Continuity conditions for finite-dimensional representations of some locally bounded groups

Necessary and sufficient conditions are obtained for the continuity of finite-dimensional representations of “almost divisible” locally pseudocompact groups in terms of the oscillations of these representations at a point. Applications to the continuity problem for finite-dimensional representations of almost connected locally compact groups and to the theory of quasirepresentations are given. To the blessed memory of Eckehart Hotzel Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by an INTAS grant.     

Norm continuous representations of locally compact groups

The structure of norm continuous representations of locally compact groups in Banach spaces is specified. Partially supported by the RFBR grant no. 08-01-00034 and by the Russian Government Program of Supporting Leading Scientific Schools under grant no. NSh-1562.2008.1. To the Memory of S. A.     

Van der Waerden continuity theorem for semisimple lie groups

As was proved by van der Waerden in 1933, every finite-dimensional locally bounded representation of a semisimple compact Lie group is continuous. This is the famous “van der Waerden continuity theorem,” and it motivated a vast literature. In particular, relationships between the assertion of the theorem (and of the inverse, in a sense, to this theorem) and some properties of the Bohr compactifications of topological groups were established, which led to the introduction and the study of certain classes of the so-called van der Waerden groups and algebras. Until now, after more than 70 years have passed, the van der Waerden theorem appears in monographs and surveys in diverse forms; new proofs were found and then simplified in important special cases. In this paper, we prove that the statement of the van der Waerden theorem remains valid for all (not necessarily compact) real semisimple Lie groups, i.e., any given finite-dimensional representation of a real semisimple Lie group is continuous if and only if this representation is locally bounded. More subtle results are also obtained. The main theorem contains several conditions equivalent to the continuity condition for a finite-dimensional representation. In particular, it is proved that every finite-dimensional representation of a real semisimple Lie group is continuous if and only if the restriction of this representation to the “compact” part, to the Abelian part, or to the nilpotent part of the Iwasawa decomposition is locally bounded, and the original van der Waerden theorem is also somewhat refined. For instance, the following assertion holds: every finite-dimensional representation of a compact semisimple Lie group is continuous if and only if the restriction of this representation to some maximal torus is locally bounded. Dedicated to the memory of George Mackey (1916–2006) Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by the INTAS grant.     

General method of calculation of any hadronic decay in the3 P 0 model

The³ P ₀ pair creation model of hadron decays is generalized to be applicable to the decay of any hadron. The wave function of the decaying hadron is expanded in terms of two clusters. The transition amplitude is derived for any combination of angular momenta, and for general wave functions in momentum space, expanded in terms of Gaussians times polynomials.Work supported by the Natural Science and Engineering Research Council of Canada, the National Science Foundation under grant # PHY-8714654, and by the Department of Energy under grant # DE-AC05-84ER40150Supported in part by the Natural Sciences and Engineering Research Council of Canada     

Quantization of triangular Lie bialgebras

We derive a universal twisting element for an arbitrary triangularγ-matrix using a simple analogue of the Fedosov quantization method. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. The work of SLL is partially supported by RFBR grant 00-02-17-956 and the grant INTAS 00-262. The work of AASh is supported by RFBR grant 02-02-06879 and Russian Ministry of Education under the grant E-00-33-184. The work of VAD is partially supported by the grant INTAS 00-561 and by the Grant for Support of Scientific Schools 00-15-96557. The work of API is partially supported by the RFBR grant 00-01-00299.     

A new formulation of the linear dynamic response to random excitations

With the damped harmonic oscillator as a model, a new formulation is presented of the stochastic response of linear dynamic systems under random excitation. The starting point of the method is the Fourier-Stieltjes representation for a general random process first introduced by Priestley. A unique feature of the present formulation is analytical compactness. This becomes a significant advantage over existing methods when one attempts to compute the covariance matrix elements which are often necessary for crossing problems.     

Rigorous entropy-energy arguments

We present a method for making rigorous various arguments which predict that certain situations are unstable because of a balance of energy vs. entropy. As applications, we give yet another proof that the two-dimensional plane rotor has no spontaneous magnetization and we make rigorous Thouless' arguments on the one-dimensional Ising model with couplingJ/n ².on leave from Departments of Mathematics and Physics, Princeton University. Research partially supported by U.S.N.S.F. under grant No. MCS-78-01885.Danforth Fellow. Research partially supported by U.S.N.S.F. under grant No. PHY 78-23952.     

Faithful representations of groups by automorphisms of topologies

It is proved that every (in general, infinite) group is the full automorphism group of some topology. Supported by RFBR under grant no. 07-01-00101-a.     

