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The aim of this paper is to give expressions for modular linear differential operators (MLDOs) of any order. In particular, we show that they can all be described in terms of Rankin-Cohen brackets and a modified Rankin-Cohen bracket found by Kaneko and Koike. We also give more uniform descriptions of MLDOs in terms of canonically defined higher Serre derivatives and an extension of Rankin-Cohen brackets, as well as in terms of quasimodular forms and almost holomorphic modular forms. The last of these descriptions involves the holomorphic projection map. The paper also includes some general results on the theory of quasimodular forms on both cocompact and non-cocompact subgroups of SL_2(\mathbb {R}), as well as a slight sharpening of a theorem of Martin and Royer on Rankin-Cohen brackets of quasimodular forms. [ABSTRACT FROM AUTHOR]

In this paper, we find some conditions under which a sequence \{\alpha (n)\} will satisfy the Laguerre inequality of any order asymptotically. Using this method, we prove that for any r and some constant c, the Maclaurin coefficients \gamma (n) of the Riemann Xi-function satisfy the Laguerre inequality of order r when n>cr^3, which provides a necessary condition for the Riemann hypothesis. We also prove that the partition function satisfies the Laguerre inequality of order r\geq 5 when n\geq 6r^4. As a consequence, it gives an affirmative answer to Wagner's conjecture on the threshold for the Laguerre inequalities of order no more than 10 for the partition function. Moreover, motivated by the study of Craven and Csordas on the complete monotonicity of the Maclaurin coefficients of entire functions in Laguerre-Pólya class, we consider the complete monotonicity of the sequences \{\alpha (n)\}. We give the criteria for the asymptotically complete monotonicity of the sequence \{\alpha (n)\} and \{\log \alpha (n)\}, respectively. With this criteria, we show that (-1)^r \Delta ^r \gamma (n)>0 for n>ce^{r^3} and (-1)^{r-1} \Delta ^r \log \gamma (n)>0 for n>cr^2. Furthermore, we propose some open problems. [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]

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]

In this paper, we study one-dimensional hyperbolic Anderson models (HAM) driven by space-time pure-jump Lévy white noise in a finite-variance setting. Motivated by recent active research on limit theorems for stochastic partial differential equations driven by Gaussian noises, we present the first study in this Lévy setting. In particular, we first establish the spatial ergodicity of the solution and then a quantitative central limit theorem (CLT) for the spatial averages of the solution to HAM in both Wasserstein distance and Kolmogorov distance, with the same rate of convergence. To achieve the first goal (i.e. spatial ergodicity), we exploit some basic properties of the solution and apply a Poincaré inequality in the Poisson setting, which requires delicate moment estimates on the Malliavin derivatives of the solution. Such moment estimates are obtained in a soft manner by observing a natural connection between the Malliavin derivatives of HAM and a HAM with Dirac delta velocity. To achieve the second goal (i.e. CLT), we need two key ingredients: (i) a univariate second-order Poincaré inequality in the Poisson setting that goes back to Last, Peccati, and Schulte (Probab. Theory Related Fields, 2016) and has been recently improved by Trauthwein (arXiv:2212.03782); (ii) aforementioned moment estimates of Malliavin derivatives up to second order. We also establish a corresponding functional CLT by (a) showing the convergence in finite-dimensional distributions and (b) verifying Kolmogorov's tightness criterion. Part (a) is made possible by a linearization trick and the univariate second-order Poincaré inequality, while part (b) follows from a standard moment estimate with an application of Rosenthal's inequality. [ABSTRACT FROM AUTHOR]

In this paper, we consider a constrained maximization problem related to planar vortex pairs with prescribed impulse. We prove existence, stability and asymptotic behavior for the maximizers, hence obtain a family of stable traveling vortex pairs approaching a pair of point vortices with equal magnitude and opposite signs. As a corollary, we get fine asymptotic estimates for Burton's vortex pairs with large impulse. We also consider a special non-concentrated case and prove a form of stability for the Chaplygin-Lamb dipole. [ABSTRACT FROM AUTHOR]

