Papers

For partitions lambda, mu, nu the Littlewood-Richardson coefficient stretches to a function P(t) = c(t*nu; t*lambda, t*mu) which, by a theorem of Derksen and Weyman, is a polynomial in t. King, Tollu and Toumazet conjectured that P has no negative coefficient. The conjecture is known only for c <= 2 and is otherwise open. We prove it for all triples whose parts number at most five. For at most four parts the proof is structural: a period-one dilation transfers the Berline-Vergne local Ehrhart formula from an integral dilate back to the rational hive polytope, and every two-dimensional transverse cone spanned by rank-four rhombus normals has positive weight (minimum 1/9). Five parts need two further ingredients. A closed-chamber transfer lemma, resting on Rassart's polynomiality of the coefficients on the closed cones of a chamber complex, reduces positivity for a fixed number of parts to full-dimensional hives. For full-dimensional five-part hives all coefficients but the linear one were already known to be positive; the linear one is where pointwise positivity of the local weights fails – an edge cone of weight -347/109824 occurs on an actual hive – and its nonnegativity is proved by an exact global certificate: a rational correction supported on 6,903 two-face types, balanced by the Minkowski closure of each two-face polygon, which makes every one of the 87,127 closed edge cones adjacent to them nonnegative. The same local method yields a rank-uniform consequence: for every full-dimensional hive polytope of any rank the top four Ehrhart coefficients are positive. We close by recording the exact general frontier and the obstructions that rule out the standard shortcuts. All finite verifications are exact and accompanied by independent replay scripts.

Let $L$ be a fixed set of positive integers. A family $\mathcal{F}\subseteq 2^{[n]}$ is called $L$-differencing if $\lvert A\setminus B\rvert\in L$ for every ordered pair of distinct members $A,B\in\mathcal{F}$. A longstanding conjecture of Frankl, proposed in 1985, asserts that every $L$-differencing family has size at most $\binom{n}{|L|}$. We resolve this conjecture asymptotically for every fixed $L$, and obtain the exact answer in the only case in which the conjectured bound could be tight. (1) If $L\ne [s]$ and $n$ is large, then every $L$-differencing family satisfies $\lvert \mathcal{F}\rvert \le \left(\frac{s}{s+1}+o_L(1)\right)\binom{n}{s}$. (2) If $L=[s]$ and $n\ge 2s-1$, then $\lvert \mathcal{F}\rvert\le\binom{n}{s}$, with equality only for $\binom{[n]}{s}$ and $\binom{[n]}{n-s}$. The first result follows by reducing directed differences to restricted Hamming distances. For the exact result, we develop a new homogeneous polynomial method, which might be of independent interest.

An additive $(n,k,d)_{q^m/q}$-code is a $\mathrm{GF}(q)$-linear subspace of $\mathrm{GF}(q^m)^n$ of $\mathrm{GF}(q)$-dimension $km$ with minimum Hamming distance $d$. We first extend the Alderson–Bruen–Silverman (ABS) model of linear codes to the additive setting: a code of length $n$ with $q^{km}$ words over an alphabet of size $q^m$ admits an ABS model if and only if it is equivalent to a nondegenerate additive code. We then ask whether an additive code that admits an extension must admit an additive extension. For linear codes ($m=1$) this is a theorem of Alderson and Gács. We characterize the additive codes admitting no additive extension as those whose associated projective system of flats is complete, and we prove that the answer to the question above is again affirmative for $(n,2,d)_{9/3}$-, $(n,2,d)_{4/2}$-, and $(n,3,d)_{4/2}$-codes. In contrast with the linear case, we show that the answer is negative in general. Scattered linear sets yield, for each square $q$, extendable additive $(n,2,d)_{q^2/q}$-codes admitting no additive extension. Further, a different method yields an extendable additive $(30,2,24)_{8/2}$-code with no additive extension. Consequently, for properly additive codes, completeness of the associated projective system does not imply maximality of the code. We conjecture that extendable $(n,2,d)_{p^2/p}$-codes, $p$ prime, always admit additive extensions.

