Papers

We consider the problem of estimating the gradient ascent line of a density originating at a given point. Going beyond mere consistency, we establish a weak convergence result for a plugin estimator based on a kernel density estimator of the density. We then leverage that result to construct a confidence region for the gradient ascent line, including by bootstrap.

In this paper, we investigate the cocompact inverse \(\sigma_k\) curvature flow in Minkowski space. We prove the longtime existence and convergence of this flow. As a consequence, quermassintegral inequalities are established.

We prove that if $M$ is a closed Riemannian surface of diameter $d$ and area $v$ with sectional curvature in the $[-1,1]$ interval, then a closed geodesic net of length $l$ has at most $f(l,d,v)$ branch points, where $f(l,d,v)=(400\bar{l})^{(180\bar{l})^4}$ for $\bar{l}= \max\{l, \frac{\exp(d)}{\min\{1, \frac{v}{4}\}}\}$ This answers a question posed by S. Becker-Kahn. We also prove that for each geodesic net in the Euclidean plane with at most n unbalanced (boundary) vertices such that all its unbalanced vertices have degree 1, the number of balanced vertices of degree $\ge 3$ (=branch points) does not exceed $(25n)^{2n^2}$. This answers a question posed in [GM] and [NP].

We consider a bound of degrees of curves by their genera in the log pair $(\mathbb{P}^{n}\times C, D)$, where $C$ is a smooth complex projective curve and $D$ is a very general smooth divisor on $\mathbb{P}^{n}\times C$. Our study is motivated by a function field analogue of the Lang-Vojta conjecture.

This note is concerned with the spreading speed of the predator component in a class of prey-predator reaction-diffusion systems with spatiotemporal heterogeneity depending on a moving variable. The main difficulty is that the full system lacks a direct comparison principle. We establish a pointwise estimate showing that the prey density is close to its carrying capacity wherever the predator density is sufficiently small. This allows us to compare solutions of the system with those of scalar Fisher-KPP equations in shifting environments. Consequently, we obtain an explicit formula for the spreading speed of the predator. In particular, for nonmonotone shifting environments, locking and nonlocal pulling phenomena may occur.

Grid-scale batteries increasingly influence outcomes in wholesale electricity markets, but their observed bid patterns remain difficult to interpret. In particular, bids that appear to reflect strategic withholding may instead arise from rational operations under price uncertainty and risk management. We develop an asset-level model of a price-taking battery that submits stepwise buy and sell bid curves in the day-ahead market under a finite set of price scenarios. The battery chooses quantity–price pairs to maximize a mean–CVaR objective subject to physical and market constraints. A direct formulation is a mixed-integer linear program, but we show that its integer decisions can be removed, yielding an exact linear programming reformulation suitable for empirical analysis. Our empirical results deliver three insights. First, withholding behavior can arise even without market power, because scarce stored energy and uncertain future prices increase the value of holding energy. Second, the effect of uncertainty depends on the state of charge: when stored energy is scarce, greater uncertainty raises sell bid prices, whereas when stored energy is abundant it can lower them. Third, risk management reshapes bid curves into layered structures that secure profitable execution across a broad set of scenarios while preserving some exposure to rare but valuable price spikes.

In this paper, we consider a class of convex composite optimization problems arising from the joint estimation of clustered multiple Gaussian graphical models. The resulting model combines a log-determinant loss term with a nonsmooth sparse clustered regularizer, which encourages both similar sparsity patterns and consistent edge values across different graphs. We first establish a necessary and sufficient condition under which the solution is block diagonal, enabling a large-scale problem to be decomposed into smaller independent subproblems and substantially reducing computational complexity. To efficiently solve this problem, we develop a proximal augmented Lagrangian method in which each subproblem is handled by a superlinearly convergent semismooth Newton method. Unlike widely used first order methods, our approach fully exploits the underlying second order information through the semismooth Newton framework, thereby achieving faster convergence and improved robustness. The efficiency and robustness of the proposed algorithm are demonstrated through comparisons with state-of-the-art methods on both synthetic and real data sets.

