Hans Kaper

Georgetown University, Mathematics and Statistics, Faculty Member

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The frozen-field approximation and the Ginzburg-Landau equations of superconductivity

by Hans Kaper and Henrik Nordborg

Practical Asymptotics, 2001

The Ginzburg-Landau (GL) equations of superconductivity provide a computational model for the stu... more The Ginzburg-Landau (GL) equations of superconductivity provide a computational model for the study of magnetic flux vortices in type-II superconductors. In this article we show through numerical examples and rigorous mathematical analysis that the GL model reduces to the frozen-field model when the charge of the Cooper pairs (the superconducting charge carriers) goes to zero while the applied field stays near the upper critical field.

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Full- and Half-Range Theory of an Indefinite Sturm-Liodville Problem

North-Holland Mathematics Studies, 1984

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Automorphism theorems and half-range theory

Progress in Nuclear Energy, 1981

ABSTRACT In the Hilbert space approach to the linear-transport equation (homogeneous slab configu... more ABSTRACT In the Hilbert space approach to the linear-transport equation (homogeneous slab configurations) the fundamental question of half-range theory concerns the existence of a bijective mapping of the underlying function space onto itself. In this article we discuss the existence and continuity properties of this mapping. We interpret the results in terms of conditions on the boundary data for half-space and finite slab boundary value problems.

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Greenhouse Effect in a Finite Anisotropically Scattering Atmosphere

Astronomy and Astrophysics

An exact analytical solution is given for the steady-state distribution of visible and infrared r... more An exact analytical solution is given for the steady-state distribution of visible and infrared radiation in a planetary atmosphere. It is assumed that the atmosphere is ffluminated from outside by visible light which provides the sole source of atmospheric heating. The atmosphere may have any finite thickness and any degree of anisotropic scattering is allowed. The solution, with either diffuse or mirror reflection of the visible radiation and infrared emission at the surface of the planet, is expressed as a linear combination of known solutions to various monoenergetic albedo problems and an infinite medium Green's function. Once the visible and infrared intensities have been found, the atmospheric temperature proffle can be readily calculated.

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Mathematical models of superconductivity

This note surveys some mathematical models that have been proposed to study the phenomenon of sup... more

Formalizing the Concept of Sound

The notion of formalized music implies that a musical composition can be described in mathematica... more The notion of formalized music implies that a musical composition can be described in mathematical terms. In this article we explore some formal aspects of music and propose a framework for an abstract approach. 1 Introduction Sounds and their attributes are traditionally defined at a relatively low level of abstraction. The terminology and basic concepts underlying the notion of a sound are tailored to Western music of the past three centuries and lack the level of abstraction needed, for example, for a significant body of recent works and some non-Western music. In this article we explore some formal aspects of music and propose a framework for a more abstract approach. As Western musicians, we are trained to think in terms of the notated score. Accordingly, we view a piece of music as a collection of individual sounds represented by dots and ovals and characterized mainly by start time, duration, and pitch. The music notation system we use today is indeed precise in defining time...

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The Ginzburg-Landau Equations Of Superconductivity In The Limit Of Weak Coupling Near The Upper Critical Field

. This article is concerned with the Ginzburg--Landau (GL) equations of superconductivity. The eq... more . This article is concerned with the Ginzburg--Landau (GL) equations of superconductivity. The equations provide a mathematical model for the study of magnetic flux vortices in superconductors. The focus is on the asymptotic case when the charge of the superconducting charge carriers (Cooper pairs) is vanishingly small and the applied magnetic field approaches the upper critical field. It is shown that the GL model reduces in the limit to the frozen-field model, where the superconducting phenomena are affected by the electromagnetic phenomena, but not vice versa. The convergence is second order in the small parameter. The analytical results are confirmed in some numerical examples. 1 Introduction Superconducting materials hold great promise for technological applications. Especially since the discovery of the so-called high-temperature superconductors in the 1980s, much research has been devoted to understanding the behavior of these new materials. While conventional superconductors...

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A collection of problems in transport theory

Transport Theory and Statistical Physics, 1975

ABSTRACT In this article we have brought together a collection of problems which were suggested a... more ABSTRACT In this article we have brought together a collection of problems which were suggested at the Fourth Conference on Transport Theory. It is hoped that the publication of these problems fulfills the dual purpose of (i) directing attention to gaps in the existing body of knowledge of transport theory, and (ii) stimulating research into new and hitherto unexplored areas of transport theory. Needless to say, the editor claims no responsibility for the correctness or incorrectness, solvability or in-solvability of any of the proposed problems. The sole responsibility rests with the proposer of each problem, whose name appears immediately after the title of the problem.

