A Note on a Triple Integral

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

Download Full-text

A Quadruple Definite Integral Expressed in Terms of the Lerch Function

Symmetry ◽

10.3390/sym13091638 ◽

2021 ◽

Vol 13 (9) ◽

pp. 1638

Author(s):

Robert Reynolds ◽

Allan Stauffer

Keyword(s):

Special Functions ◽

Definite Integral ◽

Polynomial Functions ◽

Fundamental Constants ◽

Zero Distribution ◽

Special Cases ◽

Almost All

A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch functions have an asymmetrical zero-distribution. The majority of the results in this work are new.

Download Full-text

A Quadruple Integral Involving Chebyshev Polynomials Tn(x): Derivation and Evaluation

Symmetry ◽

10.3390/sym14010100 ◽

2022 ◽

Vol 14 (1) ◽

pp. 100

Author(s):

Robert Reynolds ◽

Allan Stauffer

Keyword(s):

Zeta Function ◽

Chebyshev Polynomial ◽

Chebyshev Polynomials ◽

Zeta Functions ◽

Fundamental Constants ◽

Zero Distribution ◽

Special Cases ◽

Lerch Zeta Function ◽

Almost All

The aim of the current document is to evaluate a quadruple integral involving the Chebyshev polynomial of the first kind Tn(x) and derive in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. The zero distribution of almost all Hurwitz-Lerch zeta functions is asymmetrical. All the results in this work are new.

Download Full-text

Capacity ofκ-μShadowed Fading Channels

International Journal of Antennas and Propagation ◽

10.1155/2014/975109 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 16

Author(s):

Celia García-Corrales ◽

Francisco J. Cañete ◽

José F. Paris

Keyword(s):

Closed Form ◽

Fading Channels ◽

Ergodic Capacity ◽

Closed Form Expression ◽

Underwater Acoustic Communications ◽

Form Expression ◽

Satellite Systems ◽

Special Cases ◽

Parameter Variations ◽

Mobile Satellite

The ergodic capacity of fading channels modeled with aκ-μshadowed distribution is investigated to derive closed-form expressions. Theκ-μshadowed distribution is of particular interest because it contains, as special cases, other classical ones like one-side Gaussian, Rayleigh, Rician, Nakagami-m,κ-μ, and Rician shadowed distributions. The paper discusses the physical meaning of the distribution parameter variations and also their impact on the channel capacity. These results can be used to study the behavior of different channels like the ones in underwater acoustic communications, land mobile satellite systems, body centric communications, and other wireless communication applications. The analytical closed-form expression results are validated with numerical simulations.

Download Full-text

Explicit Baker–Campbell–Hausdorff Expansions

Mathematics ◽

10.3390/math6080135 ◽

2018 ◽

Vol 6 (8) ◽

pp. 135 ◽

Cited By ~ 7

Author(s):

Alexander Van-Brunt ◽

Matt Visser

Keyword(s):

Closed Form ◽

General Purpose ◽

Theoretical Physics ◽

Closed Form Expression ◽

Form Expression ◽

Explicit Result ◽

Special Cases ◽

Potential Applications ◽

Current Article ◽

The Creation

The Baker–Campbell–Hausdorff (BCH) expansion is a general purpose tool of use in many branches of mathematics and theoretical physics. Only in some special cases can the expansion be evaluated in closed form. In an earlier article we demonstrated that whenever [X,Y]=uX+vY+cI, BCH expansion reduces to the tractable closed-form expression Z(X,Y)=ln(eXeY)=X+Y+f(u,v)[X,Y], where f(u,v)=f(v,u) is explicitly given by the the function f(u,v)=(u−v)eu+v−(ueu−vev)uv(eu−ev)=(u−v)−(ue−v−ve−u)uv(e−v−e−u). This result is much more general than those usually presented for either the Heisenberg commutator, [P,Q]=−iℏI, or the creation-destruction commutator, [a,a†]=I. In the current article, we provide an explicit and pedagogical exposition and further generalize and extend this result, primarily by relaxing the input assumptions. Under suitable conditions, to be discussed more fully in the text, and taking LAB=[A,B] as usual, we obtain the explicit result ln(eXeY)=X+Y+Ie−LX−e+LYI−e−LXLX+I−e+LYLY[X,Y]. We then indicate some potential applications.

Download Full-text

Synthesizing of Planar and Conformal Double-Sided Parallel-Cross Dipoles FSS Based on Closed-Form Expression

IEEE Access ◽

10.1109/access.2021.3096419 ◽

2021 ◽

pp. 1-1

Author(s):

Yassine Zouaoui ◽

Larbi Talbi ◽

Khelifa Hettak ◽

Naresh K. Darimireddy

Keyword(s):

Closed Form ◽

Closed Form Expression ◽

Form Expression

Download Full-text

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

ACM SIGMETRICS Performance Evaluation Review ◽

10.1145/3453953.3453974 ◽

2021 ◽

Vol 48 (3) ◽

pp. 91-96

Author(s):

Shigeo Shioda

Keyword(s):

Distribution Function ◽

Probability Density Function ◽

Closed Form ◽

Probability Density ◽

Infinite Number ◽

Stable Distribution ◽

Random Variable ◽

Cauchy Distribution ◽

Closed Form Expression ◽

Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

Download Full-text

Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

Journal of High Energy Physics ◽

10.1007/jhep06(2021)063 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Vivek Kumar Singh ◽

Rama Mishra ◽

P. Ramadevi

Keyword(s):

Closed Form ◽

Topological Strings ◽

Chebyshev Polynomials ◽

Fibonacci Numbers ◽

Infinite Family ◽

Closed Form Expression ◽

Two Dimensional ◽

Laurent Polynomials ◽

Form Expression ◽

Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

Download Full-text