Estimates of homogenization for a parabolic equation with periodic coefficients

The asymptotic behavior of the operator exponent related to the Cauchy problem for a parabolic equation with periodic coefficients is studied either under the reduction of the periodicity cell or for large times. Estimates for the closeness of the operator exponentials (the original and the limit) with respect to the L ²-operator norm and the related H ¹-estimates are obtained under minimal assumptions concerning the smoothness of the heat matrix and of the initial data. Financially supported by RFBR under grant no. 05-01-00621.     

Anderson localization for Bernoulli and other singular potentials

We prove exponential localization in the Anderson model under very weak assumptions on the potential distribution. In one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment. In particular this solves the longstanding problem of localization for Bernoulli potentials (i.e., potentials that take only two values). In dimensions greater than one we prove localization at high disorder for potentials with Hölder continuous distributions and for bounded potentials whose distribution is a convex combination of a Hölder continuous distribution with high disorder and an arbitrary distribution. These include potentials with singular distributions.We also show that for certain Bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component.Partially supported by NSF grant DMS 85-03695Partially supported by NSF grant DMS 83-01889Partially supported by G.N.F.M. C.N.R.     

Ergosphere instability

We consider stationary asymptotically flat spacetimes having an ergosphere but with no horizon. In the framework of linear perturbation theory such configurations are unstable or marginally unstable to scalar and electromagnetic perturbations.Research supported in part by the National Science Foundation under grant MPS 74-17456 with the University of Chicago and grant MPS 74-7456 at the University of Wisconsin-Milwaukee     

The supersymmetry index and the construction of modular invariants

Relationships between the modular properties of affineG characters and the modular properties of the affine characters of regular subgroups ofG are derived by considering the branching functions that appear in calculation of the index of the Dirac-Ramond operator on super-coset models. Various applications of these relationships are described, and in particular a simple algorithm is given for generating modular invariant combinations of characters of affineG at any level by using the shift vector method on suitably chosen Lorentzian, self-dual lattices.Work supported in part by funds provided by the NSF under grant No. 87-08447 and by the U.S. Department of Energy (D.O.E) under contract # DE-AC02-76ER0306, and also by a fellowship from the Alfred P. Sloan foundation     

Computer aided design of inverted strip dielectric waveguide millimeter wave ring-resonator and coupler

A computer aided design technique has been developed for the design of millimeter-wave ring resonator and hybrid-ring coupler using inverted strip dielectric waveguides. The scattering matrix analysis for hybrid-ring coupler has also been developed. Numerical results for these components at f₀=80 GHZ with 3-dB coupling have been presented along with the variation of width, spacing, and radii with frequency.This work was supported by the faculty research committee of the University of North Dakota under research grant no. 1813/0909/2027 and partially funded by E.F. Johnson Co., Waseca, Minnesota, under grant no. 9009/8183.     

Binding of3He atoms to liquid helium

Summary We calculate the³He−⁴He chemical potential and change in volume in bulk liquid⁴He employing integral equation and Monte Carlo procedures. Detailed numerical results on these quantities and their dependence on bulk density and on the isotopic-mass difference are presented and discussed. This work has been supported by the Deutsche Forschungsgemeinschaft under grant Ri 267 and by the U.S. Department of energy under contract DE-AC 02-76 ER03077 and by the NSF under grant DMR-77-18329.     

The smooth cohomology ofN=2 supersymmetric Landau-Ginzburg field theories

We compute the smooth cohomology (both unrestricted and compactly supported) of the supercharge of an ultraviolet cutoffN=2 supersymmetric Landau-Ginzburg field theory.Supported in part by the National Science Foundation under grant DMS-9206936.Supported in part by the Department of Energy under grant DE-FG02-88ER25065.     

On multivortices in the electroweak theory II: Existence of Bogomol'nyi solutions in ℝ2

In this paper we study the Bogomol'nyi equations of the electroweak theory in the full plane. We will show that, for any distribution of the vortices, there exists a two parameter family of gauge-distinct solutions. Moreover, we also establish some sharp decay rate estimates for these solutions.Research supported in part by NSF grant DMS-88-02858 and DOE grant DE-FG02-86ER250125     

Time-displaced correlation functions in an infinite one-dimensional mixture of hard rods with different diameters

Time-displaced conditional distribution functions are calculated for an infinite, one-dimensional mixture of equal-mass hard rods of different diameters. The kinetic equation that describes the time dependence of the one-particle total distribution function is found to be non-Markovian, in contrast with the situation in systems of identical rods. The correlation function does not contain any isolated damped oscillation, except for systems of equal-diameter rods with discrete velocities. Thus, we generalize the one-component results of Lebowitz, Perçus, and Sykes, removing some nontypical features of that system.Supported by NSF grant No. MCS 75-21684 A01 (M. A.), NSF grant No. MPS 75-20638 (J. L.), and USAFOSR grant No. 73-2430 B (J. M.)John Guggenheim Fellow on sabbatical leave from Belfer Graduate School of Science, Yeshiva University, New York.