Differential calculus onISO q(N), quantum Poincaré algebra and q-gravity

We present a general method to deform the inhomogeneous algebras of theB _n,C_n,D_n type, and find the corresponding bicovariant differential calculus. The method is based on a projection fromB _n+1,C_n+1,D_n+1. For example we obtain the (bicovariant) inhomogeneousq-algebraISO _q(N) as a consistent projection of the (bicovariant)q-algebraSO _q(N=2). This projection works for particular multiparametric deformations ofSO(N+2), the so-called minimal deformations. The case ofISO _q(4) is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameterq. The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains theclassical Lorentz algebra. Only the commutation relations involving the momenta depend onq. Finally, we discuss aq-deformation of gravity based on the gauging of thisq-Poincaré algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.     

Classification of bicovariant differential calculi on quantum groups

Suppose thatq is not a root of unity. We classify all bicovariant differential calculi of dimension greater than one on the quantum groupsGL _q (N),O _q (N) andSp _q (N) for which the differentials du _j ⁱ of the matrix entriesu _j ⁱ generate the left module of first order forms. Our first classification theorem asserts that there are precisely two one-parameter families of such calculi onGL _q (N) forN3. In the limitq1 only two of these calculi give the ordinary differential calculus onGL(N). Our second main theorem states that apart from finitely manyq there exist precisely two differential calculi with these properties onO _q (N) andSp _q (N) forN4. This strengthens the corresponding result proved in our previous paper [SS2]. There are four such calculi onO _q (3). We introduce two new 4-dimensional bicovariant differential calculi onO _q (3).     

Higher order differential calculus on SL q(N)

Let be a bicovariant first order differential calculus on a Hopf algebra . There are three possibilities to construct a differential N ₀-graded Hopf algebra which contains as its first order part. In all cases is a quotient = /J of the tensor algebra by some suitable ideal. We distinguish three possible choices _u J, _s J, and _W J, where the first one generates the universal differential calculus (over ) and the last one is Woronowicz' external algebra. Let q be a transcendental complex number and let be one of the N ²-dimensional bicovariant first order differential calculi on the quantum group SL _q(N). Then for N 3 the three ideals coincide. For Woronowicz' external algebra we calculate the dimensions of the spaces of left-invariant and bi-invariant k-forms. In this case each bi-invariant form is closed. In case of 4D _± calculi on SL _q(2) the universal calculus is strictly larger than the other two calculi. In particular, the bi-invariant 1-form is not closed.     

Renormalization group study of theA+B→⊘ diffusion-limited reaction

TheA+B diffusion-limited reaction, with equal initial densitiesa(0)=b(0)=n ₀, is studied by means of a field-theoretic renormalization group formulation of the problem. For dimensiond>2 an effective theory is derived, from which the density and correlation functions call be calculated. We find the density decays in time as ford<4, with =n ₀ –Cn ₀ ^d/2 +..., whereC is a universal constant andC is nonuniversal. The calculation is extended to the case of unequal diffusion constantsD _A D _B , resulting in a new amplitude but the same exponent. Ford2 a controlled calculation is not possible, but a heuristic argument is presented that the results above give at least the leading term in an =2–d expansion. Finally, we address reaction zones formed in the steady state by opposing currents ofA andB particles, and derive scaling properties.     

Scaling function for energy-density correlations: Dispersion theory andε-expansion forT>T c

A dispersion representation for the static energy-density correlation function ² (q) ²(–q) _c =C(q,T)=A+Bt ^–h(z ²), wherez=q , t=(T—T)_c/T _c and is the correlation length, is discussed.h(z ²) is calculated to order ² in the zero-field critical region (T>T _c) for the standard isotropicn-component ⁴Ginzburg-Landau-Wilson model. Utilizing a procedure similar to that introduced by Bray for the two-point correlation function, the-expansion results are used in conjunction with an approximant for the spectral functionF(z/2) Imh(—z ²) based on the asymptotically exact short-distance expansion resulth ^–1(z ²)z ^/v[D ₀+D ₁ z ^{–(1 —)/v} +D ₂ z ^–1/v ] to predict quantitatively the full momentum dependence ofC(q,T) forT>T _c. In contrast to the two-point correlation function,C(q,T) is found to be a monotonic function as the critical temperature is approached at fixedq (forT>T _c).     

Realizability of a model in infinite statistics

Following Greenberg and others, we study a space with a collection of operatorsa(k) satisfying the q-mutator relationsa(l)a (k)a(l)= _k,l (corresponding forq=±1 to classical Bose and Fermi statistics). We show that then!×n! matrixA _n (q) representing the scalar products ofn-particle states is positive definite for alln ifq lies between –1 and +1, so that the commutator relations have a Hilbert space representation in this case (this has also been proved by Fivel and by Bozejko and Speicher). We also give an explicit factorization ofA _n (q) as a product of matrices of the form(1–q ^jT)^±1 with 1jn andT a permutation matrix. In particular,A _n (q) is singular if and only ifq ^M=1 for some integerM of the formk ²–k, 2kn.     

