Papers

In this paper, we provide a sufficient condition under which the one-dimensional local zeta function admits a holomorphic continuation to the complex plane cut along the negative real axis.

This paper provides a comprehensive treatment of inverse boundary value problems for (nonlinear) Kirchhoff plate equations under diverse general settings. We begin by establishing the global well-posedness of the nonlinear forward equations, which not only underpins the subsequent inverse analysis but also holds independent theoretical significance. The inverse problems are then examined for both passive and active measurement regimes. With a single passive boundary measurement, we establish the stable recovery of the unknown initial data. In the active regime with infinitely many boundary measurements, our results are twofold. For linear equations featuring generic time-dependent potentials-allowing for spatial unboundedness, we demonstrate the simultaneous recovery of both initial data and coefficients. For nonlinear equations, where both the nonlinearity and initial data are unknown, we develop a novel Runge approximation approach, together with carefully constructed geometric optics solutions and higher-order linearization around nonzero solutions, to prove their simultaneous determination. Furthermore, we introduce a delicate cut-off technique that provides an alternative means of addressing the scenario of vanishing initial data. Notably, the methodologies and results developed herein are readily generalizable to other boundary conditions and plate models, including the classical Euler-Bernoulli equation.

We analyse topology of random simplicial complexes in the medial regime. We show that these complexes are highly connected and have homotopy type of iterated suspensions. One of our main tools is a new combinatorial criterion for high connectivity of simplicial complexes, which is more flexible than conicity. We show that topological complexity of random simplical complexes in the medial regime is bounded above by 2 and it equals 2 for a class of homogenous medial regime random simplicial complexes, a.a.s.

This study develops an analytical and decision framework for a pooled single-server diagnostic–treatment service receiving both first-time and referred patients. A first-time patient receives diagnosis followed immediately by one of $n$ treatment modes, whereas a referred patient enters directly into the prescribed treatment mode. For the two-treatment case, closed-form probability generating functions characterize the stationary queue-length distribution; for arbitrary $n$, a matrix-geometric representation provides a computational solution. The queueing analysis is then directed to capacity control decisions. The effective load is decomposed into mode-specific workload contributions, yielding a minimum service-rate increment for a prescribed utilization target to show that utilization-equivalent interventions need not be delay-equivalent. This motivates a distribution-aware control based on the stationary probability of severe congestion. Finally, the total cost function shown to be strictly convex, providing a global minimum for restricted one-mode interventions. Analytical results are cross-validated against the Pollaczek–Khintchine relations and discrete-event simulation.

We study per-colour independent (k)-rainbow domination and its connection with independent domination in generalized prisms. Building on the known prism identity and the trivial regime above the maximum degree, we focus on the boundary case where the number of colours equals the maximum degree. For every fixed (k\ge 3), we prove that the decision problem remains NP-complete even on a highly restricted class of graphs: (C_4)-free, bipartite, ((k,2))-biregular subdivision graphs arising from simple (k)-regular graphs. The reduction gives an exact correspondence between optimal rainbow-independent dominating functions on the subdivision graph and proper (k)-edge-colourings of the original graph. We also introduce an excess parameter measuring how far the domination number lies above its natural lower bound. For cubic graphs, this excess coincides with the classical edge-colouring degree and therefore with standard resistance parameters for subcubic graphs. These results reveal a sharp one-unit threshold: above the maximum degree the problem becomes trivial for every graph, while at the boundary NP-hard instances already occur within a very narrow structural family.

We give infinite and finite presentations of the cactus Thompson group. This group was introduced as $V_{\mathrm{mock}}$ by Witzel and Zaremsky, who combined Thompson's group with cactus groups.

We first classify the potentially semistable Galois representations of a given tame inertial type using some variants of Breuil-Kisin modules. Then we construct some “model" of such Breuil-Kisin modules. From here, we build a stack of our version of Breuil-Kisin modules, and show that it is isomorphic to the Emerton-Gee stack of potentially semistable representations. This also makes it possible to construct a map from the rigid generic fiber of the potentially semistable EG stack to the analytification of the stack of Weil-Deligne representations, which “interpolates" Fontaine's recipe of associating Weil-Deligne representations to potentially semistable Galois representations. We prove the smoothness of this map, and obtain the generic formal smoothness property of the potentially semistable EG stack. This can be regarded as the global version of Kisin's regularity results on the corresponding Galois deformation rings.

