Papers

For every $\alpha\in(0,1)$ we construct an autonomous, divergence-free vector field $u \in C^\alpha_c(\mathbb{R}^2,\mathbb{R}^2)$ that does not have the weak Sard property and, nonetheless, does not induce anomalous dissipation of $L^2(\mathbb{R}^2)$ norm for solutions to the associated advection-diffusion equation. This disproves a conjecture proposed by Bagnara, Boutros, De Lellis and Mayboroda in \cite{BaBoDeMa26}.

We establish a tight stability estimate for the Minkowski asymmetry near its maximal value, improving earlier results in both the range for the admissible error and the strength of the estimate. More precisely, if an $n$-dimensional convex body $K$ has Minkowski asymmetry $s(K) \geq n-\varepsilon$ for $\varepsilon \in [0,1)$, then its Banach-Mazur distance to the $n$-simplex is at most \[ 1 + \varepsilon + \frac{\varepsilon^2}{2(1-\varepsilon)}. \] This dimension-independent estimate is sharp to the linear order in $\varepsilon$, including the constant. We apply this estimate to several problems. First, we prove a stability result for the maximal Banach-Mazur distance to the Euclidean ball, improving a previous estimate to the optimal linear order. As a key ingredient, we verify the conjecture that every convex body $K$ contains a translated copy of its volume-minimal circumscribed ellipsoid scaled down by a factor $\sqrt{n s(K)}$. Second, we prove a sharp common generalization of Schneider's higher-order Rogers-Shephard inequality and the $L_p$-Rogers-Shephard inequality, and establish a stability result of the optimal linear order. These results are based on the recent positive answer to the inequality part of the higher-order Godbersen conjecture and the accompanying proof of the $L_p$-Rogers-Shephard inequality. Finally, we improve upper bounds for the diameter of the Banach-Mazur compactum in fixed dimensions.

In this paper, we revisit the construction of optimal incentives in continuous-time principal-agent problems with drift and volatility control. Originally, a general approach relying on dynamic programming and second-order backward stochastic differential equations (2BSDEs) was developed by Cvitanić, Possamaï, and Touzi (2018) [8] to determine the optimal form of contracts in this setting. More recently, Chiusolo and Hubert (2026) [5] proposed a BSDE-based approach by introducing an alternative `contractible-volatility' problem for the principal. In addition to the proposed new method, this work highlights that the optimality result of [8] actually hinges on an assumption, stated below as Assumption 2.3, which may not hold in general. Motivated by this, we introduce in this paper a more general class of contracts, parametrised by a function $\psi$ subject to conditions that make the contract revealing for the agent and without loss of generality for the principal. We further provide two natural specifications of $\psi$: one, inspired by the BSDE approach, yielding a forcing-type contract; the other, motivated by the 2BSDE approach, correcting the duality gap when Assumption 2.3 is not satisfied.

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and let $A_n(\mathbb{K})$ be the $n$-th Weyl algebra. We prove that for every complex reflection group $G$ and every $n \geq 2$ the ring of invariant differential operators $A_n(\mathbb{K})^G$ has irreducible non-holonomic modules of Gelfand-Kirillov dimension $2n-1$, and we exhibit such modules explicitly. Through the Morita equivalence between $A_n(\mathbb{K})^G$ and the rational Cherednik algebra $H_{\mathfrak c}$ at an integral parameter, we deduce that $H_{\mathfrak c}$ has irreducible non-holonomic modules of Gelfand-Kirillov dimension $2n-1$ for $n \geq 2$. In an appendix, following a proof kindly shared by O. Mathieu, we show that $A_n(\mathbb{K})$ has irreducible modules of every Gelfand-Kirillov dimension in the interval $[n, 2n-1]$.