Wang and Ye [Adv. Math. 230 (2012), pp. 294–320] prove a Hardy–Moser–Trudinger inequality in dimension two which improves both the classical Moser–Trudinger inequality and the classical Hardy inequality in the unit disc. In this paper, we generalize their result both to the higher dimensional unit ball and to the singular weighted cases as well. More precisely, we prove that \[ \sup _{\substack {u\in W^{1,n}_0(\mathbb {B}^n),\\ \int _{\mathbb {B}^n} |\nabla u|^n dx -\left (\frac {2(n-1)}n\right)^n \int _{\mathbb {B}^n} \frac {|u|^n}{(1-|x|^2)^n} dx \leq 1}}\int _{\mathbb {B}^n} e^{(1-\frac \beta n)\alpha _n |u|^{\frac n{n-1}}} |x|^{-\beta } dx \leq C \] for any \beta \in [0,n), n\geq 3 where \alpha _n = n \omega _{n-1}^{\frac 1{n-1}} and \omega _{n-1} is the surface area of the n-1 dimensional unit sphere. The proof of Wang and Ye is based on the blow-up analysis method which seems not work in the higher dimensions. In this paper, we propose a new approach based on the method of transplantation of Green's functions to prove our inequality. As a consequence, we obtain a singular Moser–Trudinger inequality in the hyperbolic spaces which confirms affirmatively a conjecture by Mancini, Sandeep and Tintarev [Adv. Nonlinear Anal., 2 (2013), pp. 309–324]. [ABSTRACT FROM AUTHOR]

In this paper, we prove several rigidity results for compact initial data sets, in both the boundary and no boundary cases. In particular, under natural energy, boundary, and topological conditions, we obtain a global version of the main result by Galloway and Mendes [Comm. Anal. Geom. 26 (2018), pp. 63–83]. We also obtain some extensions of results by Eichmair, Galloway, and Mendes [Comm. Math. Phys. 386 (2021), pp. 253–268]. A number of examples are given in order to illustrate some of the results presented in this paper. [ABSTRACT FROM AUTHOR]

For each parabolic subgroup \mathrm {P} of the general linear group \operatorname {GL}_n(\mathbb {F}_q), a conjecture due to Lewis, Reiner and Stanton [Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), pp. 831–873] predicts a formula for the Hilbert series of the space of invariants \mathcal {Q}_m(n)^\mathrm {P} where \mathcal {Q}_m(n) is the quotient ring \mathbb {F}_q[x_1,\ldots,x_n]/(x_1^{q^m},\ldots,x_n^{q^m}). In this paper, we prove the conjecture for the Borel subgroup \mathrm {B} by constructing a linear basis for \mathcal {Q}_m(n)^\mathrm {B}. The construction is based on an operator \delta which produces new invariants from old invariants of lower ranks. We also upgrade the conjecture of Lewis, Reiner and Stanton by proposing an explicit basis for the space of invariants for each parabolic subgroup. [ABSTRACT FROM AUTHOR]

Non-Gaussian Harmonizable Fractional Stable Motion (HFSM) is a natural and important extension of the well-known Fractional Brownian Motion to the framework of heavy-tailed stable distributions. It was introduced several decades ago; however its properties are far from being completely understood. In our present paper we determine the optimal power of the logarithmic factor in a uniform modulus of continuity for HFSM, which solves an old open problem. The keystone of our strategy consists in Abel transforms of the LePage series expansions of the random coefficients of the wavelets series representation of HFSM. Our methodology can be extended to more general harmonizable stable processes and fields. [ABSTRACT FROM AUTHOR]

Let M=(M,\omega) be either S^2 \times S^2 or \mathbb {C}P^2\# \overline {\mathbb {C}P^2} endowed with any symplectic form \omega. Suppose a finite cyclic group \mathbb {Z}_n is acting effectively on (M,\omega) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism \mathbb {Z}_n\hookrightarrow Ham(M,\omega). In this paper, we investigate the homotopy type of the group Symp^{\mathbb {Z}_n}(M,\omega) of equivariant symplectomorphisms. We prove that for some infinite families of \mathbb {Z}_n actions satisfying certain inequalities involving the order n and the symplectic cohomology class [\omega ], the actions extend to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on J-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on 4-manifolds, and on the Chen-Wilczyński classification of smooth \mathbb {Z}_n-actions on Hirzebruch surfaces. [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 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, 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]