This paper analyzes the regularity of solutions to a fluid-structure interaction (FSI) problem involving a Stokesian fluid and a linear elastic solid in a two-dimensional polygonal domain, seperated by an interface which intersects the boundary of the domain. The main challenges arise from the limited solution regularity caused by geometric singularities at the domain corners. To address this, we introduce tailored weighted Sobolev spaces, denoted by $\mathbf{H}^{k,l}_{\boldsymbol{\beta}}$, and a novel solution operator framework on these spaces. This framework provides the key insight for decoupling the coupled FSI system into tractable fluid and elastic solid subproblems in the regularity analysis. As our main result, we prove the existence and uniqueness of a solution in the weighted Sobolev space $\mathbf{H}^{2,2}_{\boldsymbol{\beta}}$.

We establish a general Poisson approximation for rare local patterns in critical Bienaymé–Galton–Watson trees with offspring distribution $\mu$ in the domain of attraction of a stable law, conditioned to have a large number of vertices. A pattern is specified by a sequence-dependent mark on fringe subtrees. If marked fringe subtrees remain microscopic and nearby marked occurrences have negligible clustering, then their count is asymptotically Poisson in total variation whenever its mean remains bounded; when the mean diverges, the count satisfies a law of large numbers. The main difficulty is the global dependence created by size conditioning. We overcome it by combining the cyclic-shift representation with a refined form of the Chen–Stein bound and a bridge-removal estimate controlling the interaction between a local mark and the remainder of the conditioned random walk. For non-fringe patterns, overlapping occurrences may form clusters and the raw count need not be asymptotically Poisson. We introduce declumped indicators which select boundary witnesses of these clusters and prove a general Poisson approximation for their count. As applications, we obtain sharp asymptotics for the maximum leaf-height, equivalently the maximum protection number, and for the height of the largest complete $r$-ary tree appearing as a non-fringe subtree. Unary-chain maxima, and the maximum leaf-height when $\mu_1>0$, exhibit lattice-modulated Gumbel behavior. Complete $r$-ary patterns for $r\ge2$, and the maximum leaf-height when $\mu_1=0$, are localized on one or two consecutive integers. The results require no exponential moment and include offspring distributions with infinite variance.

We prove optimal systolic inequalities for length spaces homeomorphic to a torus of genus one or a real projective plane. In both cases, the optimal constant coincides with the constant from the (reversible) Finsler setting. This generalizes the classical results for Riemannian and Finsler surfaces. The proof of the inequality for the torus relies on an analysis of the asymptotic volume growth of the universal cover, together with a strengthening of the previously known area minimality of two-dimensional normed planes. For the inequality of the real projective plane, we similarly extend a minimality result of hemispheres. These results build upon works of Burago-Ivanov and Ivanov, respectively. In their proofs we apply recent uniformization theorems for metric disks and the theory of area-minimizing disks in metric spaces due to Lytchak-Wenger.

This paper is the first in a series devoted to the classification of automorphisms and isomorphisms of twisted Chevalley groups over commutative rings. In the present paper, we prove that every automorphism of a twisted Chevalley group of type ${}^2A_\ell$ $(\ell \geq 5)$ over a local ring in which $2$ is invertible is standard.

We present a generalization of the well-known homogeneous self-dual embedding model, which is widely used in conic optimization. The new embedding applies to a problem of minimizing the sum of two proper lower-semicontinuous convex functions and can be represented as a single inequality that uses perspectives of these functions and of their conjugates. A solution to the proposed embedding encodes a primal-dual solution to the original problem when available, or an infeasibility certificate otherwise. We then use the Douglas-Rachford algorithm to find a solution to the embedding and discuss its efficient implementation by exploiting the problem structure. The resulting algorithm recovers an existing method for solving quadratic cone programs as a special case. We demonstrate the generality and effectiveness of the algorithm on a class of convex optimization problems with non-smooth objective function and non-conic constraints.

