Papers

Let $f(n)$ be the least possible largest element of an $n$-element set of positive integers with pairwise distinct subset sums. Dubroff, Fox and Xu proved the finite lower bound \[ f(n)\ge \binom{n}{\lfloor n/2\rfloor} \] by applying a vertex-boundary estimate to the $2^{n-1}$ subsets whose sums lie below half of the total sum. We instead order all $2^n$ subsets by increasing subset-sum value. We prove that the bandwidth of this numbering is at most the largest element of the set, so the exact formula for the bandwidth of the hypercube gives the stronger finite bound \[ f(n)\ge H_n:=\sum_{j=0}^{n-1}\binom{j}{\lfloor j/2\rfloor}. \] An exact identity for $H_n$ in terms of Catalan numbers yields \[ \frac{H_n}{\binom{n}{\lfloor n/2\rfloor}} =\begin{cases} 1+\dfrac{2}{3n}+O(n^{-2}),& n\text{ even},\\[5pt] 1+\dfrac{4}{3n}+O(n^{-2}),& n\text{ odd}. \end{cases} \] For every $n\ge3$, this bound is strictly larger than the central-binomial bound. As a separate corollary, a first-order asymptotic formula for factorials rewrites this lower bound as a multiple of $\sqrt{2/\pi}\,2^n/\sqrt n$; the coefficients of $1/n$ are $5/12$ in even dimension and $7/12$ in odd dimension.

In this work, a new two-grid method presented for the elliptic partial differential equations is generalized to the time-dependent linear parabolic partial differential equations. The new two-grid waveform relaxation method uses the numerical method of lines, replacing any spatial derivative by a discrete formula, obtained here by the finite element method. A convergence analysis in terms of the spectral radius of the corresponding two-grid waveform relaxation operator is also developed. Moreover, the efficiency of the presented method and its analysis are tested, applying the two-dimensional heat equation.

In this work, a semialgebraic mode analysis (SAMA) is proposed for investigating the convergence of a multigrid waveform relaxation method applied to the Finite Element (FE) discretization of the heat equation in two and three dimensions. This analysis for finite element methods is more involved and more general than that for Finite Difference (FD) discretizations, since mass matrix must be considered. The proposed analysis results in a very useful tool to study the behaviour of the multigrid waveform relaxation method depending on the parameters of the problem.

We propose a Bayesian nonparametric mixture of regression trees with a Dirichlet process prior over tree-parameter pairs, enabling data-driven selection of ensemble size and unifying CART, BART, random forests, and boosting. A novel splitting rule driven by the posterior predictive of a Gaussian process within each terminal node generates flexible, smooth decision boundaries; remarkably, the GP density cancels exactly in the Metropolis–Hastings ratio for GROW/PRUNE moves, ensuring computational feasibility. An exact Gibbs sampler for posterior predictive inference propagates uncertainty through random tree traversal. A parallel MPI implementation distributes independent tree updates across processors, achieving adequate speedups. We prove posterior consistency at rate $n^{-1/4}$ in Hellinger distance under only continuity of the true regression function, allowing misspecification, via the identity $h(\Theta)=0$. Simulations on Friedman benchmark show near-nominal coverage (0.94 Gaussian, 0.92 Cauchy), robust to high-dimensional noise and heavy tails, outperforming BART and bagged CART. Applications to QSAR toxicity, crime, riboflavin, wheat genomics, and air quality confirm reliable credible intervals and automatic sparsity. The DP mixture offers a principled, robust, theoretically justified alternative for challenging regression with honest uncertainty quantification.

