Papers

We construct helicity-conditioned statistical states for a Whitney-discretized electromagnetic field on a closed oriented three-manifold. The simplicial de Rham complex provides exact discrete gauge symmetry, while the Whitney inner product separates exact, harmonic, and coexact sectors. The spatial Abelian Chern–Simons functional is gauge invariant and depends only on the coexact potential. After fixing harmonic modes, we introduce a helicity-biased Gaussian ensemble on the reduced electromagnetic phase space and derive explicit formulas for its admissible parameters, partition function, mean helicity, relative entropy, and Fisher information. The distribution uniquely minimizes relative entropy under a prescribed mean-helicity constraint. Its helicity susceptibility equals the variance of the discrete Chern–Simons functional and controls the local distinguishability of neighboring statistical states. Because magnetic helicity is generally not conserved under unconstrained Maxwell dynamics, these states represent conditioned inference rather than dynamical equilibrium.

We investigate identities satisfied by monoids of triangular matrices over fields and over additively idempotent semirings with 0 and 1, and exhibit several new instances in which these identities admit no finite axiomatization.

We find a bijection to prove that the set of patterns {3412, 4321} is Wilf-equivalent to the set of patterns {3412, 4231, 45321, 54312}. This provides a bijective proof of an identity recently shown by Azam and Richmond.

Among the four well-known anti-foundation axioms, $\mathsf{BAFA}$, $\mathsf{FAFA}$, $\mathsf{SAFA}$, and $\mathsf{AFA}$, the latter three are all special cases of $\mathsf{AFA}^\sim$ ($\sim$ is one of the regular bisimulations). In this paper, we generalize it to $\mathsf{AFA}^\sim$ ($\sim$ is regular and has the so-called set property), and show that this generalization is substantial by constructing new anti-foundation axioms.

The Sprott–Zeraoulia conjecture, formulated on the basis of numerical experiments, states that three-dimensional quadratic continuous-time systems with symmetric Jacobian matrices cannot exhibit chaotic behavior. We give an analytical resolution of this conjecture in terms of the principal precise notions of chaos used for continuous dynamical systems. The starting point is the classical, but decisive, fact that symmetry of the Jacobian on $\mathbb{R}^3$ is equivalent to a global gradient representation. Hence every system in the conjectured class can be written as $\dot{x}~=~\nabla V(x)$, where $V$ is a polynomial of degree at most three, and the conjecture becomes a problem about cubic polynomial gradient flows. For bounded trajectories, the monotonicity property of gradient systems first confines every omega-limit set to a critical level of $V$. When the equilibria on that level are isolated, convergence follows from the connectedness of the omega-limit set. In the general case, which may include curves or surfaces of equilibria, the Łojasiewicz gradient inequality implies finite length and convergence of every bounded forward trajectory to a single equilibrium. Consequently, compact invariant sets support no nonstationary recurrence, positive-entropy measure, topological transitivity, Smale horseshoe, or Devaney, Auslander–Yorke, Li–Yorke, mean Li–Yorke, and distributional chaos. Solutions escaping to infinity are treated through the Poincaré compactification. Finally, we connect the gradient formulation with earlier Darboux-theoretic criteria: invariant algebraic surfaces satisfy $\langle\nabla V,\nabla f\rangle=Kf$, and an invariant affine plane with constant cofactor produces an exact tangential–normal decomposition of the potential and the flow.

A nonnesting permutation of the multiset {1,1,2,2,...,n,n} is a word containing no subsequence abba with a distinct from b. Elizalde and Luo enumerated nonnesting permutations avoiding each set of two or more patterns of length three, and left open the enumeration of those avoiding the single pattern 123 (Problem 1 of their paper). We solve this problem. Writing $c_n$ for the number of nonnesting permutations of {1,1,...,n,n} avoiding 123 and $C(z)=\sum_{n\ge 0}c_nz^n$, we prove $C(z) = 1 + z(1+U)/(2 - z(1+U)(1+V))$, where $V = 1+zV^2$ and $U = 1+3z+zU^2$ are the formal power series branches with constant term 1. In particular $C$ is algebraic of degree exactly 4 over $\mathbb{Q}(z)$, and $c_n \sim \frac{9+6\sqrt{3}}{8\sqrt{\pi}}\, 6^n n^{-3/2}$. The proof constructs a generating tree for these objects whose labels consist of two nonnegative integers, and applies the kernel method twice.

