Papers

We construct high-frequency perturbations of an integrable exact symplectic twist map which preserve an analytic invariant graph with a prescribed rotation number. Their oscillation frequencies may be any positive integers tending to infinity. For every fixed Gevrey parameter $L>0$, the perturbations tend to zero in a Gevrey-$\alpha$ norm and diverge in a Gevrey-$(\alpha-\varepsilon)$ norm. The frequency class strictly contains the $\alpha$-Brjuno–Rüssmann class, and their difference has cardinality \(2^{\aleph_0}\).

We study strong asymptotics of orthogonal polynomials on an interval with varying weights, extending the framework of Totik's theorem on weighted orthogonal polynomials. Our results generalize this theorem in several directions. In particular, we allow the supports of the associated equilibrium measures to converge to a proper subinterval of the original interval of orthogonality or even collapse to a point. These extensions are motivated by applications to multiple orthogonality where such varying measures arise naturally.

Using non-$\mathbb{A}^1$-invariant motivic cohomology, we prove motivic refinements of certain known computations of the algebraic $K$-theory of singular rings, such as rings of the form $\mathbb{Z}/p^n$, $\mathbb{Z}[x]/(x^e)$, and $\mathscr{C}(X;\mathbb{C})$ for $X$ a compact Hausdorff space. These refinements are made possible by the use of integral $p$-adic Hodge theory, as a replacement for the standard use of trace methods in $K$-theory.

We develop a port-Hamiltonian framework for coupled rigid/flexible multibody systems. The rigid dynamics may be nonlinear and subject to configuration and velocity constraints, while the flexible components are described by linear port-Hamiltonian partial differential equations on one-dimensional spatial domains. The subsystems interact through boundary or distributed ports. On the flexible side, the differential operator and its domain remain fixed, whereas state dependence enters only through finite-dimensional coupling components of the Dirac structure. We show that, under a natural surjectivity condition, the port-Hamiltonian interconnection with a modulated Dirac structure of a finite-dimensional rigid subsystem again yields a modulated Dirac structure. The Hamiltonians of the subsystems add, while the internal coupling powers cancel. The framework is illustrated by a planar moving Euler–Bernoulli beam and a slider–crank mechanism with a flexible connecting member.

In 2016, Berkovich and Uncu proved that, for all nonnegative integers $i$, $j$, and $n$, the number of strict partitions of $n$ with $i$ odd-indexed odd parts and $j$ even-indexed odd parts equals the number of strict partitions of $n$ with $i$ parts congruent to $1$ modulo $4$ and $j$ parts congruent to $3$ modulo $4$. Their proof used generating functions, and they asked for a combinatorial proof. We answer their question with an explicit bijection, assembled from three classical ingredients: $2$-modular diagrams, an insertion algorithm of Chen, Gao, Ji, and Li, and Glaisher's bijection. AxiomProver autonomously formalized and verified the proof of the main theorem in Lean.

Gradient descent on a factored model $W = UV^\top$ is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not. We trace the difference to the gauge symmetry of the loss, its invariance under $(U, V) \mapsto (UQ, VQ)$. Gradient flow's low-rank mechanism is available to an optimizer only if that optimizer is gauge-equivariant, a condition necessary for the transfer but not sufficient for low-rank recovery. Gradient descent, momentum, "shared-scalar" Adam, Muon, and Shampoo satisfy it. Adam, RMSProp, and the other coordinate-wise methods do not. A structure theorem characterizes the memoryless equivariant rules as exactly the Gram-determined left preconditioners, and a transfer theorem carries gradient flow's pathwise properties to common-scalar flows. We then sort nine update rules on underdetermined matrix sensing by recovery error against the planted ground truth. A one-parameter family from coordinate-wise to shared-scalar preconditioning restores the bias monotonically, isolating anisotropy as the cause. A "spectral schedule" reconciles two opposing reports about Muon: equal-rate updates recover exactly low-rank targets but lose their edge as the spectral tail grows. In transformers, Adam separates two gauge-equivalent initializations at the first step, where the equivariant optimizers stay at float precision, and ends with the per-head invariants $W_Q^\top W_K$ 56% apart in relative Frobenius distance, a gap no per-head rotation can close. On two hyperspectral datasets at matched training loss, gradient descent cuts held-out error by 43-44% at the lowest sampling density, and at lower effective rank. Basis choice is therefore not a tuning detail but a decision about which interpolant the optimizer selects.