Let $0\leq k < n$, and let $\mathcal V\subseteq M_n(\mathbb C)$ be a complex linear subspace satisfying $\operatorname{rank}[S,T]\leq k$ for all $S,T\in\mathcal V$. Omladič, Radjavi, and Šivic proved the sharp bound $\dim\mathcal V\leq nk+\left\lfloor (n-k)^2/4\right\rfloor+1$ and conjectured a classification of the equality cases. We prove their conjecture. If equality holds, then, after a similarity and possibly transposition, $\mathcal V$ consists of all block upper-triangular matrices with arbitrary upper-left and upper-right blocks and with lower-right block in a maximal-dimensional commuting subspace of $M_{n-k}(\mathbb C)$. For $n-k\geq4$, these commuting subspaces are the classical equality cases in Schur's theorem; in dimensions $2$ and $3$, the additional equality cases also occur. At the equality dimension, the rank condition defines a projective algebraic subset of a Grassmannian. For $2\leq k\leq n-2$, we determine all of its irreducible components. If $n-k\geq4$, there are exactly two components when $n-k$ is even and exactly four when $n-k$ is odd; if $n-k\in\{2,3\}$, there are exactly two. When $n-k\geq4$, the Zariski tangent space at each such space equals the tangent space to its conjugacy orbit. When $n-k\in\{2,3\}$, the two components are obtained by varying the invariant $k$-dimensional subspace and the maximal-dimensional commuting subspace on the quotient, together with their transposes.

We study the $\mathbb{P}^1$-local motivic homotopy theory under the Nisnevich topology on $Sm_S$. We show that although the homotopy category $\mathcal{H}^{\mathbb{P}^1}(-)$ admits standard schematic functorialities, it fails six-functor gluing, thereby obstructing a full six-functor formalism. On the unstable side, we prove that the unstable $\mathbb{P}^1$-motivic localization preserves Nisnevich connectivity, despite the absence of an interval structure on $\mathbb{P}^1$. Over perfect fields, we then establish the $\mathbb{P}^1$-invariance of the higher unstable homotopy group sheaves $\pi_i^{\mathbb{P}^1}$ ($i \geq 1$) using the Bloch-Ogus-Gabber theorem. On the stable side, we prove the stable $\mathbb{P}^1$-connectivity theorem over fields. As a consequence, we obtain a homotopy $t$-structure on the stable $\infty$-category of $\mathbb{P}^1$-invariant Nisnevich spectra, and deduce the strict $\mathbb{P}^1$-invariance of stable $\mathbb{P}^1$-homotopy group sheaves.

In this paper we study $2d$-dimensional canonical systems with positive semidefinite Hamiltonians $H(t)$ satisfying $\operatorname{tr} H(t) \equiv 1$. We first establish that every such system possesses at least one nontrivial $H$-integrable solution, ensuring the existence of solutions whose energy with respect to $H$ remains finite. We then show that the space of all $H$-integrable solutions has dimension at most $d$.

We consider the damped nonlinear Klein-Gordon equation \begin{align*} \partial_t^2u-\Delta u+2\alpha\partial_tu+u-|u|^{p-1}u=0 \end{align*} on $\mathbb{R}^d$, where $\alpha>0$, $2\leq d\leq5$, and $p>2$ is in the energy-subcritical range. We classify global solutions that converge to a superposition of two positive and two negative translates of the ground state, without imposing any symmetry, coplanarity, or a priori geometric condition on their centers. We prove that every such four-soliton configuration is asymptotically either an alternating collinear configuration or an expanding rhombus with alternating signs. We further determine the precise long-time asymptotics of all four centers.

Let $p$ be a complex polynomial of degree $n\ge2$ whose zeros lie in the closed unit disk. Sendov's conjecture asserts that every zero of $p$ lies within distance one of a zero of $p'$. Mazur's recent proof, and Tao's streamlined exposition of it, reduce a hypothetical counterexample to a scalar lower bound \[ 1\le F(\eta,\alpha,n) \] together with two upper bounds for the endpoint parameter $\eta$ and the uniform restriction $0<\alpha\le 17$. We complete this scalar reduction by a two-stage optimization argument. First, $F$ is nondecreasing in $\eta$, so $\eta$ may be replaced by a piecewise polar envelope $\eta^{\ast}(\alpha)$. Writing $s=(n-1)/2$ converts the degree to a half-integer variable and gives \[ F(\eta^{\ast}(\alpha),\alpha,n) = E(\alpha,s) + \kappa(\alpha,s) \int_0^1 t^3 \widehat{\beta}(t;\alpha,s)^{\,s-\frac{3}{2}} \,dt. \] On each half-unit $\alpha$-slab, $E$ and $\kappa$ decrease with $\alpha$, whereas $\widehat\beta$ increases. Convexity in $t$ reduces all but the first mesh interval to eight explicit nodal terms. The first interval satisfies a uniform bound $3/200$. Each nodal upper term is a strictly log-concave function of the half-integer $s$, so its global discrete maximum is certified by two adjacent ratio evaluations. A fixed finite certificate over the $34$ half-unit slabs gives \[ F(\eta^{\ast}(\alpha),\alpha,n)<\frac{97}{100}<1, \] contradicting the scalar lower bound. The main contribution is a substantial reduction of the computer assistance required after the Mazur–Tao scalar reduction: all continuous optimization and the unbounded degree parameter are handled analytically, leaving only a small fixed collection of explicit one-variable inequalities.

