New rational fraction approximating formulas for the temperature integral

Three rational fraction approximations for the temperature integral have been proposed using the pattern search method. The validity of the new approximations has been tested by some numerical analyses. Compared with several published approximating formulas, the new approximations is more accurate than all approximations except the approximations proposed by Senum and Yang in the range of 5≤E/RT≤100. For low values of E/RT, the new approximations are superior to Senum-Yang approximations as solutions of the temperature integral.     

Approximation of finite-section equations by piecewise constant functions

The problem is studied of reducing the amount of discrete information required for achieving a prescribed accuracy of solving Fredholm integral equations of the first kind on a half-line. The equations are solved by the finite-section method combined with piecewise constant interpolation of the kernel and the right-hand side at uniform grid points. The approximating properties of the discretization schemes are examined, and the corresponding computational costs are analyzed.     

Locally one-dimensional difference schemes for the fractional order diffusion equation

Locally-one-dimensional difference schemes for the fractional diffusion equation in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved.     

B-(p,r)-不变凸性下广义分式规划的一个鞍点最优性准则

对于一类目标函数中有无限个分式的广义分式规划,给出一个不完全Lagrange函数,并利用已有的最优性必要条件,在B-(p,r)-不变凸性的条件下,证明了鞍点最优性准则.     

Inverse coefficient problem for a wave equation in a bounded domain

The nonlinear inverse problem for a wave equation is investigated in a three-dimensional bounded domain subject to the Dirichlet boundary condition. Given a family of solutions to the equation defined on a closed surface within the original domain, it is required to reconstruct the coefficient determining the velocity of sound in the medium. The solutions used for this purpose correspond to the acoustic medium perturbations localized in the neighborhood of a certain closed surface. The inverse problem is reduced to a linear integral equation of the first kind, and the uniqueness of the solution to this equation is established. Numerical results are presented.     

On the problem of crack system with an elliptic hole

The problem of cracks with an elliptic hole in an infinite plane is investigated. By introducing the fictitious loads on the hole edge and using the Muskhelishvili method, the problem is reduced to solving a system of mixed-type integral equations in which some are Fredholm equations but others Cauchy-type singular ones. A numerical method is suggested and can be used for the treatment of other similar equations. The numerical results for some typical examples are given, showing that the method is really effective.     

Fluids with highly directional attractive forces. IV. Equilibrium polymerization

We investigate approximation methods for systems of molecules interacting by core repulsion and highly directional attraction due to several attraction sites. The force model chosen imitates a chemical bond by providing for bond saturation when binding occurs. The dense fluid is an equilibrium mixture ofs-mers with mutual repulsion. We use a previously derived reformulation of statistical thermodynamics, in which the particle species are monomeric units with a specified set of attraction sites bonded. Thermodynamic perturbation theory (TPT) and integral equations of two types are derived. The use of TPT is illustrated by explicit calculation for a molecular model with two attraction sites, capable of forming chain and ring polymers. Successes and defects of TPT are discussed. The integral equations for pair correlations between particles of specified bonding include calculation of self-consistent densities of species. Methods of calculating thermodynamic properties from the solutions of integral equations are given.Supported by the NSF under grant No. CHE-82-11236.     

Realization of a unique time evolution unitary operator in Klein Gordon theory

The scattering theory for the Klein Gordon equation, with time-dependent potential and in a non-static space-time, is considered. Using the Klein Gordon equation formulated in the Hubert spaceL ²(R ³) and the Einstein’s relativistic equation in the spaceL ²(R ³, dx) and establishing the equivalence of the vacuum states of their linearized forms in the Hubert spaceL ²(R ³) with the help of unique symmetric symplectic operator, the time evolution unitary operatorU(t) has been fixed for the Klein Gordon equation, incorporating either the positive or negative frequencies, in the infinite dimensional Hubert spaceL ²(R ³).     

On stochastic integral representation of stable processes with sample paths in Banach spaces

Certain path properties of a symmetric α-stable process X(t) = ∫_Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed.