Papers

We propose a finite element method for Yang–Mills connections on nontrivial principal bundles with abelian structure group. Local connection forms satisfy internal jump conditions induced by the transition functions of the bundle. We discretize these forms in broken finite element exterior calculus spaces and enforce the jump conditions weakly by Lagrange multipliers. We prove well-posedness of the resulting saddle-point problem and derive an a priori convergence estimate. A numerical experiment for the Hopf fibration illustrates the method and exhibits linear convergence when the sphere is approximated by a piecewise linear mesh.

Sharma and Panda recently proved that every bipartite graph with a perfect matching has property (P); that is, it admits a non-singular real symmetric matrix with support graph G for which every vertex is a P -vertex. In this paper, we extend their result from bipartite graphs to arbitrary graphs. To this end, we introduce the notion of a P - vertex covering and define the P -vertex covering number p(G) as the minimum number of non-singular matrices in S(G) needed so that every vertex of G is a P -vertex of at least one of them. Given a maximal matching of G, we partition the vertex set into the vertices saturated by the matching and the remaining vertices, which necessarily form an independent set. We then construct separate matrices covering these two classes of vertices. We use the Implicit Function Theorem as a perturbation tool to establish the desired result.

Integral operators on the real unit interval are constructed as transported from ones on the $p$-adic unit disc via the Monna map. This gives rise to strong Markov processes on $[0, 1]$, whose paths are right-continuous and have no discontinuities other than jumps. The spectra of the corresponding diffusion operators, whose kernel functions depend on a $p$-adic distance, are calculated. Further, the solution to the Cauchy problem for their heat equations are approximated via continuous- time random walks on finite sets coming from a hierarchical partition of the unit interval induced by the $p$-adic distance. The transport of $p$-adic diffusion to the real domain via the Monna map gives rise to a simple visualisation method. Illustrations of concrete examples are undertaken in the end.

Let $C_n$ be the optimal constant with the following property. For every even, continuous, strictly positive density $f$ on $R^n$ and all origin-symmetric convex bodies $K,L\subset R^n$, the inequalities $$ \int_{K\cap\xi^\perp}f \leq \int_{L\cap\xi^\perp}f \qquad\text{for all }\xi\in S^{n-1} $$ imply $\int_Kf\leq C_n\int_Lf$. In an earlier paper the authors proved that $C_n\leq\sqrt n$. In this paper, we prove the matching lower bound $C_n\geq c\sqrt n$. To simplify the exposition, we first give a complete one-scale construction, based on earlier work of Klartag and Koldobsky, which yields $C_n\geq c\sqrt{n/\log n}$. For the sharp result, we use the random-rounding construction of Klartag and Livshyts as a black box and combine it with a spherical-averaging support-separation argument.

For a finite sequence of positive integers $\boldsymbol{a}=(a_1,\dots,a_n)$, the restricted partition function $q_{\boldsymbol{a}}(k)$ denote the number of nonnegative integer solutions to the equation $a_1x_1+a_2x_2+\cdots +a_nx_n=k$. It is proved to be a quasi-polynomial of degree $n-1$. Write $q_{\boldsymbol{a}}(k)=\sum_{j=0}^{n-1}c_j(k)k^j$ with periodic coefficient functions $c_j$, and set $b_m=\#\{i:m\mid a_i\}$. In 2008, Beck, Sam, and Woods conjectured that the minimum period of $c_j(k)$ is $\mathrm{lcm}\{m:b_m>j\}$. In this paper, we derive an exact root-of-unity formula for every coefficient function $c_j(k)$. The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of $c_j(k)$. Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.

In this work, we analyze a completely degenerate germ of a real vector field at the origin of $\mathbb{R}^3$. We construct a generic three-parameter unfolding of this germ and identify the associated local bifurcations. We prove that the phase portrait of the unfolded system contains a one-parameter family of heteroclinic orbits connecting two hyperbolic equilibrium points.

We investigate Optimization under Input Uncertainty (OIU), in which the input to the objective function, rather than the objective function itself, is subject to uncertainty. OIU appears in manufacturing processes with production tolerance, control of physical systems with actuation noise, Mixture of Experts, and Reinforcement Learning (RL). Most of the existing approaches solve OIU by using the value of the objective function but discard the information of the realized input, even though the realized input is observable in various applications. The question here is whether the discarded information of the realized input is useful to accelerate the optimization process. We affirmatively answer this question for Evolutionary Strategy (ES) by theoretically showing that the information of the realized input can reduce the variance of the gradient estimator via Rao-Blackwellization. Using the Rao-Blackwellized gradient estimator, we propose Phenotype-Accelerated Evolutionary Strategy (PAES), which is a refinement of ES for OIU. Numerical experiments show that PAES converges faster than the usual ES from simple continuous optimization problems to RL benchmarks.

