XXIII.— Dual Series Relations. V. A Generalized Schlömilch Series and the Uniqueness of the Solution of Dual Equations involving Trigonometric Series

1964 ◽  
Vol 66 (4) ◽  
pp. 258-268
Author(s):  
R. P. Srivastav
Keyword(s):  
Trigonometric Series ◽  
Arbitrary Function ◽  
Fractional Integration ◽  
Series Expansions ◽  
Uniqueness Of The Solution ◽  
Special Case

SynopsisThe methods employed in papers I–IV of this series are modified to provide the solution of certain dual equations involving trigonometric series. It is necessary to introduce a modified form of the conventional operators of fractional integration and to discuss their relation with generalized Schlömilch series expansions of an arbitrary function. These general methods are illustrated by detailed reference to a particular special case.

scholarly journals On the Bean critical-state model in superconductivity

1996 ◽  
Vol 7 (3) ◽  
pp. 237-247 ◽  
Author(s):  
L. Prigozhin
Keyword(s):  
Variational Inequalities ◽  
Critical State ◽  
Existence And Uniqueness ◽  
Critical State Model ◽  
Type Ii ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Uniqueness Of The Solution ◽  
Type Ii Superconductivity ◽  
Special Case

We consider two-dimensional and axially symmetric critical-state problems in type-II superconductivity, and show that these problems are equivalent to evolutionary quasi-variational inequalities. In a special case, where the inequalities become variational, the existence and uniqueness of the solution are proved.

scholarly journals Coalescent lineage distributions

2006 ◽  
Vol 38 (02) ◽  
pp. 405-429 ◽  
Author(s):  
Robert C. Griffiths
Keyword(s):  
Brownian Motion ◽  
Hypergeometric Functions ◽  
Common Ancestor ◽  
Recent Common Ancestor ◽  
Series Expansions ◽  
Standard Brownian Motion ◽  
Most Recent Common Ancestor ◽  
Coalescent Tree ◽  
Expected Values ◽  
Special Case

We study identities for the distribution of the number of edges at time t back (i.e. measured backwards) in a coalescent tree whose subtrees have no mutations. This distribution is important in the infinitely-many-alleles model of mutation, where every mutation is unique. The model includes, as a special case, the number of edges in a coalescent tree at time t back when mutation is ignored. The identities take the form of expected values of functions of Z t =eiX t , where X t is distributed as standard Brownian motion. Associated identities are also found for the distributions of the time to the most recent common ancestor, the time until loss of ancestral lines by coalescence or mutation, and the age of a mutation. Hypergeometric functions play an important role in the identities. The identities are of mathematical interest, as well as potentially being formulae to use for numerical integration or simulation to compute distributions that are usually expressed as alternating-sign series expansions, which are difficult to compute.

Matrix boundary-value problems for differential equations with p-Laplacian

2020 ◽  
Vol 17 (1) ◽  
pp. 41-57
Author(s):  
Olga Nesmelova
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Arbitrary Function ◽  
Algebraic System ◽  
Boundary Value ◽  
System Of Differential Equations ◽  
Differential Algebraic System ◽  
Uniqueness Of The Solution

We consider the boundary-value problem for a linear system of differential equations with matrix p-Laplacian, which is reduced to the traditional differential-algebraic system with an unknown in the form of the vector function. A generalization of various boundary-value problems for differential equations with p-Laplacian, which preserves the features of the solution of such problems, namely, the lack of uniqueness of the solution and, in this case, the dependence of the desired solution on an arbitrary function, is given.

A Family of Transients Suitable for Reproduction on a Shaker Based on the cosm(x) Window

2002 ◽  
Vol 45 (1) ◽  
pp. 178-184 ◽  
Author(s):  
David Smallwood
Keyword(s):  
Arbitrary Function ◽  
Response Spectra ◽  
Negative Peak ◽  
Special Cases ◽  
Shock Response ◽  
X Window ◽  
Temporal Moments ◽  
Special Case

A family of transients with the property that the initial and final acceleration, velocity, and displacement are all zero is derived. The transients are based on a relatively arbitrary function multiplied by window of the form cosm(x). Several special cases are discussed which result in odd acceleration and displacement functions. This is desirable for shaker reproduction because the required positive and negative peak accelerations and displacements will be balanced. Another special case is discussed which will permit the development of transients with the first five (0-4) temporal moments specified. The transients are defined with three or four parameters that will allow sums of components to be found which will match a variety of shock response spectra.