Pure partition functions of multiple SLE are characterized by null-state partial differential equations, Möbius covariance, and boundary asymptotics. After quotienting by Möbius covariance, the case of three curves is the first genuinely multivariable one: the moduli space has three independent variables, naturally represented by the three unoriented cross-ratios of the three pairs of links. We solve this Möbius-normalized three-variable problem for the two basic link-pattern types of multiple $3$-SLE$_\kappa$, namely the rainbow and neighbor patterns. Writing $\beta=4/\kappa$, we construct explicit trivariate hypergeometric-series normal forms and identify them with the corresponding pure partition functions {for all $\beta>1/2$ in both cases, equivalently for all $\kappa\in(0,8)$}. The proof is analytic. The null-state PDEs and Möbius covariance yield recursion relations for the trivariate coefficient arrays. In the rainbow case, coefficient estimates give convergence and boundary regularity on the closed cube. In the neighbor case, Pfaff systems continue the local power series to a neighborhood of $[0,1)^3$, while {boundary analysis and corner propagation} give continuity on $[0,1]^3$ {throughout the full range $\beta>1/2$}. The two-dimensional boundary degenerations are classical Appell $F_1$ and Horn $G_2$ functions. The probabilistic identification uses SLE martingale arguments and Itô calculus, together with positivity and boundary regularity. We also discuss boundary degenerations, including heuristic connections with boundary Green's functions.

We prove that a monomial ideal has projective dimension one if and only if its minimal monomial generators can be ordered so that each successive colon ideal is principal, and show that this characterization is equivalent to the monomial version of the Hilbert-Burch Lemma. Furthermore, we prove that any squarefree monomial ideal of projective dimension one with a linear resolution has the property that all its powers \(I^s\) admit linear quotients, and we provide a partial classification of such ideals.

In this paper, we establish some geometric and analytic foundations for free boundary Brakke flows. Specifically, we (i) introduce unit-regular and cyclic free boundary flows and show that they are preserved under reflections and weak limits, (ii) prove that the support of free boundary Brakke flows satisfies an avoidance principle, and (iii) introduce free boundary inner and outer flows and prove the existence of matching free boundary Brakke flows. These results serve as general tools to analyze free boundary flows through singularities, and in particular will be applied in forthcoming work with Haslhofer, where we address the mean-convex neighborhood conjecture and uniqueness conjecture for free boundary flows through (half) cylindrical singularities.

In this paper, we investigate congruences for meromorphic modular forms $F$ which have a pole at a single point $z$ in the fundamental domain of $\mathrm{SL}_2(\mathbb Z)$. For a prime $p$ with good supersingular reduction at the elliptic curve corresponding to $z$, we show that there exists a cusp form $f$ such that $F|U_p^m \equiv f|U_p^m \pmod{p^{\kappa_m}}$, where $\kappa_m=\alpha m -\beta$ with $\alpha$ only depending on the weight of $F$ and $\beta$ depending on $F$ and $p$ but is independent of $m$. In particular, if the space of cusp forms is trivial, then $F|U_p^m\equiv 0 \pmod{p^{\kappa_m}}$ vanishes $p$-adically to a high order. In order to prove these results, we use the fact that $p$ has supersingular reduction to realize $F$ as an overconvergent modular form and then utilize the theory of overconvergent forms to show the congruences.

Let $X$ be a real-valued random variable and let $p>0$. When $X$ can take negative values, $X^p$ is not generally real-valued for non-integer $p$; formulas for moments of arbitrary positive order must therefore distinguish the absolute power $|X|^p$ from the signed power $\operatorname{sgn}(X)|X|^p$. Using the layer-cake identity, we give a self-contained treatment of tail representations for the positive and negative parts of $X$, including absolute and signed-power moments, centered forms, and exact formulas on ordered discrete supports and integer lattices. The basic identities require neither a density nor continuity and apply to continuous, discrete, singular, and mixed distributions. For strictly positive random variables, we also give representations and finiteness criteria for inverse moments, together with distribution-function and Frullani–Laplace formulas for logarithmic moments. Examples illustrate two-sided support, atoms, lattice tails, and behavior near zero. The presentation places these classical identities in a common notation and states the domain and integrability conditions required in each case.

We show that on linearly locally contractible (LLC) manifolds, the beta numbers (which describe unilateral flatness) are comparable to the theta numbers (which describe bilateral flatness), quantitatively. As an application, we show that if $M\subset\mathbb{R}^n$ is a compact LLC $m$-manifold with finite Menger $p$-energy for some $p>m(m+2)$, then $M$ is in fact a $C^{1,\alpha}$ manifold. We also show that the bound $m(m+2)$ is critical by constructing, for each $n\geq 3$, an LLC $n$-sphere in $\mathbb{R}^{n+1}$ that has finite Menger $p$-energy for every $p<m(m+2)$ but is not even quasisymmetrically equivalent to the standard $n$-sphere.