Drifting models are a recent class of one-step generative models that evolve the model distribution during training using a predefined sample-based drift field. Although they avoid iterative inference, their kernel-based drift fields induce frequency-dependent training dynamics: In the linearized regime, each Fourier mode of the density residual decays at a rate determined by the kernel spectrum, leading to slow recovery of fine-scale structure. We propose Second-Order Drifting Models, which lift drifting dynamics into phase space by augmenting generated samples with artificial velocity variables. We show that the resulting density perturbations obey accelerated second-order dynamics in Fourier space, connecting drifting models to the celebrated Nesterov acceleration from optimization theory. This provides a principled mechanism for mitigating the spectral stiffness of first-order drifting while preserving one-step inference. We derive a practical semi-implicit training algorithm and evaluate it on synthetic distribution matching, sequential data generation, and robotic control. Across these settings, the second-order drifting model improves convergence behavior and achieves competitive or superior performance over first-order drifting baselines.

Let $A$ be the smallest set of positive integers containing $2$ and $3$ such that $ab-1\in A$ whenever $a,b\in A$ are distinct. We prove that $A$ has positive lower density, answering a problem of Erdős attributed to Hofstadter.

We present several representations of the Schwarzschild spacetime. Some of these are classical, such as the embedding, Tortoise, and Kruskal representations. The latter two motivate the introduction of pulsating coordinates (or the Diamond representation), a compact approach recently proposed for spherically symmetric static metrics. We then introduce the Golden representation for the Schwarzschild spacetime, where what we call the Fibonacci metric, and the golden ratio, become ubiquitous. We conclude by offering some visual and physical interpretations of this approach.

This article studies the computation of the maximal controlled invariant set (MCIS) for neural network control systems (NNCSs) with a nominal plant model and a neural network residual model. An interval-based method is developed to construct a control-affine inclusion function for the NNCS dynamics and to exploit the induced hyperplane arrangement in the input space, which reduces the verification of controlled invariance from an infinite control search to a finite set of representative evaluations. Based on this finite characterization, verification-guided algorithms iteratively construct inner and outer approximations of the the MCIS, which are naturally parallelizable. Numerical examples demonstrate the effectiveness of the proposed approach.

For $d\ge 1$, let $L(d)\in\mathbb N\cup\{\infty\}$ be the supremum of the lengths of paths in $\mathbb Z^d$ whose steps are standard basis vectors and whose vertex sets contain no collinear triple. We prove that \[ \log_2\log_2 L(d)\ge \frac{2}{5}d-O(1) \] for all sufficiently large $d$.

The subtree polynomial of a tree is the generating function that enumerates its subtrees according to their orders. Brown and Mol conjectured that every subtree root of a tree of order $n\ge 2$ lies in the disk \[ \left\{z\in\mathbb C: |z|\le 1+\sqrt[n-1]{n-1} \right\}. \] We prove this conjecture by introducing a recursive comparison method based on an extremal problem over integer compositions. We further characterize the equality case: the upper bound is attained if and only if $n$ is even and the tree is the star; in this case the unique boundary root is $-1-\sqrt[n-1]{n-1}$. We also show that every nonzero subtree root $z$ satisfies \[ |z|>\sqrt[n-1]{n-1}-1. \] The lower bound is asymptotically sharp as $n\to\infty$. For odd $n$, although the upper bound is not attained, it is asymptotically sharp as $n\to\infty$.

This paper proposes semantic-aided iterative decoding (Sem-IR) for uplink non-orthogonal transmission of a shared natural-language source. K users each hold one segment of a common sentence and superimpose low-density parity-check (LDPC) coded transmissions over an additive white Gaussian noise (AWGN) channel. At the base station, an iterative elementary signal estimator (ESE) and K parallel LDPC decoders progressively cancel inter-user interference. As high-power users pass both parity and language-plausibility checks earlier, their decoded bytes form a reliable linguistic prefix for the remaining users; a fine-tuned ByT5 byte-level language model exploits this prefix to predict byte posteriors for the unconverged user. The byte posteriors are marginalized to bit-level log-likelihood ratios and convex-combined with the LDPC posteriors inside the iterative loop. The resulting feedback closes the loop between the language model and the physical-layer iteration. Simulations show that Sem-IR outperforms orthogonal time-division access (TDMA) and the same NOMA receiver without semantic feedback in block error rate (BLER), yielding an order-of-magnitude reduction over NOMA at 8 dB.

