Search

In 1990 F. Chiarenza and M. Frasca published a paper in which they generalized a result of C. Fefferman on estimates of the integral of |bu|^{p} through the integral of |Du|^{p} for p>1. Formally their proof is valid only for d\geq 3. We present here further generalization with a different proof in which D is replaced with the fractional power of the Laplacian for any dimension d\geq 2. [ABSTRACT FROM AUTHOR]

A classical result of K. L. Chung and W. Feller deals with the partial sums S_k arising in a fair coin-tossing game. If N_n is the number of "positive" terms among S_1, S_2, ..., S_n then the quantity P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P(N_{2n+1} = r), r = 0, 1, 2, ..., 2n+1. We get to this ansatz by adaptating the Feynman–Kac methodology. [ABSTRACT FROM AUTHOR]

We are studying topological properties of the Julia set of the map F(z, p) = *** of the complex projective plane PC² to itself. We show a relation between this rational function and an uncountable family of "paper folding" plane filling curves. [ABSTRACT FROM AUTHOR]

This note is devoted to developing Morrey regularity theory for the following system of Rivière \begin{equation*} -\Delta u=\Omega \cdot \nabla u+f \qquad \text {in }B^{2}, \end{equation*} under the assumption that f belongs to some Morrey space. Our results extend the L^p regularity theory of Sharp and Topping [Trans. Amer. Math. Soc. 365 (2013), pp. 2317–2339], and also generalize a Hölder continuity result of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24] on harmonic mappings. Potential applications of our results are also possible in second order conformally invariant geometrical problems as that of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24]. [ABSTRACT FROM AUTHOR]

\Theta is the least ordinal \alpha with the property that there is no surjection f:\mathbb {R}\rightarrow \alpha. {\mathsf {AD}}_{\mathbb {R}} is the Axiom of Determinacy for games played on the reals. It asserts that every game of length \omega of perfect information in which players take turns to play reals is determined. An ideal \mathcal {I} on \omega _1 is \omega _1-dense if the boolean algebra {\wp }(\omega _1)/ \mathcal {I} has a dense subset of size \omega _1. We consider the theories, where \mathsf {CH} stands for the Continuum Hypothesis, \begin{gather*} \mathsf {ZFC} + \mathsf {CH} + \text {"There is an \omega _1-dense ideal on \omega _1.''}\\ \mathsf {ZF}+{\mathsf {AD}}_{\mathbb {R}} + \text {"\Theta is a regular cardinal.''}\end{gather*} The main result of this paper is that the first theory given above implies the existence of a class model of the second theory given above. Woodin, in unpublished work, showed that the consistency of the second equation given above implies the consistency of the first equation given above. We will also give a proof of this result, which, together with our main theorem, establish the equiconsistency of both the equations given above. As a consequence, this resolves part of question 12 of W. Hugh Woodin [ The axiom of determinacy, forcing axioms, and the nonstationary ideal , Walter de Gruyter & Co., Berlin, 1999], in particular, it shows that the theories (b) and (c) in question 12 of W. Hugh Woodin [ The axiom of determinacy, forcing axioms, and the nonstationary ideal , Walter de Gruyter & Co., Berlin, 1999] are equiconsistent. Thus, our work completes the work that was started by Woodin and Ketchersid in [ Toward AD(\mathbb {R}) from the continuum hypothesis and an \omega _1-dense ideal , ProQuest LLC, Ann Arbor, MI, 2000] some 25 years ago. We also establish other theorems of similar nature in this paper, showing the equiconsistency of the second equation given above and the statement that the non-stationary ideal on {\wp }_{\omega _1}(\mathbb {R}) is strong and pseudo-homogeneous. The aforementioned results are the only known equiconsistency results at the level of \mathsf {AD}_{\mathbb {R}} + \text {"Θ is a regular cardinal.''} [ABSTRACT FROM AUTHOR]

