A Kind of integral Representation on Riemannian Manifods

We generalized the Bochner-Martinelli integral representation to that on Riemannian manifolds. Things become quite different in such case. First we define a kind of Newtonian potential and take the interior product of its gradient to be the integral kernel. Then we prove that this kernel is harmonic in some sense. At last an integral representative theorem is proved.

INTEGRAL REPRESENTATION OF SLOWLY GROWING EQUIDISTANT SPLINES

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow a5 O (|x|~5). We present an integral representation of such splines with a distribution kernel. This repre- sentation is related to the Fourier integral of slowly growing functions. The part of the Fourier ex- ponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First. it allows us to con- struct a rich library of splines possessing the property that translations of any such spline form a ba- sis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of growing func- tion. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.

A new technique of integral representations in C^(n)

A new technique of integral representations in Cn, which is different from the well-known Henkin technique, is given. By means of this new technique, a new integral formula for smooth functions and a new integral representation of solutions of the -equations on strictly pseudoconvex domains in Cn are obtained. These new formulas are simpler than the classical ones, especially the solutions of the -equations admit simple uniform estimates. Moreover, this new technique can be further applied to arbitrary bounded domains in Cn so that all corresponding formulas are simplified.

Nevanlinna Class and Its Integral Representation or Factorization Theorem in Sector and Angular Domain

In this paper we give some sufficient conditions for analytic functions which are not identically zero and belong to Nevanlinna class in the sector and angular domain. Moreover, their integral expressions or factorization theorems are obtained.

An Integral Representation for the Weighted Geometric Mean and Its Applications

By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.

Failure Mode and Effects Analysis Based on Z-Numbers and the Graded Mean Integration Representation

Failuremode and effects analysis(FMEA)is a widely used safety assessmentmethod inmany fields.Z-number was previously applied in FMEA since it can take both possibility and reliability of information into consideration.However,the use of fuzzy weighted mean to integrate Z-valuations may have some drawbacks and is not suitable for some situations.In this paper,an improved method is proposed based on Z-numbers and the graded mean integration representation(GMIR)to deal with the uncertain information in FMEA.First,Z-numbers are constructed based on the evaluations of risk factors O,S,D for each failure mode by different experts.Second,weights of the three risk factors and experts are determined.Third,the integration representations of Z-numbers are obtained by a newmethod based on the GMIRmethod.Finally,risk priorities of the failure modes are derived considering the weights of experts and risk factors.Two examples and a case study are given to show the use of the proposed method and comparison with other methods.The results show that the proposed method is more reasonable,universal and simple in calculation.

THE REPRESENTATION OF THE SOLUTION OF STURM-LIOUVILLE EQUATION WITH DISCONTINUITY CONDITIONS

The aim of this paper is to construct the integral representation of the solution of Sturm-Liouville equation with eigenparameter-dependent discontinuity conditions at an interior point of the finite interval. Moreover, we examine the properties of the kernel function of this integral representation and obtain the partial differential equation provided by this kernel function.

Fuzzy Inventory Model under Selling Price Dependent Demand and Variable Deterioration with Fully Backlogged Shortages

The objective is to develop a model considering demand dependent on selling price and deterioration occurs after a certain period of time, which follows two-parameter Weibull distribution. Shortages are allowed and fully backlogged. Fuzzy optimal solution is obtained by considering hexagonal fuzzy numbers and for defuzzification Graded Mean Integration Representation Method. A numerical example is provided for the illustration of crisp and fuzzy, both models. To observe the effect of changes in parameters, sensitivity analysis is carried out.

EQUATION OF MATHEMATICAL PHYSICS AND OTHER AREAS OF APPLICATION: A CLASSICAL INTEGRABLE SYSTEM AND THE INVOLUTIVE REPRESENTATION OF SOLUTIONS OF THE HIGHER ORDER KAUP-NEWELL EQUATION

Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be a completely integrable system (R2N, Adp AND dq, H = H-1) with the Hamiltonian H-1 = -[A3q, p]-1/2[A2p, p][A2q, q]. while the nonlinearization of the time part leads to its N-involutive system {H(m)}. The involutive solution of the compatible fsystem (H-1), (H(m)) is mapped by into the solution of the higher order Kaup-Newell equation.

