Papers

We establish weighted bilinear summation identities for two families of Apéry-like polynomials $g_n(x)$ and $v_n(x)$. The identities express weighted sums in terms of consecutive endpoint values and, when necessary, lower moments. For $g_n(x)^2$ we obtain identities with weights $(2n+1)^r$ for $1\le r\le4$; for $v_n(x)^2$ we treat the cubic and quintic weights. Combining these formulas with congruences for the endpoint values gives supercongruences modulo $p^3$ and $p^4$, together with special evaluations modulo $p^5$ and $p^7$, where $p$ is a prime greater than $3$. In particular, $$\sum_{n=0}^{p-1}(2n+1)^3v_n\!\left(\frac52\right)^2 \equiv 6p^4-\frac{143}{3}p^6\pmod {p^7},$$ confirming a congruence conjectured by Sun. The proofs use explicit quadratic telescoping identities and $p$-adic endpoint expansions.

This text is a self-contained expository survey of the Lagrange and Markov spectra, centered on a comprehensive exposition of Markov's theorem and its generalizations. Its purpose is to provide a systematic text for learning the theory, with detailed proofs and explanations of the connections among its arithmetic, combinatorial, and geometric descriptions. The necessary background in continued fractions, quadratic irrationals, binary quadratic forms, and bi-infinite sequences is developed step by step, followed by an exposition of generalized Markov numbers, fence posets, curve lengths, and generalized Cohn matrices. One goal of this exposition is to explain the formula connecting generalized Markov numbers with the two spectra. For nonnegative integer parameters $(k_1,k_2,k_3)$, a permutation $\sigma\in\mathfrak S_3$, and a fraction label $t\in\mathbb Q_{\geq0}\cup\{\infty\}$, let $m_t$ be the associated generalized Markov number and let $k_t=k_{i_t}$ be the parameter assigned to its position $i_t\in\{1,2,3\}$. The text explains the construction of an associated finite sequence $S(t)$ of positive integers and the identity \[ \mathcal L(\alpha_{S(t)}) =\mathcal M(Q_{S(t)}) =\frac{\sqrt{((3+k_1+k_2+k_3)m_t-k_t)^2-4}}{m_t}, \] where $\alpha_{S(t)}=[\overline{S(t)}]$ and $Q_{S(t)}=(x-\alpha_{S(t)}y)(x-\alpha'_{S(t)}y)$, with the prime denoting quadratic conjugation. Here $\mathcal L$ and $\mathcal M$ denote the Lagrange and Markov constants, respectively. The survey explains how this identity relates generalized discrete Markov spectra to the classical theory and how Markov's theorem is recovered when the parameters vanish. The account also includes boundary values arising from irrational slopes and generalizations of Frobenius's uniqueness conjecture, providing a unified perspective on the classical theorem and its extensions.

We prove the Kazhdan-Lusztig correspondence for the Lie superalgebra $\mathfrak{osp}_{1|2n}$ at irrational levels, that is, we show the category $\mathrm{KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ of finite-length even ordinary modules for the affine vertex operator superalgebra of $\mathfrak{osp}_{1|2n}$ at level $k \in \mathbb{C} \setminus \mathbb{Q}$ is braided tensor equivalent to the category of finite-dimensional even weight modules for the quantum group of $\mathfrak{osp}_{1|2n}$ at parameter $q = e^{\pi i/(2k+2n+1)}$. We also prove that ${\rm KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ is braided tensor equivalent to the category ${\rm KL}_\ell^{\rm ns}(\mathfrak{so}_{2n+1})$ of finite-length ordinary modules with non-spinorial top level for the affine vertex operator algebra of $\mathfrak{so}_{2n+1}$ at level $\ell$ such that $ \frac{1}{\ell+ 2n-1} = \frac{1}{2k+2n+1} + 1 \ \ ({\rm mod}\ 2\mathbb Z).$ Consequently, by gluing vertex operator (super)algebras via tensor categories, we construct a few new families of simple conformal vertex (super)algebras, including the mixed kernel VOAs that were the missing ingredient for proving certain Feigin-Frenkel type dualities in previous work of the first-named author with Linshaw, Nakatsuka, and Sato.

