scholarly journals Some BBP-type series for polylog integrals

2021 ◽  
Vol 25 (2) ◽  
pp. 315-329
Author(s):  
Anthony Sofo
Keyword(s):  
Linear Combination ◽  
Beta Function ◽  
Zeta Functions ◽  
Type Formula ◽  
Type Series ◽  
Definite Integrals ◽  
Euler Sums ◽  
Dirichlet Beta Function ◽  
Special Case

An investigation into a family of definite integrals containing log-polylog functions will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals and may be represented as a BBP-type formula. In a special case we prove that the corresponding log integral can be represented as a linear combination of the product of zeta functions and the Dirichlet beta function.

Jones polynomial invariants for knots and satellites

1991 ◽  
Vol 109 (1) ◽  
pp. 83-103 ◽  
Author(s):  
H. R. Morton ◽  
P. Strickland
Keyword(s):  
Quantum Group ◽  
Linear Combination ◽  
Simple Formula ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Satellite Link ◽  
Representation Ring ◽  
Parallel Links ◽  
Jones Polynomials ◽  
Special Case

AbstractResults of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

Minimal surfaces with elastic and partially elastic boundary

2020 ◽  
pp. 1-22
Author(s):  
Bennett Palmer ◽  
Álvaro Pámpano
Keyword(s):  
Linear Combination ◽  
Minimal Surfaces ◽  
Elastic Boundary ◽  
Darboux Frame ◽  
Special Case

We study equilibrium surfaces for an energy which is a linear combination of the area and a second term which measures the bending and twisting of the boundary. Specifically, the twisting energy is given by the twisting of the Darboux frame. This energy is a modification of the Euler–Plateau functional considered by Giomi and Mahadevan (2012, Proc. R. Soc. A 468, 1851–1864), and a natural special case of the Kirchhoff–Plateau energy considered by Biria and Fried (2014, Proc. R. Soc. A 470, 20140368; 2015, Int. J. Eng. Sci. 94, 86–102).

scholarly journals Double Euler Sums on Hurwitz Zeta Functions

2009 ◽  
Vol 39 (6) ◽  
pp. 1869-1883 ◽  
Author(s):  
Minking Eie ◽  
Wen-Chin Liaw
Keyword(s):  
Zeta Functions ◽  
Euler Sums

scholarly journals Computing motivic zeta functions on log smooth models

2019 ◽  
Vol 295 (1-2) ◽  
pp. 427-462 ◽  
Author(s):  
Emmanuel Bultot ◽  
Johannes Nicaise
Keyword(s):  
Zeta Function ◽  
Recent Work ◽  
Explicit Formula ◽  
Essential Role ◽  
Zeta Functions ◽  
Smooth Model ◽  
Motivic Zeta Functions ◽  
Motivic Zeta Function ◽  
Special Case

Abstract We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. This formula plays an essential role in recent work on motivic zeta functions of degenerating Calabi–Yau varieties by the second-named author and his collaborators. As a further illustration, we explain how the formula for Newton non-degenerate polynomials can be viewed as a special case of our results.

Mixed cobinary trees

2018 ◽  
Vol 17 (09) ◽  
pp. 1850170
Author(s):  
Kiyoshi Igusa ◽  
Jonah Ostroff
Keyword(s):  
Short Proof ◽  
Point Of View ◽  
Catalan Number ◽  
Binary Trees ◽  
Cluster Theory ◽  
Type Formula ◽  
Isomorphism Classes ◽  
Basic Cluster ◽  
Number Formula ◽  
Special Case

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees (MCTs). We show that the number of isomorphism classes of such trees is given by the Catalan number [Formula: see text] where [Formula: see text] is the number of internal nodes. We also consider the corresponding quiver [Formula: see text] of type [Formula: see text]. As a special case of more general known results about the relation between [Formula: see text]-vectors, representations of quivers and their semi-invariants, we explain the bijection between MCTs and the vertices of the generalized associahedron corresponding to the quiver [Formula: see text]. These results are extended to [Formula: see text]-clusters in the next paper. We give one application: a new short proof of a conjecture of Reineke using MCTs.

REGULARIZATION OF GENERAL MULTIDIMENSIONAL EPSTEIN ZETA-FUNCTIONS

1989 ◽  
Vol 01 (01) ◽  
pp. 113-128 ◽  
Author(s):  
E. ELIZALDE ◽  
A. ROMEO
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Partition Function ◽  
Zeta Function ◽  
Finite Temperature ◽  
Casimir Effect ◽  
Zeta Functions ◽  
Quantum Field ◽  
Type Formula

We study expressions for the regularization of general multidimensional Epstein zeta-functions of the type [Formula: see text] After reviewing some classical results in the light of the extended proof of zeta-function regularization recently obtained by the authors, approximate but very quickly convergent expressions for these functions are derived. This type of analysis has many interesting applications, e.g. in any quantum field theory defined in a partially compactified Euclidean spacetime or at finite temperature. As an example, we obtain the partition function for the Casimir effect at finite temperature.

