Papers

We study the rate-distortion limits of online KV cache compression in autoregressive language models, formulating it as sequential Wyner-Ziv source coding on the filtration induced by the model, with the next-step query as decoder side information. Empirically, across four models spanning two families and $0.5$-$3$B parameters, we find that the next-token distribution's sensitivity to context truncation decays polynomially rather than geometrically: a power law improves on an exponential fit by an order of magnitude in extrapolation, the fitted exponent is recovered independently from a sink-plus-recent KL measurement, and the decay is verified to be free of positional-encoding artifacts by a position-preserving ablation. Under a corresponding polynomial truncation-sensitivity assumption, our main result characterizes the per-token memory requirement of suffix-only cache policies: a sliding-window scheme attains distortion $\varepsilon$ with window $w = O(\varepsilon^{-1/\alpha})$, and – under an additional two-sided Bayes-risk condition – a converse shows $w = \Omega(\varepsilon^{-1/\alpha})$ is necessary within this policy class, so the scaling is $\Theta(\varepsilon^{-1/\alpha})$ for suffix-only policies. Whether recurrent or propagating cache summaries can beat this scaling is left open. An explicit block-Markov scheme achieves the upper bound; its rate-of-convergence exponent matches the converse under additional forward-decay and regularity hypotheses (not implied by truncation sensitivity alone), and differs by a factor of two otherwise. Empirically, the polynomial law predicts the degradation curves of concrete cache policies: recency-based eviction (sliding, sink-plus-recent) suppresses distortion by roughly two orders of magnitude over random retention at equal budget, with a power-law decay in the budget.

This paper considers a safe trajectory tracking of the Stefan problem with a second-order moving boundary dynamics. The model is given by a parabolic Partial Differential Equation (PDE) defined on a time-varying domain of moving boundary governed by a second-order Ordinary Differential Equation (ODE) associated with the Neumann boundary condition. A feedforward control is designed by a series expansion approach to solve the inverse Stefan problem under given reference trajectory of the moving boundary, and the convergence of infinite series is proven. A trajectory tracking controller is derived based on an energy-shaping, which ensures the safety of the model constraint in the closed-loop system. The closed-loop system is also shown to be globally exponentially stable with respect to the tracking error by performing PDE backstepping transformation and Lyapunov analysis. Numerical simulation illustrates an effective tracking performance of the proposed method under a sinusoidal reference trajectory. Code is released at https://github.com/shumon0423/StefanTracking_ACC2026.git.

We study hypercontractivity for underdamped Langevin dynamics with a convex confining potential whose spatial Gibbs marginal satisfies a logarithmic Sobolev inequality (LSI). Unlike in the overdamped case, the noise acts only on the velocity variable, so the usual LSI-based argument does not apply. Nevertheless, under an additional tame Hessian-growth assumption on the potential, we prove that, when the spatial LSI constant is $\rho$ and the friction parameter is of order $\sqrt{\rho}$, the semigroup satisfies a Gross-type $L^p$-to-$L^q$ estimate in which the integrability exponent grows exponentially on the kinetic time scale $t \sim \rho^{-1/2}$. The central ingredient, of independent interest, is a space-time LSI, valid for controlled kinetic paths of finite action, which quantifies how dissipation in the velocity variable is transferred to the position variable. We derive it from a controlled version of the hypocoercive entropy-decay estimate, and we then restore the Gross hypercontractivity mechanism through a duality argument built on a non-reversible forward/backward interpolation of the underdamped Langevin semigroup. As a corollary, Rényi divergences of every order decay exponentially at the sharp hypocoercive rate $\mathcal{O}(\sqrt{\rho})$.

We construct Anosov flows in certain circle bundles over closed hyperbolic 3-manifolds, producing counterexamples to a conjecture of Verjovsky. Some of these 4-manifolds admit infinitely many distinct Anosov flows up to orbit equivalence. The construction is made by using Cannon-Thurston maps associated to pseudo-Anosov quasigeodesic flows in hyperbolic $3$-manifolds.

