Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness

We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p [1, ], the following unimprovable inequality holds for functions :

局部平方可积鞅的比值不等式

本文对局部平方可积鞅建立了几个比值不等式,推广了连续鞅的相应结果.     

局部平方可积鞅的比值不等式

本对局部平方可积鞅建立了几个比值不等式，推广了连续鞅的相应结果．     

On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

For 2-periodic functions and arbitrary q [1, ] and p (0, ], we obtain the new exact Kolmogorov-type inequality which takes into account the number of changes in the sign of the derivatives (x ^(k)) over the period. Here, = (r – k + 1/q)/(r + 1/p), _r is the Euler perfect spline of degree r, and . The inequality indicated turns into the equality for functions of the form x(t) = a _r (nt + b), a, b R, n N. We also obtain an analog of this inequality in the case where k = 0 and q = and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.     

Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation

We obtain a new unimprovable Kolmogorov-type inequality for differentiable 2-periodic functions x with bounded variation of the derivative x, namely

Optimal Control of Variational Inequalities with Delays in the Highest Order Spatial Derivatives

In this paper, an optimal control problem for parabolic variational inequalities with delays in the highest order spatial derivatives is investigated. The well-posedness of such kinds of variational inequalities is established. The existence of optimal controls under a Cesari-type condition is proved, and the necessary conditions of Pontryagin type for optimal controls is derived.     

Sharp Kolmogorov-Type Inequalities for Moduli of Continuity and Best Approximations by Trigonometric Polynomials and Splines

In what follows, $C$ is the space of -periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm; is the mth modulus of continuity of a function f with step h and calculated with respect to P; , ( ), ,

关于二阶导数的若干新积分不等式

根据Cauchy-Schwarz不等式,得到了C2([a,b])空间中函数的二阶导数的若干新积分不等式.     

Inequalities for a Polynomial and Its Derivative II

A well-known result of Rivlin states that if p(z) is a polynomial of degree n, such that p(z) ≠ 0 in |z| < 1, then max_{|z|=r < 1} |p(z)| ≤ ((r + 1)/2)ⁿ max_{|z| = 1} |p(z)|. In this paper, we consider the polynomial p(z) = a₀ + Σⁿ_{v = μ}a_υz^υ having all its zeros in |z| ≤ k > 1 and obtain a generalization of this result. Our result improves upon a result recently proved by Bidkham and Dewan (J. Math. Anal. Appl.166 (1992), 19-324).     

Inequalities for the Polar Derivative of a Polynomial

Let P(z) be a polynomial of degree at most n. We consider an operator D _?? which maps a polynomial P(z) into D _?? P(z) :=?nP(z)?+?(?? ? z)P??(z) and prove results concerning the estimates of |D _?? P(z)| on the disk |z|?=?R??? 1, and thereby obtain extensions and generalizations of a number of well-known polynomial inequalities.     

Some Inequalities Concerning the Polar Derivative of a Polynomial-II

In this paper, we consider the class of polynomials P（z）= anz^n＋ ∑vn=μan-vz^n-v,1≤μ≤n , having all zeros in ｜z｜≤k, k ≤1 and thereby present an alternative proof, independent of Laguerre＇s theorem, of an inequality concerning the polar derivative of a polynomial.     

Dirichlet-Type Problems for Systems of Partial Differential Equations Unresolved with Respect to the Highest Time Derivative

We establish conditions for the correct solvability of problems for systems of partial differential equations unresolved with respect to the highest time derivative with Dirichlet-type conditions with respect to time and periodic conditions with respect to space variables. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of these problems.     

On Completely Integrable Systems with Local Torus Actions

This paper concerns with symplectic topology of compact completely integrable Hamiltonian systems with only stable nondegenerate elliptic singularities. We describe all systems whose universal coverings admit actian-angle coordinates and, in particular, prove that some finite cover of a base space is diffeomorphic to a product of a convex polytope and a solvmanifold. We also construct an obstruction, vanishing of which guarantees splitting of some finite cover of a phase space as a toric variety and a torus fibering over a solvmanifold.     

关于具有可积伸张同胚的紧性

具有可积球伸张的同胚的偏差定理、等度连续性和紧性被得到．     

The Martingale Hardy Type Inequalities for Dyadic Derivative and Integral

Since the Leibniz-Newton formula for derivatives cannot be used in local fields, it is important to investigate the new concept of derivatives in Walsh-analysis, or harmonic analysis on local fields. On the basis of idea of derivatives introduced by Butzer, Schipp and Wade, Weisz has proved that the maximal operators of the one-dimensional dyadic derivative and integral are bounded from the dyadic Hardy space Hp,q to Lp,q, of weak type （L1,L1）, and the corresponding maximal operators of the two-dimensional case are of weak type （Hi, L1）. In this paper, we show that these maximal operators are bounded both on the dyadic Hardy spaces Hp and the hybrid Hardy spaces H^#p 0〈p≤1.     

Numerical Solution of Inverse Problems for a Hyperbolic Equation with a Small Parameter Multiplying the Highest Derivative

We consider two inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative. The inverse problems are reduced to systems of linear Volterra integral equations of the second kind for the unknown functions. These systems are used to prove the existence and uniqueness of the solution of the inverse problems and numerically solve them. The applicability of the methods developed here to the approximate solution of the problem on an unknown source in the heat equation is studied numerically.     

A Multipoint Problem with Complex Coefficients for Partial Differential Equations That Are Unsolvable for the Highest Derivative

Conditions of correct solvability in various functional spaces of a problem with multipoint conditions in the time variable and periodicity conditions in the space coordinate are established for partial differential equations that are unsolvable for the highest derivative with time. Metric theorems on lower estimates of small denominators that arise in the construction of solutions are proved.     

关于Polya-Szegoe不等式

本文给出两个改进的不等式,使改进后的每一个新的不等式均含有Polay-Szeg（o）两个不等式改进在内.     

平面Bonnesen型不等式

将用积分几何方法给出平面等周不等式以及Bonnesen型不等式,平面区域D的面积、周长、最大内接园半径及最小外接园半径的一些几何不等式的简单证明.