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 use the Riemann zeta distribution to give a new proof of the Erdös-Kac Central Limit Theorem. That is, if \zeta (s)=\sum _{n\ge 1} \frac {1}{n^s}, s>1, then we consider the random variable X_s with P(X_s=n)=\frac {1}{\zeta (s)n^s}, n\ge 1. In an earlier paper, the first author and Adrien Peltzer derived the analog of the Erdös-Kac Central Limit Theorem (CLT) for the number of distinct prime factors, \omega (X_s), of X_s, as s\searrow 1. In this paper we show, by means of a Tauberian Theorem, how to obtain the Central Limit Theorem of Erdös-Kac for the uniform distribution from the result for the random variable X_s. We also apply the technique to the number of distinct prime divisors of X_s that lie in an arithmetic sequence and a local CLT of the type proved by Dixit and Murty [Hardy-Ramanujan J. 43 (2020), 17–23] as well a version of the CLT for irreducible divisors of a monic polynomial over a finite field. [ABSTRACT FROM AUTHOR]

As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in TDA to articulate stability and inference theorems. Intuitively, whereas a strict interleaving between filtered spaces X and Y certifies that X and Y are approximately isomorphic, a homotopy interleaving between X and Y certifies that X and Y are approximately weakly equivalent. The main results of this paper are that homotopy interleavings induce an extended pseudometric d_{HI} on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms. To motivate these axioms, we also observe that d_{HI} (or more generally, any pseudometric satisfying these two axioms and an additional "homology bounding" axiom) can be used to formulate lifts of several fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces. Finally, we consider the problem of establishing a persistent Whitehead theorem in terms of homotopy interleavings. We provide a counterexample to a naive formulation of the result. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence and uniqueness of bounded solutions of the degenerate complex Monge-Ampère type equations on compact Hermitian manifolds and obtain the asymptotics of these solutions. As applications, we give partial answers to the Tosatti-Weinkove conjecture and Demailly-Păun conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we prove the compressible Euler limit from the Boltzmann equation with hard sphere collisional kernel and complete diffusive boundary condition in half-space by employing the Hilbert expansion which includes interior and Knudsen layers. This rigorously justifies the corresponding formal analysis in Sone's book [ Molecular gas dynamics , Birkhäuser Boston, Inc., Boston, MA, 2007] in the context of short time smooth solutions, and also generalizes the classic Caflisch's result [Comm. Pure Appl. Math. 33 (1980), pp. 651–666] to initial-boundary problem case. [ABSTRACT FROM AUTHOR]

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in \mathbb {P}^5, and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known. In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in \mathbb {P}^4 and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces. [ABSTRACT FROM AUTHOR]

In this paper, for any compact Lie group G, we show that the space of G-equivariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible. In particular, we prove the generalized Smale conjecture for spherical three-orbifolds. Moreover, for connected G, we make a classification of all PSC G-equivariant three-manifolds. [ABSTRACT FROM AUTHOR]

F-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the F-concavities preserved by the Dirichlet heat flow in convex domains on {\mathbb R}^n, and complete the study of preservation of concavity properties by the Dirichlet heat flow, started by Brascamp and Lieb in 1976 and developed in some recent papers. More precisely, we discover hot-concavity, which is the strongest F-concavity preserved by the Dirichlet heat flow; we show that log-concavity is the weakest F-concavity preserved by the Dirichlet heat flow; quasi-concavity is also preserved only for n=1; we prove that if F-concavity is strictly weaker than log-concavity and n\ge 2, then there exists an F-concave initial datum such that the corresponding solution to the Dirichlet heat flow is not even quasi-concave, hence losing any reminiscence of concavity. Furthermore, we find a sufficient and necessary condition for F-concavity to be preserved by the Dirichlet heat flow. We also study the preservation of concavity properties by solutions of the Cauchy–Dirichlet problem for linear parabolic equations with variable coefficients and for nonlinear parabolic equations such as semilinear heat equations, the porous medium equation, and the parabolic p-Laplace equation. [ABSTRACT FROM AUTHOR]