We give an explicit construction of two $2$-generated subgroups $H,K\leq \SL(3,\Z)$ whose intersection is not finitely generated. The construction takes place inside the standard parabolic subgroup $\Z^2\rtimes \SL(2,\Z)\leq \SL(3,\Z)$. The main point is to identify $H\cap K$ with the stabilizer of a point for an affine action of a free group on $\Z^2$, and then to prove, using the Schreier graph of this action, that this stabilizer is not finitely generated. Furthermore, we prove that there exists a sequence of subgroups $H_q, K_q \leq \SL(3,\mathbb{Z})$ such that $\rank(H_q)=\rank(K_q)=4$, and \[ \rank(H_q\cap K_q)\geq q+1, \] while $H_q\cap K_q$ is finitely generated.

Recall that an element $x\in R$ is complemented if there is a $y\in R$ such that $xy = 0$ and $x + y \in {\rm reg}(R)$. In a recent article [1], the authors investigated those rings for which every non-nilpotent element is complemented, calling such rings semi-complemented. As the title of the current work suggests we characterize when a commutative group $RG$ is semi-complemented

We show that perverse character varieties are (quasi-)affine. We do this in a purely stack-theoretic fashion, by exhibiting enough sections of the structure sheaf.

In this paper, we study the structure of the complete asymptotic expansion of the probability that a large combinatorial object is connected or consists of a given number of connected components. For rapidly growing labeled families of structures, the coefficients involved in these expansions are possibly negative integers. Using species theory, we interpret these coefficients as the difference between the counting sequences of two derivative species of structures. In particular, we show that this difference can be viewed as the counting sequence of the virtual species obtained with the help of an "anti-$\mathrm{SEQ}$" operator applied to the initial family of structures. Applications include $P$-angulated discrete surfaces, quadratic square-tiled surfaces, and non-orientable graph encoded manifolds, which were not reachable with our previous methods. Moving on to the weighted species, we establish the whole structure of the asymptotic expansion of the probability that a graph is connected in the Erdős-Rényi model $G(n,p)$. Here, the asymptotic coefficients are polynomials in $\frac{p}{1-p}$ and can be described both in terms of simple graphs and irreducible tournaments with ties. We also provide general asymptotic results for sequence and cycle decomposition, as well as the complete asymptotic expansion of the probability that a random labeled tournament with ties is irreducible.

Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity for learning solutions to non-linear partial differential equations (PDEs). Despite their widespread use in practical applications, a rigorous theoretical understanding of the approximation properties of RaNNs in this context remains limited. Here, we derive error bounds for RaNN approximations to time-dependent Sobolev functions and obtain a dimension-free approximation rate $\frac{1}{2}$ for sufficiently regular functions. We apply our results to two important classes of non-linear PDEs: Porous Medium Equations and Compressible Navier-Stokes Equations, showing that RaNNs are capable of efficiently approximating solutions to these complex, non-linear PDEs. Our theoretical analysis is supported by numerical experiments, showing that the obtained convergence rates extend beyond the considered setting.

We propose a semi-analytical design framework for amplitude phase shift keying (APSK) signaling for integrated sensing and communication (ISAC), focusing on i.i.d. uniform discrete input distributions for practicality and analytical tractability. First, we establish APSK design criteria in which communication performance is measured by the gap to capacity and linked to the minimum Euclidean distance, while sensing performance is characterized by the symbol-energy variance. Based on these criteria, we develop a structured APSK construction whose key parameters follow explicit scaling laws. Then we prove that this construction achieves a constant gap to capacity independent of the signal-to-noise ratio. Building upon this foundation, we further construct a parametric APSK family that bridges the communication-optimal and sensing-optimal designs, with the communication and sensing (C&S) tradeoff controlled by the number of rings and energy allocation among rings. Simulation results show that the resulting APSK family achieves C&S performance very close to the Pareto boundary achieved with time-independent, circularly symmetric, and otherwise unconstrained continuous input distributions.

