Papers

We prove that the billiard flow in any finite polygon has at least one periodic orbit. The proof by contradiction is based on a fundamental result on the dynamics of the billiard flow by Galperin, Krüger and Troubetzkoy, on the geometry of a one-parameter scaling of the natural Riemannian metric on the unit tangent bundle, and on the topology of the skeleton or cut-locus of the scaled metrics.

We investigate static, spherically symmetric (SS) spacetimes in covariant teleparallel $F(T)$ gravity sourced by nonlinear Chaplygin and polytropic fluids. Using the covariant coframe/spin-connection (CSC) formalism, we derive the corresponding field equations and conservation laws governing admissible matter distributions and nonlinear torsion sectors. A general reconstruction procedure is developed, allowing the systematic determination of teleparallel $F(T)$ models for arbitrary coframe ansätze and fluid equations of state. Focusing on power-law configurations, we obtain several classes of reconstructed solution branches, including constant-radius, compact-object-like, and wormhole-like (WH-like) branches. The Chaplygin sector naturally leads to effective dark-energy-like and exotic-matter candidate solution branches within the reconstruction framework, which may provide admissible sectors for wormhole-like reconstructed geometries, while the polytropic sector provides reconstructed branches that may serve as physically motivated candidates for future stellar-interior and compact-object models. We discuss the associated candidate horizon and throat conditions, torsion singularities, energy conditions, and local viability properties of the reconstructed branches. The resulting geometries are organized within a teleparallel invariant classification framework, highlighting the role of nonlinear torsion corrections in shaping the solution space. Overall, this work provides a unified covariant reconstruction framework for nonlinear-fluid sectors in teleparallel $F(T)$ gravity, identifying solution branches that may serve as candidates for future compact-object, stellar-interior, and wormhole studies.

Matrix-valued time series arise in a wide range of applications, such as spatio-temporal data from medical imaging and geophysics. Existing methods are mainly designed for static settings and lack adaptability to streaming and time-varying environments. Adaptive filtering techniques have also been largely limited to data with scalar or vector values, leaving adaptive forecasting for matrix-valued time series inadequately understood. To bridge these gaps, we develop an adaptive tensor regression framework that includes Matrix-on-Matrix (MoM) and Tensor-on-Matrix (ToM) formulations for streaming matrix-valued prediction. The two formulations differ in whether to directly model matrix-valued outputs or to exploit temporal structure via higher-order tensor representations. For the proposed tensor regression framework, we develop stochastic gradient descent (SGD) algorithms for online learning. We show that stacking multiple responses across time into higher-order tensors improves performance; in particular, the ToM achieves lower steady-state error and stronger denoising capability than MoM, motivating our focus on the ToM model. We further characterize the tracking behavior of SGD under time-varying dynamics. From a statistical perspective, we establish fixed-time recovery guarantees for ToM under general low-dimensional structures, including sparsity, low-rankness, and their joint sparselow-rank models.

Following previous work, the Algebra of Physical Space (APS) is used to explore Wigner-covariance and spin/helicity fields within the Constructive Standard Model (CSM) of Particle Physics. The spinor formalism of the APS is used to derive explicit Wigner-covariance of Lorentz spinors, and equivalencies with the CSM are demonstrated via the Scattering Algebra (SA). Constructive fields for spin-1/2 and spin-1 are given in the APS and the necessary maps for Wigner-covariance are proposed. The forms of these spin fields are equivalent to Pauli spinors, thereby serving as a new bridge between the CSM and the study of Quantum Information. It is further seen that spin fields of the APS are equivalent to the spin fields in the CSM. Sample Lagrangian densities of the CSM are analyzed using the methods herein, are given geometric interpretations, and are connected to traditional Pauli Theory. Finally, the hff (Higgs and two massive fermions) Lagrangian density is used to determine the first constructive scattering amplitude that is defined purely in terms of the APS. Concerningly, this leads to an anti-Hermitian action term, and this surprise is later confirmed using traditional CSM techniques. Throughout this paper, the illuminating power of Geometric Algebra is clear: Everything has a geometric interpretation, and results can be accomplished matrix-free.