We study large mass monopoles with structure group $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$ on asymptotically conical $\mathrm{G}_2$-manifolds and Calabi–Yau $3$-folds, with fixed asymptotic class. After placing the AC asymptotic theory, the variational compactness theory of Parise–Pigati–Stern, and Li's singular abelian compactness theory in a common $\Theta$-monopole framework, we prove that the mass-renormalized Yang–Mills–Higgs and intermediate energy measures converge to $8\pi\|T\|$ for a compactly supported calibrated integral codimension-three cycle $T$. This identifies the two limiting currents and shows that the variational calibration inequalities are saturated. Using this common limit as the starting point for a finer analysis, if $\mathcal S$ is the calibrated support, $\mathcal Z$ the Kuratowski upper limit of the Higgs zero sets, and $\mathcal C$ the limiting nonabelian locus, defined as the Kuratowski upper limit of Li's curvature concentration loci, then $\mathcal S\subset\mathcal Z\subset\mathcal C=\mathcal S\cup\mathcal O$, where $\mathcal O$ is precisely the obstruction to effective codimension-three monotonicity. For the cohomogeneity-one large mass families on the Bryant–Salamon $\mathrm{G}_2$-manifolds and the Stenzel Calabi–Yau $3$-fold, we prove that $\mathcal O=\varnothing$. On $X\setminus\mathcal C$ the sequence abelianizes; corrected longitudinal curvatures converge smoothly, and the remaining compactness alternatives are governed by $L^2$-harmonic $2$-forms.

The classical Blaschke–Lebesgue theorem identifies the Reuleaux triangle as the planar convex body of constant width with minimum area. We investigate this extremal problem under prescribed symmetry constraints. Specifically, we classify the minimum-area convex bodies of constant width that are invariant under a finite group $G$ of isometries of $\mathbb{R}^2$ fixing the origin. For the exceptional reflection group $D_1$, the minimizers are precisely the Reuleaux triangles invariant under the prescribed reflection. If $G$ contains the half-turn $\mathcal R_\pi$, the disk is the unique minimizer. For odd $n\geq 3$, the minimizers are regular Reuleaux $n$-gons, unique up to rotation in the cyclic case $C_n$, and exactly those satisfying the prescribed reflection symmetry in the dihedral case $D_n$.

Long context reasoning in large language models (LLMs) is usually constrained by the fact that a single inference trajectory has to simultaneously explore the context, store intermediate state, verify evidence, and produce the final answer. This becomes particularly difficult in tasks that require extraction, counting, ordering, or multi-hop reasoning, where an early mistake can propagate until the final response. In this work, we propose Chained Recursive Language Models (Chained RLM), an inference-time architecture, in which the same underlying model is called repeatedly as a sequence of fresh reasoning roots. Each root receives the original problem and context, but does not inherit the full conversational history. Instead, it receives a compact plain-text summary, a plain-text blackboard, and some durable task-specific artifacts written by predecessor roots. The motivation is to manage the context by chopping into partial tasks rather than one large inference response; in each staged computation, intermediate artifacts can be inspected, corrected, and extended by a later fresh inference by the same model. We describe the system model, handoff mechanism, artifact workspace, and evaluation protocol for this system. We study when fresh-context artifact continuation gives a measurable gain in accuracy over direct LLM answering even with recursive tool-calling.