Attention-only dynamical theories model Transformer residual directions as particles aggregating on a sphere. We extend this framework by incorporating the feed-forward network (FFN) term as a local steering field acting on each token state. The resulting theory predicts that the tangential component of the FFN field is necessary for motion in residual-direction space, that critical residual directions correspond to nonlinear projective equilibria, and that a commutator defect determines when a finite attention–FFN block can be accurately approximated by a parallel, additive flow. Across GPT-2, Pythia, Mistral, and Llama models, the extended theory improves one-step angular prediction relative to an attention-only baseline, with the contribution of the FFN increasing from GPT-2 to Llama-3-8B. Intervention experiments show that retaining only the tangential FFN component preserves most model quality, whereas retaining only the radial component causes performance to collapse. The tangential component also preserves output diversity under aggregation pressure. As a practical application, layers with small commutator defects can be approximately parallelized with only a modest increase in loss, whereas layers with large defects degrade rapidly. These findings support the interpretation of FFN layers as directional steering fields that shape Transformer residual geometry and govern the feasibility of block-level interventions.

No quasiperfect number ($\sigma(n) = 2n + 1$) is known, and its number of distinct prime factors is bounded below; the bound $\omega \ge 7$ of Hagis–Cohen has stood since 1982, obstructed by a family of “deep leaves” on which pure enumeration cannot terminate (the scan bound for the intermediate prime reaches $8 \times 10^8$, and the exponent dimension is unbounded). This paper clears that obstruction with three lemmas at the level of secondary-school algebra — a discriminant criterion, a quadratic-residue sieve, and a multilinear resolver — which eliminate the last prime $q$, the intermediate prime $p$, and the exponent dimension respectively, turning a non-terminating search into a finite decision. On this basis all 381 stems of “$3 \mid n$ and $\omega = 7$” and their $79{,}751{,}212$ deep leaves are eliminated, with the ledger closing exactly and zero solutions throughout; the complementary case “$3 \nmid n$ and $\omega = 7$” collapses to a single stem, which is eliminated directly, so that the proof does not rest on any theorem whose published record we could not independently re-verify. Together with the machine elimination of $\omega \le 6$ (Theorem B4), this yields the main theorem: \emph{any quasiperfect number, if one exists, satisfies $\omega(n) \ge 8$} — the first advance of this bound since Hagis–Cohen 1982. The full computation has been reproduced by seven separately closed ledgers across three algorithmic architectures (CPU and GPU), all with zero solutions and exact ledger closure, and the lemma layer is formalized in Lean (259 theorems, zero sorry). A 2023 preprint of Zemann reported the same bound by a different computation; our audit of its public code found a coverage gap of 35 feasible exponents, so the elimination given here is, to our knowledge, the first complete proof. Code, ledgers, and Lean sources are available from the authors.

For a holomorphic vector bundle $F$ of rank $r$ on a smooth Riemann surface $X$ we construct a trace map on the chiral homology of the chiral Clifford algebra $\CE$ attached to the purely odd bundle $E=\Pi(F\oplus F^\vee\otimes\omega_X)$. It is the chiral-algebraic realization of the rank two fermionic vertex operator superalgebra. The free fermion (bc-type) conformal field theory built from a dual pair of odd fields $ \beta_i\in F$, $\gamma^j\in F^\vee\otimes\omega_X$. We give a complete construction of this vertex operator superalgebra, its associated vertex superalgebra bundle, and the isomorphism between the latter's chiral algebra and the chiral envelope $\CE$. Using the Batalin-Vilkovisky (BV) formalism together with Feynman diagrams we prove that the resulting trace map \[ \Trch : \bigl(\widetilde\sC^{\ch}(X,\CE)_{\sQ},\, \dch_{\CE}\bigr)\longrightarrow (\OBV,-\DBV) \] is a chain map satisfying a generalized quantum master equation and is a quasi-isomorphism, generalizing to the odd/Clifford setting the trace map on chiral Weyl algebras constructed by Gui for symplectic bosons. We establish existence, homotopy uniqueness, and functoriality (including metric-independence up to chain homotopy) of the trace map, prove cyclicity of the relevant supertrace, and verify nilpotency of $\DBV$, the graded Leibniz rule, $d^2=0$ for every differential introduced. As an application we compute the trace map on a modified affine current and on a modified energy-momentum tensor, recovering, purely algebraically from the chiral chain complex, Fay's classical formulas for the variation of the fermionic (Ray-Singer) analytic torsion along the moduli of the bundle $F$ and along the moduli of the curve $X$.