Given rearrangement-invariant (r.i.) spaces $X$ and $Y$, we show that the segment multiplier, the truncated Hilbert transform, and the discrete Hilbert transform (on the associated discretized space) are bounded from $X$ to $Y$ simultaneously. Moreover, when $X = Y$ this boundedness is characterized by a condition on a pair of Boyd-type indices. We also exhibit an r.i. space on which the segment multiplier is bounded while the (non-truncated) Hilbert transform fails to be bounded.

Tensor decomposition serves as a foundational tool for feature extraction in hyperspectral image classification, a domain classically dominated by the Tucker and Canonical Polyadic decompositions. Although widely adopted, these schemes often struggle to fully encapsulate the deeply coupled geometric and intrinsic structures inherent to multidimensional hyperspectral data. Their structural reliance on rigid low-rank approximations successfully captures independent mode variations but systematically neglects complex, cross-domain spatial and spectral interdependencies. To overcome this limitation, we introduce the Holistic Multivariance Decomposition framework to achieve highly discriminative hyperspectral feature extraction. The Holistic Multivariance Decomposition provides a novel, structurally flexible tensor algorithm that explicitly models isolated spatial and spectral behaviors, alongside intricate, higher dimensional cooperative interactions. Comprehensive experimental evaluations across four benchmark hyperspectral datasets demonstrate that the proposed Holistic Multivariance Decomposition approximants consistently yield superior classification accuracy compared to conventional Tucker and Canonical Polyadic decomposition methods across diverse supervised learning algorithms. By effectively preserving essential joint multivariance features even under severe subspace compression, these results establish the Holistic Multivariance Decomposition as a robust, high fidelity computational framework for resolving complex multidimensional data structures in hyperspectral analysis.

Tutte's $4$-flow conjecture asserts that every finite bridgeless graph with no Petersen minor admits a nowhere-zero $4$-flow. Let $P$ be the Petersen graph and let $e\in E(P)$. We prove that every finite bridgeless $(P/e)$-minor-free multigraph admits a nowhere-zero $4$-flow. Since $P-e$ and $P/e$ are the two maximal proper minors of $P$, combining our result with the theorem of Thomas and Thomson for $(P-e)$-minor-free graphs shows that, for every proper minor $R$ of $P$, every finite bridgeless $R$-minor-free graph admits a nowhere-zero $4$-flow. Equivalently, every finite bridgeless graph without such a flow contains every proper minor of $P$. The proof builds on the girth-five structural framework of Thomas and Thomson together with the nonplanar extension theorem of Norin and Thomas.

We study the equality problem of generalized quasiarithmetic means for a strictly monotonic generator $f$ that is not necessarily continuous. We provide two sufficient conditions that lead to a conclusion analogous to the result of Páles and Pasteczka. We show through an example that in our case, without any extra conditions, the generator functions cannot be expected to be affine transformations of each other over the whole domain. In the remaining case, we consider the appropriate inverses of the functions, which implies that the scaling factor must coincide across the various regions of continuity.

A data-driven inverse optimization problem (DDIOP) is the problem of estimating the objective-function parameters (weights) that explain observed optimal-solution data, and it arises in many applications, including integer linear programming (ILP). It is known that, by applying gradient-based optimization methods to the suboptimality loss, the inverse optimization of ILPs can be solved exactly within finitely many oracle iterations, and that the required number of iterations is bounded as $T=O(1/\gamma(\ell_{\mathrm{sub}})^2)$ in terms of a problem-dependent geometric constant $\gamma(\ell_{\mathrm{sub}})$. However, no means of bounding $\gamma(\ell_{\mathrm{sub}})$ from below as a function of the problem size has been available, and hence the number of iterations could not be given as an explicit function of the problem size. We therefore give, when the forward problem is an integer linear program (ILP), the number of iterations sufficient for projected subgradient descent applied to the suboptimality loss to achieve exact consistency with the observed data, as a fully explicit function of the number of samples, the dimension of the features, the ranges of the features, and the structure of the constraint coefficient matrix, up to polynomial factors in the basic constants (the diameter of the weight set, the step-size parameter, and the Lipschitz constant of the suboptimality loss).

