Duality, Biorthogonal Polynomials¶and Multi-Matrix Models

The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel–Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V ₁, V ₂ in two different variables, these kernels may be expressed in terms of finite dimensional “windows” spanned by finite subsequences having length equal to the degree of one or the other of the polynomials V ₁, V ₂. The vectors formed by such subsequences satisfy “dual pairs” of first order systems of linear differential equations with polynomial coefficients, having rank equal to one of the degrees of V ₁ or V ₂ and degree equal to the other. They also satisfy recursion relations connecting the consecutive windows, and deformation equations, determining how they change under variations in the coefficients of the polynomials V ₁ and V ₂. Viewed as overdetermined systems of linear difference-differential-deformation equations, these are shown to be compatible, and hence to admit simultaneous fundamental systems of solutions. The main result is the demonstration of a spectral duality property; namely, that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters. Received: 14 September 2001 / Accepted: 18 February 2002     

Semilinear PDEs on Self-Similar Fractals

A Laplacian may be defined on self-similar fractal domains in terms of a suitable self-similar Dirichlet form, enabling discussion of elliptic PDEs on such domains. In this context it is shown that that semilinear equations such as Δu+u ^p = 0, with zero Dirichlet boundary conditions, have non-trivial non-negative solutions if 0<ν≤ 2 and p>1, or if ν >2 and 1<p< (ν+ 2)/(ν− 2), where ν is the “intrinsic dimension” or “spectral dimension” of the system. Thus the intrinsic dimension takes the r\^{o}le of the Euclidean dimension in the classical case in determining critical exponents of semilinear problems. Received: 11 December 1998 / Accepted: 22 March 1999     

Eigenvalues of Laplacian with Constant Magnetic Field on Non-Compact Hyperbolic Surfaces with Finite Area

We consider a magnetic Laplacian −Δ _A = (id + A)^* (id + A) on a non-compact hyperbolic surface M with finite area. A is a real one-form and the magnetic field dA is constant in each cusp. When the harmonic component of A satisfies some quantified condition, the spectrum of −Δ _A is discrete. In this case, we prove that the counting function of the eigenvalues of −Δ _A satisfies the classical Weyl formula, even when dA=0.     

Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces

When estimating solutions of dissipative partial differential equations in L^p-related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian −Δ, lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian (−Δ)^α, the approach of integration by parts no longer applies. In this paper, we obtain lower bounds for the integral involving (−Δ)^α by combining pointwise inequalities for (−Δ)^α with Bernstein's inequalities for fractional derivatives. As an application of these lower bounds, we establish the existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces. The generalized Navier-Stokes equations are the equations resulting from replacing −Δ in the Navier-Stokes equations by (−Δ)^α.     

Dispersion of quasi-particles in high Tc cuprates

The interplay between superconductivity (SC) and antiferromagnetism (AFM) is studied in strongly correlated systems of high T _c Cuprate superconductors. It is assumed that superconductivity arises due to BCS pairing mechanism in presence of AFM in Cu lattices of Cu-O planes. The total Hamiltonian of the system is mean field one and has been solved exactly by writing the equations of motion for the single particle Green’s functions. Equations for the appropriate single particle co-relation functions are derived and the order parameters corresponding to SC and AFM are determined. It is assumed that the Fermi energy ∈ _F = 0 and the renormalized localized f energy level coincide with the Fermi level. All the quantities in the final equation for h and Δ are made dimensionless by dividing by 2t, where t is the hopping integral. The temperature dependent values of staggered magnetic field (h) and SC gap (Δ) were determined by solving self-consistent equations for h and Δ. The quasiparticle energy bands are function of AFM gap (h), SC gap (Δ) and hybridization (V). Then the dispersion of quasi-particles are studied at different temperatures by considering temperature dependent values of h and Δ and varying other different model parameters.      

An Extension of the Beale-Kato-Majda Criterion for the Euler Equations

 The Beale-Kato-Majda criterion asserts that smooth solutions to the Euler equations remain bounded past T as long as is finite, ohgr; being the vorticity. We show how to replace this by a weaker statement, on , where Δ^j is a frequency localization around . Received: 27 February 2002 / Accepted: 29 July 2002 Published online: 14 November 2002     

Self-consistent numerical study of pure and impurity doped three-band Hubbard model

The three-band Hubbard model — both pure and with static non-magnetic impurities — has been studied within a self-consistent numerical Hartree-Fock (HF) scheme. The system shows nesting properties only in the absence of direct O-O hopping. Spin excitations in the system are gapless with the existence of a Goldstone mode in the broken-symmetry state. The variation of spinwave velocity with Cu-site Coulomb repulsion shows a (1/(2U _d)+(1/Δ)) dependence in the strong-coupling limit. Each non-magnetic impurity in the system gives rise to two gap states for a particular spin and the local moment produced is robust even at finite concentration of mobile hole doping. The gapless Goldstone mode is preserved even in case of unequal concentration of impurities on the two sublattices.     

Integrable Fredholm Operators and Dual Isomonodromic Deformations

The Fredholm determinants of a special class of integrable integral operators K supported on the union of m curve segments in the complex λ-plane are shown to be the τ-functions of an isomonodromic family of meromorphic covariant derivative operators , having regular singular points at the 2m endpoints of the curve segments, and a singular point of Poincaré index 1 at infinity. The rank r of the corresponding vector bundle over the Riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel. The matrix Riemann–Hilbert problem method is used to deduce an identification of the Fredholm determinant as a τ-function in the sense of Segal–Wilson and Sato, i.e., in terms of abelian group actions on the determinant line bundle over a loop space Grassmannian. An associated dual isomonodromic family of covariant derivative operators , having rank n= 2m, and r finite regular singular points located at the values of the exponents defining the kernel of K is derived. The deformation equations for this family are shown to follow from an associated dual set of Riemann–Hilbert data, in which the r?les of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged. The operators are analogously associated to an integral operator whose Fredholm determinant is equal to that of K. Received: 10 June 1997 / Received revised: 16 February 2001 / Accepted: 27 November 2001     

Asymptotic Statistics of Zeroes for the Lamé Ensemble

The Lamé polynomials naturally arise when separating variables in Laplace's equation in elliptic coordinates. The products of these polynomials form a class of spherical harmonics, which are joint eigenfunctions of a quantum completely integrable (QCI) system of commuting, second-order differential operators P ₀=Δ, P ₁,…,P ^{N −1} acting on C ^∞(? ^N ). These operators naturally depend on parameters and thus constitute an ensemble. In this paper, we compute the limiting level-spacings distributions for the zeroes of the Lamé polynomials in various thermodynamic, asymptotic regimes. We give results both in the mean and pointwise, for an asymptotically full set of values of the parameters. Received: 17 January 2001 / Accepted: 14 May 2001     

On amorphization and nanocomposite formation in Al-Ni-Ti system by mechanical alloying

Amorphous structure generated by mechanical alloying (MA) is often used as a precursor for generating nanocomposites through controlled devitrification. The amorphous forming composition range of ternary Al-Ni-Ti system was calculated using the extended Miedema’s semi-empirical model. Eleven compositions of this system showing a wide range of negative enthalpy of mixing (−ΔH ^mix) and amorphization (−ΔH ^amor) of the constituent elements were selected for synthesis by MA. The Al₈₈Ni₆Ti₆ alloy with relatively small negative ΔH ^mix (−0.4 kJ/mol) and ΔH^amor (−14.8 kJ/mol) became completely amorphous after 120 h of milling, which is possibly the first report of complete amorphization of an Al-based rare earth element free Al-TM-TM system (TM = transition metal) by MA. The alloys of other compositions selected had much more negative ΔH^mix and H^amor; but they yielded either nanocomposites of partial amorphous and crystalline structure or no amorphous phase at all in the as-milled condition, evidencing a high degree of stability of the intermetallic phases under the MA environment. Hence, the negative ΔH ^mix and ΔH ^amor are not so reliable for predicting the amorphization in the present system by MA     

