Papers

For each real $b$, exact upper and lower bounds on the probability $\mathsf P(X\ge b)$ over all random variables $X$ with log-concave p.d.f.'s such that $\mathsf E X=0$ and $\mathsf E X^2=1$ are obtained, as well as the best constant factor $C$ in the inequality $\mathsf P(X\ge b)\le C e^{-b}$ for all real $b\ge0$. Exact upper and lower bounds on the corresponding exponential moments (Proposition 9.4) and explicit exponentially decreasing upper bounds on the mentioned p.d.f.'s (Propositions 9.2 and 9.3) are given as well. Some general results concerning log-concave p.d.f.'s are also obtained.

We propose a neural algorithm for sampling from distributions specified by unnormalized Boltzmann densities. Our approach is based on the Jordan–Kinderlehrer–Otto scheme for the Kullback–Leibler divergence in the Wasserstein–Fisher–Rao geometry (WFR JKO scheme). Our contributions are twofold. First, we prove that, for any fixed step size, the exact WFR JKO iterates converge exponentially fast to the target as the number of iterations tends to infinity. Notably, this result requires no structural assumptions on the target, such as log-concavity or a logarithmic Sobolev inequality. Second, we develop a neural implementation of the WFR JKO scheme that parametrizes its transport and reaction components using reweighted normalizing flows. Numerical experiments on challenging multimodal targets demonstrate the promising performance of the proposed method.

Statistical-mechanical analyses predict that MacKay-Neal codes can achieve channel capacity at fixed degrees. We prove this prediction for the uncoupled $(\ell,3,3)$ MN socket ensemble for every fixed integer $\ell\ge4$. For each binary-input memoryless symmetric channel of capacity strictly greater than $3/\ell$, the ensemble-average block error probability under maximum a posteriori (MAP) decoding is $O(\log N/N)$, where $N$ is the transmitted blocklength. The actual transmitted rate converges to $3/\ell$. The result includes both punctured and transmitted variables and does not condition the sparse square matrix on invertibility. The proof establishes an entropy inequality for six bits subject to even parity by an analytic argument that reduces the domain by symmetry, restricts its interior stationary points, where both partial derivatives vanish, to the diagonal, and controls the resulting scalar functions by explicit polynomial bounds. Exact configuration counts then bound the conditional entropy on the binary symmetric channel. A standard comparison by mutual information extends the entropy bound to general symmetric channels; independent output erasures and a minimum-distance estimate for the transmitted code yield the bit and block error bounds.

The Melvin–Morton–Rozansky theorem relates the large-colour semiclassical limit of the coloured Jones polynomial to the inverse Alexander polynomial. Bar-Natan and Garoufalidis [BNG96] proved the theorem using weight systems and, more generally, established the corresponding formula for arbitrary complex semisimple Lie algebras. We give a direct proof of this semisimple Lie algebra generalization by analysing a state sum. For a complex semisimple Lie algebra $\mathfrak{g}$ and a dominant integral weight $\lambda$, we use the embedding $V_{d\lambda}\hookrightarrow V_{\lambda}^{\otimes d}$ to analyse the state sum. Under the specialization $q=e^{h/d}$, we show that, as $d\to\infty$, only the identity terms and those proportional to $E_{\alpha}\otimes F_{\alpha}$ contribute at leading order. It follows that the leading-order contribution decomposes over the positive roots, with each root contribution evaluated as in the $\mathfrak{sl}_{2}$ case. Writing $t_{\alpha}=e^{-2(\lambda,\alpha)h}$, the limit is therefore given by the product of $\Delta_{K}(t_{\alpha})^{-1}$ over the positive roots $\alpha$.