Railway crew scheduling consists of assigning a set of tasks to crew members in the form of feasible duties so as to minimise costs, and is one of the key planning problems faced by railway operators. While column generation is the dominant solution method, literature on exact branch-and-price approaches is scarce. Moreover, few publicly available benchmark instances exist. We conduct a systematic computational study of branch and price for railway crew scheduling, evaluating the contribution of branching rules, primal heuristics, and reduced cost fixing. To this end, we introduce a novel library of 100 benchmark instances, ranging in size from 90 to 3,016 tasks, each provided in a single-depot and a multi-depot variant. The instances are generated by mimicking the railway planning process, validated against data from Netherlands Railways, and publicly available with best known solutions and a solution checker. We report average optimality gaps below 1% across all instance sizes, driven primarily by the strong performance of primal heuristics, whereas branching and reduced cost fixing contribute little to closing the gap. Branch and price also outperforms a multi-commodity flow formulation, which fails to find a feasible solution on many of the smallest instances. Our findings provide support for the popularity of column generation-based primal heuristics, and show that improving the dual bound remains the main challenge for exact methods in railway crew scheduling.

We prove the realizability of genus-$0$ branch data of the form $(2^r,a \mathop{|} 3^s,b \mathop{|} 3^t,c)$ and $(2^r,a \mathop{|} 3^s \mathop{|} 3^t,b,c)$, where $a$, $b$, $c$ are not divisible by $2$, $3$, $3$ respectively. The proof uses an explicit combinatorial description of coverings of the sphere branched over $3$ points via dessins d'enfants. As a corollary, we establish realizability for a broader class of branch data with more critical values.

Quantum optimal transport (QOT) is a rapidly developing field. Among the many formulations of this adaptation of classical optimal transport (OT) to spaces of density matrices, we numerically study a family of distances based on a dynamical formulation inspired by the Benamou-Brenier OT formulation. We introduce an interior-point regularized method to compute geodesics between positive semidefinite matrices and visualize the results in terms of integral kernels and densities, inspired by quantum chemistry applications. We show that dynamical QOT may provide a good approximation to certain problems in quantum chemistry with appropriate parameter tuning. We also study the numerical properties of the distances at hand, and the convergence of the objects when the size of the matrices increases.

In a symmetric Bernoulli process, all binary strings, or “words” of the same length have the same long term frequency. However, between two such words, one may have a “frequency advantage” in the sense that in any long enough segment of the Bernoulli process, the probability that the word occurs more times than the other word is greater than the probability the other way around. To characterize the frequency advantage in the long run, the asymptotics of the difference between the two probabilities as the length of the segment of the Bernoulli process tends to infinity is derived.

Let $(X,d,\mu)$ be a complete separable metric measure space satisfying a doubling condition and a $(1,1)$-Poincaré inequality. We develop a rigorous framework unifying two lines of analysis: the isoperimetric-profile approach of Coulhon-Grigor'yan-Levin \cite{CGL2003} and the grand/small Lebesgue-Sobolev scale introduced by Fiorenza-Formica-Gogatishvili \cite{FFG2018}. An explicit profile-to-scale transform $\PhiX$, defined via an inverse integral of $\IX$, converts geometric data into grand Lebesgue parameters. Sharp, up to universal constants, embeddings $W^{1,1}(X) \hookrightarrow \mathcal{G}_X$ with explicit constants (Theorem \ref{thmmain}). A converse: controlled grand embeddings imply explicit lower bounds on $\IX$ (Theorem \ref{thmconverse}). Concrete examples in genuinely non-Euclidean settings: the Heisenberg group $\mathbb{H}^1$, a model manifold with logarithmic volume growth, and Gaussian measure on $\R^n$ treated as a locally doubling space. All arguments are carried out on general metric measure spaces without reference to charts or a smooth structure; the gradient is the upper gradient in the sense of Heinonen-Koskela, and perimeter is the outer Minkowski content.