Symmetric nonnegative matrix trifactorizations (SN-Trifactorizations) were introduced by Bukovšek-Šmigoc [Linear Algebra Appl. 2023] as a symmetric analogue of nonnegative matrix factorizations. A SN-Trifactorization of a symmetric nonnegative matrix $A$ is of the form $A = BCB^{T},$ where $B$ and $C$ are nonnegative matrices, with $C$ symmetric. The associated SNT-rank of $A$ is defined as the smallest integer $k$ for which $A$ admits such a factorization with $C \in \mathbb{R}_{+}^{k \times k}$. In this paper, we derive sharper upper bounds for the SNT-rank of the Euclidean distance matrices considered by Shitov [Linear Algebra Appl. 2025] and Bukovšek-Šmigoc [Linear Algebra Appl. 2023]. We also establish several new relationships between the rank and the SNT-rank of symmetric nonnegative matrices and show that the SNT-rank is submultiplicative with respect to the Kronecker product. Finally, motivated by a conjecture posed in the Dagstuhl Seminar Report 13082, we prove a multiplicativity result for the nonnegative rank under an additional structural assumption. We also partially resolve a conjecture of Vandaele-Gillis-Glineur-Tuyttens [J. Global Optim. 2016].

We give a non-analytic proof of the class number formula for biquadratic extensions of number fields.

We investigate a remarkable probability distribution on the symmetric group, due to Steck from the early 1970's, arising from a natural process that intertwines continuous and discrete selections for the values and positions, respectively, of a permutation. Steck used a matrix determinant to express his distribution, whereas we contribute new combinatorial formulas for it in terms of "bottom-to-top maxima" (that are simply the left-to-right maxima of the inverse) permutation statistics. These formulas specialize, in the case of the identity permutation, to a result of Pitman–Stanley from the late 1990's. We then use the Steck distribution to define a game of best choice (secretary problem variation) that incorporates a filtering process for the pool of candidates over time. We solve the model for the case where there is a single filtering step. It turns out that the probability of winning the game under optimal play is similar to the classical model ($1/e$, asymptotically), but that the interviewer must employ a different (non-positional) strategy in order to attain it. The optimal strategy depends on the relationship between interview position and the next bottom-to-top maximum value after the filtering step. Among other results, we prove that the optimal strategy always transitions from rejection to acceptance between positions $(1/e)$ and $(1/e) + (1 - 1/e)y$ as a proportion of the total candidates considered, where $y$ is the proportion of unfiltered candidates.

Let $f\colon \big(x\in (X, \Delta_X)\big)\to \big(y\in (Y, \Delta_Y)\big)$ be a finite surjective morphism between klt singularities such that $K_X+\Delta_X=f^*(K_Y+\Delta_Y)$. We show that the normalized volumes satisfy \[\widehat{\mathrm{vol}}(x, X, \Delta_X)=\mathrm{deg}(f)\cdot \widehat{\mathrm{vol}}(y, Y, \Delta_Y).\] This proves a conjecture in [LLX20, Zhu25, XZ26a].

In this paper, we introduce a novel mixed-order Rényi divergence and investigate its fundamental properties. Using this divergence, we define a family of mixed-order order-two Rényi mutual information and Rényi conditional entropy. We derive exact reliability functions of quantum soft covering and privacy amplification under the sandwiched Rényi divergence with order $\alpha\in[2,\infty)$. The former is jointly characterized by the sandwiched and mixed-order order-two Rényi mutual information quantities, while the latter is characterized by the corresponding conditional entropies. These results provide operational interpretations of the proposed mixed-order Rényi divergence. To the best of our knowledge, this is the first exact characterization of the reliability function for quantum soft covering.

It is shown that the $\mathsf{OR}$-rank (covering rank) of the $2^n \times 2^n$ unique disjointness matrix is $n^{O(1)}(3/2)^n$, hence the known lower bound $1.5^n$ turns out to be essentially tight. By the way, an upper bound $1.89^n$ is obtained for the $\mathsf{SUM}$-rank (partition rank) of this matrix.

