SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS

Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.

Necessary and Sufficient Conditions in Smooth Programming with Generalized Convexity

Hanson and Mond have grven sets of necessary and sufficient conditions for optimality in constrained optimization by introducing classes of generalized functions, called type Ⅰ functions. Recently, Bector definded univex functions, a new class of functions that unifies several concepts of generalized convexity. In this paper, additional conditions are attached to the Kuhn Tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization, under appropriate constraint qualifications.

Necessary and Sufficient Conditions for Reactive Azeotropes in Multi-reaction Mixtures in Terms of Elemental Composition

Necessary and sufficient conditions for azeotropy in reactive mixtures are derived in terms of elemental composition, which shows that in the space of elemental compositions, they take the same functional form as the conditions for azeotropy in non-reactive mixtures. The production of methyl tert-butyl ether (MTBE) is taken as an example. It is found that there are a 'pseudo' intermediate-boiling ternary reactive azeotrope at p = 101.325kPa and two 'real' ternary reactive azeotropes at p = 101.325 Pa. The introduced elemental compositions also reduce the dimensionality of the phase diagrams and provide a natural set of variables for visualization of phase behavior.

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.

NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x^2+α_2xy+α_5x^2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx^j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.

Sufficient conditions of Rayleigh-Taylor stability and instability in equatorial ionosphere

Rayleigh-Taylor （R-T） instability is known as the fundamental mechanism of equatorial plasma bubbles （EPBs）. However, the sufficient conditions of R-T instability and stability have not yet been derived. In the present paper, the sufficient conditions of R-T stability and instability are preliminarily^derived. Linear equations for small perturbation are first obtained from the electron/ion continuity equations, momentum equations, and the current continuity equation in the equatorial ionosphere. The linear equations can be casted as an eigenvalue equation using a normal mode method. The eigenvalue equation is a variable coefficient linear equation that can be solved using a variational approach. With this approach, the sufficient conditions can be obtained as follows： if the minimum systematic eigenvalue is greater than one, the ionosphere is R-T unstable; while if the maximum systematic eigenvalue is less than one, the ionosphere is R-T stable. An approximate numerical method for obtaining the systematic eigenvalues is introduced, and the R-T stable/unstable areas are calculated. Numerical experiments axe designed to validate the sufficient conditions. The results agree with the derived suf- ficient conditions.

The convergence rate and necessary-and-sufficient condition for the consistency of isogeometric collocation method

Although the isogeometric collocation(IGA-C)method has been successfully utilized in practical applications due to its simplicity and efficiency,only a little theoretical results have been established on the numerical analysis of the IGA-C method.In this paper,we deduce the convergence rate of the consistency of the IGA-C method.Moreover,based on the formula of the convergence rate,the necessary and sufficient condition for the consistency of the IGA-C method is developed.These results advance the numerical analysis of the IGA-C method.

A NECESSARY AND SUFFICIENT CONDITION FOR THEOSCILLATION OF SOLUTIONS OF LIENARD TYPE SYSTEM WITH MULTIPLE SINGULAR POINTS

In this paper, a necessary and sufficient condition for teh solution of Lienard typesystem with muliiple singular points to oscillation under the more general assumptionis given.Results of the papers [1-4] are also extended and improved in this paper.

A necessary and sufficient condition for positive solutions of singular fourth order boundary value problems

This paper deals with the existence of positive solutions for the singular fourth order boundary value problem.A necessary and sufficient condition for the existence of C3 positive solution is given by means of the monotone iterative technique.Furthermore,the uniqueness of the C3 positive solution,and the iterative sequence of the C3 positive solution are also obtained.

A NECESSARY AND SUFFICIENT FORMULATION FOR TORSION PROBLEMS OF ANISOTROPIC BODIES

Non-dimensionalized equations and boundary conditions are presented for the torsion problem of an anisotropic body. The error of the fundamental solution cited in some boundary element books Is pointed out after an examination of the fundamental solution. Furthermore,a necessary and sufficient boundary integral equation is given for the problem and compared with the conventional boundary integral equation. Numerical results show that great errors of the boundary shear stresses obtained by the conventional boundary integral equation appear with a small error of torsion stiffness. Meanwhile,the necessary and sufficient;boundary integral equation always gives accurate results.

Sufficient Conditions of Optimality for Convex Differential Inclusions of Elliptic Type and Duality

This paper deals with the Dirichlet problem for convex differential (PC) inclusions of elliptic type. On the basis of conjugacy correspondence the dual problems are constructed. Using the new concepts of locally adjoint mappings in the form of Euler-Lagrange type inclusion is established extremal relations for primary and dual problems. Then duality problems are formulated for convex problems and duality theorems are proved. The results obtained are generalized to the multidimensional case with a second order elliptic operator.

