Papers

In the article the $mm$-entropy (an entropy of a metric measure space) introduced by C. Shannon is evaluated for an $\alpha$-stable Lévy process. For $\alpha<1$ the double-sided estimates of the same order are obtained for process distribution in Skorokhod space.

We establish a Liouville type theorem for fully nonlinear uniformly elliptic equations in exterior domains in half spaces under quadratic boundary data and a quadratic growth condition, that is, any viscosity solution tends to a quadratic polynomial plus lower order terms at infinity with the rate at least $|x|^{-\alpha-n}$ for any $\alpha\in(0,1)$. As applications, this result can lead to Liouville type theorems for Monge-Amp`ere equations, k-Hessian equations and special Lagrangian equations with critical and supercritical phases.

Network operation relies on heuristics to solve many tasks rapidly and efficiently across the protocol stack. These heuristics are the result of thorough human-driven design rooted in expert knowledge of the target system and problem. Recently, approaches powered by Artificial Intelligence have shown promising results in devising solutions that outperform long-established heuristics in classical problems. We explore the possibility of applying such Automated Heuristic Design (AHD) frameworks to network environments by (i) discussing the general integration of AHD with network operation and the associated challenges, as well as (ii) proposing a practical implementation of AHD for a specific networking task, i.e., 5G decoding. Initial results show how modern AHD tools can devise heuristics for Low-Density Parity Check decoding on par with state-of-the-art solutions implemented in production systems.

Let $Q_k(x)$ be Stanley's explicit denominator for the dimer-covering generating function $F_k(x)=\sum_{n\ge0}A_{k,n}x^n$ of $k\times n$ rectangles. Stanley conjectured in 1985 that $Q_k(x)$ has only simple roots; this longstanding conjecture was recently recorded in Lai's list of open problems on tilings (see [6, Problem 33]). We disprove the conjecture by proving that $Q_{14h-1}(x)$ and $Q_{30h-1}(x)$ have repeated roots for every $h\ge1$; in particular, $k=13$ is the smallest counterexample. The construction comes from two exceptional multiplicative identities among trigonometric algebraic units. We further propose a conjecture concerning this class of trigonometric identities, which appears to be related to Robinson's problem on primitive Pell factors.

We establish scaling limit results for fluid dynamics equations driven by pseudo-transport noise. The behaviour of noise at small scales is governed by a parameter a. This extends previous results by Flandoli and Luo (2020) and Galeati (2020), which correspond to a=0 in our setting. Depending on the value of a, we prove that the noise delays the potential blow-up of both the 3D Euler and Navier-Stokes (NS) equations with high probability.

We design observer-based controllers to stabilise abstract linear boundary control systems on Hilbert spaces. Our main results introduce conditions for exponential, strong, and polynomial stability, and establish external well-posedness of the closed-loop system. We design controllers for a one-dimensional wave equation, a two-dimensional wave equation with distributed control and observation, and a non-uniform SCOLE model.

The Abu-Khzam–Langston conjecture, that is the weak-immersion analogue of Hadwiger's conjecture and a weak version of an earlier conjecture of Lescure and Meyniel, asserts that every graph $G$ contains a weak immersion of $K_{\chi(G)}$. We prove the conjecture for the class of graphs of independence number two. Along the way, we introduce a notion of cycle-matching colouring of a graph, a relaxation of edge-colouring in which colour classes induce vertex-disjoint unions of edges and odd cycles, and prove a sharpening of Vizing's theorem in this setting: every multigraph admits a cycle-matching colouring with at most $\Delta(G)$ colours.

We present the first neural probabilistic amplitude shaping that outperforms existing methods while accounting for all implementation losses, using a block-less, easily implementable sequential autoregressive encoder compatible with arithmetic distribution matching, yielding reduced rate loss and higher achievable information rates.

We prove singularity criteria for the $t$-K-stability of adjoint foliated structures. We first show that K-semistability of adjoint foliated structures implies log canonicity by extending Odaka's flag ideal characterisation of the mixed Donaldson–Futaki invariant to the adjoint foliated setting. We then prove that adjoint Calabi–Yau foliated structures are K-semistable, and klt ones are K-stable, while log canonical adjoint general type foliated structures are K-stable with respect to the canonical polarisation. We also show that K-semistable adjoint Fano foliated structures are klt. In particular, their ambient varieties are potentially klt and of Fano type.

Optimal transport is widely used to learn distributions, enforce distributional constraints, and model uncertainty. In applications, transport losses are often computed from samples through tractable representations, such as one-dimensional sorting formulas or sliced Wasserstein costs, making them practical components in training pipelines. We study parameterized objectives defined by sampled transport costs and prove graphical convergence of their subdifferentials to the subdifferential of the population objective. In particular, this ensures that standard subgradient methods consistently approach stationary points of the population-level problem. We illustrate the results in several settings, including risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems. Our analysis highlights that smooth parameterizations provide a favorable interface between statistical consistency and optimization. By contrast, transport objectives with nonsmooth costs and models may exhibit unstable derivatives in the large-sample limit.