Let G be a classical group defined over a local field F of characteristic zero. For any irreducible admissible representation \pi of G(F), which is of Casselman-Wallach type if F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F. In particular, if \pi has a generic local L-parameter, we introduce the spectral first occurrence index {\mathfrak {f}}_{\mathfrak {s}}(\pi) and the arithmetic first occurrence index {\mathfrak {f}}_{{\mathfrak {a}}}(\pi) of \pi and prove in this paper that {\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{{\mathfrak {a}}}(\pi). Based on the theory of consecutive descents of enhanced L-parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L-packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish new Maximum Principles for parabolic equations in the framework of mixed local/nonlocal operators. In particular, these results apply to the case of mixed local/nonlocal Neumann boundary conditions, as introduced by Dipierro, Proietti Lippi, and Valdinoci [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), pp. 1093–1166]. Moreover, they play an important role in the analysis of population dynamics involving the so-called Allee effect, which is performed by Dipierro, Proietti Lippi, and Valdinoci [J. Math. Biol. 89 (2024), Paper No. 19]. This is particularly relevant when studying biological populations, since the Allee effect detects a critical density below which the population is severely endangered and at risk of extinction. [ABSTRACT FROM AUTHOR]

The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond [Math. Ann. 21 (1883), no. 1, pp. 109–117]). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky [Crelle 80 (1875), pp. 1–32]) that a solution to the heat equation may not be time-analytic at t=0 even if the initial function is real analytic. Recently, it was shown by Dong and Pan [J Math. Fluid Mech. 22 (2020), no. 4, Paper No. 53]; Dong and Zhang [J. Funct. Anal. 279 (2020), no. 4, Paper No. 108563]; Zhang [Proc. Amer. Math. Soc. 148 (2020), no. 4, pp. 1665–1670] that solutions to the heat equation in the whole space, or in the half space with zero boundary value, are analytic in time under an essentially optimal growth condition. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any \delta >0, we find a solution to the heat equation on the whole plane, with exponential growth of order 2+\delta, which is nowhere analytic in time. [ABSTRACT FROM AUTHOR]

For the quadratic family f_{c}(z) = z^2+c with c in a hyperbolic component of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. In this paper we give a uniform derivative estimate of such a motion when the parameter c converges to a parabolic parameter {\hat {c}} radially; in other words, it stays within a bounded Poincaré distance from the internal ray that lands on {\hat {c}}. We also show that the motion of each point in the Julia set is uniformly one-sided Hölder continuous at {\hat {c}} with exponent depending only on the petal number. This paper is a parabolic counterpart of the authors' paper "From Cantor to semi-hyperbolic parameters along external rays" [Trans. Amer. Math. Soc. 372 (2019), pp. 7959–7992]. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence of at least two standing waves (and homoclinic solutions) for a class of time-dependent (and time-independent) discrete nonlinear Schrödinger systems or equations. The novelties of the paper are as follows. (1) Our nonlinearities are composed of three mixed growth terms, i.e., the nonlinearities are composed of sub-linear, asymptotically-linear and super-linear terms. (2) Our nonlinearities may be sign-changing. (3) Our results can also be applied to the cases of concave-convex nonlinear terms. (4) Our results can be applied to a wide range of mathematical models. [ABSTRACT FROM AUTHOR]

The chord log-Minkowski problem asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the cone-chord measure of a convex body. The chord log-Minkowski problem has been extensively studied by Guo, Xi, and Zhao [Math. Ann. (2023), DOI 10.1007/s00208-023-02721-8]; Lutwak, Xi, Yang, and Zhang [Commun. Pure Appl. Math. (2023), DOI 10.1002/cpa.22190]; Qin [Adv. Math. 427 (2023), Paper No. 109132]. In this paper, we solve the chord log-Minkowski problem when q\in (0,1), without symmetry assumptions. [ABSTRACT FROM AUTHOR]

This paper is the second in a series of four papers aiming to describe the (almost integral) Chow ring of \overline {\mathcal {M}}_3, the moduli stack of stable curves of genus 3. In this paper, we introduce the moduli stack \widetilde {\mathcal {H}}_g^r of hyperelliptic A_r-stable curves and generalize the theory of hyperelliptic stable curves to hyperelliptic A_r-stable curves. In particular, we prove that \widetilde {\mathcal {H}}_g^r is a smooth algebraic stack that can be described using cyclic covers of twisted curves of genus 0 and it embeds in \widetilde {\mathcal M}_g^r (the moduli stack of A_r-stable curves) as the closure of the moduli stack of smooth hyperelliptic curves. [ABSTRACT FROM AUTHOR]

