Singular Components Research Articles

Evidence of Mixed-Valence Hydrated Europium-Chloride Phase in Vacuum by Means of Optical and Electronic Spectroscopies

Studying the optical and electronic properties of commercial hydrated EuCl3 salts, we took advantage of the manifestation of the unfilled 4f valence shell in photoluminescence and core-level photoemission spectroscopies to unambiguously identify contributions from both Eu3+ and Eu2+ configurations. Lanthanide ion salt exhibits sharp spectral lines in good agreement with Eu3+ oxidized state. Surprisingly, the photoemission spectrum of europium 4d core level presents a singular asymmetric shape component attributed to Eu2+ species, thus revealing a significant amount of divalent europium. X-ray absorption spectroscopy confirmed the presence of Eu2+ species in proportions that depend on the origin of the salt and on the applied material processing.

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ɛ-uniformly well conditioned or ɛ-uniformly stable to perturbations of the data of the grid problem (here, ɛ is a perturbation parameter, ɛ ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ɛ-uniformly in the maximum norm at an O(N−1lnN + N0−1) rate, where N + 1 and N0 + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ɛ-uniformly well conditioned and ɛ-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ−2lnδ−1 + δ0−1); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N−1lnN and δ0 = N0−1 are the accuracies of the discrete solution in x and t, respectively.

Absence of eigenvalues for the periodic Schrödinger operator with singular potential in a rectangular cylinder

We consider the periodic Schrodinger operator on a d-dimensional cylinder with rectangular section. The electric potential may contain a singular component of the form σ(x, y)δΣ(x,y), where Σ is a periodic system of hypersurfaces. We establish that there are no eigenvalues in the spectrum of this operator, provided that Σ is sufficiently smooth and σ ∈ Lp,loc(Σ), p > d − 1.

Squirals and beyond: substitution tilings with singular continuous spectrum

AbstractThe squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of${ \mathbb{Z} }^{2} $. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalization to bijective block substitutions of trivial height on${ \mathbb{Z} }^{d} $.

Research and Design of Water Pump Diagnosis System Based on Wavelet Transform and DSP

the running status of it can be more comprehensive judged through acquisition and analysis of vibration signal for water pump rotary machinery. Therefore, the collection of vibration signal is the key to monitor equipment running status and diagnose faults. This paper introduces water pump fault diagnosis system based on DSP and wavelet transform, and the system can realize water pump vibration signal acquisition and real-time fault diagnosis. In order to improve the vibration signal to noise ratio and get relatively pure vibration signal, the vibration signal that is collected is de-noised by the wavelet de-noising technology. Due to vibration signal often contain singularity component when the pump device occur fault, therefore, the fault diagnosis information of water pump can be derived from vibration signal singularity analysis, the vibration signal is processed and analyzed by combining the modulus maximum value of wavelet transform ( WTMM ) with index, and the fault characteristic values are obtained. The knowledge base of fault model is obtained by training a large number of fault characteristic values, and suspected fault sets are obtained according to model matching method, finally diagnostic results are received by the intersection of suspected fault sets. The experiment results show that the system can effectively find the faults, distinguish faults type and identify faults degree, a kind of effective method is provided for water pump unit faults diagnosis, and has the certain instruction significance for other mechanical fault diagnosis.

Intentional Growth Training: Developing an Intervention to Increase Personal Growth Initiative

A one-week intervention (Intentional Growth Training; IGT) to increase personal growth initiative (PGI) was created and tested in a series of studies to determine its effect on PGI level. Because PGI predicts psychological well-being and depression, IGT has potential to have considerable benefit on a person's mental health. Study 1 was the initial assessment of whether IGT increased PGI greater than various control conditions. Study 2 employed an enhanced version of IGT and utilised the singular components of IGT determining which components were most impactful at increasing PGI. Study 3 was a narrative analysis of writing samples from Study 2 determining whether participant language varied by condition in use of negative or positive emotion, or personal growth-related words. In Study 1, PGI increased significantly across conditions. In Study 2, the growth activity only condition, and PGI education coupled with the growth activity, produced significant increases in PGI. In Study 3, those receiving only PGI education used more growth-related words than control conditions. The most effective IGT format appears to include both education about PGI and a growth activity. The benefits of utilising IGT are discussed.

