Papers

Given a permutation $\tau$ on $n$ letters, we consider lattices spanned by an orbit of one vector $\boldsymbol x$ in $\mathbb R^n$ under the action of $\tau$ by permutation of the coordinates. Such lattices generalize the important class of cyclic lattices and have previously been studied in~\cite{perm}, where a bound on their rank was established. We prove a sufficient condition on $\boldsymbol x$ for this bound to be achieved. We further investigate the structure of such permutation-invariant lattices, proving that they are glued by the permuted vector from the orthogonal cyclic blocks and giving a determinant formula for the lattice in terms of determinants of these blocks and the norm of the permuted vector. In the case $\boldsymbol x$ is an integer vector, these blocks are sublattices of the root lattices $A_k$ in respective dimensions with root lattices themselves and their glued direct sums also realizable by this construction. We also exhibit a glued construction of permutation-invariant algebraic integral lattices from collections of cyclic number fields. Finally, we prove a strengthened version of a previous result of~\cite{lf_ek} on a related construction of well-rounded lattices spanned by sets of algebraic conjugates.

Condensed configurations are compact incidence data obtained by grouping the cyclic flats of a matroid. We show that their inverse incidence matrices give explicit Schubert expansions and hence determine every valuative or covaluative matroid invariant. For the extended binary Golay matroid, this unexpectedly produces non-real-rooted Kazhdan-Lusztig and Z-polynomials. To our knowledge, the latter is the first counterexample in the literature.

We introduce a new class of backward logarithmic orthogonal functions and generalized backward logarithmic orthogonal functions, constructed by applying a terminal-endpoint logarithmic mapping to generalized Laguerre polynomials. These functions are designed for backward spectral approximations of problems whose solutions exhibit weak singularities at the terminal endpoint. The proposed basis functions generate non-polynomial weighted approximation spaces with nodes naturally clustered near the singular endpoint, and therefore provide an effective framework for resolving algebraic and logarithmic endpoint singularities. We develop the basic approximation theory for these backward logarithmic orthogonal functions, including recurrence relations, derivative formulas, orthogonality, Sturm–Liouville characterization, mapped Laguerre–Gauss quadrature rules, weighted projection estimates, backward Lagrange interpolation estimates, inverse inequalities, and stability properties in weighted Sobolev-type spaces defined through a terminal logarithmic pseudo-derivative. A generalized version of the basis is also introduced by incorporating an algebraic scaling parameter, which improves the flexibility of the approximation space and allows singular factors to be represented more effectively. The error analysis and numerical results show that the proposed backward logarithmic basis is particularly suitable for weakly regular functions with terminal-endpoint singularities and can recover exponential or high-order convergence rates that are typically lost when usual polynomial approximations are applied directly.

By considering arc presentations and grid diagrams of satellite knots, we prove that if K is a satellite knot with companion C and wrapping number w, then its crossing number satisfies $c(K) \geq \frac{2}{5}(w^2c(C))^{1/11}$. We also prove other similar bounds regarding the crossing number and arc index of satellite knots and links. This is partial progress on Problem 1.2 of the K3 Problem List.

Satellite communications are an essential component of sixth-generation (6G) mobile networks, which provide ubiquitous connectivity for global services. However, the satellite uplink remains a critical bottleneck for ground devices: their limited transmit power and antenna apertures result in low data rates and high packet errors. To overcome this bottleneck, this paper advocates a novel relaying paradigm termed generate-and-forward (GF) communications, where satellites exploit on-orbit generative artificial intelligence (AI) to robustly reconstruct corrupted data prior to forwarding. Specifically, we propose SpaceDiffusion, an over-the-orbit diffusion framework for satellite-assisted image transmission. The core of this framework is a channel-distortion-aware diffusion theory developed using the following approach. By formulating the recovery of compressed and lost image tokens as an inverse problem, this theory incorporates a channel-distortion correction term directly into the conventional denoising diffusion implicit model (DDIM) update. As a result, this design enables a single pretrained diffusion model to adapt dynamically to varying packet-loss patterns and compression distortions without retraining. Furthermore, we analytically characterize the progressive token-reconstruction error and derive a diffusion-step activation threshold that predicts when SpaceDiffusion is expected to outperform conventional decode-and-forward (DF) relaying. Building on these theoretical insights, we further develop an energy-aware early-exit policy to efficiently deploy SpaceDiffusion in orbit. Experimental results demonstrate that SpaceDiffusion achieves lower end-to-end latency compared to DF scheme with retransmission protocol and saves approximately 15 dB of uplink transmit power at a target perceptual quality.