The goal of this work is to introduce a notion of mean curvature for level sets of functions in non-smooth spaces with Ricci curvature bounded below, and to prove that it satisfies sharp geometric inequalities. More precisely, we define a suitable Willmore functional $\mathcal{W}$ on Sobolev functions, whose domain of finiteness is dense in $L^p$ for any $1\le p<\infty$. For any function with finite Willmore energy, we show that almost all of its level sets admit a mean curvature vector satisfying the natural integration by parts formula with respect to the tangential divergence. As a main application, we show that in ${\rm RCD}(0,N)$ spaces with Euclidean volume growth, almost every level set of the electrostatic potential possesses a mean curvature vector in the above sense. Furthermore, we prove that this vector satisfies the same sharp Willmore inequality as in the smooth setting, alongside rigidity and almost-rigidity statements. Finally, as a technical tool, we generalize the sharp isocapacitary inequality to the non-smooth setting.

We are going to prove that energy minimizing harmonic maps from a domain $\Omega$ inside an ${\rm RCD}(K,N)$ whose image lies in a small ball inside a ${\rm CAT}(\kappa)$ space are locally Lipschitz continuous. This completes the picture concerning regularity of harmonic maps between singular spaces and justifies a variant of the Bochner-Eells-Sampson inequality between singular spaces.

We establish the stability of local Riemannian Ricci curvature lower bounds along Gromov-Hausdorff convergence. A central part of our analysis is devoted to showing the stability of the parallelogram identity for weak gradients, obtained implementing the Lagrangian approach developed in [arXiv:2511.13320] in the local setting. As an application, we deduce the almost everywhere existence of Euclidean weak tangents. An important ingredient, of independent interest, is an effective local Evolution Variational Inequality along the heat flow on sufficiently small balls, with a remainder term depending on the rate of decay of the flow. This has applications to strong displacement convexity of the Entropy functional along local Wasserstein interpolations and to local essential nonbranching properties.

On spin manifolds, we give a proof of Brendle and Wang's refined positive mass theorem using the Dirac operator method. In the course of this proof, we discuss the origin of the special coefficient appearing in Brendle and Wang's spectral positivity condition for scalar curvature from the perspective of spin geometry.

We describe an implementation in Scratchpad II of linear ordinary differential operators over a differential ring, acting on a module equipped with a compatible derivation. The abstract data type facilities of the system allow such operators to be represented and manipulated as first-class objects while retaining the usual notation for operator application. For coefficients in a field, we give constructive algorithms for left and right division, greatest common divisors, least common multiples, and an extended Euclidean algorithm; the right-hand constructions may be obtained from the corresponding left-hand constructions in the opposite ring. We also discuss pseudo-division over polynomial coefficient rings and an Ore localization yielding a right field of fractions. Finally, we apply this operator arithmetic to factorization of ordinary differential equations, using the associated Riccati equation and Newton polygons to analyze possible singular parts of factors. Examples include operators with constant, elementary-function, rational-function, and matrix coefficients.

Optimization in non-convex neural network models is strongly influenced by the geometry of the solution space: sparse, isolated, point-like clusters are typically algorithmically inaccessible, whereas wide and flat regions can be found efficiently despite being relatively rare. At zero temperature this picture has been formalized in binary perceptrons through the overlap gap property (OGP), which limits algorithmic access to configurations with zero training error above a critical constraint density $\alpha_{\rm OGP}$. Here we extend this description to finite temperature, where a positive training error is allowed and statistically penalized. We first show that the frozen one-step replica-symmetry-breaking solution, dominating the zero temperature equilibrium measure, survives at any finite temperature. We furthermore derive a general criterion, based on the smoothness of the single-pattern Gibbs weight near the decision boundary, that determines when a finite-temperature relaxation of the loss removes freezing. We then extend the OGP construction to finite temperature and show that dense, algorithmically accessible regions of finite-energy configurations persist beyond $\alpha_{\rm OGP}$, up to a threshold $\alpha_{\rm OGP}(\epsilon)$ that grows with the allowed training error $\epsilon$. Finally, in the teacher-student setting, we show that these wide, finite-energy regions still retain good generalization. Using a finite energy message-passing algorithm, we demonstrate numerically that thermal noise enables effective generalization in the regime of constraint densities where both recovering the teacher and finding a zero temperature solution are computationally hard.

