World currency exchange rate cross-correlations

World currency network constitutes one of the most complex structures that is associated with the contemporary civilization. On a way towards quantifying its characteristics we study the cross correlations in changes of the daily foreign exchange rates within the basket of 60 currencies in the period December 1998–May 2005. Such a dynamics turns out to predominantly involve one outstanding eigenvalue of the correlation matrix. The magnitude of this eigenvalue depends however crucially on which currency is used as a base currency for the remaining ones. Most prominent it looks from the perspective of a peripheral currency. This largest eigenvalue is seen to systematically decrease and thus the structure of correlations becomes more heterogeneous, when more significant currencies are used as reference. An extreme case in this later respect is the USD in the period considered. Besides providing further insight into subtle nature of complexity, these observations point to a formal procedure that in general can be used for practical purposes of measuring the relative currencies significance on various time horizons.     

The first eigenvalue of the Dirac operator on locally reducible Riemannian manifolds

We prove a lower estimate for the first eigenvalue of the Dirac operator on a compact locally reducible Riemannian spin manifold with positive scalar curvature. We determine also the universal covers of the manifolds on which the smallest possible eigenvalue is attained.     

Correlated Wishart matrices and critical horizons

We discuss a practical method to determine the eigenvalue spectrum of the empirical correlation matrix. The method is based on analysis of behavior of a conformal map at a critical horizon which is defined as a border line of the physical Riemann sheet of this map. The map is a convenient representation of the Marčenko-Pastur equation.     

The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

We give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space G/K.     

An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one-form

An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold of positive scalar curvature admitting a parallel one-form is found. The possible universal covering spaces of the manifolds on which the smalles possible eigenvalue is attained are also listed. Moreover, a complete classification of the compact odd-dimensional manifolds whose universal covering space is Sⁿ⁻¹ × is given in the limiting case. All such manifolds are diffeomorphic but not necessarily isometric to Sⁿ⁻¹ × S¹.     

Darboux Transformation and Soliton Solutions for InhomogeneousCoupled Nonlinear Schrödinger Equations with Symbolic Computation

With the aid of computation, we consider the variable-coefficient coupled nonlinear Schrödinger equations with the effects of group-velocity dispersion, self-phase modulation and cross-phase modulation, which have potential applications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers. Based on the obtained nonisospectral linear eigenvalue problems (i.e. Lax pair), we construct the Darboux transformation for such a model to derive the optical soliton solutions. In addition, through the one- and two-soliton-like solutions, we graphically discuss the features ofpicosecond solitons in inhomogeneous optical fibers.     

可积开边界条件下的q形变玻色子模型

利用代数Bethe Ansatz方法在可积开边界条件下推广了q形变玻色子模型，得到可积开边界条件下此模型的哈密顿量及其本征方程.该工作可为在更小尺度下研究具有相互作用的玻色子系统提供有效的理论基础. 关键词： 代数Bethe Ansatz q形变玻色子模型')" href="#">q形变玻色子模型 开边界 可积系统     

Optimization of spectral functions of Dirichlet–Laplacian eigenvalues

We consider the shape optimization of spectral functions of Dirichlet–Laplacian eigenvalues over the set of star-shaped, symmetric, bounded planar regions with smooth boundary. The regions are represented using Fourier-cosine coefficients and the optimization problem is solved numerically using a quasi-Newton method. The method is applied to maximizing two particular nonsmooth spectral functions: the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue, both of which are generalizations of the Payne–Pólya–Weinberger ratio. The optimal values and attaining regions for n ? 13 are presented and interpreted as a study of the range of the Dirichlet–Laplacian eigenvalues. For both spectral functions and each n, the optimal attaining region has multiplicity two nth eigenvalue.     

On the investigation of nonlinear waves of the Hirota and the Maxwell–Bloch equation in nonlinear optics

In this paper, considering the Hirota and Maxwell-Bloch (H-MB) equations which is governed by femtosecond pulse propagation through two-level doped fibre system, we construct the Darboux transformation of this system through linear eigenvalue problem. Using this Daurboux transformation, we generate multi-soliton, positon, and breather solutions (both bright and dark breathers) of the H-MB equations. Finally, we also construct the rogue wave solutions of the above system.     

