Global Poincaré Inequalities on the Heisenberg Group and Applications

Let f be in the localized nonisotropic Sobolev space Wloc^1,p （H^n） on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n ＋ 2 is the homogeneous dimension of H^n. Suppose that the subelliptic gradient is gloablly L^p integrable, i.e., fH^n ｜△H^n f｜^p du is finite. We prove a Poincaré inequality for f on the entire space H^n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C0^∞（H^n） under the norm of （∫H^n ｜f｜ Qp/Q-p）^Q-p/Qp ＋ （∫ H^n ｜△H^n f｜^p）^1/p. We will also prove that the best constants and extremals for such Poincaré inequalities on H^n are the same as those for Sobolev inequalities on H^n. Using the results of Jerison and Lee on the sharp constant and extremals for L^2 to L（2Q/Q-2） Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H^n when p=2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H^n.     

ON OPPENHEIM＇S INEQUALITY

We prove several inequalies for symmetric postive semidefinite, general M-matrices and inverse Mmatrices which are generalization of the classical Oppenheim‘s Inequality for symmetric positive semidefinite matrices.     

Symmetry and Uniqueness of Solutions of an Integral System

In this paper, we study the positive solutions for a class of integral systems and prove that all the solutions are radially symmetric and monotonically decreasing about some point. Moreover, we also obtain the uniqueness result for a special case. We use a new type of moving plane method introduced by Chen-Li-Ou [1]. Our new ingredient is the use of Hardy-Littlewood-Sobolev inequality instead of Maximum Principle.     

Optimal integrability of some system of integral equations

We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in R^n ：{u（x）=1/｜x｜^α｜∫R^n v（y）^q｜y｜^β｜x-y｜^λdy,v（x）=1/｜x｜^β∫R^n u（y）^p｜y｜^α｜x-y｜^λdy.C. Jin and C. Li [Calc. Var. Partial Differential Equations, 2006, 26： 447-457] developed some very interesting method for regularity lifting and obtained the optimal integrability for p, q 〉 1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p, q ≥1 and pq ≠1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions when ｜x｜ →0 and when ｜x｜→∞.     

ESTIMATIONS OF THE CONSTANTS IN INVERSE INEQUALITIES FOR FINITE ELEMENT FUNCTIONS

In this paper, we estimate the constants in the inverse inequalities for the finite ele- ment functions. Furthermore, we obtain the least upper bounds of the constants in inverse inequalities for the low-order finite element functions. Such explicit estimates of the con- stants can be used as computable error bounds for the finite element method.     

ON THE ERROR OF QUADRATURE FORMULAE FOR CAUCHY PRINCIPAL VALUE INTEGRALS BASED ON PIECEWISE INTERPOLATION

We consider the computation of the. Cauchy principal value mtegral by quadrature formulaeof compound type, which are obtained by replacing f by a piecewise defined function F,[_f]. The behaviour of the constants m the estimates where quadrature error) is determined for fixed i and which means that not only the. order, but also the coefficient of the main term of is determined. The behaviour of these error constants is compared -with the corresponding ones obtained for the. method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Poincaré inequality for general symmetric forms

Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact embedding, and the criteria via the new Cheeger＇s constants for those inequalities are presented. Finally, the concentration or the volume growth of measures for these inequalities are studied.     

LOOMIS-WHITNEY＇S INEQUALITY ABOUT MIXED BODY

In this article, the authors establish two theorems for mixed body, which are the generalizations of the well-known Loomis-Whitney＇s inequality.     

SOBOLEV INEQUALITY ON RIEMANNIAN MANIFOLDS

Let M be an n dimensional complete Riemannian manifold satisfying the doubling volume property and an on-diagonal heat kernel estimate. The necessary-sufficient condition for the Sobolev inequality ||f||q ≤Cn,,v,p,q(||(?)f||p ||f||p) (2 ≤p < q < ∞) is given.     

On Lower Dimensional Invariant Tori in C^d Reversible Systems

In this paper, a result on the persistence of lower dimensional invariant tori in Cd reversible systems is obtained under some conditions. The theorem is proved for any d which is larger than some constants.     

