Multi-Integral Representations for Associated Legendre and Ferrers Functions

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order, including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind, including parameter values for which this function is identically zero.

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Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

Symmetry Integrability and Geometry Methods and Applications ◽

10.3842/sigma.2021.053 ◽

2021 ◽

Author(s):

Howard S. Cohl ◽

◽

Justin Park ◽

Hans Volkmer ◽

◽

...

Keyword(s):

Hypergeometric Function ◽

Complex Plane ◽

Fourier Expansion ◽

Legendre Function ◽

Legendre Functions ◽

Detailed Review ◽

Associated Legendre Function ◽

Branch Cut

We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments which correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at [1,infty). In order to complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally we give a detailed review of the 1888 paper by Richard Olbricht who was the first to study hypergeometric representations of Legendre functions.

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Effect of data quality on estimates of farm infectiousness trends in the UK 2001 foot-and-mouth disease epidemic

Journal of The Royal Society Interface ◽

10.1098/rsif.2006.0178 ◽

2006 ◽

Vol 4 (13) ◽

pp. 235-241 ◽

Cited By ~ 13

Author(s):

Nicholas J Savill ◽

Darren J Shaw ◽

Rob Deardon ◽

Michael J Tildesley ◽

Matthew J Keeling ◽

...

Keyword(s):

Mathematical Models ◽

Data Quality ◽

Foot And Mouth Disease ◽

Mouth Disease ◽

Inaccurate Data ◽

Foot And Mouth ◽

The Uk ◽

Parameter Values

Most of the mathematical models that were developed to study the UK 2001 foot-and-mouth disease epidemic assumed that the infectiousness of infected premises was constant over their infectious periods. However, there is some controversy over whether this assumption is appropriate. Uncertainty about which farm infected which in 2001 means that the only method to determine if there were trends in farm infectiousness is the fitting of mechanistic mathematical models to the epidemic data. The parameter values that are estimated using this technique, however, may be influenced by missing and inaccurate data. In particular to the UK 2001 epidemic, this includes unreported infectives, inaccurate farm infection dates and unknown farm latent periods. Here, we show that such data degradation prevents successful determination of trends in farm infectiousness.

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Determination of lipid order parameters and rotational correlation times from fluorescence depolarization experiments

FEBS Letters ◽

10.1016/0014-5793(79)80564-5 ◽

1979 ◽

Vol 108 (2) ◽

pp. 359-364 ◽

Cited By ~ 323

Author(s):

Maarten P. Heyn

Keyword(s):

Order Parameters ◽

Fluorescence Depolarization ◽

Correlation Times ◽

Lipid Order ◽

Rotational Correlation Times ◽

Rotational Correlation

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Comparative Study of Optimization Algorithms for The Optimal Reservoir Operation

MATEC Web of Conferences ◽

10.1051/matecconf/201824601003 ◽

2018 ◽

Vol 246 ◽

pp. 01003

Author(s):

Xinyuan Liu ◽

Yonghui Zhu ◽

Lingyun Li ◽

Lu Chen

Keyword(s):

Reservoir Operation ◽

Optimization Problems ◽

Optimization Algorithms ◽

Complex Problem ◽

Optimization Techniques ◽

Decision Variables ◽

Handling Methods ◽

Optimality Algorithm ◽

Parameter Values

Apart from traditional optimization techniques, e.g. progressive optimality algorithm (POA), modern intelligence algorithms, like genetic algorithms, differential evolution have been widely used to solve optimization problems. This paper deals with comparative analysis of POA, GA and DE and their applications in a reservoir operation problem. The results show that both GA and DES are feasible to reservoir operation optimization, but they display different features. GA and DE have many parameters and are difficult in determination of these parameter values. For simple problems with mall number of decision variables, GA and DE are better than POA when adopting appropriate parameter values and constraint handling methods. But for complex problem with large number of variables, POA combined with simplex method are much superior to GA and DE in time-assuming and quality of optimal solutions. This study helps to select proper optimization algorithms and parameter values in reservoir operation.