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$. A strictly increasing sequence of integers $x_1<\cdots<x_k$ is an ascending wave if its consecutive differences are nondecreasing. Let $g(n)$ be the largest integer $k$ such that every set $A\subseteq[n]$ with $|A|\ge n/2$ contains an ascending wave of length $k$. Alon and Spencer proved that \[ c_1\frac{(\log n)^2}{\log\log n}\le g(n)\le c_2(\log n)^2 \] for all sufficiently large $n$, and they conjectured that the factor $\log\log n$ in the lower bound can be removed. In this paper, we confirm their conjecture.

Memory effects and anomalous diffusion arising in models of transport in complex media, fractals, and viscoelastic materials can be described by fractional-differential equations, such as the time-fractional diffusion equation $D^\alpha_t u - \Delta_x u = 0$. The paper develops a decomposition of solutions of these equations into a product of spatial and temporal components, corresponding to a freeze-out at long times. For fractional diffusion with the Caputo derivative the spatial factor is the inverse-Laplacian $(-\Delta)^{-1}$ of the initial data, while for the Riemann-Liouville derivative it is the inverse bi-Laplacian $ (-\Delta)^{-2}$. In both cases, the temporal factor of the solution is a scaled negative power of time. This memory artifact explicitly encodes the initial data, which gives a simple and robust way to reconstruct the initial conditions in the backward-in-time inverse problem with solution values measured at long times. We derive characteristic timescales for Mittag-Leffler functions, which correspond to such factorization in anomalous diffusion. This also enables an accurate approximation of the number of real zeros of the Mittag-Leffler function. We apply these results to heat transfer on a comb, a model which manifests subdiffusion arising from the comb's fractal structure.

We show that for any real, connected, semi-simple matrix group, its Lie algebra of algebraic vector fields can be generated by three complete vector fields.

We discuss an example of a rigid non-principal ideal in a one dimensional (commutative) Gorenstein domain, which contradicts a conjecture of C. Huneke and R. Wiegand. The construction and its analysis were discovered by Codex, when prompted by one of the authors to verify the conjecture or find a counterexample. A proof of the conjecture, also discovered by Codex, is presented when the ring is equicharacteristic and its embedding dimension is at most three.

The sparse-group pliable Lasso (SGPL) extends the pliable Lasso and group pliable Lasso by combining sparse-group regularization with a predictor-level coupling penalty, enabling simultaneous group-level selection, within-group sparsity, and hierarchical structure between main effects and interactions. We propose a blockwise coordinate descent algorithm for fitting the SGPL that exploits the convexity and structure of the objective function, establish convexity and Karush–Kuhn–Tucker optimality conditions, and prove that the algorithm converges to a global minimizer. Simulation studies demonstrate competitive predictive performance and smaller interaction estimation error than the pliable Lasso and group pliable Lasso, albeit with the expected precision–recall trade-off in support recovery. We further illustrate the proposed method using a Parkinson's disease gut microbiome study and an adrenocortical carcinoma (ACC) copy-number dataset from The Cancer Genome Atlas. The Parkinson's application identifies interpretable interactions between microbial abundances and dietary variables, while the ACC application illustrates that the effectiveness of group-structured regularization depends on how well the prespecified grouping reflects the underlying signal structure.

We establish variance lower bounds for the intrinsic volumes and the face numbers of random polytopes, generated by independent samples from rotationally invariant log-concave probability measures on $\mathbb{R}^d$ with full support. Our results extend the corresponding Gaussian lower bounds of Bárány and Vu and of Bárány and Thäle to this broader class of distributions. Our proof builds on the geometric construction introduced by Bárány and Vu, for which we develop an alternative treatment based on barycentric coordinates. We combine it with local variance estimates for both intrinsic volumes and the number of faces.