Here, we investigate the motion of a thin circular quantum vortex filament near the infinite planar surface. The fluid surrounding this surface moves with a non-zero velocity $\bf{v}$, which is parallel to the surface. We study the specific features of this quantum system and show that they are quite suitable for the boundary layer theory. The developed model allows us to calculate the vortex energy spectrum, $E = E({\bf p})$, where ${\bf p}$ is the total momentum of a vortex ring. We have demonstrated that this function has complex non-trivial dependence on the velocity $\bf{v}$. It is stated that the inverse effective mass of a vortex under consideration is of a tensorial nature. In certain quantum states, the system shows the possibility of both negative and positive effective mass existing. This study employs a novel quantization method for classical closed vortex filaments, developed by the author earlier.

In this paper, we give a co-sober space $X$ whose Smyth power space $P_S(X)$ is not co-sober, which gives a negative answer to an open problem.

Let $r_n$ be the infimum of \[ \frac{\lVert P'-P'(0)P\rVert_{H^2}}{\lVert P\rVert_{H^2}} \] over all degree-$n$ polynomials $P$ satisfying $P(0)=1$ whose zeros lie in the closed unit disk. We prove the quantitative residual bound \[ r_n\geq \exp\!\bigl(-(1+o(1))\sqrt n\log n\bigr) \qquad(n\to\infty). \] As an application, we answer Erdős Problem 973 on exterior power sums in the negative, in a form quantitatively stronger than the answer first obtained by Luo, Yang, and Zhu.

We establish a unified local theory for the persistence of Sobolev regularity of vortex patch boundaries in a family of two-dimensional active scalar equations with radial convolution kernels $K(|x-y|)$. The class includes the 2D Euler equation, the generalized SQG equation in the locally integrable range $0<\beta<1$, and the quasi-geostrophic shallow water equation. Under natural assumptions on $K$ (smoothness, integrability near the origin, monotonicity, and polynomial growth), we prove that if the initial boundary belongs to $H^3(\mathbb T)$ and satisfies the arc-chord condition, then the contour dynamics equation admits a unique local solution in $C([0,T];H^3(\mathbb T))$. Under a stronger integrability condition on the kernel, we also obtain local existence of $H^2$ solutions. The proof combines Sobolev energy estimates for the contour equation with quantitative control of the arc-chord quantity.

A model governing quasi-unidirectional wave propagation at the surface of a shallow layer of water is derived from Euler equations by long wave asymptotics together with Hamiltonian reduction techniques. The balance between nonlinearity, dispersion and weak dependence on the second planar coordinate is examined and results in the so-called CH-KP equation, a mildly nonlinear version of the well known Kadomtsev-Petviashvili equation. The Hamiltonian structure of the model is obtained by Dirac reduction from the intermediate step of a fully-nonlinear, long-wave system for stratified fluids. A particular scaling is considered that reduces such system to the one-dimensional CH equation in a limiting case. Examples of weak peakon-type solutions are provided, chosen to illustrate the collision behavior supported by the limiting equation.

OpenAI announced the existence of a non-sofic group: the unit group of the binary Leavitt algebra. We exhibit a different source of examples (relying on the same technical criterion) which includes torsion-free groups.

The tangent plane scheme (TPS) is a well-established discretization of the Landau-Lifshitz-Gilbert (LLG) equation. However, rigorous a posteriori error estimates have not been established. In this work, we derive a rigorous a posteriori error estimate for the TPS based on the second-order backward differentiation formula (BDF(2)) in time and finite elements of arbitrary polynomial degree in space. The proposed estimators provide computable upper bounds for the temporal and spatial discretization errors on adaptive meshes with variable time-step sizes. This result establishes the mathematical foundation for fully adaptive algorithms for the LLG equation.

Zhao and Ho use the irreducible open cover family respected with the Scott topology to introduce the SI-compactness. It has been proved that the SI-compactness is strictly stronger than compactness. In this paper, we prove that the arbitrary product of SI-compact spaces is always SI-compact which gives a positive answer for the question posed by Zhao and Ho. Furthermore, we show that the category of all SI-compact spaces is not a reflective full subcategory of TOP.

