Surviving extrema for the action on the twisted SU(∞) one point lattice

We give a simplified construction of twist eating configurations, based on a theorem due to Frobenius. These configurations are defined through the equation:U U U ⁺ U ⁺ =exp(2in /N) withU SU(N), =1 tod andn an antisymmetric matrix with integer entries. In the (Twisted)-Eguchi-Kawai model they yield extrema some of which survive forN. Comparison is made with the Monte Carlo data of the internal energy in the small coupling region.     

A spontaneously broken quintet Baryon model for strong interactions

A gauge type model of quantum field theory for strong interactions based on a quinted of observed fields, namely the proton, neutron, , _c and _b baryon fields is proposed. Gauging the resulting global symmetry groupK= SU(3)×¹ U(1)ײ U(1)׳ U(1) for matter fields, one obtains boson-fermion field theory with eleven gauge bosons. The analysis of admissible Higgs sector indicates that the Higgs multiple consists of one adjoint and two fundamental representationsSU(3) and three scalar representations of¹ U(1),² U(1) and³ U(1). The structure of the Higgs sector implies that the original symmetry group extends to the groupK×U(2). Breaking spontaneously the obtained field theory, one converts gauge bosons into the eleven massive vector bosons which can be identified with the observed , K^*, ¯K^*, , , J/ and Y vector mesons. The surviving global symmetry is isomorphic with the symmetry groupSU(2)× ⁰ U(1)× ×¹ U(1)ײ U(1)׳ U(1) corresponding to the isospin, strangeness, baryon number, charm and beauty conservation observed in strong interactions. The surviving Higgs scalars have the same quantum numbers as , K, ¯K, , S, , , and _b mesons. The model gives a newSU(3) classification scheme for baryons without charm and beauty in terms of triplets, sextets and 15-plets. These multiplets can be identified with the observed baryons; the scheme also includes the observed Z₀ and Z₁ baryons (the experimental evidence of which is, nevertheless, still weak). The model predicts the existence and the specific quantum numbers of new mesons and baryons with charm and beauty, and provides a very simple framework for the dibaryon analysis. Since all final physical fields are massive, this model is free from infrared divergences.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

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.     

The structure of projection-valued states: A generalization of Wigner's theorem

A projection-valued state is defined to be a completely orthoadditive map from the projections on one Hilbert space into the projections on another Hilbert space, which preserves the unit. Any such mapping is shown to have the formP U ₁(P 1₁)U ₁ ^–1 U ₂(P 1₂)U ₂ ^–1 , whereU ₁ is unitary andU ₂ is antiunitary, generalizing Wigner's theorem on symmetry transformations. A physical interpretation is given and the relation to quantum logic is discussed.The contents of this paper are a portion of the author's dissertation at the University of Massachusetts at Amherst.     

Weakly Bound States in Light Hypernuclei

From the viewpoint of critical stability, we discuss the three- and four-body structure of ⁶He, ⁷He, ⁴He, and ⁴H. With the + + N three-body model, ⁶He is found to have a three-layer structure of the matter distribution: core, a skin and neutron halo. Also the level structure of ⁷He with the three-body model of ⁵He + n + n is predicted. This stimulates a new study of neutron-rich and proton-rich hypernuclei. By performing a four-body calculation with both NNN and NNN channels and with both NN and NNN channels, we show that the N-N and -N couplings are very important in critical stability of few-body hypernuclear systems.     

Some critical exponent inequalities for percolation

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation densityP (p) is discontinuous atp _c , then the critical exponent (defined by the divergence of expected cluster size, nP _n (p) (P _c –P)^– asp p _c ) must satisfy 2. (2) or (defined analogously to, but asp p _c ) and [P _n (p _c ) (n ^–1–1/) asn ] must satisfy, 2(1 – 1/). These inequalities for improve the previously known bound 1(Aizenman and Newman), since 2 (Aizenman and Barsky). Additionally, result 1may be useful, in standardd-dimensional percolation, for proving rigorously (ind>2) that, as expected,P _x has no discontinuity atp _c .     

Inverse Problem for a General Bounded Domain inR3 with Piecewise Smooth Mixed BoundaryConditions

We study the influence of a finite container on an ideal gas. The trace of theheat kernel (t) = _{= 1}exp(–t), where {} _{= 1}are the eigenvalues of the negative Laplacian – ² = – ³ _{= 1}(/x )² in the (x ¹, x ², x ³)-space,is studied for a general bounded domain with a smooth bounding surface S, where afinite number of Dirichlet, Neumann, and Robin boundary conditions on thepiecewise smooth parts S _i (i = 1, ..., n) of S are considered such that S =U _{i = 1} S _i . Some geometrical properties of (the volume, the surface area, the meancurvature, and the Gaussian curvature) are determined. Furthermore,thermodynamic quantities, particularly the energy, for an ideal gas enclosed inthe general bounded domain with Dirichlet, Neumann, and Robin conditionsare examined with the help of the asymptotic expansions of (t) for short timet. We show that these thermodynamic quantities depend on some geometricproperties of .     

