Papers

We study a class of McKean-Vlasov control problems with discrete path dependence. The coefficients and cost functional may depend on finitely many past values of the controlled state, observed at fixed deterministic times, and on their joint law. We establish well-posedness of the controlled state equation and derive necessary and sufficient optimality conditions through a stochastic Pontryagin maximum principle. The adjoint process is characterized by a backward stochastic differential equation with jumps at fixed observation times. Each jump represents the conditional sensitivity of future costs with respect to the corresponding observed state and its distribution. In the linear-quadratic case, we prove global solvability of the resulting forward-backward system by combining a continuation argument with a mean-field Riccati reduction. We finally discuss an application to time-series generation, where the terminal cost is given by a kernel-based discrepancy between the law of the sampled path and a target path distribution.

In this paper, we develop a fixed point framework for paired contractive mappings in complete $b$-metric spaces. We introduce the class of paired-Chatterjea contractions and establish a corresponding fixed point theorem. We further show that every Chatterjea-type contraction is a paired-Chatterjea contraction, whereas the converse implication does not hold, demonstrating that the paired formulation properly extends the Chatterjea-type class. Additionally, we establish a fixed point theorem for Paired contractions under the condition $\alpha\in[0,1)$, thereby extending the admissible range of the contraction constant from $[0,\frac1s)$ in Bouker et al.~(2026) to [0,1). Furthermore, we examine the Paired-Kannan fixed point result of Bouker et al.~(2026) and provide an example showing that the stated coefficient condition does not guarantee the claimed fixed point conclusion in $b$-metric spaces. Motivated by this observation, we establish a corrected Paired-Kannan fixed point theorem under a complete and appropriately formulated set of hypotheses. Under suitable contractive conditions, we prove that the contractions under consideration have fixed points if and only if they have no periodic point of prime period 2. Moreover, we show that each of these contractions has at most two fixed points. Several examples are presented to illustrate our results and distinguish the proposed paired contractive mappings from existing ones.

We study the two-dimensional incompressible magnetohydrodynamic system near the Couette equilibrium $\bigl((y,0)^{\rm T},(\beta,0)^{\rm T}\bigr)$ on $\mathbb T\times\mathbb R$, in the anisotropic regime where both viscosity and magnetic diffusivity act only in the vertical direction. For the inviscid linearized problem with $|\beta|>1/2$, we prove sharp linear-in-time growth of the vorticity and current density at the level of the time rate. In contrast, the horizontal components of the velocity and magnetic perturbations remain uniformly bounded, while the vertical components exhibit quantitative inviscid damping at the rate $\langle t\rangle^{-1}$. For the nonlinear problem, we introduce shear-adapted Fourier multipliers that simultaneously capture enhanced dissipation, critical-time effects, and echo-type resonant interactions. Under a suitable horizontal background magnetic field and a quantitative compatibility condition between the magnetic field strength, viscosity, and magnetic diffusivity, we establish global nonlinear stability for divergence-free perturbations satisfying $ \left\| \bigl( \mathbf v_{\rm in}-(y,0)^{\rm T}, \mathbf H_{\rm in}-(\beta,0)^{\rm T} \bigr) \right\|_{H^N} \leq \varepsilon_0\min\{\mu,\nu\}^{1/2}, N\geq4$. Moreover, the nonzero horizontal Fourier modes decay in a shear-adapted $H^N$ norm at the enhanced-dissipation rate $e^{-c\min\{\mu,\nu\}^{1/3}\,t},$ and the vertical velocity and magnetic components gain an additional inviscid-damping factor $\langle t\rangle^{-1}$.

The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty. We study this phenomenon for inversion over \(\mathbb F_{2^n}\), with field elements expressed in varying ordered \(\mathbb F_2\)-bases. We prove that two ordered bases induce the same coordinate inversion map if and only if they belong to the same Galois orbit. Since every orbit has size \(n\), the correspondence between ordered bases and distinct inversion maps is exactly \(n\)-to-one. We then analyze three Boolean formulations of inversion. The reference formulation has algebraic degree \(n-1\) and joint ANF leap \(1\), the mixed representation formulation has degree \(2(n-1)\) and joint ANF leap \(2\), and the complete raw formulation has degree at most \(3(n-1)\) and joint ANF leap at least \(n\). Exhaustive computations agree with the theoretical results and bounds in the cases considered. Controlled experiments with multilayer perceptrons show the same ordering in learning difficulty, while Galois orbit redundancy provides only a limited generalization benefit under the tested conditions. These results show that exact redundancy among representations can coexist with changes in Boolean structure and learning behavior when the representation is exposed as part of the input.

