish improved microlocal Kakeya-Nikodym estimates via arithmetic amplification developed by Iwaniec and Sarnak.

We study the excursion decomposition of the two-dimensional critical XOR-Ising model with either $+$ or free boundary conditions. In the first part, we construct the decomposition directly in the continuum. This construction relies on the identification of the XOR-Ising field with the cosine or sine of a Gaussian free field (GFF) $\phi$ multiplied by $\alpha = 1/\sqrt{2}$, and is obtained by an appropriate exploration of two-valued level sets of the GFF. More generally, the same construction applies to the fields $:\! \cos(\alpha \phi) \!:$ and $:\! \sin(\alpha \phi) \!:$ for any $\alpha \in (0,1)$. In the second part, we show that the continuum excursion decomposition arises as the scaling limit of the double random current decomposition of the critical XOR-Ising model on the square lattice. To this end, we exploit the rich Markovian structure of the discrete decomposition and strengthen the convergence of the double random current height function to the continuum GFF by establishing joint convergence with its cosine and sine. We conjecture that for $\alpha \in [1/2,\sqrt{3}/2)$ the continuum excursion decompositions arise as the scaling limit of those of the Ashkin-Teller polarisation field along its critical line.

In this paper, we first introduce representations of averaging pre-Lie algebras and study their matched pairs, Manin triples, and bialgebra theories. We prove that these three notions are equivalent under certain conditions. Moreover, by introducing averaging operators on quadratic Rota-Baxter pre-Lie algebras, we show that such operators give rise to averaging pre-Lie bialgebras. Then we introduce the notion of admissible classical Yang-Baxter equations in averaging pre-Lie algebras, as well as the relative Rota-Baxter operators on averaging pre-Lie algebras, and show that the relative Rota-Baxter operators on averaging pre-Lie algebras yield symmetric solutions of admissible classical Yang-Baxter equations in averaging pre-Lie algebras. Finally, we show that every averaging pre-Lie bialgebra induces an averaging Lie bialgebra.

For odd primes $p$ we consider the factors \[ A(p)=\frac{p-\chi_4(p)}{p+\chi_4(p)}, \qquad \chi_4(p)= \begin{cases} 1,&p\equiv 1\pmod 4, \\ -1,&p\equiv 3\pmod 4, \end{cases} \] and study products of $A(p)$ restricted to unions of residue classes modulo $2^m$. We give a simple criterion for the existence of a finite nonzero limit, prove a logarithmic asymptotic in the general case, and express the limiting constant in terms of Mertens-type constants in arithmetic progressions (hence in terms of Dirichlet $L$-values).

We prove the local logarithmic Brunn-Minkowski inequality for bodies of revolution. Furthermore, we give a generalization for one origin symmetric body of revolution and one body of revolution that does not need to be symmetric and restrict possible equality cases. The proof uses an operator theoretic approach together with the decomposition of spherical functions into isotypical components with respect to rotations around a fixed axis.

We introduce the Kolmogorov-Arnold Network for Dynamics (KANDy) as a zero-depth, wide neural architecture capable of discovering governing equations in chaotic and complex dynamical systems. Building on the foundation of Kolmogorov-Arnold Networks (KANs), KANDy explicitly learns governing equations by replacing sparse regression with a KAN. The synthesis of KANs and sparse regression addresses the limitations of equation discovery for KANs applied to dynamical systems and overcomes cases where sparse regression is hindered by sparsity constraints. Additionally, we show that our model, applied to the Hopf Fibration, recovers topological structure, thereby improving coherence with attractor properties. We apply our model to discrete and continuous dynamical systems, as well as to chaotic partial differential equations (PDEs). These results position KANDy as an interpretable and effective alternative for data-driven modeling of nonlinear dynamical systems.

A thermodynamic framework for asymptotic inference is developed in which sample size and parameter variance define a state space. Within this description, Shannon information plays the role of entropy, and an integrating factor organizes its variation into a first-law-type balance equation. The framework supports a cyclic inequality analogous to a reversed second law, derived for the estimation of the mean. A non-trivial third-law-type result emerges as a lower bound on entropy set by representation noise. Optimal inference paths, global bounds on information gain, and a natural Carnot-like information efficiency follow from this structure, with efficiency fundamentally limited by a noise floor. Finally, de Bruijn's identity and the I-MMSE relation in the Gaussian-limit case appear as coordinate projections of the same underlying thermodynamic structure. This framework suggests that ensemble physics and inferential physics constitute shadow processes evolving in opposite directions within a unified thermodynamic description.

In this paper we construct stable $IT_{0}$ trosion free sheaves on compactified Jacobian of an integral nodal curve. Our construction is functorial in nature and it provides an uniform method which works for both smooth and nodal curve.

We consider the problem of characterizing derived endomorphism algebras of simple objects in length categories up to quasi-isomorphism. We give such a characterization for module categories, abelian categories, exact categories, as well as, for certain differential graded analogues of them. It turns out that the property of being $d$-complicial ($d\geq 1$), in the sense of Lurie, of the involved simple-minded collections plays a central role. We also explain how this characterization can be interpreted as a coherent generation property for any minimal $A_{\infty}$-model of the derived endomorphism algebra. Along the way, we propose a notion of length exact differential graded categories and explain how they relate to length abelian $d$-truncated differential graded categories, generalizing results of Enomoto.

We study the Klein-Gordon equation in one spatial and one temporal dimension. Physically, this equation describes the wave function of a relativistic spinless boson with positive rest mass. Mathematically, this is the most elementary hyperbolic partial differential equation, after the wave equation itself. Relative to the origin, the spacetime splits according to the light cones, and we find four quarter-planes, two of which are timelike while the remaining two are spacelike. Not unexpectedly, the solutions behave quite differently in the two types of quarter-planes. It turns out that the spacelike quarter-planes exhibit a Liouville phenomenon, where insufficient growth forces the solutions to display a certain kind of symmetry, where the values on the two linear edges are in a one-to-one relation. This phenomenon shares features with the classical Liouville theorem as well as the Phragmen-Lindelof principle for harmonic functions.

\cite{tsagris2025a} proposed the generalized circular projected Cauchy (GCPC) distribution, whose special case is the wrapped Cauchy distribution. In this paper we first derive the relationship with the wrapped Cauchy distribution, and then we attempt to characterize the distribution. We establish the conditions under which the distribution exhibits unimodality. We provide non-analytical formulas for the mean resultant length and the Kullback-Leibler divergence, and analytical form for the cumulative probability function and the entropy of the GCPC distribution. We propose log-likelihood ratio tests for one, or two location parameters without assuming equality of the concentration parameters. We revisit maximum likelihood estimation with and without predictors. In the regression setting we briefly mention the addition of circular and simplicial predictors. Simulation studies illustrate a) the performance of the log-likelihood ratio test when one falsely assumes that the true distribution is the wrapped Cauchy distribution, and b) the empirical rate of convergence of the regression coefficients. Using a real data analysis example we show how to avoid the log-likelihood being trapped in a local maximum and we correct a mistake in the regression setting.

Consider a holomorphic correspondence $f$ on a compact Kähler manifold $X$ of dimension $k$. Let $1\le q\le k$ be any integer such that the dynamical degrees of $f$ satisfy $d_{q-1}<d_q$. We construct the Green currents $T_c$ of $f$ associated with the classes $c$ belonging to the dominant eigenspace for the action of $f^*$ on $H^{q,q}(X,\mathbb{R})$. We also show that the super-potential of $T_c$ is $\log$-Hölder-continuous. When $f$ has simple action on cohomology and its graph satisfies an assumption on the local multiplicity, we prove the exponential equidistribution of all positive closed currents towards the main Green current, i.e., the only one associated to the unique maximal degree $d_q$.