For a harmonic diffeomorphism between the Poincaré disks, Wan [J. Differential Geom. 35 (1992), pp. 643–657] showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to r-differentials. We study the relationship between bounded holomorphic r-differentials/(r-1)-differential and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space SL(r,\mathbb R)/SO(r) arising from cyclic/subcyclic Higgs bundles. Also, we show the equivalence between the boundedness of holomorphic differentials and having a negative upper bound of the induced curvature on hyperbolic affine spheres in \mathbb {R}^3, maximal surfaces in \mathbb {H}^{2,n} and J-holomorphic curves in \mathbb {H}^{4,2}. Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods. [ABSTRACT FROM AUTHOR]

In this paper we study the regularity properties of \Lambda-minimizers of the capillarity energy in a half space with the wet part constrained to be confined inside a given planar region. Applications to a model for nanowire growth are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Reshetikhin-Turaev invariants of knots in the three-sphere obtained from a twisted Drinfeld double of a Hopf algebra, or equivalently, the relative Drinfeld center of the crossed product \text {Rep}(H)\rtimes \text {Aut}(H). These are quantum invariants of knots endowed with a homomorphism of the knot group to \text {Aut}(H). We show that, at least for knots in the three-sphere, these invariants provide a non-involutory generalization of the Fox-calculus-twisted Kuperberg invariants of sutured manifolds introduced previously by the author, which are only defined for involutory Hopf algebras. In particular, we describe the SL(n,\mathbb {C})-twisted Reidemeister torsion of a knot complement as a Reshetikhin-Turaev invariant. [ABSTRACT FROM AUTHOR]

In this paper we use the convex integration technique enhanced by an extra iteration originally due to Källén and revisited by Kröner to provide a local h-principle for isometric embeddings in the class C^{1,1-\varepsilon } for n-dimensional manifolds in codimension \frac {1}{2}n(n+1). [ABSTRACT FROM AUTHOR]

The Nernst-Planck-Stokes (NPS) system models electroconvection of ions in a fluid. We consider the system, for two oppositely charged ionic species, on three dimensional bounded domains with Dirichlet boundary conditions for the ionic concentrations (modelling ion selectivity), Dirichlet boundary conditions for the electrical potential (modelling an applied potential), and no-slip boundary conditions for the fluid velocity. In this paper, we obtain quantitative bounds on solutions of the NPS system in the long time limit, which we use to prove (1) the existence of a compact global attractor with finite fractal (box-counting) dimension and (2) space-time averaged electroneutrality \rho \approx 0 in the singular limit of Debye length going to zero, \epsilon \to 0. [ABSTRACT FROM AUTHOR]

This paper mainly concerns the biholomorphic invariance of p-essential normality of Hilbert modules on bounded symmetric domains. By establishing new integral formulas concerning rational function kernels for the Taylor functional calculus, we prove a biholomorphic invariance result related to the p-essential normality. Furthermore, for quotient analytic Hilbert submodules determined by analytic varieties, we develop an algebraic approach to proving that the p-essential normality is preserved invariant if the coordinate multipliers are replaced by arbitrary automorphism multipliers. Moreover, the Taylor spectrum of the compression tuple is calculated under a mild condition, which gives a solvability result of the corona problem for quotient submodules. As applications, we extend the recent results on the equivalence between \infty-essential normality and hyperrigidity. [ABSTRACT FROM AUTHOR]

Let 0 < \beta \leq \alpha < 1 and \kappa > 0. I prove that there exists \eta > 0 such that the following holds for every pair of Borel sets A,B \subset \mathbb {R} with \dim _{\mathrm {H}} A = \alpha and \dim _{\mathrm {H}} B = \beta: \begin{equation*} \dim _{\mathrm {H}} \{c \in \mathbb {R}: \dim _{\mathrm {H}} (A + cB) \leq \alpha + \eta \} \leq \tfrac {\alpha - \beta }{1 - \beta } + \kappa. \end{equation*} This extends a result of Bourgain from 2010, which contained the case \alpha = \beta. The paper also contains a \delta-discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form a_{1}B + \ldots + a_{n}B. [ABSTRACT FROM AUTHOR]

This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on \mathbb {R}^n. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on \mathbb {R}^n. [ABSTRACT FROM AUTHOR]

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, L^{p}-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures. [ABSTRACT FROM AUTHOR]

It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the q-Morris constant term identity[J. Combin. Theory Ser. A 81 (1998), pp. 69–87]. This conjecture was proved and extended by Károlyi, Nagy, Petrov, and Volkov (KNPV) in 2015 [Adv. Math. 277 (2015), pp. 252–282]. In this paper, we obtain two symmetric function generalizations of the q-Baker–Forrester ex-conjecture. These include: (i) a q-Baker–Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's result. [ABSTRACT FROM AUTHOR]

In this paper, we study, for functionals having a minimax geometry on a constraint, the existence of bounded Palais-Smale sequences carrying Morse index type information. [ABSTRACT FROM AUTHOR]

This paper is to investigate the spectral properties of separable covariance matrices under elliptical populations. The separable covariance matrix model can handle both cross-row and cross-column correlations thus gain more popularity recently. Under the high-dimensional setting where the dimension p and the sample size n tend to infinity proportionally, we find the limit of the empirical spectral distribution and establish the central limit theorems (CLT) for linear spectral statistics of such kinds of sample covariance matrices. Some applications of our established CLT are also given. [ABSTRACT FROM AUTHOR]

For a semisimple Lie algebra \frak {sl}_n, the basic representation L_{\widehat {\frak {sl}_{n}}}(1,0) of the affine Lie algebra \widehat {\frak {sl}_{n}} is a lattice vertex operator algebra. The first main result of the paper is to prove that the commutant vertex operator algebra of L_{\widehat {\frak {sl}_{n}}}(l,0) in the l-fold tensor product L_{\widehat {\frak {sl}_{n}}}(1,0)^{\otimes l} is isomorphic to the parafermion vertex operator algebra K(\frak {sl}_{l},n), which is the commutant of the Heisenberg vertex operator algebra L_{\widehat {\frak {h}}}(n,0) in L_{\widehat {\frak {sl}_l}}(n,0). The result provides a version of level-rank duality. The second main result of the paper is to prove more general version of the first result that the commutant of L_{\widehat {\frak {sl}_{n}}}(l_1+\cdots +l_s, 0) in L_{\widehat {\frak {sl}_{n}}}(l_1,0)\otimes \cdots \otimes L_{\widehat {\frak {sl}_{n}}}(l_s, 0) is isomorphic to the commutant of the vertex operator algebra generated by a Levi Lie subalgebra of \frak {sl}_{l_1+\cdots +l_s} corresponding to the composition (l_1, \cdots, l_s) in the rational vertex operator algebra L_{\widehat {\frak {sl}}_{l_1+\cdots +l_s}}(n,0). This general version also resembles a version of reciprocity law discussed by Howe in the context of reductive Lie groups. In the course of the proof of the main results, certain Howe duality pairs also appear in the context of vertex operator algebras. [ABSTRACT FROM AUTHOR]

