On the Poisson Limit Theorems of Sinai and Major

Let f(ϕ) be a positive continuous function on 0 ≤ϕ≤Θ, where Θ≤ 2 π, and let ξ be the number of two-dimensional lattice points in the domain Π _R (f) between the curves r=(R+c ₁/R)f(ϕ) and r=(R+c ₂/R)f(ϕ), where c ₁<c ₂ are fixed. Randomizing the function f according to a probability law P, and the parameter R according to the uniform distribution μ _L on the interval [a ₁ L,a ₂ L], Sinai showed that the distribution of ξ under P×μ _L converges to a mixture of the Poisson distributions as L→∞. Later Major showed that for P-almost all f, the distribution of ξ under μ _L converges to a Poisson distribution as L→∞. In this note, we shall give shorter and more transparent proofs to these interesting theorems, at the same time extending the class of P and strengthening the statement of Sinai. Received: 15 June 1999 / Accepted: 11 February 2000     

Integrability ofSU(N) spin chains with elliptic exchange

For some one-parameter setH _N of linear combinations ofN(N−1)/2 elementary transpositions {P _jk} (1≤j<k≤N) at arbitrary naturalN≥3 one can construct a variety {I _m} (3≤m≤N) of operators which commute withH _N. Being applied toSU(n) spin representations of the permutation group, this proves the integrability of 1D periodic spin chains with elliptic short-range interaction. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Proximity-induced superconductivity in graphene

We propose a way of making graphene superconductive by putting on it small superconductive islands which cover a tiny fraction of graphene area. We show that the critical temperature, T _c , can reach several Kelvins at the experimentally accessible range of parameters. At low temperatures, T T _c , and zero magnetic field, the density of states is characterized by a small gap E _g ≤ T _c resulting from the collective proximity effect. Transverse magnetic field H _g (T) ∝ E _g is expected to destroy the spectral gap driving graphene layer to a kind of a superconductive glass state. Melting of the glass state into a metal occurs at a higher field H _g2(T). The article is published in the original.     

Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of N symplectically and globally coupled standard maps localized in a d=1 lattice array. The global coupling is modulated through a factor r^-α, being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0≤α≤1, and short-range (integrable) when α>1. We verify that the largest Lyapunov exponent λ_M scales as λ_M ∝ N^-κ(α), where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ_M→0). In the short-range case, κ(α) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration t_c scales as t_c ∝N^β(α), where β(α) appears to be numerically in agreement with the following behavior: β>0 for 0 ≤α< 1, and zero for α≥1. These results are consistent with features typically found in nonextensive statistical mechanics. Moreover, they exhibit strong similarity between the present discrete-time system, and the α-XY Hamiltonian ferromagnetic model.     

The Effect of the Number of Simulations on the Exponents Obtained by Finite-Size Scaling Relations of the Order Parameter and the Magnetic Susceptibility for the Four-Dimensional Ising Model on the Creutz Cellular Automaton

The four-dimensional Ising model is simulated on the Creutz cellular automaton using finite-size lattices with linear dimension 4≤L≤8. The exponents in the finite-size scaling relations for the order parameter and the magnetic susceptibility at the finite-lattice critical temperature are computed to be β=0.49(7), β=0.49(5), β=0.50(1) and γ=1.04(4), γ=1.03(4), γ=1.02(4) for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results are consistent with the renormalization group predictions of β=0.5 and γ=1. The values for the critical temperature of the infinite lattice T _c (∞)=6.6788(65), T _c (∞)=6.6798(69), T _c (∞)=6.6802(70) are obtained from the straight-line fit of the magnetic susceptibility maxima using 4≤L≤8 for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results are in very good agreement with the series expansion results of T _c (∞)=6.6817(15), T _c (∞)=6.6802(2), the dynamic Monte Carlo result of T _c (∞)=6.6803(1), the cluster Monte Carlo result of T _c (∞)=6.680(1) and the Monte Carlo using Metropolis and Wolff-cluster algorithm result of T _c (∞)=6.6802632±5×10⁻⁵.     

Soft topological objects in topological media

Topological invariants in terms of the Green’s function in momentum and real space determine properties of smooth textures within topological media. In space dimension d = 1 the topological invariant N ₃ in terms of the Green’s function (ω, k _x , x) determines the fermion number of the kink, while in space dimension d = 3 the topological invariant N ₅ in terms of the Green’s function (ω, k _x , k _y , k _z , z) determines quantization of Hall conductivity in the soliton plane within the topological insulators.     

Formation and quenching mechanisms of excited particles in an oxygen-iodine laser medium

New values of a number of kinetic constants of processes proceeding in oxygen-iodine laser media are presented. The total probabilities of formation of I₂(X, 15 ≤ v ≤ 24) and I₂(X, 25 ≤ v ≤ 47) molecules in the course of quenching of I* atoms by I₂(X) are found to be 0.9 and 0.1, respectively. The quantum yield of singlet oxygen in the reaction O(¹ D) + N₂O → N₂ + O₂(a ¹Δ) is close to 100%. The quenching rate constants of I₂(A’) by O₂, H₂O, CO₂, I₂, and Ar and of I(² P _1/2) by O(³ P), O₃, NO₂, N₂O₄, and N₂O are presented.     