Let $V$ be a graded vertex operator coalgebra and $\mathcal{C}$ the category of admissible $V$-comodules. We first give the construction of contragredient comodule of an admissible $V$-comodule. We then introduce cointertwining operators among admissible $V$-comodules. Using cointertwining operators, we define the cotensor decomposition of an admissible $V$-comodule by a universal property. Finally, we prove that the complexified Grothendieck group $\mathbb{K}[\mathcal{C}]=\mathbb{C}\otimes_{\mathbb{Z}}K[\mathcal{C}]$ is a cocommutative coassociative coalgebra. Moreover, this coalgebra is naturally dual to the fusion algebra of the category of admissible modules over the dual vertex operator algebra $V'$.

We prove that topological isomorphism of procountable groups is a universal analytic equivalence relation, answering a question of Gao, Nies, and Paolini. The same conclusion follows for non-Archimedean Polish groups. More strongly, there is one countable group $H$ for which universality already holds among inverse limits of sequences of surjective endomorphisms of $H$. Thus all the classification complexity can be carried by the bonding maps. We encode permutative equivalence of unconditional basic sequences by integer weights with prescribed symmetries. An extension of Gao, Nies, and Paolini's tree construction recovers bounded weight differences from uniform continuity while allowing these symmetries to act. Przeździecki's almost-full functor transfers the resulting inverse graph systems to groups. Identifying the stage graphs with a single graph makes the passage from weights to bonding endomorphisms continuous.

A polyhedron is a planar, $3$-connected graph. We iteratively construct all polyhedra (save for pyramids) from a unique starting graph, namely the square pyramid, via two graph transformations. This builds upon a previous construction, that starts from the full class of pyramids, and applies the same transformations. In a related result, we iteratively construct all quadrangulations of the sphere where all $4$-cycles are facial, i.e., the class of radial graphs of the polyhedra (save for antibipyramids), from a unique starting graph, namely the square antibipyramid, via a unique graph transformation. This builds upon a previous construction, that starts from the full class of antibipyramids, and applies the same transformation.

We prove linear stability of Kasner spacetimes in the direction of the Big Bang, up to an explicit finite dimensional space of non-decaying self-similar solutions. The fastest growing solution is the linearization of Taub's explicit Bianchi II solution, describing a Kasner transition. All other non-decaying solutions are inhomogeneous analogues of this, and the linearized Kasner metric. The key novelty is a notion of quasinormal modes on Kasner spacetimes, similar to quasinormal modes on stationary black holes spacetimes. All quiescent (as opposed to oscillatory) vacuum Big Bang spacetimes of dimension 4 are asymptotic to a Kasner spacetime from the perspective of a single observer at the Big Bang. They are expected to be highly unstable due to the oscillations conjectured by Belinski, Khalatnikov and Lifschitz. This paper provides a complete description of this instability on a linear level without symmetry assumptions.

