scholarly journals Analysis of Gegenbauer kernel filtration on the hypersphere

2021 ◽  
Vol 8 (11) ◽  
pp. 1-9
Author(s):  
Omenyi et al. ◽  
Keyword(s):  
Fundamental Solution ◽  
Beltrami Operator ◽  
Gegenbauer Polynomials ◽  
Wave Problem ◽  
Linear Combinations ◽  
One Dimensional ◽  
Unit Hypersphere ◽  
Finite Linear

In this study, we aim to construct explicit forms of convolution formulae for Gegenbauer kernel filtration on the surface of the unit hypersphere. Using the properties of Gegenbauer polynomials, we reformulated Gegenbauer filtration as the limit of a sequence of finite linear combinations of hyperspherical Legendre harmonics and gave proof for the completeness of the associated series. We also proved the existence of a fundamental solution of the spherical Laplace-Beltrami operator on the hypersphere using the filtration kernel. An application of the filtration on a one-dimensional Cauchy wave problem was also demonstrated.

Multi-particle Spaces

2020 ◽  
pp. 187-200
Author(s):  
Daniel Canarutto
Keyword(s):  
Operator Algebra ◽  
Particle Physics ◽  
Arbitrary Number ◽  
Exterior Algebra ◽  
Linear Combinations ◽  
Absorption And Emission ◽  
Dirac Type ◽  
Natural Generalisation ◽  
Finite Linear ◽  
Number Of Particles

The fundamental algebraic notions needed in many-particle physics are exposed. Spaces of free states containing an arbitrary number of particles of many types are introduced. The operator algebra generated by absorption and emission operators is studied as a natural generalisation of standard exterior algebra. The link between the discrete and the distributional formalisms is provided by the spaces of finite linear combinations of semi-densities of Dirac type.

scholarly journals Old and New Identities for Bernoulli Polynomials via Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Luis M. Navas ◽  
Francisco J. Ruiz ◽  
Juan L. Varona
Keyword(s):  
Fourier Series ◽  
Linear Combination ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Gegenbauer Polynomials ◽  
Linear Combinations ◽  
The Given

The Bernoulli polynomialsBkrestricted to[0,1)and extended by periodicity haventh sine and cosine Fourier coefficients of the formCk/nk. In general, the Fourier coefficients of any polynomial restricted to[0,1)are linear combinations of terms of the form1/nk. If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.

Periodic travelling interfacial hydroelastic waves with or without mass II: Multiple bifurcations and ripples

2018 ◽  
Vol 30 (04) ◽  
pp. 756-790 ◽  
Author(s):  
BENJAMIN F. AKERS ◽  
DAVID M. AMBROSE ◽  
DAVID W. SULON
Keyword(s):  
Travelling Waves ◽  
Global Bifurcation ◽  
Travelling Wave ◽  
Two Dimensional ◽  
Wave Problem ◽  
Bifurcation Theorem ◽  
One Dimensional ◽  
Spatially Periodic ◽  
Parameter Values ◽  
The Waves

In a prior work, the authors proved a global bifurcation theorem for spatially periodic interfacial hydroelastic travelling waves on infinite depth, and computed such travelling waves. The formulation of the travelling wave problem used both analytically and numerically allows for waves with multi-valued height. The global bifurcation theorem required a one-dimensional kernel in the linearization of the relevant mapping, but for some parameter values, the kernel is instead two-dimensional. In the present work, we study these cases with two-dimensional kernels, which occur in resonant and non-resonant variants. We apply an implicit function theorem argument to prove existence of travelling waves in both of these situations. We compute the waves numerically as well, in both the resonant and non-resonant cases.

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