This work presents a digital twin framework for output-feedback stabilization and parameter identification in uncertain dynamical systems. A virtual model evolves in parallel with the physical process, assimilating measurement data in real time. By design, the digital twin reconstructs the system state and generates a stabilizing feedback, while model parameters are simultaneously inferred from data of the controlled dynamics using a Bayesian approach. Numerical results for the coupled physical-virtual dynamics demonstrate how digital twins can act jointly as observers, parameter estimators, and control agents, ensuring robust performance under uncertainty.

We remove the torsion-free hypothesis from a positivity theorem of Casalaina-Martin and Zhjeqi, answering a question of them. The main result of this paper was obtained using generative AI, particularly ChatGPT 5.5 Pro and the Danus system.

We consider an optimisation problem for the elementary symmetric functions of the first three Maxwell's eigenvalues on cuboids under volume and perimeter constraint, and we show that the cube is a local minimiser. More precisely, it is the unique minimiser in an explicit cone of cuboids. The result gives a model case for the local optimisation of Maxwell's eigenvalues. On the other hand we show that such local extremality phenomena cannot be expected outside rigid geometric classes.

On a high-dimensional Poincaré ball, a Euclidean beta-type radial measure concentrates near a sphere, but hyperbolic distance amplifies the surviving radial spread, so the usual shell reduction loses part of the limiting metric. In each regime of the balance between this radial width and amplified angular separation, we determine the weak pyramid limit of the spaces rescaled at the order at which the transition between concentration and dissipation occurs. The four critical-limit regimes are the shifted-exponential railway pyramid, the diameter-at-most-one regime, the nonlinear Gaussian-transform regime, and the Gaussian regime. Each finite-branch railway mm-space generating the first pyramid has branches from a common center, a shifted exponential distribution along them, and paths between different branches through the center. We also give a sharp criterion for convergence to the diameter-at-most-one pyramid. The resulting classification identifies the precise critical scales and clarifies when angular and radial fluctuations contribute simultaneously to the limiting geometry.

We consider classical particles in the continuum in the grand canonical ensemble, with a stable, tempered and lower-regular pair potential and boundary conditions of uniformly bounded density. We prove that if the Lee–Yang zeros of the grand canonical partition function in the complex fugacity plane $z = e^{\beta\mu}$ remain bounded away from a real point $z_0 > 0$ for all sufficiently large volumes, then along cubes the thermodynamic limit and differentiation commute at $z_0$: every derivative of the finite-volume pressure in the chemical potential converges, uniformly in a neighborhood of $z_0$, to the corresponding derivative of the limiting pressure. The limiting values of all derivatives are independent of the boundary condition; in particular, the density and the particle-number variance per unit volume converge to $\beta^{-1}\partial_\mu p$ and $\beta^{-2}\partial^{2}_{\mu} p$, respectively. The result extends to the unbounded boundary conditions of Procacci and Yuhjtman for super-stable potentials in addition to Ruelle's tempered boundary conditions.

We study the interior regularity of generalized solutions to the Monge–Ampère type equation governing optimal transportation for the relativistic cost $c(x,y)=\sqrt{a^2-|x-y|^2}$ on $\mathbb{R}^n$. We construct an explicit one-parameter family of radially structured generalized solutions on a ball and exhibit, among them, a solution that is of class $C^{1,\frac{1}{2n-1}}$ but of no better Hölder class: it fails to belong to $C^{1,\beta}$ for every $\beta>\frac{1}{2n-1}$. The construction reduces the equation to a planar autonomous system whose phase variable $s=\dot r$ vanishes to order $(2n-1)$ in the base variable. As an application, in dimension two we transfer the construction to the special Lagrangian curvature equation: for every phase $\Theta\in(0,\pi/2)$ we produce a sequence of smooth graphical solutions converging uniformly to a limit of class exactly $C^{1,1/3}$. Consequently the two-dimensional special Lagrangian curvature equation admits no pure interior $C^{1,\beta}$ estimate for any $\beta>\frac{1}{3}$.

