A “Transversal” Fundamental Theorem for semi-dispersing billiards

For billiards with a hyperbolic behavior, Fundamental Theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds. A Transversal Fundamental Theorem has recently been suggested by the present authors to proveglobal ergodicity (and then, as an easy consequence, the K-property) of semidispersing billiards, in particular, the global ergodicity of systems ofN3 elastic hard balls conjectured by the celebratedBoltzmann-Sinai ergodic hypothesis. (In fact, the suggested Transversal Fundamental Theorem has been successfully applied by the authors in the casesN=3 and 4.) The theorem generalizes the Fundamental Theorem of Chernov and Sinai that was really the fundamental tool to obtainlocal ergodicity of semi-dispersing billiards. Our theorem, however, is stronger even in their case, too, since its conditions are simpler and weaker. Moreover, a complete set of conditions is formulated under which the Fundamental Theorem and its consequences like the Zig-zag theorem are valid for general semi-dispersing billiards beyond the utmost interesting case of systems of elastic hard balls. As an application, we also give conditions for the ergodicity (and, consequently, the K-property) of dispersing-billiards. Transversality means the following: instead of the stable and unstable foliations occurring in the Chernov-Sinai formulation of the stable version of the Fundamental Theorem, we use the stable foliation and an arbitrary nice one transversal to the stable one.Dedicated to Joel L. Lebowitz on the occasion of his 60th birthdayResearch partially supported by the Hungarian National Foundation for Scientific Research, grant No. 819/1     

The status of “polarized fusion”

After an introduction into the essentials of nuclear fusion reactions, being considered for future energy production in fusion reactors, different aspects of using “polarized” particles as fuel are discussed. Special emphasis is given to the four-nucleon system and the D+D reactions. The status of the theory and the experimental data are presented. Predictions for observables in the framework of existing theoretical approaches are given. The necessity of carrying out direct spin-correlated cross-section measurements is accentuated. Details of a future experiment are proposed.     

The “strong Kadison inequality” of Woronowicz

In the terminology of [4] it is shown that a normalized positive map ? from a C¹-algebra $\text{A}$ into M₂ is a sum ?=ψ₁+ψ₂, where ψ₁ is completely positive and ψ₂ is completely copositive, if and only if for all a?1 in $\text{A}$ the normalized positive map

$\text{?}{}_{\mathrm{a}}\text{=?(a}{}^2\text{)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{?(a·a)?(a}{}^2\text{)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$

satisfies the “strong Kadison inequality” of S.L. Woronowicz ([4]). p. 178).     

The “paradox” of a brake pad

By the example of a brake pad, a typical error is illustrated, i.e., the fact that the initial conditions should be reconsidered upon finding that there is no solution of the static problem in a certain region of parameters (incorrect in the sense of Hadamard). For example, it turns out that the conditions of disk rotation in the positive direction cannot be implemented in the domain of parameters leading to a “paradox.” In this region, the “brake-pad” mechanism is transformed into the “wedge stopper” mechanism.     

The “dark side” of gravitational experiments

Theoretical speculations about the quantum nature of the gravitational interaction have motivated many recent experiments. But perhaps the most profound and puzzling questions that these investigations address surround the observed cosmic acceleration, or Dark Energy. This mysterious substance comprises roughly two-thirds of the energy density of the universe. Current gravitational experiments may soon have the sensitivity to detect subtle clues that will reveal the mechanism behind the cosmic acceleration. On the laboratory scale, short-range tests of the Newtonian inverse-square law (ISL) provide the most sensitive measurements of gravity at the Dark Energy length scale, where is the observed Dark Energy density. This length scale may also have fundamental significance that could be related to the “size” of the graviton. At the University of Washington, we are conducting the world’s most sensitive, short-range test of the Newtonian ISL. Fourth Award in the 2006 Essay Competition of the Gravity Research Foundation.     

The meaning of the terms “phase” and “phase transition”

Various meanings of the terms phase and phase transition encountered in scientific literature are discussed. These terms supplement each other and cover only together all the macroscopic situations which are now denoted by this term.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 67–71, August, 1988.     

The problem with “The problem of the Pegg–Barnett phase operator”

We show that the physically absurd state obtained by Vorontsov and Rembovsky [Phys. Lett. A 254 (1999) 7] results from an unphysical choice of approximate phase measurement and not from the phase formalism itself as claimed. Thus their assertion that it is impossible to perform even an approximate measurement of phase is unfounded.     

Coupling-induced bipartite pointer states in arrays of electron billiards: quantum Darwinism in action?

We discuss a quantum system coupled to the environment, composed of an open array of billiards (dots) in series. Beside pointer states occurring in individual dots, we observe sets of robust states which arise only in the array. We define these new states as bipartite pointer states, since they cannot be described in terms of simple linear combinations of robust single-dot states. The classical existence of bipartite pointer states is confirmed by comparing the quantum-mechanical and classical results. The ability of the robust states to create "offspring" indicates that quantum Darwinism is in action.     

A review of: “The science of crystallization,microscopic interfacial phenomena”

Abstract

By W. A. Tiller, Cambridge University Press. ISBN 0-521-38827-9     

The determination of the parameters of “spherical” motion in a kerr field

The problem of the existence of class M geodesic motions on the surface r=const, all of whose mechanical parameters in the specified Kerr field are determined only from stability conditions, is posed and investigated. A system of equations which determines this class is derived and solved. It is shown that in the general case this motion does not cover the entire surface r=const and is restricted by the condition ₀ }- ₀. Simple algebraic expressions are found for all the parameters of these configurations-energy, momentum, radius, and the angle ₀-as functions of the specific angular momentuma of the Kerr field. It is shown that these motions can exist only in Kerr fields with a value of the parametera larger or equal to r_g/2. In a Kerr field with a fixed value ofa there exist only two configurations with the indicated properties. In conclusion, the properties of the M-solutions associated with the appearance of configurations with negative energies and negativeness of g₀₀ within the limits of some configurations and values ofa larger than r_g/2 are discussed. It is noted that the negative values of the energy occur only in those configurations within whose limits g₀₀ < 0. It is also pointed out that the Kerr field is free of pathological properties associated with the breakdown of causality whena > r_g/2 if one does not made the assumption, unnecessary within the framework of the problem under discussion, that the sources of the Kerr field are in the region r=0.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 17–25, May, 1979.The authors express their gratitude to A. A. Grib for useful discussions.     

The “split” iota

Within the context of a five channel mixing model the possibility is considered that the E(1420) originally seen in PP̄ annihilation at rest and originally believed to be a J^PC=1⁺⁺ state is actually a pseudoscalar state distinct from the ι(1460) seen in radiative ψ decay. The quark and glue content that this hypothesis implies and the consequences for branching ratios of the known pseudoscalars are discussed. The five channel mixing model includes the ground state pseudoscalars, their first radial excitations, and a glueball. Mixing terms include an SU(3) symmetric glueball-quark state coupling as well as the important two body intermediate state unitarity corrections. While a number of mathematically consistent solutions to the model are found, none are physically acceptable. It is concluded that the more likely scenario is that the E(1420) and the ι(1460) are the same state, and the fifth partner to the scheme lies in a higher mass range, above 1.7 GeV.