Interference constitutes the fundamental performance bottleneck in wireless networks. Meanwhile, reconfigurable intelligent surface (RIS) has emerged as a promising technique for interference mitigation by directly modifying wireless channels. In this paper, we are interested in the following problem: whether interference-free transmission (in terms of Degree-of-Freedom, DoF) can be achieved with the aid of passive RIS in the two-way K-user interference channel, which is regarded as the most severely interfered network. We show that the answer is affirmative, i.e., interference in this network can be neutralized over the air. To accomplish this goal, two prominent challenges arise: i) the unit-modulus constraint on each RIS reflecting coefficient; ii) the significant disparity between the strengths of the direct and reflective channels. To address these challenges, we exploit the high-dimensional and random nature of wireless channels. Specifically, we cast the problem within a high-dimensional convex geometric framework, which enables us to leverage the ubiquitous concentration phenomenon in high-dimensional spaces. Based on this framework, we establish both sufficient and necessary conditions on the required number of RIS elements to achieve interference-free DoF, which turns out to coincide in order sense. Furthermore, we characterize the impact of imperfect channel state information (CSI) on the achievable DoF and show that interference-free DoF remains achievable if the CSI error is below a certain threshold. Simulation results validate our theoretical findings.

We study the existence of solutions for systems of linear-quadratic vector equations with singular linear parts. We derive a sufficient condition for small right hand sides.

The segmentation of multiple degradations has been a challenging problem in the field of image segmentation. Existing level set approaches commonly adopt a length regularization term to constrain the geometric shape of the segmentation contour. However, the introduction of the length term often results in numerical instability and high computational cost. In this paper, we show that the length term is not essential under certain smoothness constraints, and theoretically prove that the presence of the length term affects the property of $|\nabla \phi|=1$. Based on the finding, we define a class of smooth images, construct the grayscale level set, and propose a fast segmentation framework for degraded images, such as heavily noisy images and intensity inhomogeneous images. The framework transforms PDE evolution into one-dimensional threshold search, which has significant advantages in computational speed, especially on large-scale images. Experiments validate the segmentation performance of the proposed framework on various degraded images.

We introduce a two-factor stochastic volatility model in which the mean-reversion speed of a Cox–Ingersoll–Ross variance process is an observable functional of a market attention field rather than a latent factor. The field admits an exact Feynman–Kac representation and generates closed-form volatility forecasts. The model nests the Heston volatility model, provides measurement-stable dynamics under observable data aggregation, and preserves its structure under an equivalent martingale measure. A mean-field analysis establishes convergence of the underlying particle system, while simulations demonstrate volatility and option-pricing differences following attention shocks.

We show that higher secant varieties of smooth projective curves have symmetric admissible determinantal representations with symmetric Ulrich sheaves of rank one if the embedding is sufficiently ample. For secant varieties of real curves, we give conditions for the representing matrices to be positive definite. This allows us, under mild (conjecturally vacuous) conditions, to represent the convex hull of the curve as a spectrahedron whenever it is a hyperbolicity cone. For secant varieties of rational normal curves, we derive very explicit representations in terms of Littlewood–Richardson coefficients. One key tool that we use are the higher Szegő kernels and the higher Scorza correspondences associated to a non-effective theta characteristic on the curve.

Interference fundamentally limits the performance of dense wireless networks, and reconfigurable intelligent surfaces (RIS) have recently emerged as a promising means of enabling interference-free transmission in the Degrees-of-Freedom (DoF) sense. This paper investigates the feasibility of achieving full DoF in a two-way K-user interference channel-a canonical interference-limited setting-by employing an active RIS. Unlike its passive counterpart, an active RIS is subject to both per-element gain constraints and a total reflection-power constraint, which renders over-the-air interference nulling equivalent to solving a constrained random linear system with coupled nonlinear constraints. By leveraging tools from high-dimensional convex geometry, we derive a tight scaling threshold on the required number of reflecting elements (REs) for full-DoF transmission. We further extend the analysis to scenarios where each RE incurs circuit power consumption under a total power budget, leading to a fundamental tradeoff between RIS transmit power and circuit power. For this setting, we establish the thresholds for both the total power and the corresponding number of REs required to achieve interference-free transmission. Simulation results validate the theoretical analysis.

We construct numerically two families of static five-dimensional Lorentzian metrics, solutions of vacuum Einstein equations with a negative cosmological constant, one with and one without a black hole region, with a metric at sections of conformal infinity which is conformal to any squashed three-dimensional sphere.

We introduce the concept of rough additive functions, which extends rough paths theory to line integrals of distributional 1-forms. In the context of gauge theory, we use this notion to define controlled gauge transformations and holonomies via RDEs. One of our main results is a rough version of Uhlenbeck compactness for distributional connections based on rough additive functions. The main ingredient is a singular elliptic PDE to obtain a Coulomb gauge. We solve this PDE using regularity structures and a method inspired by the implicit function theorem. Surprisingly, the model needed to solve the equation is determined entirely from rough additive functions, which is a simpler and geometrically more natural object. We apply our results to the Yang-Mills measure on the unit square, showing that it has a gauge-fixed representation with optimal regularity. Although we focus on the unit square in this article, we expect our results to apply to more general surfaces.

Discrete-time Poisson channels with finite intersymbol interference provide a natural model for direct-detection optical links in which multipath memory and signal-dependent shot noise appear simultaneously. Under peak and average optical-intensity constraints, we study the independent and identically distributed (i.i.d.)-constrained capacity problem of such channels. We prove that every input distribution maximizing the stationary mutual information rate within the i.i.d. input class has finite support. The proof is carried out directly on the entropy rate of the continuous-state hidden Markov output process induced by the finite-memory channel. We first establish a filtering-forgetting estimate whose constants are uniform over all admissible i.i.d. input laws. We then derive the entropy-rate first variation, construct a holomorphic extension of the corresponding influence function, and combine the Karush-Kuhn-Tucker condition with a supralinear growth argument.

This is a survey of results on incidence equivalence, a notion introduced by P$.$Griffiths around 1970 when trying to extend the classical properties of the Abel-Jacobi map for curves. Using the intermediate jacobians and the associated Abel-Jacobi maps in higher dimension, a natural question came up: is the geometrically defined incidence equivalence relation the same as Abel-Jacobi equivalence, which is of transcendental nature? I give an overview of results related to this question and to several classical conjectures that are far from resolved, such as Grothendieck's generalized Hodge conjecture. The motivation for writing this survey came from a recently observed unexpected connection of Griffiths' question to the asymptotic behaviour of the archimedean height pairing in a geometric setting.

Let $K$ and $S$ be compact Hausdorff spaces. We prove a Bollobás-type theorem for operators between complex spaces of continuous functions. More precisely, every operator that almost attains its norm at an initial function can be approximated by a norm-attaining operator whose norm-attaining function remains close to the original one. Our proof combines the iterative method developed previously in the real case with the recent construction for complex measures. In particular, our result provides a quantitative strengthening of the complex version of the Johnson-Wolfe density theorem. As a byproduct of our construction, the same conclusion can be obtained when considering only compact operators.

For any $n\ge 3$, we construct uniformly quasiregular self-maps of $\mathbb{R}^n$ of polynomial type for which the Julia set is a Cantor set.