Let $p(n)$ denote the least prime factor of $n$ and $L_C(x)=Cx^{1/2}(\log x)^2$ with $C>0$. The sum of $p(n)/n$ over composite $n$ lying in the short interval $[x,\,x+L_C(x)]$, a question raised by Erdős and Graham, is studied. (i) The constant $c=8$ in the mean asymptotic is estimated \[ S(x)=\sum_{{n<x,\ n\ \text{composite}}}\frac{p(n)}{n}=\frac{c\,x^{1/2}}{(\log x)^2}\Bigl(1+O\Bigl(\frac1{\log x}\Bigr)\Bigr) +O\Bigl(\frac{x^{1/3}}{\log x}\Bigr). \] (ii) For every fixed $C>0$, the window sums $\mu_C(x):=\sum_{x\le n\le x+L_C(x)}p(n)/n$ over composites have mean $4C$: $\frac1X\sum_{x\le X}\mu_C(x)=4C+O_C(1/\log X)$, and second moment $\frac1X\sum_{x\le X}(\mu_C(x)-4C)^2=O_C(1)$. (iii)Under the Maier-consistent bounded-fluctuation hypothesis, $\mu_C(x)=4C+O_C(1)$ holds uniformly in $x$, and the Erdős–Graham question has an affirmative answer. Unconditionally, the uniform statement remains open; proving it would require resolving short-interval prime estimates at scale $(\log y)^2$.

A common question about networked time series is whether outcomes changed because shocks transmit more strongly or because the pattern of connections changed. Standard practice inserts a recorded network into an outcome regression and reads movements of the fitted coefficient as changes in transmission strength. When the network is latent, time varying, and measured with error, this reading fails: changes in strength and changes in composition can produce the same outcome distribution at a single date, and the population coefficient moves under composition changes alone. The question becomes answerable when outcomes are analyzed jointly with repeated noisy measurements of the network, such as paired reports of bilateral trade flows. For the joint model we establish necessary and sufficient conditions for local identification, estimators of the strength and composition paths, a simultaneous confidence band for the strength path, confidence sets that remain exact under weak identification, breakdown bounds under common reporting bias, and an exactly sized test that detects changes on the observed path and attributes them to strength or to composition. Simulations assess each procedure at its stated boundary. On a mirror-reported trade panel of eighteen economies over 1995 to 2020, the diagnostics flag exactly the crisis years and the composition coordinate attached to European Union membership declines by roughly two thirds. The estimand is predictive dependence, not a causal effect.

We give three explicit constructions of the continuous Mal'cev Qp-completion of a topologically finitely generated pro-p group, using Tannakian formalism, Hopf algebras and p-adic analytic groups. We study properties of the continuous Mal'cev Qp-completion via these explicit constructions.

This paper describes an approach to variance reduction in randomized experiments using side information (covariates of the response unaffected by treatment). As with Double ML[Chernozhukov 2018], models of arbitrary complexity may be employed without concern for bias due to overfitting or regularization. For additive treatment metrics, the approach minimizes variance optimally. For more complex treatment metrics (e.g., multiplicative and ratios) the Delta Method estimate of variance is minimized. Through a variety of examples, the paper also explores modeling considerations.