The $K$-function is a fundamental summary statistic for assessing clustering or regularity of point processes in two or three dimensional Euclidean space. In practice, however, many planar point patterns arise from projecting locations of objects on a surface in three dimensional space to two dimensional space. For example, when events or objects occur in a landscape their elevation is often ignored. This can lead to erroneous conclusions regarding properties of the point process generating the point pattern. There is not a unique way to extend the classical $K$-function to point patterns on a complex surface. In this paper we propose, explore, and discuss several approaches in terms of their theoretical and computational properties. The best performing approach, coined the surface area $K$-function, can be viewed is an analogue of the classical $K$-function replacing counts of points in Euclidean balls with counts of points in surface geodesic balls. However, an important distinction is that the argument of our surface area $K$-function is area instead of radius of geodesic balls. The performances of the various surface $K$-functions are compared in applications to simulated and real data.

Grid theorems provide a fundamental link between the structure of ends and the existence of grid-like subgraphs in infinite graphs and digraphs. For every fixed width $n$, let $k(n)$ denote the least positive integer such that every digraph with an end of in-degree at least $k(n)$ contains a subdivision of the directed hexagonal grid of width $n$. Hamann and Heuer asked the exact vaule of $k(n)$. We solve this problem by proving that $$ k(n)=\left\lfloor\frac{3n}{2}\right\rfloor-1 \qquad\text{for every }n\geq4, $$ while $k(1)=1$, $k(2)=2$, and $k(3)=4$. For the upper bound, we establish a finite vacancy theorem for labelled token slides, convert it into a closed directed schedule on an auxiliary ray digraph, and lift the schedule through fresh clean linkages. For the matching lower bound, we use alternating orientations of products of a ray with a three-armed tree and prove a no-passing property for disjoint directed paths. We also determine the dual extremal parameter. Let $W(d)$ denote the largest width that is always forced by an end of in-degree $d$, with all branch rays remaining in that end, then $$W(d)=\left\lfloor\frac{2d}{3}\right\rfloor+1 \qquad\text{for every }d\geq4,$$ with $W(1)=1$ and $W(2)=W(3)=2$.

Let $\mathsf{BCT}_{\mathsf{WO}}$ assert that every well-orderable family of dense open subsets of a perfect Polish space has dense intersection, and let $\mathsf{RS}_{\mathsf{WO}}$ assert that for every nonempty countable partial order, there is a filter meeting every member of any given well-orderable family of dense subsets. Over $\mathsf{ZF}$ they are shown to be equivalent. We show that in Truss's model, they characterize those $A\subseteq2^\omega$ for which every $A$-indexed family of dense open sets has dense intersection. Assuming choice for well-orderable families of nonempty sets, we obtain an analogous characterization for total relations $R\subseteq X\times Y$ with meager vertical sections, whenever $Y$ is a surjective image of $2^\omega$. The Feferman-type model $\mathfrak N_{\aleph_1}$ studied by Truss \cite{Truss1974} satisfies these hypotheses and $\mathsf{DC}$. We study the ideal of well-orderable subsets of $2^\omega$ and its relations to other ideals in this model. We show, among other things: the well-orderable subsets of $2^\omega$ are exactly the sets which are Rothberger in every finite power, have strong measure zero, are universally null, are Marczewski null, or contain no perfect subset. The ideal of well-orderable subsets of $2^\omega$ is closed under well-ordered unions and is incomparable with the meager ideal. The equivalence with the Rothberger property does not extend to the Hurewicz property. Further more, in this model every set is Marczewski measurable, arbitrary maps into separable metric spaces have continuous perfect restrictions, and every set of positive outer measure contains a perfect subset.