Magnetic behaviour of doped dimer compounds Sr<Subscript>3</Subscript>Cr<Subscript>2−<Emphasis Type="Italic">x</Emphasis></Subscript>M<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>O<Subscript>8</Subscript> (M = V,Mn)

Wilson chains, based on a logarithmic discretization of a continuous spectrum, are widely used to model an electronic (or bosonic) bath for Kondo spins and other quantum impurities within the numerical renormalization group method and other numerical approaches. In this short note we point out that Wilson chains can not serve as thermal reservoirs as their temperature changes by a number of order ΔE when a finite amount of energy ΔE is added. This proves that for a large class of non-equilibrium problems they cannot be used to predict the long-time behavior.     

Mott-insulator-superfluid-liquid transition in a one-dimensional bosonic Hubbard model: Quantum Monte Carlo method

The values of the insulator gap Δ in one-dimensional systems of interacting bosons described by the Hubbard Hamiltonian are calculated at low temperatures by the quantum world-line Monte Carlo algorithm. The dependence of Δ on the size of the system, the temperature, and the parameters of the model is investigated. It is shown that a chain with N _a=50 sites is already sufficient to estimate the thermodynamic value of the critical quantity (t/U)_c for which a transition from the insulator into the superfluid state occurs in a commensurate system. To within the computational error, this value, (t/U)_c=0.300±0.005, agrees with the value (t/U)_c=0.304±0.002 obtained previously by the combined “exact diagonalization + renormalization-group analysis” method. The characteristic Kosterlitz-Thouless behavior of the insulator gap is demonstrated near the critical region: Δ∼exp[−b(1−t/t _c)^−1/2]. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 2, 92–96 (25 July 1996)     

Energy landscapes, lowest gaps, and susceptibility of elastic manifolds at zero temperature

We study the effect of an external field on (1 + 1) and (2 + 1) dimensional elastic manifolds, at zero temperature and with random bond disorder. Due to the glassy energy landscape the configuration of a manifold changes often in abrupt, “first order”-type of large jumps when the field is applied. First the scaling behavior of the energy gap between the global energy minimum and the next lowest minimum of the manifold is considered, by employing exact ground state calculations and an extreme statistics argument. The scaling has a logarithmic prefactor originating from the number of the minima in the landscape, and reads ΔE ₁∼L ^θ[ln(L _z L ^{- ζ})]^-1/2, where ζ is the roughness exponent and θ is the energy fluctuation exponent of the manifold, L is the linear size of the manifold, and L_z is the system height. The gap scaling is extended to the case of a finite external field and yields for the susceptibility of the manifolds ∼L ^{2D + 1 - θ}[(1 - ζ)ln(L)]^1/2. We also present a mean field argument for the finite size scaling of the first jump field, h ₁∼L ^{d - θ}. The implications to wetting in random systems, to finite-temperature behavior and the relation to Kardar-Parisi-Zhang non-equilibrium surface growth are discussed. Received December 2000 and Received in final form April 2001     

Observation of Δ+→ pπ0 decay in heavy-ion collisions

Proton-π⁰ coincidences have been measured at the beam energy of 180A MeV in the reaction Ar+Ca studied by TAPS at SIS/GSI. In the proton-π⁰ invariant mass spectrum we observe a significant excess of strength above the background obtained by event mixing. We attribute this signal to the strength distribution N _Δ of the Δ baryonic resonance. No correlation is observed in the case of deuteron-π⁰ coincidences. Assuming isotropic emission of π⁰ and Δ⁺ from a midrapidity thermal source and isospin symmetry, we determined the global N _Δ/N _πratio of 0.79 ±0.30(stat) ±0.2(syst). This value indicates that most pions produced at subthreshold energy in heavy-ion reaction are mediated by the Δ-resonance. Received: 7 April 2000 / Accepted: 29 August 2000     