Upper Bound Estimates of the Cramér Functionals for Markov Processes

We prove that the Cramér functional of a Markov process can be controlled by a function of an integral functional if the transition semigroup is uniformly integrable in L ^p . As an application of this result, a general large deviation upper bound is obtained. Then, the notation of F-Sobolev inequality is extended to general Markov processes by replacing the Dirichlet form with the Donsker–Varadhan entropy. As the other application, it is proved that the uniform integrability of a transition semigroup implies a F-Sobolev inequality.     

Kinetic constants method for the vibrational analysis of carbon suboxide and carbon subsulphide molecules

The kinetic constants method has been applied to the case of carbon suboxide and carbon subsulphide molecules to evaluate the molecular constants such as the potential energy constants, the generalised mean square amplitudes of vibration, the shrinkage constants, the Coriolis coupling constants and the centrifugal distortion constants using the recent vibrational frequencies. The values of the molecular constants evaluated by this procedure are, in general, in very good agreement with the observed and the calculated values. The Coriolis coupling constants and the centrifugal distortion constants for these molecules are reported here for the first time.     

On the best constants in the solvability conditions for the periodic boundary value problem for higher-order functional differential equations

We study the properties of the sequence of optimal constants in the conditions of unique solvability of the periodic boundary value problem for nth-order linear functional differential equations. These constants are expressed via the Euler-Bernoulli constants; simple recursion relations between them and relations with other known mathematical constants are derived.     

Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains

Summary In this paper, methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied. In order to verify solutions using computer, it is necessary to determine some constants which appear in a priori error estimations. We propose some methods for determination of these constants. In numerical examples, calculating these constants for anL-shaped domain, we verify the solution of a nonlinear elliptic equation.     

On the Koebe and Bieberbach constants for a hyperbolic planar domain

We discuss domain constants related to the classical Bieberbach and Koebe theorems. We find a class of simply connected domains for which the value of these constants is extremal and investigate the relationship between the geometry of a sequence of hyperbolic domains and the associated sequence of domain constants.     

Estimates of Hardy’s constants for tubular extensions of sets and domains with finite boundary moments

We study the behavior of constants in Hardy-type inequalities for tubular extensions of sets. We also obtain new estimates of constants for domains with finite boundary moments.     

Lebesgue Constants for Hadamard Matrices

There are many advantages in the use of Hadamard matrices in digital signal processing. However one possible disadvantage is the so-called overflow, as measured by the associated Lebesgue constants. We show that for certain classes of recursively generated Hadamard matrices, there are logarithmic upper bounds for these constants. On the other hand, for some Hadamard matrices the Lebesgue constants are of order \sqr{m}. These results have natural analogues in classical Fourier analysis.     

Calculation of Dependent Elastic Constants of Plastic Foams with a Pronounced Strut-Like Structure

Seven dependent elastic constants of monotropic plastic foams with an expressed strut-like structure are calculated. For this purpose, the basic results of the previously elaborated mathematical model for light-weight plastic foams is used. The model includes a model cell of local structure for monotropic/isotropic plastic foams and an ensemble of structural elements, which allows one to calculate the seven dependent elastic constants, taking into account the pronounced polydispersity of the structure of plastic foams. The numerical values of the constants are compared with the available experimental data, and a satisfactory agreement is found to exist. As a final result, a full set of general expressions and numerical values are obtained for all 12 elastic constants of monotropic plastic foams.     

A mechanism for pulsar radiation

The methods available for the determination of the stability constants of non-extractable complexes by solvent extraction are reviewed. The method proposed by Connick and McVey is valid only when no complexes of the auxiliary-ligand exist in the aqueous phase. This is applicable only at very low concentrations of the auxiliary-ligand. Hence, the stability constants calculated by this method decrease with increasing concentration and approach the theoretical values only at low concentrations of the auxiliary-ligand. It is essential, therefore, to apply the rigorous method incorporating the constants for the metal-auxiliary-ligand to calculate the stability constants which are independent of the changes in the experimental conditions. This is illustrated by few typical calculations.     

Force constants,generalised meansquare amplitudes of vibration,shrinkage constants and coriolis coupling coefficients of GeH3CCH and GeD3CCH

General Valence force constants have been evaluated for GeH₃CCH and GeD₃CCH. Generalised mean-square amplitudes of vibration have been obtained and the shrinkage constants have been calculated using the perpendicular mean-square amplitudes of vibration. Coriolis coupling constants have also been determined.