We consider generalizations of the classical Fourier uniqueness theorem. First, we construct a family of uniformly discrete sets $\Lambda \subset \mathbb{R}$, of uniform density one, such that the exponential system $\{e^{2\pi i\lambda x} : \lambda \in \Lambda\}$ is complete in $L^p(S)$ for every $1 \leq p < \infty$ and every measurable set $S \subset \mathbb{R}$ with $|S| < 1$. We also show that no set that is asymptotically integer can have this universality property. Additionally, for every $v \in (0,1)$, we construct a set of integer frequencies and uniform density $v$ whose exponential system is complete in $L^p(S)$ for every $1 \leq p < \infty$ and every measurable set $S \subset [0,1]$ with $|S| < v$. Finally, we prove that the Sobolev regularity condition $\alpha > \frac12$ for the existence of uniformly discrete uniqueness sets for spectra with periodic weak gaps, considered by Olevskii and Ulanovskii, is sharp. Our findings admit extensions to higher dimensions and are verified in Lean.

OpenAI~\cite{OpenAIManuscript} has recently announced a proof of finite time singularity formation for the 3D Navier-Stokes equations, in the presence of a $C^\infty$-smooth body force. The construction in~\cite{OpenAIManuscript} has a few key features, among which we single out: (i) the angular mean of the solution satisfies specific Type II bounds which are anisotropic; (ii) the solution is exactly axisymmetric in a collapsing core region. In this paper we consider solutions of the 3D Navier-Stokes equations in the presence of a real-analytic body force, and assume that these solutions satisfy properties (i) and (ii) above. We prove that such solutions are in fact regular at the putative singular point. As a consequence, in the construction of~\cite{OpenAIManuscript}, and in any construction with properties (i) and (ii) whose force remains bounded in $C^2$ up to the singular time, the force can neither vanish identically near the singular point, nor be real analytic in the space variables, locally uniformly in time. The main idea of the proof is inspired by our earlier work~\cite{CIV26} and the companion paper~\cite{CIVEulerLength}: zooming-in at the putative singularity using the anisotropic length scales provided by the a priori bounds, we arrive at ancient limits whose PDE evolution imposes additional rigidity.

We study the Anderson model on hyperbolic lattices. The graph is the Cartesian product of the Euclidean lattice with the regular hyperbolic square lattice having five squares at each vertex. We prove the existence of absolutely continuous spectrum and the absence of singular spectrum on a deterministic set of positive measure at weak disorder, and for $0<\lambda\le2$ when the single-site density is bounded below on $[-1,1]$. Truncated Cauchy distributions and controlled perturbations give purely absolutely continuous spectrum on intervals.

The interaction of moose (Alces alces) and wolf (Canis lupus) populations in ecosystems such as Isle Royale National Park is a canonical benchmark for ecological modeling of prey-predator dynamics. Mathematical modeling is a useful tool for modeling these interactions. A central challenge in this domain is the inverse problem, recovering governing system parameters from observational data. Although classical parameter estimation methods have seen considerable progress, they mostly rely on data rather than physics and therefore largely fall short in capturing the time-varying nature of ecological interactions driven by environmental fluctuations, seasonal forcing, and habitat change. This study addresses that gap by solving the inverse problem for a non-autonomous prey-predator system that incorporates theta-logistic prey growth with temporally varying intrinsic growth and natural death rates, as well as Holling type-II and ratio-dependent functional responses. A deep learning framework (self-adaptive bc-PINN with transfer learning) is employed to estimate time-dependent and constant parameters directly from population time series data (1959-2019). Before estimating the parameters, we performed a structural identifiability analysis to ensure that the model parameters are identifiable. The framework demonstrates good reconstruction and prediction results across both functional responses. The ratio-dependent model has shown the better prediction trend beyond the training data. Our framework successfully predicts the sudden decline in the moose population in 2020.