Thurston gave a simple way to construct all triangulations of the sphere for which 5 or 6 triangles meet at each vertex, using the Eisenstein integers $\mathbb{E}$. While such triangulations can be defined purely combinatorially, Thurston noticed that given such a triangulation, one can make all the triangles into flat equilateral triangles with the same edge length, and this gives the 2-sphere a flat Riemannian metric except at 12 cone points with angle deficit $\pi/3$. He showed that up to rescaling, all such Riemannian metrics arise from his procedure. He studied the moduli space $\mathcal{M}$ of all such metrics modulo rescaling, and showed that $\mathcal{M}$ is open and dense in an orbifold $\overline{\mathcal{M}} = \mathbb{PC}^{10}_+/\Gamma$. Here $\mathbb{C}^{10}_+ = \{ v \in \mathbb{C}^{10} \vert \; Q(v) > 0\}$ for some quadratic form $Q$ of signature $(1,9)$ on $\mathbb{C}^{10}$, $\mathbb{PC}^{10}_+$ is its projectivization, and $\Gamma$ is a certain discrete group of linear transformations of $\mathbb{C}^{10}$ preserving both $Q$ and the lattice $\mathbb{E}^{10} \subset \mathbb{C}^{10}$. He also showed that $\overline{\mathcal{M}}$ is the moduli space of flat Riemannian metrics on the sphere with at most $12$ cone points and angle deficits that are positive integer multiples of $\pi/3$. Here we briefly outline the basic ideas behind this work, and illustrate them with examples.

Let $\cX$ be a (projective, geometrically irreducible, non-singular) algebraic curve of genus $\ge 2$ and positive $p$-rank $\gamma(\cX)$, defined over an algebraically closed field $\mathbb{K}$ of positive characteristic $p>0$. Contrary to what occurs for the genus, no function $h(\gamma)$ exists such that $|\aut(\cX)|\le h(\gamma)$ whenever $\gamma=\gamma(\cX)$. Thus, to have a bound on $|\aut(\cX)|$ only depending on $\gamma(\cX)$, some restrictions on $\cX$ and $\aut(\cX)$ are needed. In this context, the following theorem is proven. Let $\Gamma$ be a subgroup of $\aut(\cX)$. Assume that there is a point $P\in \cX$ such that the quotient curve $\cX/S_P$ is rational, where $S_P$ denotes the Sylow p-subgroup of the stabilizer $\Gamma_P$ of $P$ in $\Gamma$. Then the following $p$-rank analog of the Riemann-Hurwitz bound \begin{equation*} |\Gamma|\le 24 \left(\frac{p}{p-1}\right)^4 \gamma(\cX)^4 \end{equation*} holds, unless a subgroup of index $\le 2$ of $\Gamma$ fixes $P$. This bound is sharp up to a constant depending only on $p$.

The mean first passage times are among the most critical characteristics of a Markov chain. In this paper, we focus on the scenario in which one or more states of a higher order ergodic Markov chain are modified to be absorbing. We prove that the resulting chain has to be absorbing. For a higher order absorbing Markov chain, we prove that the equation its fundamental tensor satisfies must be nonsingular and provide a MATLAB function fund for solving the equation. Besides, we connect each horizontal slice of the mean first passage time tensor with a fundamental tensor obtained when one state of a higher order ergodic Markov chain is modified to be absorbing, which also leads to a tensor series representation for selected mean first passage times.

We prove a duality theorem for quasi-compactly supported cohomology of quasi-coherent sheaves on rigid-analytic spaces, with respect to a smooth and Kiehl partially-proper morphism. This includes an identification of the dualizing object with volume forms. The functional analysis underlying our theory does not use condensed mathematics, but rather Ind-Banach spaces, following Ben-Bassat–Kelly–Kremnizer.

We show a universal property of the span $\infty$-category that yields a description of functors defined on this category. For this, we view the span construction as a functor from double $\infty$-categories to $\infty$-categories, and show that this functor admits a right adjoint defined by the double $\infty$-categories of squares. As an application, we obtain a model for the localization of an $\infty$-category in terms of a span $\infty$-category and also give another proof of the model given by the relative nerve. Moreover, using the span-squares adjunction, we obtain new proofs of the equivalences between different models of algebraic $K$-theory, given by the $\Q$-, the $\oS$-, the cobordism model, and the squares construction.

Magic braids are used in leatherworking to make intricate straps and bands. They appear to be impossible to make. To determine which braids can be made with this leatherworking technique, we explore their relation to several braid groups.