Data perturbation stability of difference schemes on uniform grids for a singularly perturbed convection–diffusion equation

An approach to the study of the conditioning of difference schemes and their stability to data perturbations is developed for the Dirichlet problem for a singularly perturbed convection-diffusion ordinary differential equation with the perturbation parameter e, e 2 (0;1]. We consider a standard difference scheme, which is a monotone scheme on a uniform grid, and a special scheme of the grid solution decomposition method. Constructing the special scheme, we use a decomposition of the grid solution into a regular and a singular components that are the solutions to grid subproblems considered on uniform grids. The following facts are proved for the standard scheme: (a) the scheme converges only for N-1 = o(ε) at the rate (N-1 (ε +N-1)-1), , where N +1 is the number of grid nodes, (b) the scheme is not e-uniformly well conditioned and stable to perturbations, (c) the condition number of the scheme has the order (ε-1 δ-2, where δ is the accuracy of the grid solution, δ = δst ≈ N-1 (ε +N-1)-1. In the case of convergence of the standard scheme proved theoretically, the actual accuracy of the computed solution decreases with the decrease in the parameter e under the presence of perturbations and may completely vanish for a sufficiently small e (namely, under the condition t = (lnε-1 +lnδ-2st), where t is the number of digits in the machine word). At the same time, the special scheme of the solution decomposition method converges e-uniformly in the uniform norm at the rate (N-1 lnN); in the variables e and d, the special scheme is e-uniformly well conditioned and stable to grid problem data perturbations; the condition number of the scheme of the decomposition method is of the order (δ-2 lnδ-1), δ = δdec ≈ N-1 lnN.

Fault Detection and Isolation in Inertial Measurement Units Based on -CUSUM and Wavelet Packet

The aim of this paper is to present a fault detection algorithm (FDI) based on signal processing techniques developed for an inertial measurement unit (IMU) with minimal redundancy of fiber optic gyros. In this work the recursive median filter is applied in order to remove impulses (outliers) arising from data acquisition process and parity vector operations, improving the fault detection and isolation performance. The FDI algorithm is divided into two blocks: fault detection (FD) and fault isolation (FI). The FD part of the algorithm is used to guarantee the reliability of the isolation part and is based on parity vector analysis using -CUSUM algorithm. The FI part is performed using parity space projection of the energy subbands obtained from wavelet packet decomposition. This projection is an extension of clustering analysis based on singular value decomposition (SVD) and principal component analysis (PCA). The results of the FD and FI algorithms have shown the effectiveness of the proposed method, in which the FD algorithm is capable of indicating the low-level step bias fault with short delay and a high index of correct decisions of the FI algorithm also with low-level step bias fault.

Power and Energy Characteristics of the Internal Wave Beams in a Continuously Stratified Fluid

Analytic linear theory of disturbances generated by an oscillating compact source in a viscous continuously stratified fluid was constructed. Exact solution of the emission internal waves was constructed taking into account diffusivity effects. Analy- sis is based on set of fundamental equations of incompressible flows. The linearized problem of periodic flows in a continu- ously stratified fluid, generated by an oscillating part of the inclined plane was solved by methods of singular perturbation theory. Source (rectangular or disc) oscillating linearly in a longitudinal and lateral direction. The solutions include regularly perturbed on dissipative component functions (internal waves) and singularly perturbed functions. Only one of the singular components of the flow has an analogue in the homogeneous fluid that is a periodic or Stokes’ flow. Its thickness is defined by a universal micro scale depending on kinematics viscosity coefficient and a buoyancy frequency with a factor depending on the wave slope. Other singular perturbed functions are specific for stratified flows and don’t have analogue in homogene- ous fluid. Their thickness are defined the diffusion coefficient, kinematic viscosity and additional factor depending on geome- try of the problem. We have shown that for real stratified fluid is necessary to consider influence of all dissipative factors (viscosity, stratification, diffusion).

An augmented LSQR method

The LSQR iterative method for solving least-squares problems may require many iterations to determine an approximate solution with desired accuracy. This often depends on the fact that singular vector components of the solution associated with small singular values of the matrix require many iterations to be determined. Augmentation of Krylov subspaces with harmonic Ritz vectors often makes it possible to determine the singular vectors associated with small singular values with fewer iterations than without augmentation. This paper describes how Krylov subspaces generated by the LSQR iterative method can be conveniently augmented with harmonic Ritz vectors. Computed examples illustrate the competitiveness of the augmented LSQR method proposed.

Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula

Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g>=4, a new class of vector-valued modular forms, defined on the Teichmuller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus 4 is proportional to the square root of the products of Thetanullwerte \chi_{68}, which is a proof of the recently rediscovered Klein `amazing formula'. Furthermore, it turns out that the coefficients of such a quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian locus, implying new theta relations involving the latter, \chi_{68} and the theta series corresponding to the even unimodular lattices E_8\oplus E_8 and D_{16}^+. We also find, for g=4, a functional relation between the singular component of the theta divisor and the Riemann period matrix.

Second order global uniformly convergent numerical method for a coupled system of singularly perturbed initial value problems

We consider a coupled system of singularly perturbed first order ordinary differential equations. The equations have small positive parameters of different magnitudes associated with them. A decomposition of the exact solution into smooth and singular components is constructed that is useful for the analysis of the numerical method. We apply the HODIE (High Order Difference approximation with Identity Expansions) technique to construct a second order finite difference scheme and combine this with standard backward Euler difference scheme in a special way on a piecewise-uniform Shishkin mesh to solve the system numerically. It is proved that the method is second order (up to a logarithmic factor) convergent in the maximum norm uniformly in both perturbation parameters. Numerical experiments are conducted to support the theoretical results.

On the Navier-Stokes equations: Existence theorems and energy equalities

Currently available results on the solvability of the Navier-Stokes equations for incompressible non-Newtonian fluids are presented. The order of nonlinearity in the equations may be variable; the only requirement is that it must be a measurable function. Unsteady and steady equations are considered. A lot of attention is paid to the recovery of energy balance, whose violation is theoretically admissible, in particular, in the three-dimensional classical unsteady Navier-Stokes equation. When constructing a weak solution by a limit procedure, a measure arises as a limit of viscous energy densities. Generally speaking, the limit measure contains a nonnegative singular (with respect to the Lebesgue measure) component. It is this singular component that maintains energy balance. Sufficient conditions for the absence of a singular component are studied: in this case, the standard energy equality holds. In many respects, only the regular component of the limit measure is important: in the natural form it is equal to the product of the viscous stress tensor and the gradient of a solution; if this natural form is retained, then the problem is solvable. Conditions are found for the validity of the indicated fundamental representation of the absolutely continuous component of the limit measure.

Sibuya copulas

A new class of copulas referred to as “Sibuya copulas” is introduced and its properties are investigated. Members of this class are of a functional form which was first investigated in the work of M. Sibuya. The construction of Sibuya copulas is based on an increasing stochastic process whose Laplace–Stieltjes transform enters the copula as a parameter function. Sibuya copulas also allow for idiosyncratic parameter functions and are thus quite flexible to model asymmetric dependences. If the stochastic process is allowed to have jumps, Sibuya copulas may have a singular component. Depending on the choice of the process, they may be extreme-value copulas, Lévy-frailty copulas, or Marshall–Olkin copulas. Further, as a special symmetric case, one may obtain any Archimedean copula with Laplace–Stieltjes transform as generator. Besides some general properties of Sibuya copulas, several examples are given and their properties are investigated in more detail.The construction scheme associated to Sibuya copulas provides a sampling algorithm. Further, it can be generalized, for example, to allow for hierarchical structures, or for an additional source of dependence via another copula.

Some results on shuffles of two-dimensional copulas

Using the one-to-one correspondence between two-dimensional copulas and special Markov kernels allows to study properties of T-shuffles of copulas, T being a general Lebesgue-measure-preserving transformation on [0,1], in terms of the corresponding operation on Markov kernels. As one direct consequence of this fact the asymptotic behaviour of iterated T-shuffles STn(A) of a copula A∈C can be characterized through mixing properties of T. In particular it is shown that STn(A) ((1/n)∑i=1nSTi(A)) converges uniformly to the product copula Π for every copula A if and only if T is strongly mixing (ergodic). Moreover working with Markov kernels also allows, firstly, to give a short proof of the fact that the mass of the singular component of ST(A) cannot be bigger than the mass of the singular component of A, secondly, to introduce and study another operator UT:C→C fulfilling ST○UT(A)=A for all A∈C, and thirdly to express ST(A) and UT(A) as ⁎-product of A with the completely dependent copula CT induced by T.

Investigation of multilayer, periodic, multistrip, and multislot waveguiding structures

For complex planer periodic structures, a system of spectral-domain integral equations is obtained. The unknowns in the equations are the coefficients of 2D Fourier transforms of the current density on strips and the tangential electric fields in slots. The number of dielectric layers, strips, and slots is not bounded. The system is solved by means of the Galerkin method with a basis taking into account the edge singularity. The singular (static) component of the kernel is separated and analytically transformed.

Comment on article “Is There a Metabolism of an Urban Ecosystem?” by Golubiewski

Comment on article “Is There a Metabolism of an Urban Ecosystem?” by Golubiewski

Mincle is not essential for controlling Mycobacterium tuberculosis infection

Individually and combined, Toll-like receptors (TLR)-2, -4, -9, nucleotide oligomerization domain (NOD) 2 and NALP3 contribute to the Mycobacterium tuberculosis (Mtb)-induced innate immune response only to a limited extent, particularly in terms of inducing antibacterial protection and granuloma formation in vivo. A singular essential sensory component of this initial response has not been discovered yet. Trehalose-6,6′-dimycolate (TDM), a well known mycobacterial cell wall glycolipid, is believed to be involved in these early inflammatory processes after Mtb infection. Only recently the macrophage inducible C-type lectin (Mincle) was demonstrated as an essential receptor for TDM. However, not much is known about the sensing capacity of Mincle during infection with live mycobacteria. To determine the significance of Mincle during tuberculosis (TB), we analyzed the outcome of Mtb infection in Mincle-deficient mice. Whereas in the absence of Mincle macrophages did not respond to TDM, Mincle-deficient mice were capable of mounting an efficient granulomatous and protective immune response after low and high dose infections with Mtb. Mutant mice generated a normal T helper (TH) 1 and TH17 immune response followed by the induction of efficient macrophage effector mechanisms and control of mycobacterial growth identical to wildtype mice. From our results we conclude that absence of the innate receptor Mincle can be fully compensated for in vivo in terms of sensing Mtb and mounting a protective inflammatory immune response.

Strong stability of a scheme on locally uniform meshes for a singularly perturbed ordinary differential convection-diffusion equation

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a small parameter ɛ (ɛ ∈ (0, 1]) multiplying the higher order derivative is considered. For the problem, a difference scheme on locally uniform meshes is constructed that converges in the maximum norm conditionally, i.e., depending on the relation between the parameter ɛ and the value N defining the number of nodes in the mesh used; in particular, the scheme converges almost ɛ-uniformly (i.e., its accuracy depends weakly on ɛ). The stability of the scheme with respect to perturbations in the data and its conditioning are analyzed. The scheme is constructed using classical monotone approximations of the boundary value problem on a priori adapted grids, which are uniform on subdomains where the solution is improved. The boundaries of these subdomains are determined by a majorant of the singular component of the discrete solution. On locally uniform meshes, the difference scheme converges at a rate of O(min[ɛ−1N−KlnN, 1] + N−1lnN), where K is a prescribed number of iterations for refining the discrete solution. The scheme converges almost ɛ-uniformly at a rate of O(N−1lnN) if N−1 ≤ ɛν, where ν (the defect of ɛ-uniform convergence) determines the required number K of iterations (K = K(ν) ∼ ν−1) and can be chosen arbitrarily small from the half-open interval (0, 1]. The condition number of the difference scheme satisfies the bound κP = O(ɛ−1/Kln1/Kɛ−1δ−(K + 1)/K), where δ is the accuracy of the solution of the scheme in the maximum norm in the absence of perturbations. For sufficiently large K, the scheme is almost ɛ-uniformly strongly stable.

Electromagnetic oscillations in a driven nonlinear resonator: A description of complex nonlinear dynamics

Many intriguing properties of driven nonlinear resonators, including the appearance of chaos, are very important for understanding the universal features of nonlinear dynamical systems and can have great practical significance. We consider a cylindrical cavity resonator driven by an alternating voltage and filled with a nonlinear nondispersive medium. It is assumed that the medium lacks a center of inversion and the dependence of the electric displacement on the electric field can be approximated by an exponential function. We show that the Maxwell equations are integrated exactly in this case and the field components in the cavity are represented in terms of implicit functions of special form. The driven electromagnetic oscillations in the cavity are found to display very interesting temporal behavior and their Fourier spectra contain singular continuous components. This is a demonstration of the existence of a singular continuous (fractal) spectrum in an exactly integrable system.