Papers

We study an extension of Ramanujan's identities for odd zeta values by Lim and introduce Jacobi analogues of classical Eichler integrals of Eisenstein series. In negative weight we construct explicit completions and embed these objects into a modular framework by showing that they are (singular) harmonic Maass–Jacobi forms. We further describe their non-holomorphic parts in terms of Eichler integrals, establish Ramanujan-type inversion formulas, and study their behavior under the Maass raising and lowering operators and at torsion points.

Three important problems in Fourier analysis are the Fourier restriction problem, the $L^p$-improving problem, and the Fourier decay problem. Positive results for any of these problems require a quantitative understanding of various geometric properties of the given measure, including curvature and arithmetic resonance. In this paper we establish a unified framework for providing negative results for all three problems (that is, we provide explicit obstructions to a measure satisfying certain Fourier restriction, $L^p$-improving, or Fourier decay estimates) by quantifying flat parts of the measure in the spirit of the well-known Knapp examples from harmonic analysis. Our main interest is in the application of these abstract results in various concrete settings where we use analytic and fractal geometric concepts to force `flatness'. Our framework applies generally and this allows us to unify and extend various parts of the literature. Some representative applications include: (i) we bound the Fourier dimension of the surface measure on a compact $C^2$ surface of arbitrary dimension above by the smallest ambient rank of a point on the surface; (ii) we prove that the Fourier dimension of a smooth curve in $\mathbb{R}^d$ is at most $4/(d+1)$ and so such curves cannot be Salem for $d \geq 4$ with analogous results for higher dimensional submanifolds; (iii) we obtain explicit upper bounds for the Fourier dimension of the Patterson-Sullivan measure for parabolic Kleinian group actions, as well as ergodic measures on self-affine sets; (iv) we establish novel connections between Fourier restriction/decay and a priori unrelated concepts in fractal geometry including the Assouad spectrum of projections and slices, and a strong form of tube-nullity. We establish several auxiliary results along the way, including a precise characterisation of L^2-flattening in terms of the Fourier spectrum.

We prove two new results about probability preserving actions $T:\mathbb{Z}^d \curvearrowright (X,\mu)$. First, for a function $f\in L^2(\mu)$, we provide an explicit formula for the $L^2(\mu)$-limit of the average \[\frac{1}{|Q_N^\mathcal{P}|}\sum_{v \in Q_N^\mathcal{P}} T_v f\] where $\mathcal{P}\subset \mathbb{Z}^d$ is the set of primitive vectors, i.e. those for which the greatest common divisor of its components is $1$, and $Q_N^\mathcal{P}= [-N,N]^d\cap \mathcal{P}$. Second, for a set $A\subset X$ with $\mu(A)>0$, we provide a spectral condition under which the set of $\varepsilon$-expansive directions \[\left\{ v\in \mathbb{Z}^d \, : \, \mu\left(\bigcup_{n\in \mathbb{Z}} T_{nv}A\right)>1-\varepsilon\right\}\] has lower density very close to $1$. As an application of our techniques we are also able to prove a quantitative variant of a twisted multiple recurrence theorem of Björklund, Fish and the first author (arXiv:2503.02501).

We study structure within the homotopy groups of the Davis-Januszkiewicz space DJ(K) associated with a simplicial complex K. The inclusion of each vertex in K induces a map from the two-sphere into DJ(K). These maps generate a quasi-Lie subalgebra QL(K) via the Whitehead product and a Pi-subalgebra S(K) via the Whitehead product and composition. We describe the quasi-Lie subalgebra QL(K), and show that the Pi-subalgebra S(K) coincides with the whole of the homotopy groups of DJ(K) if and only if K is a flag complex. Extensions to more general polyhedral products are also considered.

Working over a (left) coherent ring, we consider the class of weakly Ding injective complexes. These are the cycles of the exact complexes of FP-injective complexes that stay exact when applying $\Hom(A,-)$ for any FP-injective complex $A$. We study the cotorsion pair generated by the class of all such complexes, and exhibit it as part of an abelian model structure. As an application we show that when $R$ is a Ding-Chen ring, its stable chain complex category is compactly generated and triangle equivalent to the stable category of four Frobenius categories. They are the categories of all (i) complexes of Gorenstein injective modules, (ii) complexes of Gorenstein projective modules, (iii) complexes of Gorenstein flat-cotorsion modules, and (iv) complexes of Gorenstein FP-pro-injective modules.

Given a square matrix, the permanent is a determinant-like function without signs. In this paper, we study inequalities involving permanents of certain submatrices. We first focus on a family of zero-one totally nonnegative matrices that arise as anti-adjacency matrices of unit interval orders. For these matrices, we prove a collection of inequalities comparing products of permanents of consecutive principal submatrices with products of permanents of parity-selected principal submatrices. We also study a related combinatorial problem involving two families of permutations: a Young subgroup and a set of parity alternating permutations. We construct a bijective map from the first family to the second family, and verify computationally for n at most 13 that each permutation is below its image in Bruhat order. We conjecture that this property holds for all n. If true, this would imply one of the main permanent inequalities for all totally nonnegative matrices in the balanced case. More broadly, we conjecture that the full family of inequalities holds for all totally nonnegative matrices.

We prove that the reversible elementary second order cellular automaton rule 115 is periodic when started on finite initial configurations. We also study some families of finite configurations that have interesting period functions.

The uniform generation of k Hamming weight binary words, equivalent to sampling k-subsets from n elements, relies on random bits, which can be expensive. We introduce a novel paradigm, random bit recycling, and use it to generate such binary words in linear time while consuming as few random bits as possible. The resulting algorithm is nearly optimal in terms of random bit consumption, meaning that it closely matches the Shannon entropic lower bound coming from information theory.

We present an algorithm for computing the join of two elements in the weak order of the Coxeter group of type B. This extends Markowsky's algorithm for computing joins of standard permutations to signed permutations, and allows us to confirm a conjecture of Dyer concerning a geometric interpretation of these joins.

A symmetric function is called Schur-positive if it admits an expansion in the Schur basis with nonnegative coefficients. In this paper, we study the Schur-positivity of symmetric functions naturally associated with set partitions, with respect to a descent set function that considers i as descent, if i and i+1 share a block in the partition. The Schur expansion involves hook-shaped Young diagrams, and the corresponding coefficients are given by Touchard-Riordan polynomials, which enumerate matchings by their number of crossings.

We prove that the interface separating $+1$ and $-1$ spins in the near-critical planar random field Ising model (RFIM) with Dobrushin boundary conditions has a scaling limit, whose law is conformally covariant and almost surely absolutely continuous with respect to SLE$_3$. The limiting curve can be seen as a massive version of SLE$_3$ in the sense of Makarov and Smirnov, but in a random environment. We then show that the outermost spin loops of the near-critical planar RFIM with $+1$ boundary conditions have subsequential limits and that any of these limits is almost surely singular with respect to CLE$_3$. This dichotomy between absolute continuity of the single interface and singularity of the outermost loops reflects the fact that a single interface does not explore enough of the magnetization field of the near-critical RFIM to detect the singularity of this field with respect to the critical Ising magnetization field, whereas the outermost spin loops do.

The article is devoted to the problem of decay of initial stationary state for the Volterra lattice with zero boundary condition. We show that this process is asymptotically self-similar and calculate the propagation velocity of the decay wave, the leading terms of the asymptotics and corrections, in the main and transition sectors of the wave.

In this paper, we study the dynamical Prandtl equation, which plays an important role in the study of the vanishing viscosity limit of the Navier–Stokes equations. Our focus is on the (Sobolev) well-posedness regime, where the given data satisfy a crucial monotonicity condition. In this case, local classical solutions have been constructed in the pioneering works of Oleinik \cite{O68,OS99}. More recently, global weak solutions were obtained in \cite{XZ04} by Xin and Zhang, and in \cite{XZZ24} by Xin, Zhang, and Zhao, where the uniqueness and interior H"older estimates of the solutions were established (in Crocco coordinates). Using a precise description of the fundamental solution to the Kolmogorov equation in the half-space, we first obtain the H"older regularity of local weak solutions up-to-boundary. We also provide a detailed proof of higher-order regularity estimates together with $W^{2,p}$ Sobolev estimates and $H^{s}$ hypoelliptic estimates, which are nontrivial. Up-to-boundary smoothness of solutions (in Crocco coordinates) is important in order to conclude the smoothness of the Prandtl solutions in the physical variables, even in the interior. It is also physically significant for applications to the Boundary Layer Theory where the dynamical Prandtl equation is essential. Using these smoothing estimates, we then prove the global existence and regularity of classical solutions to the dynamical Prandtl equation under monotonicity assumptions, which was listed by Oleinik and Samokhin in \cite{OS99} as one of the open problems. We also develop a self-contained local existence theory using weighted energy estimates and further expand the theory of global weak solutions. The main point is to incorporate all physical types of asymptotic matching of the boundary layer with the outer flow, which is expected to be useful for applications to the Navier–Stokes equations.

Let $\mathcal{H}$ be a family of graphs. A graph is said to be $\mathcal{H}$-free if it contains no subgraph isomorphic to a graph in $\mathcal{H}$. The planar Turán number $ex_{_\mathcal{P}}(n,\mathcal{H})$ is defined as the maximum number of edges in an $\mathcal{H}$-free planar graph on $n$ vertices. In this paper, we determine the exact value of $ex_{_\mathcal{P}}(n,\{K_{4}, \Theta_{6}^{1}\})$ and a tight upper bound of $ex_{_\mathcal{P}}(n,\{K_{4}, \Theta_{6}^{2}\})$.

We show that if a graded monotone Lagrangian $L_0$ has a non-vanishing disk potential, then for every smooth isotopy $\{L_s\}_{s\in[0,1]}$ of Lagrangians starting from it and for every tame almost complex structure $J$, each $L_s$ bounds a $J$-holomorphic disk of Maslov index two. The main input is a non-archimedean analytic potential function, defined as an invariant up to analytic isomorphisms, generalizing the classical disk potential of a monotone Lagrangian. The techniques are inspired by recent developments in the Strominger-Yau-Zaslow mirror construction via family Floer theory and non-archimedean geometry. We also discuss applications such as recovering a simple case of Audin's conjecture.

Let $\Phi $ be a univalent function in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$% , $\Phi (\mathbb{D})$ is symmetric with respect to the real axis, starlike with respect to $\Phi (0)=1$, and $\Phi ^{\prime }(0)>0$. Let $\mathcal{C}% (\Phi )$ denote the class of Ma-Minda convex functions. In this article, we present the bounds on $||a_{3}|-|a_{2}||$ for Taylor's coefficients of the function $f$ in the class $\mathcal{C}(\Phi )$. We also establish the same bounds for the inverse coefficients. All the bounds we study here are sharp. We also present the conditions such that the bounds on $|a_{3}|-|a_{2}||$ and $|A_{3}|-|A_{2}||$ are invariant, where $A_{2}$ and $A_{3}$ are the first two coefficients of the Taylor series of the inverse functions of $f\in \mathcal{C}(\Phi ).$ Thus provides examples of invariance and nonvariance among the subclasses of convex functions.

Lagrangian sensors have shown promise to improve operator awareness of conditions inside a chemical reactor but three-dimensional tracking remains a mostly unsolved challenge. We explore a setup where in-silico sensors, based on a recently proposed real-world design, are tracked using data from an accelerometer and magnetometer available from a built-in inertial measurement unit. Filtering algorithms, using a bespoke dynamical model, are used to process these readings into position estimates. We compare tracking performance of an extended Kalman filter, a particle filter and the unscented Kalman filter implemented in the pykalman library. Our numerical experiments track in-silico particles moving in an analytically given three dimensional vortex as well as in the experimentally measured flow-field of a lab-scale stirred tank reactor. Using the Maxey-Riley-Gatignol equations for the movement of inertial particles as ground-truth, we demonstrate that trajectories can be reconstructed from noisy synthetic data with errors below 10%.

Scale buys interpolation; structure buys certifiable transfer. A world model's average error does not say whether a particular rollout can be trusted, or for how long. For equivariant latent world models we give a predictability certificate: a computable region spanning configuration, horizon, and resolution. Under exact equivariance, rollout error is invariant over the monoid generated by k primitive symmetries and is certified from the k generators (Theorem A); universal orbit-flatness over equivariant targets characterizes equivariance at the function level (Lemma 2), so an unconstrained architecture cannot certify the property by construction. Approximate orbit-transfer defects propagate by the finite-time Lyapunov spectrum (Theorem B): expanding channels give a logarithmic horizon $T_j(\epsilon)\sim\log(1/\epsilon)/\lambda_j$, neutral channels accumulate recurrent defect linearly, and contracting channels accumulate a bounded nonzero floor. Exact conserved charge values are certified to all horizons only at zero defect; with one-step defect $\eta$, charge-value error grows at most as $T\eta$. Empirically, on a 40-dimensional learned model a $\mathbb{Z}_N$-equivariant network recovers the full Lyapunov spectrum ($R^2=0.98$-$0.99$) where dense and recurrent baselines fail. A cone/adapted-metric certificate reads an a-priori horizon off the model's own Jacobian, tight on uniformly hyperbolic dynamics and self-abstaining elsewhere; the resulting horizon improves a budgeted re-observation decision. For public non-equivariant world models the tangent spectrum gives a training-free candidate horizon, paired with a held-out divergence cross-check that abstains or corrects when the learned loop over-promises.

This paper studies feedback Nash equilibria (FNE) in scalar discounted linear quadratic (LQ) games with $N$ players. By explicitly incorporating the discount factor, we show that finite-cost equilibria may fail to stabilize the original system, motivating a distinction between FNE and stable FNE together with a sufficient stability condition. Based on a parametric characterization of the policies, we propose numerical methods for computing all equilibria. Particular attention is devoted to the symmetric game, where a closed-form expression of the symmetric FNE and conditions for the existence of up to $M\leq2^N-2$ equilibria are derived. Numerical experiments illustrate how equilibrium multiplicity depends on the game configuration and highlight the emergence of finite-cost non-stabilizing equilibria.

This article investigates stochastic processes taking values in metric spaces that lack a topological vector space structure, a regime characterized by intricate interplay between topological, geometric, and temporal dependence structures. It is formally established that spaces admitting an isometric Hilbertian embedding constitute a strict subclass within the much broader class of metric spaces possessing the ball property. While traditional kernel methods are susceptible to geometric distortion when the underlying space cannot be isometrically embedded into a Hilbert space, we bypass such limitations by exploiting a fundamental structural property inherent to this broader class; namely, that Borel probability measures are uniquely determined by their values on balls. These separating classes provide the foundation for the subsequently introduced measure-theoretic inference methodology. We derive uniform convergence of a family of time-dependent random measures, alongside weak invariance principles for the corresponding nonstationary random fields. This framework explicitly exposes how dependence and geometric complexity influence sample path regularity. Furthermore, because the rapid decay of small-ball probabilities can prohibit the existence of limiting distributions for supremum-based discrepancy measures, we develop $L^p$-based alternatives. By directly leveraging the introduced convergence results, this approach circumvents the need for higher-order $U$-process formulations. Finally, for spaces that do admit an isometric Hilbertian embedding, and where $U$-processes naturally arise, we establish limit theory for both degenerate and nondegenerate multi-parameter $U$-processes, and demonstrate that local discrepancy tests maintain asymptotic stability under dynamic parameter regimes.

We prove the existence of Long-Range Order in a class of two-dimensional Euclidean lattice Gross-Neveu models with an even number of fermion flavors, covering three standard lattice discretizations, including naive and staggered fermions widely used in numerical studies. By performing a Hubbard-Stratonovich transformation, we map the fermionic systems to bosonic ones and establish Reflection Positivity for the resulting measures. Exploiting this structure, we combine Chessboard Estimates with a Peierls-type contour argument to prove Long-Range Order for the chirally charged fermion-mass bilinear $\overline{\psi}\psi$ at sufficiently small coupling and sufficiently large flavor number. Our analysis is robust with respect to the choice of lattice discretization and applies uniformly across different realizations of the same underlying continuum model. Moreover, we obtain uniform pointwise bounds on the bosonic two-point function, equivalently on the fermionic mass-mass correlator, showing that it is quantitatively controlled by the minimizers of the effective potential. This provides a fully rigorous and non-perturbative demonstration of Long-Range Order in lattice Gross-Neveu models and establishes a direct connection between the rigorous theory and its large-$N$ (mean-field) predictions.

Rigidity, stability and local minimizing properties of Einstein metrics as critical points of quadratic Riemannian functionals defined by $L^2$-norms of Ricci curvature, scalar curvature, Weyl curvature and Riemannian curvature have been extensively studied. However, there are non-Einstein critical points of these functionals that are not so well understood. In this paper, we study Ricci solitons, a generalization of Einstein metrics, that are critical points of a special quadratic curvature functional and analyze their rigidity.

We consider the uniqueness of the solution of the dissipative surface quasi-geostrophic equation, without assuming time-continuity and smallness of the solutions. We show that the uniqueness holds in the scale-critical Lebesgue spaces and non-homogeneous Besov spaces. The proof is based on the energy method, inspired by the approach introduced by Lions and Masmoudi (2001) in the study of uniqueness for the Navier-Stokes equations. A key ingredient of the argument is the justification of the energy inequality via the smoothing effect of the fractional heat semigroup together with an iteration scheme based on the structure of the integral equation.

Can a classically relevant stress-tensor deformation be exactly solvable, duality preserving, and physically causal? We construct an exact power-law family of nonlinear electrodynamics preserving electromagnetic duality, together with a parallel two-dimensional Lax-integrable realization. Its auxiliary geometry yields the full characteristic-cone phase diagram and a universal finite-energy fold. For the Maxwell seed, every nonzero relevant branch is acausal, whereas every causal branch is caustic-free; undeformed Maxwell theory is the only causal point in the relevant regime.

Forecasting the evolution of complex dynamical systems remains a fundamentally challenging task, primarily due to pronounced nonlinear interactions, high-dimensional state spaces, and the concomitant requirement for rigorous and reliable uncertainty quantification. Contemporary reduced-order modelling (ROM) frameworks frequently exhibit inherent trade-offs among predictive accuracy, numerical stability, and interpretability, and thus often fail to achieve an optimal balance among these competing objectives. To address these limitations, we propose a framework for forecasting complex dynamical systems via a kernel autonomous ordinary differential equation approach based on Gaussian Processes and Quadratic Order Model Reduction. Our base method, the Gaussian Process Ordinary Differential Equations model, allows accurate short-term forecasting with uncertainty quantification, and it provably converges to the real autonomous equation in the smooth case. We integrate it with quadratic order reduced-order modelling and sphere projection for learning the latent dynamics efficiently while preserving stability. Numerical experiments demonstrate that our full model outperforms ROM forecasting methods such as Extended Dynamic Mode Decomposition, Bagging Optimised Dynamic Mode Decomposition and Linear and Nonlinear Disambiguation Optimisation in terms of accuracy or computational costs. These results demonstrate the potential of the framework as a robust and stable tool for forecasting complex dynamical systems with rigorous uncertainty quantification.

We consider a scheduling problem for star observations using a telescope. Identical nights are available to observe stars that have been identified as of interest by astronomers. Each star yields a profit if it is observed; in that case, the observation must respect a visibility window and a minimum observation duration. The objective is to maximize the total profit of the observations that are actually performed. By interpreting nights as machines and stars as jobs to be processed, this problem is a classic scheduling setting of parallel machines. In the practical star-observation problem, the ability to observe depends on meteorological and atmospheric conditions, which are not known when the schedule is computed. A possible modeling is to consider that each night can be either perfect and everything is observable, either terrible and nothing is observable. The challenge is therefore to propose a ”high-quality” schedule without knowing the actual number of nights (machines) that will ultimately be available. We focus here on evaluating the worst case within the robust optimization paradigm.

