Liouville’s theorem, a landmark in ergodic theory, states that Lebesgue measure is invariant under the flow of Hamiltonian system. In this paper, we obtain a global existence of solutions for the defocusing cubic nonlinear Schrödinger equations on 2-sphere with initial data distributed according to Gibbs measure, which is invariant in some mild sense under the flow of Wick renormalized Hamiltonian. After modulating the manifold, we also improve the regularity of support space of Gibbs measure and establish the almost sure global well-posedness.

In this paper, we discuss the structural information about 2-arc-transitive (non-bipartite and bipartite) graphs of product action type. It is proved that a 2-arc-transitive graph of product action type requires certain restrictions on either the vertex-stabilizers or the valency. Based on the existence of some equidistant linear codes, a construction is given for 2-arc-transitive graphs of non-diagonal product action type, which produces several families of such graphs. Besides, a nontrivial construction is given for 2-arc-transitive bipartite graphs of diagonal product action type.

In this paper, we mainly establish some congruences involving binomial coefficients and Apéry-like numbers, for example, we prove the following result which was conjectured by Z.-H. Sun: Let p > 3 be a prime. Then

where L, M are integers and

W_n=\sum \begin{array}{c}\lfloor \frac{n}{3}\rfloor \\ k=0\end{array}\left(\begin{array}{c}2k\\ k\end{array}\right)\left(\begin{array}{c}3k\\ k\end{array}\right)\left(\begin{array}{c}n\\ 3k\end{array}\right)(-3{)}^{n-3k}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$W_n=\sum\begin{array}{c}\lfloor\frac{n}{3}\rfloor\\k=0\end{array}\left(\begin{array}{c}2k\\ k\end{array}\right)\left(\begin{array}{c}3k\\k\end{array}\right)\left(\begin{array}{c}n\\ 3k\end{array}\right)(-3)^{n-3k}$$\end{document}

are the second kind Apéry-like numbers.

In this note, we derive another kernel formula for Bessel functions attached to supercuspidal representations of GL_n(F) over p-adic fields, and obtain another Kirillov model as a corollary.

In this paper, we classify globally generated vector bundles with the first Chern class equal to 1 over Grassmannian G(d,n) (1 ≤ d ≤ n − d) over an algebraically closed field in characteristic zero, from the perspective of uniform vector bundles. In particular, we show that they are homogeneous.

A third order tensor is a three-way array which can be used to describe high dimensional data-set in dimensional reduction and clustering. In this paper, we mainly study the third order tensors. We introduce the slice identity tensor after the introduction of the slice product and Kronecker product defined on the set of third order tensors, and then we investigate the invertibility of the 3-order tensors in terms of the slice product. We also introduce the slice-diagonal tensors and investigate their spectrums.

In this paper, our goal is to introduce n-Gorenstein silting and n-(Gorenstein) FP-cosilting modules, and uncover the precise circumstances that are both required and sufficient for these modules to demonstrate the distinct features. We prove that the character module of an n-Gorenstein silting module with respect to finite-type

is n-Gorenstein cosilting. Furthermore, we give the connections between n-Gorenstein silting, n-Gorenstein tilting, n-Gorenstein star and n-Gorenstein quasi-tilting modules, and show that if a left R module T satisfies that Copres_G(Pres_Gⁿ(T)) = R-Mod, then the four above are equivalent, which more closely tie the silting, tilting and star theories in the context of Gorenstein homological algebras. We point out that the deleted n-Gorenstein projective (injective) resolutions of partial n-Gorenstein (co)silting modules are (n + 1)-Gorenstein pre(co)silting complexes. Finally, we introduce n-(Gorenstein) FP-cosilting modules. We investigate n-FP-cosilting over some extensions and obtain the relations among n-(Gorenstein) FP-cosilting, n-(Gorenstein) cosilting and Gorenstein weak n-silting modules.

In this paper, we give the cohomology of 3-Bihom-Lie algebras and we show that an α²β⁻¹-derivation is a closed 1-Bihom-cochain with the adjoint representation. As an application, we introduce abelian extensions of 3-Bihom-Lie algebras and obtain that there is a one-to-one correspondence between equivalent classes of abelian extensions and the second cohomology group by closed 2-Bihom-cochains. Moreover, we introduce the notion of a generalized derivation of 3-Bihom-Lie algebras and we construct a new 3-Bihom-Lie algebra with a generalized derivation.

The Hardy constant for a domain Ω ⊂ ℝⁿ is defined as the best constant for the Hardy inequality

In this paper we determine the Hardy constants for wedged domains. In particular we show the dependence of Hardy constant on n and the angle φ of wedged domain. Our results also reflect the influence of boundary smoothness and location of singular point on Hardy constant. Some improved Hardy inequalities on special wedged domains are also obtained.

In this article, we discuss a class of doubly perturbed distribution dependent stochastic differential equations (SDEs). We give some sufficient conditions to ensure the existence and uniqueness of solutions to the considered equations under some weak conditions, where a convex combination of the Nagumo and Osgood conditions is taken into consideration. Meanwhile, our results extend the classical hypothesis ∣α∣ + ∣β∣ < 1 on the perturbed intensity α and β to the generalized case. And then, we study the stability properties in regard to the initial value and coefficients.