Trajectory attractor of a reaction-diffusion system with a series of zero diffusion coefficients

We study a reaction-diffusion system of N equations with k nonzero and N − k zero diffusion coefficients. More exactly, the first k equations of the system contain the terms a _i Δu _i − f _j (u, v), i = 1, …, k, with the diffusion coefficient a _i > 0. The right-hand sides of the other N − k equations contain only nonlinear interaction functions −h _j (u, v), j = k + 1, …, N, with zero diffusion. Here u = (u ₁, …, u _k ) and v = (υ _k+1, …, υ _N ) are unknown concentration vectors. Under appropriate assumptions on the interaction functions f(·) and h(·), we construct the trajectory attractor of this reaction-diffusion system. We also find the trajectory attractors , δ = (δ ₁, …, δ _k ), for the analogous reaction-diffusion systems having the terms δ _j Δυ _j − h _j (u, v), j = k + 1, …, N, with small diffusion coefficients δ _j ⩾ 0 in the last N − k equations. We prove that the trajectory attractors converge to (in an appropriate topology) as δ → 0+. Dedicated to the memory of Vladimir Borovikov Partially supported by the Russian Foundation for Basic Research (projects nos. 08-01-00784 and 07-01-00500).     

Spectral Estimates for Periodic Jacobi Matrices

 We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ²(ℤ) of the form (Hψ) _n =a _n−1 ψ _n−1 +b _nψ _n +a _nψ _n+1 , where a _n=a _n+q and b _n=b _n+q are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding to H. We consider k(z) as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent over the gaps with the normalised traces of powers of H. Received: 17 April 2002 / Accepted: 1 October 2002 Published online: 13 January 2003 Communicated by B. Simon     

Evaluation of certain statistical sums

Summary Certain statistical ensembles,e.g. open chemical systems with randomly varying number of particles, are characterized by partition functions of the type ,n being a natural number anda _j ’s generalized temperatures. The state of the system is well defined if one knows the dependence ofa _j ’s on ensemble averages 〈n ^j 〉. For making the equations 〈n ^j 〉=〈n ^j 〉 (a ₁, ...,a _s) at least more accessible for numerical calculations a transformation of the partition function to a series of Fourier integrals is proposed. In the special case of the integrals can be calculated analytically transforming the statistical sum into a series of error functions.     

Universality of the Edge Distribution of Eigenvalues of Wigner Random Matrices with Polynomially Decaying Distributions of Entries

We consider an ensemble of Wigner symmetric random matrices A_n={a_ij}, i,j=1, . . . ,n with matrix elements a_ij, being i.i.d. symmetrically distributed random variables We assume that and that for p>18. We prove that the distribution of the k (k=1,2, . . . ) largest (smallest) eigenvalues has a universal limit as n→∞ (the Tracy-Widom distribution).     

Inverse resonance scattering for Jacobi operators

The Jacobi operator (Jf) _n = a _n−1 f _n−1 +a _n f _n+1 + b _n f _n on ℤ with real finitely supported sequences (a _n − 1) _n∈ℤ and (b _n ) _n∈ℤ is considered. The inverse problem for two mappings (including their characterization): (a _n , b _n , n ∈ ℤ) → {the zeros of the reflection coefficient} and (a _n , b _n , n ∈ ℤ) → {the eigenvalues and the resonances} is solved. All Jacobi operators with the same eigenvalues and resonances are also described.     

Moments and Distributions of Trajectories in Slow Random Monads

Given a finite set B (basin) with n>1 elements, which we call points, and a map M:B→B, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter, 0≤s≤1. We fix a point ⊙∈B and consider the sequence M ^t (⊙), t=0,1,2,… . A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis _n , Rec _n and Tra _n the numbers of visited, recurrent and transient points respectively. We prove that, when n tends to infinity, Vis _n and Tra _n converge in law to geometric distributions and Rec _n converges in law to a distribution concentrated at its lowest value, which is one. Now about moments. The case s=1 is trivial, so let 0≤s<1. For any natural number k there is a number such that the k-th moments of Vis _n , Rec _n and Tra _n do not exceed this number for all n. About Vis _n : for any natural k the k-th moment of Vis _n is an increasing function of n. So it has a limit when n→∞ and for all n it is less than this limit. About Rec _n : for any k the k-th moment of Rec _n tends to one when n tends to infinity. About Tra _n : for any k the k-th moment of Tra _n has a limit when n tends to infinity.     

Calculation of a Certain Determinant

The n×n determinant det[(a+j−i)Γ(b+j+i)] is evaluated. This completes the calculation of the Mellin transform of the probability density of the determinant of a random quaternion self-dual matrix taken from the gaussian symplectic ensemble. The inverse Mellin transform then gives the later probability density itself. Received: 30 August 1999 / Accepted: 29 March 2000     

Running coupling: does the coupling between dark energy and dark matter change sign during the cosmological evolution?

