Dynamical Determinants via Dynamical Conjugacies for Postcritically Finite Polynomials

We give an analogue of Levin–Sodin–Yuditskii's study of the dynamical Ruelle determinants of hyperbolic rational maps in the case of subhyperbolic quadratic polynomials. Our main tool is to reduce to an expanding situation. We do so by applying a dynamical change of coordinates on the domains of a Markov partition constructed from the landing ray at the postcritical repelling orbit. We express the dynamical determinants as the product of an (entire) determinant with an explicit expression involving the postcritical repelling orbit, thus explaining the poles in d (z).     

Long-Time tails in a random diffusion model

Letw = {w(x)xZ^d} be a positive random field with i.i.d. distribution. Given its realization, letX _t be the position at timet of a particle starting at the origin and performing a simple random walk with jump rate w^–1(X_t). The processX={X _t:t0} combined withw on a common probability space is an example of random walk in random environment. We consider the quantities _t =(d/dt) E (X _t ² –M ^–1 t and _t(w) = (d/dt)E_w(X _t ² – M^– 1t). Here E_w. is expectation overX at fixedw and E = E_w (dw) is the expectation over bothX andw. We prove the following long-time tail results: (1) lim_t t^d/2_t= V²M^d/2–3(d/2)^d/2 and (2) lim_t t^d/4 _st(w)= Z_s weakly in path space, with {Z_s:s>0} the Gaussian process with EZ_s=0 and EZ_rZ_s= V²M^d/2–4(d)^d/2 (r + s)^–d/2. HereM and V² are the mean and variance of w(0) under . The main surprise is that fixingw changes the power of the long-time tail fromd/2 tod/4. Since , with ₀ the stationary measure for the environment process, our result (1) exhibits a long-time tail in an equilibrium autocorrelation function.     

Gauge-invariant on-shellZ 2 in QED,QCD and the effective field theory of a static quark

We calculate theon-shell fermion wave-function renormalization constantZ ₂ of a general gauge theory, to two loops, inD dimensions and in an arbitrary covariant gauge, and find it to be gauge-invariant. In QED this is consistent with the dimensionally regularized version of the Johnson-Zumino relation: d logZ ₂/da ₀=i(2)^–D e ₀ ² d ^D k/k ⁴=0. In QCD it is, we believe, a new result, strongly suggestive of the cancellation of the gauge-dependent parts of non-abelian UV and IR anomalous dimensions to all orders. At the two-loop level, we find that the anomalous dimension _F of the fermion field in minimally subtracted QCD, withN _L light-quark flavours, differs from the corresponding anomalous dimension of the effective field theory of a static quark by the gauge-invariant amount

Einstein equations and Fierz-Pauli equations with self-interaction in quantum gravity

The Einstein equations can be written as Fierz-Pauli equations with self-interaction, together with the covariant Hilbert-gauge condition, where W denotes the covariant wave operator and G _ik the Einstein tensor of the metric g _ik collecting all nonlinear terms of Einstein's equations. As is known, there do not, however, exist plane-wave solutions _ik(z)with g ^ik Z,_i Z,_k=0of these equations such that what is essential to the introduction of gravitons is not satisfied in general relativity. This nonexistence corresponds with the uncertainty relation,p(g*)²(x)³h hG/ c ³ concerning the total nonlinear gravitational field g ^*ik =g ^k + ^k .     

From quantum to elliptic algebras

It is shown that the elliptic algebra at the critical level c = –2 has a multidimensional center containing some trace-like operators t(z). A family of Poisson structures indexed by a non-negative integer and containing the q-deformed Virasoro algebra is constructed on this center. We show also that t(z) close an exchange algebra when p ^m = q ^c+2 for , they commute when in addition p = q ^2k for k integer non-zero, and they belong to the center of when k is odd. The Poisson structures obtained for t(z) in these classical limits contain the q-deformed Virasoro algebra, characterizing the structures at p q ^2k as new algebras.     

V2-dependence of centrifugal constants of H2O molecules in the Morse oscillator model

It is shown in this paper that the centrifugal constantsC ^l (t) calculated in the Morse, oscillator model for the sequence from the operator H _red ^(t) , which is used in processing the rotational levels of the H₂O molecule, are linked by recurrence relations, which enables H _z ^(t) –H _z ^(O) to be given in closed form.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 56–58, October, 1982.     