We develop a cell decomposition framework for o-minimal structures equipped with a generic derivation. To a $\delta$-cell we associate source cells and finite configurations in ordinary o-minimal sorts, allowing differential-topological questions to be studied through finite jet spaces. We then introduce a metric space of definable curve germs and identify its local half-space pieces with Cartesian powers of the maximal ideal of the Hardy field of definable germs. Using this germ-space description, we prove an abstract curve selection theorem for the $\delta$-topology. In the case of closed ordered differential fields, we further describe concrete asymptotic representatives for the abstract curve germs.

We propose a novel UnCut finite element method (FEM) for the Poisson and Stokes equations on domains with curved boundaries represented by a level set function. Like the $\phi$-FEM, the method avoids numerical integration over cut subregions of boundary elements, but introduces a novel least-squares formulation that minimizes an $H^{-1}$ residual of the governing equations. This formulation ensures stability without requiring large stabilization parameters, thereby eliminating the need for user-tuned penalty parameters and improving the robustness of the computation. Optimal-order convergence of the UnCut FEM solutions is rigorously established in the $H^1$ norm for both the Poisson and Stokes equations, and numerical experiments are presented to support the theoretical analysis.

The norm map \(N:\mathcal{H}_{\mathbb{Z}_{p^k}}\to \mathbb{Z}_{p^k}\) is studied on the quaternion ring over \(\mathbb{Z}_{p^k}\), where \(p\) is an odd prime and $k\ge 1$ an integer. By means of the isomorphism \(\mathcal{H}_{\mathbb{Z}_{p^k}}\cong M_2(\mathbb{Z}_{p^k})\), quaternions are investigated using matrix methods. It is shown that the fibre size \[ a_{p^k}(m)=|\{q\in \mathcal{H}_{\mathbb{Z}_{p^k}}:N(q)=m\}| \] depends only on the \(p\)-adic valuation \(v_p(m)\) of \(m\). Explicit formulas for the fibre sizes are derived for every \(m\in\mathbb{Z}_{p^k}\): \[ a_{p^k}(m)= \begin{cases} p^{3k-2}(p^2-1), & t=0,\\[6pt] p^{3k-2-t}(p+1)(p^{t+1}-1), & 0<t<k,\\[6pt] p^{2k-1}(p^{k+1}+p^k-1), & t=k, \end{cases} \] where \(t=v_p(m)\) (with the convention \(v_p(0)=k\)). The main result of the paper is a complete solution to the local four-square problem over the ring \(\mathbb{Z}_{p^k}\): the number \(a_{p^k}(m)\) gives the exact number of representations of an arbitrary element \(m\in \mathbb{Z}_{p^k}\) as a sum of four squares, \[ x_1^2+x_2^2+x_3^2+x_4^2=m. \] The proof is purely algebraic; it relies only on matrix theory and Smith normal form, thus avoiding the abstract machinery of number theory. This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections

Generalised continuum theories couple the macroscopic deformation and the micro-/meso-scopic deformation of an underlying micro-structure. They can account for internal length-scale effects and higher-order mechanical loadings absent from classical Cauchy elasticity and has shown its efficiency in modelling metamaterials. This has led to a proliferation of higher-grade and higher-order models (e.g. strain-gradient, micromorphic, micro-polar) whose underlying kinematic reduction strategies, in particular to reduce the number of material parameters, are rarely identifiable from the free energy alone. It leaves two open issues: the lack of a systematic criterion for selecting an appropriate model, and a persistent ambiguity regarding the physical status of the local frames (”directors of matter”) used in their kinematic description. Are they physical quantities describing the change of state of the given micro-structure or arbitrary kinematic descriptors of its deformation\,? This work addresses both questions through a gauge-theoretic formulation of generalised continua in finite strains. While relying on tools and modelling choices consistent with the existing geometric literature on continuum mechanics, the present approach departs from it in its objectives, aiming at a unifying classification of available mechanical models rather than the description of a specific microstructural phenomenon such as defects. In this work, generalised configurations are defined as moving frames over classical configurations, and invariance with respect to the reference generalised configuration is shown to be a necessary and sufficient condition for a gauge invariance, recovering the micromorphic theory as the general-purpose theory of the deformation of arbitrary directors of matter. A systematic classification of first-order generalised media follows from structural group reduction, while strain-gradient continua are recovered through convected frames, and finally constrained media (e.g. couple-stress) are addressed.