We prove a long time well-posedness result for a quadratic non-linear Schrödinger equation with random well prepared initial conditions in dimension d $\ge$ 1. In some regimes, we derive the Wave Kinetic equation associated with the equation in dimension d $\ge$ 5, almost up to the kinetic time to the power d/(d + 1). Our analysis relies on the ideas introduced by Deng and Hani.

For the sinh-Gordon model (without external mass term), we show that the log-Sobolev inequality holds uniformly in the lattice and volume regularisations for all parameters in the regime in which the Gaussian multiplicative chaos has second moments. This implies that infinite-volume “massless” sinh-Gordon measures on $\mathbb{R}^2$ satisfying all Osterwalder–Schrader axioms exist and that they have exponential decay of correlations (are “massive”). The proof uses the Polchinski equation method for log-Sobolev inequalities. The required estimates on the renormalised potential are established using simple perturbation theoretic bounds, valid for absolutely monotone potentials, and an extension of the correlation inequality of Ding–Song–Sun to potentials in the GHS class. This inequality is proved also by using a variant of the Polchinski equation through the maximum principle.

For every \(n\geq2\), we prove that the complement in \(\C^n\) of a periodic closed set with compact holomorphically convex quotient in \((\C^*)^n\) is Oka. The proof combines periodic divergence-free transport with Fatou–Bieberbach basins. Applications include complements of convex tubes \(\R^n+\ii B\), products of closed annuli, and, more generally, products of planar compact sets whose complements have at most one bounded component. In particular, \(\C^2\setminus\R^2\) is Oka.

In this paper, we complete the characterization of the boundedness of higher order commutators of pointwise multiplication and a large class of singular integral operators, when the corresponding kernels satisfy certain non-degenerate conditions. This was known with the restriction that the pointwise multiplier is real-valued. We finally remove this restriction by proposing a novel complex median method. We also showcase the flexibility of this method by proving the necessity of the rectangular product BMO for the bi-commutators.

We introduce the Max-Min Covering Location Blocker Problem, a bilevel optimization problem in which a leader blocks a minimum-cost set of candidate locations so that the optimal coverage of a budget-constrained follower does not exceed a prescribed target. Each customer's coverage is determined by the least favorable open facility that can serve it. The induced follower coverage set function is nonmonotone and nonsubmodular, so the validity of interdiction cuts cannot be inherited from existing frameworks. We develop an exact nested decomposition that exploits the max-min coverage structure at both levels. At the outer level, we establish the validity of interdiction cuts from this structure and strengthen them with facility-specific coefficients bounding the coverage lost when a critical facility is removed. We solve the NP-hard follower problem by branch-and-Benders-cut after projecting out the customer coverage variables, and characterize an optimal dual solution of the separation problem in closed form, generating Benders optimality cuts in linear time without solving a linear program. Computational experiments on benchmark instances show that both ingredients substantially improve the performance of the method. With a moderate number of candidate locations, instances with 300,000 customers are solved to optimality within one hour. A comparison with two general-purpose bilevel solvers from the literature shows the proposed method to be one to three orders of magnitude faster, and to solve to optimality instances that neither of them closes. The model applies wherever the planner cannot control which facility serves a customer, as in public automated external defibrillator (AED) networks. A case study on a network using real AED data from Virginia Beach illustrates the model.

We study the distributed average consensus problem under persistent link-level disturbances modeled as a martingale difference sequence with uniformly bounded conditional second moments. Under such disturbances, the standard stochastic-approximation-based linear iteration with diminishing stepsizes drives the network to consensus on an unbiased random variable with non-vanishing variance instead of the exact initial average. To understand and resolve this limitation, we develop an anchoring-based mechanism derived from a decentralized gradient descent formulation and study the effect of incorporating a decaying anchoring term that continuously pulls each agent state toward its initial value. This perspective provides an intuitive interpretation of how state anchoring counteracts disturbance accumulation. Under standard summability conditions, we prove that the resulting algorithm achieves exact average consensus almost surely. Furthermore, this decentralized gradient perspective offers a unifying framework for several related methods and an interpretable design principle for exact average consensus under persistent disturbances.

