Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

Theory of percolation in fluids of long molecules

We use the reference interaction site model (RISM) integral equation theory to study the percolation behavior of fluids composed of long molecules. We examine the roles of hard core size and of length-to-width ratio on the percolation threshold. The critical density _c is a nonmonotonic function of these parameters exhibiting competition of different effects. Comparisons with Monte Carlo calculations of others are reasonably good. For critical exponents, the theory yields =2=2 for molecules of any noninfinite lengthL. WhenL is very large, the theory yields _cL ^–2. These predictions compare favorably with observations of the conductivity for random assemblies of conductive fibers. The threshold region where asymptotic scaling holds requires the correlation length (/ _c ) ^–v to be much larger thanL. Evidently, the range of densities in this region diminishes asL increases, requiring that density deviations from _c be no larger thanL ^–2. Otherwise, crossover behavior will be observed.     

Existence of a Constant for Finite System Extinction

We show the existence of a constant (0, ) such that if ⁿ is the extinction time for a supercritical contact process on [1, n] ^d starting from full occupancy, then {log(E[ ⁿ])}/n ^d tend to as n tends to infinity.     

Finslerian N-Spinors: Algebra

New mathematical objects called Finslerian N-spinors are discussed. The Finslerian N-spinor algebra is developed. It is found that Finslerian N-spinors are associated with an N²-dimensional flat Finslerian space. A generalization of the epimorphism SL(2, ) O ₊ (1, 3) to a case of the group SL(N,) is constructed. Particular examples of Finslerian N-spinors for N = 2, 3 are considered in detail.     

Stochastic Schrödinger operators and Jacobi matrices on the strip

We discuss stochastic Schrödinger operators and Jacobi matrices with wave functions, taking values in ^l so there are 2l Lyaponov exponents ₁..._l0 _l+1..._2l =–₁. Our results include the fact that if ₁=0 on a set positive measure, thenV is deterministic and one that says that {E|exactly 2j 's are zero} is the essential support of the a.c. spectrum of multiplicity 2j.Research partially supported by USNSF under grant DMS-8416049     

Duality for a Lorentz quantum group

We give the algebra _q ^/* dual to the matrix Lorentz quantum group _q of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form _q ^/* U _q (sl(2, )) U _q (sl(2, )). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity.     

On the isomorphism of a ‘quantum logic’ with the logic of the projections in a hilbert space

A topology is introduced in a logic using the set of pure states of . It is shown that , equipped with this topology, under suitable conditions, determines the division ring , or 2e. With the continuity of the antiautomorphism of the division ring added, it is shown that these conditions are necessary and sufficient for the projective logic to be isomorphic with the projective logic of the projections in a Hilbert space over , or 2e.P. Cotta-Ramusino gratefully acknowledges a fellowship of the Consiglio Nazionale delle Ricerche.     

Are there critical points on the boundaries of singular domains?

Generalizing a result of E. Ghys, we prove a general theorem that implies that if a rational functionf of the Riemann sphere of degree 2 leaves invariant a singular domainC (a disk or a ring) on which the rotation number off satisfies a diophantine condition, provided that on f is injective, then each boundary component ofC contains critical point off. The injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off(z)=z ⁿ +a, withn,n2 anda. We also show that the singular disks, associated to periodic elliptic points off(z)=e ^az that satisfy a diophantine condition, are unbounded in . In the end of the paper, we give a survey of the theory of iteration of entire functions of .     

Relative change of the electron density at the Cu nucleus in Cu-Ag alloys

The decay constant of⁶⁴Cu in Cu–Ag solid solutions has been measured at various Cu concentrations. Deduced values of the relative changes of electron densities (0)/(0) at the Cu nucleus are given. The observed concentration dependence of (0)/(0) is discussed in terms of a volume effect and charge transfer from Cu to Ag.     

π0 → γγ in the Spinor Strong Interaction Theory

An expression for the decay rate (⁰ ) has been derived in the frameworkof the spinor strong interaction theory, a first-principles strong interaction theoryproposed some years ago as an alternative to low-energy QCD. The startingpoint is the SO(3) gauge-invariant action for two quark mesons which has beensuccessful in accounting for confinement, ^{+ +} , e⁺, and ⁰ e⁺, nonexistenceof the Higgs boson, and other low-energy mesonic phenomena. The quasi-four-quarkmeson equations developed for the decay of a vector meson into twopseudoscalar mesons V PP has been taken over here to apply to P(⁰) VV(^{+ –}) (plus ⁺ and ^– which annihilate each other). This mechanismin principle agrees with that of the assumption of vector meson dominance inthe literature. It, together with the effect of form factors, arises naturally in theformalism and need not be assumed. Equations for the perturbed vector mesonwave functions cannot be simply solved and an assumption has been made toobtain an estimate of their magnitude. Together with a constant associated withthe strong coupling obtained earlier from V() PP(K⁺K^–), the estimated decayrate is 19.2 eV, in order-of-magnitude agreement with data (7.74 eV).     