Frames provide redundant representations that enable stable signal reconstruction under coefficient losses. In this paper, we study optimal dual frames for probabilistic erasures using the Hilbert–Schmidt norm of the associated error operators. We characterize dual frames that are optimal for $1-$erasures and establish conditions under which the canonical dual is not only optimal but also unique. We further derive lower bounds for the probabilistic reconstruction error for any $m-$ erasure and identify classes of frames for which the canonical dual remains optimal. In addition, we analyze the geometric structure of the set of optimal dual frames, showing that it is a nonempty compact convex set. These results provide new insights into robustness and optimal reconstruction in probabilistic erasure models.

In this paper, we introduce pre-vertex algebras and pre-Poisson vertex algebras as preLie-algebras in the chiral and classical pseudo-tensor categories, respectively. We show that a pre-vertex algebra gives rise to a vertex algebra, that every Rota-Baxter operator on a vertex algebra induces a pre-vertex algebra, and that every pre-vertex algebra can be embedded into a vertex algebra equipped with a Rota-Baxter operator. Moreover, we prove that pre-vertex algebras are equivalent to dendriform vertex algebras. In the Poisson setting, we demonstrate that pre-Poisson vertex algebras are obtained from Rota-Baxter operators on Poisson vertex algebras, from filtrations of pre-vertex algebras, and as classical limits of pre-vertex formal deformations of differential Zinbiel algebras. This extends the classical relationships among Rota-Baxter operators, pre-Lie algebras, and Poisson algebras to the framework of vertex algebras.

Spherical interpolation is required in numerical and geometric applications in which the unknowns are constrained to remain on the unit sphere. Spherical Interpolation of orDER $n$ (SIDER-$n$) was introduced as the high-order reconstruction component of spherical essentially non-oscillatory interpolation, where the reconstruction is built entirely from spherical linear interpolation (SLERP) operations and therefore preserves the spherical constraint exactly. This paper develops analytic first-derivative formulas for SIDER curves of arbitrary order. The central observation is that the recursive definition of SIDER can be differentiated by direct chain-rule propagation through its binary tree of SLERP operations. After deriving the total derivative of SLERP with moving endpoints, we obtain compact recursions for the derivative of SIDER-$n$, including simplified formulas at interpolation nodes and practical formulas at middle points between consecutive sampling locations. The latter are relevant when a reconstruction is evaluated halfway between data samples, as occurs in several high-order reconstruction-based numerical algorithms. The base case SIDER2 is treated explicitly, and SIDER3 and SIDER4 are used to illustrate the recursive mechanism. We also prove that the derivative is tangent to the sphere at every reconstructed point, including both sampling points and middle points. The resulting formulas extend the original SIDER/SENO framework by supplying differential information for sphere-valued reconstructions, with potential use in high-order finite-volume, ENO/WENO, SENO-type, and related methods for conservation laws and evolution problems.

In the paper [M. Keel, H. Smith, C.D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions. J. Amer. Math. Soc. 17 (2004), no. 1, 109-153], the authors prove that for the 3-D quadratic quasilinear wave equation in exterior domains with homogenous Dirichlet boundary value and small initial data of size $\varepsilon$, the lifespan ${\bar T}_{\varepsilon}$ of the smooth solution fulfills ${\bar T}_{\varepsilon}\ge e^{C/\varepsilon}$. However, for the corresponding 2-D quadratic quasilinear wave equation in exterior domains with homogenous Dirichlet or Neumann boundary value, so far it is still open whether the expected sharp lifespan $T_{\varepsilon}\ge\frac{C}{\varepsilon^2}$ holds or not. In this paper, we will solve this open question. Our main ingredients include: introducing the suitable Friedlander radiation field for the 2-D linear wave equation in exterior domains with homogenous Dirichlet or Neumann boundary value, constructing the delicate approximate solution, and establishing some crucial space-time decay estimates for the solutions of 2-D quasilinear wave equation in exterior domains. On the other hand, for the radial symmetric solutions to a class of 2-D quadratic quasilinear wave equation in exterior domains, the upper bound of the lifespan $T_{\varepsilon}\le\frac{C}{\varepsilon^2}$ is derived and the sharp constant $C$ is also determined explicitly.

In this paper, based on our previous general formulation and computational solution to Bellman's Lost-in-a-forest Problem, we provide the proof of general solution and obtained more variations and results related to this problem. This paper provides generalized formalized method connecting curve covering, lost-in-the-forest problem, and traveling salesman problem with neighborhoods. We prove the equivalence and convergence. We also provide more results of searching for two lines, connection to Wetzel's unit arc covering problem, variations with closed path, variations in three dimensions, etc. The results include general calculation equations, partial analytical results, and numerical results.