BIANALYTIC FUNCTIONS, BIHARMONIC FUNCTIONS AND ELASTIC PROBLEMS IN THE PLANE

Let the elastic body only be acted by gravity. By investigating the relations of bianalytic functions and biharmonic functions, the uniqueness and existence of the stress functions (Airy functions) are established in planar simple connected region. Moreover, the integral representation formula of the stress functions in the unit disk of the plane is obtained.

Survival probability and ruin probability of a risk model

In this paper, a new risk model is studied in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is p-thinning process. The integral representations of the survival probability are gotten. The explicit formula of the survival probability on the infinite interval is obtained in the special casc cxponential distribution.The Lundberg inequality and the common formula of the ruin probability are gotten in terms of some techniques from martingale theory.

Inhomogeneous Cauchy-Riemann equation in Octonion space

The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the Cousin problem analogue of Mittag-Laffier problem is discussed.

A RELAXATION RESULT OF FUNCTIONALS IN THE SPACE SBD(Ω)

In this article,the authors obtain an integral representation for the relaxation of the functionalF（x,u,Ω）：={∫^f（x,u（x）,εu（x））dx Ω if u∈W^1,1（Ω,R^N）, ＋∞ otherwise, in the space of functions of bounded deformation,with respect to L^1-convergence.Here Eu represents the absolutely continuous part of the symmetrized distributional derivative Eu.f（x,p,ξ）satisfying weak convexity assumption.

在Stein流形中有界域的解析簇上积分表示及其应用

In this paper,we first obtain a unified integral representation on the analytic varieties of the general bounded domain in Stein manifolds(the two types bounded domains in[3]are regarded as its special cases).Secondly we get the integral formulas of the solution of∂-equation.And we use a new and unique method to give a uniform estimate of the solution of∂-equation,which is different from Henkin's method.

Erratum to “The Riemann Hypothesis-Millennium Prize Problem” [Advances in Pure Mathematics 6 (2016) 915-920]

The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mistake. The author wishes to correct the errors in Theorem 2 of the result part.

The Riemann Hypothesis-Millennium Prize Problem

This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem-Millennium Prize Problem.

Integral formulas and the Ohsawa-Takegoshi extension theorem

We construct a semiexplicit integral representation of the canonical solution to the (-δ)-equation with respect to a plurisubharmonic weight function in a pseudoconvex domain. The construction is based on a construction related to the Ohsawa-Takegoshi extension theorem combined with a method to construct weighted integral representations due to M. Andersson.

Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle H = （Hu）u∈G^0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle （G^0, （Hu）,μ）, then dimHu = 1 （u ∈ G^0）. Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.

Higher order Poisson kernels and L^p polyharmonic boundary value problems in Lipschitz domains

In this article, we introduce higher order conjugate Poisson and Poisson kernels, which are higher order analogues of the classical conjugate Poisson and Poisson kernels, as well as the polyharmonic fundamental solutions, and define multi-layer potentials in terms of the Poisson field and the polyharmonic fundamental solutions, in which the former is formed by the higher order conjugate Poisson and the Poisson kernels. Then by the multi-layer potentials, we solve three classes of boundary value problems(i.e., Dirichlet, Neumann and regularity problems) with L^p boundary data for polyharmonic equations in Lipschitz domains and give integral representation(or potential) solutions of these problems.

Hydrogen mean force and anharmonicity in polycrystalline and amorphous ice

The hydrogen mean force from experimental neutron Compton profiles is derived using deep inelastic neutron scattering on amorphous and polycrystalline ice. The formalism of mean force is extended to probe its sensitivity to anharmonicity in the hydrogen-nucleus effective potential. The shape of the mean force for amorphous and polycrystalline ice is primarily determined by the anisotropy of the underlying quasi-harmonic effective potential. The data from amorphous ice show an additional curvature reflecting the more pronounced anharmonicity of the effective potential with respect to that of ice Ih.