Exact Inclusions of Gehring Classes in Muckenhoupt Classes

An exact bound for exponents of Muckenhoupt classes of functions containing a given Gehring class is obtained in the one-dimensional case.     

On a Problem of Dobrowolski and Williams and the Pólya-Vinogradov Inequality

In [2] E. Dobrowolski and K.S. Williams considered a problem of obtaining estimates for the sum _n=a+1 ^a+N f(n),for a certain class of functions f. One specific application of their result is a new method for estimating character sums. In particular, they obtain a form of the Pólya-Vinogradov inequality with the constant 1/(2 log 2). In this note we improve their estimates and obtain, in particular, a form of the Pólya-Vinogradov inequality with the constant 1/(3 log 3). A nice feature of our estimate is that it is obtained by a very simple argument.     

Inequalities for a Simplex and an Interior Point

We establish some inequalities for the inradius, circumradius and distances between an interior point and facets of an n-simplex, and prove a recursion inequality for the radii of the circumscribed spheres of an n-simplex and its facets.     

Cubic Diophantine Inequalities

Let λ₁, λ₂,..., λ₇ be real numbers satisfying λ _i ≥ 1. In this paper, we prove there are integers x ₁,..., x ₇ such that the inequalities |λ₁ x ³ ₁ + λ₂ x ³ ₂ + ⋯ + λ₇ x ³ ₇| < 1 and hold simultaneously. Received November 18, 1997, Accepted October 23, 1998     

半素PI-环的本质(单边)理想

本文系统研究了半素ＰＩ－环本质（单边）理想及最大商环的有关性质,从ＰＩ－理论中著名的 Ｒｏｗｅｎ非平凡相交定理出发,证明了半素 ＰＩ－环的本质（单边）理想包合一个本质两边理想（定理７,推论８）,建立了半素ＰＩ－环本质（单边）理想与其中心本质理想的内在联系（定理 ９,推论 １０）,并以此为基础,证明了半素 ＰＩ－环的最大商环是 ＰＩ的（定理 １２）,且半素 ＰＩ－环的每一正则元在其最大商环中是可逆的（定理 １４）．     

Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q-Brownian motion

In this paper, we prove the existence and uniqueness of the entropy solution for a first-order stochastic conservation law with a multiplicative source term involving a $Q$-Brownian motion. After having defined a measure-valued weak entropy solution of the stochastic conservation law, we present the Kato inequality, and as a corollary, we deduce the uniqueness of the measure-valued weak entropy solution, which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure-valued entropy solution of the stochastic conservation law, which proves the existence of the measure-valued entropy solution.     

Sharper uncertainty principles associated with Lp-norm

Motivated by the well-established phase derivative embedded technique, this study devotes to sharper uncertainty principles related to the L^p-norm type of uncertainty product, giving rise to two kinds of uncertainty inequalities that improve the classical result through providing tighter lower bounds. The conditions that truly reach these better estimates are obtained. Examples and simulations are carried out to verify the correctness of the derived results, and finally, possible applications in time-frequency analysis are also given.     

Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity

We study smoothing properties for time-dependent Schrödinger equations , , with potentials which satisfy V(x)=O(|x|^m) at infinity, m?2. We show that the solution u(t,x) is 1/m times differentiable with respect to x at almost all , and explain that this is the result of the fact that the sojourn time of classical particles with energy λ in arbitrary compact set is less than CTλ^−1/m during [0,T] when λ is very large. We also show Strichartz's inequality with derivative loss for such potentials and give its application to nonlinear Schrödinger equations.     

On the existence of solutions of variational inequalities in Banach spaces

In this paper, we study the existence of the solution of the variational inequality 〈Tx−ξ,y−x〉?0 by applying the generalized projection operator , where B is a Banach space with dual space B∗ and by using the well-known FanKKM Theorem.