Lasing in an e beam pumped Ar-N2-mixture at 406 nm

An Ar−N₂-mixture was excited by a relativistic e-beam in a longitudinal pumping geometry. Lasing is observed for the trasitions from the N₂(C) to the lower three vibrational states of N₂(B) at λ=337, 358, and 380 nm and for the first time also to the fourth vibrational state at λ=406 nm.     

Bounds on charged lepton mixing with exotic charged leptons

We examine the effects of mixing induced light heavy charged lepton neutral currents on the partial wave amplitude for the process l⁺l⁻ →ZZ (withl = e,μ or τ). By imposing the constraints that the amplitude should not exceed the perturbative unitarity limit at high energy (√s = Λ), we obtain bounds on light heavy charged lepton mixing parameter sin²(2θ _L ^a ) where θ _L ^a is the mixing angle of the ordinary charged lepton with its exotic partner. For Λ = 1 TeV, no bound is obtained on sin² (2θ _L ^a ) form _E < 0.69 TeV. However, sin² (2θ _L ^a ) ≤ 1.52×10⁻⁵ form _E = 5 TeV, sin² (2θ _L ^a ) ≤ 2.41 ×10⁻⁷ form _E = 10 TeV. Similarity for Λ = ∞ no bound is obtained on sin² (2θ _L ^a ) for m_E < 1.97 TeV and sin² (2θ _L ^a ) ≤ 0.15 form _E = 5 TeV and sin² (2θ _L ^a ) ≤ 3.88×10^-2 form _E = 10 TeV.     

Superconductor-insulator duality for an array of Josephson wires

A novel model system is proposed for the study of superconductor-insulator transitions that is a regular lattice whose each link consists of a Josephson-junction chain of N ≫ 1 junctions in sequence. The theory of such an array is developed for the case of semiclassical junctions with the Josephson energy E _J larger compared to the Coulomb energy E _C = e ²/2C of the junctions. An exact duality transformation is derived that transforms the Hamiltonian of the proposed model into a standard Hamiltonian of a JJ array. The nature of the ground state is controlled (in the absence of random offset charges) by the parameter q ≈ N ² exp with the superconductive state corresponding to small q < q _c . The values of q _c are calculated for magnetic frustrations f = 0 and f = 1/2. The temperature of the superconductive transition T _c (q) and q < q _c is estimated for the same values of f. In the presence of strong random offset charges, the T = 0 phase diagram is controlled by the parameter ; the critical value and the critical temperature at zero magnetic frustration are estimated. The text was submitted by the authors in English.     

Pseudovector heavy quarkonia in e + e −2 collisions

It is shown that BR(χ _b1(1 P)→e ⁺ e ⁻)≃3.3· 10⁻⁷ and BR(χ _c1(1 P)→e ⁺ e ⁻)≃10⁻⁸. This gives realistic possibilities for searching for the production of χ _b1(1 P) and ξ _c1(1 P) states in e ⁺ e ⁻ collisions, even on the present-day colliders, to say nothing of b and c-τ factories. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 8, 569–574 (25 April 1996)     

Liouville Type Equations with Singular Data¶and Their Applications to Periodic Multivortices¶for the Electroweak Theory

Motivated by the study of multivortices in the Electroweak Theory of Glashow–Salam–Weinberg [33], we obtain a concentration-compactness principle for the following class of mean field equations: on M, where (M,g) is a compact 2-manifold without boundary, 0 < a≤K(x)≤b, x∈M and λ > 0. We take with α _i > 0, δ _{p i} the Dirac measure with pole at point p _i ∈M, i= 1,…,m and ψ∈L ^∞(M) satisfying the necessary integrability condition for the solvability of (1)_λ. We provide an accurate analysis for solution sequences of (1)_λ, which admit a “blow up” point at a pole p _i of the Dirac measure, in the same spirit of the work of Brezis–Merle [11] and Li–Shafrir [35]. As a consequence, we are able to extend the work of Struwe–Tarantello [49] and Ding–Jost–Li–Wang [21] and derive necessary and sufficient conditions for the existence of periodic N-vortices in the Electroweak Theory. Our result is sharp for N= 1, 2, 3, 4 and was motivated by the work of Spruck–Yang [46], who established an analogous sharp result for N= 1, 2. Received: 24 September 2001 / Accepted: 7 December 2001     

Calabi-Yau manifolds of some special forms

Let M=M ₁×...×M _m be a product of Kähler C-spaces with second Betti numbers b ₂(M _i )=1 (1im). The work establishes that the complete intersections X of M produce a finite number of N-dimensional Calabi-Yau manifolds. Moreover, if b ₄(M _i )=1, then the complete intersections with vanishing first Pontrjagin classes are finitely many, as well.On the other hand, we consider hypersurfaces of weighted projective spaces and give an explicit formula for their Euler characteristics. As in the previous case, it turns out that only a finite number of these are Calabi-Yau manifolds.     