Properties of a Lattice Gas Model for 3He–4He Mixtures I: Generation of High Temperature Series

1975 ◽  
Vol 53 (10) ◽  
pp. 980-986 ◽  
Author(s):  
M. Plischke ◽  
C. J. Elliott
Keyword(s):  
High Temperature ◽  
Lattice Gas ◽  
Temperature Series ◽  
Quantum Mechanical ◽  
Series Expansions ◽  
Xy Model ◽  
Exchange Constant ◽  
Inverse Temperature ◽  
Lattice Fluid ◽  
Special Case

The Cheng–Schick model is a quantum mechanical lattice fluid for 3He–4He mixtures which reduces in one limit to the spin 1/2 XY model and in the opposite limit to a special case of the Hubbard model. The method for generating high temperature series expansions of the thermodynamic properties of the model is described in some detail. Series expansions in the inverse temperature have been obtained to order 10 for the free energy and to order 8 for the fluctuation in the long range order both for arbitrary boson and fermion fugacities and arbitrary exchange constant ratios. In the following article (Plischke and Betts) the series are analyzed and the results are compared with experiment and with other theories for 3He–4He mixtures.

scholarly journals On the Vasyunin Cotangent sums related to Riemann Hypothesis

2021 ◽  
Vol 19 ◽  
pp. 676-682 ◽  
Author(s):  
Samir Belhadj ◽  
Mouloud Goubi
Keyword(s):  
Recent Work ◽  
Generating Functions ◽  
Riemann Hypothesis ◽  
Euclidean Algorithm ◽  
Inner Product ◽  
Series Expansions ◽  
Special Case

In this work, we are interested by Vasyunin cotangent-sum V (p/q) encountered in computation of the inner product arising in the Baez-Duarte-Balazard criterion for Riemann hypothesis. By hint of generating functions theory and introduction of double Euclidean algorithm, we give series expansions of V (p/q) and the symmetric sum S (p, q) = V (p/q)+V (q/p). These calculus permit to deduce another reformulation of Vasyunin formula. This study is a complement of the recent work of M. Goubi concerning special case V (1/q).

scholarly journals Coalescent lineage distributions

2006 ◽  
Vol 38 (2) ◽  
pp. 405-429 ◽  
Author(s):  
Robert C. Griffiths
Keyword(s):  
Brownian Motion ◽  
Hypergeometric Functions ◽  
Common Ancestor ◽  
Recent Common Ancestor ◽  
Series Expansions ◽  
Standard Brownian Motion ◽  
Most Recent Common Ancestor ◽  
Coalescent Tree ◽  
Expected Values ◽  
Special Case

We study identities for the distribution of the number of edges at time t back (i.e. measured backwards) in a coalescent tree whose subtrees have no mutations. This distribution is important in the infinitely-many-alleles model of mutation, where every mutation is unique. The model includes, as a special case, the number of edges in a coalescent tree at time t back when mutation is ignored. The identities take the form of expected values of functions of Zt=eiXt, where Xt is distributed as standard Brownian motion. Associated identities are also found for the distributions of the time to the most recent common ancestor, the time until loss of ancestral lines by coalescence or mutation, and the age of a mutation. Hypergeometric functions play an important role in the identities. The identities are of mathematical interest, as well as potentially being formulae to use for numerical integration or simulation to compute distributions that are usually expressed as alternating-sign series expansions, which are difficult to compute.

scholarly journals Approximate Trigonometric Series Expansions of Some Bounded Solutions in the Tsien Problem

2019 ◽  
Vol 42 (10) ◽  
pp. 2325-2330 ◽  
Author(s):  
M. Calvo ◽  
J. I. Montijano ◽  
L. Rández ◽  
A. Elipe
Keyword(s):  
Trigonometric Series ◽  
Series Expansions ◽  
Bounded Solutions

scholarly journals Series Expansion of a Cotangent Sum Related to the Estermann Zeta Function

2021 ◽  
Vol 45 (03) ◽  
pp. 343-352
Author(s):  
M. GOUBI
Keyword(s):  
Zeta Function ◽  
Series Expansion ◽  
Series Expansions ◽  
Special Case

In this paper, we study the cotangent sum c0( ) q p related to the Estermann zeta function for the special case when the numerator is equal to 1 and get two useful series expansions of c0( 1 ) p.

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