Search results for: eigenvalues eigenvectors

Authors: Abubakar Sadiq Mensah

Abstract:

The Knowledge of the population growth of a nation is paramount to national planning. The population of a place is studied and a model developed over a period of time, Matrices is used to form model for population growth. The eigenvalue ƛ of the matrix A and its corresponding eigenvector X is such that AX = ƛX is calculated. The stable age distribution of the population is obtained using the eigenvalue and the characteristic polynomial. Hence, estimation could be made using eigenvalues and eigenvectors.

Keywords: eigenvalues, eigenvectors, population, growth/stability

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Authors: Malika Yaici, Kamel Hariche, Tim Clarke

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The relationship between eigenstructure (eigenvalues and eigenvectors) and latent structure (latent roots and latent vectors) is established. In control theory eigenstructure is associated with the state space description of a dynamic multi-variable system and a latent structure is associated with its matrix fraction description. Beginning with block controller and block observer state space forms and moving on to any general state space form, we develop the identities that relate eigenvectors and latent vectors in either direction. Numerical examples illustrate this result. A brief discussion of the potential of these identities in linear control system design follows. Additionally, we present a consequent result: a quick and easy method to solve the polynomial eigenvalue problem for regular matrix polynomials.

Keywords: eigenvalues/eigenvectors, latent values/vectors, matrix fraction description, state space description

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Authors: Chimezie P. Nnanwa, Thomas C. Urama, Patrick O. Ezepue

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In this paper we use Random Matrix Theory to analyze the eigen-structure of the empirical correlations of 82 stocks which are consistently traded in the Nigerian Stock Exchange (NSE) over a 4-year study period 3 August 2009 to 26 August 2013. We apply the Marchenko-Pastur distribution of eigenvalues of a purely random matrix to investigate the presence of investment-pertinent information contained in the empirical correlation matrix of the selected stocks. We use hypothesised standard normal distribution of eigenvector components from RMT to assess deviations of the empirical eigenvectors to this distribution for different eigenvalues. We also use the Inverse Participation Ratio to measure the deviation of eigenvectors of the empirical correlation matrix from RMT results. These preliminary results on the dynamics of asset price correlations in the NSE are important for improving risk-return trade-offs associated with Markowitz’s portfolio optimization in the stock exchange, which is pursued in future work.

Keywords: correlation matrix, eigenvalue and eigenvector, inverse participation ratio, portfolio optimization, random matrix theory

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Authors: Brando Vagenende, Marie-Anne Guerry

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For stochastic matrices of any order, the geometric description of the convex set of eigenvalues is completely known. The purpose of this study is to investigate the subset of the monotone matrices. This type of matrix appears in contexts such as intergenerational occupational mobility, equal-input modeling, and credit ratings-based systems. Monotone matrices are stochastic matrices in which each row stochastically dominates the previous row. The monotonicity property of a stochastic matrix can be expressed by a nonnegative lower-order matrix with the same eigenvalues as the original monotone matrix (except for the eigenvalue 1). Specifically, the aim of this research is to focus on the properties of eigenvalues of monotone matrices. For those matrices up to order 3, there already exists a complete description of the convex set of eigenvalues. For monotone matrices of order at least 4, this study gives, through simulations, more insight into the geometric description of their eigenvalues. Furthermore, this research treats in a geometric and algebraic way the properties of eigenvalues of monotone matrices of order at least 4.

Keywords: eigenvalues of matrices, finite Markov chains, monotone matrices, nonnegative matrices, stochastic matrices

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Authors: Krish Jhurani

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The research paper presents an in-depth analysis of symmetry properties in linear algebraic systems under the operation of non-canonical scalar multiplication structures, specifically semirings, and near-rings. The objective is to unveil the profound alterations that occur in traditional linear algebraic structures when we replace conventional field multiplication with these non-canonical operations. In the methodology, we first establish the theoretical foundations of non-canonical scalar multiplication, followed by a meticulous investigation into the resulting symmetry properties, focusing on eigenvectors, eigenspaces, and invariant subspaces. The methodology involves a combination of rigorous mathematical proofs and derivations, supplemented by illustrative examples that exhibit these discovered symmetry properties in tangible mathematical scenarios. The core findings uncover unique symmetry attributes. For linear algebraic systems with semiring scalar multiplication, we reveal eigenvectors and eigenvalues. Systems operating under near-ring scalar multiplication disclose unique invariant subspaces. These discoveries drastically broaden the traditional landscape of symmetry properties in linear algebraic systems. With the application of these findings, potential practical implications span across various fields such as physics, coding theory, and cryptography. They could enhance error detection and correction codes, devise more secure cryptographic algorithms, and even influence theoretical physics. This expansion of applicability accentuates the significance of the presented research. The research paper thus contributes to the mathematical community by bringing forth perspectives on linear algebraic systems and their symmetry properties through the lens of non-canonical scalar multiplication, coupled with an exploration of practical applications.