This paper is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two complex parameters. Understanding their parameter spaces is key to understanding families with more asymptotic values, just as understanding quadratic polynomials was for rational maps more generally. The first such families studied were the one-dimensional slices of the exponential family, \exp (z) + a, and the tangent family \lambda \tan z. The exponential case exhibited phenomena not seen for rational maps: Cantor bouquets in both the dynamic and parameter spaces, and no bounded hyperbolic components. The tangent case, with its two finite asymptotic values \pm \lambda i, is closer to the rational case, a kind of infinite degree version of the latter. In this paper, we consider a general family that interpolates between \exp (z) + a and \lambda \tan z. Our new family has two asymptotic values and a one-dimensional slice for which one of the asymptotic values is constrained to be a pole, the "polar asymptotic value" of the title. We show how the dynamic and parameter planes for this slice exhibit behavior that is a surprisingly delicate interplay between that of the \exp (z) + a and \lambda \tan z families. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the nonlinear least squares problem: \min _{{\boldsymbol {x}}\in \mathbb {H}^d}\|\lvert A{\boldsymbol {x}}\rvert -{\boldsymbol {b}}\| where A\in \mathbb {H}^{m\times d}, {\boldsymbol {b}}\in \mathbb {R}^m with \mathbb {H}\in \left \{\mathbb {R},\mathbb {C}\right \} and consider the uniqueness and stability of solutions. This problem arises in applications such as phase retrieval and absolute value rectification neural networks. While several results have been developed to characterize the uniqueness and stability of solutions when {\boldsymbol {b}}=\lvert A{\boldsymbol {x}}_0\rvert for some {\boldsymbol {x}}_0\in \mathbb {H}^d, no existing results address the case where {\boldsymbol {b}} is arbitrary. In this paper, we investigate the uniqueness and stability of solutions for the more general case where {\boldsymbol {b}} is not necessarily equal to \lvert A{\boldsymbol {x}}_0\rvert for any {\boldsymbol {x}}_0\in \mathbb {H}^d. We prove that for any matrix A\in \mathbb {H}^{m\times d}, there is always a vector {\boldsymbol {b}}\in \mathbb {R}^m for which the solution to the nonlinear least squares problem is not unique. However, we show that such "bad" vectors {\boldsymbol {b}} are negligible in practice; specifically, if {\boldsymbol {b}}\in \mathbb {R}_{ }^m does not lie in some measure zero set, then the solution is unique. Furthermore, we establish certain conditions under which the solution is guaranteed to be unique. Regarding the stability of solutions, we prove that the solution is not uniformly stable. However, if we restrict the vectors {\boldsymbol {b}} to a convex set where the solution to the least squares problem is unique, then the solution becomes stable. To the best of our knowledge, our results represent the first theoretical results of the uniqueness and stability of solutions for the nonlinear least squares problem. [ABSTRACT FROM AUTHOR]

The principal tool of computation used in this paper is classical Gaussian quadrature on the interval [0,1], which happens to be particularly effective here. Explicit expressions are found for the derivatives of the Ramanujan integral in question, and it is proved that the latter is completely monotone on (0,\infty). As a byproduct, known series expansions for incomplete gamma functions are examined with regard to their convergence properties. The paper also pays attention to another famous integral, the Euler integral — better known as the gamma function — revitalizing a largely neglected part of the function, the part corresponding to negative values of the argument, which plays a prominent role in our work. [ABSTRACT FROM AUTHOR]

Immediately following the commentary below, this previously published article is reprinted in its entirety: Cathleen Synge Morawetz, "The mathematical approach to the sonic barrier". [ABSTRACT FROM AUTHOR]

Let \mathrm {L}^2_a(\mathbb {D}) be the classical Bergman space and let M_h denote the operator of multiplication by a bounded holomorphic function h. Let B be a finite Blaschke product of order n. An open question proposed by R. G. Douglas is whether the operators M_B on \mathrm {L}^2_a(\mathbb {D}) similar to \oplus _1^n M_z on \oplus _1^n \mathrm {L}^2_a(\mathbb {D})? The question was answered in the affirmative, not only for Bergman space but also for many other Hilbert spaces with reproducing kernel. Since the operator M_z^* is in Cowen-Douglas class B_1(\mathbb {D}) in many cases, Douglas question can be reformulated for operators in B_1(\mathbb {D}), and the answer is affirmative for many operators in B_1(\mathbb {D}). A natural question occurs for operators in Cowen-Douglas class B_n(\mathbb {D}) (n>1). In this paper, we investigate a family of operators, which are in a norm dense subclass of Cowen-Douglas class B_2(\mathbb {D}), and give a negative answer. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic profiles of zero points for the spatial variable of the solutions to the heat equation. By giving suitable conditions for the initial data, we prove the existence of zero points by extending the high-order asymptotic expansion theory for the heat equation. This reveals a previously unknown asymptotic profile of zero points diverging at O(t). In a one-dimensional spatial case, we show the zero point's second and third-order asymptotic profiles in a general situation. We also analyze a zero level set in high-dimensional spaces and obtain results that extend the results for the one-dimensional spatial case. [ABSTRACT FROM AUTHOR]