Nontrivial extensions of a representation of the Poincaré group with mass and helicity zero by its tensorial product

Let U(a, ) be a representation of the Poincaré group with mass and helicity zero, realized in the space of C -functions with compact support on ³, without the origin. Let U ⁽²⁾(a, ) denote the tensorial product of U(a, ) by itself. We explicitly determine the cocycles of extension of U(a, ) by U ⁽²⁾(a, ) and we prove that the nontrivial cohomology is indexed by (u(),),u()D ¹ _]0,3\,.     

On two “self-creation” cosmologies

An attempt to produce a continuous creation theory by adapting the Brans-Dicke theory is described. The universe is seen to be created out of self-contained gravitational, scalar, and matter fields. However, the solution of the one-body problem reveals unsatisfactory characteristics of the theory, and in particular the principle of equivalence is severely violated. A second theory is described which retains the attractive features of the first theory and which does not fall foul of its objections. There do exist empirical tests for the theory which are described and which will require further examination. In the limit this theory approaches general relativity in every respect.Notation ² gf the invariant d'Alembertian - t ₀ Hubble time - H Hubble's constant - scalar field - coupling constant - T _M energy-momentum tensor of matter and nongravitational energy, and nonscalar field energy - T _M energy-momentum tensor of scalar field energy - T _M covariant form - T _M contravariant form - T _M mixed form - T _M T _M trace - T _{M ;} covariant differentiation - contravariant differentiation - T Ricci tensor - R curvature scalar (in tensorial equations) - Kronecker symbol - () a function of used in the text - density - p pressure - g the metric tensor - R(t) scale factor (in cosmological equations) - U the fluid 4-velocity (covariant) - U the fluid 4-velocity (contravariant) - functions differentiated once with respect to time ( , differenciated twice) - k the Robertson-Walker curvature constant=+1, 0, or –1 - proportional to - g gravitational coefficient - parameter - angle of deflection, or coordinate - angle of precession or coordinate - angle of precession - G _v the force density - ³( – _n (t)) the Dirac delta function - proper time - K an unknown function definingG - total angle of deflection - r ₀ minimum radius of approach of a light ray grazing the sun     

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     

The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions

Existence and uniqueness results are established for solutions to the Becker-Döring cluster equations. The density is shown to be a conserved quantity. Under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined. Denoting the set of equilibrium solutions byc ⁽⁾, 0 _s , the principal result is that if the initial density _{0 s} then the solution converges strongly toc ^(o), while if ₀ > _s the solution converges weak* toc ^(s). In the latter case the excess density ₀– _s corresponds to the formation of larger and larger clusters, i.e. condensation. The main tools for studying the asymptotic behaviour are the use of a Lyapunov function with desirable continuity properties, obtained from a known Lyapunov function by the addition of a special multiple of the density, and a maximum principle for solutions.     

Differential algebras for quantum groups of type B, C, and D

We study higher order bicovariant differential calculi on the quantum groups O_q(N) and Sp _q (N). We show that the second antisymmetrizer exterior algebra _u is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars biinvariant 1-form. Moreover is central in _u and _u is an inner differential calculus. We show that the quadratic dual to the left-invariant algebra _{s L} is isomorphic to the reflection equation algebra. Let be an arbitrary left-covariant first order differential calculus. We show that the dimension of the space of left-invariant 2-forms in the universal exterior algebra equals the number of linearly independent quadratic-linear relations in the quantum tangent space.     

The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus

We consider a billiard in the punctured torus obtained by removing a small disk of radius >0 from the flat torus ², with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity only. Let (), and respectively R(), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when ² is being identified with [0,1)². We prove that the probability measures on [0, ) associated with the random variables and R are weakly convergent as 0⁺ and explicitly compute the densities of the limits. Research partially supported by ANSTI grant C6189/2000.     

Kinetic equation in the kinetic region of the dilute and nonuniform electron plasma

Mori's scaling method is used to derive the kinetic equation for a dilute, nonuniform electron plasma in the kinetic region where the space-time cutoff (b, t _c) satisfies _Dbl _f , _D t _{c f} , with _D the Debye length, _D ^–1= _p the plasma frequency, andl _f and _f the mean free path and time, respectively. The kinetic equation takes account of the nonuniformity of the order ofl _f and _D for the single-and the two-particle distribution function, respectively. Thus the Vlasov term associated with the two-particle distribution function is retained. This kinetic equation is deduced from the kinetic equation in the coherent region obtained by Morita, Mori, and Tokuyama, where the space-time cutoff of the coherent region satisfies _Dbr ₀, _Dt _{c 0}, withr ₀ the Landau length and ₀ the corresponding time scale.     