We prove that the energy ${\mathcal E}(G)$ of any simple graph $G$ of order $n\ge5$ satisfies \[ {\mathcal E}\ge r(G)+\bar d(G)-1, \] where $r(G)$ and $\bar d(G)$ denote, respectively, the rank of the adjacency matrix and the average degree of $G$. We also characterize all extremal graphs. As consequences, our result settles five previously conjectured lower bounds for the energy of nonsingular graphs in their stated ranges, namely \[ \begin{gathered} {\mathcal E}(G)\ge n-1+\bar d(G),\qquad {\mathcal E}(G)\ge\Delta(G)+\delta(G),\qquad {\mathcal E}(G)\ge2\sqrt{\bar d(G) (n-1)},\qquad {\mathcal E}(G)\ge\frac{M_1(G)}{m},\qquad {\mathcal E}(G)\ge\frac{M_1(G)}{2m}+\frac{2m}{n}, \end{gathered} \] where $m$ is the number of edges, $\Delta(G)$ and $\delta(G)$ are the maximum and minimum degrees, and the first Zagreb index $ M_1(G)$ is the sum of degree squares.

Let $\mathcal{X}$ be a local orbifold Calabi-Yau threefold whose coarse space $X$ has transverse $A_{N}$ singularities along a smooth non-compact curve. We define orbifold Gopakumar-Vafa invariants, and we prove they are integers and satisfy a finiteness property. We compute our invariants for local orbifold $K3$ surfaces, where we prove an orbifold version of the classical Yau-Zaslow formula: we show that for an orbifold $K3$ surface $\mathcal S$, the genus zero Gopakumar-Vafa invariant of $\mathcal{S}$ in a class of square $2n$ is the Euler characteristic of the Hilbert scheme of $n+1$ points on the singular surface $S$.

We prove that $S^2\times S^3$ admits a Riemannian metric with positive sectional curvature. We view it as a principal circle bundle over $S^2\times S^2$. A diagonal Cheeger deformation of the base and a connection whose curvature form vanishes on the remaining flat tori yield a nonnegatively curved connection metric whose zero-curvature planes are the horizontal lifts of the tangent planes to those tori. We then perturb this metric by the real part of a global complex-valued symmetric $2$-tensor. Differentiation along the circle fibers produces a trace-free first variation of the second fundamental form on local horizontal lifts of the flat tori. The Gauss equation converts this into a positive second-order curvature term that dominates as the fibers shrink. A quantitative lower bound for the Hessian in directions normal to the set of zero-curvature planes extends this positivity to nearby planes. The metric and the proof are discovered by the Odin Automatic AI Research Agent.

Overparameterized networks often have ill-conditioned, saddle rich loss landscapes. They are usually trained with first-order methods that struggle on those landscapes, while many second-order methods stay infeasible at scale. Some scalable cubic methods replace the Hessian by a computationally cheap approximation and drop the exact block curvature. This paper introduces ARC-block, a blockwise adaptive regularization with cubics (ARC) optimizer. For each parameter tensor it minimizes a cubic model over the true block Hessian, with a per-block adaptive cubic constant and a monotone guard on the full loss. Large tensors are handled matrix-free in a Lanczos-built Krylov subspace. ARC-block's CubicKrylov subsolver's descent is monotone with a per-block cubic decrease bound. For exact steps, and for Krylov steps that meet the Cartis–Gould–Toint residual test, we provide an $\mathcal{O}(\sqrt{B}\varepsilon^{-3/2})$ sweep bound above a cross-block coupling floor. Another subsolver, ARC-$\varphi_1$, uses a trust-controlled exponential relaxation step. Among the cubic Newton methods evaluated here, these are the only ones whose steps stay exact on every parameter block (within the Krylov subspace), including on a $91.4$M-parameter implicit neural representation (INR). Run to convergence on FINER 2D image-fitting INR network, ARC-$\varphi_1$ reaches $133.5$ dB peak signal-to-noise ratio (PSNR), with no post peak collapse. Tuned Adam peaks at $78.2$ dB at $2{,}629$ s (step ${\sim}625$k, about $44$ minutes), further steps degrade its PSNR. On the same benchmark, SOAP, L-BFGS, and the scalable diagonal cubic method AdaCubic stay at or below Adam's $78.2$ dB peak. Over the benchmark's 20 images, run to a plateau rule, ARC-$\varphi_1$'s per-image PSNR has median $132.4$ and mean $131.5\pm3.3$ dB against tuned Adam's $72.8\pm4.5$ dB.

We show that, in the ring of virtual species, \[ \mathcal{I}=\sum_{m\geq 0}(-1)^m\prod_{i=1}^{m}\bigl((E^{-1})^i-1\bigr), \] where $\mathcal{I}$ is the species of interval orders and $E^{-1}$ is the multiplicative inverse of the species $E$ of sets. The right-hand side is the virtual species of signed ballot matrices introduced by Claesson and Hannah. They showed that its signed cardinality counts labeled interval orders. We strengthen this to a species identity, which we prove twice: first algebraically and then bijectively, using a natural sign-reversing involution. The cycle index series of $\mathcal{I}$ specializes to the generating series for labeled and unlabeled interval orders. We describe the automorphism group of an interval order as a Young subgroup and prove the identity $\mathcal{I}=\mathcal{R}\circ E_+$, where $\mathcal{R}$ is the species of rigid interval orders. We also show that Glaisher's T-number $T_n$ counts the $24$-colored interval orders on $[n]$ in which no isolated element has color $24$.

We develop a stochastic framework for a broad class of heterogeneous SIR epidemic models. In the finite-population construction, each initially susceptible individual is assigned a fixed nonnegative susceptibility, and infection occurs when the accumulated population-level infection pressure exceeds an individual random threshold. Infectious periods are independent and exponentially distributed with a common recovery rate. For any susceptibility distribution with finite mean, we prove a uniform-on-compact law of large numbers for the susceptible, infectious, removed, and cumulative infection-pressure processes. In the limit, an individual with susceptibility lambda remains susceptible under cumulative pressure x with probability exp(-x lambda). It follows that the susceptible fraction is given by the Laplace transform of the initial susceptibility distribution, while the incidence rate is governed by the mean susceptibility among those who remain susceptible. The resulting limits recover several familiar heterogeneous SIR systems, including the classical power-law model, and also yield other closed nonlinear incidence forms. The framework therefore provides a unified probabilistic foundation for deterministic epidemic models with persistent individual heterogeneity.

Ion channels and pumps drive ion-selective currents through cell membranes at localized sites, yet a cell's electrical state is routinely summarized by a single transmembrane voltage. Combining theory and numerical simulations, we resolve the spatiotemporal dynamics of charge reorganization driven by a localized current on a spherical membrane vesicle. At early times, the response is insensitive to membrane geometry: as in the case of a flat membrane (arXiv:2407.11947; arXiv:2508.14001), the transmembrane voltage decays in a monopolar fashion, varying inversely with distance from the source, and crosses over to a dipolar tail that scales as the inverse cube of distance. Under sustained current, this monopolar response spreads outward from the source. Because the vesicle is closed, this response cannot persist indefinitely; once the monopolar front traverses the entire vesicle, the subsequent charging dynamics is dominated by a spatially uniform mode corresponding to capacitive charging of the membrane. We further decompose the bulk potentials into an electrostatic image-charge component that generates the bulk electric fields and a spatially uniform capacitive mode that can be represented as an equivalent circuit. We also derive a nonlocal cable equation governing the transmembrane voltage dynamics and show that the uniform mode is its long-time solution. This work provides a first-principles basis for the electrophysiological simplification of an electrotonically compact cell.

Write P(N) for the assertion that every principal minor of the Fourier matrix of order N is nonzero. The square-free principal-minor conjecture was previously known for a few uniform small-multiplier families, including several families with two prime factors; broader higher-factor results were non-uniform, apart from isolated exact verifications. We prove a near-complete prime-multiplier theorem for the composite base 10: P(10p) holds for every prime p \notin {2,5,11}. The proof begins with the complete cyclotomic norm spectrum of the principal minors of the order-ten Fourier matrix. Its rational-prime support is {2,3,5,11,31}. Ordinary finite-characteristic lifting settles the norm-safe multipliers. A known small-multiplier theorem handles one norm-exceptional case; another is settled by retaining the prime-ideal chart in which each carrier degenerates. This yields a flag-selective lifting lemma and an active-rank norm budget. The chart analysis also isolates the limitation at the remaining square-free exception: active carriers can cover every local chart, so the first-order argument stops. We discuss this obstruction, the non-uniformity of fixed-base lifting, and the difficulties in passing to general square-free orders. All finite calculations are exact and have been independently confirmed.

We resolve a question posed by Harvey and Lawson concerning the uniqueness of continuous viscosity solutions to the Dirichlet problem for the Lagrangian mean curvature equation on the unit ball with continuous boundary data. For every dimension $n\geq 2$, we construct a continuous phase $\psi \colon \overline{B_1}\to(-n\frac{\pi}{2},n\frac{\pi}{2})$ and continuous boundary data $g \colon \partial B_1\to \mathbb{R}$ for which the Dirichlet problem \[ \sum_{i=1}^n \arctan\lambda_i(D^2u)=\psi(x)\quad\text{in }B_1, \qquad u=g\quad\text{on }\partial B_1, \] admits a continuum of distinct continuous viscosity solutions.

For $n\geq 0$, Stirling-Ramanujan constants $S_n$ are the Ramanujan summation of the divergent series $\sum_{k\geq 1} k^n\log k$. These constants are exponential periods over the exponential base field $\mathbb{E}=\mathbb{Q}(t,e^{-t})$. We generalize this result to a broad class of General Stirling-Ramanujan constants. Given polynomials $P$ and $Q$, with $\Re Q(s)>0$ for $\Re s >0$, the constant $S(P,Q)$ is the Ramanujan summation of the series $\sum_{k\geq 1} P(k)\log Q(k)$. They are exponential periods over $\mathbb{K}_P((\omega)) (t,e^{-t}, (e^{-\omega t}))$ where $\mathbb{K}_P$ is the field of definition of $P$ and $(\omega)$ are the zeros of $Q$. These constants are related to the derivatives $\zeta'_H(-n,w)$ of Hurwitz zeta function at negative integers, which we prove are exponential periods over $\mathbb{E} \left ( w,e^{-wt}\right )$. They can be expressed using Bendersky Gamma functions $\hat \Gamma_n$ and the values $\log \hat \Gamma_n(w)$ for $\Re w >0$ are exponential periods over $\mathbb{E} \left ( w,e^{-wt}\right )$. The logarithms of the determinants of the Laplacian on spheres and lens spaces are exponential periods over $\mathbb{E}$. The values $\zeta(2n+1)/\pi^{2n}$ are exponential periods over $\mathbb{E}$. For a periodic function $\chi:\mathbb{Z}\to \mathbb{C}$, we extend Ramanujan summation to the twisted series $\sum_{k\geq 1} \chi(k) P(k)\log Q(k)$ and define General Twisted Stirling-Ramanujan constants $S_{\chi}(P,Q)$. We derive integral formulas proving that they are exponential periods over $\mathbb{E}(\chi )=\mathbb{Q}(\chi)(t,e^{-t})$. For $n\geq 0$, $L_\chi'(-n)$ and $L_\chi(n+1)/\pi^n$ are given in terms of these constants and are exponential periods over $\mathbb{E}(\chi )$.

We present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approach parameterizes the underlying vector field using a neural architecture and trains it by minimizing a multi-step prediction loss over a finite horizon. To ensure numerical tractability, we optimize a mean absolute error objective averaged across prediction steps, and progressively increase the horizon during training. A key feature of this formulation is that it enforces consistency under repeated composition of the learned dynamics. As a result, the identified models exhibit significantly improved stability compared with approaches based on one-step regression of the vector field. When combined with sparsity-promoting regularization, this leads to parsimonious models that generalize beyond the training data. We demonstrate accurate recovery of systems exhibiting a wide range of behaviors, including stable and unstable fixed points, periodic orbits, and chaotic attractors. For chaotic systems, while long-term trajectory prediction is inherently limited by sensitivity to initial conditions, we show that multi-step training yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents. Moreover, we establish theoretical bounds linking trajectory error to statistical accuracy, providing a step toward a principled explanation for this behavior.

This paper investigate the conforming virtual element method for the optimal control problem governed by the linear poroelasticity equation. Both distributed control and Neumann boundary controls are considered with the optimize-then-discretize approach. We establish the well-posedness of the discrete state and adjoint problems in three-field formulation by deriving the first-order optimality condition. The optimal a priori error estimates are developed through a novel poroelastic projection for both control problems, and these estimates are uniform with respect to the relevant physical parameters of the model problem. Moreover, the numerical experiments are carried out using the primal dual active set strategy, and demonstrate the support for the derived theoretical results.