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Lecture Scheduling Automation Using Genetic Algorithm

International Journal of Engineering and Advanced Technology - Regular Issue ◽

10.35940/ijeat.e1210.0585c19 ◽

2019 ◽

Vol 8 (5C) ◽

pp. 1444-1447

Keyword(s):

Genetic Algorithm ◽

Genetic Algorithms ◽

Test Results ◽

Scheduling Problems ◽

Scheduling System ◽

Parameter Values

This paper aims produce an academic scheduling system using Genetic Algorithm (GA) to solve the academic schedule. Factors to consider in academic scheduling are the lecture to be held, the available room, the lecturers and the time of the lecturer, the suitability of the credits with the time of the lecture, and perhaps also the time of Friday prayers, and so forth. Genetic Algorithms can provide the best solution for some solutions in dealing with scheduling problems. Based on the test results, the resulting system can automate the scheduling of lectures properly. Determination of parameter values in Genetic Algorithm also gives effect in producing the solution of lecture schedule

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NMR Determination of Order Parameters in Quadrupolar Glasses

25th Congress Ampere on Magnetic Resonance and Related Phenomena ◽

10.1007/978-3-642-76072-3_28 ◽

1990 ◽

pp. 60-61

Author(s):

W. Wiotte ◽

S. Elschner ◽

J. Petersson ◽

R. Blinc

Keyword(s):

Order Parameters

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Langevin field equations: Properties and renormalization

Quantum Field Theory and Critical Phenomena ◽

10.1093/oso/9780198834625.003.0035 ◽

2021 ◽

pp. 857-874

Author(s):

Jean Zinn-Justin

Keyword(s):

Integral Representation ◽

Langevin Equation ◽

Gauge Theories ◽

Homogeneous Spaces ◽

Brst Symmetry ◽

Integral Representations ◽

Constraint Equations ◽

Dynamic Action ◽

Geometric Origin ◽

Field Integral

Langevin equations for fields have been proposed to describe the dynamics of critical phenomena, or as an alternative method of quantization, which could be useful in cases where ordinary methods lead to difficulties, like in gauge theories. Some of their general properties will be described here. For a number of problems, in particular related to perturbation theory, it is more convenient to work with an action and a field integral than with the equation directly, because standard methods of quantum field theory (QFT) then become available. For this purpose, one can associate a field integral representation, involving a dynamic action to the Langevin equation. The dynamic action can be interpreted as generated by the Langevin equation, considered as a constraint equation. Quite generally, the integral representation of constraint equations, including stochastic equations, leads to an action that has a Slavnov–Taylor (ST) symmetry and, in a different form, has an anticommuting type Becchi–Rouet–Stora–Tyutin (BRST) symmetry, a symmetry that involves anticommuting parameters. This symmetry has no geometric origin, but is merely a consequence of associating a specific form of integral representations to the constraint equations. This symmetry is used in a number of different examples to prove the renormalizability of non-Abelian gauge theories, or to prove the geometric stability of models defined on homogeneous spaces under renormalization. In this chapter, it is used to prove Ward-Takahashi (WT) identities, and to determine how the Langevin equation renormalizes.

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Positive Forms on Nuclear *-Algebras and Their Integral Representations

Canadian Journal of Mathematics ◽

10.4153/cjm-1990-023-3 ◽

1990 ◽

Vol 42 (3) ◽

pp. 410-469 ◽

Cited By ~ 4

Author(s):

Alain Bélanger ◽

Erik G. F. Thomas

Keyword(s):

Integral Representation ◽

Positive Cone ◽

Approximate Identity ◽

Integral Representations ◽

Regularity Conditions ◽

Direct Integral ◽

Positive Form ◽

Representations Of Algebras ◽

Almost All ◽

Order Intervals

Abstract.The main result of this paper establishes the existence and uniqueness of integral representations of KMS functionals on nuclear *- algebras. Our first result is about representations of *-algebras by means of operators having a common dense domain in a Hilbert space. We show, under certain regularity conditions, that (Powers) self-adjoint representations of a nuclear *-algebra, which admit a direct integral decomposition, disintegrate into representations which are almost all self-adjoint. We then define and study the class of self-derivative algebras. All algebras with an identity are in this class and many other examples are given. We show that if is a self-derivative algebra with an equicontinuous approximate identity, the cone of all positive forms on is isomorphic to the cone of all positive invariant kernels on These in turn correspond bijectively to the invariant Hilbert subspaces of the dual space This shows that if is a nuclear -space, the positive cone of has bounded order intervals, which implies that each positive form on has an integral representation in terms of the extreme generators of the cone. Given a continuous exponentially bounded one-parameter group of *-automorphisms of we can define the subcone of all invariant positive forms satisfying the KMS condition. Central functionals can be viewed as KMS functionals with respect to a trivial group action. Assuming that is a self-derivative algebra with an equicontinuous approximate identity, we show that the face generated by a self-adjoint KMS functional is a lattice. If is moreover a nuclear *-algebra the previous results together imply that each self-adjoint KMS functional has a unique integral representation by means of extreme KMS functionals almost all of which are self-adjoint.

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