New look at Gleason's theorem for signed measures

It is shown that the Gleason theorem holds not only for a finite but also for ann-finite signed measurem, wheren is a cardinal, defined on all closed subspaces of a Hilbert space whose dimension is a nonmeasurable cardinal 2, ifm is bounded from below on all one-dimensional subspaces.     

Random walk with persistence

The exact analytic result is obtained for the Fourier transform of the generating functionF(R,s)= _n=0 s ⁿ P(R,n), whereP(R,n) is the probability density for the end-to-end distanceR inn steps of a random walk with persistence. The moments R ²(n), R ⁴(n), and R ⁶(n) are calculated and approximate results forP(R,n) and R ^–1(n) are given.     

Static anisotropic and isotropic electric dipole polarizabilities of Na n F n-1 clusters

Recently, electric susceptibilities of Na_nF_n-1 clusters have been measured by deflecting a molecular beam in an inhomogeneous electric field. The analysis of the deflection of a cluster by the electric field needs the calculation of the electric properties. We present the calculation of the static anisotropic and average dipolar polarizabilities within a model in which the Na_nF_n-1 clusters ( 2 n 23) are treated as one electron embedded in the field of n ions Na⁺ and of n - 1 ions F^-. The accuracy of the results is evaluated and discussed on small clusters (n 5) in comparison with ab initio calculations. The relationship between the polarizabilities, the electronic localization and the geometric structure is discussed. Received 10 September 2002 Published online 3 July 2003     

The Action of √−Δ on Weighted Sobolev Spaces

The action of – on distributions is examined within the context of weighted Sobolev spaces. The results obtained are as follows: (1) – is a continuous map of R ⁿ ), the space of rapidly decreasing functions, to L ^{2, s} (R ⁿ for any s < n/2 +1; (2) if k R and s > –n/2 – 1, then – is a continuous map from H ^{k, s} (R ⁿ ), the weighted Sobolev space, to H ^{k–1, t} (R ⁿ for some t. The results are optimal in a sense.     

Crossover scaling function for the free energy

A cubic field, coupling tos|s|², inn-component spin models induces a bicritical crossover fromn-isotropic to Ising like (m=1) critical behaviour for 1<n<, but to classical behaviour in the limitn. By following the analysis of Nelson and Domany, the bicritical scaling function for the free energy ind dimensions is obtained correct to order =4–d and for general (m,n). The mechanism responsible for the breakdown of hyperscaling in the classical behaviour is discussed.     

A theorem on the first heteroclinic tangency in two-dimensional maps. Orientation-preserving cases

Using the properties of the Jordan curve, the following theorem on the heteroclinic tangency in orientation-preserving two-dimensional maps is proved: LetT :R ² R ² be a one-parameter family ofC ¹ diffeomorphisms andJ=DetDT be such that 0<J1 or 1J<. LetW _u ⁿ be the unstable manifold of a hyperbolicn-cycle andW _s ^m the stable manifold of a hyperbolicm-cycle. Suppose that for< _c ,W _u ⁿ andW _s ^m have no common points, and that for> _c ,W _u ⁿ andW s/m have a transversal heteroclinic point. Then at= _c ,W _u ⁿ andW _s ^m are in the first asymptotic heteroclinic tangency except for the following three cases: (1)n=m; both cycles are without reflection. (2)m=2n; then- andm-cycles are with and without reflection, respectively; (3)n=2m; then- andm-cycles are without and with reflection, respectively.     

Phase uniqueness and correlation length in diluted-field Ising models

The diluted-field Ising model, a random nonnegative field ferromagnetic model, is shown to have a unique Gibbs measure with probability I when the field mean is positive. Our methods involve comparisons with ordinary uniform field Ising models. They yield as a corollary a way of obtaining spontaneous magnetization through the application of a vanishing random magnetic field. The correlation lengths of this model defined as (lim _n-(1/n) log ₀; _n)^-1, wheren is the site on the first coordinate axis at distancen from the origin and ₀; _n is the origin ton two-point truncated correlation function, is non-random. We derive an upper bound for it in terms of the correlation length of an ordinary nonrandom model with uniform field related to the field distribution of the diluted model.     

