Precise estimate of the 2-capacity of a condenser

For quite a large class of condensers E (including, in particular, all space annuli) greatest lower bounds for their 2-capacity are obtained in terms of the Newtonian capacity of certain sets associated with E. A class of condensers for which equality is achieved in the bound is described completely.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 253–257, February, 1990.     

Modulus,functional, and potential characteristics of condensers in a domain; relations between them

A series of new statements about relations between the known modulus, functional, and potential characteristics of space condensers are proved. A theorem on the continuity of the moduli of special families of curves is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 604–613, May, 1992.     

Interior Capacities of Condensers in Locally Compact Spaces

The study deals with the theory of interior capacities of condensers in a locally compact space, a condenser being treated here as a finite collection of arbitrary sets with sign + 1 or − 1 prescribed such that the closures of oppositely signed sets are mutually disjoint. We are motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in \mathbb Rⁿ\mathbb R^n and closed condensers). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing new equivalent definitions to the capacity), but now always solvable (e.g., even for nonclosed condensers). For all positive definite kernels satisfying Fuglede’s condition of consistency between the strong and vague (= weak*) topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior equilibrium measures associated with a set. We describe those solutions and the corresponding equilibrium constants, analyze their uniqueness and continuity, and point out their characteristic properties. Such results are new even for classical kernels in \mathbb Rⁿ\mathbb R^n, which is important in applications.     

On holomorphic representations of discrete Heisenberg groups

We define a class of monomial not necessary unitary representations of discrete Heisenberg groups. Their moduli space is a complex-analytic manifold. There exist characters of the representations which are automorphic forms on the moduli space.     

Siegel coordinates and moduli spaces for morphisms of Abelian varieties

We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincaré decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in the decomposition. For a given type of morphisms the moduli variety is irreducible, and is obtained from a product of Siegel spaces modulo the action of a discrete group. Mathematics Subject Classification (2000) 14K20     

Deformation of Plates of Small Condensers and Belinskii's Problem

We study the homeomorphic embeddings of a compact set K, a union of nondegenerate continua, into which preserve the conformal moduli of all condensers whose plates are continua in K. Using a result by V. N. Dubinin together with the estimates for the conformal moduli of infinitesimal condensers, we prove that Belinskii's conjecture (that such a mapping extends to a Mobius automorphism of the whole space ), corroborated by the author in 1990 for n=2 is also valid for n>2 if the compact set in question is regular at some collection of (n+2) points. This essentially strengthens the previous result of the author (1992) in which regularity was required at each point of the compact set.     

Web‐spline‐based finite element approximation of some quasi‐newtonian flows: Existence‐uniqueness and error bound

This article deals with the web‐spline‐based finite element approximation of quasi‐Newtonian flows. First, we consider the scalar elliptic p‐Laplace problem. Then, we consider quasi‐Newtonian flows where viscosity obeys power law or Carreau law. We prove well‐posedness at the continuous as well as the discrete level. We give some error bounds for the solution of quasi‐Newtonian flow problem based on the web‐spline method. Finally, we provide the numerical results for the p‐Laplace problem. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 54–77, 2015     

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

In this paper, we propose a discrete duality finite volume (DDFV) scheme for the incompressible quasi‐Newtonian Stokes equation. The DDFV method is based on the use of discrete differential operators which satisfy some duality properties analogous to their continuous counterparts in a discrete sense. The DDFV method has a great ability to handle general geometries and meshes. In addition, every component of the velocity gradient can be reconstructed directly, which makes it suitable to deal with the nonlinear terms in the quasi‐Newtonian Stokes equation. We prove that the proposed DDFV scheme is uniquely solvable and of first‐order convergence in the discrete L²‐norms for the velocity, the strain rate tensor, and the pressure, respectively. Ample numerical tests are provided to highlight the performance of the proposed DDFV scheme and to validate the theoretical error analysis, in particular on locally refined nonconforming and polygonal meshes.     

The discrete maximal operator in metric spaces

We study the action of the so-called discrete maximal operator on Newtonian, Hölder and BMO spaces on metric measure spaces equipped with a doubling measure and a Poincaré inequality. The discrete maximal operator has better regularity properties than the standard Hardy-Littlewood maximal operator and hence is a more flexible tool in this context.     