Let \(C_n\) be the minimum cost of a perfect matching in an \(n\times n\) matrix of independent uniform random variables. We prove that \[ \sqrt n\{C_n-\zeta(2)\} \ \Longrightarrow\ \mathcal N\bigl(0,4\zeta(2)-4\zeta(3)\bigr). \] The proof begins with an exact change of variables based on a uniformly rooted shortest-path selection of an optimal dual potential. After the unused reduced costs are integrated out, a reference law separates the rows conditionally on the potential field, while the ordered potential gaps become independent exponentials. The only residual dependence is a directed-tree factor. Ordering the potentials turns its zero–one support into a Ferrers matrix, whose matrix-tree determinant is triangular. A singular inverse-degree estimate and exact normalization then yield total-variation convergence to the reference law. Finally, a conditional triangular-array central limit theorem accounts for row noise, and a second triangular array accounts for the linear response of the potential field. The strategy used here is likely to be applicable to other problems.

We prove a Heintze–Karcher type inequality for a large class of log-convex weights in Euclidean and Hyperbolic spaces. As a consequence we obtain an Alexandrov theorem for $\lambda$-self expanders.

Slime Trail is a two-player combinatorial game in which the players alternately move a shared token to an adjacent vertex, permanently removing each vertex the token leaves, while attempting to reach a goal node. Ferland and Burke (2017) proved that Slime Trail is PSPACE-complete on arbitrary planar graphs and asked whether the same holds for the grid version actually used in play. We resolve this open problem by proving that Cardinal Grid Slime Trail, that is, Slime Trail on a square grid with four-directional movement, is PSPACE-complete. We adapt their QBF reduction to the grid setting, designing grid-compatible gadgets that respect the degree-4 bound and the parity constraints of the integer lattice. We further show the construction extends, under a 45-degree rotation, to the eight-directional variant.

The Schiffer conjecture states that if a smooth domain $\Omega \subset \mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian which is constant at the boundary, then the domain is a ball. It is intimately related to Pompeiu's problem, stating that if a nonzero function integrates zero over any rigid motion of $\Omega$, then $\Omega$ is a ball. We disprove both conjectures in $\mathbb{R}^2$, constructing infinitely many planar domains $\Omega$ which are not balls and satisfy the conditions above. Our domains are $N$-fold symmetric, with $N$ sufficiently large. Our approach is based on a novel strategy of considering a relaxed problem where $N$ can be any real number (which corresponds to the Schiffer problem only when $N$ is a natural number). We then apply bifurcation theory to this relaxed problem, showing that the size of the local bifurcation branch can be taken independently of $N$. This result allows us to conclude that branches starting with $N$ sufficiently close to an integer reach integer values of $N$.

The product of two Cartesian quaternions can be written as a quadratic form based on a 4x4 matrix made of the imaginary numbers i,j,k introduced by Hamilton. We generalized this matrix to non-Cartesian bases, and showed that the matrix is made of the metric and the cross tensors. We called it crossmetric tensor. Its symbols are ia,ib,ic; they are the elementary crystallographic quaternions. We determined the crossmetric tensors for the six crystal families. We also showed that any unit crystallographic quaternion can be geometrically represented by an infinity of pairs of oriented planes intersecting along the vectorial component of the quaternion such that the angle between them is the semiangle of the rotation. The composition of quaternions follows the intuitive source-target rule. The elementary quaternions ia,ib,ic are geometrically represented by pair of perpendicular and oriented planes intersecting along the axis a,b,c, respectively. Three other complementary quaternions were also introduced. They are the pairs of planes ma, mb, mc. The elementary quaternions follow Hamilton rules on the squares of imaginary numbers. The complementary quaternions follow Hamilton rules on the bi and tri-products. For Cartesian basis, the quaternions i,j,k appear as a specific case of crystallographic quaternions. They are formed by the pairs of perpendicular faces of the cube mx, my, mz.

We consider a stochastic process on $\mathbb{Z}^d$ for $d \geq 1$. Given a translation invariant and ergodic starting configuration of finite clusters, each cluster $C$ performs a continuous time simple random walk with rate $|C|^{-\alpha}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. In all dimensions, we show that if $\alpha\ge 0$, there is almost surely no spontaneous creation of an infinite cluster within finite time. Moreover, for any $\alpha\le-1-2/d$ there is a finite-time blowup almost surely. In the regime $\alpha\in(-1,0)$ we show that the behavior greatly depends on the initial configuration. In addition, in dimension one, we establish the exact phase diagram.