Keywords: eigenspaces, eigenvectors, invariant subspaces, near-rings, non-canonical scalar multiplication, semirings, symmetry properties

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Authors: Abdon Choque-Rivero

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The Jacobi system with the Dirichlet boundary condition is considered on a half-line lattice when the coefficients are real valued. The inverse problem of recovery of the coefficients from various data sets containing the so-called transmission eigenvalues is analyzed. The Marchenko method is utilized to solve the corresponding inverse problem.

Keywords: inverse scattering, discrete system, transmission eigenvalues, Marchenko method

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Authors: Taysir Soliman

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One of the current popular clustering techniques is Spectral Clustering (SC) because of its advantages over conventional approaches such as hierarchical clustering, k-means, etc. and other techniques as well. However, one of the disadvantages of SC is the time consuming process because it requires computing the eigenvectors. In the past to overcome this disadvantage, a number of attempts have been proposed such as the Power Iteration Clustering (PIC) technique, which is one of versions from SC; some of PIC advantages are: 1) its scalability and efficiency, 2) finding one pseudo-eigenvectors instead of computing eigenvectors, and 3) linear combination of the eigenvectors in linear time. However, its worst disadvantage is an inter-class collision problem because it used only one pseudo-eigenvectors which is not enough. Previous researchers developed Deflation-based Power Iteration Clustering (DPIC) to overcome problems of PIC technique on inter-class collision with the same efficiency of PIC. In this paper, we developed Parallel DPIC (PDPIC) to improve the time and memory complexity which is run on apache spark framework using sparse matrix. To test the performance of PDPIC, we compared it to SC, ESCG, ESCALG algorithms on four small graph benchmark datasets and nine large graph benchmark datasets, where PDPIC proved higher accuracy and better time consuming than other compared algorithms.

Keywords: spectral clustering, power iteration clustering, deflation-based power iteration clustering, Apache spark, large graph

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Authors: Michael Danilov, Sergei Iskakov, Vladimir Mazurenko

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The present paper describes a high-performance parallel realization of an exact diagonalization solver for quantum-electron models in a shared memory computing system. The proposed algorithm contains a storage format for efficient computing eigenvalues and eigenvectors of a quantum electron Hamiltonian matrix. The results of the test calculations carried out for 15 sites Hubbard model demonstrate reduction in the required memory and good multiprocessor scalability, while maintaining performance of the same order as compressed row storage.

Keywords: sparse matrix, compressed format, Hubbard model, Anderson model

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Authors: Ezgi Kaya, A. Dilek Maden

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A topological index is a number related to graph which is invariant under graph isomorphism. In theoretical chemistry, molecular structure descriptors (also called topological indices) are used for modeling physicochemical, pharmacologic, toxicologic, biological and other properties of chemical compounds. Let G be a graph with n vertices and m edges. For a given edge uv, the quantity nu(e) denotes the number of vertices closer to u than v, the quantity nv(e) is defined analogously. The vertex PI index defined as the sum of the nu(e) and nv(e). Here the sum is taken over all edges of G. The energy of a graph is defined as the sum of the eigenvalues of adjacency matrix of G and the Laplacian energy of a graph is defined as the sum of the absolute value of difference of laplacian eigenvalues and average degree of G. In theoretical chemistry, the π-electron energy of a conjugated carbon molecule, computed using the Hückel theory, coincides with the energy. Hence results on graph energy assume special significance. The Laplacian matrix of a graph G weighted by the vertex PI weighting is the Laplacian vertex PI matrix and the Laplacian vertex PI eigenvalues of a connected graph G are the eigenvalues of its Laplacian vertex PI matrix. In this study, Laplacian vertex PI energy of a graph is defined of G. We also give some bounds for the Laplacian vertex PI energy of graphs in terms of vertex PI index, the sum of the squares of entries in the Laplacian vertex PI matrix and the absolute value of the determinant of the Laplacian vertex PI matrix.