Real cubic surfaces and real hyperbolic geometry

The moduli space of stable real cubic surfaces is the quotient of real hyperbolic four-space by a discrete, nonarithmetic group. The volume of the moduli space is 37π²/1080 in the metric of constant curvature ?1. Each of the five connected components of the moduli space can be described as the quotient of real hyperbolic four-space by a specific arithmetic group. We compute the volumes of these components. To cite this article: D. Allcock et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On capacity characteristics of condensers

We introduce and study a one-parameter family of capacity characteristics of condensers in ? ^p ,p ≥ 3, that contains some known capacities as elements extremal with respect to the parameter. We establish new relations between the capacity characteristics of condensers and sets.     

Structure and Rigidity in (Gromov) Hyperbolic Groups and Discrete Groups in Rank 1 Lie Groups II

We borrow the Jaco-Shalen-Johannson notion of characteristic sub-manifold from 3-dimensional topology to study cyclic splittings of torsion-free (Gromov) hyperbolic groups and finitely generated discrete groups in rank 1 Lie groups. Our JSJ canonical decomposition is a fundamental object for studying the dynamics of individual automorphisms and the automorphism group of a torsion-free hyperbolic group and a key tool in our approach to the isomorphism problem for these groups [S3]. For discrete groups in rank 1 Lie groups, the JSJ canonical decomposition serves as a basic object for understanding the geometry of the space of discrete faithful representations and allows a natural generalization of the Teichmüller modular group and the Riemann moduli space for these discrete groups. Submitted: September 1996, revised version: April 1997     

Approximation properties of a class of linear operators

This work focuses on a class of linear positive operators of discrete type. We present the relationship between the local smoothness of functions and the local approximation. Also, the degree of approximation in terms of the moduli of smoothness is established, and the statistical convergence of the sequence is studied. Copyright © 2013 John Wiley & Sons, Ltd.     

Moduli spaces of local systems and higher Teichmüller theory

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.     

Hybrid control of a multicellular converter

In this paper a new switching scheme for modelling multicellular converters is proposed. We consider a converter with two cells which is represented as a hybrid system with four modes of operation. The operation modes of the system are governed by the adjustable reference voltage and a reference current, which are calculated by means of an energy balance principle. This study is justified by the hybrid behavior of the system considered, i.e., it presents discrete and continuous variables which represent the state of the switches and the evolution of the current and the terminal voltage of the floating condensers, respectively. The synthesis of the control is based on the determination of the switching sequences, ensuring the stability and the integrity of the switches.     

The Building Game: From Enumerative Combinatorics to Conformational Diffusion

We study a discrete attachment model for the self-assembly of polyhedra called the building game. We investigate two distinct aspects of the model: (i) enumerative combinatorics of the intermediate states and (ii) a notion of Brownian motion for the polyhedral linkage defined by each intermediate that we term conformational diffusion. The combinatorial configuration space of the model is computed for the Platonic, Archimedean, and Catalan solids of up to 30 faces, and several novel enumerative results are generated. These represent the most exhaustive computations of this nature to date. We further extend the building game to include geometric information. The combinatorial structure of each intermediate yields a systems of constraints specifying a polyhedral linkage and its moduli space. We use a random walk to simulate a reflected Brownian motion in each moduli space. Empirical statistics of the random walk may be used to define the rates of transition for a Markov process modeling the process of self-assembly.     

Realization of cascades on surfaces with finitely many moduli of topological conjugacy

A method for constructing cascades on surfaces is developed, which makes it possible to model structurally unstable discrete dynamical systems with finitely many orbits of heteroclinic tangency and preset moduli of topological conjugacy.     

An extension based on qR-integral for a sequence of operators

The paper deals with a sequence of linear positive operators introduced via q-Calculus. We give a generalization in Kantorovich sense of its involving qR-integrals. Both for discrete operators and for integral operators we study the error of approximation for bounded functions and for functions having a polynomial growth. The main tools consist of the K-functional in Peetre sense and different moduli of smoothness.     

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

We compute the discrete affine group of Schlesinger transformations for isomonodromic deformations of a Fuchsian system of second-order differential equations. These transformations are treated as isomorphisms between the moduli spaces of logarithmic sl(2)-connections with given eigenvalues of the residues on ¹. The discrete structure is computed with the use of the modification technique for bundles with connections. The result generalizes the well-known classical computations of symmetries of the hypergeometric equation, the Heun equation, and the sixth Painlevé equation.     

The Schwarzian Derivative of a p-Valent Function

Mathematical Notes - We show that a comparison of the capacities of suitable condensers gives an inequality for the Schwarzian derivatives of a holomorphic p-valent function defined on the unit...