In this paper, we study the rigidity under $K{\ddot{a}}hler$ deformation of the complex structure of odd Lagrangian Grassmannians, i.e., the Lagrangian case $Gr_{\omega}$(n, 2n+1) of odd symplectic Grassmannians. To obtain the global deformation rigidity of the odd Lagrangian Grassmannian, we use results about the automorphism group of this manifold, the Lie algebra of infinitesimal automorphisms of the affine cone of the variety of minimal rational tangents and its prolongations.

For positive integers d and n, let $[n]^d$ be the set of all vectors ($a_1,a_2,{\cdots},a_d$), where ai is an integer with $0{\leq}a_i{\leq}n-1$. A subset S of $[n]^d$ is called a Sidon set if all sums of two (not necessarily distinct) vectors in S are distinct. In this paper, we estimate two numbers related to the maximum size of Sidon sets in $[n]^d$. First, let $\mathcal{Z}_{n,d}$ be the number of all Sidon sets in $[n]^d$. We show that ${\log}(\mathcal{Z}_{n,d})={\Theta}(n^{d/2})$, where the constants of ${\Theta}$ depend only on d. Next, we estimate the maximum size of Sidon sets contained in a random set $[n]^d_p$, where $[n]^d_p$ denotes a random set obtained from $[n]^d$ by choosing each element independently with probability p.

The zero-divisor graph of a noncommutative ring R, denoted by ${\Gamma}(R)$, is a graph whose vertices are nonzero zero-divisors of R, and there is a directed edge from a vertex x to a distinct vertex y if and only if xy = 0. Let $R=M_2(F_q)$ be the $2{\times}2$ matrix ring over a finite field $F_q$. In this article, we investigate the automorphism group of ${\Gamma}(R)$.

We prove an Omori-Yau maximum principle on Alexandrov spaces which do not have Perelman singular points and satisfy the infinitesimal Bishop-Gromov condition.

All rings are commutative with $1{\neq}0$ and n is a positive integer. Let ${\phi}:{\Im}(R){\rightarrow}{\Im}(R){\cup}\{{\emptyset}\}$ be a function where ${\Im}(R)$ denotes the set of all ideals of R. We say that a proper ideal I of R is ${\phi}$-n-absorbing primary if whenever $a_1,a_2,{\cdots},a_{n+1}{\in}R$ and $a_1,a_2,{\cdots},a_{n+1}{\in}I{\backslash}{\phi}(I)$, either $a_1,a_2,{\cdots},a_n{\in}I$ or the product of $a_{n+1}$ with (n-1) of $a_1,{\cdots},a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of ${\phi}$-n-absorbing primary ideals.

In the present note, we deal with $L^2$ harmonic 1-forms on complete submanifolds with weighted $Poincar{\acute{e}}$ inequality. By supposing submanifold is stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^2$ harmonic 1-forms, which are some extension of the results of Kim and Yun, Sang and Thanh, Cavalcante Mirandola and $Vit{\acute{o}}rio$.

We present regularity conditions for suitable weak solutions of the Navier-Stokes equations with slip boundary data near the curved boundary. To be more precise, we prove that suitable weak solutions become regular in a neighborhood boundary points, provided the scaled mixed norm $L^{p,q}_{x,t}$ with 3/p + 2/q = 2, $1{\leq}q$ < ${\infty}$ is sufficiently small in the neighborhood.

In this paper, we study piecewise Noetherian (resp., piecewise w-Noetherian) properties in several settings including flat (resp., t-flat) overrings, Nagata rings, integral domains of finite character (resp., w-finite character), pullbacks of a certain type, polynomial rings, and D + XK[X] constructions.

This article gives upper bounds on the number of Fourier-Jacobi coefficients that determine a paramodular cusp form in degree two. The level N of the paramodular group is completely general throughout. Additionally, spaces of Jacobi cusp forms are spanned by using the theory of theta blocks due to Gritsenko, Skoruppa and Zagier. We combine these two techniques to rigorously compute spaces of paramodular cusp forms and to verify the Paramodular Conjecture of Brumer and Kramer in many cases of low level. The proofs rely on a detailed description of the zero dimensional cusps for the subgroup of integral elements in each paramodular group.

In this paper, we first introduce two one-parameter relaxation (OPR) iterative methods for solving singular saddle point problems whose semi-convergence rate can be accelerated by using scaled preconditioners. Next we present formulas for finding their optimal parameters which yield the best semi-convergence rate. Lastly, numerical experiments are provided to examine the efficiency of the OPR methods with scaled preconditioners by comparing their performance with the parameterized Uzawa method with optimal parameters.

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t] and define a random vector $Z_n:C[0,t]{\rightarrow}{\mathbb{R}}^n$ by $Zn(x)=(\int_{0}^{t_1}h(s)dx(s),{\cdots},\int_{0}^{t_n}h(s)dx(s))$, where 0 < $t_1$ < ${\cdots}$ < $t_n$ < t is a partition of [0, t] and $h{\in}L_2[0,t]$ with $h{\neq}0$ a.e. In this paper we will introduce a simple formula for a generalized conditional Wiener integral on C[0, t] with the conditioning function $Z_n$ and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $F(x)=f(\int_{0}^{t}e(s)dx(s))$ for $x{\in}C[0,t]$, where $f{\in}L_p(\mathbb{R})(1{\leq}p{\leq}{\infty})$ and e is a unit element in $L_2[0,t]$. Finally we express the generalized analytic conditional Feynman integral of F as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of $L_2[0,t]$ used in the transformation is independent of e and the conditioning function $Z_n$ does not contain the present positions of the generalized Wiener paths.