Entropy functionals of Kolmogorov-Sinai type and their limit theorems

Two functionals and are introduced forC ^*-dynamical systems with invariant states and stationary channels. It is shown that the Kolmogorov-Sinai-type theorems hold for these functionals and . Our functionals and are set within the framework of quantum information theory and generalize a quantum KS entropy by CNT and the mutual entropy by Ohya.     

Three-hadron form factor via quark-triangle loop

An explicit construction in a manifestly Lorentz-invariant and cyclically symmetric form is given for a three-hadron (H) coupling through their respective (q ) constituents interacting via a quark triangle loop under the most general conditions of unequal mass kinematics atboth the elementary quark and composite-hadron levels. The only ansatz employed is that theHq vertex function _H at each vertexi is a function of _i , the relative momentumtransverse to the corresponding hadron momentumP _i, an assumption which can be justified from several independent angles. The resulting structure , withD( ) being a model-independent function, has proved crucial for an explicit integration over the time-like component of the loop momentum, and thus provided an analytical, cyclically symmetric structure free of overlapping pole singularities. The applicational potential of this quantity is outlined.     

Self-avoiding walks on percolation clusters at criticality and lattice animals

We use the real-space renormalization group method to study the critical behavior of self-avoiding walks (SAWs) on both site percolation clusters at percolation threshold and site lattice animals in a square lattice. The correlation length exponents of SAWs are found to be on the percolation clusters atp _c and _SAW ^LA =0.804 on lattice animals. These results are compared with Kremer's suggestion of modified Flory formula where is the fractal dimension of the fractal object.     

On the time evolution automorphisms of the CCR-algebra for quantum mechanics

In ordinary quantum mechanics for finite systems, the time evolution induced by Hamiltonians of the form is studied from the point of view of *-automorphisms of the CCRC*-algebra (see Ref. [1, 2]). It is proved that those Hamiltonians do not induce *-automorphisms of this algebra in the cases: a) and b)V L (,dx) L ¹ (,dx), except when the potential is trivial.     

Partial transposition and nonextendible maps

The wide class of nonextendible positive maps inC ^*-algebras [1, 3] is studied. An example is given showing that the set of nonextendible positive maps betweenC ^*-algebras and is not closed inL( ) in the weak topology.     

The large-scale limit of Dyson's hierarchical vector-valued model at low temperatures. the marginal casec = \sqrt 2

In this paper we construct the equilibrium states of Dyson's vector-valued hierarchical model with parameter at low temperatures and describe their large-scale limit. The analogous problems for <c<2 and 1<c< were solved in our papers [1] and [2]. In the present case the large-scale limit is similar to the case <c<2, i.e. it is a Gaussian self-similar field with long-range dependence in the direction orthogonal to and a field consisting of independent Gaussian random variables in the direction parallel with the magnetization. The main difference between the two cases is that now the normalizing factor in the direction of the magnetization contains, beside the square-root of the volume, a logarithmic term too.Dedicated to Roland Dobrushin     

On the blow-up of solutions of the 3-D Euler equations in a bounded domain

It is shown that if [0, ) is the maximal interval of existence of a smooth solutionu of the incompressible Euler equations in a bounded, simply connected domain R ³, then , where =×u is the vorticity. Crucial to this result is a special estimate proven in of the maximum velocity gradient in terms of the maximum vorticity and a logarithmic term involving a higher norm of the vorticity.     

CP-violating effects inZ decays to τ leptons

We calculate CP-odd correlations inZ decays to leptons, . These correlations are sensitive to the weak dipole moment of the . With 10⁷ producedZ particles and with observation of the decay channels and v we estimate that can be determined with an accuracy of about (1 s.d.).     

Morse and Melnikov functions for NLS Pde's

The theory of the focusing NLS equation under periodic boundary conditions, together with the Floquet spectral theory of its associated Zakharov-Shabat linear operator , is developed in sufficient detail for later use in studies of perturbations of the NLS equation. Counting lemmas for the non-selfadjoint operator , are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discD in the complex eigenvalue plane. The radius of the discD is controlled by theH ¹ norm of the potential . For this integrable NLS Hamiltonian system, unstable tori are identified, and Backlund transformations are then used to construct global representations of their stable and unstable manifolds-whiskered tori for the NLS pde.The Floquet discriminant is used to introduce a natural sequence of NLS constants of motion, [ , where _c ^j denotes thej ^th critical point of the Floquet discriminant ()]. A Taylor series expansion of the constants , with explicit representations of the first and second variations, is then used to study neighborhoods of the whiskered tori. In particular, critical tori with hyperbolic structure are identified through the first and second variations of , which themselves are expressed in terms of quadratic products of eigenfunctions of . The second variation permits identification, within the disc D, of important bifurcations in the spectral configurations of the operator . The constant , as the height of the Floquet discriminant over the critical point _c ^j , admits a natural interpretation as a Morse function for NLS isospectral level sets. This Morse interpretation is studied in some detail. It is valid globally for the infinite tail, , which is associated with critical points outside the discD. Within this disc, the interpretation is only valid locally, with the same obstruction to its global validity as to a global ordering of the spectrum. Nevertheless, this local Morse theory, together with the Backlund representations of the whiskered tori, produces extremely clear pictures of the stratification of NLS invariant sets near these whiskered tori-pictures which are useful in the study of perturbations of NLS. Finally, a natural connection is noted between the constants of the integrable theory and Melnikov functions for the theory of perturbations of the NLS equation. This connection generates a simple, but general, representations of the Melnikov functions.Funded in part by AFOSR-90-0161 and by NSF DMS 8922717 A01     

The microstructure of electrodeposited amorphous or microcrystalline Fe-P alloys

Dependent on is produced through electrodeposition in three modifications: microcrystalline if , amorphous if and in a mixed state containing both components if . The interpretation of results of large and small angle X-ray scattering, of measurements of the coercivity and of the mechanical density shows that the three modifications consist of two phases. The structure of the grain boundary phase in the microcrystalline alloys differs considerably from that of the amorphous phase. This arises from the dependence of the atomic volume on the average phosphorus content . The volume fraction of the crystalline component decreases with increasing up to . But even in alloys revealing X-ray patterns without any peaks coming from crystalline particles small angle X-ray scattering indicates the existence of metal-rich inclusions occupying some percent of the total volume.     

