Different Orthocomplementations on the Subspace Lattice of a Finite-Dimensional Complex Vector Space

In this paper we prove, by using real closed fields and model theory, the followingresult: for any integer n 3, there exist, on the lattice of all subspaces on thevector space C ⁿ , 2(^2K0)orthocomplementations leading to nonisomorphic structuresof orthomodular lattices.     

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 .     

Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem

We prove that theq-state Potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) wheneverq>2r. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay forq7), triangular lattice (q11), hexagonal lattice (q4), and Kagomé lattice (q6). The proofs are based on the Dobrushin uniqueness theorem.     

On the extremality of the disordered state for the Ising model on the Bethe lattice

We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k2 is extremal if and only if is less or equal to the spin glass transition value, given by tanh( _c ^SG = 1/k.The work was partially supported by the NSF grant DMS 9504513.     

Strong electric field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential

We consider the Schrödinger operator with electric potential V, which decays at infinity, and magnetic potential A. We study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated under the essential-spectrum lower bound. We concentrate on the cases of rapidly decaying V (e.g. V L ^m/2( ^m ) for m 3) and arbitrary A, or slowly decaying V (i.e. V(x |x|^– , (0,2), as |x| ) and magnetic potentials A corresponding to constant magnetic fields B = curl A.Partially supported by the Ministry of Culture, Science and Education under Grant No. 52.     

Characterizations of classical and quantum logics

In classical logic (Boolean algebras) probability systems involving correlations are fully characterized by the system of generalized Bell inequalities. On the other hand, probability systems with pairwise correlations on orthomodular lattices (OML) representing quantum logics are so general that the only inequalities that hold universally are the trivial inequalities 0p _i1, 0p _ijmin {p _i,p _j}. In this paper it is shown that every correlation sequence p=(p₁,...,p _n,...,P _ij,...) satisfying the above inequalities can be represented by a probability measure on an orthomodular latticeL admitting a full set of {0,1}-valued probability measures with the additional property that isL ortho-Arguesian.     

Gap exponents for percolation processes with triangle condition

A study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region. It is show that the gaps are given by _t =2 fort=2, 3,. Scaling theory predicts thatP _p (¦C ₀¦S(p))–(p _c –p) andE _p (1/¦C ₀¦; ¦C ₀¦S(p))–(p _c –p)³, whereS(p) is the typical cluster size. It is found that (p _c –p)P _p (|C ₀S(p) ^1–)(p _c –p)^1–2 and (p _c –p)³E _p (1/|C ₀|;|C ₀|S(p) ^1–))(p _c –p)^3–4.     

Lower bounds on the width of Stark-Wannier type resonances

We prove that the Schrödinger operator –d ²/dx²+Fx+W(x) onL ²(R) withW bounded and analytic in a strip has no resonances in a region ImE–exp(–C/F).     

Thermodynamic formalism and localization in Lorentz gases and hopping models

The thermodynamic formalism expresses chaotic properties of dynamical systems in terms of the Ruelle pressure (). The inverse-temperature-like variable allows one to scan the structure of the probability distributin in the dynamic phase space. This formalism is applied here to a lorentz lattice gas. where a particle moving on a lattice of sizeL ^d collides with fixed scatterers placed at random locations. Here we give rigorous arguments that the Ruelle pressure in the limit of infinite systems has two branches joining with a slope discontinuity at =1. The low- and high- branches correspond to localization of trajectories on respectively the most chaotic (highest density) region and the most deterministic (lowest density) region, i.e. () is completely controlled by rare fluctuations in the distribution of scatterers on the lattice. and it dose not carry and information on the global structure of the static disorder. As approaches unity from either side, a localization-delocalization transition leads to a state where trajectories are extended and carry information on transprot properties. At finiteL the narrow region around =1 where the trajectories are extended scales as (InL)^–2. where depends on the sign of 1–, ifd>1, and as (L InL)^–1 ifd=1. This result appears to be general for diffusive systems with static disorder, such as random walks in random environments or for the continuous Lorentz gas. Other models of random walks on disordered lattices, showing the same phenomenon, are discussed.     

On the ground-state threshold in random two-dimensional Ising ±J models

For square, triangular, and for hexagonal lattices there is numerical and theoretical support that the ground-state thresholdp _c between ferro- and paramagnetism in random 2D Ising ±J models, withp as the concentration of antiferromagnetic bonds, is identical top ^*which is characterized by minimal matching properties of frustrated plaquettes. From square lattices of size 100×100 we have got p_c,sq<0.117 by simulations which produced average groundstate magnetizations per spin by means of exact minimal matchings. Moreover, from the squareL×L-lattices treated (L = 10, 20, 50, 100) we obtained the estimatep _c,sq 0.1 which is in agreement with the Grinstein estimatep _c,sq 0.099 andp _c,sq 0.105 by Freund and Grassberger.     

Compositions of Sasaki projections

In an orthomodular lattice (abbreviated OML) L, a Sasaki projection is a mappinga _x(a)=x(x va) fromL toL, wherexL. We study compositions of finite numbers of Sasaki projections and of the same Sasaki projections composed in inverse order. By using the Baer-semigroup of all finite compositions of Sasaki projections, we establish a new characterization of kernels of congruences in OMLs and a generalization of the parallelogram law for dimension OMLs. Our results are also related to quantum measurements via Pool's definition of the change of the support of a state after a measurement.     

