The Chern classes of Sobolev connections

AssumeF is the curvature (field) of a connection (potential) onR ⁴ with finiteL ² norm . We show the chern number (topological quantum number) is an integer. This generalizes previous results which showed that the integrality holds forF satisfying the Yang-Mills equations. We actually prove the natural general result in all even dimensions larger than 2.     

Scaling and thermal equilibrium of hadrons fromZ 0 decays

An analysis of secondary hadrons fromZ ⁰ (91) decays of the LEP experiments indicates that the scaling of distributions implies the same temperatureT=0.261 GeV for all the secondaries. The multiplicities ofZ ⁰,K ⁰, ,..., and computed with their quark contents and the sameT agree with the data. The ratio of to the phase-space covered by the rapidity distribution, depends only on the energy, , fore ⁺ e ^– annihilations and collisions as well.     

Navier-Stokes equations and area of interfaces

We present new a priori estimates for the vorticity of solutions of the three dimensional Navier-Stokes equations. These estimates imply that theL ¹ norm of the vorticity is a priori bounded in time and that the time average of the 4/(3+) power of theL ^4/(3+) spatial norm of the gradient of the vorticity is a priori bounded. Using these bounds we construct global Leray weak solutions of the Navier-Stokes equations which satisfy these inequalities. In particular it follows that vortex sheet, vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the Navier-Stokes equations. Next we apply these inequalities in conjunction with geometric measure theoretical arguments to study the two dimensional Hausdorff measure of level sets of the vorticity magnitude. We obtain a priori bounds on an average such measure, >. When expressed in terms of the Reynolds number and the Kolmogorov dissipation length , these bounds are

Modified Weyl theory and extended elementary objects

To represent extension of objects in particle physics, a modified Weyl theory is used by gauging the curvature radius of the local fibers in a soldered bundle over space-time possessing a homogeneous space G/H of the (4, 1)-de Sitter group G as fiber. Objects with extension determined by a fundamental length parameter R₀ appear as islands D_(i) in space-time characterized by a geometry of the Cartan-Weyl type (i.e., involving torsion and modified Weyl degrees of freedom). Farther away from the domains D_(i), space-time is identified with the pseudo-Riemannian space of general relativity. Extension and symmetry breaking are described by a set of additional fields ( , given as a section on an associated bundle over space-time B with structural group = G D(1), where D(1) is the dilation group. Field equations for the quantities defining the underlying bundle geometry and for the fields are established involving matter source currents derived from a generalized spinor wave function. Einstein's equations for the metric are regarded as the part of the -gauge theory related to the Lorentz subgroup H of G exhibiting thereby the broken nature of the -symmetry for regions outside the domains D_(i).Talk presented at the International Conference on Field Theory and General Relativity held at Utah State University, Logan, Utah, June 26–July 2, 1988.     

On the Modified Korteweg–De Vries Equation

We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg–de Vries equation , with initial data . We assume that the coefficient is real, bounded and slowly varying function, such that , where . We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space . In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395–418), here we exclude the condition that the integral of the initial data u ₀ is zero. We prove the time decay estimates and for all , where . We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.     

Tunneling Across an Inhomogeneous Delta-Barrier

The authors deal with the tunneling of electrons across an inhomogeneous delta-barrier defined by the potential energy (where 0$$ " align="middle" border="0"> and 0$$ " align="middle" border="0"> are two constants). In particular, the perpendicular incidence of an electron with a given value of the wave vector is considered. The electron is forward-scattered into the region behind the barrier (region 2: 0$$ " align="middle" border="0"> ), i. e. the wave function is composed of plane waves with all wave vectors such that and \left. 0 \right)} $$ " align="middle" border="0"> ) (where ). Therefore, if 0$$ " align="middle" border="0"> , the wave function of the electron is represented as , where . An approximate formula is derived for the amplitude . The authors pay a special attention to the flow density and calculate this function in two cases: 1. for the plane and 2. for high values of is the diffraction angle). The authors discuss the relevance of their diffraction problem in a prospective quantum-mechanical theory of the tunneling of electrons across a randomly inhomogeneous Schottky barrier.     

Chargino-neutralino production inp\bar p-collisions for the left-right supersymmetric model

An extension of the Supersymmetric Standard Model to the supersymmetricSU(2) _L×SU(2) _R×U(1) _B–L model is considered. The gauge group contains a bidoublet and triplet Higgs field. We investigate the possibility of detecting chargino and neutralino production in -collisions atCDF, namely . A numerical analysis is performed for , tan 1 and a lower bound on the lightest supersymmetric particle (LSP) mass, GeV. Using conservative assumptions ofM _WR300 GeV andgg _Lg _R, we find the cross sections are: , and pb, and thus _L 24 _L at TeV. Cross sections are also given for larger values of the center of mass energy up to those available at the SSC. The results are compared with the prompt-lepton background of theW _{L, R} ^– decays from . Both decays for bosons show Jacobian peaks for (p _T150 GeV forM _WR) at =90°. Furthermore the chargino signature unlike the promptlepton background is symmetric under the Jacobian peak. We also exhibit the dependence of the angular distribution of the chargino on the c.m. angle forp _T40 GeV, 150 GeV at TeV.     