In this paper we explore the extent to which the algebraic structure of a monoid M determines the topologies on M that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or T_1 topologies; Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids and inverse monoids. If M is a topological monoid such that every homomorphism from M to a second countable topological monoid N is continuous, then we say that M has automatic continuity. We show that many well-known, and extensively studied, monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid \mathbb {N} ^\mathbb {N}; the full binary relation monoid B_{\mathbb {N}}; the partial transformation monoid P_{\mathbb {N}}; the symmetric inverse monoid I_{\mathbb {N}}; the monoid \operatorname {Inj}(\mathbb {N}) consisting of the injective transformations of \mathbb {N}; and the monoid C(2^{\mathbb {N}}) of continuous functions on the Cantor set 2^{\mathbb {N}}. The monoid \mathbb {N} ^\mathbb {N} can be equipped with the product topology, where the natural numbers \mathbb {N} have the discrete topology; this topology is referred to as the pointwise topology. We show that the pointwise topology on \mathbb {N} ^\mathbb {N}, and its analogue on P_{\mathbb {N}}, is the unique Polish semigroup topology on these monoids. The compact-open topology is the unique Polish semigroup topology on C(2 ^\mathbb {N}), and on the monoid C([0, 1] ^\mathbb {N}) of continuous functions on the Hilbert cube [0, 1] ^\mathbb {N}. The symmetric inverse monoid I_{\mathbb {N}} has at least 3 Polish semigroup topologies, but a unique Polish inverse semigroup topology. The full binary relation monoid B_{\mathbb {N}} has no Polish semigroup topologies, nor do the partition monoids. At the other extreme, \operatorname {Inj}(\mathbb {N}) and the monoid \operatorname {Surj}(\mathbb {N}) of all surjective transformations of \mathbb {N} each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on \mathbb {N} ^\mathbb {N}, P_{\mathbb {N}}, and \operatorname {Inj}(\mathbb {N}) coincide with the pointwise topology; and we characterise the Zariski topology on B_{\mathbb {N}}. Along the way we provide many additional results relating to the Markov topology, the small index property for monoids, and topological embeddings of semigroups in \mathbb {N}^{\mathbb {N}} and inverse monoids in I_{\mathbb {N}}. Finally, the techniques developed in this paper to prove the results about monoids are applied to function clones. In particular, we show that: the full function clone has a unique Polish topology; the Horn clone, the polymorphism clones of the Cantor set and the countably infinite atomless Boolean algebra all have automatic continuity with respect to second countable function clone topologies. [ABSTRACT FROM AUTHOR]

In this paper we study \times _0-products of Lannér diagrams. We prove that every \times _0-product of at least four Lannér diagrams with at least one diagram of order \geq 3 is superhyperbolic. As a corollary, we obtain that known classifications exhaust all compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of simplices. We also consider compact hyperbolic Coxeter polytopes whose every Lannér subdiagram has order 2. The second result of this paper slightly improves recent Burcroff's upper bound on the dimension of such polytopes to 12. [ABSTRACT FROM AUTHOR]

This paper is part of a program to establish the existence theory for the conformally invariant Dirac equation \[ D_g\psi =f(x)|\psi |_g^{\frac 2{m-1}}\psi \] on a closed spin manifold (M,g) of dimension m\geq 2 with a fixed spin structure, where f:M\to \mathbb {R} is a given function. The study on such nonlinear equation is motivated by its important applications in Spin Geometry: when m=2, a solution corresponds to an isometric immersion of the universal covering \widetilde M into \mathbb {R}^3 with prescribed mean curvature f; meanwhile, for general dimensions and f\equiv constant, a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant. Comparing with the existing issues, the aim of this paper is to establish multiple existence results in a new geometric context, which have not been considered in the previous literature. Precisely, in order to examine the dependence of solutions of the aforementioned nonlinear Dirac equations on geometrical data, concrete analyses are made for two specific models on the manifold (S^m,g): the geometric potential f is a perturbation from constant with g=g_{S^m} being the canonical round metric; and f\equiv 1 with the metric g being a perturbation of g_{S^m} that is not conformally flat somewhere on S^m. The proof is variational: solutions of these problems are found as critical points of their corresponding energy functionals. The emphasis is that the solutions are always degenerate: they appear as critical manifolds of positive dimension. This is very different from most situations in elliptic PDEs and classical critical point theory. As corollaries of the existence results, multiple distinct embedded spheres in \mathbb {R}^3 with a common mean curvature are constructed, and furthermore, a strict inequality estimate for the Bär-Hijazi-Lott invariant on S^m, m\geq 4, is derived, which is the first result of this kind in the non-locally-conformally-flat setting. [ABSTRACT FROM AUTHOR]

