Hermite--Pad\'e polynomials and analytic continuation: new approach and some results

We discuss a new approach to realization of the well-known Weierstrass's programme on efficient continuation of an analytic element corresponding to a~multivalued analytic function with finite number of branch points. Our approach is based on the use of Hermite--Pad\'e polynomials

On one example of a Nikishin system

The paper puts forward an example of a~Markov function $f=\operatorname{const}+\widehat{\sigma}$ such that the three functions $f,f^2$ and $f^3$ form a Nikishin system. A conjecture is proposed that there exists a~Markov function $f$ such that, for each $n\in\mathbb N$, the system $f,f^2,\dots,f^n$ constitutes a~Nikishin system. Bibliography:~20~titles

On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

For an interval $E=[a,b]$ on the real line, let $\mu$ be either the equilibrium measure, or the normalized Lebesgue measure of $E$, and let $V^{\mu}$ denote the associated logarithmic potential. In the present paper, we construct a function $f$ which is analytic on $E$ and possesses four branch points of second order outside of $E$ such that the family of the admissible compacta of $f$ has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals $V^{-\mu}$.Comment: 28 pages, submitted to Jounal of Approximation Theor

Embedding AC Power Flow in the Complex Plane Part I: Modelling and Mathematical Foundation

Part I of this paper embeds the AC power flow problem with voltage control and exponential load model in the complex plane. Modeling the action of network controllers that regulate the magnitude of voltage phasors is a challenging task in the complex plane as it has to preserve the framework of holomorphicity for obtention of these complex variables with fixed magnitude. The paper presents two distinct approaches to modelling the voltage control of generator nodes. Exponential (or voltage-dependent) load models are crucial for accurate power flow studies under stressed conditions. This new framework for power flow studies exploits the theory of analytic continuation, especially the monodromy theorem for resolving issues that have plagued conventional numerical methods for decades. Here the focus is on the indispensable role of Pade approximants for analytic continuation of complex functions, expressed as power series, beyond the boundary of convergence of the series. The zero-pole distribution of these rational approximants serves as a proximity index to voltage collapse. Finally the mathematical underpinnings of this framework, namely the Stahl's theory and the rate of convergence of Pade approximants are explained.Comment: 14 pages. Initial Submission: March 26th, 2016 Updated: July 18th, 2016 (addition: Second approach for PV bus modeling). arXiv admin note: text overlap with arXiv:1504.0324

Embedding AC Power Flow with Voltage Control in the Complex Plane : The Case of Analytic Continuation via Pad\'e Approximants

This paper proposes a method to embed the AC power flow problem with voltage magnitude constraints in the complex plane. Modeling the action of network controllers that regulate the magnitude of voltage phasors is a challenging task in the complex plane as it has to preserve the framework of holomorphicity for obtention of these complex variables with fixed magnitude. Hence this paper presents a significant step in the development of the idea of Holomorphic Embedding Load Flow Method (HELM), introduced in 2012, that exploits the theory of analytic continuation, especially the monodromy theorem for resolving issues that have plagued conventional numerical methods for decades. This paper also illustrates the indispensable role of Pad\'e approximants for analytic continuation of complex functions, expressed as power series, beyond the boundary of convergence of the series. Later the paper demonstrates the superiority of the proposed method over the well-established Newton-Raphson as well as the recently developed semidefinite and moment relaxation of power flow problems.Comment: 9 pages, 10 figures, 7 table

Electronic properties and Fermi surface for new Fe-free layered superconductor BaTi2Bi2O from first principles

Very recently, as an important step in the development of layered Fe-free pnictide-oxide superconductors, the new phase BaTi2Bi2O was discovered which has the highest TC (about 4.6 K) among all related non-doped systems. In this Letter, we report for the first time the electronic bands, Fermi surface topology, total and partial densities of electronic states for BaTi2Bi2O obtained by means of the first-principles FLAPW-GGA calculations. The inter-atomic bonding picture is described as a high-anisotropic mixture of metallic, covalent, and ionic contributions. Besides, the structural and electronic factors, which can be responsible for the increased transition temperature for BaTi2Bi2O (as compared with related pnictide-oxides BaTi2As2O and BaTi2Sb2O), are discussed.Comment: 7 pages, 3 figure

Trace formulas for a class of Jacobi operators

In this paper we study a class of Jacobi operators, such that each operator is generated by the unit Borel measure with a support consisting of a finite number of intervals on the real line R and a finite number of points in C, located outside the convex hull of the intervals and symmetrically with respect to R. In such a class of operators we have obtained the asymptotic behavior of the diagonal Green's function and trace formulas for sequences of coefficients corresponding to a given operator. Bibliography: 34 titles.Comment: in Russia

Structural, elastic and electronic properties of Ir-based carbides-antiperovskites Ir3MC (M = Ti, Zr, Nb and Ta) as predicted from first-principles calculations

Structural, elastic, electronic properties and the features of inter-atomic bonding in hypothetical Ir-based carbides-antiperovskites Ir3MC (M=Ti, Zr, Nb and Ta), as predicted from first-principles calculations, have been investigated for a first time. Their elastic constants, bulk, shear and Young`s moduli, compressibility, Poisson`s ratio, Debye temperature have been evaluated, and their stability, character of elastic anisotropy, brittle / ductile behavior, as well as electronic structure have been explored in comparison with binary carbides MC having NaCl-type structure. Authors hope that the presented results will be useful for future synthesis of these phases, as well as for extending the knowledge about the group of antiperovskite-type promising materials

On the limit zero distribution of type I Hermite-Pad\'e polynomials

In this paper are discussed the results of new numerical experiments on zero distribution of type I Hermite-Pad\'e polynomials of order $n=200$ for three different collections of three functions $[1,f_1,f_2]$. These results are obtained by the authors numerically and do not match any of the theoretical results that were proven so far. We consider three simple cases of multivalued analytic functions $f_1$ and $f_2$, with separated pairs of branch points belonging to the real line. In the first case both functions have two logarithmic branch points, in the second case they both have branch points of second order, and finally, in the third case they both have branch points of third order. All three cases may be considered as representative of the asymptotic theory of Hermite-Pad\'e polynomials. In the first two cases the numerical zero distribution of type I Hermite-Pad\'e polynomials are similar to each other, despite the different kind of branching. But neither the logarithmic case, nor the square root case can be explained from the asymptotic point of view of the theory of type I Hermite-Pad\'e polynomials. The numerical results of the current paper might be considered as a challenge for the community of all experts on Hermite-Pad\'e polynomials theory.Comment: Bibliography: 79 items; figures: 72 item

Some numerical results on the behavior of zeros of the Hermite-Pad\'e polynomials

We introduce and analyze some numerical results obtained by the authors experimentally. These experiments are related to the well known problem about the distribution of the zeros of Hermite--Pad\'e polynomials for a collection of three functions $[f_0 \equiv 1,f_1,f_2]$. The numerical results refer to two cases: a pair of functions $f_1,f_2$ forms an Angelesco system and a pair of functions $f_1=f,f_2=f^2$ forms a (generalized) Nikishin system. The authors hope that the obtained numerical results will set up a new conjectures about the limiting distribution of the zeros of Hermite--Pad\'e polynomials.Comment: Bibliography: 71 titles; 79 picture