Kotelskiy, Watson and Zibrowius show that the Bar-Natan invariant $\unicode{x0414}(T)$ associated to a $4$-ended oriented tangle $T$ extends to a twisted complex $\unicode{x0414}^\infty(T)$ over an $A_\infty$-enhancement of the Bar-Natan algebra. We prove that $\unicode{x0414}^\infty(T)$ itself is a tangle invariant, settling a conjecture from their work. We deduce this from an invariance statement for a matrix multifactorization $\mathcal M(T)$, generalizing a construction of Ballinger to the coefficient ring $\mathbb Z[G]$. This multifactorization is well defined up to a notion of 1-homotopy equivalence. The proof rests on explicit formulas for special deformation retracts of Koszul matrix factorizations, which realize delooping in this setting and may be of independent interest.

Entropy-regularized optimal transport (EOT) has become a key tool in optimal transport, providing both theoretical insight and efficient computation via the Sinkhorn algorithm. We study the first-order \(\Gamma\)-expansion of EOT as the regularization parameter \(\varepsilon\) vanishes, by rewriting EOT as the minimization of a relative entropy with respect to a Gibbs measure \(\gibbs\) associated with the duality gap energy. Using a mild extension of the Laplace method, we prove the convergence of the Gibbs measures and derive this \(\Gamma\)-expansion for non-degenerate ground costs, under suitable assumptions on the duality gap, the contact set and the marginals. This yields at the same time an entropic selection principle and an expansion of the EOT cost up to \(o(\varepsilon)\) for this class of ground costs, generalizing existing results.

We establish general conditions for the weak convergence of regular one-dimensional diffusion processes when their scale functions, speed measures, state spaces and boundary conditions may substantially vary. Our approach is based on the classical Itô–McKean representation of regular diffusions via time changes and scale transformations of a single Brownian motion. This reduces many stochastic convergence problems to deterministic questions concerning the convergence of integrals, monotone functions, compositions, and generalized inverses. As a main application, we investigate various homogenization regimes for diffusions in media with closely spaced semipermeable membranes, where the resulting limiting processes include not only reflected and skew Brownian motions, but also skew-sticky Brownian motions and snapping-out diffusions. This yields a representation of a snapping-out diffusion as a function of a single Brownian motion, without introducing an auxiliary sequence of independent exponential random variables.

We construct an explicit binary linear system game that separates $C_{qa}$, the closure of the set of finite dimensional quantum correlations, from $C_{qc}$, the set of commuting operator correlations. The game admits a perfect commuting operator strategy, while every correlation in $C_{qa}$ has a success probability strictly less than 1. This provides a concrete counterexample to Tsirelson's problem in its approximation form. The defining system has 1417152 equations in 1889684 variables, with exactly three nonzero coefficients per equation and a single nonzero entry on the right hand side. We also compute the classical value exactly. The complete system is specified in Lean 4, and the separation and exact classical value are formalized using Mathlib.

We study the exact $L^p$ norms for a number of maximal operators, which are prominent in harmonic analysis, semigroup theory, and probability. We prove that a strongly continuous positive symmetric sub-Markovian semigroup $(T_t)_{t\ge0}$ on a nonatomic $\sigma$-finite measure space $(X,\mathcal F,\mu)$ has maximal $L^p$ norm $p'=p/(p-1)$ for every $1<p<\infty$ whenever $f\mapsto T_{t_0}(\mathbf 1_Ef)$ is compact on $L^2(\mu)$ for some $t_0>0$ and some measurable set $E$ of positive measure. The lower bound follows by approximating finite dyadic conditional expectations at suitable semigroup times; the upper bound is Stein's maximal inequality in its sub-Markovian form. Important examples of such maximal operators we discuss include maximal functions of various semigroups (heat, Poisson, Ornstein–Uhlenbeck, Schrödinger, Dunkl, Laguerre, Jacobi) as well as the centered Hardy-Littlewood maximal operator on $\mathbb{R}.$ In particular, the heat and Poisson maximal operators on every complete connected Riemannian manifold of positive dimension without boundary have exact $L^p$ norm $p'$, with no curvature or stochastic completeness assumptions. Combining our approach with previous work of the first and last authors we also prove that $p'$ is the limit, as $d\to \infty,$ of the $L^p$ norms $B(p,d)$ of the centered Hardy-Littlewood maximal operators on $\mathbb{R}^d.$ On the other hand, we show that the norm of this maximal operator is strictly larger than $p',$ in dimensions $d\ge2.$