Zermelo's algorithm is a classical method for computing the maximum likelihood estimator in the Bradley–Terry (BT) model, but its convergence can be slow in practice. To accelerate computation, Newman introduced a family of Zermelo-type fixed-point iterations parameterized by $\alpha$, with Zermelo's algorithm recovered at $\alpha=1$. Empirical evidence suggests that the choice $\alpha=0$ often converges substantially faster, making it a promising alternative, yet the mechanism underlying this acceleration remains elusive. This paper provides theoretical insight into this phenomenon through a systematic local convergence analysis. We derive closed-form expressions for local convergence factors under synchronous and asynchronous updates and analyze their dependence on $\alpha$ via spectral analysis of the associated Jacobian matrices. For synchronous updates, we show that the algorithm may fail to converge when $\alpha<1$, and its local convergence factor is quasi-convex in $\alpha$ under the population BT model. In contrast, asynchronous updates are always locally convergent, and their local convergence factor is provably monotonically increasing in $\alpha$ under the population BT model of consistently ordered bipartite comparison graphs, establishing the optimality of $\alpha=0$ in this setting. We further establish asymptotic approximation results for the population convergence factors under the BT model, justifying their practical relevance. Numerical experiments on synthetic and real-world datasets confirm the theory. Our analysis complements existing convergence results and shows that the acceleration of $\alpha=0$ arises not only from the parameter choice but, more importantly, from the use of asynchronous updates.

We study pathwise uniqueness for cyclic catalytic stochastic differential equations whose state-dependent square-root diffusion coefficients are non-Lipschitz and degenerate on the boundary. The approach is the direct construction of a strong solution using a Malliavin compactness criterion. The key is the development of a new family of boundary-sensitive weighted Malliavin estimates for the tangent processes of the smooth approximations. Pathwise uniqueness then follows from the dual Yamada-Watanabe argument together with the weak uniqueness available in the literature.

Meandering hyperbolicity, introduced by Kapovich, Kim, and Lee, extends classical hyperbolicity beyond the setting of word-hyperbolic groups. In this paper, we prove that every meandering-hyperbolic action is locally semi-rigid in the $C^0$ topology. This extends the previously established stability theory in the Lipschitz $C^0$ topology to the full $C^0$ topology. Consequently, we recover the $C^0$-local semi-rigidity of both boundary actions of word-hyperbolic groups and cocompact lattices in semisimple Lie groups from a single dynamical principle.

We improve an explicit upper bound on the height of the modular polynomial $\Phi_N$ given by Breuer, Gómez, and Pazuki. We also prove new explicit upper bounds on the height of modular polynomials for the cube root of $j$ and the Weber modular function $\mathfrak f$.

Closed-loop generative selection has become a workhorse of computational drug discovery: a learned generative model proposes candidate molecules, a fitness oracle scores them, the best are kept, and the model is retrained on this elite set before the next round. Despite its wide use, the method has lacked a rigorous convergence theory, largely because retraining the model each round breaks the Markov property on which classical evolutionary-algorithm analysis relies. We develop a self-contained theory of convergence and expected running time for this class of algorithms. By recovering a Markov structure on an enlarged state space, we show that elitism makes the search absorbing, and we prove almost-sure convergence together with a runtime bound that decomposes the search into the time spent escaping each fitness level. We then analyse the role of the model's memory—how much of the past it is trained on. When learning improves steadily with more data, deeper memory never hurts; when it does not, an exit-time analysis pinpoints the optimal memory depth and shows that excess memory can actually slow convergence. The theory extends to multi-objective search and to noisy oracles: we quantify how many repeated evaluations certify progress under light-tailed noise, and how robust estimators restore guarantees under heavy tails. Recast in terms of oracle evaluations - the true bottleneck in drug design - the analysis yields a concrete, evaluation-minimal strategy. Areproducible study confirms the predictions, including the surprising cost of excess memory. We close with three open problems.