Formulas For Elements Research Articles

A fast algorithm for computing a matrix transform used to detect trends in noisy data

A recently discovered universal rank-based matrix method to extract trends from noisy time series is described in Ierley and Kostinski (2019) but the formula for the output matrix elements, implemented there as an open-access supplement MATLAB computer code, is O(N4), with N the matrix dimension. This can become prohibitively large for time series with hundreds of sample points or more. Based on recurrence relations, here we derive a much faster O(N2) algorithm and provide code implementations in MATLAB and in open-source JULIA. In some cases one has the output matrix and needs to solve an inverse problem to obtain the input matrix. A fast algorithm and code for this companion problem, also based on the recurrence relations, are given. Finally, in the narrower, but common, domains of (i) trend detection and (ii) parameter estimation of a linear trend, users require, not the individual matrix elements, but simply their accumulated mean value. For this latter case we provide a yet faster O(N) heuristic approximation that relies on a series of rank one matrices. These algorithms are illustrated on a time series of high energy cosmic rays with N>4×104. Program summaryProgram Title: Pfromdata, QofP, mbasisandcoeffs, nonzerop, Qavgapprox, PofQ, mexact, CodeTestingCPC Library link to program files:http://dx.doi.org/10.17632/mkcxrky9jc.1Licensing provisions: MITProgramming language: MATLAB and JuliaNature of problem: An order-rank data matrix and its transform to a stable form are used repeatedly to detect and/or extract trends from noisy data. An efficient yet accurate calculation of the matrix transform is therefore required.Solution method: We introduce and apply an analytic recursion relation, which speeds up the execution of the matrix transform from O(N4) arithmetic operations to O(N2). Since this matrix transform is called often during optimization, our improvement allows for far shorter optimization times, for a given sample size. For example, a transform whose time is extrapolated to an unrealistic 75 days on a Dell personal laptop computer with a 2.2 GHz quad-core AMD processor running 32 bit MATLAB version R2015b on 64 bit Windows 10 (N=5000), now takes a fraction of a second.

Anisotropic turbulence-induced statistical properties evolution of a radially polarized partially coherent pulsed beam on propagation

In this study, we introduce a new kind of partially coherent vector beam with Gaussian distribution in the spatial, temporal and correlation components named a radially polarized partially coherent Gaussian Schell-model pulsed (GSMP) beam as a natural extension of scalar Gaussian Schell-model pulsed beam. Analytical formulas for the two-frequency cross-spectral density matrix elements of this beam are derived based on the extended Huygens–Fresnel theory. Within the frame of extend anisotropic non-Kolmogorov spectrum, the effects of asymmetry anisotropic turbulence eddies on the statistical properties of a radially polarized partially coherent GSMP beam are investigated numerically. Our work improves the classical theory of stochastic electromagnetic beams on propagation and may be useful for applications involving free-space optical communications.

An expansion formula for type A and Kronecker quantum cluster algebras

We introduce an expansion formula for elements in quantum cluster algebras associated to type A and Kronecker quivers with principal quantization. Our formula is parametrized by perfect matchings of snake graphs as in the classical case. In the Kronecker type, the coefficients are q-powers whose exponents are given by a weight function induced by the lattice of perfect matchings. As an application, we prove that a reflectional symmetry on the set of perfect matchings satisfies Stembridge's q=−1 phenomenon with respect to the weight function. Furthermore, we discuss a relation of our expansion formula to generating functions of BPS states.

An Analogue of the Perelomov-Popov Formula for the Lie Superalgebra q(N)

In this short note we study the center of the universal enveloping algebra of the strange Lie superalgebra Q(N). We obtain an analogue of the well known Perelomov-Popov formula (1968) for central elements of this algebra - an expression of the central characters through the highest weight parameters.