Logical independence in quantum logic

The projection latticesP(₁),P(₂) of two von Neumann subalgebras ₁, ₂ of the von Neumann algebra are defined to be logically independent if A B0 for any 0AP(₁), 0BP(₂). After motivating this notion in independence, it is shown thatP(₁),P(₂) are logically independent if ₁ is a subfactor in a finite factor andP(₁),P(₂ commute. Also, logical independence is related to the statistical independence conditions called C^*-independence W^*-independence, and strict locality. Logical independence ofP(₁,P(₂ turns out to be equivalent to the C^*-independence of (₁,₂) for mutually commuting ₁,₂ and it is shown that if (₁,₂) is a pair of (not necessarily commuting) von Neumann subalgebras, thenP(₁,P(₂ are logically independent in the following cases: is a finite-dimensional full-matrix algebra and ₁,₂ are C^*-independent; (₁,₂) is a W^*-independent pair; ₁,₂ have the property of strict locality.     

Direct observation of the second-order Stark effect of theX-B transition in molecular iodine

The second-order Stark shift of the components of the hyperfine structure of the transition¹ _g ⁺ ( = 0,j = 13, 15) ³ _ou ⁺ ( = 43,j = 12, 16) (of molecular iodine have been studied by means of saturated absorption spectroscopy in an external cell with the I₂ vapour located in an electric field. The anisotropic polarizabilities of the upper and lower levels together with the difference between the isotropic polarizabilities of the levels of the transition have been obtained.     

The modular group and super-KMS functionals

Every normal, faithful, self-adjoint functional on a von Neumann algebraA canonically determines a one-parameter-weakly continuous *-automorphism group (the analog of the modular group) and a canonical ₂ grading onA, commuting with . We show that the functional satisfies the weak super-KMS property with respect to and Furthermore, we prove that and are the unique pair of a-weakly continuous one-parameter *-automorphism group and a grading of the algebra, commuting with each other, with respect to which is weakly super-KMS. The above results thus provide a complete extension of the theory of Tomita and Takesaki to the nonpositive case.Supported in part by the National Science Foundation under Grant DMS-8922002.     

One-dimensional harmonic lattice caricature of hydrodynamics

We derive the hydrodynamic (Euler) approximation for the harmonic time evolution of infinite classical oscillator system on one-dimensional lattice ¹ It is known that equilibrium (i.e., time-invariant attractive) states for this model are translationally invariant Gaussian ones, with the mean 0, which satisfy some linear relations involving the interaction quadratic form. The natural parameter characterizing equilibrium states is the spectral density matrix function (SDMF)F(), [– , ). Time evolution of a space profile of local equilibrium parameters is described by a space-time SDMFF(t;x, ) t, xR ¹. The hydrodynamic equation forF(t; x, ) which we derive in this paper means that the normal mode profiles indexed by are moving according to linear laws and are mutually independent. The procedure of deriving the hydrodynamic equation is the following: We fix an initial SDMF profileF(x, ) and a familyP ,>0 of mean 0 states which satisfy the two conditions imposed on the covariance of spins at various lattice points: (a) the covariance at points close to the value ^–1 x in the stateP is approximately described by the SDMFF(x, ); (b) The covariance (on large distances) decreases with distance quickly enough and uniformly in. Given nonzerotR ¹, we consider the states P _–1 ,>0, describing the system at the time moments ^–1 t during its harmonic time evolution. We check that the covariance at lattice points close to ^–1 x in the state P _–1 is approximately described by a SDMFF(t;x, ) and establish the connection betweenF(t; x, ) andF(x,).     

The lanthanideβ″ aluminas

The entire sodium ion content of sodium alumina (Na_1.67Mg_0.67Al_10.33O₁₇) can be replaced with a variety of lanthanide ions by simple diffusion reactions at moderate temperatures (500–700°C). Lanthanide alumina crystals are hard, clear, chemically stable, and have well-defined crystal structures. The fluorescence spectrum of Nd³⁺ in alumina is similar to that in YAG. The lifetime of the⁴ F _3/2 state of Nd³⁺ in completely-exchanged alumina (350s at 10²¹ Nd³⁺ cm^–3) is about 45% longer than in YAG (240s at 10²⁰Nd³⁺ cm^–3). The lanthanide aluminas may be of considerable interest as new phosphor and laser host materials.     

A discrete spectrum of solutions of the wave equation with strong cubic nonlinearity

The nonlinear wave equation, _tt –+³=0, has many solutions that are periodic in time and localized in space, all with infinte energies. The search for spherically symmetric solutions that are well represented by the simple approximation, (r, t)A(r) sin t, leads to a discrete spectrum of solutions{ _N (r, t; )}. The solutions are nonlinear wavepackets, and they can be regarded as particles. The asymptotic theory () of the motion of the guiding center of theNth wavepacket, in the presence of a specified potential, is characterized by an infinite mechanical mass and an infinte interaction mass, and they are compatible. The rest mass in the classical relativistic mechanics of guiding centers ism ₀ c ²= _N ; i.e. the spectrum { _N } determines a spectrum of Planck's constants.On leave (1972–73) Université de Paris VI, Département de Mécanique, 75 Paris 5^e, France.