The general position problem in graphs seeks the largest set of vertices such that no three vertices lie on a common geodesic. Its counting refinement, the general position polynomial $\psi(G)$, asks for all such possible sets. In this paper, We describe general position sets for several classes of graphs and provide explicit formulas for the general position polynomials of complete multipartite graphs. We specialize to balanced complete multipartite graphs and show that for part size $r\le 4$, the polynomial $\psi(K_{r,\dots,r})$ is log-concave and unimodal for all numbers of parts, while for larger $r$, counterexamples show that these properties fail. Finally, we analyze the corona $G\circ K_1$ and prove that unimodality of $\psi(G)$ is retained for numerous natural classes (paths, edgeless graphs, combs). This contributes to an open problem, but the general case remains unknown. Our findings support the parallel between general position polynomials and classical position-type parameters, and identify balanced multipartite graphs and coronas as promising testbeds for additional research.

A family $\mathcal{G}$ of sets is a weak copy of a poset $P$ if there is a bijection $f:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ implies $f(p)\subseteq f(q)$. If $f$ satisfies $p\leqslant q$ if and only if $f(p)\subseteq f(q)$, the $\mathcal{G}$ is a strong copy of $P$. We study the anti-Ramsey numbers $\mathrm{ar}(n,P), \mathrm{ar^*}(n,P)$, the maximum number of colors used in a coloring of $2^{[n]}$ that does not admit a rainbow weak or strong copy of $P$, respectively. We establish connections to the well-studied extremal numbers $\mathrm{La}(n,P)$ and $\mathrm{La^*}(n,P)$ and determine asymptotically $\mathrm{ar^*}(n,T)$ for all tree posets $T$ and $\mathrm{ar^*}(n,O_{2k})$ for all crown posets $O_{2k}$.

We develop a structural and quantitative framework for analyzing the Collatz map through modular dynamics, valuation statistics, and combinatorial decomposition of trajectories into bursts and gaps. We establish several exact and asymptotic results, including an affine scrambling structure for odd-to-odd dynamics, structural decay of residue information, and a quantitative bound on the per-orbit contribution of expanding primitive families via a phantom gain analysis. In particular, we prove that the average phantom gain remains strictly below the contraction threshold under uniform distribution, with a robust extension under bounded total-variation discrepancy. Building on these components, we reduce the convergence of Collatz orbits to an explicit orbitwise regularity condition: agreement between time averages and ensemble expectations for truncated observables, together with a tail-vanishing condition. Under this condition, formulated in terms of weak mixing or controlled discrepancy, the orbit converges. Accordingly, the present work should be interpreted as a structural and conditional reduction of the Collatz conjecture, rather than a complete proof. It isolates the remaining obstruction as a single orbitwise upgrade from ensemble behavior to pointwise control, while establishing several independent exact results that may be of separate interest.

The subtour relaxation of the traveling salesman problem (TSP) plays a central role in approximation algorithms and polyhedral studies of the TSP. A long-standing conjecture asserts that the integrality gap of the subtour relaxation for the metric TSP is exactly 4/3. In this paper, we extend the exact verification of this conjecture for small numbers of vertices. Using the framework introduced by Benoit and Boyd in 2008, we confirm their results up to n=10. We further show that for n=11 and n=12, the published lists of extreme points of the subtour polytope are incomplete: one extreme point is missing for n=11 and twenty-two extreme points are missing for n=12. We extend the enumeration of the extreme points of the subtour polytope to instances with up to 14 vertices in the general case. Restricted to half-integral vertices, we extend the enumeration of extreme points up to n=17. Our results provide additional support for the 4/3-Conjecture. Our lists of extreme points are available on the public bonndata repository (this https URL).

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.

We construct a cellular sheaf from any feedforward ReLU neural network by placing one vertex for each intermediate quantity in the forward pass and encoding each computational step - affine transformation, activation, output - as a restriction map on an edge. The restricted coboundary operator on the free coordinates is unitriangular, so its determinant is $1$ and the restricted Laplacian is positive definite for every activation pattern. It follows that the relative cohomology vanishes and the forward pass output is the unique harmonic extension of the boundary data. The sheaf heat equation converges exponentially to this output despite the state-dependent switching introduced by piecewise linear activations. Unlike the forward pass, the heat equation propagates information bidirectionally across layers, enabling pinned neurons that impose constraints in both directions, training through local discrepancy minimization without a backward pass, and per-edge diagnostics that decompose network behavior by layer and operation type. We validate the framework experimentally on small synthetic tasks, confirming the convergence theorems and demonstrating that sheaf-based training, while not yet competitive with stochastic gradient descent, obeys quantitative scaling laws predicted by the theory.

We expand our novel computational method for unit commitment (UC) to include long-horizon planning. We introduce a fast novel algorithm to commit hydro-generators, provably accurately. We solve problems with thousands of generators at 5 minute market intervals. We show that our method can solve interconnect size UC problems in approximately 1 minute on a commodity hardware and that an increased planning horizon leads to sizable operational cost savings (our objective). This scale is infeasible for current state-of-the-art tools. We attain this runtime improvement by introducing a heuristic tailored for UC problems. Our method can be implemented using existing continuous optimization solvers and adapted for different applications. Combined, the two algorithms would allow an operator operating large systems with hydro units to make horizon-aware economic decisions.

Let $K$ be a number field with algebraic closure $\overline{K}$ and let $S$ be a finite set of places of $K$ that contain all the archimedean places. For an integer $d \ge 2$, consider the unicritical polynomial family $f_{d,c}(z) = z^d + c$. Recently, Benedetto and Ih studied the distribution of post-critically finite parameters $c$ that are $S$-integral relative to a fixed point $\alpha \in \overline{K}$ such that $f_{d, \alpha}$ is not post-critically finite. In this paper, we study the quantitative aspects of their result. In particular, under some additional assumptions we establish quantitative bounds on the number of $S$-integral post-critically finite parameters in the generalized Mandelbrot set $\mathcal{M}_{d, v}$ relative to a non post-critically finite parameter $\alpha$ as $\alpha$ varies over number fields of bounded degree.

This paper establishes a natural quantum counterpart of weak equilibration for statistical ensembles in integrable systems. For quantum systems with pure point spectrum, single-time expectation values under unitary evolution are typically quasiperiodic, and hence generally do not admit a pointwise limit as $t\to\infty$. To overcome this difficulty, we introduce a weighted time-averaging procedure and prove that the resulting averaged dynamics converge to the diagonal (dephased) equilibrium state. We further illustrate and validate the theoretical result through a three-spin quantum integrable model.

Many entropy-conservative and entropy-stable (summarized as entropy-preserving) methods for hyperbolic conservation laws rely on Tadmor's theory for two-point entropy-preserving numerical fluxes and its higher-order extension via flux differencing using summation-by-parts (SBP) operators, e.g., in discontinuous Galerkin spectral element methods (DGSEMs). The underlying two-point formulations have been extended to nonconservative systems using fluctuations by Castro et al. (2013, doi:https://doi.org/10.1137/110845379) with follow-up generalizations to SBP methods. We propose specific forms of entropy-preserving fluctuations for nonconservative hyperbolic systems that are simple to interpret and allow an algorithmic construction of entropy-preserving methods. We analyze necessary and sufficient conditions, and obtain a full characterization of entropy-preserving three-point methods within the finite volume framework. This formulation is extended to SBP methods in multiple space dimensions on Cartesian and curvilinear meshes. Additional properties such as well-balancedness extend naturally from the underlying finite volume method to the SBP framework. We use the algorithmic construction enabled by the chosen formulation to derive several new entropy-preserving schemes for nonconservative hyperbolic systems, e.g., the compressible Euler equations of an ideal gas using the internal energy equation and a dispersive shallow-water model. Numerical experiments show the robustness and accuracy of the proposed schemes.

We give an account of mutation of theta functions in cluster scattering diagrams, starting with a notion of mutation that is related to, but different from, the notion of mutation defined by Gross, Hacking, Keel, and Kontsevich. This different approach to mutation leads to several applications. Three of the applications simplify the process of computing structure constants for multiplication of theta functions, and these are used in another paper on cluster scattering diagrams of affine type. Notable in these three applications is the appearance of mutation symmetries and dominance regions. The other two applications have to do with pointed reduced bases, a variation on the pointed bases of Fan Qin. We give a characterization of pointed reduced bases analogous to Qin's characterization of pointed bases. All of these applications take place in a version of Gross, Hacking, Keel, and Kontsevich's canonical algebra that can be constructed for an arbitrary exchange matrix.

Let $X$ be a projective hypersurface that is not a cone. The symmetrizer group of $X$ is an algebraic group parametrizing hypersurfaces whose Jacobian ideal coincides with that of $X$. We show that if the locus of points in $X$ with multiplicity $d-1$ does not contain a line, then the dimension of the nilpotent part of the Lie algebra associated to the symmetrizer group is at most $2$, and the dimension of the symmetrizer group is bounded by $\dim X + 2$. To achieve this, we investigate the relation between a class of singularities on $X$ with highly degenerate tangent cones and the unipotent part of its symmetrizer group.

Duality is a foundational tool in robust and distributionally robust optimization (RO and DRO), underpinning both analytical insights and tractable reformulations. The prevailing approaches in the literature primarily rely on saddle-point arguments, Lagrangian techniques, and conic duality. In contrast, this paper applies perturbation duality in the sense of Fenchel--Rockafellar convex analysis and demonstrates its effectiveness as a general and unifying methodology for deriving dual formulations in RO and DRO. We first apply perturbation duality to a recently proposed DRO framework that unifies phi-divergence and Wasserstein ambiguity sets through optimal transport with conditional moment constraints. We establish the associated dual representation without imposing compactness assumptions previously conjectured to be necessary, instead introducing alternative conditions motivated by perturbation analysis and leveraging the Interchangeability Principle. We then revisit the concept of robust duality -- commonly described as ``primal-worst equals dual-best'' -- and show that perturbation-based formulations provide a unified and transparent characterization of this principle. In particular, we develop a bifunction-based representation that encompasses existing formulations in the literature and yields concise and general proofs, substantially simplifying recent results. This work positions perturbation duality as a versatile and underutilized framework for RO and DRO, offering both conceptual unification and technical generality across a broad class of models.

Two distinct knots are said to be friends if their complements, filled along the 0-slope, produce diffeomorphic 3-manifolds. In this article, we develop a practical algorithm, implemented using SnapPy and Regina, to search for a friend of a given knot. As an application, we construct a census of simple knots that admit friends and use these data to formulate conjectures about knot friends.

We study Virasoro constraints for Gromov-Witten theory of a product variety when one factor has semi-simple quantum cohomology.

We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles $\mathsf{RT}^{!\alpha}_k$ asserting that every $k$-coloring of the exactly $\alpha$-large subsets of an infinite $X \subseteq \mathbb{N}$ admits an infinite homogeneous set, where $\alpha$-largeness is defined via systems of fundamental sequences in the style of Ketonen and Solovay. For each countable ordinal $\alpha < \Gamma_0$ and each $k \geq 2$, we prove over $\mathsf{RCA}_0$ that the hierarchy of theorems $\mathsf{RT}^{!\a}_k$ corresponds exactly to the hierarchy of systems axiomatized by closure under transfinite Turing jumps, yielding a fine-grained classification between $\mathsf{ACA}_0$ and $\mathsf{ATR}_0$. Our results extend previous work on the case $\alpha=\omega$ and provide a uniform correspondence between countable indecomposable ordinals below $\Gamma_0$ and natural Ramsey-like theorems.

We report on some computational experiments related to the trivial units property and unique product property for group rings of torsion-free groups. These properties are related to Kaplansky's unit and zero-divisor conjectures. Our investigations include a classification of certain symmetric non-trivial units in the binary group ring of the Hantzsche-Wendt group; this group was used in Gardam's refutal of Kaplansky's unit conjecture. We also exhibit and investigate a new candidate group that fails the unique units property but may satisfy the trivial unit property. No examples of groups with these properties are known to date.

We investigate the quasilinear elliptic Hamilton--Jacobi--Bellman equation in $\mathbb{R}^{N}$ \begin{equation*} -\frac{1}{2}\Delta u+\frac{1}{p}\,|\nabla u|^{p}+u=f(x),\qquad p>1, \end{equation*}% within the optimal growth and uniqueness framework introduced by Bensoussan et al. (1984) for the case $N=1$ and Alvarez (1996) as well as Bensoussan and Frehse (1992) for the case $N\geq 1$. A central result of the paper shows that when $f$ is radially symmetric, the unique admissible solution is itself radially symmetric. The proof relies on the invariance of the operator under rotations, the uniqueness principle, and an averaging argument over the orthogonal group. In addition, we provide explicit barrier constructions, monotone approximation schemes, and local regularity estimates that guarantee the existence of classical solutions. As an illustration, we derive an exact quadratic solution in the case $% f(x)=|x|^{2} $ and $p=2$. Finally, we connect the PDE to its stochastic control interpretation and establish the extension of Alvarez's uniqueness theory to the regime-switching system of Hamilton--Jacobi--Bellman equations, proving the global well-posedness in the multi-regime setting.

We consider square-integrable functionals of Poisson point processes for which the variance upper bound provided by the classical Poincaré inequality is suboptimal, a phenomenon known as superconcentration. In this paper, we establish a rigorous mathematical equivalence between superconcentration and the chaotic behaviour of the functional, and certain associated random sets, under perturbations driven by the Ornstein-Uhlenbeck semigroup on the Poisson space. Leveraging the Malliavin-Stein method, we develop general variance identities and bounds for Poisson functionals, providing a unified framework to prove superconcentration, particularly for geometric functionals that can be expressed as a sum of local score functions. We apply our results to rigorously establish superconcentration and the chaotic behaviour in some models of stochastic geometry. Specifically, we analyse horizontal box-crossing indicators in certain critical continuum percolations, as well as the number of vertices with small degrees and the number of isolated $\Gamma$-components in random geometric graphs in the dense regime.

We study the regularity and branching of strictly abnormal minimizing geodesics in sub-Riemannian geometry. We construct examples of real-analytic sub-Riemannian manifolds admitting minimizing geodesics that lose regularity at an interior point of their domain and exhibit branching, thereby resolving longstanding open questions.

We study an axial dispersion tubular reactor model governed by a nonlinear parabolic equation with Robin-type boundary conditions and boundary feedback control. We derive sufficient conditions for the exponential stability of the steady-state solution of the closed-loop system and provide an explicit estimate of the decay rate. In addition, numerical simulations are presented to illustrate the sharpness of the obtained decay rate for different choices of the feedback gain parameter.

A cutting-plane model for a nonsmooth function is the maximum of several first-order expansions centered at different points. Using such a model in a bundle method leads to linear convergence (of serious steps) to a minimum. In smooth optimization, superlinear convergence can be achieved by using higher-order models. We show that the same is true for the nonsmooth case, i.e., we show that cutting-plane models involving higher-order expansions can be used to achieve superlinear convergence in nonsmooth optimization. We first formally define higher-order cutting-plane models for lower-$C^2$ functions and derive an error estimate. Afterwards, we construct a trust-region bundle method based on these models that achieves local superlinear convergence of serious steps, and overall superlinear convergence for certain finite max-type functions. Finally, we verify the superlinear convergence in numerical experiments.