Keywords: energy, Laplacian energy, laplacian vertex PI eigenvalues, Laplacian vertex PI energy, vertex PI index

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Authors: Dia I. Abu-Al-Nadi, M. J. Mismar, T. H. Ismail

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A technique for estimating the direction-of-arrival (DOA) of unknown number of source signals is presented using the eigen-approach. The eigenvector corresponding to the minimum eigenvalue of the autocorrelation matrix yields the minimum output power of the array. Also, the array polynomial with this eigenvector possesses roots on the unit circle. Therefore, the pseudo-spectrum is found by perturbing the phases of the roots one by one and calculating the corresponding array output power. The results indicate that the DOAs and the number of source signals are estimated accurately in the presence of a wide range of input noise levels.

Keywords: array signal processing, direction-of-arrival, antenna arrays, Eigenvalues, Eigenvectors, Lagrange multiplier

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Authors: Soumitr Sanjay Dubey, Shubhajit Roy Chowdhury, Rahul Shrestha

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In this paper, we have presented a novel approach of calculating eigenvalues of any matrix for the first time on Field Programmable Gate Array (FPGA) using Triangular Systolic Arra (TSA) architecture. Conventionally, additional computation unit is required in the architecture which is compliant to the algorithm for determining the eigenvalues and this in return enhances the delay and power consumption. However, recently reported works are only dedicated for symmetric matrices or some specific case of matrix. This works presents an architecture to calculate eigenvalues of any matrix based on QR algorithm which is fully implementable on FPGA. For the implementation of QR algorithm we have used TSA architecture, which is further utilising CORDIC (CO-ordinate Rotation DIgital Computer) algorithm, to calculate various trigonometric and arithmetic functions involved in the procedure. The proposed architecture gives an error in the range of 10−4. Power consumption by the design is 0.598W. It can work at the frequency of 900 MHz.

Keywords: coordinate rotation digital computer, three angle complex rotation, triangular systolic array, QR algorithm

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Authors: Serifenur Cebesoy, Yelda Aygar, Elgiz Bairamov

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In this study, we examine some spectral properties of non-selfadjoint matrix-valued difference equations consisting of a polynomial type Jost solution. The aim of this study is to investigate the eigenvalues and spectral singularities of the difference operator L which is expressed by the above-mentioned difference equation. Firstly, thanks to the representation of polynomial type Jost solution of this equation, we obtain asymptotics and some analytical properties. Then, using the uniqueness theorems of analytic functions, we guarantee that the operator L has a finite number of eigenvalues and spectral singularities.

Keywords: asymptotics, continuous spectrum, difference equations, eigenvalues, jost functions, spectral singularities

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Authors: Shaowei Sun

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Let G be a graph with vertex set V(G)={v_1,v_2,...,v_n} and edge set E(G). For any vertex v belong to V(G), let d_v denote the degree of v. The normalized Laplacian matrix of the graph G is the matrix where the non-diagonal (i,j)-th entry is -1/(d_id_j) when vertex i is adjacent to vertex j and 0 when they are not adjacent, and the diagonal (i,i)-th entry is the di. In this paper, we discuss some bounds on the largest and the second smallest normalized Laplacian eigenvalue of trees and graphs. As following, we found some new bounds on the second smallest normalized Laplacian eigenvalue of tree T in terms of graph parameters. Moreover, we use Sage to give some conjectures on the second largest and the third smallest normalized eigenvalues of graph.

Keywords: graph, normalized Laplacian eigenvalues, normalized Laplacian matrix, tree

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Authors: Thomas Chinwe Urama, Patrick Oseloka Ezepue, Peters Chimezie Nnanwa

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This paper investigates the universal financial dynamics in two dominant stock markets in Sub-Saharan Africa, through an in-depth analysis of the cross-correlation matrix of price returns in Nigerian Stock Market (NSM) and Johannesburg Stock Exchange (JSE), for the period 2009 to 2013. The strength of correlations between stocks is known to be higher in JSE than that of the NSM. Particularly important for modelling Nigerian derivatives in the future, the interactions of other stocks with the oil sector are weak, whereas the banking sector has strong positive interactions with the other sectors in the stock exchange. For the JSE, it is the oil sector and beverages that have greater sectorial correlations, instead of the banks which have the weaker correlation with other sectors in the stock exchange.