We introduce a unified framework for supervised and unsupervised learning through density estimation using reproducing kernel Hilbert spaces (RKHSs). To this end, we introduce the notion of a nested RKHS, that is, an increasing sequence of RKHSs whose union is dense in an ambient $L^2$-space. This construction allows us to exploit the computational structure of RKHS methods at each finite approximation level while preserving the approximation power of the entire $L^2$-space. We first study statistical learning in a fixed RKHS. In both unsupervised and supervised settings, we establish strong convergence results for empirical averages of random elements taking values in an RKHS continuously embedded in $L^2$. We then use the nested RKHS structure to combine these statistical convergence results with the density of the union of the RKHSs in the ambient $L^2$-space. We apply this construction to probability density estimation and binary Bayesian classification. For density estimation, we construct explicit empirical estimators of the orthogonal projections of the target density onto the RKHSs and establish their almost sure convergence. A diagonal selection procedure then yields a sequence of estimators converging almost surely to the target density in $L^2$. For binary Bayesian classification, we formulate the approximation of the Bayes target as the minimization of suitable population and empirical risk functionals. We prove consistency of the empirical minimizers and use the nested structure to obtain convergence towards the Bayes target in $L^2$. We further provide a geometric interpretation in terms of hyperplanes and weighted distances in the ambient $L^2$-space. Finally, we show that the population risk admits a direct probabilistic interpretation in terms of the probability of misclassification.

Let $M^{n+1}$ be a closed, connected smooth manifold, where $n\geq 1$. For each prescribed volume fraction $s\in(0,1)\setminus\{\frac12\}$, we prove that the isoperimetric region of volume $s \operatorname{Vol}_g(M)$ is unique for a generic set of smooth Riemannian metrics $g$. At half volume, a generic metric has exactly two minimizers, a region $E$ and its complement $E^c$. As consequences, for every $m>0$, uniqueness holds for a generic set of metrics $g$ satisfying $m<\operatorname{Vol}_g(M)$, and it also holds for a generic set of pairs $(g,m)$ with $0<m<\operatorname{Vol}_g(M)$. The proof is variational and requires neither boundary regularity nor nondegeneracy of the constrained Jacobi operator. In particular, the results apply even when isoperimetric boundaries are singular.

This article introduces mutual evaluation of a replicable task worker and a critic that incentivizes truthful reporting, both modeled as strategic agents. The critic chooses a finite-valued rule that induces an evaluation score on joint report laws. Their common payoff is analyzed through regret relative to the unrestricted critic envelope. The critic rule is distinct from the evaluation score. This class enables a peer-free information elicitation mechanism using conditionally independent replications of a worker on the same task. This replication-loop mechanism implements a type-agreement payoff using same-task replications and new-task samples. In contrast to the peer-prediction and scoring-rule literature, implementations are shown that produce unbiased Pearson and Shannon information scores without requiring peers, a ground-truth reference, or likelihood-ratio estimation. A valid binary critic also can be represented by shared finite type annotations of worker returns. One runtime restriction is that the number of required replicas is random and can depend on the critic rule. Other timing effects, such as commitment and reoptimization, yield distinct incentives, connecting the framework to variational peer prediction. This mechanism class illustrates why strategic considerations matter for both critic and worker agents.

We prove new small-ball inequalities for boolean matrix-series. The leading example is $\mathbb E_s[{\text{det}(I-S^2)^\beta\,\mathbf 1_{\{\|S\|<1\}}}]\ge e^{-O(\beta\tau)}$, which holds for boolean matrix-series $S=\sum_i s_iA_i$ formed using symmetric matrices $A_1,\dots,A_n$ and uniformly random signs $s\in\{\pm1\}^n$. Specifically, this inequality holds for all $\beta\ge1$ with $\tau=\sum_i\text{Tr} A_i^2$, as soon as the maximum of $(\text{Tr} A_i^2)_{i=1}^n$ and a certain variance term are bounded above by universal constants. The proof combines the Gaussian reciprocal estimate of (Akbas and Sra 2026), the directional-variation signing theorem of (Guo, Fang, and Lu 2026), and a replica argument that turns existence into a Gibbs law on good signings. Most notably, boolean small-ball delivers a new, interlacing-free proof of Kadison-Singer (most general case); it also recovers Matrix Spencer and Komlós as quick corollaries, while yielding more than six almost immediate proofs of an assortment of discrepancy theoretic problems.