For elliptic curves $E/\mathbb{Q}$ in short Weierstrass form \[ E=E(a_4,a_6): y^2=x^3+a_4x+a_6, \] we derive a multivariable Euler product generating function which encodes the Tamagawa product $Tam(E)=\prod_p c_p(E)$. Using this generating function, we compute limiting distributions, exact covariances, and moment and tail bounds for four important statistics on the Tamagawa number; for example we show that more than half of all curves in this height ordering have trivial Tamagawa product, about $42.2\%$ have exactly one prime with nontrivial local Tamagawa number, and only about $6.8\%$ have exactly two such primes. The product is obtained by specializing an Euler product indexed by local reduction data that we derive from Tate's algorithm. The results in this paper were autoformalized in Lean by AxiomProver.

An optimal stopping problem is monotone when its stopping region is preserved under subsequent transitions, so that the one-step look-ahead stopping time is optimal. We develop a finite-horizon discrete-time theory of multiple stopping that separates general recursive dynamic programming from this stronger monotone structure. Extending the classical martingale-system approach, we construct a recursively defined optimal vector of stopping times without assuming monotonicity. If monotonicity holds at every exercise level, this vector coincides with the recursive vector of one-step look-ahead stopping times. A bounded counterexample shows that monotonicity of the original single-stopping problem alone is insufficient, whereas monotonicity propagates to every level when the level-one one-step look-ahead function is deterministic. We apply the theory to multiple-selection last-success problems. For the classical independent odds setting, a continuous-time Poisson embedding gives an alternative proof of the multiple-stopping lower bound and clarifies the associated threshold constants. We also examine random horizons, Markov-dependent trials, and an unknown success probability

We develop q-ary Guessing Random Additive Noise Decoding (GRAND) for linear codes over a q-ary alphabet. The decoder works on a sorted symbol-likelihood array and uses three local child generation rules to generate symbol deviation patterns. We then prove that these rules induce a monotone rooted spanning tree of the full row-index space, so best-first traversal gives maximum-likelihood (ML) decoding under unlimited search. Reed-Solomon simulations verify ML agreement and show the finite-budget performance-complexity tradeoff.

Fluid antenna systems (FASs) offer significant spatial flexibility, yet securing them against eavesdropping is critical for practical FAS deployment in military, satellite, and internet-of-things networks. Although large language model (LLM)-assisted genetic algorithms (LLM-GAs) can address this secure FAS port selection problem, whether further algorithmic improvement is possible warrants deeper investigation. To this end, we propose a memetic algorithm based on reflective evolution (ReEvo). Unlike the state-of-the-art LLM-GAs, which design only crossover or mutation operators with an LLM, our algorithm leverages an LLM to evolve dedicated crossover, mutation, and local-search operators offline. These operators are then embedded into a memetic search framework, thereby obviating any online LLM queries during execution. Simulation results at equal generation counts demonstrate that our proposed algorithm achieves a higher secure sum-rate than the conventional GA and the state-of-the-art LLM-GAs.

This article generalizes the de-initialization framework, proposed by (Roberts and Rosenthal, 2001) for total variation distance, to Wasserstein distances. In essence, de-initialization captures the phenomenon that in the presence of certain structural features, the convergence behaviour of a Markov chain can be reduced to that of a "simpler" stochastic process. Our results allow to measure this in terms of a general Wasserstein distance. We apply them to analyse the Wasserstein convergence behaviour of Gibbs sampling, linchpin variable sampling and slice sampling.