In this paper, we introduce the concept of graded m-nil clean ring to extend the existing notion of graded nil-clean ring introduced in [10]. We explore fundamental properties of these rings, emphasizing the interplay between the identity component and the graded structure. We investigate certain conditions under which graded group rings, graded matrix rings and graded amalgamated algebras inherit the m-nil clean property from their components. Specifically, we establish sufficient conditions for graded group rings and graded matrix rings over commutative m-nil clean rings to be graded m-nil clean rings.

K. S. S. Nambooripad introduced an interesting class of categories known as normal categories, which are categories with subobjects, morphisms admitting factorization and having sufficiently many cones. These normal categories plays fundamental role in the study of structure of regular semigroups. In [6] we discussed the category of chain bundles and category of chains. In the present paper revisits classical notions of bundles, including fibre bundles, vector bundles, and principal G-bundles, and discuss the chain categories arising from the category of bundles. Moreover, its is verified that these categories are categories with subobjects.

We investigate the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces. We show that the covariance operator of a weak second-order Radon probability measure on such a space is uniquely determined by a symmetric positive-definite function. In the Gaussian case, we characterize those functions that arise from covariance operators in terms of $\gamma$-radonifying operators. Building on these results, we extend Driscoll's theorem to the reproducing kernel Banach space setting.

Let $r_1$ and $r_2$ be positive integers with $r_1 \le r_2$. A graph $G$ is called $2$-DCC vertex $[r_1,r_2]$-pancyclic if, for any two distinct vertices of $G$ and any integer $\ell \in [r_1,r_2]$, there exist two vertex-disjoint cycles of lengths $\ell$ and $|V(G)|-\ell$, respectively, containing the two vertices separately. In this paper, we investigate the two-disjoint-cycle-cover vertex pancyclicity of the split-star network $S_n^2$. We prove that $S_n^2$ is $2$-DCC vertex $[3,n!/2]$-pancyclic for $n\ge4$.

Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in the unit disc $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. For $0<p<\infty$ and $f\in\mathcal{H}(\mathbb{D})$, let $M_p^p(r,f)=\int_0^{2\pi}|f(re^{i\theta})|^p \frac{d\theta}{2\pi}$ and $M_\infty(r,f) = \sup_{|z|=r}|f(z)|$. For $0<p<q\leq \infty$, Hardy and Littlewood proved the prevalent inequality $$M_q(r,f)\le C(p,q)\frac{M_p(\rho,f)}{(\rho-r)^{\frac{1}{p}-\frac{1}{q}}}$$ for $0\leq r<\rho\leq 1$ and $f\in\mathcal{H}(\mathbb{D})$. In this paper, we obtain an improvement of this well-known inequality which is employed to characterize the symbols $g\in\mathcal{H}(\mathbb{D})$ such that the analytic paraproducts $T_gf(z)=\int_0^z f(\zeta)g'(\zeta)\,d\zeta$, $S_gf(z)=\int_0^z f'(\zeta)g(\zeta)\,d\zeta$ and $M_gf(z)=f(z)g(z)$, are bounded between two different mixed-norm spaces $A^{p,q}_\omega=\{ g\in \mathcal{H}(\mathbb{D}): \int_0^1 M_p^q(r,g) \omega(r)\,dr<\infty\}$ induced by a radial doubling weight $\omega$. En route to the proof of these characterizations, we consider an open Carleson measure problem posed by Luecking and we solve it in a meaningful particular case.

Many prediction tasks in computational science and engineering become one-to-many after coarse graining and partial observation. In such settings, deterministic surrogates trained by squared loss may learn a well-defined mathematical object, the conditional mean, while still missing the task-relevant variability in the underlying conditional law. In this work, we formulate this limitation as the conditional-mean barrier and develop a diagnostic framework for identifying it in fitted scientific machine-learning surrogates. The framework combines residual-feature orthogonality and effect-size diagnostics to distinguish deterministic underfitting from irreducible conditional variability. We also make explicit a simple consequence of paired squared loss: stochastic outputs do not by themselves overcome the barrier, because the objective penalizes model variance and drives the predictor back to the conditional mean. The diagnosis therefore yields a modeling prescription: when residual variability matters, the loss must score richer features of the conditional law rather than a point prediction. Reproducible numerical studies on a controlled two-branch law and a two-scale Lorenz-96 closure problem show how the diagnostic identifies the barrier, how deterministic closures can suppress collective fluctuation statistics in rollout, and how a minimal likelihood-based stochastic-scale model can recover substantially more variability.