We consider random orthonormal polynomials \begin{equation*} F_{n}(x)=\sum _{i=0}^{n}\xi _{i}p_{i}(x), \end{equation*} where \xi _{0}, ..., \xi _{n} are independent random variables with zero mean, unit variance and uniformly bounded (2+\varepsilon) moments, and (p_n)_{n=0}^{\infty } is the system of orthonormal polynomials with respect to a fixed compactly supported measure on the real line. Under mild technical assumptions satisfied by many classes of classical polynomial systems, we establish universality for the leading asymptotics of the average number of real roots of F_n, both globally and locally. Prior to this paper, these results were known only for random orthonormal polynomials with Gaussian coefficients (see D. D. Lubinsky, I. E. Pritsker, and X. Xie [Proc. Amer. Math. Soc. 144 (2016), pp. 1631–1642]) using the Kac-Rice formula, a method that does not extend to the generality of our paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a poset isomorphism between two support \tau-tilting posets. We take several information from combinatorial properties of support \tau-tilting posets and give an analogue of Happel-Unger's reconstruction theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds (M^{n},g) with boundary and with dimension n<8 that was established by McCormick [Proc. Amer. Math. Soc. 146 (2018), pp. 4039–4046]. First, we show that any asymptotically flat static (M^{n},g) which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems. As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension n < 8 under the natural condition of Schwarzschild stability. This generalizes an earlier result of Miao [Classical Quantum Gravity 22 (2005), pp. L53–L59]. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes, e.g. see V. Agostiniani and L. Mazzieri [Comm. Math. Phys. 355 (2017), pp. 261–301], C. Cederbaum and G. J. Galloway [J. Math. Phys. 62 (2021), p. 22], and S. Raulot [Classical Quantum Gravity 38 (2021), p. 22]. [ABSTRACT FROM AUTHOR]

Let G be a reductive group over a field of prime characteristic. An indecomposable tilting module for G whose highest weight lies above the Steinberg weight has a character that is divisible by the Steinberg character. The resulting "Steinberg quotient" carries important information about G-modules, and in previous work we studied patterns in the weight multiplicities of these characters. In this paper we broaden our scope to include quantum Steinberg quotients, and show how the multiplicities in these characters relate to algebraic Steinberg quotients, Weyl characters, and evaluations of Kazhdan-Lusztig polynomials. We give an explicit algorithm for computing minimal characters that possess a key attribute of Steinberg quotients. We provide computations which show that these minimal characters are not always equal to quantum Steinberg quotients, but are close in several nontrivial cases. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE. [ABSTRACT FROM AUTHOR]

Let \mu and \nu be probability measures in the complex plane, and let p and q be independent random polynomials of degree n, whose roots are chosen independently from \mu and \nu, respectively. Under assumptions on the measures \mu and \nu, the limiting distribution for the zeros of the sum p+q was computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021), p. 124719] as n \to \infty. In this paper, we generalize and extend this result to the case where p and q have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of \mu and \nu, scaled by the limiting ratio of the degrees of p and q. Additionally, our approach provides a complete description of the limiting distribution for the zeros of p + q for any pair of measures \mu and \nu, with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment. [ABSTRACT FROM AUTHOR]

A chain complex can be viewed as a representation of a certain self-injective quiver with relations, Q. To define Q, include a vertex q_n and an arrow q_n \xrightarrow {\partial } q_{n-1} for each integer n. The relations are \partial ^2 = 0. Replacing Q by a general self-injective quiver with relations, it turns out that some of the key properties of chain complexes generalise. Indeed, consider the representations of such a Q with values in {}_{A}\mspace {-1mu}\operatorname {Mod} where A is a ring. We showed in earlier work that these representations form the objects of the Q-shaped derived category , \mathcal {D}_{Q}(A), which is triangulated and generalises the classic derived category \mathcal {D}_{}(A). This follows ideas of Iyama and Minamoto. While \mathcal {D}_{Q}(A) has many good properties, it can also diverge dramatically from \mathcal {D}_{}(A). For instance, let Q be the quiver with one vertex q, one loop \partial, and the relation \partial ^2 = 0. By analogy with perfect complexes in the classic derived category, one may expect that a representation with a finitely generated free module placed at q is a compact object of \mathcal {D}_{Q}(A), but we will show that this is, in general, false. The purpose of this paper, then, is to compare and contrast \mathcal {D}_{Q}(A) and \mathcal {D}_{}(A) by investigating several key classes of objects: Perfect and strictly perfect, compact, fibrant, and cofibrant. [ABSTRACT FROM AUTHOR]