We study the shuffle algebra realization of the positive subalgebra $Y_n^{>}(\mathbb{k})$ of the Yangian associated to $\mathfrak{sl}_n$ over an algebraically closed field $\mathbb{k}$ of characteristic $p>2$. In contrast to the characteristic zero case, the natural homomorphism from $Y_n^{>}(\mathbb{k})$ to the modular shuffle algebra $W^{(n)}(\mathbb{k})$ is not an isomorphism. We determine its kernel and image, showing that the kernel is precisely the ideal generated by the $p$-center of $Y_n^{>}(\mathbb{k})$, while the image consists of elements satisfying an additional wheel condition related to the characteristic $p$, thus providing a shuffle algebra realization for the restricted Yangian $Y_n^{>,[p]}$. The proof relies on the specialization maps approach and the construction of the small Yangian $\bar{y}^{>}_n(\mathbb{k})$, obtained by the reduction modulo $p$ method from an integral form $\mathbf{Y}_n^>$ of the Yangian $Y_n^{>}$ associated to $\mathfrak{sl}_n$ over $\mathbb{C}$.

We propose a globally convergent trust-region bundle method for minimizing lower-$C^2$ functions using higher-order cutting-plane models. Under certain growth assumptions on the objective around its minimum, the method is able to compute infinitely many trust regions of decreasing size that contain the minimum. We show that these growth assumptions are satisfied for certain finite max-type functions with sharp or quadratic growth. Enclosing the minimum in this way can be used to initialize local superlinearly convergent methods, which we demonstrate in numerical experiments.

The observable distance $d_{\mathrm{conc}}$ based on measure concentration and the box distance $\Box$ based on collapsing theory are extended to geometric data sets introduced by Hanika--Schneider--Stumme. On the set $\mathcal{D}$ of isomorphism classes of geometric data sets, $d_{\mathrm{conc}}$ is non-separable and $\Box$ is complete and non-separable. We introduce the class $\mathcal{D}/\mathcal{L}$ of $\mathcal{L}$-compact geometric data sets in $\mathcal{D}$, for a monoidal subfamily $\mathcal{L}$ of 1-Lipschitz functions $\operatorname{Lip}_1(\mathbb{R})$, and prove its $\Box$-completeness and separability. We then construct a natural compactification of $(\mathcal{D}/\mathcal{L}, d_{\mathrm{conc}})$ by means of \emph{$\mathcal{L}$-pyramids} when $\mathcal{L}$ contains the clipping family. We further prove a complete limit formula for the observable diameter of $\operatorname{Lip}_1(\mathbb{R})$-pyramids, and show that applying our construction to Hanika--Schneider--Stumme's embedding is compatible with the compactification and preserves the polynomial-time computability of the observable diameter.

In this paper, we prove energy and Morawetz estimates for solutions to Teukolsky equations in spacetimes with metrics that are perturbations, compatible with nonlinear applications, of Kerr metrics in the full subextremal range. The Teukolsky equations are written in tensorial form using the non-integrable formalism in \cite{GKS22}, and we follow the approach in \cite{Ma} of relying on a Teukolsky wave/transport system. The estimates are proved by extending the ideas from our earlier result \cite{MaSz24} on the corresponding problem for the scalar wave, notably the use of $r$-foliation-adapted microlocal multipliers for the wave part, and by incorporating techniques from \cite{Ma} to control the linear coupling terms between the components of the Teukolsky wave/transport system. Additionally, in order to adapt the methodology of \cite{MaSz24} to tensorial waves, we introduce a well-suited regular scalarization procedure which is of independent interest. This result, alongside our companion paper \cite{MaSz24}, is an essential step towards extending the current proof of Kerr stability in \cite{GCM1} \cite{GCM2} \cite{KS:Kerr} \cite{GKS22} \cite{Shen}, valid in the slowly rotating case, to a complete resolution of the Kerr stability conjecture, i.e., the statement that the Kerr family of spacetimes is nonlinearly stable for all subextremal angular momenta.

Quantum simulation has wide applications in quantum chemistry and physics. Recently, scientists have begun exploring the use of randomized methods for accelerating quantum simulation. Among them, a simple and powerful technique, called qDRIFT, is known to generate random product formulas for which the average quantum channel approximates the ideal evolution. qDRIFT achieves a gate count that does not explicitly depend on the number of terms in the Hamiltonian, which contrasts with Suzuki formulas. This work aims to understand the origin of this speed-up by comprehensively analyzing a single realization of the random product formula produced by qDRIFT. The main results prove that a typical realization of the randomized product formula approximates the ideal unitary evolution up to a small diamond-norm error. The gate complexity is already independent of the number of terms in the Hamiltonian, but it depends on the system size and the sum of the interaction strengths in the Hamiltonian. Remarkably, the same random evolution starting from an arbitrary, but fixed, input state yields a much shorter circuit suitable for that input state. In contrast, in deterministic settings, such an improvement usually requires initial state knowledge. The proofs depend on concentration inequalities for vector and matrix martingales, and the framework is applicable to other randomized product formulas. Our bounds are saturated by certain commuting Hamiltonians.

We study the consensus formation for an agents based model, generalizing that originally proposed by Krause \cite{Kr}, by allowing the communication channels between any couple of agents to be switched on or off randomly, at each time step, with a probability law depending on the proximity of the agents' opinions. Namely, we consider a system of agents sharing their opinions according to the following updating protocol. At time $t+1$ the opinion $X_{i}\left( t+1\right) \in\left[ 0,1\right] $ of any agent $i$ is updated at the weighted average of the opinions of the agents communicating with it at time $t.$ The weights model the confidence level an agent assigns to the opinions of the other agents and are kept fixed by the system dynamics, but the set of agents communicating with any agent $i$ at time $t+1$ is randomly updated in such a way that the agent $j$ can be chosen to belong to this set independently of the other agents with a probability that is a non increasing function of $\left\vert X_{i}\left( t\right) -X_{j}\left(t\right) \right\vert .$ This condition models the fact that a communication among the agents is more likely to happen if their opinions are close. We prove that if the agent's communication graph at time one, conditionally on the initial believes' configuration, is sufficiently connected, the system reaches consensus at geometric rate, i.e., more precisely, as the time tends to infinity the agents' opinions will reach the same value geometrically fast. We also discuss the consensus formation for a system of infinitely many agents. In particular we analyze the evolution of the empirical average of the agents' opinions in the limit as the size of the system tends to infinity and characterize its fixed points in terms of agents' consensus proving that this is reached geometrically fast with the same rate computed for the finite system.

We establish the weak convergence of the intensity of a nearly-unstable Hawkes process with heavy-tailed kernel. Our result is used to derive a scaling limit for a financial market model where orders to buy or sell an asset arrive according to a Hawkes process with power-law kernel. After suitable rescaling the price-volatility process converges weakly to a rough Heston model. Our convergence result is stronger than previously established ones that have either focused on light-tailed kernels or the convergence of integrated volatility process. The key is to establish the tightness of the family of rescaled volatility processes. This is achieved by introducing a new methods to establish the $C$-tightness of càdlàg processes based on the classical Kolmogorov-Chentsov tightness criterion for continuous processes.

Importance sampling (IS) is an efficient stand-in for model refitting in performing (LOO) cross-validation (CV) on a Bayesian model. IS inverts the Bayesian update for a single observation by reweighting posterior samples. The so-called importance weights have high variance -- we resolve this issue through adaptation by transformation. We observe that removing a single observation perturbs the posterior by $\mathcal{O}(1/n)$, motivating bijective transformations of the form $T(\theta)=\theta + h Q(\theta)$ for $0<h\ll 1.$ We introduce several such transformations: partial moment matching, which generalizes prior work on affine moment-matching with a tunable step size; log-likelihood descent, which partially invert the Bayesian update for an observation; and gradient flow steps that minimize the KL divergence or IS variance. The gradient flow and likelihood descent transformations require Jacobian determinants, which are available via auto-differentiation; we additionally derive closed-form expressions for logistic regression and shallow ReLU networks. We tested the methodology on classification ($n\ll p$), count regression (Poisson and zero-inflated negative binomial), and survival analysis problems, finding that no single transformation dominates but their combination nearly eliminates the need to refit.