We define almost affine vector rank-metric codes as subsets $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ whose canonical projections have cardinalities that are powers of $q^m$, and prove that they naturally induce $q$-matroids. We establish that the operations of puncturing and shortening correspond to restriction and contraction of the $q$-matroid, and show that the rank-weight and formal dual distance distributions are determined by the induced $q$-matroid. We briefly discuss applications to perfect $q$-matroid ports in linear network coding, and show that disconnected $q$-matroids need not induce disconnected ports. Finally, we show that certain Additive Generalized Twisted Gabidulin codes yield direct examples of strictly almost affine rank-metric codes, alongside a separate construction derived from proper finite semifields.

The parareal algorithm is a powerful parallel-in-time integration method that accelerates the numerical solution of evolution equations by iteratively combining a fine propagator and a coarse propagator. Although the convergence of the parareal algorithm has been extensively studied, most existing analyses assume that the fine propagator is either an exact solver or a single-step method. In this paper, we construct and analyze a parareal algorithm for solving parabolic equations, where the fine propagator is based on the two-step backward differentiation formula (BDF2), while the coarse propagator remains a single-step method. We propose a novel approach to design an effective correction for the initialization steps and establish linear convergence of the iteration. Numerical results fully support the theoretical findings, show clear improvements over existing multistep parareal strategies, and indicate that the proposed approach extends effectively to higher-order BDF methods and to nonlinear problems.

We analyze a stochastic 5-neighbor cellular automaton with several conserved quantities, including the particle density. By examining the eigenvalue problem of the associated transition matrix, we derive an explicit formula for the stationary distribution on each irreducible component, in which the weight of each configuration is expressed in terms of the numbers of occurrences of two specific local patterns. This analysis further allows us to theoretically derive the dependence of the mean flux on the conserved quantities. In particular, we recover the mean flux formula in the deterministic case by taking the zero-noise limit of the system.

We prove Hadwiger's Conjecture for $\{\text{co-claw}, \text{co-gem}\}$-free graphs and $\{\text{fork}, \text{antifork}\}$-free graphs, where the co-claw is the disjoint union of a triangle and a vertex, the co-gem is the disjoint union of a 4-vertex path and a vertex, the fork is obtained from $K_{1,3}$ by subdividing one of the edges, and the antifork is the complement of the fork. The $\{\text{co-claw}, \text{co-gem}\}$-free graphs include the complements of line graphs of triangle-free multigraphs, and thus our results imply Hadwiger's Conjecture for these graphs. In fact, we prove a stronger result: every $\{\text{co-claw}, \text{co-gem}\}$-free graph $G$ has a $K_{\chi(G)}$-model where each branch set has size at most 2, and every $\{\text{fork}, \text{antifork}\}$-free graph $G$ has a $K_{\chi(G)}$-model where at most one branch set has size greater than 2.

The Virtual Element Method (VEM) is a well-established framework for solving partial differential equations on polygonal and polyhedral meshes. In this paper, we introduce a novel hybrid VEM that integrates both conforming and nonconforming virtual spaces. We apply this formulation to a three-dimensional linear elasticity problem, providing rigorous theoretical analysis to demonstrate optimal convergence rates. Furthermore, we explore the extension of this approach to domains with curved boundaries.

In this article, we show the surjectivity of word maps w from SU(2)* SU(2) to SU(2) induced by several families of words in the free group of rank 2. Also, we prove the surjectivity of certain word maps on SL(2,C).

In this thesis we study embedding formalisms for $\mathcal{N}$-extended anti-de Sitter (AdS) superspaces in four and five dimensions. Specifically, building on earlier work in the four-dimensional case, we develop bi-supertwistor realisations for these AdS superspaces as well as their harmonic and projective extensions. We also describe the precise correspondence between such global approaches to AdS superspaces and their realisations within the supergravity setting. Finally, we present applications of our formalism, including new models for superparticles propagating in AdS superspaces in diverse dimensions.

A defining set of a Latin square is a partially filled-in Latin square which completes to no other Latin square of the same order. We introduce the concept of a $k$-strong defining set, in which if less than $k$ entries are deleted, the property of being a defining set is retained. Equivalently, a $k$-strong defining set intersects every Latin trade in the Latin square at least $k$ times. In the addition table for integers modulo $n$, when $n$ is even we determine the minimum size of a $k$-strong defining set for any $k$. For odd $n$ we give a construction for a minimally $2$-strong defining set. We furthermore give computational results for Latin squares of small orders.

In this paper, we characterize three classes of smooth projective surfaces: relatively minimal elliptic surfaces with $\chi\ge 1$, rational surfaces $X$ satisfying that $-K_X$ iis nef and $\kappa(-K_X)\ge 1$,, and projective K3 surfaces on which every integral pseudoeffective divisor admits an integral Zariski decomposition.