Given a universal factorization algebra $\mathcal{A}$, we constructed in our previous paper a derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, together with chiral modules $\\mathfrak{Z}_{\mathcal{A}}^+$ and $\mathfrak{Z}_{\mathcal{A}}^-$ associated to a puncture, and a chiral bimodule $\mathfrak{Z}_{\mathcal{A}}$ associated to a node. Furthermore, these constructions assemble to a factorization $\mathcal{A}$-module over any family of nodal punctured curves. In this paper, we show that in the case where $\mathcal{A}$ is constructed from a quasi-conformal vertex algebra $V$, the zeroth homology algebra $H^0\mathfrak{Z}_{\mathcal{A}}$ is naturally isomorphic to Zhu's associative algebra $A(V)$, and we identify $H^0\mathfrak{Z}_{\mathcal{A}}$ with the bimodule underlying the mode-transition algebra of Damiolini-Gibney-Krashen. We also give an explicit description of the smoothing module $H^0\tilde{\mathfrak{Z}}_{\mathcal{C}}$ which describes the deformation of $H^0\mathfrak{Z}_{\mathcal{A}}$ which we attach to a smoothing family of a nodal curve. We therefore get a geometric interpretation of the Zhu algebra and the mode-transition algebra, as the integration of a factorization algebra over a certain compactification of configuration spaces of punctured nodal curves.

This paper presents an a-priori and a-posteriori error analysis for the time-dependent case of a coupled continuum pipe-fow in three dimensions on anisotropic meshes (i.e. elements with very large aspect ratio). Our investigation includes nonconforming discretizations as well as different elements and different versions of Crouzeix-Raviart fnite element methods. Under suitable regularity assumptions on the exact solution, we derive optimal a priori error estimates in the H^1-like norm. Leveraging residual elements, local error indicators and a global estimator are generated, demonstrating reliability and efficiency. The efficiency of error estimators holds unconditional, though its reliability involves the alignment measure notion. Mesh requirements are established to ensure the optimality of the error estimator.

In this work we investigate the role of neural architectures as implicit functional priors in control problems governed by ordinary differential equations. Rather than focusing on highly complex problems, our objective is to investigate architecture-dependent effects in controlled dynamical systems within the simplest physically interpretable settings possible. In particular, we study a controlled linear RLC electrical circuit and a nonlinear Duffing-type dynamical system. Both systems are analyzed first through classical optimal-control formulations and later through PINN-based approaches. We compare different combinations of multilayer perceptrons (MLPs) and Fourier-based KAN-like architectures, and analyze their influence on the resulting controls. The numerical experiments suggest that different architectural choices systematically generate qualitatively distinct controls, even under identical governing equations, loss functionals, initial and target states, training parameters and physical constraints. Significant differences appear in the spectral structure, smoothness, energy distribution, and phase-space behavior of the learned solutions. A central observation of this work is the emergence of a functional specialization phenomenon when the neural architectures are allowed sufficient freedom to shape the structure of the learned controls. More specifically, in the systems considered here, Fourier-based architectures tend to produce trajectories with richer oscillatory content, whereas smoother low-frequency-biased architectures tend to generate more regular and energetically efficient controls. This suggests that different functional components of the control problem may be handled more efficiently by different neural architectures, leading to an implicit specialization between state representation and control generation.

Machine-assisted mathematical discovery has been a long-standing challenge in machine learning and artificial intelligence. In recent years, we have seen tremendous progress with generative AI, yet its contribution to automated discovery in advanced mathematical research has been limited. One of the most difficult benchmarks in this context is the Birch test, which asks whether a machine can discover truly novel and non-trivial mathematical structures without human intervention. In this work, we particularly focus on the branch of automated conjecture discovery. We use HypothesiX, an automated conjecture mining agent and analyse its generated conjectures related to the distribution of twin primes to verify the conditions of the Birch test. Furthermore, note that automated discovery is now operating at scale, but verifying its non-triviality still depends on human evaluation. We propose a benchmark to quantify the non-triviality of machine-generated conjectures using the Mahalanobis distance within an embedding cluster of selected known mathematical conjectures. We also note that this quantified benchmark can be used as an error indication signal to localise the incorrectness of a new mathematical statement, which autoformalisers fail to verify due to their limitations in proof discovery capability.