Let p \geq 7 be a prime number. Let S(3) denote the third Morava stabilizer algebra. In recent years, Kato-Shimomura and Gu-Wang-Wu found several families of nontrivial products in the stable homotopy ring of spheres \pi _* (S) using H^{*,*} (S(3)). In this paper, we determine all nontrivial products in \pi _* (S) of the Greek letter family elements \alpha _s, \beta _s, \gamma _s and Cohen's elements \zeta _n which are detectable by H^{*,*} (S(3)). In particular, we show \beta _1 \gamma _s \zeta _n \neq 0 \in \pi _*(S), if n \equiv 2 mod 3, s \not \equiv 0, \pm 1 mod p. [ABSTRACT FROM AUTHOR]

In this paper, we show that the p-adic limit of a family of Eisenstein series on the exceptional domain where the exceptional group of type E_{7,3} acts is an ordinary modular form for a congruence subgroup. [ABSTRACT FROM AUTHOR]

In this paper we construct a bounded wandering domain with the property that, in a sense we make precise, nearly all of its forward iterates are contained within a bounded domain. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the computations of the p-rank of curves in two different setups. We first work with complete intersection varieties in \mathbf {P}^n \text { for } n\ge 2 and compute explicitly the action of Frobenius on the top cohomology group. In case of curves and surfaces, this information suffices to determine if the variety is ordinary. Next, we consider curves on more general surfaces with p_g(S) = 0 = q(S) such as Hirzebruch surfaces and determine p-rank of curves on Hirzebruch surfaces. [ABSTRACT FROM AUTHOR]

The classical Hard Lefschetz theorem (HLT), Hodge-Riemann bilinear relation theorem (HRR) and Lefschetz decomposition theorem (LD) are stated for a power of a Kähler class on a compact Kähler manifold. These theorems are not true for an arbitrary class, even if it contains a smooth strictly positive representative. Dinh-Nguyên proved the mixed HLT, HRR and LD for a product of arbitrary Kähler classes. Instead of products, they asked whether determinants of Griffiths positive k\times k matrices with (1,1)-form entries in \mathbb {C}^n satisfy these theorems in the linear case. This paper answered their question positively when k=2 and n=2,3. Moreover, assume that the matrix only has diagonalized entries, for k=2 and n\geqslant 4, the determinant satisfies HLT for bidegrees (n-2,0), (n-3,1), (1,n-3) and (0,n-2). In particular, for k=2 and n=4,5 with this extra assumption, the determinant satisfies HRR, HLT and LD. Two applications: First, a Griffiths positive 2\times 2 matrix with (1,1)-form entries, if all entries are \mathbb {C}-linear combinations of the diagonal entries, then its determinant also satisfies these theorems. Second, on a complex torus of dimension \leqslant 5, the determinant of a Griffiths positive 2\times 2 matrix with diagonalized entries satisfies these theorems. [ABSTRACT FROM AUTHOR]

The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau (see their paper in the references), and was given the name \lq\lq classical freeness" by Jethro van Ekeren and Reimundo Heluani [Comm. Math. Phys. 386 (2021), no. 1, pp. 495-550] in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), pp. 1689–1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{m|2r}) for all positive integers m,n,r satisfying -\frac {m}{2} + r +n+1 > 0. In particular, it holds for the rational vertex superalgebras L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{1|2r}) for all positive integers r,n. [ABSTRACT FROM AUTHOR]

In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a C^2 open neighborhood of Finsler metrics all of whose closed geodesics are hyperbolic. For surfaces this result holds also for Riemannian metrics. This follows from a recent result of Contreras and Mazzucchelli [Duke Math. J. 173 (2024), pp. 347–390]. Furthermore, geodesic flows of Riemannian or Finsler metrics on surfaces are C^2 stably ergodic if and only if they are Anosov. [ABSTRACT FROM AUTHOR]