Data assimilation (DA) uses Bayesian inference to update the state of a numerical forecast model with observed data. In this study, we propose a fundamentally different, unified approach to atmospheric data assimilation. We use latent video flow-matching to sample temporally consistent trajectories from a prior trained using ERA5 reanalysis (69 variables over an 8-day window). We also use posterior sampling to assimilate real observation sources, such as those from the NOAA Integrated Global Radiosonde Archive and the Integrated Surface Database. Because the prior generates a continuous trajectory, it naturally propagates information between observed and unobserved frames. Therefore, we can perform various DA tasks, such as filtering and smoothing, simply by changing the observed frames. Moreover, we generate full-state ensemble forecasts directly from sparse observations, achieving performance competitive with state-of-the-art observation-to-forecast models.

Partial orthomodular lattices are an intermediate class of algebras between ortholattices and orthomodular lattices. In this short note, we obtain a duality for partial orthomodular lattices via a subcategory of the category of spectral spaces, independently of the Axiom of Choice.

Random surfaces on a hard substrate often exhibit entropic repulsion, wherein the surface is propelled upwards to allow entropically preferable downward fluctuations. A particularly rich class of examples arises from the low-temperature 3D Ising model. A powerful approach to studying such surfaces is through their level curves, which form a family of non-intersecting random curves. In [CIW18, CIW19], an ensemble of Brownian lines with geometrically increasing area tilts was proposed as a putative limiting model in this case. This model falls outside the scope of techniques based on integrable or SDE structures, which have been key ingredients in the study of the Airy line ensemble. A particularly intriguing question about such line ensembles concerns their mixing properties when viewed as a Markov process, and in particular the rate of decay of correlations in time. For the Airy line ensemble, this decay is known to be inverse quadratic. The first quantitative bound on the decay of correlations in the area-tilted model, established in [CG25], was slower than polynomial in time. An earlier result [DLZ24] had established positivity of the spectral gap for the finite-line version of the ensemble, without quantitative bounds. This left open the important question of the true decay rate of correlations for the infinite ensemble. Settling this question for sufficiently large area-tilt strength, corresponding to sufficiently low temperature for the 3D Ising model, we prove exponential decay of correlations for the infinite ensemble and a uniform (in the number of lines) positive spectral gap for the finite ensemble. Our proof is based on establishing a precise form of separation of scales between curves of different indices, using a novel probabilistic approach involving embedding supercritical branching processes in the line ensemble.

We introduce a purely operator-theoretic framework for quantitative homogenization that bypasses the traditional reliance on large-scale spatial regularity and probabilistic assumptions. Inspired by Tartar's vision of a point-free theory, we derive explicit norm resolvent estimates using only the algebraic structure of multiscale operators and the abstract geometry of Hilbert spaces. In this framework, the effective macroscopic dynamics and the abstract corrector emerge naturally from an orthogonal decomposition of the state space, governed algebraically by a Schur complement. To quantify the convergence rate, we introduce a frequency-splitting technique and a flux corrector obtained by inverting the microscopic derivative on the residual field; this controls the error for regular macroscopic data in terms of explicit, $\varepsilon$-independent operator norms. This abstract perspective places on one footing stationary, non-stationary, periodic, quasi-periodic, and stochastic homogenization. We demonstrate that the physical distinctions between these media—and their respective convergence rates—are captured by the behavior of the spectral measure of the microscopic derivative operator, evaluated on the residual field, near zero frequency; in the periodic case the rate $O(\varepsilon)$ is proved, while in the stochastic case the rates are conditional on a density-of-states hypothesis. Furthermore, our results apply for multiscale systems that do not uphold subadditivity of energy quantities or do not have a well-defined Brillouin zone.