Regularity of weak solutions to a two-dimensional modified Dirac-Klein-Gordon system of equations

We show that solutions to the modified Dirac-Klein-Gordon system in standard notation

Characterizations of Boolean algebras of idempotents

The notions of the left (right) Jordan groupoids are introduced. IfR is an associative^* ring with the identity and ifU(R) [resp.P(R)] denotes the set of all idempotents (resp. projections) of the^* ringR, then the operationsp q =p – 2pq – qp + 4qpq andp q =q – 2pq – 2qp + 4pqp ifp, q U(R) [resp.p, q P(R)] are the nonassociative linear operations inU(R) [resp. inP(R)]. The present paper shows that the operations and are associative iffpq=qp forp, q U(R) [resp.p, q P(R)]. As a corollary it follows from this that the orthomodular poset (U(R), , 0, 1,) is a Boolean algebra [which is commutative, i.e.,pq= qp, p, q U(R)] iff (U(R), , 0, 1,) or (U(R), , 0, 1,) are Jordan associative groupoids. Similar results hold for (P(R), ,0, 1, ).     

Rational rotation numbers for maps of the circle

We consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms, that is maps like

On the magnetic anisotropy of CuO

The anisotropy of the magnetic susceptibility of CuO has been measured. In both, the paramagnetic and the antiferromagnetic state the susceptibility can be described to a good approximation by

Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann–Hilbert problem approach is used to derive the leading-order asymptotics as of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation ( NLSE), , with finite-density initial data

Bound States in a Locally Deformed Waveguide: The Critical Case

We consider the Dirichlet Laplacian for astrip in with one straight boundary and a width , where $f$ is a smooth function of acompact support with a length 2b. We show that in the criticalcase, , the operator has nobound statesfor small .On the otherhand, a weakly bound state existsprovided . In thatcase, there are positive c ₁,c ₂ suchthat the corresponding eigenvalue satisfies for all sufficiently small.     

Bounded speed of propagation for solutions to radiative transfer equations

The Radiative Transfer Equation is the nonlinear transport equation

Lyapunov exponents of the Schrödinger equation with quasi-periodic potential on a strip

We prove that all the non-negative Lyapunov exponents of difference Schrödinger equation

Dirichlet Forms and Dirichlet Operators for Gibbs Measures of Quantum Unbounded Spin Systems: Essential Self-Adjointness and Log-Sobolev Inequality

For each [0, 1] we consider the Dirichlet form and the associated Dirichlet operator for the Gibbs measure of quantum unbounded spin systems interacting via superstable and regular potential. The Gibbs measure is related to the Gibbs state of the system via a (functional) Euclidean integral procedure. The configuration space for the spin systems is given by We formulate Dirichlet forms in the framework of rigged Hilbert spaces which are related to the space . Under appropriate conditions on the potential, we show that the Dirichlet operator is essentially self-adjoint on the domain of smooth cylinder functions. We give sufficient conditions on the potential so that the corresponding Gibbs measure is uniformly log-concave (ULC). This property gives the spectral gap of the Dirichlet operator at the lower end of the spectrum. Furthermore, we prove that under the conditions of (ULC), the unique Gibbs measure satisfies the log-Sobolev inequality (LS). We use an approximate argument used in the study of the same subjects for loop spaces, which in turn is a modification of the method originally developed by S. Albeverio, Yu. G. Kondratiev, and M. Röckner.     

A nonlinear instability for 3×3 systems of conservation laws

The phenomenon of nonlinear resonance provides a mechanism for the unbounded amplification of small solutions of systems of conservation laws. We construct spatially 2-periodic solutionsu ^N C ([0,t _N ] × witht _N bounded, satisfying

Equivalence of Deviau's, Hestenes', and Parra's Formulations of Dirac Theory

Daviau showed the equivalence of matrix Dirac theory, formulated within a spinor bundle , to a Clifford algebraic formulation within space Clifford algebra Pauli algebra (matrices) biquaternions. We will show, that Daviau's map : is an isomorphism. It is shown that Hestenes' and Parra's formulations are equivalent to Daviau's Clifford algebra formulation, which uses outer automorphisms. The connection between different formulations is quite remarkable, since it connects the left and right action on the Pauli algebra itself viewed as a bi-module with the left (resp. right) action of the enveloping algebra . The isomorphism established in this article and given by Daviau's map does clearly show that right and left actions are of similar type. This should be compared with attempts of Hestenes, Daviau, and others to interprete the right action as the iso-spin freedom.