Elliptic Eigenvalue Problems with Large Drift and Applications to Nonlinear Propagation Phenomena

This paper is concerned with the asymptotic behaviour of the principal eigenvalue of some linear elliptic equations in the limit of high first-order coefficients. Roughly speaking, one of the main results says that the principal eigenvalue, with Dirichlet boundary conditions, is bounded as the amplitude of the coefficients of the first-order derivatives goes to infinity if and only if the associated dynamical system has a first integral, and the limiting eigenvalue is then determined through the minimization of the Dirichlet functional over all first integrals. A parabolic version of these results, as well as other results for more general equations, are given. Some of the main consequences concern the influence of high advection or drift on the speed of propagation of pulsating travelling fronts.The third author was partially supported by a NSF grant.     

Eigenvalues of the Kählerian Dirac Operator

We get estimates on the eigenvalues of the Kählerian Dirac operator in terms of the eigenvalues of the scalar Laplace–Beltrami operator. In odd complex dimension, these estimates are sharp, in the sense that, for the first eigenvalue, they reduce to Kirchberg's inequality.     

Formation of Superheavy Nuclei in Massive Fusion Reactions

We recently performed a series of improvement on evaluation of eigenvalues without complicated iterations.In this work we first discuss evaluation of the lowest eigenvalue for given systems,by which one conveniently obtains the value of the lowest eigenvalue based on the dimension and width of given matrix.We also discuss a strong correlation between eigenvalues and diagonal matrix elements for large matrices,by which one is able to predict eigenvalues approximately without iterations.     

Critical value computation for H∞-Riccati difference equations

In designing finite horizon discrete time H_∞ controllers, the associated H_∞-Riccati difference equations must be solved. But the Riccati equation has a non-negative solution only when γ⁻² is small enough. So it is important to get the upper bound of the parameter, i.e., the critical value that ensures the existence of the solution to the Riccati equation. The solution sequence of the Riccati difference equation can be constructed by the conjoined basis of an associated linear Hamiltonian difference system. Based on this expression and the Hamiltonian difference system eigenvalue theorems, the equivalence between the critical value and the first order eigenvalue of the linear Hamiltonian difference system is presented. Since the critical value is also shown to be the fundamental eigenvalue of a generalized Rayleigh quotient, an extended form of Wittrick-Williams algorithm is presented to search this value.     

Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations

To explore band structures of three-dimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell equations. The solutions of these eigenvalue problems cannot be computed effectively unless a suitable combination of eigenvalue solver and preconditioner is chosen. Taking eigenvalue problems due to Yee’s scheme as examples, we propose using Krylov–Schur method and Jacobi–Davidson method to solve the resulting eigenvalue problems. For preconditioning, we derive several novel preconditioning schemes based on various preconditioners, including a preconditioner that can be solved by Fast Fourier Transform efficiently. We then conduct intensive numerical experiments for various combinations of eigenvalue solvers and preconditioning schemes. We find that the Krylov–Schur method associated with the Fast Fourier Transform based preconditioner is very efficient. It remarkably outperforms all other eigenvalue solvers with common preconditioners like Jacobi, Symmetric Successive Over Relaxation, and incomplete factorizations. This promising solver can benefit applications like photonic crystal structure optimization.     

Density matrix for an electron confined in quantum dots under uniform magnetic field and static electrical field

Using unitary transformations, this paper obtains the eigenvalues and the common eigenvector of Hamiltonian and a new-defined generalized angular momentum (L_z) for an electron confined in quantum dots under a uniform magnetic field (UMF) and a static electric field (SEF). It finds that the eigenvalue of L_z just stands for the expectation value of a usual angular momentum l_z in the eigen-state. It first obtains the matrix density for this system via directly calculating a transfer matrix element of operator \exp( -\beta H) in some representations with the technique of integral within an ordered products (IWOP) of operators, rather than via solving a Bloch equation. Because the quadratic homogeneity of potential energy is broken due to the existence of SEF, the virial theorem in statistical physics is not satisfactory for this system, which is confirmed through the calculation of thermal averages of physical quantities.     

Diffusion model for annealing algorithm with perturbation quadratic approximation

We have dealt with the simulated annealing algorithm as a suitable method for finding the global minimum of a functionf(x) of one variable. The convergence properties of this algorithm have been investigated via diffusion equation solved at a fixed temperature. The convergence rate is mainly governed by the first subdominant eigenvalue of the generator. We have tried to calculate approximately this eigenvalue for arbitrary minimized functionf(x). These calculations lead us to propose some parameter for controlling the temperature.     