The tube algebra, which carries the structure of a $C^*$ weak Hopf algebra, is a fundamental tool for characterizing topological excitations in topological phases. In this work, we generalize the tube algebra framework to codimension-2 defects in $(2+1)$D gapped phases described by Turaev–Viro–Barrett–Westbury TQFTs, with particular emphasis on Levin–Wen string-net models. We show that such codimension-2 defects are described by a defect tube algebra, which naturally carries the structure of a comodule algebra over the weak Hopf tube algebra associated with the topological excitations. Boundary and domain wall defects are then characterized by the representations of the corresponding boundary and domain wall defect tube algebras. In particular, a domain wall defect tube algebra can be regarded as a generalized Drinfeld double of the tube algebras associated with the two adjacent boundary defects. More generally, for a domain wall joining $N$ bulk phases, the associated domain wall defect tube algebra can be viewed as an $N$-tuple algebra. This perspective extends naturally to general $k$-defects, namely codimension-2 defects joining $k$ codimension-1 defects, for which the resulting generalized tube algebra carries the structure of a multicomodule algebra.

The classical Durfee-square argument gives a combinatorial proof of the pentagon identity for the quantum dilogarithm. Just as the pentagon identity is associated with the $A_2$ quiver, Keller's identity is associated with the square-product quiver $A_2\square A_2$. Previous proofs of Keller's identity use cluster categories or spectral sequences in rapid-decay equivariant cohomology. We give three proofs of Keller's identity: a generating-function proof, an explicit weight-preserving bijection on colored partitions, and a standard-monomial proof using a four-colored arc algebra. Their common mechanism is an iterated Durfee decomposition: two possible pairings give horizontal and vertical decompositions, while a third binary step accounts for the coupling factor. These constructions provide a “superpotential analogue” of the Durfee-square argument.

In this paper, we study proper and complete edge-colorings of Johnson graphs $J(n,2)$, also called $n$-triangular graphs. They are isomorphic both to the 2-token graphs of complete graphs and to the line graphs of complete graphs. A $t$-edge-coloring of a graph $G$ is a function that assigns one color from $\{1,2,\ldots,t\}$ to each edge. Such a coloring is called proper if no two incident edges receive the same color, and complete if every pair of distinct colors appears on a pair of incident edges. The achromatic index, denoted by $\alpha_2(G)$, is the largest integer $t$ for which $G$ admits a proper and complete $t$-edge-coloring. We establish new lower and upper bounds for $\alpha_2(J(n,2))$, provide explicit proper and complete edge-colorings attaining the lower bounds, and determine the exact value of $\alpha_2(J(n,2))$ for several values of $n$.

Eichler gave an elementary but involved proof of the Baker-Campbell-Hausdorff formula, based on the associativity of the product of three exponentials. We show that by considering the associativity of the product of four exponentials, one can bypass the complications of Eichler's proof to arrive at a shorter proof that is both elementary and straightforward. As a corollary of the proof, a recursive scheme for the Baker-Campbell-Hausdorff formula is derived, in which explicit Bernoulli numbers are notably absent.

We first construct the differential refinement of singular homology, verifying that it satisfies the axioms dual to their cohomological counterparts. Then, we define the differential cap product, leading to Poincaré duality. The essential uniqueness of both differential homology and the cap product is proven. Moreover, we construct the relative, non-compact, and local coefficient versions, so that we can state Poincaré and Lefschetz dualities for every smooth manifold with our without boundary. Afterwards, we develop the analogous refinement of any rationally-even homology theory, with the same properties. We also define a "differentiation map" for fibre bundles, dual to the integration map in cohomology. We conclude by sketching some possible applications of this theory in mathematical physics.

We show that a finite number of time-frequency shifts of a Schwartz function can be linearly dependent. This disproves the so-called HRT conjecture of Heil, Ramanathan, and Topiwala. In particular, we provide an example consisting of 12 time-frequency shifts.

In this paper we consider a wide class of weighted means of random variables which converge weakly to the Dickman distribution. This is inspired by a result proved in [3]. Then we prove a large deviation principle for the sequence of these weighted means. In particular we recover a result proved in [8]. Moreover the generalized framework of this paper allows to consider suitable Neyman Type A distributed random variables.

We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. Let $X=G/K$ be a symmetric space of noncompact type and real rank at least $2$. We prove a general vanishing theorem for the normalized torsion in first homology along sequences of torsion-free lattices in $G$. In particular, if $G$ is simple, we get \[ \dfrac{\log |H_1(M_n;\mathbb{Z})_{\operatorname{tors}}|}{\mathrm{vol}(M_n)} \longrightarrow 0 \] for any sequence of distinct manifolds $M_n = \Gamma_n \backslash X$. This answers a question of Abért, Gelander, and Nikolov, and confirms the degree-one vanishing with trivial integral coefficients predicted by a conjecture of Bergeron and Venkatesh in the higher-rank setting. In addition, we get quantitative bounds with respect to the minimal injectivity radius for both the torsion in first homology and the minimal number of generators of $\Gamma$. Finally, we prove the analogous statements for affine buildings.

The iterate convergence of Bregman projected gradient method (BPGM) has remained a long-standing open problem, especially for the widely adopted Shannon entropy kernel. Existing convergence results are often limited, relying on Lipschitz continuity of the kernel's gradient or restrictive conditions on the objective function. In this paper, we develop a novel convergence analysis framework for the BPGM with the Shannon entropy kernel, yielding strong convergence results for a broad class of objective functions under linear constraints. The cornerstone of our framework is a new concept called scaled Kurdyka-Łojasiewicz (SKŁ) property, which captures the local growth behavior of a function under the Bregman geometry. We show that the SKŁ property ensures the iterate convergence of BPGM and holds for all continuous subanalytical functions. Furthermore, we prove that the BPGM sequence exhibits linear convergence if the problem possesses an SKŁ\ exponent of $1/2$. We then furnish the examples of functions with the SKŁ\ exponent $1/2$ by proving that the SKŁ\ exponent $1/2$ is implied by the KŁ exponent $1/2$ under strict complementarity and local Lipschitz continuity of the objective's gradient. Building on these novel results, our work takes a first step towards resolving the open problem of BPGM iterate convergence.

We prove $a_n(S) \le e\,(n+1)\, s_n(S)$ for every s-number sequence $(s_n)$, every bounded linear operator $S$ between normed spaces, and every $n \in \mathbb{N}_0$, where $a_n$ are the approximation numbers, which are the largest s-numbers. This is sharp up to the constant and settles conjectures of Mityagin, Henkin, Carl and Pietsch dating back to 1963. We also extend it to widths of convex sets and discuss optimality there. The proof is elementary.

We classify, up to isomorphism, gradings by finite cyclic groups on classical simple Lie algebras $\mathfrak{g}$ over an algebraically closed field of arbitrary characteristic. Using the smoothness of the automorphism group scheme $\operatorname{\mathbf{Aut}}\mathfrak{g}$ and correspondence between $\mathbb Z_m$-gradings and morphisms $\boldsymbol{\mu}_m\to\operatorname{\mathbf{Aut}}\mathfrak{g}$, we express the classification as an orbit problem for certain Weyl-type groups. More generally, for the affine group scheme $\mathbf G$ associated to a semisimple algebraic group $G$ and the constant group scheme $\mathbf \Gamma_0$ associated to a subgroup $\Gamma_0$ of the automorphism group of the based root datum of $G$, we consider the classification of morphisms $\boldsymbol{\mu}_m\to \mathbf G \rtimes \mathbf\Gamma_0$ up to conjugation by $G \rtimes \Gamma_0$. We show that the classification in characteristic $p$ is the same as in characteristic $0$ except that, in characteristic $p$, only elements of $\Gamma_0$ whose order is prime to $p$ can occur. For $\mathbb{Z}_m$-gradings on $\mathfrak{g}$, this extends the classification by Kac coordinates to arbitrary characteristic, with the caveat that only diagram automorphisms of order prime to $p$ are allowed and, if $p=2$ or $3$, the type of $\operatorname{Aut}\mathfrak{g}$ is not always the same as the type of $\mathfrak{g}$.