Correlation functions and the Goldstone picture for the hierarchical classical vector model at low temperatures in three or more dimensions

Low-temperature properties of the one-and two-point correlation functions are obtained for the pure state classical vector model in a hierarchical formulation. We consider theZ ^d lattice model (d3) where the single-site spin variableR ^v has a density proportional to for large. We obtain the pure state one- and two-point functions by introducing a uniform magnetic field which goes to zero as the volume goes to infinity. Using renormalization group methods, we generate a sequence of effective actions and spin variable and determine the spontaneous magnetization (one-point function parallel to the field). We confirm the Goldstone picture by showing that the truncated two-point function has the canonical massless decay x–y^–(d–2) x,yZ^d in the directions perpendicular to the field. We show a faster decay in the parallel direction and for larged that the decay is x-y^–(d+2).Research support by CNPq, Brazil.     

A study of perfect wetting for Potts and Blume-Capel models with correlation inequalities

The so-called perfect wetting phenomenon is studied for theq-state,d2 Potts model. Using a new correlation inequality, a general inequality is established for the surface tension between ordered phases ( ^a,b ) and the surface tension between an ordered and the disordered phases ( ^a,f ) for any even value ofq. This result implies in particular at the transition point _t where the previous phases coexist forq large. This inequality is connected to perfect wetting at the transition point using thermodynamic considerations. The same kinds of results are derived for the Blume-Capel model.     

Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems

We consider stochastic processes, with finite, in which spin flips (i.e., changes of S ^t _x ) do not raise the energy. We extend earlier results of Nanda–Newman–Stein that each site x has almost surely only finitely many flips that strictly lower the energy and thus that in models without zero-energy flips there is convergence to an absorbing state. In particular, the assumption of finite mean energy density can be eliminated by constructing a percolation-theoretic Lyapunov function density as a substitute for the mean energy density. Our results apply to random energy functions with a translation-invariant distribution and to quite general (not necessarily Markovian) dynamics.     

On the Davey–Stewartson and Ishimori Systems

We study the initial value problem for the two-dimensional nonlinear nonlocal Schrödinger equations i u_t + u = N(v), (t, x, y) R³, u(0, x, y) = u₀(x, y), (x, y) R² (A), where the Laplacian = ² _x + ² _y, the solution u is a complex valued function, the nonlinear term N = N₁ + N₂ consists of the local nonlinear part N₁(v) which is cubic with respect to the vector v=(u,u_x,u_y,\overline{u},\overline{u}_{x},\overline{u}_{y}) in the neighborhood of the origin, and the nonlocal nonlinear part N₂(v) =(v, ^{– 1} _x K_x(v)) + (v, ^{– 1} _y K_y(v)), where (, ) denotes the inner product, and the vectors K_x (C⁴(C⁶; C))⁶ and K_y (C⁴(C⁶; C))⁶ are quadratic with respect to the vector v in the neighborhood of the origin. We assume that the components K⁽²⁾ _x = K⁽⁴⁾ _x 0, K⁽³⁾ _y = K⁽⁶⁾ _y 0. In particular, Equation (A) includes two physical examples appearing in fluid dynamics. The elliptic–hyperbolic Davey–Stewartson system can be reduced to Equation (A) with , and all the rest components of the vectors K_x and K_y are equal to zero. The elliptic–hyperbolic Ishimori system is involved in Equation (A), when , and . Our purpose in this paper is to prove the local existence in time of small solutions to the Cauchy problem (A) in the usual Sobolev space, and the global-in-time existence of small solutions to the Cauchy problem (A) in the weighted Sobolev space under some conditions on the complex conjugate structure of the nonlinear terms, namely if N(eⁱ v) = eⁱ N(v) for all R.     