In recent years, the Norlund sum and its applications have attracted considerable attention, especially in mathematics and other applied sciences. Because this sum plays an important role in mathematical modeling involving differential equations and difference equations, as well as in the theory of infinite series and integral transforms. Therefore, one of the most important goals of this article is to present a new approach to generating function and to derive new formulas including the sum of (inverse) Laplace transforms, the Norlund sum, the alternative Hurwitz zeta function, and the inverse Catalan sum formula with related operators. Another aim is to establish the relationship between the inverse Catalan sum formula and the derivative operator, and to give new formulas and relations, including the zeta function, the Bernoulli numbers, and the Laplace transform. Furthermore, applications are given including the relationship between these operators and the Dirichlet L-function. Some new formulas for certain classes of infinite series, including the Dirichlet L-function, the zeta function, Euler numbers of the second kind, and the Apostol-Bernoulli polynomials of higher-order.

This work investigates how toxin-mediated interactions and directed movements shape the emergence of coherent structures in plant-herbivore systems. The analysis focuses on a two-compartment model enclosing a toxin-dependent functional response and a cross-diffusion term that represents ecologically plausible herbivores' movement towards, or away from, vegetation. Two distinct dynamical regimes arise depending on toxicity strength. Under weak toxicity, the system admits at most one biologically feasible coexistence equilibrium, which may lose stability through a Hopf bifurcation generating small-amplitude temporal oscillations. Under strong toxicity, the nonlinear functional response becomes non-monotonic, allowing for multiple coexistence equilibria and abrupt regime shifts. The influence of cross-diffusion on stability is also examined, identifying the conditions under which Turing instabilities and mixed spatiotemporal patterns occur. Near the corresponding bifurcation thresholds, Stuart-Landau amplitude equations are derived via weakly nonlinear analysis, providing a unified framework for the modulation of oscillatory, stationary, and combined Turing-Hopf modes. Numerical simulations corroborate the theoretical predictions, illustrating transitions from spatially uniform states to oscillations, spatial patterns, and mixed behaviour. Overall, this manuscript highlights how chemical defences, nonlinear feedbacks, and movement strategies jointly determine the emergence, selection, and robustness of coherent structures in plant-herbivore systems.

We establish a refined singularity estimate for nonnegative n-superharmonic functions. For complete noncompact locally conformally flat manifolds with nonnegative Ricci curvature, we analytically characterize the volume growth, verify Yau's conjecture on the Cohn-Vossen inequality, and prove a sharp gap theorem that removes all auxiliary assumptions from earlier works.

In this paper we deal with the boundedness of certain generalized sequence-to-function Hardy-Littlewood-Pólya-type operators. We completely characterize the $l^{p}-L^{q}$ boundedness of these operators for all $(p, q)\in [1, \infty]\times[1, \infty]$. These results complement some previous results in the literature. Our method relies on the ideas of generalized Schur's tests developed by Okikiolu, Sinnamon, Tao and Zhao.

Motivated by the recent interest in multipoint Padé approximants in the physics community, we discuss Chebyshev rational functions and show how they give rise to multipoint Padé approximants in exactly the same way that Chebyshev polynomials produce Padé approximants. We present recurrence relations for Chebyshev rational functions, as well as the underlying continued fraction of Thiele type (also known as $R_{II}$ type). Finally, we provide numerical evidence illustrating the effects of noise on this interpolation scheme and show that a phenomenon similar to that recently observed by Costin, Dunne, and Meynig for Padé approximants also occurs in the multipoint setting.

Motivated by variation problems for the Néron–Tate height pairing, we construct an adelic Poincaré bundle on the Néron model of an abelian scheme over a curve. This extends the adelic Poincaré bundle of Yuan–Zhang, which encodes the fiberwise Néron–Tate height pairing.

We show that the toric ideal of the Fano matroid polytope does not have a quadratic Gröbner basis. This resolves in the negative a strong version of White's conjecture from matroid theory. This result was found independently by De Loera, Ferroni, Morales, and Rambau. Our approach is based on a new characterization of regular unimodular flag triangulations, which reduces the problem to an instance of SMT involving boolean and real variables. We then use an SMT solver to prove unsatisfiability. Using this approach, we also show that all 8-element matroids which do not have the Fano matroid or its dual as a minor, with the possible exception of the matroid $T_8$, have toric ideals which admit quadratic Gröbner bases.

We get estimates on the Gauss-Kuzmin-Wirsing constant for continued fractions.