Some continuous functions related to corner polyhedra,II

The group problem on the unit interval is developed, with and without continuous variables. The connection with cutting planes, or valid inequalities, is reviewed. Certain desirable properties of valid inequalities, such as minimality and extremality are developed, and the connection between valid inequalities for P(I, u ₀) and P _- ⁺ (I, u ₀) is developed. A class of functions is shown to give extreme valid inequalities for P _- ⁺ (I, u ₀) and for certain subsetsU ofI. A method is used to generate such functions. These functions give faces of certain corner polyhedra. Other functions which do not immediately give extreme valid inequalities are altered to construct a class of faces for certain corner polyhedra. This class of faces grows exponentially as the size of the group grows.     

Two-sided inequalities for the Struve and Lommel functions

Abstract

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial di?erential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function ₁F₂(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function ₁F₂(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function.     

When separability of the space of ultra-extremal functions is preserved

Abstract

The ultrametrically injective hull T_X of an ultrametric space (X, d) is investigated by viewing it as the space of ultra-extremal functions over X. It turns out that the ultra-extremal functions are also ultra-Ka?etov functions, satisfying two inequalities derived from the strong triangle inequality. We shall compare the ultra-extremal functions with some classes of functions defined with the help of one of the two inequalities from the definition of ultra-Kat?tov functions. We shall consider the question of when separability of the space of ultra-extremal functions is preserved.     

Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of ? ⁿ . The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the p-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.     

Hardy-Sobolev type inequalities on the H-type group

Motivated by the idea of Badiale and Tarantello who have found Hardy-Sobolev inequalities on Rⁿ, a class of Hardy-Sobolev type inequalities on H-type groups is proved via a new representation formula for functions. Extremal functions realizing equality in the inequalities are discussed by refined Concentration-Compactness principles. Finally, some sharp constants for Hardy type inequalities are given. The project supported by National Natural Science Foundation of China, Grant No. 10371099.     

Weighted integral and integro-differential inequalities

A general weighted integral inequality for two continuous functions on an interval [a,b] is presented. The equality conditions are given. This result implies the new inequalities for the incomplete beta and gamma functions as well as the related estimates for the confluent hypergeometric function, error function, and Dawson's integral. Also it implies various weighted integro-differential inequalities, those of the Opial type included, and some inequalities which involve the Erdélyi–Kober and Riemann–Liouville fractional integrals.     

On the Bernstein Inequality for Rational Functions with a Prescribed Zero

We extend some results of Giroux and Rahman (Trans. Amer. Math. Soc.193(1974), 67–98) for Bernstein-type inequalities on the unit circle for polynomials with a prescribed zero atz=1 to those for rational functions. These results improve the Bernstein-type inequalities for rational functions. The sharpness of these inequalities is also established. Our approach makes use of the Malmquist–Walsh system of orthogonal rational functions on the unit circle associated with the Lebesgue measure.     

Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn

The main results of this paper concern sharp constants for the Moser‐Trudinger inequalities on spheres in complex space ?ⁿ. We derive Moser‐Trudinger inequalities for smooth functions and holomorphic functions with different sharp constants (see Theorem 1.1). The sharp Moser‐Trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the CR setting. Though there is a close connection in spirit between inequalities proven here on complex spheres and those on the Heisenberg group for functions with compact support in any finite domain proven earlier by the same authors [17], derivation of the sharp constants for Moser‐Trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the Heisenberg group. Variants of Moser‐Onofri‐type inequalities are also given on complex spheres as applications of our sharp inequalities (see Theorems 1.2 and 1.3). One of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients (see Theorem 1.4). © 2004 Wiley Periodicals, Inc.     

On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1

We present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H ¹(Ω), indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. We consider a polygonal domain Ω in two or three space dimensions and its shape-regular simplicial triangulation. The nonconforming approximations of H ¹(Ω) consist of functions from H ¹ on each element such that the mean values of their traces on interelement boundaries coincide. The key idea is to extend the proof of the discrete Poincaré–Friedrichs inequalities for piecewise constant functions used in the finite volume method. The results have applications in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.     