The Drinfeld compactification \overline {\operatorname {Bun}}{}_B' of the moduli stack \operatorname {Bun}_B' of Borel bundles on a curve X with an Iwahori structure is important in the geometric Langlands program. It is closely related to the study of representation theory. In this paper, we construct a resolution of singularities of it using a modification of Justin Campbell's construction of the Kontsevich compactification. Furthermore, the moduli stack {\operatorname {Bun}}_B' admits a stratification indexed by the Weyl group. For each stratum, we construct a resolution of singularities of its closure. Then we use this resolution of singularities to prove a universally local acyclicity property, which is useful in the quantum local Langlands program. [ABSTRACT FROM AUTHOR]

The paper introduces a geometrically unfitted finite element method for the numerical solution of the tangential Navier–Stokes equations posed on a passively evolving smooth closed surface embedded in \mathbb {R}^3. The discrete formulation employs finite difference and finite elements methods to handle evolution in time and variation in space, respectively. A complete numerical analysis of the method is presented, including stability, optimal order convergence, and quantification of the geometric errors. Results of numerical experiments are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing an arbitrary generalized singular value of a Grassman or real matrix pair and a triplet of associated generalized singular vectors. Based on the QR factorization, the problem is reformulated as two novel trace maximization problems, each of which has double variables with unitary constraints or orthogonal constraints. Theoretically, we show that the arbitrarily prescribed extreme generalized singular values and associated triplets of generalized singular vectors can be determined by the global solutions of the constrained trace optimization problems. Then we propose a geometric inexact Newton–conjugate gradient (Newton-CG) method for solving their equivalent trace minimization problems over the Riemannian manifold of all fixed-rank partial isometries. The proposed method can extract not only the prescribed extreme generalized singular values but also associated triplets of generalized singular vectors. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Finally, numerical experiments on both synthetic and real data sets show the effectiveness and high accuracy of our method. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn–Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in L^1(\mathbb {R}) of the numerical solution. Our results partially answer positively to the open problem posed by J. Cui and J. Hong [J. Differential Equations 269 (2020), pp. 10143–10180] on computing the density of the exact solution numerically. [ABSTRACT FROM AUTHOR]

In this paper, we study ultra-weak discontinuous Galerkin methods with generalized numerical fluxes for multi-dimensional high order partial differential equations on both unstructured simplex and Cartesian meshes. The equations we consider as examples are the nonlinear convection-diffusion equation and the biharmonic equation. Optimal error estimates are obtained for both equations under certain conditions, and the key step is to carefully design global projections to eliminate numerical errors on the cell interface terms of ultra-weak schemes on general dimensions. The well-posedness and approximation capability of these global projections are obtained for arbitrary order polynomial space based on a wide class of generalized numerical fluxes on regular meshes. These projections can serve as general analytical tools to be naturally applied to a wide class of high order equations. Numerical experiments are conducted to demonstrate these theoretical results. [ABSTRACT FROM AUTHOR]

Interacting particle or agent systems that exhibit diverse swarming behaviors are prevalent in science and engineering. Developing effective differential equation models to understand the connection between individual interaction rules and swarming is a fundamental and challenging goal. In this paper, we study the data-driven discovery of a second-order particle swarming model that describes the evolution of N particles in \mathbb {R}^d under radial interactions. We propose a learning approach that models the latent radial interaction function as Gaussian processes, which can simultaneously fulfill two inference goals: one is the nonparametric inference of the interaction function with pointwise uncertainty quantification, and the other is the inference of unknown scalar parameters in the noncollective friction forces of the system. We formulate the learning problem as a statistical inverse learning problem and introduce an operator-theoretic framework that provides a detailed analysis of recoverability conditions, establishing that a coercivity condition is sufficient for recoverability. Given data collected from M i.i.d trajectories with independent Gaussian observational noise, we provide a finite-sample analysis, showing that our posterior mean estimator converges in a Reproducing Kernel Hilbert Space norm, at an optimal rate in M equal to the one in the classical 1-dimensional Kernel Ridge regression. As a byproduct, we show we can obtain a parametric learning rate in M for the posterior marginal variance using L^{\infty } norm and that the rate could also involve N and L (the number of observation time instances for each trajectory) depending on the condition number of the inverse problem. We provide numerical results on systems exhibiting different swarming behaviors, highlighting the effectiveness of our approach in the scarce, noisy trajectory data regime. [ABSTRACT FROM AUTHOR]