Principal stratification provides a causal inference framework for investigating treatment effects in the presence of a post-treatment variable. Principal strata play a key role in characterizing the treatment effect by identifying groups of units with the same or similar values for the potential post-treatment variable at all treatment levels. The literature has focused mainly on binary post-treatment variables. Few papers considered continuous post-treatment variables. In the presence of a continuous post-treatment, a challenge is how to identify and characterize meaningful coarsening of the latent principal strata that lead to interpretable principal causal effects. This paper introduces the Confounders-Aware SHared atoms BAyesian mixture (CASBAH), a novel approach for principal stratification with binary treatment and continuous post-treatment variables. CASBAH leverages Bayesian nonparametric priors with an innovative hierarchical structure for the potential post-treatment outcomes that overcomes some of the limitations of previous works. Specifically, the novel features of our method allow for (i) identifying coarsened principal strata through a data-adaptive approach and (ii) providing a comprehensive quantification of the uncertainty surrounding stratum membership. Through Monte Carlo simulations, we show that the proposed methodology performs better than existing methods in characterizing the principal strata and estimating principal effects of the treatment. Finally, CASBAH is applied to a case study in which we estimate the causal effects of US national air quality regulations on pollution levels and health outcomes.

Coded computing has emerged as a promising framework for tackling significant challenges in large-scale distributed computing, including the presence of slow, faulty, or compromised servers. In this approach, each worker node processes a combination of the data, rather than the raw data itself. The final result then is decoded from the collective outputs of the worker nodes. However, there is a significant gap between current coded computing approaches and the broader landscape of general distributed computing, particularly when it comes to machine learning workloads. To bridge this gap, we propose a novel foundation for coded computing, integrating the principles of learning theory, and developing a framework that seamlessly adapts with machine learning applications. In this framework, the objective is to find the encoder and decoder functions that minimize the loss function, defined as the mean squared error between the estimated and true values. Facilitating the search for the optimum decoding and functions, we show that the loss function can be upper-bounded by the summation of two terms: the generalization error of the decoding function and the training error of the encoding function. Focusing on the second-order Sobolev space, we then derive the optimal encoder and decoder. We show that in the proposed solution, the mean squared error of the estimation decays with the rate of $\mathcal{O}(S^3 N^{-3})$ and $\mathcal{O}(S^{\frac{8}{5}}N^{\frac{-3}{5}})$ in noiseless and noisy computation settings, respectively, where $N$ is the number of worker nodes with at most $S$ slow servers (stragglers). Finally, we evaluate the proposed scheme on inference tasks for various machine learning models and demonstrate that the proposed framework outperforms the state-of-the-art in terms of accuracy and rate of convergence.

We investigate three causality-violating spacetimes: Misner space (including Kip Thorne's "moving wall" model), the pseudo-Schwarzschild spacetime, and a new model introduced here, the pseudo-Reissner-Nordström spacetime. Despite their different physical origins -- ranging from a flat vacuum solution to a black-hole-type vacuum solution to a non-vacuum model requiring exotic matter -- all three share a common warped-product structure, $2$-dimensional cylindrical base metrics of Eddington-Finkelstein type, and fundamental causal features such as Cauchy and chronology horizons, acausal regions, and analogous geodesic behaviour. Building on a conjecture first proposed in 2016, we present a formal proof that the three models are pairwise isocausal on their universal covers and on suitable causally regular regions of their compactified forms. The proof is constructive, providing explicit causal bijections on the covers and identifying a concrete deck-equivariance criterion governing descent to the compactified spacetimes: if the equivariance degree satisfies $|k|=1$ the models are globally isocausal, whereas if $|k|>1$ or equivariance fails, then at most a one-way causal relation holds between the compactified models. These results supply a rigorous causal classification linking these spacetimes, placing them within a unified Misner-type family and providing a framework for extending the classification to other causality-violating solutions.

Let $G=(V,E)$ be an undirected unweighted multi-graph and $S\subseteq V$ be a subset of vertices. A set of edges with the least cardinality whose removal disconnects $S$, that is, there is no path between at least one pair of vertices from $S$, is called a Steiner mincut for $S$ or simply an $S$-mincut. Connectivity Carcass is a compact data structure storing all $S$-mincuts in $G$ announced by Dinitz and Vainshtein in an extended abstract by Dinitz and Vainshtein in 1994. The complete proof of various results of this data structure for the simpler case when the capacity of $S$-mincut is odd appeared in the year 2000 in SICOMP. Over the last couple of decades, there have been attempts towards the proof for the case when the capacity of $S$-mincut is even, but none of them met a logical end. We present the following results. - We present the first complete, self-contained exposition of the connectivity carcass which covers both even and odd cases of the capacity of $S$-mincut. - We derive the results using an alternate and much simpler approach. In particular, we derive the results using submodularity of cuts -- a well-known property of graphs expressed using a simple inequality. - We also show how the connectivity carcass can be helpful in efficiently answering some basic queries related to $S$-mincuts using some additional insights.

Species complexes are groups of closely related populations exchanging genes through dispersal. We study the dynamics of the structure of species complexes in a class of metapopulation models where demes can exchange genetic material through migration and diverge through the accumulation of new mutations. Importantly, we model the ecological feedback of differentiation on gene flow by assuming that the success of migrations decreases with genetic distance, through a specific function $h$. We investigate the effects of metapopulation size on the coherence of species structures, depending on some mathematical characteristics of the feedback function $h$. Our results suggest that with larger metapopulation sizes, species form increasingly coherent, transitive, and uniform entities. We conclude that the initiation of speciation events in large species requires the existence of idiosyncratic geographic or selective restrictions on gene flow.

Transformer LLMs have been shown to exhibit strong reasoning ability that scales with inference-time compute, most prominently through token-space "thinking" chains of thought. A growing line of work pushes extra computation into the model's latent space, which we term Auxiliary Latent-Space Computation (ALSC). Existing ALSC methods largely fall into three buckets: (i) token-mediated latent rollouts, (ii) residual/activation steering, and (iii) memory (KV) compression. An underexplored alternative is memory consolidation/reconsolidation, two processes in the brain that are responsible for stabilising newly formed memory traces, and, upon recall, transiently rendering established traces plastic such they can integrate new contextual information before restabilising. In Transformer LLMs, this can be seen as analogous to performing in-place rewrites of new KV segments, and rewrites of recalled past segments. In this work, we give a theoretical justification as to why memory (re)consolidation via KV cache rewrites is beneficial for improved reasoning. We do this through the lens of Information Bottleneck (IB) theory, which posits that model generalisation emerges from an optimal balance between input information compression and retention of predictive information in latent representations. We then introduce the Bottlenecked Transformer, which augments a backbone LLM with a Cache Processor, an auxiliary Transformer that performs periodic, non-causal, in-place KV rewrites at newline-delimited reasoning step boundaries. The Processor consolidates recently written KV entries and reconsolidates a small, top-k attention-selected set of prior entries. We evaluate our Bottlenecked Transformer architecture on math reasoning benchmarks. Our model sees consistent performance gains over vanilla Transformers and pause-token augmented baselines, with gains of up to +6.6pp for selected tasks/backbones.