An approach to generate three-term recurrence relations with arbitrary degree steps is proposed for orthogonal polynomials. Specifically, given any class of orthogonal polynomials $\{Q_{p}(x)\}_{p=0}^{\infty}$ defined by Favard's theorem, we employ the adjacent members $Q_{p}(x)$ and $Q_{p-1}(x)$ to compute $Q_{p+s}(x)$ of high degree and the one of low degree $Q_{p-t}(x)$, where $(s,t)$ are parameters for degree step adjustment. The coefficients of both relations are analyzed, revealing novel properties that enable the derivation of three-term recurrence relations with respect to $Q_{p+s}(x)$, $Q_{p}(x)$ and $Q_{p-t}(x)$ by eliminating $Q_{p-1}(x)$. Furthermore, in addition to the standard recursive formula, which is characterized by degree increase, the formulas for degree decrease and end-to-middle directions are also formulated. Moreover, explicit recurrence relations with 2-degree steps are presented for Hermite, Gegenbauer and Legendre polynomials. The computation precision of the proposed recurrence relations is also compared with that of the standard ones.

Theorems by Bray-Brendle-Neves and Zhu about positive scalar curvature metrics on products of a 2-sphere and an n-torus suggests that positive scalar curvature suggests and appropriate topological assumptions should lead to the existence of a topologically non trivial 2-spheres of small area, which can be stated as upper bound on the 2-systole of such manifolds. Recent progress have been made in this direction by Sha and Tsiamis under the additional Kähler assumption while Checcini-Hirsh-Ziedler, Stryker and Tsiamis showed similar upper bounds on the stable 2-systole using index theoretic methods. We prove here similar inequalities for some Kähler manifold which control the relative sizes of the representatives of a well chosen set of homology classes. For instance on $(\mathbb{CP}^1x\mathbb{CP}^1 , \omega)$ with a positive scalar curvature Kähler metric we quantitavely show that largeness of one factor imposes smallness of the other one.

In this paper, we develop a unified geometric framework for the propagation dynamics of multidimensional nonlocal Fisher–KPP equations with asymmetric dispersal. In dimensions $N\geqslant2$, asymmetry leads to a multidimensional propagation geometry determined by the full family of minimal wave speeds, and the intrinsic spreading set may even fail to contain the origin. We show that the mean dispersal drift $\mathbf p=\int_{\mathbb R^N}x\mathcal J(x)dx$ provides a natural center for the spreading geometry, yielding a new drift-centered radial description. For unbounded initial supports, we prove that the spreading set $\mathcal W$ is given by the Minkowski-sum formula. Although $\mathcal W$ need not be convex, it remains star-shaped with respect to $\mathbf p$ and admits a Freidlin–Gärtner-type variational characterization. We further establish locally truncated Hausdorff convergence under a hypothesis on directions, and a global $o(t)$-Hausdorff approximation under a finite-thickness assumption on the initial support. Thus, our results extend the classical Freidlin–Gärtner picture to genuinely biased multidimensional nonlocal propagation.

We investigate three clean-like properties for arbitrary rings, for endomorphism rings of abelian groups and for matrix rings over finite fields. Specifically, we study and establish when a ring is weakly strongly $k$-nil-clean for some fixed natural number $k\geq 2$, when the matrix ring is either quasi $2$-nil-clean or quasi $3$-nil-clean, and when the endomorphism ring is weakly clean. Our theorems improve substantially on some results due to Goldsmith-Vámos in Rend. Sem. Mat. Univ. Padova (2007), Breaz et al. in Linear Algebra & Appl. (2013), Koşan-Zhou in Front. Math. China (2016), Su et al. in J. Algebra & Appl. (2027), and some other existing results in this current topic.