We study stochastic gradient descent with random reshuffling for finite sums \[ F(x)=\frac1n\sum_{i=1}^n f_i(x). \] For fresh reshuffling with a constant component stepsize, if each $f_i$ has an $L$-Lipschitz gradient and the average $F$ is $\mu$-strongly convex with a Lipschitz-continuous Hessian, we prove the last-epoch rate \[ \mathbb E[F(y_K)-F(x_\star)] =\widetilde O\!\left(T^{-2}+n^2T^{-3}\right), \qquad T=nK, \] matching the known quadratic lower bound in its $(n,K)$-dependence. The components may be nonconvex, and no componentwise Hessian continuity or separate bounded-iterate assumption is required. More generally, a $\nu$-Hölder-continuous average Hessian adds only $\widetilde O(n^{1+\nu}T^{-2-2\nu})$, so every $\nu\ge 1/2$ preserves the quadratic rate. Under convex components, a decreasing-stepsize result removes the large-epoch requirement and recovers the same two-term scale once $nK$ exceeds the condition-number scale. We also analyze epoch-wise ProxRR for $\mathcal P=F+\psi$. Writing $x^\dagger$ for the composite minimizer and $\beta_\star=\|\nabla F(x^\dagger)\|$, we prove \[ \mathbb E\|y_K-x^\dagger\|^2 =\widetilde O\!\left( \frac{\beta_\star^2}{K^2} +T^{-2}+n^2T^{-3} +n^{1+\nu}T^{-2-2\nu} \right). \] For $\nu\ge 1/2$, we show that the $\beta_\star^2/K^2$ splitting term is unavoidable and obtain a matching lower bound up to logarithms in the stated constant-stepsize regime.

We prove that every one-relator group is sofic, answering a question of Nate Brown. More generally, every group admitting a reducible presentation without proper powers is sofic, with one proper-power relation allowed at the final reducible step. The proof starts from the endpoint-preserving edge replacements of Poulin–Wróbel. We retain the free-group word carried by each terminal replacement and use it to define an \(F(S)\)-valued cocycle repair; in its relative form, the repair preserves the previously chosen generator coordinates while controlling the new relator defect. Coinduction transports the lower cocycle through successive one-relator-product extensions, allowing the construction to be iterated along reducible presentations. For a final relator \(w^m\), cyclic translates of the repaired set amplify a zero-defect set of measure close to \(1/m\) to one of measure close to one. A cocycle criterion then converts arbitrarily small relator defect into finite permutation approximations via a treeable skew-product orbit relation.

Uniform exponential tracking is addressed for nonstationary trajectories of the nonlinear Kuramoto–Sivashinsky equation subject to a constant input delay. The reference belongs to a family of complete trajectories contained in the global attractor and need not be stationary, periodic, slowly varying, or generated by a finite-dimensional exosystem. The delayed input is represented by a first-order transport equation coupled with the fourth-order tracking-error dynamics. A predictor–backstepping transformation compensates for the temporal mismatch between command generation and actuation by mapping the augmented closed-loop system into the nominal delay-free error dynamics driven by the outgoing trace of a homogeneous transport subsystem. This subsystem vanishes after one delay interval. Uniform attractor bounds permit the feedback parameters and stability constants to be selected independently of the initial time and the reference trajectory. Global well-posedness and uniform exponential stability are established in the augmented state space. Numerical results show that, for the considered configuration, uncompensated delayed feedback amplifies the tracking error, whereas predictor compensation restores sustained decay. Predictor-consistency, finite-time-extinction, and discretization-refinement diagnostics support the numerical implementation.

The Integrated Healthcare Timetabling Competition 2024 (IHTC-2024) defined a benchmark that integrates surgical case planning, patient admission scheduling and nurse-to-room assignment. Its instance set remains live: the organizers keep a public table of best-known upper bounds that is open to new submissions. We report five improved upper bounds on that table, for instances i02 (1263, previously 1264), i05 (12744, previously 12760), m01 (3380, previously 3384), m03 (6692, previously 6697) and m04 (3315, previously 3318). All five solutions were checked with the official IHTP validator and carry no hard-constraint violations. The solutions are not built from scratch: each is derived from the corresponding solution published on the benchmark page, holding its admission days, room assignments and operating-theater assignments fixed, and reoptimizing only the nurse-to-room-and-shift layer to proven optimality. The improvements are small in magnitude, between 0.075% and 0.125%, but they carry a structural observation: on the instances where the nurse layer can be closed exactly, it closes within seconds on all but one of them, and several of the published state-of-the-art solutions fall short of optimality in that layer.

Soft Guessing Random Additive Noise Decoding queries corrections in an observation-dependent order. For a known channel, the matched order ranks corrections by nonincreasing conditional posterior probability. Under channel uncertainty, admissible channels may induce matched orders that assign different ranks to the same correction at the same observation. We instead use a posterior-envelope order, defined by the pointwise supremum of codebook-independent uniform-input correction posteriors. For fixed measurable posterior representatives and a measurable envelope, this order minimizes, at each observation, the worst-case logarithm of the rank–posterior product over admissible channel–correction pairs with positive posterior probability. For uniform-subset codebooks, this finite-block minimax property does not by itself determine the ensemble-average error exponent, which depends on the distribution of the realized correction rank. For i.i.d. correction–observation pairs with a finite correction alphabet whose joint distribution is dominated by the product of counting measure on that alphabet and a $\sigma$-finite measure on the observation space, Arimoto conditional Rényi entropy characterizes the matched rank-moment spectrum. An auxiliary posterior-envelope order preserves this spectrum on $[0,1]$ when its posterior family contains the corresponding conditional power tilts and has subexponential conditional Shtarkov complexity on observation sets whose complement probabilities decay exponentially faster than the reciprocal correction-space cardinality. Under additional differentiability and entropy nondegeneracy conditions, the auxiliary order attains the matched uniform-subset ensemble-average error exponent at every fixed normalized code rate strictly between zero and one. For uniform-input memoryless channels, this exponent equals Gallager's uniform-input random-coding exponent.

For a finite group $G$, let $\rho^{*}(G)$ be the set of primes dividing some sub-class size of $G$, and let $\sigma^{*}(G)$ be the maximum number of distinct prime divisors of a sub-class size. Motivated by the conjugacy-class version of Huppert's $\rho$–$\sigma$ problem, we prove that \[ |\rho^{*}(G)|<2\sigma^{*}(G) \] for every finite nonnilpotent group $G$. This answers \cite[Remark~5.3]{QY} and extends to arbitrary finite nonnilpotent groups the coefficient-$2$ bound proved in \cite[Theorem~1.7]{QY} for solvable nonnilpotent groups. The coefficient $2$ is best possible. The proof does not use the classification of finite simple groups.

We determine the smallest square size at which the operator norm and the completely bounded norm of a Schur multiplier on $S^4$ can differ. More precisely, for $A=\begin{bmatrix}1&2&0\\1&-2&0\\2&2\mathrm{i}&0\end{bmatrix}$ we prove \[ \|M_A\|_{S^4_3\to S^4_3}^2\leq\frac{1119}{250} <\left(\frac{18983532}{947485}\right)^{1/2} \leq \|\mathrm{Id}_{S^4_2}\otimes M_A\|_{S^4_6\to S^4_6}^2. \] We also prove that no $2\times2$ Schur multiplier can have unequal operator and completely bounded norms on $S^4$, so square size three is optimal. In addition, we determine the exact ordinary norm of the previous multiplier analytically and give an independent computer-assisted certification of a rational upper bound.