Exact order of Hoffman’s error bounds for elliptic quadratic inequalities derived from vector-valued Chebyshev approximation

In this paper, we introduce the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities. Elliptic quadratic inequalities are closely related to Chebyshev approximation of vector-valued functions (including complex-valued functions). The set of Chebyshev approximations of a vector-valued function defined on a finite set is shown to be Hausdorff strongly unique of order exactly 2 ^s for some nonnegative integer s. As a consequence, the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities is exactly 2 ^-s for some nonnegative integer s. The integer s, called the order of deficiency (which is computable), quantifies how much the Abadie constraint qualification is violated by the elliptic quadratic inequalities. Received: April 15, 1999 / Accepted: February 21, 2000?Published online July 20, 2000     

On some Turán-type inequalities for q-special functions

Intrinsic inequalities involving Turán-type inequalities for some q-special functions are established. A special interest is granted to q-Dunkl kernel. The results presented here would provide extensions of those given in the classical case.     

Generalized mixed integer rounding inequalities: facets for infinite group polyhedra

We present a generalization of the mixed integer rounding (MIR) approach for generating valid inequalities for (mixed) integer programming (MIP) problems. For any positive integer n, we develop n facets for a certain (n + 1)-dimensional single-constraint polyhedron in a sequential manner. We then show that for any n, the last of these facets (which we call the n-step MIR facet) can be used to generate a family of valid inequalities for the feasible set of a general (mixed) IP constraint, which we refer to as the n-step MIR inequalities. The Gomory Mixed Integer Cut and the 2-step MIR inequality of Dash and günlük  (Math Program 105(1):29–53, 2006) are the first two families corresponding to n = 1,2, respectively. The n-step MIR inequalities are easily produced using periodic functions which we refer to as the n-step MIR functions. None of these functions dominates the other on its whole period. Finally, we prove that the n-step MIR inequalities generate two-slope facets for the infinite group polyhedra, and hence are potentially strong.      

Some nonlinear dynamic inequalities on time scales

The aim of this paper is to investigate some nonlinear dynamic inequalities on time scales, which provide explicit bounds on unknown functions. The inequalities given here unify and extend some inequalities in (B G Pachpatte, On some new inequalities related to a certain inequality arising in the theory of differential equation, J. Math. Anal. Appl. 251 (2000) 736–751).     

Classes caractéristiques des intersections complètes

We prove various inequalities for characteristic classes of smooth projective varieties which are complete intersections. We relate such classes to symmetric functions of the degrees of the hypersurfaces that intersect onX, by means of symmetric functions that are invariant by translation, and use positivity properties and inequalities between Newton classes. This extends previous results of H. S. Tai.

     

Correlation inequalities for statistical models of lattice gas

We consider models of statistical mechanics of the type of lattice gas with attractive interaction of general kind. We propose a method for obtaining inequalities that connect multipoint correlation functions of different order. This method allows one, on the one hand, to strengthen similar inequalities, which can be obtained within the framework of the FKG method, and on the other hand, to obtain new inequalities. We introduce the notion of duality for models of lattice gas. We show that if, under the transformation p ⇒ 1 - p, the correlation inequalities for a model with attraction turn into the corresponding inequalities that are also satisfied, then the correlation functions of the dual model also satisfy the latter inequalities. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 765–773, June, 1998.     

Opial type inequalities for cosine and sine operator functions

Various L _p form Opial type inequalities are given for cosine and sine operator functions with applications.     

Poincaré like inequalities for semigroups, cosine and sine operator functions

Here we present Poincaré type general L _p inequalities regarding semigroups, cosine and sine operator functions.     

PROBABILISTIC OSTROWSKI TYPE INEQUALITIES

New very general univariate and multivariate probabilistic Ostrowski type inequalities are established, involving ‖·‖_∞ and ‖·‖ _p , p≥1 norms of probability density functions. Some of these inequalities provide pointwise estimates to the error of probability distribution function from the expectation of some simple function of the engaged random variable. Other inequalities give upper bounds for the expectation and variance of a random variable. All are done over finite domains. At the end are given applications, especially for the Beta random variable.     

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L ^p norms of $ \left\vert \nabla f\right\vert We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and G?tze. We also get inequalities between the L ^p norms of and moreover L ^p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L ^p norm by the Schatten norm C _p .     

Bernstein–Nikolskii and Plancherel–Polya inequalities in Lp ‐norms on non‐compact symmetric spaces

By using Bernstein‐type inequality we define analogs of spaces of entire functions of exponential type in L_p (X), 1 ≤ p ≤ ∞, where X is a symmetric space of non‐compact. We give estimates of L_p ‐norms, 1 ≤ p ≤ ∞, of such functions (the Nikolskii‐type inequalities) and also prove the L_p ‐Plancherel–Polya inequalities which imply that our functions of exponential type are uniquely determined by their inner products with certain countable sets of measures with compact supports and can be reconstructed from such sets of “measurements” in a stable way (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)