Keywords: random matrix theory, cross-correlations, emerging markets, option pricing, eigenvalues eigenvectors, inverse participation ratios and implied volatility

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Authors: V. T. Ngo, D. M. Xie

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Frequently, in the design of machines, some of parameters that directly affect the rotor dynamics of the machines are not accurately known. In particular, bearing stiffness support is one such parameter. One of the most basic principles to grasp in rotor dynamics is the influence of the bearing stiffness on the critical speeds and mode shapes associated with a rotor-bearing system. Taking a rig shafting as an example, this paper studies the lateral vibration of the rotor with multi-degree-of-freedom by using Finite Element Method (FEM). The FEM model is created and the eigenvalues and eigenvectors are calculated and analyzed to find natural frequencies, critical speeds, mode shapes. Then critical speeds and mode shapes are analyzed by set bearing stiffness changes. The model permitted to identify the critical speeds and bearings that have an important influence on the vibration behavior.

Keywords: lateral vibration, finite element method, rig shafting, critical speed

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Authors: Ayşe Dilek Maden

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For a given a simple connected graph, we present some new bounds via a new approach for a special topological index given by the sum of the real number power of the non-zero normalized Laplacian eigenvalues. To use this approach presents an advantage not only to derive old and new bounds on this topic but also gives an idea how some previous results in similar area can be developed.

Keywords: degree Kirchhoff index, normalized Laplacian eigenvalue, spanning tree, simple connected graph

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Authors: Alfi Y. Zakiyyah, Hanni Garminia, M. Salman, A. N. Irawati

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In the terminology of the hypergraph, there is a relation with the terminology graph. In the theory of graph, the edges connected two vertices. In otherwise, in hypergraph, the edges can connect more than two vertices. There is representation matrix of a graph such as adjacency matrix, Laplacian matrix, and incidence matrix. The adjacency matrix is symmetry matrix so that all eigenvalues is real. This matrix is a nonnegative matrix. The all diagonal entry from adjacency matrix is zero so that the trace is zero. Another representation matrix of the graph is the Laplacian matrix. Laplacian matrix is symmetry matrix and semidefinite positive so that all eigenvalues are real and non-negative. According to the spectral study in the graph, some that result is generalized to hypergraph. A hypergraph can be represented by a matrix such as adjacency, incidence, and Laplacian matrix. Throughout for this term, we use Laplacian matrix to represent a complete tripartite hypergraph. The aim from this research is to determine second smallest eigenvalues from this matrix and find a relation this eigenvalue with the connectivity of that hypergraph.

Keywords: connectivity, graph, hypergraph, Laplacian matrix

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Authors: Getachew T. Sedebo, Stephan V. Joubert, Michael Y. Shatalov

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Conventional configuration of the vibratory disc gyroscope is based on in-plane non-axisymmetric vibrations of the disc with a prescribed circumferential wave number. Due to the Bryan's effect, the vibrating pattern of the disc becomes sensitive to the axial component of inertial rotation of the disc. Rotation of the vibrating pattern relative to the disc is proportional to the inertial angular rate and is measured by sensors. In the present paper, the authors investigate a possibility of making a 3D sensor on the basis of both in-plane and bending vibrations of the disc resonator. We derive equations of motion for the disc vibratory gyroscope, where both in-plane and bending vibrations are considered. Hamiltonian variational principle is used in setting up equations of motion and the corresponding boundary conditions. The theory of thin shells with the linear elasticity principles is used in formulating the problem and also the disc is assumed to be isotropic and obeys Hooke's Law. The governing equation for a specific mode is converted to an ODE to determine the eigenfunction. The resulting ODE has exact solution as a linear combination of Bessel and Neumann functions. We demonstrate how to obtain an explicit solution and hence the eigenvalues and corresponding eigenfunctions for annular disc with fixed inner boundary and free outer boundary. Finally, the characteristics equations are obtained and the corresponding eigenvalues are calculated. The eigenvalues are used for the calculation of tuning conditions of the 3D disc vibratory gyroscope.

Keywords: Bryan’s effect, bending vibrations, disc gyroscope, eigenfunctions, eigenvalues, tuning conditions

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Authors: Roger Yu

Abstract:

A numerical scheme has been developed to solve wave equations for chaotic systems such as stadium-shaped cavity. The same numerical method can also be used for finding wave properties of rectangle cavities with randomly placed obstacles. About 30k eigenvalues have been obtained accurately on a normal circumstance. For comparison, we also initiated an experimental study which determines both eigenfrequencies and eigenfunctions of a stadium-shaped cavity using pulse and normal mode analyzing techniques. The acoustic cavity was made adjustable so that the transition from nonchaotic (circle) to chaotic (stadium) waves can be investigated.