We develop a geometric theory of congruence spectra for semirings. We define the congruence spectrum for a semiring as the set of prime congruences on it, equip it with the Zariski topology, and construct a structure sheaf. For a class of semirings including $\boldsymbol{B}$-algebras, we establish quasi-compactness of their congruence spectra. We then prove reconstruction results for congruence spectra of quotients $K/\boldsymbol{E}(V)$, where $V$ is a subset of Euclidean space, $K$ is a tropical Laurent polynomial semiring, a tropical Laurent polynomial function semiring, or a tropical rational function semifield, and $\boldsymbol{E}(V)$ is the congruence of pairs of elements of $K$ that agree on $V$. The global sections of the structure sheaf recover $K/\boldsymbol{E}(V)$, and the corresponding algebraic and geometric categories are contravariantly equivalent. For finite unions of $\boldsymbol{R}$-rational polyhedral sets $V$, we completely classify the prime congruences on $K / \boldsymbol{E}(V)$. As an application, we study rational function semifields of tropical curves with parallel rays. We show that their congruence spectra recover the underlying curves together with the parallelism of rays and their intrinsic metrics.

We determine the positive scalar curvature Kähler cone of every minimal compact Kähler surface of Kodaira dimension $-\infty$. By the Enriques–Kodaira classification, every such surface is either $\mathbb P^2$ or a geometrically ruled surface $\mathbb P(E)\to\Sigma_g$, where $E$ is a rank-two holomorphic vector bundle over a compact Riemann surface of genus $g$. On $\mathbb P^2$, every Kähler class admits a Kähler metric of positive scalar curvature, whereas on $\mathbb P(E)$, every Kähler class of positive total scalar curvature admits a metric of positive scalar curvature if and only if $g\le1$ or $E$ is slope-semistable. We further prove that if a smooth compact Kähler surface admits a birational morphism onto a ruled surface $\mathbb P(E)\to\Sigma_g$, where $g\ge2$ and $E$ is slope-unstable, then it contains a Kähler class of positive total scalar curvature admitting no positive scalar curvature Kähler metric. Finally, for every smooth compact Kähler surface $X$ birational to $Y=\mathbb P^2$ or $Y=\mathbb P(E)$, we show that Kähler classes admitting positive scalar curvature metrics on $Y$ are exactly the pushforwards of classes admitting such metrics on $X$.

We determine the radii of concavity for the classes $\mathcal{S}_e^*$ and $\mathcal{C}_e$ of univalent functions associated with the exponential mapping $e^z$. Under the geometric framework of Avkhadiev–Wirths for conformal mappings with unbounded convex complements of opening angle $\pi A$ ($A \in (1,2]$), the radii are characterized by explicit transcendental equations. Sharpness is established globally by constructing explicit univalent extremal functions related to the disk automorphism $\omega_0(z) = -z$, and the strict monotonicity of these radii with respect to the parameter $A$ is verified numerically.

Aircraft outer mould lines are often heavily constrained early by payload requirements, manufacturability limits, and low-observability considerations. The remaining aerodynamic design space is typically small, highly constrained, and difficult to represent using a concise and physically meaningful set of design variables. A structural modal parameterisation method (MPM) is introduced for such problems. The method constructs a tunable pseudo-structure and solves its eigenvalue problem to obtain a modal basis, from which a selected subset of modes parameterises the geometry. Boundary conditions, stiffness distribution, density, thickness, and added masses are treated as intentional basis-design variables that shape the admissible deformation space prior to optimisation. The resulting design coordinates are independent of any specific flow solver, making the approach broadly applicable. For tightly constrained aerodynamic shape optimisation problems, it provides a practical balance of parameter conciseness, robust constraint awareness, and geometric flexibility.

Chronic wasting disease (CWD) is a fatal prion disease affecting deer, elk, moose, reindeer, muntjac, and other cervids. Because free-ranging cervid populations face environmental variability and randomness, deterministic models may miss important dynamics like stochastic fade-out. We develop a stochastic Susceptible-Infectious-Environmental model using differential equations with reflection to ensure the susceptible class remains non-negative. We examine how environmental variability influences cervid populations as CWD pressure and control measures increase. For the deterministic model, we derive the basic reproduction number as the sum of direct and environmental contributions, showing the endemic phase arises at R0=1. For the stochastic system, we establish local well-posedness, positivity, and the disease-free law. The top Lyapunov exponent for invasion remains unaffected by reflection. We evaluate CWD mitigation using a deep reinforcement learning agent trained with Proximal Policy Optimization in a hybrid action space, comparing hunting, decontamination, and combined strategies. In the deterministic case, hunting alone can control the disease but reduces the population by about 58%, while decontamination requires sustained effort. The combined policy more than doubles the cervid population and nearly eliminates infection and contamination. In the stochastic case, the policy contains the disease in about 80% of runs, with 10% experiencing large outbreaks; effectiveness decreases as noise increases. Across all scenarios, the agent consistently emphasizes environmental decontamination, the key control method.

Complete permutation polynomials play an important role in cryptography, particularly in the design of cryptographic primitives such as the Lai–Massey scheme and S-boxes. We generalize a result of Sun, Li, Guo, and Qu (2021) by characterizing the complete permutation behavior of the mapping $\Psi(X)=M(X+\psi(AX))$ over $\mathbb{F}_q^n$, where $\mathbb{F}_q$ is a finite field of $q$ elements with $q$ being a prime power, $M\in GL(n, \mathbb{F}_q)$, $GL(n, \mathbb{F}_q)$ is the general linear group of order $n$ over $\mathbb{F}_q$, $A_{m \times n}$ is a full-rank matrix over $\mathbb{F}_q$, and $\psi=(\psi_1,\psi_2,\ldots,\psi_n)$ with each component function $\psi_i:\mathbb{F}_q^m\to\mathbb{F}_q$. Furthermore, we establish criteria for the permutation and complete permutation properties of the mapping $F(X)=T(X+B^tf(AX))$ over $\mathbb{F}_{q}^n$, $f: \mathbb{F}_{q}^{m} \rightarrow \mathbb{F}_{q}^{n-m}$, $T \in GL(n, \mathbb{F}_q)$, $A_{m \times n}$ and $B_{(n-m)\times n}$ are full-rank matrices over $\mathbb{F}_q$, $B^t$ represents the transpose of the matrix $B$, and $0<m<n$ are integers. These results also generalize an earlier result of Gravel and Panario (2023), who showed that any arbitrary function $f$ from $\mathbb{F}_q^m$ to $\mathbb{F}_q^{\,n-m}$ can be extended to a bijection over $\mathbb{F}_{q}^n$ through the mapping $F(X)=T(X+B^tf(AX))$, under the condition $AB^t=0$. Here we do not impose the restriction that $AB^t=0$.

The "Riemann-type integrals" of the title are not a single new integral, but the family of classical constructions – Riemann, Riemann-Stieltjes, Young-Kondurar, complex-analytic, and Itô – unified by one elementary method. We give a simple argument computing the Riemann integral of a polynomial directly, establishing the fundamental theorem of calculus for polynomials and then continuous functions. The same idea extends to the real line, the complex plane, the multidimensional case, the Riemann-Stieltjes integral, and the Itô integral of a polynomial of a Wiener process (more generally, a continuous semimartingale), recovering the classical Itô formula. The contribution is methodological: a single algebraic telescoping identity, paired with an estimate on a quadratic remainder term, drives every case. What changes is only the remainder's fate: it vanishes identically for polynomials on $\mathbb{R}$; it vanishes in the limit under a quadratic-variation condition $Q(\pi)\to0$ on $\mathbb{C}$ and for Hölder paths; and for a continuous semimartingale it survives as the quadratic variation producing the Itô correction – except for a planar Brownian motion, or more generally a conformal martingale, where an algebraic cancellation inside the complex square removes it again. We regard this last point – that the Itô correction and its cancellation in the conformal case are two instances of the same quadratic remainder, not unrelated facts – as the paper's most distinctive observation. The manuscript has three parts: Part I develops the core method on $\mathbb{R}$ and $\mathbb{C}$; Part II extends it deterministically to the multidimensional, Riemann-Stieltjes, and Hölder (Young-Kondurar) cases; Part III extends it to the stochastic setting.

Let $n\geq 1$ be an odd integer, set $m=(n-1)/2$ and let $M$ be an $n\times n$ matrix whose coefficients are of the form $M_{i,j}=aij+bi+cj+d$ where $0\leq i,j\leq n-1$. Then we prove that for all squares centered at the central coefficient $M_{m,m}$, the mean value of the square equals $M_{m,m}$.

A symmetric distribution of charges or currents typically produces an electromagnetic field having the same symmetry group. However, under special circumstances we show that a discrete symmetry of the source distribution leads to a larger continuous symmetry for the resulting field in restricted regions. The conditions for this promotion from discrete to continuous symmetry are that the fields be affine linear and the symmetry of the source distribution act irreducibly on the ambient space. We illustrate this effect with many examples. This phenomenon is pedagogically interesting both as a source of inspiration in electro- and magnetostatics and as an accessible example of group representation theory.

We write down generators and relations for a minimalist version of the nilHecke category which allows us to categorify the divided powers of a generating functor. We then classify these minimalist counterparts of the nilHecke category, subject to suitable assumptions. A minimalist version of the odd nilHecke category is introduced as well.

A finite simple graph $G$ is called a cograph if it does not contain the path on four vertices $P_4$ as an induced subgraph. It is classically known that the family of cographs are well-quasi-ordered by the induced subgraph relation \cite{D}. In preceding work of Knudsen and the third author \cite[Theorem 7.2]{KR}, it was shown that this well-quasi-order statement admitted a categorification, which allowed those authors to prove universal finite generation statements about the homology groups of configuration spaces on cographs \cite[Theorem 1.5]{KR}. In this work, we expand \cite[Theorem 7.2]{KR} to be compatible with the family of polynomial rings on the vertex sets of cographs. By consequence, we are able to prove a number of universality results related with edge and toric ideals of these polynomial ring, partially generalizing and expanding upon work of Kahle \cite{kahle2019binomial}. We also conclude strong restrictions on the kinds of topologies that can arise from graph complexes and anchored configuration spaces associated to cographs, as well as combinatorial constraints on the possible combinatorics of hyperplane arrangements of cographs.

Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\mathcal{O}_S^\times$ for its unit group. Let $\pi : X \rightarrow \mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\operatorname{Red}(\pi)$ the set of $\alpha \in \mathbb{P}^1(K)$ such that the fibre $\pi^{-1}(\alpha)$ is reducible (i.e. the Galois action on the fibre is intransitive). Hilbert's Irreducibility Theorem asserts that $\operatorname{Red}(\pi)$ is contained in a thin subset of $\mathbb{P}^1(K)$. In this paper we give an explicit description of $\mathcal{O}_S^\times \cap \operatorname{Red}(\pi)$. As an application we prove the following result inspired by a classical theorem of Pólya and Siegel. Let $p_1,\dotsc,p_s$ be rational primes. Let $f \in \mathbb{Q}[x]$.Then the following are equivalent: - There are infinitely many tuples $(e_1,\dotsc,e_s) \in \mathbb{N}^s$ such that the polynomial $f(x)-p_1^{e_1} \cdots p_s^{e_s}$ is reducible. - $f=p_1^{a_1} \cdots p_s^{a_s} g^\ell$ (with $\ell$ prime) or $f=-4 p_1^{a_1} \cdots p_s^{a_s} g^4$ for some $g \in \mathbb{Q}[x]$ and some integers $a_1,\dotsc,a_s$.

This paper considers a downlink communication framework comprising a simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-aided by orthogonal time frequency space (OTFS) and non-orthogonal multiple access (NOMA) technologies. Further, delay-Doppler mobility in such frameworks renders classical alternating optimization impractical for per-coherence interval reconfiguration. To mitigate such issues, the STAR-RIS phase-shift and energy-splitting design is formulated as a constrained, non-convex sum-rate maximization problem with closed-form maximum ratio transmission beamforming and fixed NOMA power allocation. To circumvent the per-interval re-optimization burden, a deep reinforcement learning (DRL) approach is adopted that maps observed channel realizations to STAR-RIS configurations through a single forward pass. Specifically, Beta-Space Soft Actor-Critic (SAC-BSE), a maximum entropy DRL agent, is proposed. Simulation results, with two NOMA-multiplexed users on each STAR-RIS branch, confirm rapid convergence, limit the sum-rate degradation to roughly 10% across a 128-fold user-speed range, and yield consistent gains over OTFS-only, NOMA-only, STAR-RIS-only, fixed-split, and mode-switching baselines as transmit power and the number of STAR-RIS elements increase.

I present a matrix-free symmetric interior penalty discontinuous Galerkin method for adaptively refined Cartesian meshes on GPUs. At non-matching interfaces, auxiliary shadow cells represent the adjacent coarse polynomial on the fine level. This approach converts non-matching interfaces into matching faces, permitting uniform face evaluation throughout the mesh. I prove that this construction is equivalent to the standard non-matching formulation. Furthermore, this representation yields a local geometric multigrid method in which frozen shadows provide inter-level boundary data and carry residual contributions to coarser levels. A primal–dual pairing eliminates shadow assembly from the Krylov iteration. Numerical experiments with cubic elements ($p=3$) on an NVIDIA A100 show stable multigrid convergence under increasing refinement depth and efficient GPU execution.

We introduce a reduced-order learning framework that couples a convolutional autoencoder to a structured latent neural ordinary differential equation. A learned scalar generates the latent dynamics through an antisymmetric operator and a positive-definite dissipative operator. The antisymmetric component permits motion along scalar level sets, while the dissipative component decreases the scalar. Adding a coercive quadratic term gives bounded continuous-time latent trajectories and approach to the scalar's critical set under the stated assumptions. The scalar provides a structural constraint rather than an approximation of physical energy. The encoder, decoder, scalar, and operators are trained jointly on short snapshot windows without time-derivative supervision, physical-energy labels, or dissipation penalties. At inference, one initial state is encoded and evolved continuously in latent space, with decoding only at requested times. We demonstrate the framework on micromagnetic magnetization trajectories from NIST $\mu$MAG Standard Problem 4 and compare it with dissipative-only dynamics and an unstructured latent neural differential equation. Repeated training runs show competitive observed-horizon accuracy, while selected longer rollouts illustrate improved retention of approximately stationary states relative to the unstructured baseline. These results motivate scalar-generated latent dynamics as an inductive bias for reduced-order prediction and other learning tasks where bounded evolution and equilibrium behavior are important.

The factorization problem is a classical topic in the theory of polynomial rings. In this paper, we obtain an irreducibility criterion for central polynomials over division algebras. An example demonstrating the computation of factors for a central polynomial is also presented.

Time-dependent density functional theory, though exact in theory, is in practice applied in an adiabatic approximation using exchange-correlation functionals with only local temporal dependence. Simultaneously, the exact correlation potential formally depends on, among other quantities, the time-history of electron densities. Here we develop a framework to learn neural network models of correlation functionals that feature explicit memory-dependence. Our framework includes two different approaches, both linked through their use of adjoint-based optimization. One approach decouples the learning of the functional from inversion to find ground truth values of the correlation potential. The other approach learns the functional directly without requiring inversion. We apply these methods to modeling the electron dynamics of two-electron systems in two spatial dimensions. In both cases, our methods yield correlation functionals with low test set propagation error, outperforming standard local density and generalized gradient approximation functionals by one to two orders of magnitude. Overall, our framework points at one strategy to move beyond the adiabatic approximation and develop memory-dependent correlation functionals that yield accurate propagation for excited state and/or non-equilibrium dynamics.

We solve an open problem posed by Peter Fritz in \cite{Fritz} by proving that there are uncountably many C-Post complete congruential modal logics which are neighborhood complete. The method is algebraic: we construct an uncountable sequence of varieties of modal algebras which are minimal in the lattice of all subvarieties of modal algebras and are generated by a single complete and atomic modal algebra.

We give a degree-one Pieri formula for integral nonsymmetric Jack polynomials. In direct analogy with the symmetric case, its coefficients are described by a combinatorial device called jeu de fleches, a filling of composition diagrams. The proof is self-contained and uses reflection and raising recursions which appear to be new