Stochastic broadening of signals in distorsionless transmission lines

The paper presents a theory of a stochastic continuous transmission line in which the series inductanceLΔ[1+l(x)], series resistanceRΔ[1+r(x)], shunt capacitanceCΔ[1+c(x)], and shunt conductanceGΔ[1+g(x)] are defined as Gaussian random functions. (The continuous line is considered as a limiting case of a lumped transmision line.) The non-negative random functionsL(x),R(x),C(x), andG(x) are chosen as delta-correlated, i.e. their correlation function is of the formΘδ(x′ −x″) whereΘ is a 4×4 positive definedx-independent matrix. Propagation of a signal of Gaussian shape is analyzed. A special attention is devoted to the so-called distorsionless lines defined by the deterministic conditionR/L=G/C. As a consequence of the stochasticity of the functionsl(x),r(x),c(x), andg(x), transmitted signals do become distorted: they become broadened. An explicit formula for this broadening is derived. This work has been supported by the Slovak Grant Agency VEGA under contract No. 1/4319/97.     

Interplay of superconductivity and antiferromagnetism in SmRh4B4

A microscopic theory of interplay of superconductivity and antiferromagnetism in rare earth ternary systems is developed from first principles for less than half filledf atomic shells. Self consistent equations for the superconducting order parameter Δ and magnetic order parameter Γ, are derived using a Green’s function technique and equation of motion method. The theory is applied to explain the experimental results in the antiferromagnetic superconductor SmRh₄B₄. The present model explains true coexistence of superconductivity and antiferromagnetism and the suppression of superconductivity by antiferromagnetism. The behaviour of superconducting order parameter (Δ), magnetic order parameter (Γ), the specific heat, the density of states, free energy and critical field (H _c) is also studied for the system SmRh₄B₄.     

Some consequences of neutral current systematics

Two kinds of general consequences of the ΔS=0 weak hadron neutral current independent of a gauge model are presented. Firstly are results which depend on the quark parton model. These involve bounds among neutrino inclusive cross-section and a bound onQ(Z, N) in terms of these inclusive cross-sections. Secondly are results which are independent of the quark-parton model and depend only on the SU(3) structure of the most general ΔS=0 neutral current. These tests of isopin and speciallyU-spin properties of the current are given forv+N→v+hadron+anything,v+N→v+baryon+meson ande ⁺ e ⁻→baryon+anti-baryon. In addition some conjectures are made with regard to the semi-inclusive neutrino-reactions using the quark parton model.     

Analysis of Δ<Superscript>0</Superscript>(1232) production in collisions of protons with carbon nuclei at 4.2 GeV/c

The production of Δ⁰(1232)-resonances in p+¹²C collisions at 4.2 GeV/c was analyzed with 4π acceptance. The mass distribution of Δ⁰(1232) was reconstructed using an angular criterion. The fraction of charged π ⁻-mesons coming from Δ⁰(1232) decay was estimated and compared to those obtained in earlier works. The momentum, transverse momentum, kinetic energy, and rapidity distributions as well as invariant cross sections of Δ⁰(1232)-resonances were reconstructed in the laboratory frame. The mean kinematical characteristics of the reconstructed Δ⁰(1232) were compared to those of participant protons in experiment and within some of the models. The freeze-out temperature of Δ⁰(1232) estimated in the present analysis was compared with those obtained using different methods for Δ(1232) produced with other sets of colliding nuclei at various incident energies. The relative number of nucleons excited to Δ⁰ (1232) at freeze-out conditions in p+¹²C collisions was estimated.     

On the Quantum Invariants for the Spherical Seifert Manifolds

We study the Witten–Reshetikhin–Turaev SU(2) invariant for the Seifert manifolds S ³/Gamma where Γ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of modular forms with half-integral weight, and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the Eichler integral. We further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to Γ.