Andrei Borisovich Bogatyrev

• Dr.Sci in Math&Physics, 2003
• Professor at Russian Academy of Sciences

We consider the cell decomposition of the moduli space of real genus $2$ curves with marked point on the unique real oval. The cells are enumerated by certain graphs, whose weights describe the complex structure on the curve. We show that the collapse of an edge in a graph results in a root-like singularity of the natural map from the weights on gr...

Pell-Abel equation is a functional equation of the form P^{2}-DQ^{2} = 1, with a given polynomial D free of squares and unknown polynomials P and Q. We show that the space of Pell-Abel equations with the fixed degrees of D and of primitive solution P is a complex manifold and we compute the number of its connected components. As application we comp...

Рассматривается клеточное разбиение пространства модулей вещественных кривых рода $2$ с отмеченной точкой на единственном вещественном овале. Клетки перечисляются определенными графами, веса которых описывают комплексную структуру на кривой. Показано, что стягивание ребра графа приводит к корневой особенности естественного отображения из весов граф...

Предлагается новый полуаналитический метод вычисления характеристик течения в пористой среде, ограниченной водонепроницаемой преградой в виде прямоугольной ломаной. Основой метода является применение римановых тэта-функций. Он позволяет с машинной точностью находить решение задач фильтрации, а также некоторых их характеристик, не решая задачу в цел...

We consider the cell decomposition of the moduli space of real genus two curves with a marked point on the only real oval. The cells are enumerated by certain graphs with their weights describing the complex structure on a curve. We show that collapse of the edge of the graph results in a root like singularity of the natural mapping from the graph...

Any ramified holomorphic covering of a closed unit disc by another such a disc is given by a finite Blaschke product. The inverse is also true. In this note we give two explicit constructions for a holomorphic ramified covering of a disc by other bordered Riemann surface. The machinery used here strongly resembles the description of magnetic config...

Making use of two different analytical–numerical methods for capacity computation, we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers. These two methods are based respectively on the use of the Lauricella function (Bezrodnykh and Vlasov, 2002; Bezrodnykh, 2016 [64,65]) and Riemann theta...

The best uniform rational approximation of the \emph{sign} function on two intervals separated by zero was explicitly solved by E.I. Zolotar\"ev in 1877. This optimization problem is the initial step in the staircase of the so called approximation problems for multiband filters which are of great importance for electrical engineering. We show that...

The best uniform rational approximation of the sign function on two intervals was explicitly found by Russian mathematician E.I. Zolotarëv in 1877. The progress in math eventually led to the progress in technology: half a century later German electrical engineer and physicist W. Cauer on the basis of this solution has invented low- and high-pass el...

Изучается отображение периодов, действующее из пространства модулей вещественных гиперэллиптических кривых в евклидово пространство. Отображение возникает при анализе чебышeвской конструкции, используемой при условной оптимизации равномерной нормы многочленов и рациональных функций. Разложение пространства модулей на многогранники, перечисляемые пл...

The magnetic textures on nanoscale possess topological features due to the continuity of the magnetization vector field and its boundary conditions. In thin planar nanoelements, where the dependence of the magnetization across the thickness is inessential, the textures can be represented as a soup of 2-d topological solitons, corresponding to magne...

It is well known that a ramified holomorphic covering of a closed unitary disc by another such a disc is given by a finite Blaschke product. The inverse is also true. In this note we give an explicit description of holomorphic ramified coverings of a disc by other bordered Riemann surfaces. The problem of covering a disc by an annulus arises e.g. i...

The best uniform rational approximation of the \emph{sign} function on two intervals was explicitly found by Russian mathematician E.I. Zolotar\"ev in 1877. The progress in math eventually led to the progress in technology: half a century later German electrical engineer and physicist W.Cauer has invented low- and high-pass electrical filters known...

A new analytical method for the conformal mapping of rectangular polygons with a straight angle at infinity to a half-plane and back is proposed. The method is based on the observation that the SC integral in this case is an abelian integral on a hyperelliptic curve, so it may be represented in terms of Riemann theta functions. The approach is illu...