Given $s\in(1,2]$, define $$H_s[0,1]=\{f\in C[0,1]:{\dim}_HG_f([0,1])=s\}$$ and $$\overline{B}_s[0,1]=\{f\in C[0,1]:\overline{{\dim}}_BG_f([0,1])=s\}.$$ The main goal of this paper is to study the $(\alpha,\beta)$-lineability/spaceability of the sets $H_s[0,1]$ and $\overline{B}_s[0,1]$. As a principal result, we prove that $H_s[0,1]$ is $(p,\mathfrak{c})$-spaceable for $p=1,2$ and also $(n,n+m)$-lineable for any $m,n\in\mathbb{N}$. This partially answers a question raised by Liu et al. concerning the Hausdorff dimension of graphs of continuous functions. Furthermore, for a cardinal number $\alpha$, we prove that $\overline{B}_s[0,1]$ is $(\alpha,\mathfrak{c})$-spaceable if and only if $\alpha<\aleph_0$. This completely resolves an open question raised by Liu et al. concerning the upper box dimension of graphs of continuous functions.

The traditional conjecture that RR-flux is quantized in stable K-cohomology fails to account for the presence of NS-brane sources: These impose nonlinear relations – reductions of the famous quadratic relation on M-brane flux – that can only be captured by unstable nonabelian cohomology theories. Here we consider a deformation of unstable K-theory which properly quantizes the fluxes coupling to D0/D2/NS5-branes, find a twisted version that quantizes also the fluxes coupling to NS1/D4-branes, and show that this oxidizes to a proper electromagnetic quantization of M-brane fluxes.

We develop a novel randomized conjugate gradient least squares (RCGLS) method for solving least-squares problems, in which iterative sketching is employed at each step to reduce the dimension and hence the computational cost. In particular, we propose a new perspective on the classical CGLS method, where the next descent direction is determined via a constraint correction problem associated with the gradient. Based on this insight, we replace the gradient with a randomized coordinate gradient that naturally satisfies the variance reduction property, leading directly to the proposed RCGLS method. We prove that RCGLS converges linearly in expectation, with a better convergence bound compared to the randomized coordinate descent method. Furthermore, we investigate an implementation of the method that avoids full-dimensional vector operations, which are the major bottleneck of vanilla RCGLS for sparse matrices and render it impractical. We also show how to apply the RCGLS method to solve the ridge regression problem, yielding a lightweight, parallelizable, and accelerated method for such problems. Numerical experiments are provided to confirm our results.

In this paper, we will discuss several radii problems related to Wright Function involving four parameters.

Two-stage stochastic linear optimization is known to be #P-hard when all involved random variables are independently and uniformly distributed over intervals, even with fixed recourse. We show that this problem is in fact #P-hard in the strong sense. More surprisingly, #P-hardness persists even when the random vector is one-dimensional, i.e., uniformly distributed over a single interval. To obtain this result, we show that computing the area of a two-dimensional polytope given by a compact extended formulation is #P-hard. Furthermore, for a problem in standard form, we prove strong #P-hardness in case the number of second-stage constraints is fixed, while fixing the number of second-stage variables leads to a weakly #P-hard problem. Finally, if both the dimension of the random vector and the number of second-stage constraints are fixed, the problem turns out to be tractable.

We investigate slicing properties of $m$-subharmonic functions in product domains $\Omega = \Omega' \times \Omega'' \subset \mathbb{C}^n = \mathbb{C}^p \times \mathbb{C}^{n-p}$, where $p, m, n$ are integers satisfying $1 \leq p \leq m-1 < n-1$.\\ Given an $m$-subharmonic function $v$ on $\Omega$, we prove the existence of a pluripolar subset $E \subset \Omega'$ such that, for every $x' \in \Omega' \smallsetminus E$, the slice $v_{|\{x'\}\times \mathbb{C}^{n-p}}$ is well defined and $(m - q_{m,p})$-subharmonic on $\Omega''$, where $q_{m,p}$ denotes the smallest integer greater than or equal to $\frac{mp}{n}$.\\ Moreover, we show that, outside a negligible subset of $\Omega'$, the $m$-Lelong number of $v$ at $(x', x'')$ coincides, up to a multiplicative constant, with the $(m - q_{m,p})$-Lelong number of the slice $v_{|\{x'\}\times \Omega''}$ at $x''$.