For $r> 2$, we study the moduli space parameterising fixed degree morphisms $\mathbb P^r\to X$ where $X$ is a smooth hypersurface of low degree. More precisely, we prove the following result: let $n\geq 2, e\geq 1$ and $X\subset \mathbb P^{n-1}$ a smooth degree $d\geq 2$ hypersurface over an algebraically closed field of characteristic zero or greater than $d$, then $\mathrm{Mor}_e(\mathbb P^r, X)$ is irreducible of the expected dimension if \[ n> 2^d(d-1)\binom{de+r-1}{r-1}. \] Our result extends the result of Browning-Yamagishi for $r=2$ via multiblock Weyl differencing.

We study a singular Dirichlet problem driven by a regularized $p(x)$-Laplacian under a prescribed modular constraint. Using growth and differentiability estimates for the regularized energy, together with coercivity and modular compactness, we construct nonnegative constrained minimizers for a family of nonsingular approximating functionals. Under the admissibility of signed constrained variations, each minimizer satisfies an approximate Euler–Lagrange equation with a uniquely determined constraint multiplier. An explicit uniform boundary lower barrier and a compatible weight condition, combined with a variable exponent Hardy inequality, provide global dual control of the singular sources and convergence against every zero-trace Sobolev test function. For these approximate minimizing pairs, the singular-limit result gives subsequential strong Sobolev convergence of the states and convergence of their multipliers, while preserving positivity and the prescribed modular. The limiting pair is therefore a positive normalized weak solution of the singular problem, obtained by removing the reaction regularization while keeping the gradient regularization fixed.

We report improved global-minimum configurations for the classical Thomson problem of $N=60$, $61$, $92$, and $99$ repulsive Coulomb charges confined to a disk. By combining the quenched molecular dynamics (QMD) method with the fixed-border heuristic introduced by Amore and Zarate, we systematically obtain configurations with energies $E_{\mathrm{QMD}}(60)=2159.3584240930$, $E_{\mathrm{QMD}}(61)=2237.19264190$, $E_{\mathrm{QMD}}(92)=5358.35353314$, and $E_{\mathrm{QMD}}(99)=6254.83029083$, which improve upon the previously best-known values. For $N=60$, the Voronoi diagram of our configuration differs from the one reported earlier. The symmetries for $N=61$ $(C_{2})$ and $N=99$ $(D_{1})$ are confirmed and are consistent with the known symmetry patterns for these configurations. For $N=92$, whereas the previously reported Voronoi diagram has lower symmetry, our solution exhibits a clear $D_{1}$ (axial) symmetry of defects. The results are highly reproducible across multiple independent runs, providing strong evidence that these configurations are robust global-minimum candidates.

We consider the problem of testing Granger causality in time series, specifically, whether the future outcome $Y_{t+1}$ and the exposure history $\bar X_t$ are conditionally independent given the history of $\bar Y_t$ and a set of confounding variables $\bar Z_t$ up to time $t$. This testing procedure distinguishes true causal effects from associations driven by common external processes, supporting reliable decision-makings in applications such as finance, neuroscience, and climate science. While traditional approaches assume a linear vector autoregressive (VAR) model and are vulnerable to misspecification, we instead address a model-free version of the problem. We propose nonlinearly regressing both the outcome and exposure on the joint history of $Y$ and $Z$, and calculating a test statistic based on the sample covariance of residuals, we call the Generalised Temporal Covariance Measure (GTCM). To account for heteroscedasticity and improve power against local alternatives, we incorporate variance weights and employ a data-adaptive test based on polynomial lag expansions.The type I error control of the test relies on the relatively weak assumption that user-chosen regression procedures estimate conditional means at a sufficiently fast rate that is slow enough to accommodate nonparametric settings. By further assuming stability of the regression procedures and weak dependence in the time series, we can utilise the entire dataset to estimate the conditional means without splitting the time series into subsets.

