Sharp Asymptotics for the Neumann Laplacian with Variable Magnetic Field: Case of Dimension 2

The aim of this paper is to establish estimates of the lowest eigenvalue of the Neumann realization of on an open bounded subset with smooth boundary as B tends to infinity. We introduce a “magnetic” curvature mixing the curvature of ∂Ω and the normal derivative of the magnetic field and obtain an estimate analogous with the one of constant case. Actually, we give a precise estimate of the lowest eigenvalue in the case where the restriction of magnetic field to the boundary admits a unique minimum which is non degenerate. We also give an estimate of the third critical field in Ginzburg–Landau theory in the variable magnetic field case. Submitted: June 26, 2008., Accepted: November 28, 2008.     

Model Selection for Simplicial Approximation

In the computational geometry field, simplicial complexes have been used to describe an underlying geometric shape knowing a point cloud sampled on it. In this article, an adequate statistical framework is first proposed for the choice of a simplicial complex among a parametrized family. A least-squares penalized criterion is introduced to choose a complex, and a model selection theorem states how to select the “best” model, from a statistical point of view. This result gives the shape of the penalty, and then the “slope heuristics method” is used to calibrate the penalty from the data. Some experimental studies on simulated and real datasets illustrate the method for the selection of graphs and simplicial complexes of dimension two.     

Weyl-Eddington-Einstein affine gravity in the context of modern cosmology

We propose new models of the “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposed method for obtaining the geometry using the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein theory with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) meson, and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the Lagrangian determines further details of the theory, for example, the nature of the fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, dark energy must also arise. The basic parameters of the theory (cosmological constant, mass, possible dimensionless constants) are theoretically indeterminate, but in the framework of modern “multiverse” ideas, this is more a virtue than a defect. We consider further extensions of the affine models and in more detail discuss approximate effective (“physical”) Lagrangians that can be applied to the cosmology of the early Universe.     

On the evolution equation for magnetic geodesics

Orbits of charged particles under the effect of a magnetic field are mathematically described by magnetic geodesics. They appear as solutions to a system of (nonlinear) ordinary differential equations of second order. But we are only interested in periodic solutions. To this end, we study the corresponding system of (nonlinear) parabolic equations for closed magnetic geodesics and, as a main result, eventually prove the existence of long time solutions. As generalization one can consider a system of elliptic nonlinear partial differential equations (PDEs) whose solutions describe the orbits of closed p-branes under the effect of a “generalized physical force”. For the corresponding evolution equation, which is a system of parabolic nonlinear PDEs associated to the elliptic PDE, we can establish existence of short time solutions.     

Interaction of Two Charges in a Uniform Magnetic Field: II. Spatial Problem

The interaction of two charges moving in ℝ³ in a magnetic field B can be formulated as a Hamiltonian system with six degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotation symmetry, we reduce this system to one with three degrees of freedom. For special values of the conserved quantities, choices of parameters or restriction to the coplanar case, we obtain systems with two degrees of freedom. Specialising to the case of Coulomb interaction, these reductions enable us to obtain many qualitative features of the dynamics. For charges of the same sign, the gyrohelices either “bounce-back”, “pass-through”, or exceptionally converge to coplanar solutions. For charges of opposite signs, we decompose the state space into “free” and “trapped” parts with transitions only when the particles are coplanar. A scattering map is defined for those trajectories that come from and go to infinite separation along the field direction. It determines the asymptotic parallel velocities, guiding centre field lines, magnetic moments and gyrophases for large positive time from those for large negative time. In regimes where gyrophase averaging is appropriate, the scattering map has a simple form, conserving the magnetic moments and parallel kinetic energies (in a frame moving along the field with the centre of mass) and rotating or translating the guiding centre field lines. When the gyrofrequencies are in low-order resonance, however, gyrophase averaging is not justified and transfer of perpendicular kinetic energy is shown to occur. In the extreme case of equal gyrofrequencies, an additional integral helps us to analyse further and prove that there is typically also transfer between perpendicular and parallel kinetic energy.      