Motivated by a recent article of R. Vanlaere (2026), we investigate the spectral asymptotics of a singular harmonic oscillator arising from the radial reduction of the magnetic Schrödinger operator on the unit disk in the presence of a constant magnetic field and an Aharonov-Bohm flux. The corresponding eigenvalue problem is equivalent to the study of the zeros of the Kummer confluent hypergeometric function with respect to its first parameter. Combining now standard semiclassical methods with the uniform asymptotic theory of Whittaker and confluent hypergeometric functions developed by Dunster (1989), Gabutti-Gatteschi (2001) and improved quite recently by Dunster (2026), we obtain a comprehensive description of the different spectral regimes. When the boundary lies in the classically forbidden region, we derive the exponentially small asymptotic correction to the eigenvalue caused by the tunneling effect. For energy levels above the tunneling regime, we derive a shifted Bohr-Sommerfeld asymptotics for highly excited states at a fixed magnetic field. Finally, for the transition regime where the energy level touches the boundary, we derive a three-term asymptotic expansion of the eigenvalue. In an appendix, we translate these spectral results into explicit asymptotic formulas for the zeros of the Kummer function.

We provide a general upper bound on multicolor ordered Ramsey numbers in terms of the interval chromatic number and the degeneracy of an ordered graph. We extend previous results by Conlon, Fox, Lee, and Sudakov (2017) by showing that for every $n$-vertex ordered graph $G^<$ with degeneracy $d\geq2$, and interval chromatic number $\chi$, its $q$-color ordered Ramsey number satisfies $r_<(G^<;q) \in n^{O(d^{q-1}\lceil \log{\chi}\rceil^{q-1})}$ for every $q \geq 2$. For fixed parameters $q,d,\chi$, the resulting estimate is polynomial in $n$. For triangle-free ordered graphs $G^<$, we also provide the stronger estimate $n^{O(q^2d {\lceil \log \chi \rceil}^{q-1})}$. It also follows from a recent result by Li (2026) that our upper bound is almost tight for ordered matchings.

We give a counterexample showing that an asymptotic mean value property for $p$-harmonic functions in $\mathbb R^d$ does not necessarily hold in the pointwise sense for $d\geq 3$ and $p>2$ near $2$. The proof uses a perturbation argument from $p=2$ and the Implicit Function Theorem to establish the counterexample. We also use a computer assisted proof to give an explicit range of values of $p$ for which the counterexample holds. Finally, we also establish a counterexample for the pointwise asymptotic variational mean value property.

We study the arithmetic of the realizations for hypergeometric motives attached to ${}_3F_2(1)$ hypergeometric series defined over number fields, focusing on the case where the étale realization is reducible and irregular. We then use motivic techniques to prove new evaluation formulas for certain ${}_3F_2(1)$ series, understood as a period of the hypergeometric motive. In addition, we provide examples which are closely related to the exact values of $L$-functions for modular forms in special cases.

In this paper, we study the Abelian maximal pattern complexity $p_{\alpha}^{\ast \mathrm{ab}}(k)$, introduced by Kamae, Widmer and Zamboni, of infinite words $\alpha \in \mathbb{A}^{\mathbb{N}_{0}}$ over finite alphabets $\mathbb{A}$. For recurrent aperiodic words, we determine a lower bound and prove its sharpness. We further give a structural characterization of the words with minimal Abelian maximal pattern complexity. In the general case, we prove that an infinite word $\alpha $ is aperiodic if and only if $\binom{p_{\alpha }^{\ast \mathrm{ab}}(k)}{2}\geq k$ for every $k.$ For aperiodic words over $\ell \geq 2$ letters, each occurring infinitely often, we further prove that $p_{\alpha }^{\ast \mathrm{ab}}(k)\geq m$ whenever $\binom{m}{2}\leq (\ell -1)(k-\ell +2)$, for all $m,k$. Together with a matching construction, this shows that the minimum Abelian maximal pattern complexity in this class is $\sqrt{2(\ell -1)k}+O_{\ell }(1)$. We call a word an Abelian pattern Sturmian word if, at every $k$, its Abelian maximal pattern complexity is the least positive integer $m$ satisfying $\binom{m}{2}\geq k$. We show that a word is Abelian pattern Sturmian if and only if, after relabeling its alphabet, it is the characteristic word of an infinite set $E\subset \mathbb{N}_{0}$ for which the bipartite graph on two disjoint copies of $\mathbb{N}_{0}$, with a left vertex $r$ adjacent to a right vertex $s$ exactly when $r+s\in E$, is a forest.