We introduce general methods to derive lower tail large deviation principles, for a variety of nonlinear problems of combinatorial interest. These methods are especially effective in "critical" regimes, where the nonlinearities are not sparse enough for Poissonian lower tails (e.g., Janson's inequality does not provide a sharp tail bound), but the situation is not so dense that combinatorial effects dominate (e.g., we cannot directly apply the theory of hypergraph containers or the relative entropy framework of Kozma-Samotij). In particular, these are regimes where one expects interesting phase transitions to occur. Our results have a number of consequences related to subgraphs in random graphs, answering various questions of Warnke and Jenssen-Perkins-Potukuchi-Simkin. For example, consider a random graph $G\sim \mathbb G(n,p)$, and let $L_\triangle(n,p)=\log\Pr[G\text{ is triangle-free}]$. The first-order asymptotics of $L_\triangle(n,p)$ have long been known in all parameter ranges except the critical regime where $p$ has order of magnitude $1/\sqrt n$. We are able to fill this gap: for any fixed $c>0$, we show that $n^{-3/2}L_\triangle(n,c/\sqrt n)$ converges to a limit expressed in terms of a two-parameter variational problem, which has a single phase transition at $c\approx 4.341$. In contrast, we show that there is no such phase transition for the probability that $G$ is $C_{2\ell}$-free, for any fixed even cycle $C_{2\ell}$. We also obtain some applications in combinatorial design theory. Addressing conjectures of Glock-Kühn-Lo-Osthus, Kelly, and Kwan-Sah-Sawhney-Simkin, we obtain new estimates on the number of order-$n$ Steiner triple systems with no Pasch configuration and the number of order-$n$ Latin squares with no $2\times 2$ Latin subsquare (both tight up to a factor of $\exp(o(n^2))$).

Let $X_\sigma$ be the affine normal toric variety associated with a full-dimensional strongly convex rational polyhedral cone $\sigma$, and let $m$ be a primitive degree with slice $P_m=\sigma\cap[m=1]$. We construct an explicit flat algebraic family whose completion is universal for equivariant deformations in all degrees $-jm$, $j\ge1$, simultaneously. We prove that the irreducible components of the reduced base correspond bijectively to maximal lattice-friendly Minkowski decompositions of $P_m$, and describe the induced family on each component. When $m\in\sigma^\vee$, we also obtain a formally universal equivariant family for the pair $(X_\sigma,V(\chi^m))$. These results extend earlier miniversality and component theorems to arbitrary affine normal toric varieties and arbitrary primitive degrees.

For a convex body $K$ its parallel section function in the direction $\theta\in S^{n-1}$ is defined by $$A_{K,\theta}(t) = \mathrm{vol}_{n-1} \left( K \cap \{ x \in \mathbb{R}^n ~ | ~ \langle x, \theta \rangle = t \} \right) .$$ We study to what extent the body $K$ is determined by partial information about these functions. As applications, we characterize convex bodies with locally separable section functions, establish partial results on the homothety conjecture for bodies of flotation, and answer a question of Barker and Larman concerning the determination of convex bodies from section functions at infinitely many distances from the origin.

Let $C_1,\ldots,C_k$ be the convex hulls of independent partial-sum processes in $\mathbb{R}^d$ whose increments are exchangeable within each process. We express the expected mixed volume $\mathbb{E} V_d(C_1[m_1],\ldots,C_k[m_k])$, $m_1+\cdots+m_k=d$, through mean absolute determinants of disjoint block sums of the increments; no general-position assumption is needed. For random walks with i.i.d. integrable increments the block sums are independent, and the formula extends the expected-volume formula of Barndorff-Nielsen and Baxter and of Vysotsky and Zaporozhets to mixed volumes. We then prove a continuous-time counterpart: for independent Lévy processes with finite first moments, the expected mixed volume of the closed convex hulls of their paths is an explicit integral of mean absolute determinants over a product of simplices. For symmetric stable processes the integral is evaluated in terms of the associated zonoids. As a geometric application, we compute the mean mixed volume of random projections of mutually orthogonal canonical orthoschemes.

We introduce the Domination Reciprocal Mean Square index \(DRMS\), a new domination-degree-based topological index obtained by replacing the ordinary degree in the reciprocal mean square index with the domination degree. We establish elementary bounds in terms of graph size and extremal domination degrees, derive a one-parameter family of sharp lower bounds via the power mean, and obtain sharp bounds in terms of several well-known domination-degree-based indices, with a full characterization of the equality cases. We also compute closed-form expressions for \(DRMS\) on several standard graph families and for the corona product of two graphs.