Let $G=(V,E)$ be a finite loopless directed graph. Each directed two-step diamond identifies its two pairs of opposite edges, and the connected components of the resulting auxiliary graph on $E$ define a canonical partition $\Pi_{\mathrm{opp}}(G)$. We ask whether a finite relational certificate built from the blocks of an edge partition identifies this canonical partition. We prove the functoriality and a universal coarsening property of $\Pi_{\mathrm{opp}}(G)$, count its complete fixed-profile fibre and its exact radius-one transposition neighbourhood, and reduce the parity condition in the certificate to an odd closed walk in a directed $\mathbb Z_2$-gain graph. In a fixed catalogue, four labelled target-positive rows, representing three isomorphism types, each admits a noncanonical positive partition obtained by a single cross-block transposition. In one row a stronger positive comparison partition also preserves every per-vertex, fixed-role incoming and outgoing count; it arises from a 12-edge alternating trade. The corresponding incidence fibre is a singleton in the other three rows. A deterministic 64-index family of comparison partitions yields 78 bounded positives among 256 nonidentity partitions. On the same profile fibre, one member has a complete 885-element relation semigroup and no write-preserve-use certificate, giving an exact negative. Thus the certificate has genuine but intermediate selectivity: it is neither determined by block sizes nor specific to the canonical target. The claims are finite statements in computational combinatorics and are supported by explicit, independently checkable certificates.

Motivated by equivariant neural networks, we study piecewise linear equivariant maps between finite-dimensional real representations of compact groups. We show that all genuinely non-linear piecewise linear behaviour is confined to the subspaces on which the identity component of the group acts trivially, while equivariance forces linearity on the corresponding orthogonal complements. As a consequence, we obtain a compact-group analogue of the finite-group existence criterion of Gibson–Tubbenhauer–Williamson for non-zero equivariant piecewise linear maps between irreducible representations, with the identity component giving rise to a rigidity phenomenon absent from the finite-group case.

Newtonian mechanics is usually written on a spacetime of fixed geometry. Here we build that geometry, introducing a propagator and a bilinear form only as the description demands. Within this reconstruction the Principle of Inertia emerges as a theorem rather than an independent postulate. Inertial motion is blind to the scale of the form and to the torsion the propagator defines, and that blindness is a geometric precursor of the universality of free fall, before gravity enters. Newton's Second Law is a compatibility between two derivatives along a curve. The force exceeds the mass times the lowered acceleration by an exact term, and the law survives arbitrary torsion. Requiring force-free motion to be inertial along curves parametrised by arc length restricts the torsion to be totally antisymmetric. The remaining torsion is a three-form which we postulate the particle carries along its worldline. Its contraction with the velocity annihilates that velocity, so the supplementary condition of spinning-particle mechanics holds identically, and we propose that contraction as a classical spin. The algebra is familiar from relativistic spin hydrodynamics; the assignment, to the best of our knowledge, has not been proposed elsewhere. Since inertial motion resolves neither the mass nor the spin, the predictive content lies in the transport law we postulate, which turns the spin against a parallel-transported frame at a rate fixed by the acceleration alone and by no coupling constant. That rate becomes a laboratory number only once the arc-length parameter is identified with a clock, which we leave open. The dynamics asks non-degeneracy of the form, so this spacetime is not Newton–Cartan's Galilean one, and gravitation lies outside the construction. We prove every mathematical result, make three constitutive postulates and one physical conjecture.

Electrodiffusion has been extensively studied in the isothermal setting, whereas the mathematical theory of thermally coupled electrodiffusion remains comparatively underdeveloped. We investigate a non-isothermal electrodiffusion model describing the evolution of multiple ionic species with different diffusivities and valences in a two-dimensional incompressible viscous fluid. The coupling to a spatially and temporally varying temperature gives rise to a nonlinear and nonlocal system with logarithmic nonlinearities in the ionic fluxes. We establish local well-posedness for strictly positive initial concentrations and prove global well-posedness when the initial temperature is close to a homogeneous state by developing a new entropy structure tailored to thermodiffusive effects. No smallness assumption is imposed on the initial ionic concentrations or fluid velocity. To overcome the singularity of the logarithmic terms, we develop a novel cutoff-mollification regularization, derive uniform logarithmic estimates, and prove persistence of strict positivity of the ionic concentrations. This positivity removes the singular behavior of the logarithmic nonlinearities and is essential for the uniqueness argument. These results provide a rigorous mathematical framework for the analysis of non-isothermal electrohydrodynamics systems.

Let $E$ be a finite set, and let $V \subseteq \mathbb{C}^E$ be a linear subspace that is not contained in any coordinate hyperplane. The closure of $V$ in the product of projective lines $(\mathbb{P}^1)^E$ is a singular variety $Y_V$ known as the matroid Schubert variety (or arrangement Schubert variety). We use operational Chow cohomology to prove that every line bundle on $Y_V$ is the restriction of a line bundle on $(\mathbb{P}^1)^E.$ We then give combinatorial characterizations of when $Y_V$ is Gorenstein and $\mathbb{Q}$-Gorenstein, respectively. We provide examples of linear subspaces $V$ such that $Y_V$ is Gorenstein but not smooth, $\mathbb{Q}$-Gorenstein but not Gorenstein, and not $\mathbb{Q}$-Gorenstein, respectively.

Given a real-valued function $f$, let $\mathcal{N}_f(\delta, Q)$ be the number of rational points with denominators at most $Q\ge 1$ in the $(\delta/Q)$-tubular neighbourhood of the graph of the function $f$. A heuristic predicts that the number of such points grows like the area of the neighbourhood provided that $\delta$ is big enough (in a suitable sense). Considerable efforts have been committed to prove this heuristic for regular curves. This culminated in the works by Vaughan & Velani~(2006) and by Huang~(2015) establishing an asymptotic expansion for $\mathcal{N}_f(\delta, Q)$ provided that $\delta\gg Q^{-1+\epsilon}$ for some $\epsilon>0$ when the map $f$ is, among other assumptions, twice continuously differentiable. The present work deals with the thus-far unexplored regime where minimal regularity conditions are imposed on the curve. More precisely, it is concerned with the case where the map $f$ is an a.s. realisation of the graph of Brownian motion. The main result establishes the existence of an almost sure asymptotic expansion for the counting function for all values of $\delta$, with the exception of a critical regime, thereby going well beyond the theory currently available for regular curves. A key ingredient in the proof is the derivation of the area heuristic, which relies on establishing the a.s. asymptotics of the area of the tubular neighbourhood of the graph of Brownian motion. This result has two main consequences: firstly, it completes the counting aspect of the theory of Diophantine approximation on the graph of Brownian motion initiated by Sprindžuk (1979). Secondly, it hints at the existence of a theory unifying the analysis of rational points near a curve on the one hand and, on the other, its local Hölder regularity and fine-scale oscillations. It thus builds a seemingly new bridge between Number Theory and Multifractal Analysis.

Trajectory optimization computes dynamically feasible motions that enable autonomous systems to accomplish complex tasks while satisfying operational and environmental constraints. This tutorial presents OpenSCvx, an open-source Python framework that bridges the gap between high-level problem specification and efficient numerical optimization. Rather than requiring users to derive solver-specific mathematical formulations, OpenSCvx provides a symbolic modeling interface that automatically constructs and solves trajectory optimization problems from modular descriptions of objectives, dynamics, and constraints. Beyond simplifying problem formulation, OpenSCvx supports (i) continuous-time constraint modeling, (ii) temporal and logical specifications, (iii) automatic vectorization for scalable and batched optimization, and (iv) a modular architecture that enables new algorithms, models, and solver backends to be incorporated with minimal effort. These capabilities allow researchers and practitioners to rapidly prototype, solve, and extend state-of-the-art trajectory optimization methods.

It is shown that the solutions to a wide class of semi-linear parabolic equations with analytic (holomorphic) nonlinearity, with spatially periodic boundary conditions, are analytic in time with values in the Gevrey class of analytic functions with respect to the spatial variable.

We give a new intrinsic construction of the continuous tropical Prym variety, originally due to Röhrle and Zakahrov. In particular, given a harmonic double cover of graphs we explicitly construct the tropical Prym as integral torus. Moreover, we show that the tropical Prym–Torelli map from the moduli space of harmonic double covers to the moduli space of principally polarised tropical abelian varieties is continuous.

We introduce new properties of Haagerup-type for Fell bundles over discrete groups and discuss their relationship with Haagerup properties for the associated $C^*$-algebras. In particular, our results on Fell bundles associated with twisted unital $C^*$-dynamical systems extend previous results known in the untwisted case.

The revenue maximization problem in service systems with strategic customers is a central problem in the thread of Rational Queueing literature. The standard modeling assumption is that customers act independently and, therefore, the situation is modeled as a game among the customers and the service provider. In the present work, we focus on the case where the customers are grouped and controlled by their leader and the situation becomes a game between the customers' leader and the service provider. We examine the effect of customers' cooperation for the join-or-balk dilemma at the unobservable single-server Markovian queue and present the impact on the social welfare and service provider's revenue.

We explore fast convergence of the Riemannian Frank–Wolfe method for smooth geodesically convex optimization over compact feasible sets. Hadamard manifolds are the main setting. On general complete manifolds, the analysis accounts for all feasible minimizing geodesics. We consider the open loop step-size $\eta_k=a/(k+a)$, which only uses the iteration index. Under a local Hölderian error bound and local length-normalized directional scaling, every $a>2$ gives the eventual rate $O(k^{-1/(1-\theta)})$ for $\theta\in(0,1/2]$. An interior-ball condition yields $O(k^{-2})$ for strongly geodesically convex objectives. Under an exact Riemannian scaling inequality and a uniform positive lower bound on the gradient norm, every $a\geq2$ gives $O(k^{-a})$ after an explicit threshold index. The same rate holds for the smallest Frank–Wolfe gap over the most recent half of the iterates. For geodesic balls of radius \(R<\pi/2\) in the unit sphere, we establish the scaling inequality with $\alpha_R=\tfrac12\cot R$, yielding $O(k^{-a})$ primal error and recent-window gap rates. We also analyze the standard gap-feedback short step under the local error-bound conditions, obtaining $O(k^{-1/(1-2\theta)})$ for $\theta<1/2$ and a linear rate for $\theta=1/2$. Numerical experiments illustrate the predicted rates and compare iteration-only and feedback-based step selection.

Target Wake Time (TWT), introduced in IEEE 802.11ax, lets an access point schedule exactly when each station wakes, transmits, and dozes. Existing TWT schedulers optimize energy or throughput, treating information freshness at best as a constraint and offering no performance guarantees. We design the wake schedule itself for freshness: minimize the weighted average Age of Information (AoI) over stations subject to per-station energy budgets, where the decision variables are the TWT triples (wake interval, offset, service period duration). We derive a renewal-exact AoI model for TWT under per-SP block fading and validate it against packet-level 802.11ax simulation with ~1% mean error. We show that, unlike preemptive scheduling, non-preemptive TWT packing can be infeasible at schedule density 1, and identify the granularity condition under which a small-first best-fit packer provably succeeds. Around this we build Harmonic-Greedy, a scheduler combining a convex relaxation, anchor-optimized power-of-two rounding, and a best-of-uniform safeguard, and prove it is a constant-factor approximation: 4/ln 2 ~= 5.77 under a mild granularity assumption and 6/ln 2 ~= 8.66 unconditionally. We implement the complete system in ns-3 – a TWT wake/doze mechanism integrated with the power-save architecture, plus the scheduler – and show that it is the only scheduler that stays near a relaxation lower bound across all regimes: against a strong energy-greedy baseline it ties when per-station energy floors already pin the periods, and wins by 4-36% exactly where the schedule density is binding and must be redistributed by AoI weight or channel quality rather than by energy budget – the regime our analysis identifies.

We describe GPT-5.6 Sol Pro's short proof that $2\cdot d_{GH}(S^1,S^{2k+1}) \le \tfrac{2\pi k}{2k+1}$. This is the odd-dimensional case of a tight upper bound by Harrison and Jeffs on the Gromov–Hausdorff distance between the circle and spheres.

Every $n$-vertex plane triangulation admits a polychromatic red–blue vertex coloring with discrepancy at most $n-2\ceil{n/3}\le\floor{n/3}$, resolving a conjecture of Asayama and Matsumoto. The proof follows by combining Loyola et al. 2026 and Kawarabayashi et al. 2026. For $n\ge6$ and $n\not\equiv5\pmod6$, we prove the sharper bound $n-2\ceil{(n+2)/3}$. An exhaustive census of all $9{,}150$ non-isomorphic sphere triangulations with $4\le n\le12$ verifies the bounds and also confirms the sharper estimate at $n=11$, the first order in the remaining open residue class.