Quantum-Logics-Valued Measure Convergence Theorem

In this paper, the following quantum-logic valued measure convergence theorem is proved: Let (L ₁, 0, 1) be a Boolean algebra, (L ₂, , , 0, 1) be a quantum logic and { _n : n N} be a sequence of s-bounded (L ₂, , , 0, 1)-valued measures which are defined on (L ₁, 0, 1). If for each a (L ₁, 0, 1), { _n (a)} _{n N} is an order topology Cauchy sequence, when {v(a)} convergent to 0, { _n (a)} is order topology convergent to 0 for each n N, where v is a nonnegative finite additive measure which is defined on (L ₁, 0, 1), then when {v(a)} convergent to 0, { _n (a)} are order topology convergent to 0 uniformly with respect to n N.     

Instability of the paramagnetic state towards incommensurate magnetic order in the 2-d Hubbard model

We determine the instability line separating the paramagnetic phase in the phase diagram of the 2-d Hubbard Model from a phase with incommensurate magnetic order. A mean-field approximation of the Kotliar-Ruckenstein slave boson representation is used to calculate the wave-vector dependent magnetic susceptibility. For largeU/t the instability occurs at a densityn0.37, and a wave-vector close toq=(0,). The dependence ofq onU andn is also given.     

Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity

The computational complexity of internal diffusion-limited aggregation (DLA) is examined from both a theoretical and a practical point of view. We show that for two or more dimensions, the problem of predicting the cluster from a given set of paths is complete for the complexity class CC, the subset of P characterized by circuits composed of comparator gates. CC-completeness is believed to imply that, in the worst case, growing a cluster of size n requires polynomial time in n even on a parallel computer. A parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths, and then corrects defects in the guessed cluster through a nonlocal annihilation process. The parallel running time of the relaxation algorithm for two-dimensional internal DLA is studied by simulating it on a serial computer. The numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n. Thus the computational resources needed to grow large clusters are significantly less on average than the worst-case analysis would suggest. For a parallel machine with k processors, we show that random clusters in d dimensions can be generated in ((n/k+logk)n ^2/d ) steps. This is a significant speedup over explicit sequential simulation, which takes (n ^1+2/d ) time on average. Finally, we show that in one dimension internal DLA can be predicted in (logn) parallel time, and so is in the complexity class NC.     

Charge accumulation layer on the <Emphasis Type="Italic">n</Emphasis>-GaN (0001) surface with ultrathin Ba coatings

The adsorption of Ba on the n-type GaN(0001) surface is studied. It is found that submonolayer Ba coatings induce cardinal changes in the electronic properties of the surface with the formation of a charge accumulation layer in the region of the near-surface band bending. The excitation of the Ba/n-GaN system by light from the region of GaN transparency results in photoemission. The lowest value of the work function corresponds to ～1.90 eV at a Ba coverage of ～0.4 ML. Two surface bands induced by Ba adsorption are found in the surface photoemission spectra.     

Some Questions of the Nevanlinna Theory for the Complex Half-Plane

In this paper we give variants of the logarithmic derivative lemma and the second main theorem for the complex half-plane. Here the second main theorem is a special case of a theorem concerning polynomial functions of the coordinates of a holomorphic curve f : C ₊ P ⁿ . This theorem is a variant for C ₊ of a theorem of A. Eremenko and M. Sodin, who considered an entire curve f : C ₊ P ⁿ . The proofs are realized because of the introduction of natural classes of subharmonic, -subharmonic, holomorphic, meromorphic functions in C ₊ . Such functions we shall call just functions.     

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.     

Propagation and spreading of unstable perturbations in electron-hole plasmas subject to crossed fields

A model is developed for electrostatic drift instabilities which arise in inhomogeneous electron-hole plasmas subject to crossed fieldsE ₀B ₀. The instabilities are initiated by gradients in the equilibrium plasma densityn ₀. Using two-fluid magnetohydrodynamics and linear perturbation theory the dispersion relation of local density oscillations is calculated for arbitrary inhomogeneous equilibrium distributions and plasma densities. For cases wheren ₀B ₀ andn ₀ ⁺ n ₀ ^- it is found that the propagation direction of maximal gain, , is the bisectrix of the angle between (–n ₀) and (E ₀ ×B ₀) and that stable and unstable configurations are distinguished by the angle between ₀ andn ₀. A local density perturbation built by superpositions of the plane waves, and initially chosen radially symmetric, broadens unisotropically. In the direction transverse to the broadening is anomaleously enhanced, as compared to the broadening by diffusion in the stable case. The results are referred to experimental observations of low-frequency instabilities reported in [1].     