In this paper, we consider the data-driven discovery of Lyapunov functions for autonomous dynamical systems. We represent the Lyapunov function as an expression tree of fixed depth and formulate the Lyapunov discovery task as a constrained self-supervised symbolic regression problem. The constraints model the output of the Lyapunov function for a given input as well as the Lyapunov stability conditions. This modeling approach makes no a priori assumptions about the functional form of the Lyapunov function, is inherently interpretable since the function is obtained in a symbolic form, and, in principle, can be applied to any continuous dynamical system. We also develop a tailored branch-and-bound-and-check solution approach to efficiently solve the resulting learning task. Applications to several case studies show the ability of the proposed approach to discover Lyapunov functions.

Liu and Wan introduced T-adic exponential sums as a way to interpolate all character sums with character having p-power conductor. In this paper, we generalize their T-adic theory to partial T-adic exponential sums. We prove that the associated L-functions are T-adic meromorphic, and as a consequence, give a p-adic proof of rationality for all partial L-functions of characters with p-power conductor. We also give Newton-over-Hodge estimates.

We study noncommutative analogues and integrable discretisations of nonlinear Schrödinger (NLS)-type systems associated with reduction groups. In particular, we consider the Ablowitz–Kaup–Newell–Segur (AKNS) system, the Kaup–Newell derivative NLS system, and the Mikhailov–Shabat–Yamilov deformation of the derivative NLS system together with their Darboux–Bäcklund transformations and associated lattice equations. We derive the continuum limits of previously constructed integrable lattice systems and recover the corresponding NLS-type partial differential equations. We then construct a noncommutative deformation of the Mikhailov–Shabat–Yamilov system and show that, unlike the AKNS and Kaup–Newell cases, its Lax representation requires the introduction of nonlocal variables. Furthermore, we derive Darboux–Bäcklund transformations and integrable discretisations for the noncommutative derivative NLS and deformation derivative NLS systems in the form of vertex–bond lattice equations. We also construct explicit solutions for a six-point derivative NLS-type lattice equation and for its noncommutative analogue.

Dyck paths and certain black/white labelling of nodes leads to the white-height. Using generating functions and tricks of the trade, we establish that the average white-height among Dyck paths of half length $n$ is asymptotic to $\frac12\sqrt{\pi n}$ for two different models. These are appealing results that could be presented to students to learn the trade. A last section links walks with double steps and 2-Motzkin paths. For them, the white-height is the natural concept. Folks who might be offended by the notion of white-height might choose their own colours that they like.

We consider Schrödinger operators with dynamically defined potentials that are anti $\alpha$-Hölder and generated by hyperbolic dynamics. We prove an anti Lipschitz estimate on functions related to the orbit of $\vec{e}_1$ under an associated projectivized cocycle. After, we apply the estimate on subshifts of finite type and maps with one expanding direction on $\mathbb{T}^d$ where $d\geq 1$. In particular, the application leads to uniformly positive Lyapunov exponents with sufficiently large coupling constants.

We study the Laplace equation with Dirichlet and Neumann boundary conditions posed on multi-arcs, i.e., collections of open arcs meeting at junction points. We begin by introducing a scale of Sobolev spaces constructed using the Sobolev spaces on open arcs as main building block and extend the definition of trace operators. We reformulate the boundary value problems using boundary integral formulations. We then establish a well-posed integral formulation for the Dirichlet problem, which can be discretized using standard numerical methods. We further investigate the singular behavior of the solution densities at branch points through numerical experiments and observe that these singularities are comparable to the corner singularities arising in polygonal domains. For the Neumann problem, we show that the associated hypersingular operator is not necessarily invertible on classical Sobolev spaces and provide numerical evidence that solutions may develop jump discontinuities at branch points.

To construct obstructions to the stability of rational maps with non-summable critical points in their Julia sets, we introduce automorphic measures with complex eigenvalues for rational maps on the Riemann sphere. In particular, these measures extend the classical notions of quasi-invariant and conformal measures by allowing the respective Radon–Nikodym derivative to be complex-valued and proportional to a multiplicative cocycle. \[ j_{(s,t)}(R) = |R'|^{s} \left(\frac{|R'|}{R'}\right)^{t}, \] which plays the role of a generalized automorphy factor in the sense of group actions. The existence of such measures reveals a close connection between geometric and dynamical properties of rational maps. We show that the existence of certain automorphic measures, particularly unimodular measures and their associated vector fields, implies instability of the corresponding rational map. Specifically, for a weakly dissipative rational map admitting a $( -1, 1)$-unimodular measure, there exists an integer $q \ge 1$ such that the map is $q$-unstable. This result generalizes earlier instability criteria involving pseudoconformal measures and connects the presence of such measures to the failure of structural stability. Furthermore, we establish ergodic and combinatorial conditions ensuring the existence of unimodular measures or vector fields – most notably, through bounded recurrence, bounded velocity of arguments, and relations with the Milnor–Thurston kneading theory. These criteria provide a unified framework linking automorphic measures, Lyapunov spectra, and the geometric deformation spaces of rational maps.

The Weil-Petersson (WP) volumes of the (compactified) moduli space of ${N}=1$ supersymmetric Riemann surfaces with Neveu-Schwarz (NS) boundaries are frequently discussed in the literature. Such surfaces can also have marked points called Ramond (R) punctures, where the superconformal structure degenerates. Computing the volumes when these R punctures are included is more challenging for the usual differential and algebraic geometry approaches, and they are therefore less well explored. In particular, the spectral curve describing the inclusion of R punctures is apparently unknown, so far. However, the right random matrix model approach can handle the NS and~R sectors on an equal footing. Such a construction is presented, showing how to use a recently developed technique to readily compute many closed-form formulae for $V^{(2m)}_{g,n}(\{b_i\})$, the WP volumes for genus $g$ with $n$ NS-boundaries of geodesic lengths $b_i$ ($i{=}1,\ldots,n$), and $2m$ R-punctures. Several striking relations between volumes (and subsectors thereof) emerge naturally in this approach. Moreover, the hitherto missing spectral curve is presented, and its use for (re-)deriving the $V^{(2m)}_{g,n}(\{b_i\})$ is demonstrated by using topological recursion.

The superstring amplitude's Mellin transform in energy is computed at fixed momentum transfer. In the forward limit, it is shown that this transformed amplitude reduces to the Riemann zeta function, while for $t\neq 0$ it represents a deformation of $\zeta$. This object exhibits remarkable mathematical properties as a result of physical attributes of the string amplitude. For integer subtractions, the dispersion relation defined by the Mellin transform yields the effective field theory expansion of the string amplitude order by order in $s$ at finite $t$, for which a new closed-form expression is derived.

We prove, to all orders, maximal transcendentality of the strongly coupled large-N Principal Chiral Model in the double-scaling regime introduced in our earlier work. We also prove that, after a natural shift of the coupling constant, the coefficients of the vacuum-energy expansion are expressed purely as polynomials in odd zeta values with rational coefficients. The first 35 explicitly computed orders reveal further number-theoretic regularities, pointing to hidden structure beyond maximal transcendentality.

Could two worlds localised on neighbouring branes communicate through gravity alone? We investigate this question in a minimal higher-dimensional framework in which Standard Model fields are confined to our brane while gravity propagates through the bulk. From the brane-to-brane graviton propagator we derive the retarded transfer kernel of the inter-brane link and identify the transition from evanescent to propagating Kaluza-Klein modes. The central idea is to give the Kaluza-Klein tower a new role: not only as a spectrum of massive gravitational states, but as a set of communication carriers. Below the first KK threshold the channel is effectively four-dimensional and is mediated only by the massless graviton. Above threshold, massive KK modes open as additional propagating subchannels, and information may be encoded in their occupation pattern, relative phases, and arrival-time structure as well as in ordinary signal variables. The compactification determines the KK masses, wavefunctions, brane overlap factors, and propagation phases, which together define a multi-input multi-output (MIMO) channel matrix. In the resolved-mode limit, the tower yields approximate parallel subchannels, to which standard information-theoretic notions such as capacity bounds, water-filling, effective rank, and sparse occupancy codes apply. The production and detection of such signals are highly model-dependent and not assumed to be feasible with known technology. Nevertheless, the channel structure is well defined: a neighbouring brane-world could be separated from us by a microscopic distance in the compact space while remaining hidden because the only shared interaction is gravity. The first observable signature may not be a deliberate message, but the spectral and modal structure of the Kaluza-Klein tower itself, revealing partial information about the geometry of a nearby hidden world.