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C^\ast-algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C^\ast-algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space. [ABSTRACT FROM AUTHOR]

Given a fixed \alpha \in (0,1), we study the inverse problem of recovering the isometry class of a smooth closed and connected Riemannian manifold (M,g), given the knowledge of a source-to-solution map for the fractional Laplace equation (-\Delta _g)^\alpha u=f on the manifold subject to an arbitrarily small observation region \mathcal O where sources can be placed and solutions can be measured. This can be viewed as a non-local analogue of the well known anisotropic Calderón problem that is concerned with the limiting case \alpha =1. In this paper, we solve the non-local problem under the assumption that the a priori known observation region \mathcal O belongs to some Gevrey class while making no geometric assumptions on the inaccessible region of the manifold, namely M\setminus \mathcal O. Our proof is based on discovering a connection to a variant of Carlson's theorem in complex analysis that reduces the inverse problem to Gel'fand inverse spectral problem. [ABSTRACT FROM AUTHOR]

In this paper, we construct a compactification of the space of Bridgeland stability conditions on a smooth projective curve, as an analogue of Thurston compactifications in Teichmüller theory. In the case of elliptic curves, we compare our results with the classical one of the torus via homological mirror symmetry and give the Nielsen–Thurston classification of autoequivalences using the compactification. Furthermore, we observe an interesting phenomenon in the case of the projective line. [ABSTRACT FROM AUTHOR]

The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of C_{0}-semigroups \left (\mathcal {V}(t)\right)_{t \geqslant 0} in L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d}) governing conservative linear kinetic equations on the torus with general scattering kernel \boldsymbol {k}(v,v') and degenerate (i.e. not bounded away from zero) collision frequency \sigma (v)=\int _{\mathbb {R}^{d}}\boldsymbol {k}(v',v)\boldsymbol {m}(\mathrm {d}v'), (with \boldsymbol {m}(\mathrm {d}v) being absolutely continuous with respect to the Lebesgue measure). We show in particular that if N_{0} is the maximal integer s \geqslant 0 such that \begin{equation*} \frac {1}{\sigma (\cdot)}\int _{\mathbb {R}^{d}}\boldsymbol {k}(\cdot,v)\sigma ^{-s}(v)\boldsymbol {m}(\mathrm {d}v) \in L^{\infty }(\mathbb {R}^{d}), \end{equation*} then, for initial datum f such that \displaystyle \int _{\mathbb {T}^{d}\times \mathbb {R}^{d}}|f(x,v)|\sigma ^{-N_{0}}(v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v) <\infty it holds \begin{equation*} \left \|\mathcal {V}(t)f-\varrho _{f}\Psi \right \|_{L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})}=\dfrac {{\varepsilon }_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho _{f}≔\int _{\mathbb {R}^{d}}f(x,v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v), \end{equation*} where \Psi is the unique invariant density of \left (\mathcal {V}(t)\right)_{t \geqslant 0} and \lim _{t\to \infty }{\varepsilon }_{f}(t)=0. We in particular provide a new criteria of the existence of invariant density. The proof relies on the explicit computation of the time decay of each term of the Dyson-Phillips expansion of \left (\mathcal {V}(t)\right)_{t \geqslant 0} and on suitable smoothness and integrability properties of the trace on the imaginary axis of Laplace transform of remainders of large order of this Dyson-Phillips expansion. Our construction resorts also on collective compactness arguments and provides various technical results of independent interest. Finally, as a by-product of our analysis, we derive essentially sharp "subgeometric" convergence rate for Markov semigroups associated to general transition kernels. [ABSTRACT FROM AUTHOR]

In this paper we prove that among all convex domains of the plane with two axes of symmetry, the maximizer of the first non-trivial Neumann eigenvalue \mu _1 with perimeter constraint is achieved by the square and the equilateral triangle. Part of the result follows from a new general bound on \mu _1 involving the minimal width over the area. Our main result partially answers to a question addressed in 2009 by R. S. Laugesen, I. Polterovich, and B. A. Siudeja. [ABSTRACT FROM AUTHOR]