People make strategic decisions many times a day - during negotiations, when coordinating actions with others, or when choosing partners for cooperation. The resulting dynamics can be studied with learning theory and evolutionary game theory. These frameworks explore how people adapt their decisions over time, in light of how effective their strategies have been. The outcomes of such learning processes depend on how sensitive individuals are to the performance of their strategies. When they are more sensitive, they systematically favor strategies they deem more successful. When they are less sensitive, their learning process is noisier and more erratic. Traditionally, most models treat this sensitivity as a fixed parameter - like the "selection strength" parameter in evolutionary models. Instead, we study how strategies and sensitivities co-evolve. We find that the co-evolutionary endpoints depend on both the type of strategic interaction and the learning rule employed. In prisoner's dilemmas, we often observe sensitivities to increase indefinitely. But in snowdrift and stag-hunt games, sensitivities often converge to a finite value, or we observe evolutionary branching altogether. These results shed light on how evolution might shape learning mechanisms for social behavior. They suggest that noisy learning does not need to be a by-product of cognitive constraints. Instead, it can serve as a means to gain strategic advantages.

Understanding the generalization properties of neural networks on simple input-output distributions is key to explaining their performance on real datasets. The classical teacher-student setting, where a network is trained on data generated by a teacher model, provides a canonical theoretical test bed. In this context, a complete theoretical characterization of fully connected one-hidden-layer networks with generic activation functions remains missing. In this work, we develop a general framework for such networks with large width, yet much smaller than the input dimension. Using methods from statistical physics, we derive closed-form expressions for the typical performance of both finite-temperature (Bayesian) and empirical risk minimization estimators in terms of a small number of order parameters. We uncover a transition to a specialization phase, where hidden neurons align with teacher features once the number of samples becomes sufficiently large and proportional to the number of network parameters. Our theory accurately predicts the generalization error of networks trained on regression and classification tasks using either noisy full-batch gradient descent (Langevin dynamics) or deterministic full-batch gradient descent.

Recent studies suggest that context-aware low-rank approximation is a useful tool for compression and fine-tuning of modern large-scale neural networks. In this type of approximation, a norm is weighted by a matrix of input activations, significantly improving metrics over the unweighted case. Nevertheless, existing methods for neural networks suffer from numerical instabilities due to their reliance on classical formulas involving explicit Gram matrix computation and their subsequent inversion. We demonstrate that this can degrade the approximation quality or cause numerically singular matrices. To address these limitations, we propose a novel inversion-free regularized framework that is based entirely on stable decompositions and overcomes the numerical pitfalls of prior art. Our method can handle possible challenging scenarios: (1) when calibration matrices exceed GPU memory capacity, (2) when input activation matrices are nearly singular, and even (3) when insufficient data prevents unique approximation. For the latter, we prove that our solution converges to a desired approximation and derive explicit error bounds.

The synergy between integrated sensing and communication (ISAC) and reconfigurable intelligent surfaces (RISs) unlocks novel applications and advanced services for next-generation wireless networks, yet also introduces new security challenges. In this study, a novel dual target-mounted RISs-assisted ISAC scheme is proposed, where a base station with ISAC capability performs sensing of two unmanned aerial vehicle (UAV) targets, one of which is legitimate and the other is eavesdropper, while communicating with the users through an RIS mounted on the legitimate UAV target. The proposed scheme addresses dual security threats posed by a hostile UAV target: eavesdropping on legitimate user communications and random interference attacks launched by a malicious RIS mounted on this eavesdropper UAV target, aiming to disrupt secure transmissions. Moreover, malicious RIS interference is also optimized for a worst-case scenario, in which both the channel state information (CSI) and the transmit beamforming of the base station are assumed to be fully compromised by a malicious RIS-mounted eavesdropper UAV. A non-convex optimization problem maximizing the secrecy rate of the users is formulated, and a semi-definite relaxation (SDR)-based two-stage solution is developed to optimize the transmit beamforming matrix of the base station and the phase shift coefficients of the legitimate RIS. Extensive computer simulations are conducted to evaluate the robustness of the proposed solution under various system configurations. The proposed system's communication performance is assessed using the secrecy rate metric, while the sensing performance is evaluated through the signal-to-interference-plus-noise ratio and the Cramer-Rao bound (CRB) for angle-of-departure (AoD) estimation of the eavesdropper UAV target.

Boolean networks are a widely used modeling framework in systems biology for studying gene regulation, signal transduction, and cellular decision-making. Empirical studies indicate that biological Boolean networks exhibit a high degree of canalization, a structural property of Boolean update rules that stabilizes dynamics and constrains state transitions. Despite its central role, existing software packages provide limited support for the systematic generation of Boolean functions and networks with prescribed canalization properties. We present BoolForge, a Python toolbox for the random generation and analysis of Boolean functions and networks, with a particular focus on canalization. BoolForge enables users to (i) generate random Boolean functions with specified canalizing depth, layer structure, and related constraints; (ii) construct Boolean networks with tunable topological and functional properties; and (iii) analyze structural and dynamical features including canalization measures, robustness, modularity, and attractor structure. By enabling controlled generation alongside analysis, BoolForge facilitates ensemble-based investigations of structure-dynamics relationships, benchmarking of theoretical predictions, and construction of biologically informed null models for Boolean network studies. Availability and Implementation: BoolForge is implemented in Python ($\geq$3.10) and can be installed via \texttt{pip install boolforge}. Source code and documentation are available at this https URL. A PDF tutorial compendium is provided as Supplementary Material.

In previous work arXiv:2407.13556, we proposed an extended phase space structure at null infinity accommodating large gauge symmetries for sub$^n$-leading soft theorems in Yang-Mills, via dressing fields arising in the Stueckelberg procedure. Here, we give an explicit boundary action controlling the dynamics of these fields. This allows for a derivation from first principles of the associated charges, together with an explicit renormalization procedure when taking the limit to null and spatial infinity, matching with charges proposed in previous work. Using the language of fibre bundles, we relate the existence of Stueckelberg fields to the notion of extension/reduction of the structure group of a principal bundle, thereby deriving their transformation rules in a natural way, thus realising them as Goldstone-like objects. Finally, this allows us to give a geometric picture of the gauge transformation structure at the boundary, via a loop group coming from formal expansions in the coordinate transversal to the boundary.

Stochastic optimal control methods often struggle in complex non-convex landscapes, frequently becoming trapped in local optima due to their inability to learn from historical trajectory data. This paper introduces Memory-Augmented Potential Field Theory, a unified mathematical framework that integrates historical experience into stochastic optimal control. Our approach dynamically constructs memory-based potential fields that identify and encode key topological features of the state space, enabling controllers to automatically learn from past experiences and adapt their optimization strategy. We provide a theoretical analysis showing that memory-augmented potential fields possess non-convex escape properties, asymptotic convergence characteristics, and computational efficiency. We implement this theoretical framework in a Memory-Augmented Model Predictive Path Integral (MPPI) controller that demonstrates significantly improved performance in challenging non-convex environments. The framework represents a generalizable approach to experience-based learning within control systems (especially robotic dynamics), enhancing their ability to navigate complex state spaces without requiring specialized domain knowledge or extensive offline training.

We study $N$ run-and-tumble particles (RTPs) in one dimension interacting via a double-well potential $W(r)=-k_0 \, r^2/2+g \, r^4/4$, which is repulsive at short interparticle distance $r$ and attractive at large distance. At large time, the system forms a bound state where the density of particles has a finite support. We focus on the determination of the total density of particles in the stationary state $\rho_s(x)$, in the limit $N\to+\infty$. We obtain an explicit expression for $\rho_s(x)$ as a function of the ''renormalized" interaction parameter $k=k_0-3m_2$ where $m_2$ is the second moment of $\rho_s(x)$. Interestingly, this stationary solution exhibits a transition between a connected and a disconnected support for a certain value of $k$, which has no equivalent in the case of Brownian particles. Analyzing in detail the expression of the stationary density in the two cases, we find a variety of regimes characterized by different behaviors near the edges of the support and around $x=0$. Furthermore, we find that the mapping $k_0\to k$ becomes multi-valued below a certain value of the tumbling rate $\gamma$ of the RTPs for some range of values of $k_0$ near the transition, implying the existence of two stable solutions. Finally, we show that in the case of a disconnected support, it is possible to observe steady states where the density $\rho_s(x)$ is not symmetric. All our analytical predictions are in good agreement with numerical simulations already for systems of $N = 100$ particles. The non-uniqueness of the stationary state is a particular feature of this model in the presence of active (RTP) noise, which contrasts with the uniqueness of the Gibbs equilibrium for Brownian particles. We argue that these results are also relevant for a class of more realistic interactions with both an attractive and a repulsive part, but which decay at infinity.