A report on the Fourth Conference on Transport Theory

Transport Theory and Statistical Physics, 1975

ABSTRACT The Fourth Conference on Transport Theory was held on the campus of Virginia Polytechnic... more ABSTRACT The Fourth Conference on Transport Theory was held on the campus of Virginia Polytechnic Institute and State University at Blacksburg, Virginia, during the week of March 10, 1975. (Previous conferences were held in Blacksburg, Virginia, 1969; Los Alamos, New Mexico, 1971;and Blacksburg, Virginia, 1973.) As explained in the report on the third conference this fourth conference was again a full-size meeting, with some twenty participants, the proceedings of which will be published. The conference was supported by a grant from the National Science Foundation. The funding was on a somewhat smaller: scale than for the 1969 and 1971 conferences (which were funded by the U.S. Atomic Energy Commission), which accounts for the smaller attendance at this year&#39;s meeting. The organizers of the conference were R. L. Bowden and William Greenberg, (VPI&amp;SU).

A review of: “Topics in differential and integral equations and operator theory”

Transport Theory and Statistical Physics, 1986

ABSTRACT Edited by I. Gohberg; translated from the Russian by A. Jacob; OT7, Birkhauser Verlag, B... more

Ordinary Differential Equations in $R^n $—Problems and Methods (L. C. Piccinini, G. Stampacchia and G. Vidossich)

SIAM Review, 1985

Spectral Representation of an Unbounded Linear Transformation Arising in the Kinetic Theory of Gases

SIAM Journal on Mathematical Analysis, 1979

Boundary Value Problems of Mixed Type Arising in the Kinetic Theory of Gases

SIAM Journal on Mathematical Analysis, 1979

The linear integro-differential transport equation is interpreted in functional form as a relatio... more

Asymptotic Behavior of the Solution of the Integral Transport Equation in Slab Geometry

SIAM Journal on Applied Mathematics, 1977

Elementary Solutions of the Reduced Three-Dimensional Transport Equation

Journal of Mathematical Physics, 1969

By treating one of the space dimensions exactly and approximating the other two by the exp (−iB·r... more By treating one of the space dimensions exactly and approximating the other two by the exp (−iB·r) assumption, which is suggested by asymptotic transport theory, it is possible to reduce the three-dimensional transport equation to an equation that is of one-dimensional form and that still contains details of the complete three-dimensional angular distribution. In this paper we develop the method of elementary solutions for the reduced transport equation in the case of time-independent, monoenergetic neutron transport in homogeneous media with isotropic scattering. The spectrum of the transport operator consists of a pair of discrete points if B2 is sufficiently small and a continuum which occupies a two-dimensional region in the complex spectral plane. The eigenfunctions possess full-range and half-range orthogonality and completeness properties, which are proved via the solution of two-dimensional integral equations using the theory of boundary-value problems for generalized analytic functions. As applications we solve the Green&#x27;s function for an infinite homogeneous prism and the albedo operator for a semi-infinite homogeneous prism. Also discussed are possible generalizations of the method to more complicated forms of the reduced transport equation.

A constructive approach to the solution of a class of boundary value problems of mixed type

Journal of Mathematical Analysis and Applications, 1978

In this article we discuss the solution of boundary value problems which are described by the lin... more

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Integral operators generated by exponential integrals

Journal of Mathematical Analysis and Applications, 1977

Some results for integral operators whose kernels are the exponential integrals are presented.

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Asymptotic evaluation of two families of integrals

Journal of Mathematical Analysis and Applications, 1977

In this paper the integrals f&7) = sz exp[-(t + T)]tY(ln t)"(t + T)-' dt and gmU = J-T exp[-/ t -... more In this paper the integrals f&7) = sz exp[-(t + T)]tY(ln t)"(t + T)-' dt and gmU = J-T exp[-/ t -7 j]tY(ln t)"(t -T)-' dt are investigated for positive real values of the variable 7. Here, m is a nonnegative integer, Y is a complex variable with Re(v) > -I. Both integrals are related to the complex integral CJ~,,(Z) = Jf exp[-(t -z)]t-Y(ln t)"(t -z)-' dt with 0 Q Re(y) < 1, the

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The initial-value transport problem for monoenergetic neutrons in an infinite slab with delayed neutron production

Journal of Mathematical Analysis and Applications, 1967

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Numerical evaluation of the slab albedo problem solution in one-speed anisotropic transport theory

Journal of Computational Physics, 1970

ABSTRACT In this paper the normal mode expansion technique is employed to obtain numerical result... more ABSTRACT In this paper the normal mode expansion technique is employed to obtain numerical results for the solution of the slab albedo problem in one-speed transport theory with anisotropic scattering. Methods are presented by which one may compute the angular distribution of particles anywhere in a slab of finite thickness for any degree of anisotropy of the scattering function. Many numerical examples are given and, from these examples, it is demonstrated how the slab albedo problem solution varies with the problem parameters, viz., the degree of anisotropy of the scattering function, the angle of incidence of the source beam, the slab thickness and the multiplication factor.

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