Mirković–Vilonen (MV) polytopes play a key role in the representation theory of reductive algebraic groups, while the geometric behavior of prime MV polytopes under Minkowski addition remains a subtle open problem. This paper focuses on type A and regards Schubert matroid polytopes as fundamental prime MV building blocks. Using the crystal structure on MV polytopes, we strengthen Sanchez's compatibility condition and establish a necessary and sufficient condition: the positive Minkowski sum of such polytopes is again an MV polytope precisely when the indexing family is weakly separated. Working within discrete convex analysis, we relate discrete concave tropical Plücker functions to concave extensions on the hypercube and the resulting generalized matroid subdivisions, showing that weak separation is equivalent to the stability of these subdivisions under common refinement. We further clarify the intrinsic connection between our subdivision constructions and the hypersimplex matroid subdivisions developed by Early, providing a natural flag-type generalization of his classical results. We briefly discuss generalized positroids and generalized polypositroids, and identify the MV fan $\mathcal{MV}$ as the secondary fan of hypercube generalized positroid subdivisions. Accordingly, maximal weakly separated sets correspond to maximal cones in $\mathcal{MV}$ and produce the finest such subdivisions. This work unifies MV polytope theory with tropical matroid geometry, advances the understanding of compatibility phenomena in MV combinatorics, and offers new perspectives at the interface of representation theory and combinatorics.

This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as standard normal variates. Additionally, the paper provides a clear exposition of the developed methodology. To establish our results, we develop a novel approach utilizing the classical Cauchy's bounds for the zeros of a deterministic polynomial with complex coefficients. We also corroborate our analytical results with extensive simulations. The methodology developed in the paper can potentially be applied to a broad class of problems regarding bounds and the distribution of zeros in the theory of random polynomials.

For every dimension $n\ge3$, we construct a family $\Sigma_n(a,b)$ of smooth complete fans with Picard number five and fixed combinatorial type. For integers $a,b\ge1$, the minimum number of new rays in a smooth projective refinement is $\min(a,b)$, attained by ordinary invariant codimension-two blow-ups. If projective simplicial refinements with maximal-cone multiplicity at most an integer $\mu\ge1$ are allowed, the minimum is $\lceil\min(a,b)/\mu\rceil$. The threefold lower bound depends only on subdivisions of four two-dimensional cones. Toric wedges preserve this minimum while raising the dimension, and a selective support-function criterion gives the matching upper bound. We also construct a projectivization algorithm natural under lattice isomorphisms. For smooth input, it principalizes an intrinsic monomial ideal by ordinary blow-ups with smooth, possibly disconnected centers. Running a fixed functorial principalization algorithm over $\mathbb{Q}$ gives a sequence of fans valid over every field. An intrinsic resolution procedure treats arbitrary complete rational fans. Finally, Oda's threefold shows why symmetry can prevent a canonical ordering of the components of a center, even when their simultaneous blow-up is canonical.

In this paper, we study impartial achievement games and impartial avoidance games introduced by Anderson and Harary. Using the criteria of maximal subgroups, we study the game for Frobenius groups and non-abelian groups with all abelian subgroups.

Domain decomposition methods (DDMs) provide a unifying framework for the scalable numerical solution of partial differential equations. Originating from Schwarz's alternating method, they have evolved into a rich family of algorithms that combine local robustness with global convergence acceleration and natural parallelism. Over the past decades, domain decomposition has played a central role in enabling large-scale simulations in numerous applications. This chapter presents an overview of modern DDMs, with a particular emphasis on scalable preconditioning techniques for challenging problems, including indefinite and high-frequency regimes. We revisit the fundamental concepts - overlapping decompositions, partition of unity, additive and restricted Schwarz formulations - and explain their algebraic interpretations. We then clarify their role as preconditioners in Krylov subspace solvers and discuss the necessity of coarse space corrections for scalability. Beyond a the survey aspect, the chapter distills key theoretical insights and practical design principles that have emerged over the past twenty years. Special attention is given to robust coarse spaces (GenEO, DtN-based approaches) and high-performance implementations. The goal is to provide both a coherent overview of the field and a concise, practice-oriented guide for readers seeking to understand and apply domain decomposition methods without navigating the entire literature.