Gowers, Green, Manners, and Tao (Annals '25) recently resolved Marton's polynomial Freiman-Ruzsa conjecture. We give an algorithmic counterpart to their result: given uniform sampling and membership-oracle access to a set $A \subseteq \mathbb{F}_2^n$ with doubling constant at most $K$, our algorithm outputs a subspace of size at most $|A|$ whose $K^{O(1)}$ translates cover $A$. The algorithm runs in $\textsf{poly}(n,K)$ time. As applications, we obtain polynomial-time algorithms for a variety of learning problems, including quadratic Goldreich-Levin, improper agnostic tomography of stabilizer states, and tomography of quantum states with bounded stabilizer extent.

We investigate the sensitivity of solutions to multi-dimensional scalar balance laws with respect to perturbations of the initial data analytically and numerically. Reliable first-order sensitivity information is essential in gradient-based optimization and inverse problems constrained by hyperbolic balance laws. In hyperbolic problems, such perturbations affect both the smooth components of the solution and the locations of shocks. Consequently, classical difference quotients of the solution operator generally fail to converge in $L^1$, even in one space dimension. Although generalized tangent-vector techniques have been developed for one-dimensional problems, extending these ideas to multiple space dimensions is considerably more challenging because it requires a geometric description of hypersurfaces of discontinuity. We represent perturbations of hypersurfaces of discontinuity by normal displacements and account for the induced variation of the normal direction. This representation yields evolution equations for the components of a generalized tangent vector. Based on this calculus, we develop a numerical method for computing first-order variations with respect to the initial data. Numerical experiments in two space dimensions confirm the expected first-order accuracy of the resulting approximation.

We develop a new approach to study Bernoulli percolation in dimensions $d>6$. The key idea is to approximate open paths at probability $p'>p$ by a Markov chain of pointed $p$-open clusters. This allows us to transfer sharp information on the two point function from $p$ to $p'$. This inductively gives good asymptotic estimates on the two point function as $p\uparrow p_c$. As a main application, we show that having critical meanfield behavior is a semi-decidable problem. Along the way, we also prove sharp asymptotics for the susceptibility and the average radius of gyration, and prove a local central limit theorem for the slightly subcritical two-point function.

We prove local regularity for axisymmetric suitable weak solutions of the 3D Navier-Stokes equations, which are smooth before the terminal time $t=0$, and satisfy Type~II pointwise bounds at a vanishing length scale $\ell(t)$. We say $\ell(t)$ is an Euler length if it is non-increasing, satisfies a doubling condition, and if $\ell(t)\to0$ and $(-t)/\ell(t)^2\to0$ as $t\to0^-$. This includes power laws $\ell(t)=(-t)^\gamma$ with $0<\gamma<1/2$, and logarithmic enlargements of the parabolic length. Our main result shows that local bounds of the type $|u(\cdot,t)| \leq C\ell(t)/(-t)$ and $|\nabla^2 u(\cdot,t)|\leq C/((-t)\ell(t))$ for all $t\in (-1,0)$ imply regularity. The proof adapts the circulation and potential-vorticity argument of our earlier paper~\cite{CIV26} to ancient limits obtained from the rescaled vorticity system by zooming in. We show that the second derivative a priori assumption may be replaced by a Hölder bound on the azimuthal vorticity and a corresponding bound on its potential vorticity.

We prove that hypercontractivity implies entropy contraction for a single quantum channel, without a detailed balance condition. For primitive quantum Markov semigroups that are KMS-symmetric with respect to a faithful invariant state \(\sigma\), we obtain the modified Log-Sobolev bound $\alpha_1\geq \frac{\lambda}{(2+\log\|\sigma^{-1}\|_\infty)}$ where $\lambda$ is the spectral gap. This removes the assumption of \(L_p\)-regularity for the comparison through the log-Sobolev constant. As an application, we show that the hypercontractivity of an average of two conditional expectations implies the approximate tensorization of relative entropy.

In this paper we study a weighted generalization of the Poisson-Voronoi tessellation called the Poisson-Laguerre tessellation, where the nuclei of the generating Poisson process additionally carry independent non-negative random weights. For each cell we can define its inradius as the radius of the largest ball centered at the nucleus and contained in the cell. We consider point processes of nuclei, weights and inradii, where the nuclei are taken from growing observation windows and the processes are suitably rescaled and shifted to see the behavior of large inradii. We prove convergence in distribution to suitable Poisson processes, obtaining as corollaries the asymptotic behavior of the maximal inradii. Our results cover dimension $d \geq 3$ and bounded random weights, dimension $d = 2$ and random weights with suitable finite exponential moments as well as dimension $d \geq 2$ and random weights following a power-law distribution. Particularly, we observe a different behavior of the Poisson-Laguerre tessellation in the planar and higher dimensional cases when the weights are bounded. The proofs of our results are based on suitable Poisson process approximation techniques and a careful investigation of the geometry of large cells.

Location estimation exhibits markedly different finite-sample behavior across noise distributions: regular families typically yield root-\(n\) rates, whereas compactly supported laws may admit faster, boundary-driven rates. We question whether a single estimator, without knowledge of the density's shape, can adapt to the instance-wise optimal estimation rate, as an oracle that knows the underlying location family can. For a known location family with symmetric log-concave noise density \(f\), the optimal location estimation error with sample size \(n\) under failure probability \(\delta\) is known to be Le Cam's two-point rate: \[ \sup\left\{r>0:\mathsf{H}^2\left(f_0, f_{2r}\right)\lesssim \frac{\log(1/\delta)}{n}\right\}. \] When the location family is unknown, we propose a shape-agnostic estimator that attains this oracle benchmark simultaneously over all symmetric unimodal densities with non-decreasing hazard rates, a class strictly broader than symmetric log-concave distributions. We establish that the Hellinger-driven two-point rate can be characterized solely by a multiscale function of dyadic quantile gaps. This new structural connection between Hellinger divergence and quantile geometry motivates a simple estimation procedure that aggregates sample mid-summaries with carefully designed data-dependent weights. The resulting estimator is finite-sample instance-optimal and runs in only \(O(\log(n))\) time on sorted samples.

We introduce a new method that brings the combinatorics of hyperplane arrangements into the study of representation zeta functions of compact Lie groups. For the Witten zeta function $\zeta_\Phi(s)$ associated with a root system $\Phi$, our method yields elegant formulas for $\zeta_\Phi(0)$ and $\zeta_\Phi'(0)$ in terms of the exponents of various parabolic subsystems of $\Phi$. Such formulas do not appear to be readily accessible through the conventional analytic techniques in the literature. More generally, the method applies to a broad family of conical zeta functions, expressing these two special values through the Möbius function of the intersection poset of the associated hyperplane arrangement.

We prove existence and uniqueness of a strong initial trace for every bounded kinetic solution of a stochastic scalar conservation law, although no initial value is prescribed and no nondegeneracy condition is imposed on the flux. The core of the proof is pathwise. After fixing a realization, the rescaled stochastic terms and kinetic measure vanish in the blow-up limit, so Panov's compactness argument applies; degenerate flux intervals are handled by recursive dimension reduction. The stochastic setting creates several additional difficulties. The martingale identities must remain valid on one common full-probability set throughout the reductions, which requires a parameterized stochastic-Fubini construction. Moreover, the pathwise trace is not automatically measurable because its exceptional sets may depend on the realization. Deterministic time averages and right-continuity of the filtration yield a jointly measurable initial trace, with local strong convergence along essential times, both almost surely and in mean.