In this paper we put forward a running coupling scenario for describing the interaction between dark energy and dark matter. The dark sector interaction in our scenario is free of the assumption that the interaction term Q is proportional to the Hubble expansion rate and the energy densities of dark sectors. We only use a time-variable coupling b(a) (with a the scale factor of the universe) to characterize the interaction Q. We propose a parametrization form for the running coupling b(a)=b ₀ a+b _e (1−a) in which the early-time coupling is given by a constant b _e , while today the coupling is given by another constant, b ₀. For investigating the feature of the running coupling, we employ three dark energy models, namely, the cosmological constant model (w=−1), the constant w model (w=w ₀), and the time-dependent w model (w(a)=w ₀+w ₁(1−a)). We constrain the models with the current observational data, including the type Ia supernova, the baryon acoustic oscillation, the cosmic microwave background, the Hubble expansion rate, and the X-ray gas mass fraction data. The fitting results indicate that a time-varying vacuum scenario is favored, in which the coupling b(z) crosses the noninteracting line (b=0) during the cosmological evolution and the sign changes from negative to positive. The crossing of the noninteracting line happens at around z=0.2–0.3, and the crossing behavior is favored at about 1σ confidence level. Our work implies that we should pay more attention to the time-varying vacuum model and seriously consider the phenomenological construction of a sign-changeable or oscillatory interaction between dark sectors.     

Static charged spheres with anisotropic pressure in general relativity

We report a generalization of our earlier formalism [Pramana, 54, 663 (1998)] to obtain exact solutions of Einstein-Maxwell’s equations for static spheres filled with a charged fluid having anisotropic pressure and of null conductivity. Defining new variables: w=(4π/3)(ρ+ε)r ², u=4πξr ², v _r=4πp _r r ², v _⊥=4πp _⊥ r ²[ρ, ξ(=−(1/2)F ₁₄ F ¹⁴), p _r, p _⊥ being respectively the energy densities of matter and electrostatic fields, radial and transverse fluid pressures whereas ε denotes the eigenvalue of the conformal Weyl tensor and interpreted as the energy density of the free gravitational field], we have recast Einstein’s field equations into a form easy to integrate. Since the system is underdetermined we make the following assumptions to solve the field equations (i) u=v _r=(a ²/2κ)r ⁿ⁺², v _⊥=k ₁ v _r, w=k ₂ v _r; a ², n(>0), k ₁, k ₂ being constants with κ=((k ₁+2)/3+k ₂) and (ii) w+u=(b ²/2)r ⁿ⁺², u=v _r, v _⊥−v _r=k, with b and k as constants. In both cases the field equations are integrated completely. The first solution is regular in the metric as well as physical variables for all values of n>0. Even though the second solution contains terms like k/r ² since Q(0)=0 it is argued that the pressure anisotropy, caused by the electric flux near the centre, can be made to vanish reducing it to the generalized Cooperstock-de la Cruz solution given in [14]. The interior solutions are shown to match with the exterior Reissner-Nordstrom solution over a fixed boundary. Dedicated to Prof. F A E Pirani.     

Spectrum of Non-Hermitian Heavy Tailed Random Matrices

Let (X _jk ) _j,k ≥ 1 be i.i.d. complex random variables such that |X _jk | is in the domain of attraction of an α-stable law, with 0 < α < 2. Our main result is a heavy tailed counterpart of Girko’s circular law. Namely, under some additional smoothness assumptions on the law of X _jk , we prove that there exist a deterministic sequence a _n ~ n ^1/α and a probability measure μ _α on \mathbbC{\mathbb{C}} depending only on α such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix (a_n^-1X_jk)_{1 ￡ j,k ￡ n}{(a_n^{-1}X_{jk})_{1\leq j,k\leq n}} converges weakly to μ _α as n → ∞. Our approach combines Aldous & Steele’s objective method with Girko’s Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous’ Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of μ _α . In contrast with the Hermitian case, we find that μ _α is not heavy tailed.     

Condensation in Nongeneric Trees

We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical Galton-Watson tree with mean offspring probability m<1. We calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like (1−m)N where N is the size of the tree. These trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2β−2 if the weight w _n of a vertex of degree n is asymptotic to n ^−β .     

Reentrant transitions in the van Hemmen model of a spin glass

The van Hemmen model of a spin glass, which is an Ising model with random couplings J_ij between sites i and j equal to J₀ + J(ξ_iη_j + ξ_jη_i), where (ξ_i, η_i) are independent, identically distributed random variables, is studied in the pair approximation of the cluster variation method. For the family of probability distributions (1 − p)δ(ξ_i − a) + pδ(ξ_i) + (1 − p)δ(ξ_i + a), where p is varied, phase diagrams are constructed. They are qualitatively different from the mean-field phase diagrams and display a competition between tendencies towards spin-glass and towards ferromagnetic ordering, which results in reentrant transitions. It is argued that the observed effects are not accidental but are borne by the competition of bonds of the underlying lattice system.     