Let $A$ be an ergodic linear automorphism of the torus $\T^N$ whose center space has dimension two. We prove a dichotomy result that for every $f \in \mathcal{U}$, where $\mathcal{U}$ is the $C^{22}$ neighborhood of $A$ inside volume preserving diffeomorphisms on $\T^N$, exactly one of two alternatives holds. Either $f$ has the accessibility property, or $f$ is topologically conjugate to $A$ by a homeomorphism homotopic to the identity which simultaneously conjugates every homeomorphism commuting with $f$ to an affine automorphism of the torus. No irreducibility of the characteristic polynomial of $A$ is assumed. The alternatives exclude each other because a partially hyperbolic diffeomorphism topologically conjugate to $A$ is never accessible, so within $\mathcal{U}$ the failure of accessibility is equivalent to topological conjugacy to $A$. In the second case the centralizer of $f$ in the group of homeomorphisms of the torus is isomorphic to the group of affine transformations commuting with $A$. When the characteristic polynomial of $A$ is irreducible we show that this group is a finite extension of the unit group of an order in the number field generated by an eigenvalue of $A$, hence virtually free abelian of rank $r_1 + r_2 - 1$. The characteristic polynomial is allowed to be reducible, in which case the same description holds with the linear parts ranging over the centralizer of $A$ in $\GL(N,\Z)$. Our method uses an elementary density property of the projection of the lattice to the center space, which replaces irreducibility and which we establish for every ergodic automorphism with no restriction on the dimension of the center space.

Let $F_k(G)$ denote the $k$-token graph of a finite graph $G$. The token graph reconstruction conjecture asks whether, for any $1\le k<|V(G)|$, $F_k(G)\cong F_k(H)$ implies $G\cong H$. This paper proves that the conjecture holds for almost every graph.

The purpose of this paper is to study the existence and convergence of $\phi$-best proximity points in Banach spaces. A strong convergence theorem is established by employing a shrinking projection algorithm designed to compute common $\phi$-best proximity points of a proximally weakly suppressive mapping. Finally, we address a projection scheme to solve split $\phi$-best proximity point and variational problem in a Banach space. These contributions provide new tools for solving nonlinear problems involving non-self mappings.

We study the deterministic norm-query complexity of computational problems in finite-dimensional vector spaces over $\mathbb{Q}_p$ equipped with an arbitrary ultrametric norm. For orthogonalization, we prove that no uniform finite query bound depending only on the dimension exists: for every deterministic algorithm that produces an $N$-orthogonal basis for every ultrametric norm $N$, the number of norm queries is unbounded as $N$ varies. We then study the Longest Vector Problem (LVP) for a rank-$m$ $p$-adic lattice. By adapting a brute-force search to the general norm-query setting and eliminating the scalar redundancy among nonzero coefficient vectors modulo $p$, we obtain an algorithm using exactly $(p^m-1)/(p-1)$ norm queries for $m\ge 2$, and prove that no deterministic norm-query algorithm can use fewer queries in the worst case. Finally, we consider the Closest Vector Problem (CVP). Apart from the trivial cases in which no norm query is needed, we prove that the deterministic worst-case norm-query complexity of the CVP is unbounded, even when the lattice and the target vector are fixed.

We prove that if the exceptional set $E_p$ for the $p$-adic Littlewood conjecture is non-empty, then its logarithmic Hausdorff dimension is at least one. More precisely, whenever $E_p$ is non-empty, it has positive Hausdorff measure with respect to the gauge function $ h(r)=\frac{1}{\log(1/r)}. $ In particular, every non-empty $E_p$ has the cardinality of the continuum. We obtain stronger conclusions for the $t$-adic Littlewood conjecture over a finite field $\mathbb F_q$. For every prime power $q$, non-emptiness of the exceptional set $E_q^{(t)}$ implies that its $1/\log(1/r)$-Hausdorff measure is infinite. Moreover, when $q$ is odd, we refine the recent construction of Lai and Sprang~\cite{LaiSprang2026} and prove that \[ \mathcal H^{h_{A_q}}(E_q^{(t)})=\infty, \] where \[ h_{A_q}(r)=\frac{1}{(\log(1/r))^{A_q}}, \qquad A_q=\frac{q-1}{2}\log_2(q-1). \] In characteristic two, the corresponding conclusion with exponent one remains conditional on the existence of a counterexample.