Keywords: quantum dot, chaos, numerical method, eigenvalues

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Authors: Benedict Ita, Peter Okoi

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In this work, the concept of the extended Nikiforov-Uvarov technique is discussed and employed to obtain the exact bound state energy eigenvalues and the corresponding normalized eigenfunctions of the inverse square root potential. With expressions for the exact energy eigenvalues and corresponding eigenfunctions, the expressions for the expectation values of the inverse separation-squared, kinetic energy, and the momentum-squared of the potential are presented using the Hellmann Feynman theorem. For visualization, algorithms written and implemented in Python language are used to generate tables and plots for l-states of the energy eigenvalues and some expectation values. The results obtained here may find suitable applications in areas like atomic and molecular physics, chemical physics, nuclear physics, and solid-state physics.

Keywords: Schrodinger equation, Nikoforov-Uvarov method, inverse square root potential, diatomic molecules, Python programming, Hellmann-Feynman theorem, second order differential equation, matrix algebra

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Authors: Seyed Ahmad Mojallal

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Let G be a connected threshold graph of order n with m edges and trace T. In this talk we give a lower bound on Laplacian energy in terms of n, m, and T of G. From this we determine the threshold graphs with the first four minimal Laplacian energies. We also list the first 20 minimal Laplacian energies among threshold graphs. Let σ=σ(G) be the number of Laplacian eigenvalues greater than or equal to average degree of graph G. Using this concept, we obtain the threshold graphs with the largest and the second largest Laplacian energies.

Keywords: Laplacian eigenvalues, Laplacian energy, threshold graphs, extremal graphs

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Authors: Ahmed Salam, Haithem Benkahla

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Recently, an efficient backward-stable algorithm for computing eigenvalues and vectors of a symmetric and Hamiltonian matrix has been proposed. The method preserves the symmetric and Hamiltonian structures of the original matrix, during the whole process. In this paper, we revisit the method. We derive a way for implementing the reduction of the matrix to the appropriate condensed form. Then, we construct a novel version of the implicit QR-algorithm for computing the eigenvalues and vectors.

Keywords: block implicit QR algorithm, preservation of a double structure, QR algorithm, symmetric and Hamiltonian structures

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Authors: Wang Xingbo

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Computation of determinant in the form |I-X| is primary and fundamental because it can help to compute many other determinants. This article puts forward a time-reducible approach to compute determinant |I-X|. The approach is derived from the Newton’s identity and its time complexity is no more than that to compute the eigenvalues of the square matrix X. Mathematical deductions and numerical example are presented in detail for the approach. By comparison with classical approaches the new approach is proved to be superior to the classical ones and it can naturally reduce the computational time with the improvement of efficiency to compute eigenvalues of the square matrix.

Keywords: algorithm, determinant, computation, eigenvalue, time complexity

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Authors: Saieed Akbari, Yousef Bagheri, Amir Hossein Ghodrati, Sima Saadat Akhtar

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Let G be a simple graph with the vertex set V (G) and with the adjacency matrix A (G). The energy E (G) of G is defined to be the sum of the absolute values of all eigenvalues of A (G). Also let n and m be number of edges and vertices of the graph respectively. A regular graph is a graph where each vertex has the same number of neighbours. Given a graph G, its line graph L(G) is a graph such that each vertex of L(G) represents an edge of G; and two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint in G. In this paper we show that for every regular graphs and also for every line graphs such that (G) 3 we have, E(G) 2nm + n 1. Also at the other part of the paper we prove that 2 (G) E(G) for an arbitrary graph G.

Keywords: eigenvalues, energy, line graphs, matching number

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Authors: Egor Stadnichuk

Abstract:

Relativistic runaway electron avalanches (RREA) are a widely accepted source of thunderstorm gamma-radiation. In regions with huge electric field strength, RREA can multiply via relativistic feedback. The relativistic feedback is caused both by positron production and by runaway electron bremsstrahlung gamma-rays reversal. In complex multilayer thunderstorm electric field structures, an additional reactor feedback mechanism appears due to gamma-ray exchange between separate strong electric field regions with different electric field directions. The study of this reactor mechanism in conjunction with the relativistic feedback with Monte Carlo simulations or by direct solution of the kinetic Boltzmann equation requires a significant amount of computational time. In this work, a theoretical approach to study feedback mechanisms in RREA physics is developed. It is based on the matrix of feedback operators construction. With the feedback matrix, the problem of the dynamics of avalanches in complex electric structures is reduced to the problem of finding eigenvectors and eigenvalues. A method of matrix elements calculation is proposed. The proposed concept was used to study the dynamics of RREAs in multilayer thunderclouds.