We connect different degrees of smoothness of real-valued periodic functions to the number of qubits required for their high-precision Fourier-basis amplitude encodings. Our central observation is that the Gevrey hierarchy, which stratifies the space between smooth and analytic functions, provides a natural class for high-precision quantum algorithms. We use Gevrey regularity to analyze a quantum algorithm for solving general linear partial differential equations (PDEs) with periodic boundary conditions. We then extend the solver to PDEs with Dirichlet boundary conditions using thin Gevrey collars and a regularized inverse. Our resource bounds specify precision-dependent Fourier cutoffs and account for restricted discrete stability, elementary gate costs, and restriction to the Dirichlet domain. Finally, we apply our approach to isotropic and anisotropic Poisson equations with periodic and Dirichlet boundary conditions.

We study random Bernoulli measures $\mu_{\mathbf{p}}$ in a random environment $\mathbf{p}=(p_1,p_2,\ldots)$, where $p_n$ are independent and identically distributed on $(0,1)$. For almost every environment $\mathbf{p}$, we establish several almost sure properties of $\mu_{\mathbf{p}}$: its local dimension, $L^q$ dimensions, Rajchman property, mutual singularity with any fixed measure, and a dichotomy for normal numbers of its typical points.

Clustering mixed tabular data requires a unified metric space to bridge the inherent heterogeneity between continuous numerical measurements and discrete categorical symbols. Traditionally, algorithms rely entirely on dataset-internal statistics to estimate categorical relationships, which confines the learned metric to empirical co-occurrences and ignores conceptually obvious yet statistically unobserved affinities. Although LLMs offer external world knowledge, applying their text-centric reasoning to highly abstract tabular concepts presents significant challenges. Bridging this modality gap to construct a semantically complete metric typically requires embedding LLMs into iterative metric learning loops to dynamically optimize cross-modality representations. This incurs intractable computational overhead, forcing a compromise between semantic enrichment and scalability. Therefore, we propose GRACE, an LLM-grounded framework for scalable mixed-data clustering. GRACE shifts semantic acquisition to the attribute-value level via a multi-perspective LLM querying strategy, mapping heterogeneous values into knowledge-informed descriptions. Crucially, this one-shot grounding extracts general-purpose semantic representations that embed heterogeneous attributes into a unified space, decoupling expensive LLM invocation from iterative optimization. Furthermore, GRACE cross-validates these external semantics against dataset-internal statistical evidence to ensure alignment with the dataset-specific cluster structure. Ultimately, GRACE matches the scalability of conventional statistics-driven baselines while achieving superior clustering accuracy and conceptual interpretability over 11 competing methods. The source code is available at https://github.com/develop-yang/GRACE-GRACE-A

For a prime $p$ and $k\in\mathbb{F}_p$, the generalized Markoff graph $G_{p,k}$ is an undirected graph whose vertices are the solutions over the finite field $\mathbb{F}_p$ of the normalized Markoff equation \[ x_1^2+x_2^2+x_3^2=x_1x_2x_3+k, \] where two vertices are adjacent if they differ by a Vieta involution. The Markoff graph $G_p$ is obtained from $G_{p,0}$ by removing the origin. The structure of $G_p$ has been the subject of extensive study; in particular, a major breakthrough of Bourgain, Gamburd, and Sarnak established that $G_p$ contains a giant connected component. Combined with Chen's remarkable divisibility theorem, this implies that $G_p$ is connected for all sufficiently large primes $p$. In the same paper, Bourgain, Gamburd, and Sarnak further asked whether the family $\{G_p\colon \text{primes }p\geq 5\}$ forms an expander family. This motivates us to investigate the robustness of connectivity in the Markoff graphs. The main result of this paper is proved in the general setting: for every prime $p\geq5$ and every $k\in\mathbb{F}_p\setminus\{4\}$, each connected component $C$ of $G_{p,k}$ with $|V(C)|\geq 3$ is $2$-connected. Reducing to the case $k=0$, we conclude that if the Markoff graph $G_p$ is connected, then it is in fact $2$-connected. Consequently, the Markoff graph $G_p$ is $2$-connected for all sufficiently large primes $p$. This is sharp in the sense that $G_p$ is not $3$-connected for any prime $p\geq 7$.