Zero level of a purely magnetic two-dimensional nonrelativistic Pauli operator for SPIN-1/2 particles

We study the manifold of complex Bloch-Floquet eigenfunctions for the zero level of a two-dimensional nonrelativistic Pauli operator describing the propagation of a charged particle in a periodic magnetic field with zero flux through the elementary cell and a zero electric field. We study this manifold in full detail for a wide class of algebraic-geometric operators. In the nonzero flux case, the Pauli operator ground state was found by Aharonov and Casher for fields rapidly decreasing at infinity and by Dubrovin and Novikov for periodic fields. Algebraic-geometric operators were not previously known for fields with nonzero flux because the complex continuation of “magnetic” Bloch-Floquet eigenfunctions behaves wildly at infinity. We construct several nonsingular algebraic-geometric periodic fields (with zero flux through the elementary cell) corresponding to complex Riemann surfaces of genus zero. For higher genera, we construct periodic operators with interesting magnetic fields and with the Aharonov-Bohm phenomenon. Algebraic-geometric solutions of genus zero also generate soliton-like nonsingular magnetic fields whose flux through a disc of radius R is proportional to R (and diverges slowly as R → ∞). In this case, we find the most interesting ground states in the Hilbert space L ₂ (ℝ ² ).     

Fibrewise completion and unstable Adams spectral sequences

We describe a tower of spaces whose inverse limit is a “fiberwise completion” of a fibrationE →B, and study the resulting spectral sequence converging to the homotopy groups of the space of lifts of a mapX →B. This is used to give a proof of the “generalized Sullivan conjecture”. All three authors were supported in part by the National Science Foundation.     

Realizability of greedy algorithms

We discuss new models of an “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposal to specify the space-time geometry by the use of the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, of other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein gravity with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) vector field (vecton), and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the geometric Lagrangian determines further details of the theory, for example, the nature of the vector and scalar fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, which are discussed here, dark energy must also arise. We mainly focus on intricate relations between geometry and dynamics while only very briefly considering approximate cosmological models inspired by the geometric approach.     

Asymptotics of the optimal time in a singular perturbed linear time-optimal problem

We consider an abstract attainability problem under constraints of asymptotic character and describe a general approach to constructing “nonsequential” attraction sets in the space of generalized elements formalized as finitely additive measures. We also study the existence and structure of an asymptotic formula universal in the range of “asymptotic constraints” and topologies of the space of generalized elements in the case when the space of ordinary solutions is not necessarily compactifiable.     

Root subsystems of loop extensions

We completely classify the real root subsystems of root systems of loop algebras of Kac–Moody Lie algebras. This classification involves new notions of “admissible subgroups” of the coweight lattice of a root system Ψ, and “scaling functions” on Ψ. Our results generalise and simplify earlier work on subsystems of real affine root systems.     

Parallel computing in the statistical system Jasp

Modern statistical data analysis often requires powerful computers. Parallel computing is a technique to realize such a computer system. Although many “low-level” software technologies for parallel computing have been developed, they are not easy to use for statisticians who are accustomed to “high-level” statistical languages. In this paper, we describe high-level parallel computing functions in a statistical analysis system called Jasp. We implemented them mainly considering ease of use.     

Quantum Backreaction (Casimir) Effect II. Scalar and Electromagnetic Fields

Casimir effect in most general terms may be understood as a backreaction of a quantum system causing an adiabatic change of the external conditions under which it is placed. This paper is the second installment of a work scrutinizing this effect with the use of algebraic methods in quantum theory. The general scheme worked out in the first part is applied here to the discussion of particular models. We consider models of the quantum scalar field subject to external interaction with “softened” Dirichlet or Neumann boundary conditions on two parallel planes. We show that the case of electromagnetic field with softened perfect conductor conditions on the planes may be reduced to the other two. The “softening” is implemented on the level of the dynamics, and is not imposed ad hoc, as is usual in most treatments, on the level of observables. We calculate formulas for the backreaction energy in these models. We find that the common belief that for electromagnetic field the backreaction force tends to the strict Casimir formula in the limit of “removed cutoff” is not confirmed by our strict analysis. The formula is model dependent and the Casimir value is merely a term in the asymptotic expansion of the formula in inverse powers of the distance of the planes. Typical behaviour of the energy for large separation of the plates in the class of models considered is a quadratic fall-of. Depending on the details of the “softening” of the boundary conditions the backreaction force may become repulsive for large separations. Communicated by Klaus Fredenhagen submitted 9/09/04, accepted 1/07/05     