On spaces associated with invariant divisors on Galois covers of Riemann surfaces and their applications

Let $f:X \to S$ be a Galois cover of Riemann surfaces, with Galois group $G$. In this paper we analyze the $G$-invariant divisors on $X$, and their associated spaces of meromorphic functions, differentials, and $q$-differentials. We generalize the trace formula for non-trivial elements of $G$ on $q$-differentials, as well as the Chevalley--Weil Formula. When $G$ is Abelian or when the genus of $S$ is 0 we prove additional results, and we also determine the non-special $G$-invariant divisors when both conditions are satisfied.

Propagation properties of radially polarized multi-Gaussian Schell-model beams in oceanic turbulence.

By utilizing the extended Huygens-Fresnel principle, we derive the analytical formulas for the cross-spectral density matrix elements of a radially polarized multi-Gaussian Schell-model (RPMGSM) beam propagating in oceanic turbulence. Effects of beam parameters and oceanic turbulence parameters on the propagation properties of RPMGSM beams are investigated in detail by numerical simulation. Our results show that the RPMGSM beam with larger beam order M has an advantage over the radially polarized Gaussian Schell-model beam for reducing turbulence-induced degradation. Compared with temperature-induced turbulence fluctuation, the salinity-induced turbulence fluctuation makes a greater contribution to the influence on propagation properties of RPMGSM beams.

Product Formula, Global Relations, and Polyadic Numbers

In the paper, the role of the product formula for nonzero elements of an algebraic number field of finite degree over the field of rationals in several problems of number theory is discussed.

Boundary matrices for the higher spin six vertex model

In this paper we consider solutions to the reflection equation related to the higher spin stochastic six vertex model. The corresponding higher spin R-matrix is associated with the affine quantum algebra Uq(sl(2)ˆ). The explicit formulas for boundary K-matrices for spins s=1/2,1 are well known. We derive difference equations for the generating function of matrix elements of the K-matrix for any spin s and solve them in terms of hypergeometric functions. As a result we derive the explicit formula for matrix elements of the K-matrix for arbitrary spin. In the lower- and upper- triangular cases, the K-matrix simplifies and reduces to simple products of q-Pochhammer symbols.

Trace formula and inverse nodal problem for a conformable fractional Sturm-Liouville problem

ABSTRACTIn this paper, we have developed the spectral theory for a conformable fractional Sturm-Liouville problem with boundary conditions which include conformable fractional derivatives of order α, , and prove a completeness theorem and an expansion theorem. We obtain the canonical infinite product constructed by the zero set of the characteristic function of . Also, we calculate the regularized trace formula of the eigenvalues and investigate the inverse nodal problem for this problem with real-valued coefficients on a finite interval. The oscillation of the eigenfunctions for sufficiently large n is established, and an asymptotic formula for elements constructed by of nodal points is obtained. The uniqueness theorem is proved, and an effective procedure for solving the inverse problem is given. Finally, we present two examples to illustrate our theoretical findings.

Calculation of Elements of Spin Groups Using Method of Averaging in Clifford’s Geometric Algebra

We present a method of computing elements of spin groups in the case of arbitrary dimension. This method generalizes Hestenes method for the case of dimension 4. We use the method of averaging in Clifford's geometric algebra previously proposed by the author. We present explicit formulas for elements of spin group that correspond to the elements of orthogonal groups as two-sheeted covering. These formulas allow us to compute rotors, which connect two different frames related by a rotation in geometric algebra of arbitrary dimension.

Reordering in a multi-parametric family of algebras

This article is devoted to a multi-parametric family of associative complex algebras defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems. General reordering and nested commutator formulas for arbitrary elements in these families are presented, generalizing some well-known results in mathematics and physics. A generalization of this family in three generators is also considered.

Electromagnetic fractional multi-Gaussian Schell-model beams and their statistical properties

A recently introduced class of scalar fractional multi-Gaussian Schell-model beams is extended to electromagnetic domain. The conditions on source parameters guaranteeing that the source generates a physical beam are derived. Analytical formulas for the cross-spectral density matrix elements of the electromagnetic fractional multi-Gaussian Schell-model (EM FMGSM) beams propagating in non-Kolmogorov turbulence are derived. The behaviour of the statistical characteristics, including the spectral density, the degree of coherence and the degree of polarization, of the beam on propagation in free space and non-Kolmogorov turbulence are investigated. It is shown that by choosing appropriate parameters of the beam and the turbulence, we can alleviate influence of the turbulent atmospheric effects effectively. Our findings may be useful in material surface processing and free-space optical communications.

Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence.

Based on the extended Huygens-Fresnel principle, analytical formulas for the cross-spectral density matrix elements of a partially coherent electromagnetic rotating elliptical Gaussian vortex (PCEREGV) beam through non-Kolmogorov turbulence are derived and the corresponding second-order statistics are investigated numerically. The numerical calculations of the PCEREGV beam are first developed to validate the correctness of the analytical results. Our findings show that the transverse spectrum density with an initially elliptical annular shape splits into two light spots during the beam propagation. By choosing appropriate parameters of the beam and the turbulence, we can maximize the spectrum density and control the spectral states of coherence and polarization so as to alleviate the influence of the turbulent atmosphere effectively.

Transition sum rules in the shell model

An important characterization of electromagnetic and weak transitions in atomic nuclei are sum rules. We focus on the non-energy-weighted sum rule (NEWSR), or total strength, and the energy-weighted sum rule (EWSR); the ratio of the EWSR to the NEWSR is the centroid or average energy of transition strengths from an nuclear initial state to all allowed final states. These sum rules can be expressed as expectation values of operators, in the case of the EWSR a double commutator. While most prior applications of the double-commutator have been to special cases, we derive general formulas for matrix elements of both operators in a shell model framework (occupation space), given the input matrix elements for the nuclear Hamiltonian and for the transition operator. With these new formulas, we easily evaluate centroids of transition strength functions, with no need to calculate daughter states. We apply this simple tool to a number of nuclides, and demonstrate the sum rules follow smooth secular behavior as a function of initial energy, as well as compare the electric dipole (E1) sum rule against the famous Thomas-Reiche-Kuhn version. We also find surprising systematic behaviors for ground state electric quadrupole (E2) centroids in the $sd$-shell.

Seidel’s morphism of toric $4$-manifolds

Following McDuff and Tolman's work on toric manifolds [McDT06], we focus on 4-dimensional NEF toric manifolds and we show that even though Seidel's elements consist of infinitely many contributions, they can be expressed by closed formulas. From these formulas, we then deduce the expression of the quantum homology ring of these manifolds as well as their Landau-Ginzburg superpotential. We also give explicit formulas for the Seidel elements in some non-NEF cases. These results are closely related to recent work by Fukaya, Oh, Ohta, and Ono [FOOO11], Gonz\'alez and Iritani [GI11], and Chan, Lau, Leung, and Tseng [CLLT12]. The main difference is that in the 4-dimensional case the methods we use are more elementary: they do not rely on open Gromov-Witten invariants nor mirror maps. We only use the definition of Seidel's elements and specific closed Gromov-Witten invariants which we compute via localization. So, unlike Alice, the computations contained in this paper are not particularly pretty but they do stay on their side of the mirror. This makes the resulting formulas directly readable from the moment polytope.

Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence

Recently, we introduced a new class of radially polarized beams with multi-cosine Gaussian Schell-model(MCGSM) correlation function based on the partially coherent theory (Tang et al., 2017). In this manuscript, we extend the work to study the statistical properties such as the spectral density, the degree of coherence, the degree of polarization, and the state of polarization of the beam propagating in isotropic turbulence with a non-Kolmogorov power spectrum. Analytical formulas for the cross-spectral density matrix elements of a radially polarized MCGSM beam in non-Kolmogorov turbulence are derived. Numerical results show that lattice-like intensity pattern of the beam, which keeps propagation-invariant in free space, is destroyed by the turbulence when it passes at sufficiently large distances from the source. It is also shown that the polarization properties are mainly affected by the source correlation functions, and change in the turbulent statistics plays a relatively small effect. In addition, the polarization state exhibits self-splitting property and each beamlet evolves into radially polarized structure upon propagation.

Bremsstrahlung from few-electron ions

We have developed the new approach which can be used for accurate evaluations of the bremsstrahlung cross-sections in the case of few-electron atoms and ions. Our approach is based on the explicit formula for the electron density distribution in such systems. We derive the closed analytical formula for the matrix elements which are needed for highly accurate computations of atomic form-factors of two-electron atoms and ions. Multiple bremsstrahlung from few-electron ions is considered. We also discuss the energy loss due to bremsstrahlung in a plasma which contains multi-charged ions and free electrons. Bremsstrahlung from a high-temperature plasma is considered as well as its role in the high-temperature burn-up of deuterium plasma.

MQCT. I. Inelastic Scattering of Two Asymmetric-Top Rotors with Application to H2O + H2O.

A mixed quantum/classical theory (MQCT) for the inelastic collision of two asymmetric-top rotor molecules is developed. In this method, the quantum state-to-state transitions between the rotational states of molecules (internal) are treated quantum mechanically using the time-dependent Schrodinger equation, whereas their relative translational motion (responsible for scattering) is treated classically, using the average trajectory approach. Two versions of the formula for transition matrix elements are presented: a straightforward approach that uses numerical multidimensional quadrature over all the internal degrees of freedom and a more standard analytic approach that uses the expansion of the PES over the basis set of spherical harmonics. Adaptation to the case of identical molecules scattering is presented and is applied to the rotational excitation of two water molecules, H2O + H2O, using the PES from recent literature. Calculations of collisional excitation from the ground state of the system into a number of low-lying excited rotational states are carried out in a broad range of energies. Analysis of computed opacity functions shows a rather unusual scattering regime, dominated by a strong anisotropic long-range interaction (dipole-dipole). The coupled-states (CS) approximation is tested and found to agree semiquantitatively (within a factor of 2) with the fully coupled version of the method. Differential cross sections for the elastic scattering indicate a very narrow forward scattering peak.

Sensitivity Analysis of Networks with Fractional Elements

Fractional circuits have attracted the extensive attention of scholars and researchers all around the world in view of their superior performance and extra design freedom compared with conventional circuits. On the one hand, conventional circuit theory could be extended with the development of fractional circuits; on the other hand, the newly developed fractional circuits constitute a new challenge for the analysis and synthesis methods of conventional theory. As is known to all, almost no physical circuit can perform exactly in accordance with the design in view of the practical considerations such as the tolerance of circuit elements, environmental effects and aging problems. Therefore, the analysis of the effects of element parameter variations on circuit performance is of interest and should be considered during the design stage. Hence, it is necessary to conduct a sensitivity analysis for networks with fractional elements. In this paper, the adjoint network method is generalized to analyze the sensitivity of networks with fractional elements. The sensitivity formulae for the multi-port fractional elements are derived with the help of Tellegen’s theorem, and the time domain formulae are also derived. Some linear circuits and nonlinear circuits that contain nonlinear resistors are taken as examples to demonstrate the validity and accuracy of the formulae.

Matrix method of constructing the differential equations of motion of an exoskeleton and its control

Abstract Two mathematical models of rods of variable length from which an exoskeleton can be created, providing comfortable movement of a human in it owing to duplication of the properties of a motion-support apparatus, are considered. Their structure is elucidated on the basis of an analysis of the differential equations of motion, allowing for representing them in vector-matrix form. General regularities of the construction of the matrix elements entering into the system of differential equations of motion are established and generalizing formulae for the matrix elements are obtained. A new matrix method of constructing the differential equations of motion is presented and illustrated by a specific example. This system of equations is solved numerically. The possibility of reinforcing the control actions for control of the exoskeleton motion with a human inside it is considered.