scholarly journals A NEW APPROACH TO THE PARAMETRIZATION OF THE CABIBBO–KOBAYASHI–MASKAWA MATRIX

2001 ◽  
Vol 16 (09) ◽  
pp. 1645-1652 ◽  
Author(s):  
V. GUPTA
Keyword(s):  
Cp Violation ◽  
Linear Combination ◽  
Quark Mass ◽  
New Approach ◽  
Ckm Matrix ◽  
Unit Matrix ◽  
Mass Matrices ◽  
The Matrix ◽  
Special Case

The CKM-matrix V is written as a linear combination of the unit matrix I and a matrix U which causes intergenerational-mixing. It is shown that such a V results from a class of quark-mass matrices. The matrix U has to be Hermitian and unitary and therefore can depend at most on four real parameters. The available data on the CKM-matrix including CP-violation can be reproduced by [Formula: see text]. This is also true for the special case when U depends on only 2 real parameters. Also, for such a V the invariant phase Φ≡ϕ12+ϕ23-ϕ13, satisfies a criterion suggested for "maximal" CP-violation.

scholarly journals Analytical Solution to Improper Integral of Divergent Power Functions Using The Riemann Zeta Function

2019 ◽  
Author(s):  
Siamak Tafazoli ◽  
Farhad Aghili
Keyword(s):  
Closed Form Solution ◽  
Zeta Functions ◽  
Bernoulli Numbers ◽  
Form Solution ◽  
Binomial Coefficients ◽  
Solution Technique ◽  
Power Functions ◽  
Definite Integrals ◽  
Improper Integral ◽  
Riemann Zeta

This paper presents an analytical closed-form solution to improper integral $\mu(r)=\int_0^{\infty} x^r dx$, where $r \geq 0$. The solution technique is based on splitting the improper integral into an infinite sum of definite integrals with successive integer limits. The exact solution of every definite integral is obtained by making use of the binomial polynomial expansion, which then allows expression of the entire summation equivalently in terms of a weighted sum of Riemann zeta functions. It turns out that the solution fundamentally depends on whether or not $r$ is an integer. If $r$ is a non-negative integer, then the solution is manifested in a finite series of weighted Bernoulli numbers, which is then drastically simplified to a second order rational function $\mu(r)=(-1)^{r+1}/(r+1)(r+2)$. This is achieved by taking advantage of the relationships between Bernoulli numbers and binomial coefficients. On the other hand, if $r$ is a non-integer real-valued number, then we prove $\mu(r)=0$ by the virtue of the elegant relationships between zeta and gamma functions and their properties.

scholarly journals Development of a Digital Model of a Beta Function Based on Mellin’s Integral Transforms

2021 ◽  
Vol 2096 (1) ◽  
pp. 012108
Author(s):  
A M Makarov ◽  
A S Ermakov ◽  
E A Pisarenko ◽  
V A Ryndiuk
Keyword(s):  
Signal Processing ◽  
High Speed ◽  
Beta Function ◽  
Integral Transforms ◽  
Digital Model ◽  
Computer Algorithms ◽  
Unknown Phase ◽  
Phase Sequence ◽  
Logarithmic Functions ◽  
Special Case

Abstract The implementation of a digital model of the beta function for use in computer algorithms is a time-consuming task. This is due to the complexity of the high-precision representation of its integrand functions, which require a large number of intermediate operations, which entails a large load on the computational power. Purpose: Development of the basic theoretical provisions of the integral Mellin transform in relation to the theory of signal processing against the background of noise and research of their discrete representation. Results: It is shown in the paper that the beta function can be considered as a special case of Mellin’s integral transforms. Based on this statement, a mathematical model of the beta function was developed. Using the properties of parametrically periodic oscillations belonging to the class of trigonometric-logarithmic functions, it was possible to create a digital model for representing the beta function. Practical relevance: Based on the established digital model can be realized a high-speed algorithm for calculating the beta function with a given accuracy. Such algorithms can serve as a basis for creating signal processing programs in order to detect wideband phase-shift keyed signals against a background of noise with an unknown phase sequence. An example of using such algorithms is the search for Wi-Fi bugs.

A Note on Relations Between the Zeta-Functions of Galois Coverings of Curves Over Finite Fields

1990 ◽  
Vol 33 (3) ◽  
pp. 282-285 ◽  
Author(s):  
Amilcar Pacheco
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Group Ring ◽  
Algebraic Curve ◽  
Zeta Functions ◽  
Finite Subgroup ◽  
Rational Group ◽  
Galois Coverings ◽  
Curves Over Finite Fields ◽  
Special Case

AbstractLet C be a complete irreducible nonsingular algebraic curve defined over a finite field k. Let G be a finite subgroup of the group of automorphisms Aut(C) of C. We prove that certain idempotent relations in the rational group ring Q[G] imply other relations between the zeta-functions of the quotient curves C/H, where H is a subgroup of G. In particular we generalize some results of Kani in the special case of curves over finite fields.

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