We formulate a relationship between finite-order rondle invariants with respect to triple-point modifications and the lower central series of subgroups of a pure twin group. Using our formulation, we construct infinitely many infinite sequences of prime plane curves such that, for each sequence, the curves share the same invariants up to any fixed order.

We prove a Fredholm determinantal identity for the tilted Toeplitz minor $$ D_{N}^{\xi,\theta}(\varphi):= \det\bigl[(\theta_{i}\xi_{j}\varphi)_{i-j}\bigr]_{i,j=1}^{N}, $$ generalizing the Borodin-Okounkov-Geronimo-Case (BOGC) identity to oblique splittings of the Hardy space. The tilts $\xi_{j},\theta_{i}$ enter only through an oblique projection that multiplies the trace-class kernel $K$ inside the Fredholm determinant; the BOGC operator $A=I-K$ constructed from $\varphi$ is unchanged. Baik-Liao-Liu (arXiv:2603.01964) and Liu-Tripathi (arXiv:2604.24747) have recently shown that the same tilted Toeplitz minor admits a contour Fredholm-determinantal representation, in connection with the periodic Totally Asymmetric Simple Exclusion Process (TASEP). In the periodic TASEP application of Baik-Liao-Liu, the formula plays an important role in identifying the periodic KPZ fixed point with general initial data. Our formula is a companion to their Fredholm determinant and readily reduces to the original BOGC identity. The one-sided tilted Toeplitz minor (that is, when all $\theta_i=1$) admits a bialternant form recovering Schur and Grothendieck polynomials as special cases. A Cauchy-Binet expansion realizes $D_{N}^{\xi,\theta}$ as a restricted sum over partitions of products of Jacobi-Trudi type determinants, generalizing Gessel's theorem. In the pure-shift setting this specializes to a skew Schur expansion. Finally, for finite Laurent exponential symbols, we record explicit resolvent-block flow identities and formulate the associated finite-dimensional closure problem. We also illustrate a possible asymptotic application leading to finite-rank perturbations of the Airy kernel.

Integrated sensing, communication, and computation (ISCC) is emerging as a unified design paradigm for future vehicular networks that require joint environment perception, safety-critical information exchange, and latency-sensitive task processing. In New Radio Vehicle-to-Everything (NR-V2X) Mode 2, autonomous resource selection is performed through sensing-based semi-persistent scheduling (SB-SPS), which is effective for distributed communication resource reservation but does not explicitly consider sensing-resource demand, task-induced computation workload, and the additional latency introduced by mobile edge computing (MEC) offloading. This paper develops multi-agent proximal policy optimization-based SB-SPS (MAPPO-SPS), an ISCC-aware cross-layer scheduler that jointly adapts SB-SPS reservation, radio-resource partitioning, and overflow-driven computation-offloading decisions at control epochs. The scheduling problem is formulated as a cooperative partially observable Markov game and solved using MAPPO with centralized training and decentralized execution (CTDE). Simulation results show that MAPPO-SPS achieves a balanced tradeoff among CRLB-based sensing accuracy, packet reception ratio (PRR), effective throughput, energy consumption, and end-to-end delay.

We study finite set-theoretic intersections of tropical curves with tropical hypersurfaces. An elementary divisor argument bounds their cardinality by the weighted sum of the Newton support function over the unbounded directions of the curve, without a transversality hypothesis. For tropical lines the bound is $h_\Delta(1,\dots,1)+\sum_i h_\Delta(-e_i)=n!\,\MV(\Delta,\Sigma,\dots,\Sigma)$. Explicit examples show that neither the $\ell^1$-diameter nor the support cardinality minus one gives a corresponding bound for tropical lines. We also record width refinements of the classical-line estimate of Lang, Shapiro and Shustin. A self-contained account of the standard transverse multiplicity formulas explains the role of saturated normal lattices and mixed-cell volumes. Finally, applying the tropical Bernstein theorem to locally independent subsystems gives a mixed-volume bound for the regular transverse locus of a redundant system. An example shows why a fixed subsystem need not compute the count on the whole redundant intersection.