We remove the log-concavity assumption on the weight in the analytic weighted Yau–Tian–Donaldson correspondence established in [arXiv:2406.10939, arXiv:2407.09929, arXiv:2503.22183] and discuss applications to Sasaki and conformally Kähler Einstein–Maxwell geometries.

Bayesian additive regression trees (BART) can require many splits to approximate boundaries misaligned with the predictor axes. RoBART assigns each tree a rotation shared by all internal nodes, retaining axis-aligned splits in rotated coordinates and constant leaves. We jointly propose a Givens rotation sequence and cutpoints on the resulting grid by Metropolis-Hastings and establish reversibility with respect to the conditional posterior with leaf means integrated out. For additive functions with component-specific rotations and anisotropic Hölder smoothness, we prove posterior contraction in empirical $L_2$ distance and for the noise standard deviation. Under the stated prior, design, and grid conditions, with fixed numbers of predictors, trees, and components and no more components than trees, the rate is a sum of componentwise rates determined by smoothness and the number of rotated coordinates used. We also establish a posterior contraction lower bound showing that there exist functions for which RoBART adapts to the intrinsic dimension but axis-aligned BART does not.

Levin asked what can be said about the locus of points lying on a given curve in $\mathbb{G}_m^n$ that have a non-zero integer multiple on another given curve in $\mathbb{G}_m^n$ for $n \geq 3$. We give a definite answer to the abelian analogue of Levin's question, proving what is predicted by the Zilber–Pink conjecture in this case. An important ingredient in our proof is a strengthening of a height inequality by Vojta and Rémond.

As DNA-based data storage evolves, protecting user privacy during data retrieval has become increasingly important. We study sequential random sampling DNA private information retrieval (SRS DNA PIR) with multiple colluding random sampling servers, where the database is partitioned into servers of equal size. We investigate the tradeoff between the download cost, defined as the expected number of queries, and the privacy leakage, measured by mutual information. We derive lower bounds on this tradeoff, including a bound given by an optimization problem. This bound is tight when each server stores two files, and we construct schemes that attain it. For servers of any size, we construct schemes that apply a single-server scheme to a randomly selected subset of servers.

Let $G$ be a simple abelian variety of dimension $g \in \mathbb{N}$ defined over $\mathbb{Q}^\mathrm{alg}$ and let $C_1, C_2 \subseteq G^N(\mathbb{C})$ be irreducible closed algebraic curves with $N \geq 3$. Further assume that at least one of $C_1$ and $C_2$ is not defined over $\mathbb{Q}^\mathrm{alg}$. Suppose that there does not exist an algebraic subgroup $H_1 \subseteq G^N(\mathbb{C})$ of dimension $g$ such that $C_1 \subseteq H_1$ and that there does not exist an algebraic subgroup $H_2 \subseteq G^N(\mathbb{C})$ of dimension $2g$ such that $C_1 \cup C_2 \subseteq H_2$. Denoting $\mathcal{N} = \{n \in \mathbb{N} \ | \ [n]C_1 \subseteq C_2\}$, we prove that $\bigcup_{n \in \mathbb{N} \setminus \mathcal{N}}\{x \in C_1 \ | \ nx \in C_2\}$ is finite.

Links are disjoint unions of circles smoothly embedded in $S^3$. We use reinforcement learning and Bayesian optimisation to obtain new upper bounds on several link invariants that are not known to be algorithmically computable: the slice genus and the unknotting number for links, and the strong slice genus for algebraically split links. We also compute lower bounds using known invariants. Combining the upper and lower bounds, we obtain new exact values in many cases. Our unknotting agents can reproduce the non-additivity of the unknotting number for several counterexamples due to Brittenham and Hermiller, in some cases finding new unknotting trajectories.