We study the asymptotic behavior of the optimal stabilization prefactor for the free Schrödinger equation, optimized over all bounded stationary linear feedbacks. For a prescribed decay rate $\delta>0$, let $\widehat{C}_S(\delta)$ denote this optimal prefactor. Assuming that the uncontrolled region contains a nonempty open set and that the observability cost satisfies $C(T)\le C_0e^{C_0/T}$ for small $T>0$, we prove $$ e^{c\sqrt{\delta}}\le \widehat{C}_S(\delta)\le e^{C\sqrt{\delta}}, \qquad \delta\gg1. $$ The lower bound is based on the uncontrolled open hole and a quantitative cutoff construction adapted to the Laplacian, while the upper bound combines an exponentially weighted Gramian with the small-time observability cost $C(T)\lesssim e^{C/T}$. The results apply to controlled Schrödinger equations in three settings: flat tori, including a strip-control example beyond the geometric control condition; the whole space $\mathbb{R}^n$ with exterior-ball control; and bounded smooth domains satisfying the geometric control condition for generalized geodesics.

This paper refines the notion of cyclic Massey products to the bi-graded setting, just as quadruple ABC Massey products refine the notion of quadruple Massey products. The result, we call “cyclic ABC Massey products,” are in general non-trivial and contain information different from the quadruple ABC Massey products.

We study degenerations of orbits in varieties of chain complexes of projective modules. We show that degenerations are characterized in terms of certain admissible short exact sequences in the exact category of complexes of projectives, analogously to work from Riedtmann and Zwara. We then extend our study to the homotopy category. In this context, complexes of projectives with a given $g$-vector form an ind-variety, and we prove that degenerations of orbits are characterized by the existence of certain distinguished triangles in the category. This links to the algebraic definition of degeneration for triangulated categories introduced by Jensen, Su and Zimmermann.

We prove that the Dirichlet-to-Neumann map associated to the Schrödinger operator $-\Delta+q$ on a planar bounded Lipschitz domain uniquely determines an arbitrary complex-valued potential $q\in L^p$, $p>1$, whenever zero is not a Dirichlet eigenvalue. Our argument uses Nachman's d-bar equation only at sufficiently large complex frequencies, where exceptional points are absent. A new annular frequency estimate converts the first asymptotic coefficient of the complex geometric optics solutions into a pointwise bound for a primitive of the potential difference, reducing the inverse problem to a unique-continuation argument.

Let $\Omega\subset\mathbb R^2$ be a bounded smooth domain. We prove that the weak Dirichlet-to-Neumann map for $-\Delta+V$ uniquely determines every complex-valued potential $V\in L^p(\Omega)$, $p>1$, provided that zero is not a Dirichlet eigenvalue. This extends the previously known range $p>4/3$ to all $p>1$. The proof uses Bukhgeim's quadratic-phase solutions and an average of Alessandrini's identity over the phase center. After separating the Neumann-series tails, we show that each remaining Born term tends to zero. The fixed-order estimates combine bounds for the absolute kernels with oscillatory cancellation and a duality argument for the product of the two Cauchy transforms.

We show functional inequalities with logarithmic weights as a critical case of Caffarelli-Kohn-Nirenberg inequalities. Original CKN inequalities have polynomial weights and are shown via scaling argument. In this paper, we develop this argument into it for nonlinear scaling. Our inequalities are generalizations of critical Hardy type inequalities.

Large-scale and high-dimensional discrete optimal transport often involves a tension between scalability and exact assignment feasibility: many scalable approaches optimize relaxed, factorized, or sparse transport plans rather than maintaining a one-to-one assignment throughout optimization. We introduce SWAP, an iterative solver that operates directly in permutation space for equally weighted point clouds. Starting from any feasible assignment, SWAP uses geometry-aware proposals to identify cost-decreasing local rearrangements. Consequently, every iterate is a valid permutation, so the one-to-one constraints are satisfied exactly throughout the optimization without rounding or feasibility correction. SWAP avoids constructing an $N\times N$ cost or coupling matrix and requires only $O(Nd)$ memory. We establish monotone descent, introduce a family of cycle-stability conditions, prove that SWAP reaches pairwise stability almost surely under generic nonparallel conditions, and derive sufficient conditions under which pairwise stability implies global optimality. Experiments on an ImageNet assignment with 640,500 points in 2,048 dimensions, synthetic benchmarks, and MERFISH spatial alignment demonstrate that SWAP achieves lower transport costs on challenging large-scale and high-dimensional problems, with substantially reduced runtime and memory compared with

In this paper, we investigate energy conservation for weak solutions to the two-component Camassa-Holm system. We derive a local energy balance equation in the sense of distributions with an explicit defect term. As an application, we determine that the Onsager exponent is 1, and as a consequence, we also obtain the uniqueness of weak solutions.

The key-value (KV) cache of autoregressive transformers can be viewed as a fourth-order tensor spanning attention heads, tokens, features, and grouped layers. We measure the singular-value spectra of all four mode unfoldings on Mistral-7B-v0.3 and LLaMA-2-13B and compare four standard tensor decompositions: Tucker, CP, tensor train, and t-SVD, at matched storage. The spectra partition the four axes into two classes. The token and feature modes carry low-rank structure, particularly for keys. The head and layer modes are nearly full-rank and resist compression at any practical error level. Among the four decompositions, Tucker achieves the lowest reconstruction error at every compression ratio from $2\times$ to $5\times$, because it can leave the full-rank modes untouched. Comparisons with two-dimensional unfolding baselines show that the preferred representation differs between keys and values: 2D methods achieve lower key error, while four-way Tucker achieves lower value error at matched storage. A mode-pinning theorem certifies the full-rank preservation from the measured spectra alone. Two further spectral properties affect the compressible modes without touching the full-rank ones: values reach a higher error floor than keys at every ratio, and post-RoPE keys lose $41\%$ - $64\%$ of their pre-RoPE compressibility on both models.

For every fixed integers $r \ge 4$ and $2 \le k \le r-2$, we construct $r$-dimensional cubical sheaf complexes whose degree-$k$ CSS codes have positive constant rate, linear distance, and constant soundness, with bounded row and column weights. Taking $r=4$ and $k=2$ gives a family of asymptotically good binary qLTCs. At the core of our construction is a uniform product-expansion theorem for explicit Reed-Solomon codes on norm-one evaluation sets. The key point is that the expansion constant stays bounded away from zero as the local code lengths grow. We place these codes on arithmetic cubical complexes, obtaining constant local expansion for both the resulting sheaf and its dual. Together with the local-to-global framework of Dinur, Lin, and Vidick (FOCS 2024) and sheaf duality, this gives linear distance and constant soundness, while an asymmetric choice of local code dimensions gives positive rate. The resulting codes are explicit and polynomial-time computable.

We study two-dimensional steady gravity water waves with general vorticity in a homogeneous, inviscid fluid of finite depth, subject to a surface pressure that tends to distinct constants upstream and downstream. Starting from a fixed supercritical shear flow, we construct two global branches of monotone fronts connecting distinct laminar states, with increasing and decreasing profiles corresponding to the two signs of the pressure transition. Small-amplitude fronts are first obtained via the implicit function theorem and then extended by analytic global continuation.

