## Absolutely Convergent Series

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### Alexey Kuznetsov - One of the best experts on this subject based on the ideXlab platform.

• ##### On the density of the supremum of a stable process
arXiv: Probability, 2011
Co-Authors: Alexey Kuznetsov
Abstract:

We study the density of the supremum of a strictly stable L\'evy process. As was proved recently in F. Hubalek and A. Kuznetsov "A Convergent Series representation for the density of the supremum of a stable process" (Elect. Comm. in Probab., 16, 84-95, 2011), for almost all irrational values of the stability parameter $\alpha$ this density can be represented by an Absolutely Convergent Series. We show that this result is not valid for all irrational values of $\alpha$: we construct a dense uncountable set of irrational numbers $\alpha$ for which the Series does not converge Absolutely. Our second goal is to investigate in more detail the important case when $\alpha$ is rational. We derive an explicit formula for the Mellin transform of the supremum, which is given in terms of Gamma function and dilogarithm. In order to illustrate the usefulness of these results we perform several numerical experiments and discuss their implications. Finally, we state some interesting connections that this problem has to other areas of Mathematics and Mathematical Physics, such as q-Series, Diophantine approximations and quantum dilogarithms, and we also suggest several open problems.

• ##### a Convergent Series representation for the density of the supremum of a stable process
Electronic Communications in Probability, 2011
Co-Authors: Friedrich Hubalek, Alexey Kuznetsov
Abstract:

We study the density of the supremum of a strictly stable Levy process. We prove that for almost all values of the index $\alpha$ - except for a dense set of Lebesgue measure zero - the asymptotic Series which were obtained in Kuznetsov (2010) "On extrema of stable processes" are in fact Absolutely Convergent Series representations for the density of the supremum.

### Bin‐hao Jiang - One of the best experts on this subject based on the ideXlab platform.

• ##### Expansion of Spherical Wave Functions for Electro-Magnetic Fields of a Vertical Magnetic Dipole over an Anisotropic Half-Space
Chinese Journal of Geophysics, 2020
Co-Authors: Bin‐hao Jiang
Abstract:

We derive an accurate and analytical expression for electromagnetic fields of a vertical magnetic dipole over an uniaxial medium half-space. By using the expansion of spherical wave functions for a cylindrical wave function and with the help of the theory of hypergeometric functions, the Sommerfeld type integral contained in the electromagnetic fields is expressed as a rapidly and Absolutely Convergent Series of spherical wave functions; and the coefficients of the Series are the Legendre Polynomials with the complex argument for the constitutive parameter. The present results have explicit mathematical and physical interpretations and no the restriction of the locations of dipole source and observation points, medium parameters and frequency. The expression can be used conveniently to calculate and analyze electromagnetic fields at any points.

### Liu Yongtan - One of the best experts on this subject based on the ideXlab platform.

• ##### Expansion of Spherical Wave Functions for Hertz Potential of a Vertical Magnetic Dipole over Conducting Medium Plane
2020
Co-Authors: Liu Yongtan
Abstract:

An accurate and efficient method for computing Sommerfeld type integrals is crucial in the analysis of the electromagnetic waves for radiators and scatters in the presence of a conducting half-space.In this paper,an analytical approach is described for the evaluation of the Hertz potential of a vertical magnetic dipole over the finitely conducting medium plane.By using the expansion of spherical wave functions for a cylindrical wave function and the techniques of the hypergeometric functions,the Sommerfeld integral contained in the potential is expressed as a rapidly and Absolutely Convergent Series of spherical wave functions;and the coefficients of the Series are cast into the Legender polynomials with complex argument for the medium parameter.The obtained result has the advantage that it holds without the restrictions on the locations of dipole source and observation points,ranges of medium parameters and frequency and so on,and can conveniently be employed to calculate and analyze the electromagnetic fields.

### Aldo Procacci - One of the best experts on this subject based on the ideXlab platform.

• ##### Absence of Phase Transitions in a Class of Integer Spin Systems
Journal of Statistical Physics, 2020
Co-Authors: Thiago Morais, Aldo Procacci
Abstract:

We exhibit a class of integer spin systems whose free energy can be written in term of an Absolutely Convergent Series at any temperature. This class includes spin systems on ℤ d interacting through infinite range pair potential polynomially decaying at large distances r at a rate 1/r d+e with e>0. It also contains the Blume-Emery-Griffiths model in the disordered phase at large values of the crystal field.

• ##### Effects of Boundary Conditions on Irreversible Dynamics
Annales Henri Poincaré, 2018
Co-Authors: Aldo Procacci, Benedetto Scoppola, Elisabetta Scoppola
Abstract:

We present a simple one-dimensional Ising-type spin system on which we define a completely asymmetric Markovian single spin-flip dynamics. We study the system at a very low, yet nonzero, temperature, and we show that for free boundary conditions the Gibbs measure is stationary for such dynamics, while introducing in a single site a $$+$$ + condition the stationary measure changes drastically, with macroscopical effects. We achieve this result defining an Absolutely Convergent Series expansion of the stationary measure around the zero temperature system. Interesting combinatorial identities are involved in the proofs.

• ##### Absence of Phase Transitions in a Class of Integer Spin Systems
Journal of Statistical Physics, 2009
Co-Authors: Thiago Morais, Aldo Procacci
Abstract:

We exhibit a class of integer spin systems whose free energy can be written in term of an Absolutely Convergent Series at any temperature. This class includes spin systems on ℤ^ d interacting through infinite range pair potential polynomially decaying at large distances r at a rate 1/ r ^ d + ε with ε >0. It also contains the Blume-Emery-Griffiths model in the disordered phase at large values of the crystal field.

• ##### Polymer Gas Approach to N-Body Lattice Systems
Journal of Statistical Physics, 1999
Co-Authors: Aldo Procacci, Benedetto Scoppola
Abstract:

We give a simple proof, based only on combinatorial arguments, of the Kotecký–Preiss condition for the convergence of the cluster expansion. Then we consider spin systems with long-range N -body interactions. We prove directly, using the polymer gas representation, that the pressure may be written in terms of an Absolutely Convergent Series uniformly in the volume when the interaction is summable in a suitable sense. We also give an estimate of this radius of convergence. In order to get the proof we use a method introduced by Cassandro and Olivieri in the early 1980s. We apply this method to various concrete examples.

### E T Shavgulidze - One of the best experts on this subject based on the ideXlab platform.

• ##### Perturbation theory with Convergent Series for calculating physical quantities specified by finitely many terms of a divergent Series in traditional perturbation theory
Theoretical and Mathematical Physics, 2020
Co-Authors: V V Belokurov, Yu P Solov'ev, E T Shavgulidze
Abstract:

A new high-accuracy method is suggested for calculating physical quantities for which only finitely many terms of the divergent Series in a traditional perturbation theory are known. The method is based on approximating the desired quantity with the sum of finitely many terms of an Absolutely Convergent Series. As an example, the β-function in the ϕ 4 4 model and the critical exponent α characterizing the behavior of the He4 heat capacity near the phase transition point are calculated.

• ##### method of approximate evaluation of path integrals using perturbation theory with Convergent Series i
Theoretical and Mathematical Physics, 1996
Co-Authors: V V Belokurov, Yu P Solovev, E T Shavgulidze
Abstract:

We propose a method of evaluating path integrals such that the integral is approximated with any accuracy by summations of a finite number of terms of an Absolutely Convergent Series.