Building on a new regularized continuous formulation which avoids spurious solutions of the maximum $s$-plex problem, we develop a tailored block-coordinate first-order method that combines away-step Frank-Wolfe updates on the standard simplex with linear optimization over the fractional $b$-matching polytope. We establish finite stabilization of the auxiliary variables, convergence to stationary points, and finite support identification under suitable conditions. Numerical experiments on standard benchmark graphs show that, also in practice, regularization avoids the support-identification failures observed with the original cubic formulation and enables the proposed method to quickly find high-quality $s$-plexes.

We construct chiral extensions of the cotangent Yangian using vertex algebroids. At zero level, the construction yields a vertex algebra with a compatible coproduct, whose Zhu reduction recovers the cotangent Yangian of Abedin and Niu. We also construct vertex algebra extensions at non-zero levels and study level-additive coproducts. The zero-level vertex algebra admits two natural quasi-conformal structures. We compute the corresponding sphere one-point conformal blocks and equip each block space with an associative product induced by the coproduct. For the first quasi-conformal structure, the blocks space can be identified with the dual of a certain Lie algebroid homology; for the second, we describe the block algebra and its classical limit in terms of mini-twistor geometry.

Suppose $f_1, \dots, f_{n+1}$ are elements of a regular ring $R$, where $n := \dim R$. The first author proved that $f_1^n \cdots f_{n+1}^n \in (f_1^{n+1}, \dots, f_{n+1}^{n+1})R$; this uses the Brian\c con-Skoda theorem — no simpler proof is known, as far as we are aware. We show here that this result is optimal in many respects. On the other hand, when the $f_i$ are homogeneous general polynomials of a fixed degree in $R := \mathbb{F}[x_1,\dots,x_n]$, for $\mathbb{F}$ a field of characteristic zero, we prove that $f_1 \cdots f_{n+1} \in (f_1^2, \dots, f_{n+1}^2)R$.

We study rational approximation to logarithms of positive rational numbers and give a separated-weight determinant argument for $\mu(\log r)=2$ whenever $r\in\mathbb Q_{>0}\setminus\{1\}$. The geometric input is an interpolation theorem for logarithmic jets at points with distinct multiplicative coordinates. Its successive weight thresholds are independent of the interpolation centres; the eventual degree threshold may depend on them. A local intersection-length bound and a curve inequality lead to surjectivity through a blow-up and Serre vanishing. For a fixed rational argument, this produces a nonzero rational determinant. Clearing denominators gives an arithmetic lower bound, while an exact row translation and repeated Taylor indices yield a contradictory analytic upper bound. The consequences include nonzero rational linear combinations of such logarithms, rational affine changes, and logarithms of positive numbers with a rational positive integer power. We also give a bounded-fibre interpolation extension, an explicit bound for the final analytic degree, and a counterexample to interpolation under the volume condition alone.

Let $n\ge2$, $0<s<1$, and $p,q>0$. We prove a Liouville theorem for the fractional Lane–Emden system \[ (-\Delta)^s u=v^p,\qquad (-\Delta)^s v=u^q, \qquad u, v>0 \qquad\text{in }\R^n. \] More precisely, we show that the system does not have a strictly positive entire solution whenever \[ \frac1{p+1}+\frac1{q+1}>\frac{n-2s}{n}. \] No radial symmetry, global boundedness, finite-energy assumption, or prescribed decay at infinity is required. Our proof adapts the localized potential-kernel virial and source-layer interval-pressure method recently introduced for the classical Hénon–Lane–Emden system. The basic reason the method also works in the fractional setting is simple: the Riesz kernel associated with $(-\Delta)^s$ has the same logarithmic derivative structure as the Newton kernel, with $n-2$ replaced by $n-2s$. However, two genuinely nonlocal points require special treatments. When $2s\le1$, the individual kernel-gradient terms are not locally absolutely integrable, so the two equations must be combined before the diagonal limit is taken. When $2s>n-1$, the kernel restricted to a line is not integrable at infinity, and the interval comparison must therefore be formulated using nonnegative extended integrals. After these issues are resolved, a localized virial identity, a one-dimensional interval-pair inequality, and an averaged source-layer estimate yield a universal local energy bound. The subcritical scaling then forces the same energy to diverge, giving the desired contradiction.