A (2+1)-dimensional fermion in the Coulomb and magnetic field backgrounds

In the problem of a two-dimensional hydrogen-like atom in a magnetic field background, we construct quasi-classical solutions and the energy spectrum of the Dirac equation in a strong Coulomb field and in a weak constant homogeneous magnetic field in 2+1 dimensions. We find some “exact” solutions of the Dirac and Pauli equations describing the “spinless” fermions in strong Coulomb fields and in homogeneous magnetic fields in 2+1 dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 105–118, April, 1999.     

A dynamical system connected with an inhomogeneous 6-vertex model

A completely integrable dynamical system in discrete time is studied by methods of algebraic geometry. The system is associated with factorization of a linear operator acting in the direct sum of three linear spaces into a product of three operators, each acting nontrivially only in the direct sum of two spaces, and subsequently reversing the order of the factors. There exists a reduction of the system, which can be interpreted as a classical field theory in the 2+1-dimensional space-time, whose integrals of motion coincide, in essence, with the statistical sum of an inhomogeneous 6-vertex free-fermion model on the 2-dimensional kagome lattice (here the statistical sum is a function of two parameters). This establishes a connection with the “local,” or “generalized,” quantum Yang-Baxter equation. Bibliography:10 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Published inZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 178–196. Translated by I. G. Korepanov.     

Nonequilibrium statistical operator in the generalized molecular hydrodynamics of fluids

We discuss the important role of the Zubarev nonequilibrium statistical operator method in the generalized molecular hydrodynamics of fluids. Using this method allows developing a consistent approach of generalized collective excitations for simple, ion, polar, magnetic, and some other fluids. We construct a nonequilibrium statistical operator and derive the corresponding transport equations for a system that relaxes and passes into the state of molecular hydrodynamics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 91–101, January, 2008.     

A generalization of Malho’s method for obtaining large cardinal numbers

Mahlo used a method by which fixed points of an enumeration of regular cardinals were employed to get a hierarchy of “large cardinals.” He also employed a second method which, in a certain sense, is much stronger than the first. Here the methods are investigated and generalized and the relations between them are clarified. This stronger method turns out to be a kind of “least upper bound” to all “fixed-points operations.” Possibilities of strengthening these processes in a natural way are pointed out.     

Monomial conditions on prime rings

The study of pivotal monomials (and related conditions) is continued and extended, with the aim of studying carefully a situation generalizing Martindale's theory of prime rings with generalized polynomial identity. This is used to describe various classes of rings in terms of simple elementary sentences. The focus is on prime “Johnson” rings, which play a crucial role in our characterizations. It turns out that these rings can be characterized in terms of generalized pivotal monomials, thereby yielding a theory similar to that of [17]. An erratum to this article is available at .     

Lebesgue numbers and Atsuji spaces in subsystems of second-order arithmetic

We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to ; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is compact” are equivalent to ; the statement “every Lebesgue space is Atsuji” is provable in ; the statement “every Atsuji space is Lebesgue” is provable in . We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to . Received: February 2, 1996     

Conservation theorems for magnetic field and vorticity in two-dimensional conducting fluid flows

The classical conservation theorems for magnetic force lines, magnetic flux through a fluid surface, and intensity of magnetic vector tubes are generalized to plane flows of a finitely conducting fluid in an orthogonal magnetic field. The Helmholtz and Kelvin vorticity conservation theorems are generalized for plane motion of a viscous conducting fluid in an orthogonal magnetic field and the Bernoulli integral is derived. The Bernoulli integral is also generalized for plane motion of viscous ideally conducting fluid in a longitudinal magnetic field. Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 2, pp. 46–49, 1994.