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.     

On the survival probability of a random walk in a finite lattice with a single trap

We consider the survival of a random walker in a finite lattice with periodic boundary conditions. The initial position of the random walker is uniformly distributed on the lattice with respect to the trap. We show that the survival of a random walker, U _n>, can be exactly related to the expected number of distinct sites visted on a trap-free lattice by U _n=1–S _n/N ^D (*) whereN ^D is the number of lattice points inD dimensions. We then analyze the behavior of S_n in any number of dimensions by using Tauberian methods. We find that at sufficiently long times S _n decays exponentially withn in all numbers of dimensions. InD = 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid forN ²N 1 whenD = 1 andN lnN n 1 whenD = 2. No such crossover exists when Z3. The form of (*) suggests that the single trap approximation is indeed a valid low-concentration limit for survival on an infinite lattice with a finite concentration of traps.     

Complex quantum group,dual algebra and bicovariant differential calculus

The method used to construct the bicovariant bimodule in ref. [CSWW] is applied to examine the structure of the dual algebra and the bicovariant differential calculus of the complex quantum group. The complex quantum group Fun _q (SL(N, C)) is defined by requiring that it contains Fun _q (SU(N)) as a subalgebra analogously to the quantum Lorentz group. Analyzing the properties of the fundamental bimodule, we show that the dual algebra has the structure of the twisted product Fun _q (SU(N))Fun _q (SU(N)) _reg ^* . Then the bicovariant differential calculi on the complex quantum group are constructed.     

Linking numbers of closed manifolds at random in ℝ n,inductances and contacts

We give here new results of topology and integral geometry concerning the Gauss linking number I of closed manifolds inn-dimensional space. The rigid manifolds have arbitrary shapes and dimensions, and are statistically at random positions in ⁿ . Generalizing Pohl's work, for two closed manifoldsC ₁ ^r ,C ₂ ^s , of respective dimensionsr ands, with 0rn–1, andr+s+1=n, we consider the kinematic linking integralI=<I ²(x,O)d ⁿ x>, of the square linking number I ofC ₁ ^r andC ₂ ^s , over the group of Euclidean motions of one manifold (translationsx, rotationsO). Introducing a new tensorial method, and using group theory, we show quite generally thatI=num. fact. , where is a length variable and whereA , (=1, 2) are characteristic functions associated with the manifoldC only. We study functionsA and of a manifoldC ^r , of dimensionr, in all cases 0rn–1.A always exists.A(0) givesC's area, whereas equals the interior volume of a hypersurfaceC. is found to exist and not to vanish only if 2 dimC+1=n andn=3+4q=3, 7, 11 ...A and are explicitly calculated for segments andr-spheresS ^r . As an application the topological excluded volume of a gas of nonlinked spheresS ^r moving in ^2r+1 is calculated. We generalize toN manifoldsC , =1, ...,N, linked successively to each other and forming a ring. The cyclic product of their linking numbers is integrated over the group of motions of the manifolds. It is shown to factorize completely in Fourier space, with special algebraic rules, over the set of 2N characteristic functionsA , , associated with theC 's. The same algebra of characteristic functions is shown to describe a larger class of topology and electromagnetism properties: a new theorem is given for a family of Euclidean group integrals involving the random linking numbers, mutual inductances and contact distributions ofN manifolds.     

Geometric expansion of the boundary free energy of a dilute gas

We consider a dilute classical gas in a volume ^–1 which tends to ^d by dilation as 0. We prove that the pressurep(^–1) isC ^q in at =0 (thermodynamic limit), for anyq, provided the boundary isC ^q and provided the Ursell functionsu _n (x ₁, ...,x _n) admit moments of degreeq and have nice derivatives.     

Spin glasses with long-range interaction at high temperatures

We study the Ising andN-vector spin glasses with exchange couplings J=(J _ij ;i, jZ ^d ), which are independent random variables with EJ_ij=0 andEJ ⁿ _ij ⁿ n!¦i–j¦ ^–nd , forn, some finite constant >0, and >1/2. For sufficiently small, we show that forE-a.a.J there is a weakly unique, extremal, infinite-volume Gibbs measure _J for which the expectation of a single (component of) spin vanishes and which has the cluster property inL ₂(E) with the same decay as interaction. This work is based on results and methods of Fröhlich and Zegarlinski.     

Radiactiv corrections to deeply virtual compton scattering

The non-commuting matrix elements of matrices from the quantum group GL _q(2;C) with q = being the n-th root of unity are given a representation as operators in Hilbert space with help of C ₄ ⁽ⁿ⁾ generalized Clifford algebra generators.The case of q C, |q| = 1 is treated parallelly.     

Generalized q-Modified Laguerre Functions

In this paper we define a new q-special function A _n (x, b, c; q). The new function is a generalization of the q-Laguerre function and the Stieltjes–Wigert function. We deduced all the properties of the function A _n (x, b, c; q). Finally, lim _q1 A _n ((1 – q)x, –, 1;q) gives L _n ^(,)(x,q), which is a -modification of the ordinary Laguerre function.     