An isomorphism between the Para-Fermi algebra and the algebra O(n,n+1)

It is shown that the relations which define the para-Fermi creation and annihilation operatorsb _i ,a _i (i = 1,...,n) can be considered as commutation relations of the algebra 0(n, n+1). It turns out that every representation of 0(n, n+1) determines a representation of the para-Fermi algebra and vice versa.     

Similarity parameters evaluation in the Po Valley during convective situations with Sodar data

Summary In well-developed convection, the surface sensible-heat flux can be evaluated from Sodar data; by using these flux values, it is possible to estimate some scale parameters in the mixed layer (likeθ _* andw _*). The use ofθ _*,w _* and the height of the mixed layerz _i, obtainable from other sources (i.e. radiosoundings), allows to verify some similarity relationships. We analysed the vertical profiles of the ratioσ _w ² /w _* ² : we found that it follows the theoretical behaviour depending onz/z _i if we use numerical coefficients slightly different from those usually referred to in the literature.     

Inverse problem for the Grad-Shafranov equation with affine right-hand side

For a given domain ω ⋐ ℝ² with boundary γ = ∂ω, we study the cardinality of the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) of pairs of numbers (a, b) for which there is a function u = u _(a,b): ω → ℝ such that ∇² u(x) = au(x) + b ⩾ 0 for x ∈ ω, u| _γ = 0, and ||∇u(s)| − Φ(s) ⩽ η for s ∈ γ. Here η ⩾ 0 stands for a very small number, Φ(s) = |∇(s)| / ∫ _γ |∇v| d γ, and v is the solution of the problem ∇² v = a ₀ v + 1 ⩾ 0 on ω with v| _γ = 0, where a ₀ is a given number. The fundamental difference between the case η = 0 and the physically meaningful case η > 0 is proved. Namely, for η = 0, the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) contains only one element (a, b) for a broad class of domains ω, and a = a ₀. On the contrary, for an arbitrarily small η > 0, there is a sequence of pairs (a _j , b _j ) ∈ $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) and the corresponding functions u _j such that ‖f _u ^j+1‖ − ‖f _u ^j ‖ > 1, where ‖f _u ^j = max _x∈ω |f _u ^j (x)| and f _u ^j (x) = a _j u _j (x) + b _j . Here the mappings f _u ^j : ω → ℝ necessarily tend as j → ∞ to the δ-function concentrated on γ.     

Central Limit Theorems for Nonlinear Hierarchical Sequences of Random Variables

Let a random variable x ₀ and a function f:[a, b] ^k [a, b] be given. A hierarchical sequence {x _n :n=0, 1, 2,...} of random variables is defined inductively by the relation x _n =f(x _{n–1, 1}, x _{n–1, 2}..., x _{n–1, k} ), where {x _{n–1, i} :i=1, 2,..., k} is a family of independent random variables with the same distribution as x _n–1. We prove a central limit theorem for this hierarchical sequence of random variables when a function f satisfies a certain averaging condition. As a corollary under a natural assumption we prove a central limit theorem for a suitably normalized sequence of conductivities of a random resistor network on a hierarchical lattice.     

The Variational Principle for a Class of Asymptotically Abelian C*-Algebras

Let (A,α) be a C^*-dynamical system. We introduce the notion of pressure P _α(H) of the automorphism α at a self-adjoint operator H∈A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra ? of A such that for all pairs a,b∈? the C^*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α ^j (a),b]= 0 for |j|≥p. For systems in this class we prove the variational principle, i.e. show that P _α(H) is the supremum of the quantities h _φ(α) −φ(H), where h _φ(α) is the Connes–Narnhofer–Thirring dynamical entropy of α with respect to the α-invariant state φ. If H∈A, and P _α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h _φ(α), and any state of finite maximal entropy is a trace. Received: 19 April 2000 / Accepted: 14 June 2000     

Quadratic spherical bessel functions and the inverse scattering problem

In the course of inverting the partial-wave Born approximation, a new expression for the inverse function ofj _l ² (ρ) was obtained. Using this result, one can also derive two expressions involving the binomial coefficients. Finally, a particular differential operator whose effect onj _l ² (ρ) was previously investigated by Mavromatis and Jalal is shown to have similar effects onn _l ² (ρ) andn _l (ρ)j _l (ρ).     

Phase Transitions in the Dynamics of Slow Random Monads

Let us have a finite set B (basin) with n>1 elements, which we call points, and a map M:B→B. Following Vladimir Arnold, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter in [0,1].