A new analytical method for calculation the characteristics of the flow in porous medium bounded by a rectangular polygonal path is proposed. The method is based on the usage of high genus Riemann theta functions.

Making use of two different analytical-numerical methods for capacity computation, we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers. These two methods are based respectively on the use of the Lauricella function and Riemann theta functions. We apply these results to benchmark the perf...

A large enough piece of ferromagnet is usually not magnetized uniformly, but develops a magnetization texture. In thin films these textures can be doubly-periodic. Such are the well known magnetic bubble domains and the recently observed "skyrmion" magnetization textures in MnSi. In this paper we develop a theory of periodic magnetization textures,...

We use the language of real meromorphic differentials from the theory of Klein surfaces to describe the metastable states of multiply connected planar ferromagnetic nanoelements that minimize the exchange energy and have no side magnetic charges. These solutions still have sufficient internal degrees of freedom, which can be used as Ritz parameters...

A novel analytical approach to the synthesis of electrical (e.g., analogue, digital or microwave) filters is proposed. This approach allows to obtain the lowest possible order filters with given involved specification including, e.g., many pass- and stopbands, narrow transition bands, high attenuation at the stopbands and low magnitude oscillations...

Known properties of Chebyshev polynomials are the following: they have simple critical points with only two (finite) critical values. Those properties uniquely determine the named polynomials modulo affine transformations of dependent and independent variables. A similar property of Zolotarev fractions: simple critical points and only four critical...

We present a universal closed formula in terms of theta functions for the Logcapacity of several segments on a line. The formula for two segments was obtained by N. Achieser (1930); three segments were considered by T. Falliero and A. Sebbar (2001).

Для описания метастабильных состояний многосвязных плоских ферромагнитных наноэлементов, минимизирующих обменную энергию и не имеющих боковых магнитных зарядов, используется язык вещественных мероморфных дифференциалов из теории поверхностей Клейна. Эти решения все же имеют достаточное количество внутренних степеней свободы, которые можно использов...

A novel analytical approach to the synthesis of electrical (e.g. analogue, digital or microwave) filters is proposed. This approach allows to obtain lowest possible degree filters with given involved specification including e.g. many pass and stop bands, narrow transition bands, high attenuation at the stop bands and low magnitude oscillations at t...

In this paper we use the language of real meromorphic differentials from the theory of Klein surfaces to describe the metastable states of multiply connected planar ferromagnetic nanoelements which minimize the exchange energy and have no side magnetic charges. Those solutions still have enough internal degrees of freedom which may serve as the Rit...

Here we consider an interplay between the topology of the magnetization texture (which is a topological soliton, or Skyrmion) in a planar magnetic nano-element and the topology of the element itself (its connectivity). We establish the existence of a set of constraints, coupling these topologies, which are specific for multiply connected elements a...

We study the period mapping from the moduli space of real hyperelliptic curves to a Euclidean space. The mapping arises in the analysis of Chebyshev’s construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of the moduli space into polyhedra labelled by planar graphs allows us t...

In this note we present a universal formula in terms of theta functions for the Log- capacity of several segments on a line. The case of two segments was studied by N.I.Akhiezer (1930); three segments were considered by A.Sebbar and T.Falliero (2001).

Flat magnetic nano-elements are an essential component of current and future spintronic devices. By shaping an element it is possible to select and stabilize chosen metastable magnetic states, control its magnetization dynamics. Here, using a recent significant development in mathematics of conformal mapping, complex variable based approach to the...

For the evaluation and inversion of abelian integrals we show that the image of the Abel-Jacobi map of genus less than 5 hyperelliptic curve in its Jacobian is the intersection of shifted theta divisors with specified shifts. Therefore the image is a solution of a (slightly overdetermined) set of equations in the Jacobian.