On the equilibrium states in quantum statistical mechanics

Representations of theC*-algebra of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential are studied. Both for finite and for infinite systems it is shown that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of onto its commutant. This means that there is an equivalent anti-linear representation of in the commutant. The relation of these properties with the Kubo-Martin-Schwinger boundary condition is discussed.     

Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra

Let be a complex simple Lie algebra. We show that whent is not a root of 1 all finite dimensional representations of the quantum analogU _t are completely reducible, and we classify the irreducible ones in terms of highest weights. In particular, they can be seen as deformations of the representations of the (classical)U .     

Calculation of proton polarization in deuteron disintegration with longitudinally polarized electrons

For the reaction dependences of the proton polarization on the deuteron electromagnetic current and on the final-state interaction are studied. The possibility of separation of the structure functions contributing to this observable is discussed. Polarizations of protons in are calculated for the kinematics of a proposed measurement in the MIT-Bates research program.     

Limits of states

Estimates for vector representations of states are used to prove that {C _n C ₀} is strong-operator convergent toC ₀, whereC _n is the universal central support of _n and { _n } is a sequence of states of aC*-algebra converging in norm to ₀. States of of a given type are shown to form a norm-closed convex subset of the (norm) dual of . The pure states of form a norm-closed subset of the dual.With partial support of the National Science Foundation (USA)     

On the Schrödinger equation and the eigenvalue problem

If _k is thek ^th eigenvalue for the Dirichlet boundary problem on a bounded domain in ⁿ , H. Weyl's asymptotic formula asserts that , hence . We prove that for any domain and for all . A simple proof for the upper bound of the number of eigenvalues less than or equal to - for the operator –V(x) defined on ⁿ (n3) in terms of is also provided.Research partially supported by a Sloan Fellowship and NSF Grant No. 81-07911-A1     

Nonlinear superposition for Liouville's equation in three spatial dimensions

We derive in 3+0 dimensions exact solutions of Liouville's equation ²=exp , by applying the Bäcklund transformation technique in conjunction with the principle of nonlinear superposition. The procedure, which is later extended to 3+1 dimensions, yield, as a byproduct, particular solutions of ² and ² =exp (² +² ).     

Preferred Invariant Symplectic Connections on Compact Coadjoint Orbits

Let be the Haag--Kastler net generated by the (2) chiral current algebra at level 1. We classify the SL(2, )-covariant subsystems by showing that they are all fixed points nets ^H for some subgroup H of the gauge automorphisms group SO(3) of . Then, using the fact that the net ₁ generated by the (1) chiral current can be regarded as a subsystem of , we classify the subsystems of ₁. In this case, there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem .     

Ellipsoidal bag model for heavy quark system

The ellipsoidal bag model is used to describe heavy quark systems such asQ ,Q g andQ ² . Instead of two step model, these states are described by an uniform picture. The potential derived from the ellipsoidal bag forQ is almost equivalent to the Cornell potential. For aQ ² system with large quark pair separation, an improvement of 70 MeV is obtained comparing with the spherical bag.     

Geometro-stochastic quantization of a theory for extended elementary objects

The geometro-stochastic quantization of a gauge theory based on the (4,1)-de Sitter group is presented. The theory contains an intrinsic elementary length parameter R of geometric origin taken to be of a size typical for hadron physics. Use is made of a soldered Hilbert bundle over curved spacetime carrying a phase space representation of SO(4, 1) with the Lorentz subgroup related to a vierbein formulation of gravitation. The typical fiber of is a resolution kernel Hilbert space constructed in terms of generalized coherent states related to the principal series of unitary irreducible representations of SO(4, 1), namely de Sitter horospherical waves for spinless particles characterized by the parameter . The framework is, finally, extended to a quantum field-theoretical formalism by using bundles with Fock space fibers constructed from .Supported in part by NSERC Research Grant No. A5206.     

Synthesis and spectroscopic studies of 4-Formyl-4′-N,N-dimethylamino-1,1′-biphenyl: The unusual red edge effect and efficient laser generation

The synthesis and photophysics of 4-formyl-4-N,N-dimethylamino-1,1-biphenyl are reported. The emission spectrum in various solvent polarities demonstrates solvatochromism, indicating that the fluorescence originates from an electronically excited species with a strong charge transfer character. The change in [ _max(absorption) – _max(emission)] varies from 1500 cm^–1 inn-heptane to as much as 7500 cm^–1 in acetonitrile. In protic solvents, the unusual excitation energy-dependent steady-state emission (red edge effect), resulting from solvent dielectric relaxation, was observed in media with a low viscosity. The large Stokes-shifted and high-yield fluorescence led to the observation of the efficient lasing action. The frequency tunability of the laser output is strongly solvent dependent, generating a new charge transfer laser dye in the blue-green region.