We establish functional quantization and empirical Wasserstein rates for continuous Itô processes under the supremum norm. We assume that the initial condition and the drift and diffusion integrands are controlled by a time-uniform random upper bound with a finite $\rho$-moment for some $\rho>1$. Under this assumption, the $n$-point $L^q$ quantization error is at most $C(\log n)^{-1/2}$ for every $1\leq q<\rho$. The known Brownian lower bound shows that the exponent $1/2$ is optimal over this class. Our proof combines an adaptive dyadic time partition with localization according to the size of the integrands. A general transfer principle yields the sharp mean rate $(\log N)^{-1/2}$ for the $p$-Wasserstein distance between the empirical law of $N$ independent copies and their common path law, together with nonasymptotic deviation bounds, whenever $1\leq p<\rho$. Applications include empirical path-law estimates for path-dependent SDEs and a path-space limit-theory estimate for path-dependent McKean–Vlasov interacting particle systems with common noise.

The Lambert $W$ (LW) model for the cosmic fluid equation of state was proposed by S. Saha and K. Bamba in 2019-2020. A recent (early 2026) paper by Dubey, et al, carries out a new fit of the LW model to observational data, in order to estimate the model's parameters. That paper exhibits a formula for the logarithm of the relative Hubble factor at redshift $z$, $\ln(H(z)/H_0)$, which is based upon a numerical integration. That integral can be evaluated analytically, and we present the details in this working paper. The resulting analytic expression for $H(z)/H_0$ may be convenient for exploration of the LW model. We present two approximation formulas for the Lambert $W$ function, highly accurate for redshifts $z$ between 0 and 3, which may be useful for researchers who do not have ready access to a function library with the Lambert $W$ function. We also present a simplified proof of the analytic formula for another logarithmic integral which may be useful for exploring some variants of the LW model.

For any symmetrizable generalized intersection matrix (GIM) $C$, we construct an acyclic valued quiver $(Q,\mathbf{d})$ endowed with an involution $\theta$. Let $\mathcal{D}$ be the bounded derived category of finite-dimensional representations of $(Q,\mathbf{d})$, and let $\Sigma$ stand for the suspension functor of $\mathcal{D}$. We show that the orbit category $\mathcal{D}/(\theta\circ\Sigma)$ carries a canonical triangulated structure and is $2$-periodic. Applying Peng–Xiao's construction to this orbit category, we prove that the GIM algebra $\operatorname{gim}(C)$ is isomorphic to the integral Ringel–Hall Lie algebra associated with $\mathcal{D}/(\theta\circ\Sigma)$. As a further application of the above machinery, we investigate elliptic Lie algebras of types $D_4^{(1,1)}$, $E_6^{(1,1)}$, $E_7^{(1,1)}$ and $E_8^{(1,1)}$. For each elliptic Dynkin diagram, we define a finite-dimensional algebra $A$ by taking an appropriate quotient of the acyclic quiver $Q$ attached to the GIM matrix $C$. From the resulting $2$-periodic triangulated categories, we build the corresponding Ringel–Hall Lie algebras, and establish a surjective Lie algebra homomorphism from each elliptic Lie algebra to its integral Ringel–Hall counterpart. This map is conjectured to be injective, and its injectivity on real root spaces is confirmed.

We study the clamped plate eigenvalue problem. We show that clamped plate eigenfunctions can be approximated in the $L^2$-sense by their oscillatory components, which solve the Helmholtz equation, with an error that decays exponentially fast in the frequency on any compact subdomain. Additionally, we establish an upper bound on the boundary localization of clamped plate eigenfunctions analogous to the one observed for membrane eigenfunctions.

We construct by convex integration a nonzero singular weak solution of the one-dimensional periodic cubic nonlinear Schrödinger equation that is compactly supported in time, has zero initial datum, and satisfies \[ u\in\bigcap_{\alpha<1/6} C_t^0 H_x^\alpha \cap \bigcap_{1\leq p<3} C_t^0 L_x^p. \] The key new ingredient is a two-carrier perturbation: its zero-output interaction cancels the previous error, without introducing any zero Fourier mode, which is crucial in the cubic NLS. Whenever a singular solution belongs to \(C_t^0L_x^3\), this nonlinearity agrees with the ordinary product \(|u|^2u\). Together with unconditional uniqueness at \(H^{1/6}\), this makes the threshold sharp within the singular solution class considered here.

We study the problem of testing independence between the coordinate processes of a jointly stationary ergodic binary process from finite observations. We prove that no test is pointwise consistent in probability: any procedure whose power tends to one against every dependent law must have nonvanishing false-positive probability on some independent stationary ergodic law along infinitely many sample sizes. Quantitatively, for some jointly stationary ergodic law with independent coordinate processes, the false-positive probability has limsup at least $1/2$. Thus, even under stationarity and ergodicity, independence cannot be consistently decided from increasingly long finite samples. The proof combines a diagonal construction with rare markers that create detectable dependence at selected scales while converging to a limiting process whose coordinates are independent.