In 1950, Erdős posed a question known as the minimum modulus problem on covering systems for \mathbb {Z}, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough [Ann. of Math. (2) 181 (2015), no. 1, pp. 361–382] in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed 10^{16}. Recently, Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [Invent. Math. 228 (2022), pp. 377–414] developed a versatile method called the distortion method and significantly reduced Hough's bound to 616,000. In this paper, we apply this method to present a proof that the smallest degree of the moduli in any covering system for \mathbb {F}_q[x] of multiplicity s is bounded by a constant depending only on s and q. Consequently, we successfully resolve the minimum modulus problem for \mathbb {F}_q[x] and disprove a conjecture by Azlin [ Covering Systems of Polynomial Rings Over Finite Fields , University of Mississippi, Electronic Theses and Dissertations. 39, 2011]. [ABSTRACT FROM AUTHOR]

The objective of this paper is to characterize the structure of the set \Theta for a continuous ergodic upper probability \mathbb {V}=\sup _{P\in \Theta }P \Theta contains a finite number of ergodic probabilities; Any invariant probability in \Theta is a convex combination of those ergodic ones in \Theta; Any probability in \Theta coincides with an invariant one in \Theta on the invariant \sigma-algebra. The last property has already been obtained in Cerreia-Vioglio, Maccheroni, and Marinacci [Proc. Amer. Math. Soc. 144 (2016), pp. 3381–3396], which first studied the ergodicity of such capacities. As an application of the characterization, we prove an ergodicity result, which improves the result of Cerreia-Vioglio, Maccheroni, and Marinacci [Proc. Amer. Math. Soc. 144 (2016), pp. 3381–3396] in the sense that the limit of the time means of \xi is bounded by the upper expectation \sup _{P\in \Theta }E_P[\xi ], instead of the Choquet integral. Generally, the former is strictly smaller. [ABSTRACT FROM AUTHOR]

In this paper, we study and characterize the nilpotent global centers of a generalized polynomial Kukles system with degree three. A sufficient and necessary condition of global centers is established under certain parametric conditions. [ABSTRACT FROM AUTHOR]

Let \mathcal M be a semifinite factor with a fixed faithful normal semifinite tracial weight \tau such that \tau (I)=\infty. Denote by \mathscr P(\mathcal M,\tau) the set of all projections in \mathcal M and \mathscr P^{\infty }(\mathcal M,\tau)=\{P\in \mathscr P(\mathcal M,\tau): \tau (P)=\tau (I-P)=\infty \}. In this paper, as a generalization of Uhlhorn's theorem, we establish the general form of orthogonality preserving maps on the Grassmann space \mathscr P^{\infty }(\mathcal M,\tau). We prove that every such map on \mathscr P^{\infty }(\mathcal M,\tau) can be extended to a Jordan *-isomorphism \rho of \mathcal M onto \mathcal M. [ABSTRACT FROM AUTHOR]

We continue our study on the global dynamics of a non- local reaction-diffusion-advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where the species depend solely on light for their metabolism. In our previous works, we proved that system (1.1) is a strongly monotone dynamical system with respect to a non-standard cone, and some competitive exclusion results were obtained. In this paper, we aim to demonstrate the existence of coexistence steady state as well as competitive exclusion. Our results highlight that advection in dispersal strategy can lead to transitions between various competitive outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we study the global dynamics of epidemic network models with standard incidence or mass-action transmission mechanism, when the dispersal of either the susceptible or the infected people is controlled. The connectivity matrix of the model is not assumed to be symmetric. Our main technique to study the global dynamics is to construct novel Lyapunov type functions. [ABSTRACT FROM AUTHOR]

This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point potentials. In particular, we study three-point analogues of the sphere packing problem and the optimization problem for p-frame energies based on three points. It turns out that both problems are inherently connected to the problem of nearly orthogonal sets by Erdős. As the outcome, we provide a new solution of the Erdős problem from the three-point packing perspective. We also show that the orthogonal basis uniquely minimizes the p-frame three-point energy when 0

This paper studies the combinatorial Calabi flow for circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. By using a Lyapunov function, we show that the flow exists for all time and converges exponentially fast to a circle pattern metric with prescribed attainable curvatures. This provides an algorithm to search for the desired circle patterns. [ABSTRACT FROM AUTHOR]

