The spectral properties of many‐electron atomic Hamiltonians and the method of configuration interaction. II. Compactness proof associated with an infinite system of linear equations for two‐electron atoms

1979 ◽  
Vol 20 (5) ◽  
pp. 943-952
Author(s):  
M. H. Choudhury
Keyword(s):  
Configuration Interaction ◽  
Spectral Properties ◽  
Infinite System ◽  
Linear Equations ◽  
System Of Linear Equations

On the Stresses in a Notched Strip

1952 ◽  
Vol 19 (2) ◽  
pp. 141-146
Author(s):  
Chih-Bing Ling
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Numerical Results ◽  
Theoretical Solution ◽  
Experimental Results ◽  
System Of Linear Equations ◽  
Transverse Bending ◽  
Longitudinal Tension

Abstract In a previous paper by the author (1), a theoretical solution for a notched strip under longitudinal tension is given. The result demands the solution of an infinite system of linear equations. A considerable amount of labor is involved in solving such a system. It seems, however, that the labor can be diminished by adapting to the solution a process known as the promotion of rank. In this paper such a process is described and then applied to solve the problem of a notched strip under transverse bending. The solution of this problem seems also to be new. The numerical results obtained are compared graphically with the experimental results available.

THE ELECTROSTATIC FIELD OF TWO EQUAL CIRCULAR CO-AXIAL CONDUCTING DISKS

1949 ◽  
Vol 2 (4) ◽  
pp. 428-451 ◽  
Author(s):  
E. R. LOVE
Keyword(s):  
Integral Equation ◽  
Explicit Solution ◽  
Existence And Uniqueness ◽  
Infinite System ◽  
Linear Equations ◽  
Neumann Series ◽  
Standard Theory ◽  
System Of Linear Equations ◽  
Electrostatic Problem ◽  
Spheroidal Harmonics

Abstract In the earliest discussion of this problem Nicholson (1) expressed the potential as a series of spheroidal harmonics with coefficients satisfying an infinite system of linear equations, and gave a formula for an explicit solution; but this formula appears to be meaningless and its derivation to contain serious errors. In the present paper, starting tentatively from Nicholson's infinite system of linear equations, a much simpler, though still implicit, specification of the potential is developed; this involves a Fredholm integral equation the existence and uniqueness of whose solution are deducible from standard theory. The specification so obtained for the potential is shown rigorously to satisfy the differential equation and boundary conditions of the electrostatic problem. The Neumann series of the integral equation is shown to converge to its solution, so that the potential, and other aspects of the field, can be explicitly formulated and thus computed. The errors in Nicholson's process of solving his system of equations are exhibited in detail, and it is concluded that attempts to carry through that process without error cannot lead to an explicit solution.

A new approach to (s, S) inventory systems

1993 ◽  
Vol 30 (4) ◽  
pp. 898-912 ◽  
Author(s):  
Jian-Qiang Hu ◽  
Soracha Nananukul ◽  
Wei-Bo Gong
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Inventory Systems ◽  
Inventory Level ◽  
Maclaurin Series ◽  
System Of Linear Equations ◽  
New Approach ◽  
Recursive Procedures ◽  
Recursive Equations ◽  
Infinite Linear

In this paper, we consider period review (s, S) inventory systems with independent and identically distributed continuous demands and full backlogging. Using an approach recently proposed by Gong and Hu (1992), we derive an infinite system of linear equations for all moments of inventory level. Based on this infinite system, we develop two algorithms to calculate the moments of the inventory level. In the first one, we solve a finite system of linear equations whose solution converges to the moments as its dimension goes to infinity. In the second one, we in fact obtain the power series of the moments with respect to s and S. Both algorithms are based on some very simple recursive procedures. To show their efficiency and speed, we provide some numerical examples for the first algorithm.(s, S) INVENTORY SYSTEMS; DYNAMIC RECURSIVE EQUATIONS; INFINITE LINEAR EQUATIONS; MACLAURIN SERIES

The eigenvalue problem for infinite systems of linear equations

1977 ◽  
Vol 82 (2) ◽  
pp. 269-273 ◽  
Author(s):  
F. P. Sayer
Keyword(s):  
Eigenvalue Problem ◽  
Infinite System ◽  
Linear Equations ◽  
Theoretical Work ◽  
Rectangular Plates ◽  
System Of Linear Equations ◽  
Usual Procedure ◽  
Systems Of Linear Equations ◽  
Image Position

Given an infinite system of linear equationswhere the aij depend on a parameter λ, the eigenvalue problem is to determine values of λ for which xj (j = 1, 2, …) are not all zero. This problem (Taylor (3) and Vaughan (4)) can arise in the vibration of rectangular plates. Little theoretical work, however, appears to have been done concerning the existence and determination of the eigenvalues. The usual procedure (see (3) and (4)) is to consider a truncated or reduced system of N equations and find the values of λ for which the determinant of the N × N matrix [aij] vanishes. If a particular λ tends to a constant value as N is increased then this value is assumed to be an eigenvalue. The question therefore arises as to what happens if no limit exists. Can we assert that there are no eigenvalues? By constructing an appropriate example we show that the non-existence of a limit does not imply the non-existence of eigenvalues. In order to construct our example we first establish a result concerning the Legendre polynomials.

A new approach to (s, S) inventory systems

1993 ◽  
Vol 30 (04) ◽  
pp. 898-912
Author(s):  
Jian-Qiang Hu ◽  
Soracha Nananukul ◽  
Wei-Bo Gong
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Inventory Systems ◽  
Inventory Level ◽  
Maclaurin Series ◽  
System Of Linear Equations ◽  
New Approach ◽  
Recursive Procedures ◽  
Recursive Equations ◽  
Infinite Linear

In this paper, we consider period review (s, S) inventory systems with independent and identically distributed continuous demands and full backlogging. Using an approach recently proposed by Gong and Hu (1992), we derive an infinite system of linear equations for all moments of inventory level. Based on this infinite system, we develop two algorithms to calculate the moments of the inventory level. In the first one, we solve a finite system of linear equations whose solution converges to the moments as its dimension goes to infinity. In the second one, we in fact obtain the power series of the moments with respect to s and S. Both algorithms are based on some very simple recursive procedures. To show their efficiency and speed, we provide some numerical examples for the first algorithm. (s, S) INVENTORY SYSTEMS; DYNAMIC RECURSIVE EQUATIONS; INFINITE LINEAR EQUATIONS; MACLAURIN SERIES

scholarly journals Method of Infinite System of Equations for Problems in Unbounded Domains

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Dang Quang A ◽  
Tran Dinh Hung
Keyword(s):  
Numerical Experiments ◽  
Infinite System ◽  
Linear Equations ◽  
Unbounded Domains ◽  
Uniform Grid ◽  
System Of Equations ◽  
System Of Linear Equations ◽  
One Dimensional ◽  
Infinite Domains ◽  
Artificial Boundaries

Many problems of mechanics and physics are posed in unbounded (or infinite) domains. For solving these problems one typically limits them to bounded domains and find ways to set appropriate conditions on artificial boundaries or use quasi-uniform grid that maps unbounded domains to bounded ones. Differently from the above methods we approach to problems in unbounded domains by infinite system of equations. In this paper we present starting results in this approach for some one-dimensional problems. The problems are reduced to infinite system of linear equations. A method for obtaining approximate solution with a given accuracy is proposed. Numerical experiments for several examples show the effectiveness of the offered method.

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