Generalized smoothness, such as (L0, L1)-smoothness, have recently attracted considerable attention due to their ability to model optimization problems arising in modern machine and deep learning, where the classical Lipschitz assumptions of the gradient is often violated. At the same time, computing exact gradients may be impractical or computationally expensive in many applications. In this work, we study convex (L0, L1)-smooth optimization (for normalized gradient method we consider quasi-convex problems too) under access only to a normalized approximation recently proposed Comparison Oracle, which returns an inexact normalized gradient in linear time with a bounded absolute error. Within this framework, we develop comparison-oracle variants of Normalized Gradient Descent and Gradient Descent with Polyak stepsizes. We establish explicit upper bounds on the approximation error that guarantee convergence and derive convergence rates for all proposed methods. Unlike existing analyses, our results require neither classical smoothness assumptions nor access to exact gradients or their exact normalized counterparts. Finally, numerical experiments corroborate the theoretical findings.

We study feature bagging through the lens of algorithmic stability. Feature bagging is an ensemble strategy that aggregates base learners trained on randomly subsampled feature subsets, possibly in a data-dependent manner. We introduce feature instability (FI), the feature-axis analogue of instance instability (II), which measures sensitivity to removing a single feature. Smaller values of II or FI correspond to stronger stability, and our experiments show that FI captures generalization-relevant information complementary to II. Within this framework, we analyze feature bagging in both a parametric linear model and a model-free setting inspired by recursive feature subsampling in random forests. In both settings, we establish formal guarantees showing that feature bagging improves the relevant stability relative to its non-bagged counterpart, with larger improvements under more aggressive subsampling. We further show that a modest number of bagging rounds is sufficient to approach the infinite-bagging stability level.

Real-world systems, from climate models to power grids, often fluctuate due to sensor noise, drift, or environmental variability, yet standard chaos diagnostics assume fixed parameters and asymptotic horizons. We introduce a finite-time framework for two-dimensional maps under independent, identically distributed parameter noise. First, we prove that the maximal finite-time Lyapunov exponent converges, after centering and scaling, to a Gaussian law whose mean and variance depend explicitly on the map's Jacobian statistics. Second, we develop an attractor separation algorithm that uses FTLE histograms and a geometry-based classifier to partition phase space into chaotic and periodic regions under noise. Third, we validate our theory numerically on the noisy Domenicali map, demonstrating Gaussian FTLE distributions, predictable shifts in Kaplan-Yorke dimension, and a sharp noise threshold for basin escape. Finally, we estimate the critical noise level $\sigma_c$ at which attractor coalescence occurs, using three complementary numerical methods to bound it above and to pinpoint an estimate.

For graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the minimum integer $N$ such that every $N$-vertex graph contains $F$ or its complement contains $H$. If $F$ is connected and $|F|\ge\sigma(H)$, a construction of Burr gives $R(F,H)\ge(\chi(H)-1)(|F|-1)+\sigma(H)$, where $\sigma(H)$ denotes the minimum order of a color class in a proper $\chi(H)$-coloring of $H$. Burr proved that this bound is attained for $F=C_n$ when $n$ is sufficiently large. Allen, Brightwell and Skokan conjectured that equality already holds whenever $n\geq |H| \chi(H)$, while Haslegrave, Hyde, Kim and Liu subsequently proved it whenever $n\ge C|H|\log^4\chi(H)$. Pokrovskiy and Sudakov conjectured that the optimal linear condition $n\geq C|H|$ suffices; this conjecture was also highlighted by Montgomery in his 2026 ICM survey (see Conjecture 9.2). In this paper, we resolve this conjecture by proving that there is an absolute constant $C>0$ such that $R(C_n,H)=(\chi(H)-1)(n-1)+\sigma(H)$ for every nonempty graph $H$ and every $n\ge C|H|$. This gives the first bound linear in $|H|$, and is best possible up to a constant factor. Our proof builds on the framework of Haslegrave, Hyde, Kim and Liu, and combines some new ideas in expansion and switching cycle lengths.