Quark-antiquark states and their radiative transitions in terms of the spectral integral equation: Charmonia

Earlier by the authors (Yad. Fiz. 70, 68 (2007)), the bƃ states were treated in the framework of the spectral integral equation, together with simultaneous calculations of radiative decays of the considered bottomonia. In the present paper, such a study is carried out for the charmonium states. We reconstruct the interaction in the c-c sector on the basis of the data for the charmonium levels with J _PC = 0⁻⁺, 1⁻⁻, 0⁺⁺, 1⁺⁺, 2⁺⁺, 1⁺⁻ and radiative transitions ψ(2S) → γχ _c0(1P), γχ _c1(1P), γχ _c2(1P), γχ _c(1S) and χ _c0(1P), χ _c1(1P), χ _c2(1P) → γJ/ψ. The c-c levels and their wave functions are calculated for the radial excitations with n ≤ 6. Also, we determine the c-c component of the photon wave function using the e ⁺ e ⁻-annihilation data: e ⁺ e ⁻ → J/ψ(3097), ψ(3686), ψ(3770), ψ(4040), ψ(4160), ψ(4415) and perform the calculations of the partial widths of the two-photon decays for the n = 1 states η _c0(1S), χ _c0(1P), χ _c2(1P) → γγ and n = 2 states η _c0(2S) → γγ, χ _c0(2P) → γγ. We discuss the status of the recently observed c-c states X(3872) and Y(3941): according to our results, the X(3872) can be either χ _c1(2P) or η _c2(1D), while Y(3941) is χ _c2(2P). The text was submitted by the authors in English.     

Evaluation of Effective Resistances in Pseudo-Distance-Regular Resistor Networks

The effective resistance or two-point resistance between two nodes of a resistor network is the potential difference that appears across them when a unit current source is applied between the nodes as terminals. This concept arises in problems which deal with graphs as electrical networks including random walks, distributed detection and estimation, sensor networks, distributed clock synchronization, collaborative filtering, clustering algorithms and etc. In the previous paper (Jafarizadeh et al. in J. Math. Phys. 50:023302, 2009) a recursive formula for evaluation of effective resistances on the so-called distance-regular networks was given based on the Christoffel-Darboux identity. In this paper, we consider more general networks called pseudo-distance-regular networks or QD type networks, where we use the stratification of these networks and show that the effective resistances between a given node, say α, and all of the nodes β belonging to the same stratum with respect to α, are the same. Then, based on the spectral techniques, for those α,β’s which satisfy L^-1_aa=L^-1_bbL^{-1}_{\alpha\alpha}=L^{-1}_{\beta\beta} (L ⁻¹ is the pseudo-inverse of the Laplacian of the network), an analytical formula for effective resistances R_ab^(m)R_{\alpha\beta^{(m)}} (the equivalent resistance between terminals α and β, so that β belongs to the m-th stratum with respect to α) is given in terms of the first and second orthogonal polynomials associated with the network. From the fact that in distance-regular networks, L^-1_aa=L^-1_bbL^{-1}_{\alpha\alpha}=L^{-1}_{\beta\beta} is satisfied for all nodes α,β of the network, the effective resistances R_ab^(m)R_{\alpha\beta^{(m)}} for m=1,2,…,d (d is diameter of the network which is the same as the number of strata) are calculated directly, by using the given formula.     

Quantum information entropies of ultracold atomic gases in a harmonic trap

The position and momentum space information entropies of weakly interacting trapped atomic Bose–Einstein condensates and spin-polarized trapped atomic Fermi gases at absolute zero temperature are evaluated. We find that sum of the position and momentum space information entropies of these quantum systems containing N atoms confined in a D( ≤ 3)-dimensional harmonic trap has a universal form as S_t^(D) = N(a D - b lnN) S_\mathrm{t}^{(D)} = N(a D - b \ln N) , where a ≃ 2.332 and b = 2 for interacting bosonic systems and a ≃ 1.982 and b = 1 for ideal fermionic systems. These results obey the entropic uncertainty relation given by Beckner, Bialynicki-Birula and Myceilski.     

Criticality of the mean-field spin-boson model: boson state truncation and its scaling analysis

The spin-boson model has nontrivial quantum phase transitions at zero temperature induced by the spin-boson coupling. The bosonic numerical renormalization group (BNRG) study of the critical exponents β and δ of this model is hampered by the effects of boson Hilbert space truncation. Here we analyze the mean-field spin boson model to figure out the scaling behavior of magnetization under the cutoff of boson states N _b . We find that the truncation is a strong relevant operator with respect to the Gaussian fixed point in 0 < s < 1/2 and incurs the deviation of the exponents from the classical values. The magnetization at zero bias near the critical point is described by a generalized homogeneous function (GHF) of two variables τ = α – α _c and x = 1/N _b . The universal function has a double-power form and the powers are obtained analytically as well as numerically. Similarly, m(α = α _c ) is found to be a GHF of ϵ and x. In the regime s > 1/2, the truncation produces no effect. Implications of these findings to the BNRG study are discussed.