Universal Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains

In this paper we study the eigenvalues of the buckling problem on domains in a unit sphere. We obtain universal bounds on the (k + 1)^th eigenvalue in terms of the first k eigenvalues independent of the domains. Partially supported by FEMAT. Partially supported by CNPq, Pronex and Proex.     

Propagating waves and edge vibrations in anisotropic composite cylinders

The entire dispersive spectra of a cylinder with cylindrical anisotropy are determined from three different algebraic eigenvalue problems deducible from the same finite element formulation. The displacement vector v in this version of the finite element method has the form f(r) exp i(εz + nθ + ωt) with the radial dependence f(r) taken as quadratic interpolation polynomials. Therefore, this discretization procedure allows a cylinder with radially inhomogeneous material properties to be modeled. The three different algebraic eigenvalue problems that emerge depend on whether the axial wave number ε or the natural frequency ω is regarded as the eigenvalue parameter and on the real, purely imaginary or complex nature of ε. For ε specified as real, an eigenvalue problem results for the natural frequencies ω_i for waves propagating along the z-direction of a cylinder of infinite extent. When ε is specified to be purely imaginary, then an algebraic eigenvalue problem governing the edge vibrations (end modes) of a semi-infinite cylinder is obtained. The third eigenvalue problem can be obtained by considering ω to be prescribed and regarding ε as the eigenvalue parameter. The algebraic eigenvalue problem that results is quadratic in the eigenvalue parameter and admits solutions for ε which may be real, purely imaginary or complex. Complex ε's correspond to edge vibrations in a cylinder which are exponentially damped trigonometric wave forms. Moreover, for the case ω = 0, the eigenvalue analysis yields ε as the characteristic inverse decay lengths for systems of elastostatic self-equilibrated edge effects in the context of St. Venant's principle. All the eigenvalue problems are solved by efficient techniques based on subspace iteration. Examples of two four-layer angle-ply cylinders are presented to illustrate this comprehensive finite element analysis.     

Time series analysis and Monte Carlo methods for eigenvalue separation in neutron multiplication problems

A time series approach has been applied to the nuclear fission source distribution generated by Monte Carlo (MC) particle transport in order to calculate the non-fundamental mode eigenvalues of the system. The novel aspect is the combination of the general technical principle of projection pursuit for multivariate data with the neutron multiplication eigenvalue problem in the nuclear engineering discipline. Proof is thoroughly provided that the stationary MC process is linear to first order approximation and that it transforms into one-dimensional autoregressive processes of order one (AR(1)) via the automated choice of projection vectors. The autocorrelation coefficient of the resulting AR(1) process corresponds to the ratio of the desired mode eigenvalue to the fundamental mode eigenvalue. All modern MC codes for nuclear criticality calculate the fundamental mode eigenvalue, so the desired mode eigenvalue can be easily determined.     

Investigating the dispersive properties of the three-dimensional photonic crystals with face-centered-cubic lattices containing epsilon-negative materials

In this paper, dispersive properties of three-dimensional (3D) photonic crystals (PCs) with face-centered-cubic (fcc) lattices composed of the isotropic positive-index materials and epsilon-negative materials are theoretically investigated based on a modified plane wave expansion (PWE) method. The eigenvalue equations of such structure (spheres with epsilon-negative materials inserted in the dielectric background) are deduced. The band structures can be obtained by solving such nonlinear eigenvalue equations. It can be obviously seen that a photonic band gap (PBG), a flat band region, and two stop band gaps (SBGs) in the Г-X and Г-L directions appear, respectively. The results show that the upper edges of flat band region cannot be tuned by any parameters except for the electronic plasma frequency. The first PBG and the first SBGs above the flat band region in the Г-X and Г-L directions for the 3D PCs can be modulated by the filling factor, relative dielectric constant and electronic plasma frequency, respectively. However, the damping factor has no effect on the locations of the first PBG and the first SBGs above the flat band region in the Г-X and Г-L directions. These results may provide theoretical instructions to design the future optoelectronic and communication devices containing epsilon-negative materials.