A continuum theory of superconductivity

Abstract. A continuum theory of superconductivity is formulated for a mixture consisting of three species: a superelectron fluid, a conducting fluid, and a conducting elastic solid. Each one of the three species is subject to their own electro-magnetic (E-M) fields and motions. Irreversible thermodynamics are used to obtain constitutive equations. Field equations, boundary and initial conditions are given. A special case is obtained, suitable for mathematical analysis and applications. The Pippard theory of superconductivity is shown to be a special case of the present theory.     

A continuum theory of dense suspensions

A continuum theory is introduced for viscous fluids carrying dense suspensions (such as blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertia that make the mixture an anisotropic fluid whose viscosity changes with motion and orientation of suspensions. The microinertia balance law coupled with the equations of motion of an anisotropic fluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutive equations are obtained in terms of the frame-independent tensors. In a special case, a theory of bar-like suspensions is obtained. The field equations, boundary and initial conditions are given for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory is demonstrated with the solution of the channel flow problem. The mean viscosity of the fluid with suspensions is determined. The motions of suspensions down flow are demonstrated.     

A continuum theory of dense suspensions

A continuum theory is introduced for viscous fluids carrying dense suspensions (such as blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertia that make the mixture an anisotropic fluid whose viscosity changes with motion and orientation of suspensions. The microinertia balance law coupled with the equations of motion of an anisotropic fluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutive equations are obtained in terms of the frame-independent tensors. In a special case, a theory of bar-like suspensions is obtained. The field equations, boundary and initial conditions are given for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory is demonstrated with the solution of the channel flow problem. The mean viscosity of the fluid with suspensions is determined. The motions of suspensions down flow are demonstrated.     

The mathematical theory of defects in a Cosserat continuum

We give a survey of the theory of dislocations and disclinations in moment media. We study the theory of incompatible deformations of three- and two-dimensional Cosserat continua. In the context of a differential-geometric approach we give a physical interpretation of the geometric quantities in terms of the continuous theory of defects.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 34–40.     

On continuum incidence problems related to harmonic analysis

We consider averaging operators over curves and surfaces satisfying the rotational curvature condition of Phong and Stein. Using combinatorial arguments restricted weak-type versions of the standard estimates are obtained under weaker assumptions on the regularity of these surfaces. We also consider maximal functions over circles in the plane and obtain a simpler proof of Wolff's endpoint result.     

Plastic deformation of bicrystals within continuum dislocation theory

We analyze the evolution of the plastic distortion and the nucleation and accumulation of dislocations within a model bicrystal with one active slip system in each single crystal (symmetric with respect to the interface), which is subject to prescribed displacements of plane–strain shear and extension, and we present closed–form analytical solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the solutions of stochastic initial-value problems in continuum mechanics

This work deals with the analysis of a class of initial-value problems for nonlinear hyperbolic equations with stochastic process coefficients and four time-space independent variables. A perturbation procedure is developed, leading to a sequence of stochastic, partial differential equations which are solved by application of the Adomian decomposition method. A pertinent application in mechanics is also considered, and quantitative results are presented and discussed.     

Initial-value problems in potential theory

Potential theory characteristically leads to partial differential equations under boundary conditions. It is shown that these boundary-value problems can be converted directly into equivalent initial-value problems. These initial-value problems are shown to lead to a number of standard and nonstandard numerical procedures.This research was supported by the National Institutes of Health under Grants Nos. GM-16197-01 and GM-16437-01 and by the Atomic Energy Commission under Contract No. AT(11-1)-113, Project No. 19.     

Discrete problems in probability theory

A short survey is given of some directions in probability theory that have developed most intensively in recent years. Separable statistics and criteria for the verification of statistical hypotheses based on them, various schemes for distributing particles among cells, and problems connected with estimating the unknown size of a finite collection are considered.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 22, pp. 3–60, 1984.     

Extremal problems in graph theory

The aim of this note is to give an account of some recent results and state a number of conjectures concerning extremal properties of graphs.     

Unsolved problems in number theory

Periodica Mathematica Hungarica - 68 unsolved problems and conjectures in number theory are presented and brie y discussed. The topics covered are: additive representation functions, the...     

Open problems in pluripotential theory

We propose a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry. The list includes both local questions as well as issues related to the compact complex manifold setting.     

Some problems in approximation theory

This dissertation consists of three parts. The first part is devoted to the approximation of a fixed element by a fixed approximating set. In the second part we study approximation by finite sets (Chapter I, Secs. 3–6) and by finite-dimensional subspaces (Chapter II, Secs. 7–15). Selected problems with geometric (Secs. 1–3) and variational (Secs. 4–6) content are considered in the third chapter.The author's summary of dissertation for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended on December 18, 1970, at a session of the Scientific Council of the Mechanics-Mathematics Faculty of Moscow State University. The official opposition was made up of Academician A. N. Kolmogorov, K. I. Babenko, and S. B. Stechkin.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 593–607, May, 1971.     

Application of nonlinear continuum dislocation theory to antiplane shear

We apply the nonlinear dislocation theory to the problem of antiplane constrained shear in a single crystal with one slip system. By taking dissipation into account, the relaxed energy functional has to be minimized. We show that, up to a threshold strain, no dislocations are nucleated and therefore the plastic slip is zero. Since this threshold value depends on the width of the specimen, a size effect takes place. The stress strain curve turns out to be a hysteresis loop exhibiting the work hardening due to the dislocation pile-up. It is shown that the Bauschinger effect holds true. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Qualitative properties of a continuum theory for thin films

We discuss qualitative aspects of a continuum theory for thin films rigorously derived in [B. Schmidt, On the passage from atomic to continuum theory for thin films, preprint 82/2005, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig]. The stored energy density is examined for convexity properties and limiting behavior under large and small strains. A study of the dependence of the theory on relaxation parameters leads to the result that the scale of convergence used in [B. Schmidt, On the passage from atomic to continuum theory for thin films, preprint 82/2005, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig] is the only scale for which a limiting theory that also accounts for atomic relaxation effects is non-trivial.     

Random expected utility theory with a continuum of prizes

This note generalizes Gul and Pesendorfer’s random expected utility theory, a stochastic reformulation of von Neumann–Morgenstern expected utility theory for lotteries over a finite set of prizes, to the circumstances with a continuum of prizes. Let [0, M] denote this continuum of prizes; assume that each utility function is continuous, let \(C_0[0,M]\) be the set of all utility functions which vanish at the origin, and define a random utility function to be a finitely additive probability measure on \(C_0[0,M]\) (associated with an appropriate algebra). It is shown here that a random choice rule is mixture continuous, monotone, linear, and extreme if, and only if, the random choice rule maximizes some regular random utility function. To obtain countable additivity of the random utility function, we further restrict our consideration to those utility functions that are continuously differentiable on [0, M] and vanish at zero. With this restriction, it is shown that a random choice rule is continuous, monotone, linear, and extreme if, and only if, it maximizes some regular, countably additive random utility function. This generalization enables us to make a discussion of risk aversion in the framework of random expected utility theory.