We investigate the real algebraic complexity of contours of amoebas associated with algebraic hypersurfaces and complete intersections in complex algebraic tori. Motivated by the foundational estimates of Lang–Shapiro–Shustin \cite{LSS}, we develop a toric and logarithmic approach relating contour geometry to logarithmic Gauss maps, mixed volumes, and Newton polytope geometry. We prove that the logarithmic criticality equations admit substantially sharper asymptotic behavior than the classical Pfaffian bounds, reducing the expected growth from order $d^{2n}$ to order $d^n$. We further formulate a conjectural asymptotic theory for complete intersections based on logarithmic Grassmannian geometry and Schubert degeneracy loci. The work establishes new connections between amoeba theory, real algebraic geometry, tropical geometry, and logarithmic toric geometry.

Random sampling is a fundamental tool in modern machine learning and numerical linear algebra for reducing the computational cost of large-scale matrix problems. Existing analyses, however, rely primarily on subspace embedding guarantees, which do not precisely characterize the statistical bias of nonlinear random oblique projections induced by sampling, which arises ubiquitously in subsampled least squares and fast low-rank approximation methods. Because (pseudo)inversion is nonlinear, these random oblique projections can be systematically biased even when the underlying sketch is unbiased, thereby introducing hidden bias into downstream least squares and low-rank approximation solutions. In this work, we develop a unified non-asymptotic theory for random oblique projections in high dimensions. We show that standard random sampling schemes generally induce a systematic statistical bias overlooked by classical subspace embedding-style analyses, and we propose a principled debiasing framework to correct it. We illustrate the power of the theory through two canonical applications. For subsampled least squares, we obtain sharp bias–variance characterizations, reveal previously unrecognized statistical suboptimality in widely used sampling schemes, and identify when debiasing yields provable improvements. For fast CUR decomposition, we develop a debiased approach with improved approximation accuracy. Numerical experiments further validate our theoretical findings.

We study nonsmooth composite optimization over embedded submanifolds defined by nonlinear equality constraints. We propose a feasibility-safeguarded inexact proximal linearized method whose iterations require neither projection nor retraction onto the manifold. A computable smooth-dual residual certifies the approximate tangential step and controls the errors in both tangency and composite linearization consistency. The step is followed by a normal correction evaluated at the displaced trial point. To coordinate the resulting optimality and feasibility errors, a nonmonotone backtracking procedure reduces the tangential stepsize faster than the normal stepsize and accepts only corrected points within a prescribed constraint-violation region. Under the stated regularity and level-boundedness assumptions, we establish a uniform backtracking bound, positive lower bounds on the accepted stepsizes, boundedness of the primal and multiplier sequences, stationarity of all cluster points, and an $O(\varepsilon^{-2})$ outer iteration complexity bound. The analysis does not assume compactness of the manifold or a separate global distance-to-manifold error bound. For a specified parameter regime guaranteeing full tangential steps, a KL assumption on an auxiliary function yields full-sequence convergence. Numerical experiments on sparse principal component analysis and sparse spectral clustering illustrate the performance of the method.

This manuscript studies a special case of the Hurwitz enumeration problem: for branched covers from genus g compact Riemann surface to the Riemann sphere, with three branch points, and require the branching data at one of the branch points to be of the partition (m 1^{n-m}), obtain a formula of Hurwitz number. The Hurwitz enumeration problem can be transformed into enumeration of a class of unrooted hypermaps. We first provide a enumeration formula for rooted hypermaps, thereby obtaining the weighted Hurwitz numbers. Next give the quantitative relationship between the enumeration of unrooted hypermaps and that of rooted hypermaps in the general setting. Finally, combining these two results, we obtain a formula for the unrooted hypermaps or the Hurwitz numbers in the special setting.

Brief proofs of classical results of Lie on finite dimensional subalgebras of vector fields in two and three variables are outlined. The results for algebras of maximal rank for vector fields in $\mathbb{C}^N$ – $N$ arbitrary – are also given.