scholarly journals Quantum periods and TBA-like equations for a class of Calabi-Yau geometries

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bao-ning Du ◽  
Min-xin Huang
Keyword(s):  
Large Class ◽  
Bethe Ansatz ◽  
Difference Equations ◽  
The Other ◽  
Elementary Functions ◽  
Quantum Dilogarithm ◽  
Thermodynamic Bethe Ansatz ◽  
Dilogarithm Function ◽  
Quantum Operators

Abstract We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.

scholarly journals Green's function for the Laplace–Beltrami operator on a toroidal surface

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120479 ◽  
Author(s):  
Christopher C. Green ◽  
Jonathan S. Marshall
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Vortex Dynamics ◽  
Stereographic Projection ◽  
Three Dimensional ◽  
Beltrami Operator ◽  
Toroidal Surface ◽  
Torus Surface ◽  
Dilogarithm Function ◽  
Prime Function

Green's function for the Laplace–Beltrami operator on the surface of a three-dimensional ring torus is constructed. An integral ingredient of our approach is the stereographic projection of the torus surface onto a planar annulus. Our representation for Green's function is written in terms of the Schottky–Klein prime function associated with the annulus and the dilogarithm function. We also consider an application of our results to vortex dynamics on the surface of a torus.

scholarly journals FRACTAL INDEX, CENTRAL CHARGE AND FRACTONS

2000 ◽  
Vol 15 (31) ◽  
pp. 1931-1939 ◽  
Author(s):  
WELLINGTON DA CRUZ ◽  
ROSEVALDO DE OLIVEIRA
Keyword(s):  
Hausdorff Dimension ◽  
Central Charge ◽  
Statistical Parameter ◽  
Conformal Anomaly ◽  
Fermi Velocity ◽  
Conformal Field ◽  
Fractal Index ◽  
Fractal Statistics ◽  
Universal Classes ◽  
Dilogarithm Function

We introduce the notion of fractal index associated with the universal class h of particles or quasiparticles, termed fractons which obey specific fractal statistics. A connection between fractons and conformal field theory (CFT)-quasiparticles is established taking into account the central charge c[ν] and the particle-hole duality ν↔1/ν, for integer-value ν of the statistical parameter. In this way, we derive the Fermi velocity in terms of the central charge as [Formula: see text]. The Hausdorff dimension h which labeled the universal classes of particles and the conformal anomaly are therefore related. Following another route, we also established a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.

scholarly journals On the l.c.m. of shifted Fibonacci numbers

2021 ◽  
pp. 1-10
Author(s):  
Carlo Sanna
Keyword(s):  
Rational Number ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Random Variables ◽  
Periodic Sequence ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Number Formula ◽  
Dilogarithm Function

Let [Formula: see text] be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that [Formula: see text] where [Formula: see text] is the least common multiple and [Formula: see text] is the golden ratio. We prove that for every periodic sequence [Formula: see text] in [Formula: see text] there exists an effectively computable rational number [Formula: see text] such that [Formula: see text] Moreover, we show that if [Formula: see text] is a sequence of independent uniformly distributed random variables in [Formula: see text] then [Formula: see text] where [Formula: see text] is the dilogarithm function.

New integral representations of the polylogarithm function

2006 ◽  
Vol 463 (2080) ◽  
pp. 897-905 ◽  
Author(s):  
Djurdje Cvijović
Keyword(s):  
Bernoulli Polynomials ◽  
Integral Representations ◽  
The Other ◽  
Important Special Case ◽  
Polylogarithm Function ◽  
Physical Problems ◽  
The Riemann Zeta Function ◽  
Dilogarithm Function ◽  
Riemann Zeta ◽  
Special Case

Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Li s ( z ). The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm Li s ( z ) for any complex z for which | z |<1. Two are valid for all complex s , whenever Re  s >1. The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is a positive integer. Our earlier established results on the integral representations for the Riemann zeta function ζ (2 n +1), n ∈ N , follow directly as corollaries of these representations.

Some properties associated to a certain class of starlike functions

2019 ◽  
Vol 69 (6) ◽  
pp. 1329-1340 ◽  
Author(s):  
Vali Soltani Masih ◽  
Ali Ebadian ◽  
Sibel Yalçin
Keyword(s):  
Function Theory ◽  
Starlike Functions ◽  
Sharp Inequality ◽  
Open Unit ◽  
Geometric Function Theory ◽  
Open Unit Disk ◽  
Geometric Function ◽  
The Family ◽  
Logarithmic Coefficients ◽  
Dilogarithm Function

Abstract Let 𝓐 denote the family of analytic functions f with f(0) = f′(0) – 1 = 0, in the open unit disk Δ. We consider a class $$\begin{array}{} \displaystyle \mathcal{S}^{\ast}_{cs}(\alpha):=\left\{f\in\mathcal{A} : \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1+\left(\alpha-1\right) z-\alpha z^2}, \,\, z\in \Delta\right\}, \end{array}$$ where 0 ≤ α ≤ 1/2, and ≺ is the subordination relation. The methods and techniques of geometric function theory are used to get characteristics of the functions in this class. Further, the sharp inequality for the logarithmic coefficients γn of f ∈ $\begin{array}{} \mathcal{S}^{\ast}_{cs} \end{array}$(α): $$\begin{array}{} \displaystyle \sum_{n=1}^{\infty}\left|\gamma_n\right|^2 \leq \frac{1}{4\left(1+\alpha\right)^2}\left(\frac{\pi^2}{6}-2 \mathrm{Li}_2\left(-\alpha\right)+ \mathrm{Li}_2\left(\alpha^2\right)\right), \end{array}$$ where Li2 denotes the dilogarithm function are investigated.

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