Hamilton Spaces of Order k ≥ 1

A suitable dual for the k-acceleration bundle(T ^k M, ^k ,M) is the fiberedbundle (T ^k–1 M× _M T*M). The mentioned bundle carries a canonicalpresymplectic structure and k canonical Poisson structures. By means of thisdual we define the notion of Hamilton spaces of orderk, whose total spaceconsists of points x of the configuration spaceM, accelerations of order 1,...,k – 1, y ⁽¹⁾,...,y ^(k–1), and momenta p. Some remarkable Hamiltonian systemsare pointed out. There exists a Legendre mapping from the Lagrange spaces oforder k to the Hamilton space of order k.     

SL(n+1, C) strata and orbits in the solution space of Euclidean CP n models

We further study the action of SL(n+1, C) on the space of finite action solutions of the bidimensional Euclidean CP ⁿ models. We decompose the space of k-instantons into strata. Each stratum in characterized by an integer m with 0mmin(k, n) which can be calculated from the instanton by purely algebraic means. The k-instantons with m=n are called generic. Their stratum is shown to be dense in the space of k-instantons when kn. The isotropy subgroups for each stratum are identified.

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Résumé Nous poursuivons l'étude de l'action de SL(n+1, C) sur l'espace des solutions à action finie du modèle CP ⁿ sur l'espace euclidien bi-dimensionnel. L'espace des k-instantons est décomposé en strates. Chaque strate est caractérisée par un entier m tel que 0 mmin(k, n) et qui peut être calculé à partir de l'instanton par des méthodes purement algébriques. Les k-instantons avec m=n sont dits génériques. Leur strate est dense dans l'espace des k-instantons (lorsque k n). Les sous-groupes d'isotropie de chacune des strates sont identifiés.

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Supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Fonds FCAR pour l'aide et le soutien à la recherche.     

Factoriality of <Emphasis Type="Italic">q</Emphasis>-Gaussian von Neumann Algebras

We prove that the von Neumann algebras generated by n q-Gaussian elements, are factors for n2.     

Kinetic formulation of the isentropic gas dynamics andp-systems

We consider the 2×2 hyperbolic system of isentropic gas dynamics, in both Eulerian or Lagrangian variables (also called thep-system). We show that they can be reformulated as a kinetic equation, using an additional kinetic variable. Such a formulation was first obtained by the authors in the case of multidimensional scalar conservation laws. A new phenomenon occurs here, namely that the advection velocity is now a combination of the macroscopic and kinetic velocities. Various applications are given: we recover the invariant regions, deduce newL estimates using moments lemma and proveL –w* stability for 3.     

Conformational completion of Lorentz symmetric spaces

Every simply connected Lorentz symmetric space of dimensionn3 has a symmetric quotient which can be conformally imbedded in a quadric of the projective space ⁿ⁺¹ (R).     

Crossover finite-size scaling at first-order transitions

In a recent paper we developed a method which allows one to control rigorously the finite-size behavior in long cylinders near first-order phase transitions at low temperature. Here we apply this method to asymmetric transitions with two competing phases, and to theq-state Potts model as a typical model of a temperature-driven transition, whereq low-temperature phases compete with one high-temperature phase. We obtain the finite-size scaling of the firstN eigenvalues (whereN is the number of competing phases) of the transfer matrix in a periodic box of volumeL × ... ×L ×t, and, as a corollary, the finite-size scaling of the shape of the order parameter in a hypercubic box (t=L), the infinite cylinder (t=), and the crossover regime from hypercubic to cylindrical scaling. For the two-phase case (N=2 we find that the crossover length _L is given by O(L^w)exp(L^v), where is the inverse temperature, is the surface tension, and w=1/2 if v+1=2 whilew=0 if v+1 >2. For the standard Ising model we also consider free boundary conditions, showing that _L=exp[L^v+O(L^v– 1)] for any dimension v+12. For v+1=2 we finally discuss a class of boundary conditions which interpolate between free (corresponding to the interpolating parameter g=0) and periodic boundary conditions (corresponding to g=1), finding that _L=O(L^w)exp(L ^v) withw=0 forg=0 andw=1/2 for 0<g1.     

Stimulated IR- and vacuum-UV emission following two-photon-excitation of molecular hydrogen using an ArF laser

Hydrogen molecules are excited by two 193 nm-photons [X¹ _g ⁺ E,F¹ _g ⁺ ;Q₀₂(1)] generated with a commerical oscillator-amplifier ArF laser. Stimulated emission is subsequently observed in the IR (=752.4 nm to 836.6 nm;E, FB) and in the VUV (=134.0 nm to 160.9 nm;BX) with a VUV conversion efficiency of 0.2%. The rotational fine-structure, the saturation behaviour, pump depletion as well as SRS-onset on and off two-photon resonance are investigated.     

Expansion of the frequency criterion of absolute stability for controllable systems with one nonlinear nonstationary element

It is demonstrated that for nondegenerate systems having a nonstationary nonlinearity the frequency condition k^–1 + Re W(i) 0 (k < ) ensures exponential absolute stability.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 3, pp. 319–322, March, 1972.