Chemical reactions as dynamical systems on the interval

We consider the most general chemical reaction of the typen ₁ A ₁+...+n _N A _N m ₁ B ₁+...+m _M B _M whereN,M1,n ₁,...,n _N andm ₁,...,m _M are positive integers defining the stoichiometry, andA ₁,...,A _N andB ₁,...,B _M are the names of chemicals or ions. We assume that _i=1 ^N n_i= _j=1 ^M m_j. The time evolution of the concentrations is given by the law of mass action and leads to a dynamical system (with discrete or continuous time) which is governed by a polynomial map of the interval [B, C], where B0 and C1. We define the physically meaningful range for the parameters of the map, and we show that, within such a range, the map has a unique fixed point, which is stable and a global attractor, with the exception of one particular case, where bifurcation is observed.     

Critical exponentβ for hyperfine fields in nickel

The magnetic hyperfine fields for⁶³Ni,⁶⁶Cu, and⁶⁷Zn nuclei in nickel metal have been measured by means of perturbed-ray angular distribution techniques at different temperatures up to 1 K below the Curie temperature,T _C . The temperature dependence of the fields can be very well fitted by (1—T/T _C ) with best values=0.322(16) for⁶³NiNi, = 0.427(42) for⁶⁶CuNi, and=0.427(14) for⁶⁷ZnNi respectively. The differences between these exponents indicate that there could be probe atom dependent deviations from proportionality between hyperfine field and bulk magnetization in the critical region.Work performed in partial fulfillment of the requirements for a doctorate in physics at the Freie Universität, Berlin     

Acoustomagnetoelectric effects at meso-ultrasonic frequencies in anisotropic semiconductors in arbitrary (classical) magnetic fields

Anisotropic acoustomagnetoelectric (AME) effects at meso-ultrasonic frequencies are calculated analytically in semiconductors with an anisotropic mobility () in arbitrary classical magnetic fields. For Bq(q is the ultrasonic wave vector) and an arbitrary direction of q two transverse components of the AME field (E ^B q E _y ^B ) occur in the crystal, and the longitudinal acoustoelectric field changes under the action of a longitudinal magnetic field (E _q ^B =E _q ^B -E _q ⁰ ),E ^B is even, and E ^B is odd in B; for B 1 the component E _y ^B E ^B /B, andE ^B and E _q ^B are independent of B and can be commensurate with the zero-field acoustoelectric field E _q ⁰ if the anisotropy of is large (hexagonal ZnS and ZnO or n-Ge highly compressed along [111]). The transverse AME field E _st ^B is calculated in the configuration E _st ^B qBE _st ^B (standard AMEeffect). For B >> 1 the field B 1E _st ^B B ^–3, so thatE ^B , E _y ^B , and _q ^B can be greater than E _st ^B here. The acoustoelectric analog of the Grabner effect (E _G ^B ), i.e., the component of the AME field along a transverse magnetic field (E _G ^B Bq) is also calculated. For pB > 1 the componentE _G ^B B ^–3.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 93–97, June, 1989.     

An application of Bell's inequalities to a quantum state extension problem

Using techniques from the study of quantum violations of Bell's inequalities, we give examples of three C ^*-algebras A, B, C, and states ₁₂ on A B, and ₂₃ on B C, which agree on B, but do not have a common extension to A B C. This situation cannot occur in classical probability, i.e. for commutative algebras.     

q-analog ofA m−1 ⊕A n−1 ⊂A mn−1

A natural embeddingA _m–1 A _n–1 A _mn–1 for the corresponding quantum algebras is constructed through the appropriate comultiplication on the generators of each of theA _m–1 andA _n–1 algebras. The above embedding is proved in theq-boson realization by means of the isomorphism between theA _q ^– (mn ~ ⁿ A _q ^– (m) ~ ^m A _q ^– (n)algebras.     

The tortuosity of occupied crossings of a box in critical percolation

We consider the length of an occupied crossing of a box of size [0,n]×[0, 3n] ^D–1 (in the short direction) in standard (Bernoulli) bond percolation on ^D at criticality. Let ¦s _n¦ be the length of the shortest such crossing. It is believed that ¦s _n¦ ^1+c in some sense for somec>0. Here we show that if the correlation length(p) satisfies (p)p _c}–p) ^– for some <1, then with a probability tending to 1, ¦s _n¦>/C ₁ n ^1/(logn)^–(1–)/. The assumption (p)C ₃(p _c–p)^– with <1 has been rigorously established^(1,2) for largeD, but cannot hold⁽³⁾ forD=2. In the latter case, let ¦l _n¦ be the length of the lowest occupied crossing of the square [0,n]². We outline a proof ofP _pc(¦l_n¦ n ^1+c)n ^– for somec, >0. We also obtain a result about the length of optimal paths in first-passage percolation.