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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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