Series expansion analysis of a tetrahedral cluster spin chain

Using analytical series expansion by continuous unitary transformations we study the magnetic properties of a frustrated tetrahedral spin- chain. Starting from the limit of isolated tetrahedra we analyze the evolution of the ground state energy and the elementary triplet dispersion as a function of the inter-tetrahedral coupling. The quantum phase diagram is evaluated and is shown to incorporate a singlet product, a dimer, and a Haldane phase. Comparison of our results with those from several other techniques, such as density matrix renormalization group, exact diagonalization, bond-operator theory and other numerical series expansion are provided and convincing agreement is found.     

Atomic Decomposition of Hardy Spaces Associated with Certain Laguerre Expansions

Let L _n ^a (x), n=0,1,…, be the Laguerre polynomials of order a>−1. Denote ℓ _n ^a (x)=(n!/Γ(n+a+1))^1/2 L _n ^a (x)e ^−x/2. Let

A singular perturbation problem with discontinuous data in a cuboid

We analyse the asymptotic behaviour of the solution of a 3Dsingularly perturbed convection–diffusion problem withdiscontinuous Dirichlet boundary data defined in a cuboid. Wewrite the solution in terms of a double series and we obtainan asymptotic approximation of the solution when the singularparameter 0. This approximation is given in terms of a finitecombination of products of error functions and characterizesthe effect of the discontinuities on the small -behaviour ofthe solution in the singular layers.     

Asymptotic expansions and convergence rates for Euler’s approximations of semigroups

We provide optimal bounds for errors in Euler’s approximations of semigroups in Banach algebras and of semigroups of operators in Banach spaces. Furthermore, we construct asymptotic expansions for such approximations with optimal bounds for remainder terms. The sizes of errors are controlled by smoothness properties of semigroups. In this paper we use Fourier–Laplace transforms and a reduction of the problem to the convergence rates and asymptotic expansions in the Law of Large Numbers. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09. This paper was written in 2004. In the interim, several related articles were published; let us mention [14, 13, 15].     

G-functions and multisum versus holonomic sequences

The purpose of the paper is three-fold: (a) we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of Gevrey type-1 with rational exponents, (b) we construct a class of G-functions that come from enumerative combinatorics, and (c) we give a counterexample to a question of Zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms. The proofs utilize the notion of a G-function, introduced by Siegel, and its analytic/arithmetic properties shown recently by André.     

Univariate modified Fourier methods for second order boundary value problems

We develop and analyse a new spectral-Galerkin method for the numerical solution of linear, second order differential equations with homogeneous Neumann boundary conditions. The basis functions for this method are the eigenfunctions of the Laplace operator subject to these boundary conditions. Due to this property this method has a number of beneficial features, including an condition number and the availability of an optimal, diagonal preconditioner. This method offers a uniform convergence rate of , however we show that by the inclusion of an additional 2M basis functions, this figure can be increased to for any positive integer M.      

Asymptotic expansions and higher order properties of semi-parametric estimators in a system of simultaneous equations

Asymptotic expansions are made for the distributions of the Maximum Empirical Likelihood (MEL) estimator and the Estimating Equation (EE) estimator (or the Generalized Method of Moments (GMM) in econometrics) for the coefficients of a single structural equation in a system of linear simultaneous equations, which corresponds to a reduced rank regression model. The expansions in terms of the sample size, when the non-centrality parameters increase proportionally, are carried out to O(n⁻¹). Comparisons of the distributions of the MEL and GMM estimators are made. Also, we relate the asymptotic expansions of the distributions of the MEL and GMM estimators to the corresponding expansions for the Limited Information Maximum Likelihood (LIML) and the Two-Stage Least Squares (TSLS) estimators. We give useful information on the higher order properties of alternative estimators including the semi-parametric inefficiency factor under the homoscedasticity assumption.     

The magnetic properties of oxide spinel Li0.5Fe2.5−2xAlxCrxO4 solid solutions

The exchange interactions (J_BB and J_AB are the intra and the inter-sublattice exchange interactions between neighbouring spins, respectively) are obtained by using the general expressions of canting angle and critical temperature obtained by mean field theory of Li_0.5Fe_2.5−2xAl_xCr_xO₄. The expression of magnetic energy of Li_0.5Fe_2.5−2xAl_xCr_xO₄ is obtained for different spin configurations and dilution x. The saturation magnetisation of Li_0.5Fe_2.5-2xAl_xCr_xO₄ is obtained with different values of dilution x. The magnetic phase diagram of Li_0.5Fe_2.5-2xAl_xCr_xO₄ materials is obtained by high temperature series expansions (HTSEs). The critical exponent associated with the magnetic susceptibility of Li_0.5Fe_2.5−2xAl_xCr_xO₄ is deduced.     

MgAl2O4∶Er^3+,Yb^3+荧光粉的制备及其上转换发光性能

以尿素为沉淀剂,采用低温水热法结合煅烧过程制备出MgAl2O4∶Er^3+,Yb^3+上转换荧光粉,并对样品的结构、微观形貌及上转换发光性能予以表征。结果表明,随尿素加入量的增大,产物主形貌由六角片状结构向纳米棒状转变,经1100℃煅烧可得纯相镁铝尖晶石结构,且Er^3+和Yb^3+能有效进入MgAl2O4晶格并占据Mg^2+位置形成均匀固溶体。在980 nm光激发下,MgAl2O4∶1.0%(n/n)Er^3+,x%(n/n)Yb^3+(x=0~8.0)荧光粉表现出在524、545 nm处绿光以及658 nm处的强红光发射,红绿光强度均在5.0%(n/n)Yb^3+掺杂时达到最大,但红绿光强度比却在7.0%(n/n)Yb^3+掺杂时达到最大值5.2,这归因于Er^3+-Er^3+之间交叉弛豫(CR)在红光发射过程中所起的重要作用。通过控制荧光粉中Yb^3+的掺杂量,能初步实现对于黄绿光色度的有效调控。