We begin this chapter by listing areas of science and technology where we come across problems relating to optimization of the uniform norm. After that we investigate least deviation problems using methods of convex analysis. We deduce a generalized alternation principle which completely characterizes solutions of such problems. In giving the defin...

For an efficient use of the Chebyshev representation for extremal polynomials we must investigate how the periods of the abelian differential ηM behave as functions of a point M in the moduli space. In this chapter we develop a combinatorial geometric approach to the investigation of the period map. To curves M in the moduli space we shall assign i...

The analytic approach [25, 43] to the same problem put forward below required about a second of calculations on a considerably less powerful processor. Our analysis of the optimization Problem B in Chap. 1 enables us to figure out the characteristic features of the solution. We know that the optimal stability polynomial has many alternance points,...

Chebyshev and his students Zolotarëv, the brothers V. A. and A. A. Markov, Korkin, and Posse reduced extremal problems for polynomials to Pell’s equation, a geometric interpretation of which is suggested in the following construction.

In this chapter we study the structure of the set of curves M associated with real polynomials of degree n by means of the Chebyshev correspondence.

We have already seen that to investige properties of the Chebyshev representation and, in particular, of Abel’s equations we must consider the space of real hyperelliptic curves. For a fixed genus g this space consists of several components, which are distinguished by another topological invariant of a real curve, the number k of (co)real ovals on...

1 Least deviation problems.- 2 Chebyshev representation of polynomials.- 3 Representations for the moduli space.- 4 Cell decomposition of the moduli space.- 5 Abel's equations.- 6 Computations in moduli spaces.- 7 The problem of the optimal stability polynomial.- Conclusion.- References.

We propose an analytical approach to the conformal mapping of (rectangular) polygons based on the theory of Riemann surfaces and theta functions.

An effective representation is obtained for rational functions all of whose critical points, apart from , are simple and their corresponding critical values lie in a four-element set. Such functions are described using hyperelliptic curves of genus . The classical Zolotarëv fraction arises in this framework for . Bibliography: 8 titles.

J.Ritt has investigated the structure of complex polynomials with respect to superposition. In particular, he listed all the polynomials admitting different double decompositions into indecomposable polynomials. The analogues of Ritt theory for rational functions were constructed just for several particular classes of the said functions, say for La...

We give an explicit multi-parametric construction for Jenkins-Strebel differentials on real algebraic curves. Roughly speaking, the square of any real holomorphic abelian differential subjected to certain linear restrictions will be a JS quadratic differential. Comment: 4 pages, 1 fig

We discuss two constructive approaches to the solution of problems of polynomial approximation in the uniform (Chebyshev) norm and also attainable estimates of the solution norms for initial value problems for Hermitian systems of differential and algebraic equations.

Eigenvalue problem for Poincare-Steklov-3 integral equation is reduced to the solution of three transcendential equations for three unknown numbers, moduli of pants. The complete list of antisymmetric eigenfunctions of integral equation in terms of Kleinian membranes is given.

A new efficient variational formula for the Kleinian prime form (factor) in the frame of the Schottky model of Riemann surfaces is presented. We also give an elementary explanation for the choice of the sign in the transformation formula for the prime factor.

Various physical and mathematical settings bring us to a boundary value problem for a harmonic function with spectral parameter in the boundary conditions. One of those problems may be reduced to a singular 1D integral equation with spectral parameter. We present a constructive representation for the eigenvalues and eigenfunctions of this integral...

Antisymmetric solutions of Poincaré-Steklov Integral Equations are presented. Constructive visual representations for all so-called antisymmetric solutions is obtained by using methods of complex geometry and combinatorics, in which R(t) is a real-valued rational function of degree 3 with separated real critical values different from the endpoints...

Today the techniques related to Riemann surfaces are widely used in different branches of mathematics, theoretical physics, industry and even in medicine. Unfortunately, many achievements of the theory remain on paper since theoretical formulae contain special functions that few scholars try to compute. In the present note we describe several usefu...