Training reinforcement learning (RL) policies from scratch is costly: it requires careful reward and environment design, extensive tuning, and substantial computation. Yet many control problems already have a functional but suboptimal policy available as a baseline. This paper proposes a method for embedding such a baseline into the RL training process, simultaneously improving training efficiency relative to from-scratch methods and producing a learning policy that outperforms the baseline. At each step, the method arbitrates between the baseline policy and a trainable learning policy, initially relying strongly on the baseline policy and then progressively transferring agency to the learning policy. By the end of training, the learning policy is a standalone neural network that operates without baseline policy support. The paper formalizes what it means for the baseline policy to be functional: under this policy, the agent reaches a goal set and remains there with high probability. The proposed arbitration mechanism is designed to exploit this property during training, yielding high goal-reaching rates right from the beginning of training. A theoretical analysis provides a formal interpretation of this behavior under stated assumptions and extends it to the final baseline-free regime, where explicit lower bounds are derived for the goal-reaching probability of the standalone learning policy. Empirical results on continuous-control benchmarks show that the proposed method achieves returns that match or exceed those of competitive approaches, while maintaining the highest goal-reaching rates throughout training among the compared methods – including in the final stage, where the learning policy operates without any baseline support.

We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives. A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer, on which a non-linear scalar activation function is applied, and then returns the output into a Banach space via some linear readouts. By proving a weighted Nachbin theorem, we establish a universal approximation theorem for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. This leads us to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. As a further application, we show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.

In this note, we use a simple argument to show the existence of large values of conjecturally sharp size for Dirichlet $L$-functions attached to primitive characters of fixed order at $\sigma\in (1/2, 1]$. More precisely, for every fixed integer $g\geq 2$ we prove the existence of a primitive character $\chi$ of order $g$ and conductor $Q\asymp x$ such that $|L(1,\chi)| \geq e^\gamma\left(\log\log x+\log\log\log x-\log(2\log g)+o(1)\right). $ We also show that for every fixed $1/2<\sigma<1$ there exists a primitive character $\chi$ of order $g$ and conductor $Q\asymp x$ such that $\log |L(\sigma,\chi)| \geq \left(C_g(\sigma)+o(1)\right) (\log x)^{1-\sigma}(\log\log x)^{-\sigma}, $ for some explicit positive constant $C_g(\sigma).$ Previously, such bounds were known only conditionally on the Generalized Riemann Hypothesis, and even then only in the special cases $g=2$ and $g=3$.

Standard diffusion models typically use a single time-homogeneous Gaussian terminal distribution as the reference law for generation. While this choice is analytically convenient and empirically powerful, it provides little explicit structure for data concentrated near low-dimensional manifolds, where different regions of the data distribution may correspond to distinct local geometric or semantic factors. As a result, the reverse model must recover manifold-level structure almost entirely from an unstructured terminal reference distribution. We propose PTL-Diffusion, a proof-of-concept diffusion framework whose forward noising process converges to a nonconstant periodic family of Gaussian terminal laws rather than to a single invariant law. Unlike a phase-conditioned DDPM, where phase information only enters the denoising network while the forward process remains unchanged, PTL-Diffusion embeds phase structure directly into the forward noising dynamics. The proposed construction remains close to standard denoising diffusion models: for a periodically forced Ornstein–Uhlenbeck-type forward process, we derive closed-form forward marginals, the limiting periodic Gaussian terminal family, and explicit Gaussian reverse posteriors, enabling standard noise-prediction training. We also introduce an invariant-average regularization term coupling the phase-conditioned reverse dynamics through the averaged periodic reference law. Experiments on torus and cylinder point-cloud benchmarks and the Olivetti face dataset show that PTL-Diffusion improves manifold-level distributional matching over matched DDPM baselines, reducing phase-conditioned errors, feature-space covariance errors, and nearest-neighbour manifold distances. These results suggest structured terminal reference laws as a promising direction, while motivating more expressive phase constructions and larger-scale evaluations.