We determine the minimum hyperbolic area of Teichmuller curves arising from holomorphic quadratic differentials on closed Riemann surfaces of genus two. The minimum is 3\pi/5, and it is attained precisely by quadratic differentials q=\omega^2 for which the translation surface (X,\omega) lies in the GL_2^+(R)-orbit of the double-pentagon translation surface. Equivalently, the extremal projective Veech group is the triangle group \Delta(2,5,\infty). The proof combines a small-area classification of noncompact hyperbolic orbifolds with a derivative-preserving affine descent construction for nonsquare quadratic differentials. The three possible nonsquare zero patterns are then excluded by arithmetic, marked-point, and covering obstructions.

We prove that the Diophantine problem, i.e. the problem of solving equations with coefficients, in one-relator groups and one-relation monoids is undecidable in general. Specifically, we show that for all $d \geq 2$, the Diophantine problem for the Baumslag-Gersten group $\mathcal{B}_d = \operatorname{BG}(1,d)$ and for a closely related one-relation monoid $\mathcal{M}_d$ are both undecidable. Both results are obtained by an encoding of a recent undecidability result due to Rybalov.

Let $\mathcal H=(V,E)$ be a locally finite simple hypergraph, equip $V$ with the hyperpath metric, and consider the equal-edges random walk \cite{CoupetteEtAl2023}. For adjacent vertices $x$ and $y$, we prove that the idleness function $\alpha\mapsto\kappa_\alpha^{\mathcal H}(x,y)$ is piecewise affine and with no more than three affine pieces. A separate mass-balance argument gives linearity on $[1/2,1]$ for every locally finite simple hypergraph and, consequently, a limit-free expression for the Lin–Lu–Yau curvature. In the $r$-uniform linear case, the hypergraph walk agrees exactly with the simple random walk on its 2-section. This reduction transfers the sharp endpoint intervals of Bourne, Cushing, Liu, Münch, and Peyerimhoff \cite{BourneEtAl2018}.

This article introduces a new implementation for computing a numerical irreducible decomposition for a system of polynomial equations. The implementation is part of the software package HomotopyContinuation.jl written in the programming language Julia.

Let $R_r(k)$ denote the diagonal $r$-color graph Ramsey number. We prove that there exist absolute constants $c,K>0$ such that \[ R_r(k)\le \exp\!\left(-c\frac{k}{r^2\log^4(2r)}\right)r^{rk} \] for every $r\ge2$ and every $k\ge Kr^2\log^6(2r)$. The proof combines a positive-coefficient root filter of variable order with a retained-spine refinement of the multicolor book method.b

This paper investigates the problems of stability analysis and nonsingular sliding mode stabilization for exact prescribed-time control. By exploiting the isochronism of simple harmonic motion, this paper establishes a novel exact prescribed-time control framework. First, a novel Lyapunov analysis method for exact prescribed-time control is proposed, based on which the exact prescribed-time stabilization of scalar systems is achieved. It is theoretically established that for arbitrary non-zero initial conditions, the settling time is exactly equal to the prescribed time. Next, the framework is extended to a novel sliding mode control law. It guarantees exact prescribed-time convergence for almost all initial conditions even in the presence of external disturbances. Notably, the proposed control law is nonsingular across the entire state space and maintains uniformly bounded gains. Finally, simulation results validate the effectiveness of the proposed methods.

We study a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family $$ P_m(x_1,\ldots,x_m,u) = \prod_{j=1}^m(1+x_j) + u\prod_{j=1}^m(1-x_j), \qquad m\geq1. $$ Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for $\mathrm m_{\mathbb D}(P_m)$. We prove that, for every fixed $m$, the value $\pi^m\mathrm m_{\mathbb D}(P_m)$ belongs to the algebra of level-$4$ cyclotomic multiple polylogarithm values, and we evaluate the first nontrivial case $m=2$ explicitly in terms of $\operatorname{Li}_4(1/2)$, $\zeta(3)$, Catalan's constant, $\pi$, and $\log2$. We also derive an explicit three-term asymptotic expansion for $\mathrm m_{\mathbb D}(P_m)$ as $m\to\infty$.

We study fractional diffusion equations on compact metric graphs, where the nonlocal dynamics is governed by fractional powers of the shifted Kirchhoff-Laplacian. Building on recent advances in the analysis of fractional operators on metric graphs, we establish a rigorous mathematical framework and propose a fully discrete scheme based on backward Euler time-stepping and finite element discretization. To approximate the action of the fractional operator, we employ rational approximations, reducing the problem to a sequence of sparse elliptic solves for efficient implementation. We derive error estimates for the temporal, spatial, and rational discretizations, and confirm convergence through numerical experiments.