Linear inverse problems are prevalent in geophysics, signal processing, image restoration, and medical imaging. Mathematically, they can be formulated as $Gm = d$. When $G$ is ill-conditioned or singular, the problem becomes ill-posed and requires regularization methods or generalized inverse methods for a stable solution. However, both types of methods encounter significant computational difficulties for large-scale linear inverse problems. In this paper, we establish a spectral-domain pseudo-inverse construction method for a class of linear inverse problems that can be diagonalized by unitary matrices. The core idea is to construct a stable pseudo-inverse operator directly in the transform domain, starting from the unitary diagonalization structure of the matrix. We first provide the analytic Singular Value Decomposition (SVD) of this class of matrices, clarifying the correspondence between spectral decomposition and SVD. On this basis, we define spectral-domain regularization filtering factors, construct a stable spectral-domain pseudo-inverse operator, and prove its bounded stability as well as its consistency in converging to the Moore–Penrose generalized inverse. This construction is numerically equivalent to zeroth-order Tikhonov regularization and converges to the Moore–Penrose generalized inverse as $\alpha \to 0^+$, but its methodological path differs from both. This paper reveals that the stable generalized inverse of a class of structured matrices can be directly constructed from their spectral decomposition, providing an efficient solution method for large-scale structured inverse problems. The Fourier transform case is a special instance of this method when the unitary matrix is taken as the discrete Fourier transform matrix.

Magnitude, denoted by $\operatorname{Mag}(X)$, is a real-valued invariant of compact metric spaces whose large-scale growth reflects their geometry. Willerton showed that, for a compact homogeneous Riemannian manifold $X$, $\operatorname{Mag}(tX)$ grows like $t^{\dim X}$, with the volume of $X$ appearing in its leading asymptotic terms. We study a homogeneous Moran Cantor set $E$ equipped with the Euclidean metric $d$ and with its coding ultrametric $d_u$, writing $E_u=(E,d_u)$. We prove that the upper and lower growth exponents of $\operatorname{Mag}(tE_u)$, called the magnitude dimensions of $E_u$, coincide respectively with the upper and lower Euclidean box dimensions of $E$. In the self-similar case with constant contraction ratio $r$, we obtain $\operatorname{Mag}(tE_u)=t^s/\widetilde{p}(\log t)+o(t^s)$ as $t\to\infty$, where $s$ is the Hausdorff dimension of $E$ and $\widetilde{p}$ is a positive smooth function of period $-\log r$. The harmonic mean of the leading coefficient $1/\widetilde{p}$ is $m\log m/((m-1)\Gamma(s+1))$, giving a fractal analogue of Willerton's leading-order asymptotics with a log-periodic, rather than constant, coefficient.

We study the inradius and inverse inradius of the convex hull of standard planar Brownian motion. Our main goals are to improve the existing bounds on the expected values of these quantities and to establish variance bounds. The main contributions of the paper, however, are the following intermediate results we obtain in order to compute our bounds. We find closed-form expressions for the expected values of the minimum and maximum of the ranges of two independent one-dimensional Brownian motions. Besides improving the upper bound on the expected inradius, this result also gives an explicit expression for a quantity that has been investigated repeatedly in the literature and for which only a numerical evaluation has been known so far. Furthermore, we study the exit time of planar Brownian motion from an unbounded region that we call the hourglass domain and compute its first two moments. This result leads to a nearly fivefold improvement of the previous upper bound on the expected inverse inradius and plays an essential role in establishing the corresponding upper bound on the variance.

In 1978 Babai raised the question whether all semiminimal Cayley graphs have bounded chromatic number. In this paper we show that semiminimal Cayley graphs of abelian and generalized dihedral groups have circular chromatic number at most 4, thus extending a result of Barajas and Serra.