We prove two positivity conjectures proposed by Guo for alternating sums and factorial ratios built from Gaussian coefficients. The first result proves the positivity of the odd $q$-super Catalan numbers \[ C_{m,n}(q)=\frac{[2m+1]![2n]!}{[m+n+1]![m]![n]!}. \] The proof uses the positivity theorem of Warnaar and Zudilin for the usual $q$-super Catalan numbers, together with two recurrences obtained from a double application of the $q$-Chu–Vandermonde summation. The second result proves Guo's conjectural strengthening of his alternating-sum positivity theorem, replacing the exponent coefficient $2r-1$ by every odd coefficient $2b-1$, $1\leq b\leq r$. Its proof combines a $q\mapsto q^{-1}$ reciprocity with a finite deletion recurrence.

Although irregular vectors for the Virasoro algebra are widely used in modern mathematical physics, a rigorous existence and uniqueness theorem in arbitrary rank has not been available in the literature. In this paper, we develop an algebraic framework, based on Virasoro differential operators on the parameter space, which gives such a theorem for arbitrary integer and half-integer ranks. A key ingredient is the construction of a canonical operator \(L_*\) from the coefficient matrix of the vector-field part of a truncated Virasoro realization. This operator closes the recursive system by isolating the derivative with respect to the highest irregular parameter. Using this mechanism, we prove the existence and uniqueness of formal irregular vectors of arbitrary integer rank. We then construct the truncated Virasoro vector fields required in the half-integer-rank setting and prove the existence and uniqueness of the corresponding half-integer-rank formal irregular vectors. We also prove that, after a scalar gauge normalization, the canonical solutions satisfy the full lower Virasoro deformation equations. These results provide an algebraic foundation for the rigorous construction of irregular conformal blocks built from higher-rank irregular vectors. After passing to eigenvalue coordinates, the vector-field part of the half-integer construction is identified with the differential realizations appearing in the literature, while the zeroth-order terms are explained by scalar gauge freedom.

We study the asymptotic dynamics of multi-bubble solutions to the focusing energy-critical wave equation in five dimensions. Assuming that the solution asymptotically decomposes into a finite superposition of spatially separated bubbles with comparable scales, we prove a rigidity result that describes the precise long-time behavior of these scales. More precisely, we show that all scaling parameters are necessarily of order $t^{-2}$, and that the corresponding renormalized modulation vector converges to a connected component of a finite-dimensional algebraic set determined by the limiting spatial configuration of the bubbles. This algebraic system encodes the strong interactions between the polynomial tails of the bubbles and governs the effective asymptotic dynamics of the multi-bubble regime.

A perturbed family of interval exchange maps (FIEMs) provides a natural two-\linebreak{}dimensional area-preserving extension of interval exchange maps, with each IEM parameterized by an action variable $y$. Such families arise, for example, as models for iso-energy return maps of perturbed pseudointegrable Hamiltonian impact systems. These maps inherit a time-reversal symmetry, motivating the study of symmetric FIEMs. In the unperturbed case, the dynamics are generically uniquely ergodic for almost every value of $y$, while a dense set of action values supports periodic intervals. Exploiting time-reversal symmetry, we characterize these intervals and show that symmetric periodic orbits correspond to their midpoints. Under perturbation, the action variable is no longer conserved and generically periodic intervals break into isolated elliptic or hyperbolic periodic orbits. For sufficiently small perturbations, symmetric periodic orbits persist and can be located by a one-dimensional search along symmetry lines. Associated bifurcations generating symmetric and asymmetric periodic orbits are described and connected to those of the standard map, viewed here as a perturbed family of two-interval exchange maps.

In online convex optimization (OCO), a decision-maker is confronted with an unknown environment and seeks to play an optimal sequence of decisions on a short time-scale using only past information. Recent advances in second-order OCO methods have demonstrated tighter regret bounds and improved empirical performance over traditional first-order methods. However, this performance comes at a cost: a matrix inversion is now required, which scales with the cube of the size of the problem. In this work, we propose sketching to mitigate this limitation. Specifically, we present the online sketched Newton-Raphson method (OSNR) which preserves the tight regret bounds obtained with second-order methods while presenting a strict computational improvement in terms of complexity. We discuss three application scenarios of OSNR: online root finding, unconstrained OCO, and time-varying equality-constrained OCO, and present their respective regret and a constraint violation bound for the latter. In all three applications, OSNR achieves sublinear dynamic regret bounds. For the equality-constrained case, the extension OSNR with equality constraints OSNR-EC is shown to yield sublinear cumulative constraint violation. Finally, we illustrate the performance of OSNR and OSNR-EC on two numerical examples, viz., online position tracking and optimal power flow, and observe that OSNR and OSNR-EC exhibit high performance even at low sampling rates.