Electrical properties of GaAs layers irradiated by H4 ions

The quantities(D) and(T) are studied in n- and p-GaAs, irradiated at T = 300°K by H⁺ ions (5 MeV). It is shown that the resistance of lightly doped GaAs specimens increases from original values of ₀ to 10⁹ ·cm upon irradiation by H⁺ ions (5 MeV) to integral fluxes up to D^* – 10¹⁵ H⁺/cm². For D > D^* the layer resistance decreases from 10⁹ ·cm to 1 ·cm at 300°K. It was found that all the GaAs specimens intensely irradiated by H⁺ ions had p-type conductivity near 300°K. Isochronic annealing of radiation defects was studied in the temperature interval 20–700°C.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 39–43, January, 1982.     

Noncommutative Geometry and the Regularization Problem of 4D Quantum Field Theory

We give a noncommutative version of the complex projective space ² and show that scalar QFT on this space is free of UV divergencies. The tools necessary to investigate quantum fields on this fuzzy ² are developed and several possibilities to introduce spinors and Dirac operators are discussed.     

Determination of the electron density at the Cu nucleus in Cu-Au alloys

The half-life of⁶⁴Cu in Cu–Au solid solutions has been measured as a function of the Cu concentration. Relative changes of electron densities (0)/(0) at the Cu nucleus are deduced. The observed nonlinear concentration dependence of (0)/(0) is discussed in terms of volume and charge transfer effects.     

A borel-weil-bott approach to representations of sl q (2, ℂ)

We use a quite concrete and simple realization of sl _q (2, ) involving finite difference operators. We interpret them as derivations (in the noncommutative sense) on a suitable graded algebra, which gives rise to the noncommutative scheme ¹ II ^1* as the counterpart of the standard ¹ = Sl(2, )/B.     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

The eightfold way to color geometrodynamics

Color models of strong interactions are generalized to aGL(8,) ^f GL(8,) ^c gauge theory incorporating space-time curvature and Cartan's torsion. Following Salam, the dynamics is determined by an Einstein-Dirac-type Lagrangian. The resulting field equations are anonlinear (due to the torsion) Heisenberg-Pauli-Weyl equation for the fundamental spinor fields and a generalized Einstein equation for the background metric of hadronic dimensions. According to this model baryonic quarks are confined ingeon (black soliton)-type objects by the tensor gluons ofstrong gravity. This approach also leads to a black soliton mass formula which is in qualitative agreement with part of the baryon spectrum. Hadronic mesons are interpreted as gluon strings trapped in a multiconnected space-time. Interrelations of color geometrodynamics with other bag models are pointed out. Finally, the conceptual origin of this space-time foundation of quark confinement is presented.     

Hamiltonian collapsing of irrational Lagrangian submanifolds with small first Betti number

Among the main symplectic invariants of a closed Lagrange submanifoldL of the cotangent of ⁿ is the tubular radiusR(L) defined as the smallest tube D(r) × ^n–1 of ⁿ T^{* n} in whichL can be pushed by an Hamiltonian diffeotopy of ⁿ. We show here, using pseudo-holomorphic techniques, that such a submanifold cannot collapse if the first Betti number ofL is smaller than 3 and if the Maslov class ofL does not vanish; in other words,R(L) is then strictly positive and one can actually give an explicit lower bound in terms of the Liouville and Maslov classes ofL.Partially supported by Research Grants NSERC OGP0092913 and FCAR EQ3518     

Classical solutions ofℂP n−1 models and their generalizations

We discuss finite action solutions ofP ^n–1 models, of their Grassmannian generalizations and of a classical fermion — boson model based on the supersymmetric extension of theP ^n–1 model.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.I would like to thank Dr. A. M. Din for his collaboration with me on all the topics discussed here. Also I would like to thank E. F. Corrigan, D. B. Fairlie, M. Günyadin, J. Lukierski, I. Singer, R. Stora and many others for discussions and constructive criticisms and Professor J. Niederle for his invitation to present this talk at the symposium and for hospitality in Czechoslovakia.     

Growth of correlation in the Hopfield model

Introducing a finite correlation ₀ between any two learned patterns (others remaining uncorrelated), we observe in a numerical simulation that the Hopfield model stores these two patterns with correlation _f such that _f0 for any loading capacity. The patterns are memorized perfectly (with _f= ₀) up to -0.05 for finite correlations ₀ not exceeding a value _c(), where _c() decreases continuously to zero at -0.05.     

Metrics are Clifford algebra involutions

Four-dimensional space-time, all relevant inner products, and some of the groups leaving these inner products invariant are manufactured from more basic algebraic ingredients, all inside the 8-dimensional Pauli algebra : (1) Euclidean 3-spaceE ³, (2) Minkowski 4-spaceM ⁴, (3) complex 4-space ⁴, and all three metrics and all three inner products. The groupsSO(3;) SO(3; 1;) SO (4;) are obtained as images of twofold covering maps of subgroups of or their direct product. A method of embedding in the Clifford algebraC(1;n–1) ofn-dimensional Minkowski space is given for anyn4. Furthermore, all three groups act not only on the relevant vector spaces, but on all ofC(1;n–1), leaving setwise invariant.