Surface tension,step free energy,and facets in the equilibrium crystal

Some aspects of the microscopic theory of interfaces in classical lattice systems are developed. The problem of the appearance of facets in the (Wulff) equilibrium crystal shape is discussed, together with its relation to the discontinuities of the derivatives of the surface tension (n) (with respect to the components of the surface normaln) and the role of the step free energy ^step(m) (associated with a step orthogonal tom on a rigid interface). Among the results are, in the case of the Ising model at low enough temperatures, the existence of ^step(m) in the thermodynamic limit, the expression of this quantity by means of a convergent cluster expansion, and the fact that 2^step(m) is equal to the value of the jump of the derivative / (when varies) at the point =0 [withn=(m ₁ sin ,m ₂ sin , cos )]. Finally, using this fact, it is shown that the facet shape is determined by the function ^step(m).     

Spectroscopic manifestations of phase changes in CaAr n clusters: finite size effects

The finite temperature optical spectroscopy of CaAr_n clusters in the range 6 n 146 is investigated using a Diatomics-In-Molecule (DIM) Hamiltonian and classical parallel tempering Monte Carlo simulations. The absorption spectrum is calculated in the vertical approximation at various temperatures between 2 K and 50 K. Several typical situations are reported. CaAr₆ shows a strong thermal broadening and shift of the spectral lines, possibly associated with isomerization. CaAr₁₃ only shows some broadening. CaAr₃₇ exhibits features corresponding to coexisting isomers at low temperature. Finally, the abrupt changes in the absorption spectrum in CaAr₁₄₆ at about 20 K are indicative of surface diffusion.     

Toeplitz algebras and Rieffel deformations

We establish a representation theorem for Toeplitz operators on the Segal-Bargmann (Fock) space ofC ⁿ whose symbols have uniform radial limits. As an application of this result, we show that Toeplitz algebras on the open ball inC ⁿ are strict deformation quantizations, in the sense of M. Rieffel, of the continuous functions on the corresponding closed ball.     

Critical dynamics of φ2n with random fields in the tree approximation

In the tree approximation, the critical relaxational dynamics of ²ⁿ with quenched random fields is calculated by renormalization near the upper critical dimensiond _u=4+2/(n–1). Although the relation between the characteristic frequency _c q and the wave-numberq is as usual,unusual results are obtained for the asymptotic decay of the order-parameter correlation functionS(q,t).Dedicated to Prof. Dr. W. Döring on the occasion of his 75th birthday     

Hyperbolic initial-boundary-value problem for magnetic flux compression by plane liners

Based on the (relativistic) Maxwell equations with displacement current E/t, the initial-boundary-value problem for the compression of an initially homogeneous magnetic fieldB={0,B(x,t),0} between a fixed liner atx=0 and a detonation-driven liner atx=s(t) is solved analytically. By homogenizing the boundary conditions at the moving boundary, the transient electromagnetic fields are shown to be a superposition of quasistatic elliptic (E/t=0) and hyperbolic (E/t0) wave solutions. The wave equation is solved by a Fourier expansion in time-dependent eigenfunctionsf _n =f _n [nx/s(t)] for the variable region 0xs(t), where the Fourier amplitudes _n (t) are determined by coupled differential equations of second order. It is concluded that the conventional elliptic flux compression theories (E/t=0) hold approximately for nonrelativistic liner speeds , whereas the hyperbolic theory (E/t0) is valid for arbitrary liner speeds .     

Differential Equations Compatible with KZ Equations

We define a system of dynamical differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra g. These are equations on a function of n complex variables z _i taking values in the tensor product of n finite dimensional g-modules. The KZ equations depend on the dual variable in the Cartan subalgebra of g. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions.