Results 261 to 270 of about 401,736 (305)
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1975
The student is probably already familiar with the result that the sum of the infinite geometric progression: 1 + x + x 2 + x 3 + ... + x r + ... is equal to 1/(1 — x), as long as the common ratio x is numerically less than 1. We may thus write: $$ \frac{1}{{1 - x}} = 1 + x + {x^2} + {x^3} + ... + {x^r} + ...\left( {\left| x \right| < 1} \right) $$
Brian Knight, Roger Adams
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The student is probably already familiar with the result that the sum of the infinite geometric progression: 1 + x + x 2 + x 3 + ... + x r + ... is equal to 1/(1 — x), as long as the common ratio x is numerically less than 1. We may thus write: $$ \frac{1}{{1 - x}} = 1 + x + {x^2} + {x^3} + ... + {x^r} + ...\left( {\left| x \right| < 1} \right) $$
Brian Knight, Roger Adams
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1992
Abstract Exact power series expansions for thermodynamic functions have in the past proved an invaluable aid to understanding the critical behaviour of insoluble models. Indeed, the first suggestions of power law singularities at criticality were based on such analyses.
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Abstract Exact power series expansions for thermodynamic functions have in the past proved an invaluable aid to understanding the critical behaviour of insoluble models. Indeed, the first suggestions of power law singularities at criticality were based on such analyses.
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Series Expansions of Generalized Matrix Products
Proceedings of the 44th IEEE Conference on Decision and Control, 2006We consider generalized products of random matrices. They arise in discrete event systems (DES), such as queueing networks or stochastic Petri nets, where they are used to express the state transition dynamic. Instances of such DES are those whose state dynamic can be modelled through a matrix-vector multiplication in conventional, max-plus and min ...
Haralambie Leahu, Bernd Heidergott
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Expansions in Series of Products of Eigenfunctions
Results in Mathematics, 1996Consider \(m\)-fold products \(\eta = \eta_1 \dots \eta_m\) of special functions. Each of the factors satisfies a differential equation (of order \(\geq 1\) \([\geq 2\), in general]). A general method is introduced so that this product is the first component of a solution of a first order system of differential equations (Theorem 2.1). The construction
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Expansions in series of appell polynomials
Mathematical Notes of the Academy of Sciences of the USSR, 1969We consider questions connected with the problem of expanding an entire function in a series of Appell polynomials {Pn(t)}, when the entire function is of growth no higher than first order and of normal type σ, under the assumption that certain functions A(t) and B(t) are regular in the disc ¦t¦
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On the Expansion of the Partition Function in a Series
The Annals of Mathematics, 19431. A geometric property of the Farey series, discovered by L. R. Ford (1) is used in this note for the construction of a new path of integration to replace the circle carrying the Farey dissection, first introduced by Hardy and Ramanujan in their classical paper (2).
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Expansions in Series of Polynomials
Journal of the London Mathematical Society, 1968openaire +2 more sources
Asymptotic Series and Expansions
2003The method of perturbations can be used to develop approximate solutions to differential equations, which have nonlinearities or variable coefficients so that an exact solution cannot be constructed. Questions of convergence of the perturbation expansion in terms of a suitable small parameter are abandoned if such an expansion can provide a useful ...
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ON AN EXPANSION IN EXPONENTIAL SERIES
The Quarterly Journal of Mathematics, 1956openaire +1 more source