We show that Reed-Solomon codes of dimension $k$ and block length $n$ over any finite field $\mathbb{F}$ can be deterministically list decoded from agreement $\sqrt{(k-1)n}$ in time $\text{poly}(n, \log |\mathbb{F}|)$. Prior to this work, the list decoding algorithms for Reed-Solomon codes, from the celebrated results of Sudan and Guruswami-Sudan, were either randomized with time complexity $\text{poly}(n, \log |\mathbb{F}|)$ or were deterministic with time complexity depending polynomially on the characteristic of the underlying field. In particular, over a prime field $\mathbb{F}$, no deterministic algorithms running in time $\text{poly}(n, \log |\mathbb{F}|)$ were known for this problem. Our main technical ingredient is a deterministic algorithm for solving the bivariate polynomial factorization instances that appear in the algorithm of Sudan and Guruswami-Sudan with only a $\text{poly}(\log |\mathbb{F}|)$ dependence on the field size in its time complexity for every finite field $\mathbb{F}$. While the question of obtaining efficient deterministic algorithms for polynomial factorization over finite fields is a fundamental open problem even for univariate polynomials of degree $2$, we show that additional information from the received word can be used to obtain such an algorithm for instances that appear in the course of list decoding Reed-Solomon codes.

We study the problem of denoising when only the noise level is known, not the noise distribution. Independent noise $Z$ corrupts a signal $X$, yielding the observation $Y = X + \sigma Z$ with known $\sigma \in (0,1)$. We propose \emph{universal} denoisers, agnostic to both signal and noise distributions, that recover the signal distribution $P_X$ from $P_Y$. When the focus is on distributional recovery of $P_X$ rather than on individual realizations of $X$, our denoisers achieve order-of-magnitude improvements over the Bayes-optimal denoiser derived from Tweedie's formula, which achieves $O(\sigma^2)$ accuracy. They shrink $P_Y$ toward $P_X$ with $O(\sigma^4)$ and $O(\sigma^6)$ accuracy in matching generalized moments and densities. Drawing on optimal transport theory, our denoisers approximate the Monge--Ampère equation with higher-order accuracy and can be implemented efficiently via score matching. Let $q$ denote the density of $P_Y$. For distributional denoising, we propose replacing the Bayes-optimal denoiser, $$\mathbf{T}^*(y) = y + \sigma^2 \nabla \log q(y),$$ with denoisers exhibiting less-aggressive distributional shrinkage, $$\mathbf{T}_1(y) = y + \frac{\sigma^2}{2} \nabla \log q(y),$$ $$\mathbf{T}_2(y) = y + \frac{\sigma^2}{2} \nabla \log q(y) - \frac{\sigma^4}{8} \nabla \!\left( \frac{1}{2} \| \nabla \log q(y) \|^2 + \nabla \cdot \nabla \log q(y) \right)\!.$$

We consider the effective theory of heterotic strings in two spacetime dimensions, in a double field theory-inspired formalism, manifestly consistent with $T$-duality in string theory. Restricting the gauge group to a single $\mathrm{U}(1)$, we study the charged black hole solution and perform a precise analysis of the properties of the dual geometry with the $\mathrm{O}(1,2; \mathbb{R})$-valued generalized metric. We comment on some aspects related to singularities and gauge dependence. We show that the classification program for higher derivative corrections can also be applied to the heterotic case. We further elucidate how a previously proposed solution to the equations of motion, parametrized in a manner fully non-perturbative in $\alpha'$, can be extended to the scenario with $r$ abelian fields and the corresponding $\mathrm{O}(1,1+r; \mathbb{R})$ symmetry. We discuss some novel features of the solution for charged black holes.

In this paper, we adapt the Discrete Variable (DV)-Circuit Quantum-Classical Physics-Informed Neural Network (QCPINN) and apply it for the first time to four typical reservoir seepage models. These include the pressure diffusion equation for heterogeneous single-phase flow, the nonlinear Buckley-Leverett (BL) equation for simplified two-phase waterflooding, the convection-diffusion equation for compositional flow considering adsorption, and the fully coupled pressure-saturation two-phase oil-water seepage equation for heterogeneous reservoirs with exponential permeability distribution. The QCPINN integrates classical preprocessing/postprocessing networks with a DV quantum core, leveraging quantum superposition and entanglement to enhance high-dimensional feature mapping while embedding physical constraints to ensure solution consistency. We test three quantum circuit topologies (Cascade, Cross-mesh, Alternate) and demonstrate through four numerical experiments that QCPINNs achieve higher prediction accuracy than classical PINNs. Specifically, the Alternate topology outperforms others in heterogeneous single-phase flow, BL equation simulations and heterogeneous fully coupled pressure-saturation two-phase flow, while the Cascade topology excels in compositional flow with convection-dispersion-adsorption coupling. The Cross-mesh topology shows competitive early-stage convergence and accuracy across scenarios with balanced performance in coupled two-phase flow. Our work verifies the feasibility of QCPINN for reservoir engineering applications, bridging the gap between quantum computing research and industrial practice in oil and gas engineering.

We explore the sum over topologies in AdS$_3$ quantum gravity and its relationship with the statistical interpretation of the boundary theory. We formulate a statistical version of the conformal bootstrap that systematizes the universal statistical properties of high-energy CFT$_2$ data. We identify a series of surgery moves on bulk manifolds that precisely reflect the requirements of typicality and crossing symmetry of the boundary ensemble. These surgery moves generate a large number of bulk manifolds that have to be included in any reasonable definition of the gravitational path integral. We show that this procedure generates only on-shell (hyperbolic) manifolds, although it does not produce all of them. These proofs rely on structure theorems of 3-manifolds, which non-trivially interact with the requirements of the statistical boundary ensemble. We illustrate the application of this procedure with many examples, such as Euclidean wormholes, twisted $I$-bundles and handlebody-knots. Our findings reveal a large space of possible choices of which manifolds can be included in the gravitational path integral, reflecting a wide range of possible statistical ensembles consistent with crossing symmetry and typicality.

We study a structured bi-level optimization problem where the upper-level objective is a smooth function and the lower-level problem is policy optimization in a Markov decision process (MDP). The upper-level decision variable parameterizes the reward of the lower-level MDP, and the upper-level objective depends on the optimal induced policy. Existing methods for bi-level optimization and RL often require second-order information, impose strong regularization at the lower level, or inefficiently use samples through nested-loop procedures. In this work, we propose a single-loop, first-order actor-critic algorithm that optimizes the bi-level objective via a penalty-based reformulation. We introduce into the lower-level RL objective an attenuating entropy regularization, which enables asymptotically unbiased upper-level hyper-gradient estimation without solving the unregularized RL problem exactly. We establish the finite-time and finite-sample convergence of the proposed algorithm to a stationary point of the original, unregularized bi-level optimization problem through a novel lower-level residual analysis under a special type of Polyak-Lojasiewicz condition. We validate the performance of our method through experiments on a GridWorld goal position problem and on happy tweet generation through reinforcement learning from human feedback (RLHF).

Collective excitations of Fractional Quantum Hall (FQH) liquids at long wavelengths are thought to be of a generally covariant geometric nature, governed by area-preserving diffeomorphisms ($\mathrm{SDiff}$). But current analyses rely solely on the corresponding perturbative $w_\infty$ Lie algebra. We argue this is insufficient: We identify a non-perturbative construction of the effective Maxwell-Chern-Simons quantum field theory which carries unitary $\mathrm{SDiff}$ equivariance. But this turns out to be non-differentiable, suggesting underappreciated subtleties when the usual Hilbert space truncation is removed.

Applied work under interference typically models outcomes as functions of own treatment and a low-dimensional exposure mapping of others' treatments, even when that mapping may be misspecified. We ask what policy object such exposure-based procedures target. Taking the marginal policy effect as primitive, we show that any researcher-chosen exposure mapping induces a unique pseudo-true outcome model: the best approximation to the underlying potential outcomes within the class of functions that depend only on that mapping. This yields a decomposition of the marginal policy effect into exposure-based direct and spillover effects, and each component optimally approximates its oracle counterpart, with a sign-preserving interpretation under monotonicity. We then study a structured misspecification setting in which outcomes depend on both network spillovers and a global equilibrium channel, while the analyst may model only one. In this setting, we obtain a sharper asymptotic decomposition into direct, local, and global components, implying that existing estimators recover their respective oracle channel-specific effects even when the other channel is present but omitted from the maintained this http URL analysis also yields phase transitions in convergence rates and higher-order expansions for Z-estimators. A semi-synthetic experiment calibrated to a large cash-transfer study illustrates the empirical relevance of the framework.

This thesis documents a voyage towards truth and beauty via formal verification of theorems. To this end, we develop libraries in Lean 4 that present definitions and results from diverse areas of MathematiCS (i.e., Mathematics and Computer Science). The aim is to create code that is understandable, believable, useful, and elegant. The code should stand for itself as much as possible without a need for documentation; however, this text redundantly documents our code artifacts and provides additional context that isn't present in the code. This thesis is written for readers who know Lean 4 but are not familiar with any of the topics presented. We manifest truth and beauty in three formalized areas of MathematiCS (optimization theory, matroid theory, and the theory of grammars). In the pursuit of truth, we focus on identifying the trusted code in each project and presenting it faithfully. We emphasize the readability and believability of definitions rather than choosing definitions that are easier to work with. In search for beauty, we focus on the philosophical framework of Roger Scruton, who emphasizes that beauty is not a mere decoration but, most importantly, beauty is the means for shaping our place in the world and a source of redemption, where it can be viewed as a substitute for religion.

Recently, Fendley et al. (2025) [arXiv:2511.04674] revealed a new simple way to demonstrate the integrability of XYZ Heisenberg model by constructing a one-parameter family of integrals of motion in the matrix product operator (MPO) form with bond dimension 4. In this work, I report on the discovery of two-parameter families of MPOs that commute with Heisenberg spin chain Hamiltonian in case of various anisotropies (XXX, XXZ, XX, XY and XYZ). These solutions are connected by taking appropriate limits. For all cases except XYZ, I also write down Floquet charges of two-step Floquet protocols corresponding to the Trotterization. I describe a symbolic algebra approach for finding such integrals of motion and speculate about possible generalizations and applications.

Four distinct admissibility geometries govern sequential and distribution-free inference: Blackwell risk dominance over convex risk sets, anytime-valid admissibility within the nonnegative supermartingale cone, marginal coverage validity over exchangeable prediction sets, and Cesàro approachability (CAA) admissibility, which reaches the risk-set boundary via approachability-style arguments rather than explicit priors. We prove a criterion separation theorem: the four classes of admissible procedures are pairwise non-nested. Each geometry carries a different certificate of optimality: a supporting-hyperplane prior (Blackwell), a nonnegative supermartingale (anytime-valid), an exchangeability rank (coverage), or a Cesàro steering argument (CAA). Martingale coherence is necessary for Blackwell admissibility and necessary and sufficient for anytime-valid admissibility within e-processes, but is not sufficient for Blackwell admissibility and is not necessary for coverage validity or CAA-admissibility. All four criteria can be viewed through a common schematic template (minimize Bayesian risk subject to a feasibility constraint), but the decision spaces, partial orders, and performance metrics differ by criterion, making them geometrically incompatible. Admissibility is irreducibly criterion-relative.

We study torus knot invariants in the lens space $S^{3}/\mathbb{Z}_{p}$ within Chern--Simons theory. Using the surgery and modular description of lens spaces, we derive a general expression for the invariant of an $(\alpha,\beta)$ torus knot in this background. In the large-$N$ limit these invariants simplify and acquire a universal form: the invariant of an $(\alpha,\beta)$ torus knot in $S^{3}/\mathbb{Z}_{p}$ can be expressed in terms of the invariant of the $(\alpha,\alpha+p\beta)$ torus knot in $S^{3}$. After an appropriate redefinition of knot variables, the generating functions of these invariants exhibit a structure analogous to quiver partition functions. Since the associated quiver is independent of the rank $N$ and level $k$ of Chern--Simons theory, the large-$N$ result provides a direct way to identify the underlying quiver, allowing us to determine the quiver structure associated with torus knots in $S^{3}/\mathbb{Z}_{p}$.

We present a formulation of an optimal control problem for a two-dimensional diffusion process governed by a Fokker-Planck equation to achieve a nonequilibrium steady state with a desired circulation while accelerating convergence toward the stationary distribution. To achieve the control objective, we introduce costs for both the probability density function and flux rotation to the objective functional. We formulate the optimal control problem through dimensionality reduction of the Fokker-Planck equation via eigenfunction expansion, which requires a low-computational cost. We demonstrate that the proposed optimal control achieves the desired circulation while accelerating convergence to the stationary distribution through numerical simulations.

Markov decision processes (MDPs) is viewed as an optimization of an objective function over certain linear operators over general function spaces. A new existence result is established for the existence of optimal policies in general MDPs, which differs from the existence result derived previously in the literature. Using the well-established perturbation theory of linear operators, policy difference lemma is established for general MDPs and the Gauteaux derivative of the objective function as a function of the policy operator is derived. By upper bounding the policy difference via the theory of integral probability metric, a new majorization-minimization type policy gradient algorithm for general MDPs is derived. This leads to generalization of many well-known algorithms in reinforcement learning to cases with general state and action spaces. Further, by taking the integral probability metric as maximum mean discrepancy, a low-complexity policy gradient algorithm is derived for finite MDPs. The new algorithm, called MM-RKHS, appears to be superior to PPO algorithm due to low computational complexity, low sample complexity, and faster convergence.

Population-based learning paradigms, including evolutionary strategies, Population-Based Training (PBT), and recent model-merging methods, combine fast within-model optimisation with slower population-level adaptation. Despite their empirical success, a general mathematical description of the resulting collective training dynamics remains incomplete. We introduce a theoretical framework for neural network training based on two-time-scale population dynamics. We model a population of neural networks as an interacting agent system in which network parameters evolve through fast noisy gradient updates of SGD/Langevin type, while hyperparameters evolve through slower selection--mutation dynamics. We prove the large-population limit for the joint distribution of parameters and hyperparameters and, under strong time-scale separation, derive a selection--mutation equation for the hyperparameter density. For each fixed hyperparameter, the fast parameter dynamics relaxes to a Boltzmann--Gibbs measure, inducing an effective fitness for the slow evolution. The averaged dynamics connects population-based learning with bilevel optimisation and classical replicator--mutator models, yields conditions under which the population mean moves toward the fittest hyperparameter, and clarifies the role of noise and diversity in balancing optimisation and exploration. Numerical experiments illustrate both the large-population regime and the reduced two-time-scale dynamics, and indicate that access to the effective fitness, either in closed form or through population-level estimation, can improve population-level updates.