Let \Gamma \subseteq \mathbb {N} be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of \Gamma which depends only on the width of \Gamma, that is, the difference between the largest and the smallest generator of \Gamma. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width. [ABSTRACT FROM AUTHOR]

Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of product and permutation categories (PROPs) and product and braid categories (PROBs) with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action. [ABSTRACT FROM AUTHOR]

The space \mathbf {H}^{4,2} of vectors of norm -1 in \mathbb {R}^{4,3} has a natural pseudo-Riemannian metric and a compatible almost complex structure. The group of automorphisms of both of these structures is the split real form \mathsf {G}_2'. In this paper we consider a class of holomorphic curves in \mathbf {H}^{4,2} which we call alternating. We show that such curves admit a so called Frenet framing. Using this framing, we show that the space of alternating holomorphic curves which are equivariant with respect to a surface group is naturally parameterized by certain \mathsf {G}_2'-Higgs bundles. This leads to a holomorphic description of the moduli space as a fibration over Teichmüller space with a holomorphic action of the mapping class group. Using a generalization of Labourie's cyclic surfaces, we then show that equivariant alternating holomorphic curves are infinitesimally rigid. [ABSTRACT FROM AUTHOR]

In this paper, we prove effective quantitative decorrelation of values of two Hecke eigenforms as the weight goes to infinity. As consequences, we get an effective version of equidistribution of mass and zeros of certain linear combinations of Hecke eigenforms. [ABSTRACT FROM AUTHOR]

In this paper, the author solves the long term open problem of Kerzman on sup-norm estimate for Cauchy-Riemann equation on polydisc in n-dimensional complex space. The problem has been open since 1971. He also extends and solves the problem on product domains \Omega ^n, where \Omega is any bounded domain in \mathbb {C} with C^{1,\alpha } boundary for some \alpha >0. [ABSTRACT FROM AUTHOR]

In this work, we prove a compactness theorem on the space of all Hamiltonian stationary Lagrangian submanifolds in a compact symplectic manifold with uniform bounds on area and total extrinsic curvature. This generalizes the compactness theorems in Chen and Warren [J. Differential Geom. 126 (2024), pp. 65–97] and Chen and Ma [Var. Partial Differential Equations 60 (2021), Paper No. 75, 23]. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a certain family of holomorphic correspondences on ... x ... and prove their equidistribution properties. In particular, for any correspondence in this family we prove that the naturally associated multivalued map F is such that for any a ∈ ..., we have that (Fn)*(δa) converges to a probability measure μF for which F*(μF) = μF d where d is the degree of F. This result is used to show that the minimal Hutchinson invariant set, introduced by P. Alexandersson, P. Brändén, and B. Shapiro [An inverse problem in Pólya--Schur theory. I. Non-degenerate and degenerate operators, preprint, 2024], of a large class of operators and for sufficiently large n exists and is the support of the aforementioned measure. We prove that under a minor additional assumption, the minimal Hutchinson-invariant set is a Cantor set. [ABSTRACT FROM AUTHOR]

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical Schur–Weyl duality and Soergel's Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B, and module categories of endomorphism algebras of generator-relative cogenerators over B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n so that the endomorphism algebra of a generator-cogenerator provides an n-faithful cover, in the sense of Rouquier, of B. The degree n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple modules of B. We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category \mathcal {O} establishing an integral version of Soergel's Struktursatz. We show that deformations of the combinatorial Soergel's functor have better homological properties than the classical one. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of k-Hessian equations, we can deduce an Obata-type formula and a Liouville-type theorem by integration by parts. [ABSTRACT FROM AUTHOR]

In the present paper, we study the shifted hypergeometric function f(z)=z_{2}F_{1}(a,b;c;z) for real parameters with 0

In this paper, we make the first attempt to figure out the differences on Hölder regularity in time of solutions and conserved physical quantities between the ideal electron magnetohydrodynamic equations concerning Hall term and the incompressible Euler equations involving convection term. It is shown that the regularity in time of magnetic field B is C_{t}^{\frac {\alpha }2} provided it belongs to L_{t}^{\infty } C_{x}^{\alpha } for any \alpha >0, its energy is C_{t}^{\frac {2\alpha }{2-\alpha }} as long as B belongs to L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0<\alpha <1 and its magnetic helicity is C_{t}^{\frac {2\alpha +1}{2-\alpha }} supposing B belongs to L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0<\alpha <\frac 12, which are quite different from the classical incompressible Euler equations. [ABSTRACT FROM AUTHOR]