Low density flux fluids in high —T c superconductors: The influence of disorder

We analyze the influence of thermal and frozen-in disorder on the flux line (FL) density close to the lower critical fieldH _c1. Arguments which consider the steric repulsion of fluctuating FLs give with the roughness exponent of a single FL andd the space dimensionality. We show by a phenomenological scaling approach and a renormalization group treatment, that this is correct only fordd _c =2/–1, i.e. for . Ford>d _c the steric FL repulsion at scales more than some critical one is irrelevant and . For disordered superconductorsd _c =2 and ford=2, 3. We also found the melting line for a FL lattice in the presence of frozen-in impurities close toH _c1.     

Unified fields of analytic space-time-energy

An analytic gravitational fieldZ (Z ^y ) is shown to include electromagnetic phenomena. In an almost flat and almost static complex geometryds ² =zdzdz of four complex variables z=t, x, y, x the field equationsR Rz = –(U U – Z ) imply the conventional equations of motion and the conventional electromagnetic field equations to first order if =(Z ^v) and =(z ) are expressed in terms of the conventional mass density function , the conventional charge density function , and a pressurep as follows: v=const=p/c ²–10^–29 gm/cm³.     

Estimates of logarithmic Sobolev constat for finite-volume continuous spin systems

LetM be a compact, connected Riemannian manifold (with or without boundary); we study the logarithmic Sobolev constant for stochastic Ising models on . Let {} be a sequence of cubes inZ ^d ; we show that the logarithmic Sobolev constant for the finite systems onM ^A shrinks at most exponentially fast in ||^(d-1)/d (d2), which is sharp in order for the classical Ising models withM=[–1, 1]. Moreover, a geometrical lemma proved by L. E. Thomas is also improved.     

Accurate determination of wavenumbers for iodine molecular lines in the red spectral region

The hyperfine structure of various absorption lines of molecular iodine with wavenumbers between 12980 and 13890 cm^–1 has been resolved using Doppler-free polarization spectroscopy. The wavenumbers of theo-component of 17 rovibrational lines of I₂ due to the transitionB ³ _ou ⁺ –X¹ _g ⁺ with even rotational quantum numbers have been determined with an accuracy of 0.001 cm^–1. A comparison of the centers of gravity of these 17 lines with the values of the iodine atlas of Gerstenkorn et al. yields a difference of thus corroborating the data of the iodine atlas in the red region within limits of error.     

Improved Peierls Argument for High-Dimensional Ising Models

We consider the low-temperature expansion for the Ising model on , with ferromagnetic nearest neighbor interactions in terms of Peierls contours. We prove that the expansion converges for all temperatures smaller than Cd(log d)^–1, which is the correct order in d.     

Some New Results on the Kinetic Ising Model in a Pure Phase

We consider a general class of Glauber dynamics reversible with respect to the standard Ising model in ^d with zero external field and inverse temperature strictly larger than the critical value _c in dimension 2 or the so called slab threshold in dimension d 3. We first prove that the inverse spectral gap in a large cube of side N with plus boundary conditions is, apart from logarithmic corrections, larger than N in d = 2 while the logarithmic Sobolev constant is instead larger than N ² in any dimension. Such a result substantially improves over all the previous existing bounds and agrees with a similar computations obtained in the framework of a one dimensional toy model based on mean curvature motion. The proof, based on a suggestion made by H. T. Yau some years ago, explicitly constructs a subtle test function which forces a large droplet of the minus phase inside the plus phase. The relevant bounds for general d 2 are then obtained via a careful use of the recent –approach to the Wulff construction. Finally we prove that in d = 2 the probability that two independent initial configurations, distributed according to the infinite volume plus phase and evolving under any coupling, agree at the origin at time t is bounded from below by a stretched exponential , again apart from logarithmic corrections. Such a result should be considered as a first step toward a rigorous proof that, as conjectured by Fisher and Huse some years ago, the equilibrium time auto-correlation of the spin at the origin decays as a stretched exponential in d = 2.     

Higher-order susceptibilities of the regular and the random Ising model on the Cayley tree. I

Explicit expressions for the fourth-order susceptibility (⁴), the fourth derivative of thebulk free energy with respect to the external field, are given for the regular and the random-bond Ising model on the Cayley tree in the thermodynamic limit, at zero external field. The fourth-order susceptibility for the regular system diverges at temperature T _c ⁽⁴⁾ = 2k _B ^–1 J/ln{1+2/[(z–1)^3/4–1]}, confirming a result obtained by Müller-Hartmann and Zittartz [Phys. Rev. Lett. 33:893 (1974)]; Herez is the coordination number of the lattice,J is the exchange integral, andk _B is the Boltzmann constant. The temperatures at which ⁽⁴⁾ and the ordinary susceptibility (²) diverge are given also for the random-bond and the random-site Ising model and for diluted Ising models.     