We study pairs consisting of a rational isosceles triangle and a rational triangle having a prescribed angle $\theta$, with equal perimeter and equal area. Writing $\rho=\cos(\theta)\in\mathbb{Q}$, we obtain a two-parameter description of the solutions and, for fixed $\rho$, a family of curves $C_\rho$ which are generically of genus $2$. We determine the two singular specializations: there is no genuine pair for $\rho=\frac{1}{2}$, while $\rho=\frac{47}{49}$ admits infinitely many genuine pairs. We completely parametrize two natural one-parameter families in which the $\theta$-triangle is isosceles, and prove several density results for values of $\rho$ admitting one or more genuine pairs. For thirty-five explicit values of $\rho$, Chabauty–Coleman computations determine $C_\rho(\mathbb{Q})$ completely and show that the corresponding pair is unique up to homothety. We also study multiplicity when one triangle is fixed; the condition for a scalene $\theta$-triangle to have two isosceles partners leads to an elliptic $K3$ surface and yields a dense set of such values of $\rho$. Finally, we construct infinite families with at least four genuine pairs and exhibit examples carrying at least five, and in two cases at least six, genuine pairs.

We classify all $D(u^2)$-triples of triangular numbers whose smallest element is $T_{4u-1}$ or $T_{4u}$. We study the regularity of $D(N)$-triples of triangular numbers. We also show that infinitely many positive integers $u$ admit an all-triangular $D(u^2)$-quadruple. For every positive integer $k$, triangular number $T_{16k^2-1}$ belongs to infinitely many $D(u^2)$-quadruples of triangular numbers with distinct positive integer parameters $u$.

This paper has two aims. First, we study nowhere-zero flows in countably infinite graphs, observing the different roles played by two types of bridge. Second, we ask how typical graphs with such bridges are. Since there is no canonical probability measure on countable graphs, we explore an analytic approach and study which graph classes are nowhere dense and which are meager for two different definitions of a distance. We characterize in this sense graphs with a bridge (separating a finite component, or two infinite ones), $D$-edge-colorable graphs, etc.

Data assimilation estimates a dynamical state from partial and noisy observations. Classical ensemble filters are efficient but restrict analysis updates through finite sample covariance and affine Gaussian distribution. We introduce the Flow Ensemble Filter (FlowEF), which uses conditional flow matching to transport the forecast ensemble from a classical baseline filter to an analysis ensemble. FlowEF uses a localized Gaussian source during training, transports forecast ensemble members from a baseline filter at deployment, and conditions its velocity field on ensembles from that baseline filter and the observation. The proposed model therefore learns a nonlinear update while mapping each baseline ensemble independently. For sparsely observed dynamical systems, FlowEF improves both deterministic and probabilistic metrics over all four classical ensemble filters. It also achieves the best performance among the state-of-the-art generative data assimilation models.

We define a Bergman barycentric extension of boundary homeomorphisms of the complex unit ball by \[ E_B(f)(z)=\barB(f_*\sigma_z), \] where \(\barB\) is the Busemann barycenter for the Bergman metric and \(\sigma_z\) is the visual, equivalently Poisson–Szegő, measure based at \(z\). We prove well-definedness, prescribed boundary values, full \(\Aut(\B^n)\)-naturality, and interior real-analyticity. For every CR-quasisymmetric boundary homeomorphism, the Bergman barycentric extension is a quasi-isometry of complex hyperbolic space; moreover \(E_B(f^{-1})\) is a coarse inverse of \(E_B(f)\). Under sufficiently small positive CR cross-ratio distortion we obtain a sharper Tukia-type theorem: for every \(M>1\), the extension is a real-analytic diffeomorphism satisfying \[ M^{-1}d_B(x,y)\le d_B(E_B(f)(x),E_B(f)(y))\le M d_B(x,y). \] In contrast, for \(n\ge2\) there are smooth CR-orientation-preserving CR-quasisymmetric boundary diffeomorphisms whose barycentric extensions are non-injective and, after normalization, have singular differential. Thus large-scale quasi-isometric control persists on the full CR-quasisymmetric class, whereas local non-degeneracy requires stronger boundary control.