We prove a filtration-level conversion from uniform fixed-time testing to all-sample one-sided error control and uniform future-tail control. Given an adapted deterministic binary test whose worst-case fixed-time error tends to zero, the construction selects update times with prescribed summable error budgets and holds each selected decision until the next update. For every prescribed significance level, the construction yields a test that controls Type I error at every sample size; from each update time onward, the probability of any subsequent error is bounded uniformly over each hypothesis class by the remaining budget. Hence the test is uniformly consistent and eventually correct almost surely under both hypotheses. Applying the conversion to finite-alphabet coordinate processes proves the first implication in Ryabko's Conjecture 5.2.

Let $X$ be a Gushel–Mukai threefold and let $\operatorname{Ku}(X)$ be its Kuznetsov component. We show that the Bridgeland moduli spaces of semistable objects in $\operatorname{Ku}(X)$ are normal. Moreover, under certain numerical conditions, if the numerical class is primitive and $X$ is general, we prove that the corresponding Bridgeland moduli space is integral.

We prove that strict higher Du Bois singularities are invariant under small deformations. Using this, we prove a base change theorem for the relative Du Bois complex with strict higher Du Bois fibers, answering a question of Kovács–Taji. We exhibit failures of deformation invariance and base change for $1$-Du Bois fibers, showing that the strictness condition is essentially sharp. We also generalize the local-freeness theorem of Friedman–Laza beyond the local complete intersection setting for families over an arbitrary base, and show the unobstructedness of deformations of strict-$1$-Du Bois canonical Calabi-Yau varieties.

The factorisation of graph polynomials often reflects combinatorial decomposition. For a nonempty loopless graph $G$, we first prove that the two-variable interlace polynomial $q(G;x,y)$, introduced by Arratia, Bollobás and Sorkin, is irreducible over $\mathbb{C}[x,y]$ if and only if $G$ is connected, exactly paralleling the classical irreducibility theorem for the Tutte polynomial. The loopless hypothesis is essential: we construct an infinite family of connected looped graphs whose two-variable interlace polynomials are reducible. For a nonempty graph $G$, we prove that Courcelle's multivariate interlace polynomial $C_G(u,v;\mathbf{x},\mathbf{y})$ is irreducible over $\mathbb{C}[u,v,x_a,y_a:a\in V(G)]$ if and only if $G$ is connected.

Let $n\geq 2$, $1\leq \ell\leq n-1$, and $1<p<\infty$. We prove that every $v\in \dot W^{n/p,p}(\mathbb{R}^n;\Lambda^\ell)$ has a representative $u\in \dot W^{n/p,p}\cap L^\infty$ with $du=dv$ and $\max{|u|{\dot W^{n/p,p}},|u|{L^\infty}}\lesssim |v|{\dot W^{n/p,p}}$. Equivalently, $d[\dot W^{n/p,p}\Lambda^\ell]=d[(\dot W^{n/p,p}\cap L^\infty)\Lambda^\ell]$ with equivalent quotient norms. The proof reduces the selection problem to an endpoint graph estimate for a Riesz potential and the exact Hodge projection. Its main analytic input is a finite-dimensional-input Maz'ya–$\Phi$ inequality for operator-valued homogeneous kernels. For the Riesz/Hodge pair, a nonlinear spherical profile built from the projected kernel has exact atomic cancellation and is coercive by the identity $\int{S^{n-1}}P(\theta),d\bar{\sigma}=(\ell/n)\operatorname{Id}$. Frequency-localized graph closure and Hahn–Banach duality then return a bounded representative.

In his recent breakthrough work [JASA, 2021], Chatterjee proposed a rank-based correlation coefficient $\xi(X,Y)$ to measure the dependence of a random variable $Y$ on $X$. Unlike classical measures such as Pearson, Spearman, or Kendall, $\xi$ satisfies $\xi=0$ if and only if $X$ and $Y$ are independent, and $\xi=1$ if and only if $Y$ is a measurable function of $X$, without requiring monotonicity or linearity. This paper proposes a refined measure of $\xi(X,Y)$ and investigates its theoretical properties. We derive several equivalent representations, construct an estimator, and establish its strong consistency as well as its asymptotic distribution. A natural extension of the proposed framework is also presented. The Monte Carlo simulations show that the proposed estimator outperforms Chatterjee's rank correlation in terms of finite-sample performance, particularly in controlling Type I error under the null.