An effective method for finding the polynomial approximating the exponential function with order 3 at the origin and deviating from 0 by at most 1 on the longest interval of the real axis is put forward. This problem is reduced to the solution of four equations on a 4-dimensional moduli space of algebraic curves. A numerical realization of this met...

The classical Chebyshev and Zolotarev polynomials are the first ranks of the hierarchy of extremal polynomials, which are typical solutions of problems on the conditional minimization of the uniform norm over a space of polynomials. In the general case such polynomials are connected with hyperelliptic curves the genus of which labels the ranks of t...

For the description of extremal polynomials (that is, the typical solutions of least deviation problems) one uses real hyperelliptic curves. A partitioning of the moduli space of such curves into cells enumerated by trees is considered. As an application of these techniques the range of the period map of the universal cover of the moduli space is e...

More than a hundred years ago H.Poincare and V.A.Steklov considered a problem for the Laplace equation with spectral parameter in the boundary conditions. Today similar problems for two adjacent domains with the spectral parameter in the conditions on the common boundary of the domains arises in a variety of situations: in justification and optimiz...

The construction of stable explicit multistage Runge-Kutta methods in 1950–1960 stumbled over a certain extremal problem for polynomials. The solution to this problem is known as the optimal stability polynomial and its computation is notoriously difficult. We propose a new method for the effective evaluation of optimal stability polynomials which...

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By using Ramanujan's q-extension of the Euler integral representation for the gamma function, we derive the Mellin integral transforms for the families of the discrete q-Hermite II, the Al-Salam-Carlitz II, the big q-Laguerre, the ...

A hierarchy of extremal polynomials described in terms of real hyperelliptic curves of genus g\geqslant0 is constructed. These polynomials depend on g integer-valued and g continuous parameters. The classical Chebyshëv polynomials are obtained for g=0 and the Zolotarëv polynomials for g=1.

A complex-geometric theory of the Poincaré-Steklov integral equation is developed. Solutions of this equation are effectively represented and its spectrum is localized

We consider a parameterization of the set of polynomialsT n(E, x) whose deviation from zero is the least on a systemE consisting of several intervals on the real axis. We point out a new way for obtaining the equations which describe the boundary of the maximum set of the least deviationE + ⊃E. We describe the geometry of the variety of all possibl...

We consider the Poincaré-Steklov singular integral equation obtained by reducing a boundary value problem for the Laplace operator with a spectral parameter in the boundary condition to the boundary. It is shown that this equation can be restated equivalently in terms of the classical Riemann monodromy problem. Several equations of this type are so...

An exact analytical solution is given to a problem of relative arrangement of two molecules which minimizes the weighted sum of squared interatomic distances.

A cell decomposition of the space of polynomials T{sub n}(E,x) of least deviation from zero on a system E of several closed intervals of the real axis is discussed. An effective method for calculating the T{sub n} in each cell making use of automorphic functions is put forward.

P. L. Chebyshev posed a problem of finding a polynomial of least deviation on the dosed set E of the real axis. He himself obtained the solution of this problem for E = [-1,1]. The solution of the least deviation problem for E = [- l, a] U [b, 1], -1 < a < 4 < 1, followed from Zolotarev's works [14, 24, 25], and for many special cases it was obtain...

Eigenvalues and eigenfunctions are explicitly found for a family of singular integral equations. It is shown how their discrete spectrum becomes continuous as the equation degenerates.

This paper deals with a generalized spectral problem with two Poincaré-Steklov boundary influence operators for a pair of adjacent domains. This problem arises when constructing iteration algorithms for the methods of disintegration and composition of domains and the fictitious domain method.

A generalized spectral problem p1 = λp 2is considered for two Poincaré - Steklov operators [3,4] corresponding to two adjacent domains in a plane. A sufficient condition for the problem's spectrum to be discrete is suggested. A continuous spectrum is known to impair the convergence of domain-decomposition iterative methods. A Poincaré-Steklov opera...