Necessary Conditions and Situational Variables of Payments for Ecosystem Services

Clarifying the necessary conditions for the emergence of payments for ecosystem services (PES) and the situational variables that affect PES is the basis for their interpretation, prediction, and selection. This article proposes an analytical framework for the emergence of PES and argues that the key to determining whether PES can occur and whether a selected PES program is appropriate is to evaluate the net gain. When payers anticipate that a PES program will provide a satisfactory number of ES and a net gain over the opportunity cost and will cover all costs, it is assumed that the program will be implemented. When it is difficult to accurately evaluate the net gain of PES, the situational variables that affect the costs and benefits need to be examined. The group characteristics, ES characteristics, spatial and temporal contacts between the suppliers and demanders, correlation with private goods and additionality are important situational variables that affect the emergence and choice of PES.

Some Necessary and Sufficient Condition of Multivariate Random Variable Satisfy Normal Distribution

In this paper,we extended some results of article[1],obtain some sufficient and necessary condition which multivariate random variable satisfy normal distribution.

A NECESSARY AND SUFFICIENT CONDITION FOR SINGULAR NONLINEAR SECOND-ORDER BOUNDARY VALUE PROBLEMS TO HAVE POSITIVE SOLUTIONS

In this paper the following result is obtained: Suppose f(x,u,v) is nonnegative, continuous in ( a, b)×R +×R +; f may be singular at x=a (and/or x=b ) and v=0; f is nondecreasing on u for each x,v, nonincreasing on v for each x,u; there exists a constant q∈(0,1) such that t qf(x,t -1 u,tu)f(x,u,u)λ qf(x,λ -1 u,λu),0<t<1<λ, u∈R +. Then a necessary and sufficient condition for the equation u″+f(x,u,u)=0 on the boundary condition αu(a)-βu′(a)=0, γ(b)+δu′(b)=0 to have C 1(I) nonzero solutions is that 0<∫ b af(x,e(x),e(x))dx<∞, where α,β,γ,δ are nonnegative real numbers, Δ=(b-a)αγ+αδ+βγ>0, e(x)=G(x,x), G(x,y) is Green's function of above mentioned boundary value problem (when f(x,u,v)≡0). Received September 9,1996. Revised March 31,1997. 1991 MR Subject Classification: 34B.

OPTIMALITY CONDITIONS AND DUALITY RESULTS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS WITH THE MULTIPLE INTERVAL-VALUED OBJECTIVE FUNCTION

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a （weakly） LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a （weakly） LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.

Sufficient Optimality Conditions for Multiobjective Programming Involving (V, ρ)h,ψ-type Ⅰ Functions

New classes of functions namely (V, ρ)_(h,φ)-type I, quasi (V, ρ)_(h,φ)-type I and pseudo (V, ρ)_(h,φ)-type I functions are defined for multiobjective programming problem by using BenTal's generalized algebraic operation. The examples of (V, ρ)_(h,φ)-type I functions are given. The sufficient optimality conditions are obtained for multi-objective programming problem involving above new generalized convexity.

New sufficient conditions for general linear SISO Takagi-Sugeno fuzzy systems as universal approximators

By the best approximation theory, it is first proved that the SISO （single-input single-output） linear Takagi-Sugeno （TS） fuzzy systems can approximate an arbitrary polynomial which, according to Weierstrass approximation theorem, can uniformly approximate any continuous functions on the compact domain. Then new sufficient conditions for general linear SISO TS fuzzy systems as universal approximators are obtained. Formulae are derived to calculate the number of input fuzzy sets to satisfy the given approximation accuracy. Then the presented result is compared with the existing literature＇s results. The comparison shows that the presented result needs less input fuzzy sets, which can simplify the design of the fuzzy system, and examples are given to show its effectiveness.

SIMPLE SUFFICIENT CONDITIONS OF NONSINGULAR H-MATRICES

1 IntroductionH-matrices have important role in numerical analysis and matrix theory,etc.In thispaper,several practical criteria for H-matrices are obtained,to which the matrix is notnecessary be a diagonally dominant matrix,and the numbers of rows which are not diago-nal dominant can be n-1.We introduce the concept of α-diagonally dominant matrix,andby using the concept to investigate H-matrices.

Study on Intention and Forming Conditions of SMT-MG

Surface Mount Technology Manufacturing Grid (SMT-MG) is the application of the grid technology in the SMT product manufacturing.In this paper,as the new manufacturing mode of SMT product,SMT-MG was brought forward.Then the concept and intention of the SMT-MG were analyzed.And from the three aspects of sufficient condition,necessary condition and stability condition,the static and dynamic conditions under which the SMT-MG was formed were mainly discuss.Finally,it was proved that the formation and development of SMT-MG is feasible and necessary.

Necessary Optimality Conditions for Multi-Objective Semi-Infinite Variational Problem

In this paper, necessary optimality conditions for a class of Semi-infinite Variational Problems are established which are further generalized to a class of Multi-objective Semi-Infinite Variational Problems. These conditions are responsible for the development of duality theory which is an extremely important feature for any class of problems, but the literature available so far lacks these necessary optimality conditions for the stated problem. A lemma is also proved to find the topological dual of as it is required to prove the desired result.