Instability of the molecular structure of monobenzoporphin to the alternation of the macrocycle bond lengths and its manifestation in the electronic spectra

Quantum-chemical calculations of the geometric structure of the molecules of monobenzoporphin (H₂ MBP) and monobenzoporphin with methyl and ethyl substituents in the five-member rings (H₂MBPm) have been carried out by the restricted and unrestricted Hartree-Fock methods with the AM1 Hamiltonian (AM1 RHF and AM1 UHF methods). The calculation of the above-indicated molecules by the AM1 RHF method without restrictions on their symmetry has given, for them, a planar structure with an alternation of the lengths of the bonds along the 18-member azacyclopolyene and the symmetry C _1h for their aromatic part. The calculation of the transitions to the excited electron Q states in such a structure by the CNDO/S method has shown that these states are characterized by large hypsochromic shifts (~3000–4000 cm^–1 ) relative to the Q levels of porphin (H₂P), which is in contradiction with the experimental data, according to which these shifts are bathochromic and comprise = –330 cm ^–1 and = –750 cm^–1. Optimization of the geometry of the H₂ MBP and H₂MBPm molecules by the AM1 UHF method gives, for them, a structure with equal lengths of the bonds along the 18-member azacyclopolyene with a symmetry differing insignificantly from the D _2h symmetry; elements of the structure with a lower symmetry and an alternation of the lengths of the bonds are retained in the condensed pyrrolenine and benzene rings. The calculation of the shifts of the Q levels in the H₂MBPm molecule of this geometry relative to the analogous levels in H₂P has shown that they are bathochromic and equal to = –520 cm^–1, and the RHF calculation with optimization of the geometry of the molecule and restrictions on the effective symmetry D _2h of the 18-member azacyclopolyene has given = –350 cm^–1 and = –430 cm^–1. The restrictions imposed on the C ₂ symmetry of the H₂MBP molecules by the RHF method are inadequate to equalize the lengths of the bonds along the 18-member azacyclopolyene. The calculations of the energy of the B levels of the monobenzoporphyrins considered also lend credence to their geometric structure with equal lengths of the bonds along the 18-member azacyclopolyene.Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 71, No. 6, pp. 712–721, November–December, 2004.This revised version was published online in April 2005 with a corrected cover date.     

q-Analogue of the Watanabe Unitary Transform Associated to the q-Continuous Gegenbauer Polynomials

We associate a family of Hilbert spaces H _q ^2;(D) of analytic functions on the unit disk D=z :|z|<1 the q-continuous Gegenbauer polynomials C _n (x;q) on the interval]–1;1[ and give a q-analogue of the unitary integral transform that Watanabe constructed from the Hilbert space L ²(]–1;1[;(1–x ²)^– dx onto the weighted Hilbert space H ^2;(D).     

Direction-direction correlations of oriented polymers

We calculate the direction-direction correlations between the tangent vectors of an oriented self-avoiding walk (SAW). LetJ (x) andJ ^v (0) be components of unit-length tangent vectors of an oriented SAW, at the spatial pointsx and 0, respectively. Then for distances |x| much less than the average distance between the endpoints of the walk, the correlation function ofJ (x) withJ ^v (0) has, ind dimensions, the form . The dimensionless amplitudek(d) is universal, and can be calculated exactly in two dimensions by using Coulomb gas techniques, where it is found to bek(2)=12/25 ². In three dimensions, the -expansion to second order in together with the exact value ofk(2)in two dimensions allows the estimatek(3)=0.0178±0.0005. In dimensionsd4, the universal amplitudek(d) of the direction-direction correlation functions of an oriented SAW is the same as the universal amplitude of the direction-direction correlation functions of an oriented random walk, and is given byk(d)= ²(d/2)/(d–2) ^d .