Keywords: terrestrial Gamma-ray flashes, thunderstorm ground enhancement, relativistic runaway electron avalanches, gamma-rays, high-energy atmospheric physics, TGF, TGE, thunderstorm, relativistic feedback, reactor feedback, reactor model

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Authors: Suleiman Bashir Adamu, Lawan Sani Taura

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We study the role of boundary conditions via -symmetric quantum mechanics, where denotes parity operator and denotes time reversal operator. Using the one-dimensional Schrödinger Hamiltonian for a free particle in an infinite square well, we introduce symmetric boundary conditions. We find solutions of the 1D Klein-Gordon equation for a free particle in an infinite square well with Hermitian boundary and symmetry boundary conditions, where in both cases the energy eigenvalues and eigenfunction, respectively, are obtained.

Keywords: Eigenvalues, Eigenfunction, Hamiltonian, Klein- Gordon equation, PT-symmetric quantum mechanics

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Authors: Xianwei Zheng, Yuan Yan Tang

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Classical window Fourier transform has been widely used in signal processing, image processing, machine learning and pattern recognition. The related Gabor transform is powerful enough to capture the texture information of any given dataset. Recently, in the emerging field of graph signal processing, researchers devoting themselves to develop a graph signal processing theory to handle the so-called graph signals. Among the new developing theory, windowed graph Fourier transform has been constructed to establish a time-frequency analysis framework of graph signals. The windowed graph Fourier transform is defined by using the translation and modulation operators of graph signals, following the similar calculations in classical windowed Fourier transform. Specifically, the translation and modulation operators of graph signals are defined by using the Laplacian eigenvectors as follows. For a given graph signal, its translation is defined by a similar manner as its definition in classical signal processing. Specifically, the translation operator can be defined by using the Fourier atoms; the graph signal translation is defined similarly by using the Laplacian eigenvectors. The modulation of the graph can also be established by using the Laplacian eigenvectors. The windowed graph Fourier transform based on these two operators has been applied to obtain time-frequency representations of graph signals. Fundamentally, the modulation operator is defined similarly to the classical modulation by multiplying a graph signal with the entries in each Fourier atom. However, a single Laplacian eigenvector entry cannot play a similar role as the Fourier atom. This definition ignored the relationship between the translation and modulation operators. In this paper, a new definition of the modulation operator is proposed and thus another time-frequency framework for graph signal is constructed. Specifically, the relationship between the translation and modulation operations can be established by the Fourier transform. Specifically, for any signal, the Fourier transform of its translation is the modulation of its Fourier transform. Thus, the modulation of any signal can be defined as the inverse Fourier transform of the translation of its Fourier transform. Therefore, similarly, the graph modulation of any graph signal can be defined as the inverse graph Fourier transform of the translation of its graph Fourier. The novel definition of the graph modulation operator established a relationship of the translation and modulation operations. The new modulation operation and the original translation operation are applied to construct a new framework of graph signal time-frequency analysis. Furthermore, a windowed graph Fourier frame theory is developed. Necessary and sufficient conditions for constructing windowed graph Fourier frames, tight frames and dual frames are presented in this paper. The novel graph signal time-frequency analysis framework is applied to signals defined on well-known graphs, e.g. Minnesota road graph and random graphs. Experimental results show that the novel framework captures new features of graph signals.

Keywords: graph signals, windowed graph Fourier transform, windowed graph Fourier frames, vertex frequency analysis

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Authors: R. Vigneswaran, S. Thilakanathan

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Variable time-stepping algorithms for solving dynamical systems performed poorly for long time computations which pass close to a fixed point. To overcome this difficulty, several authors considered phase space error controls for numerical simulation of dynamical systems. In one generalized phase space error control, a step-size selection scheme was proposed, which allows this error control to be incorporated into the standard adaptive algorithm as an extra constraint at negligible extra computational cost. For this generalized error control, it was already analyzed the forward Euler method applied to the linear system whose coefficient matrix has real and negative eigenvalues. In this paper, this result was extended to the linear system whose coefficient matrix has complex eigenvalues with negative real parts. Some theoretical results were obtained and numerical experiments were carried out to support the theoretical results.

Keywords: adaptivity, fixed point, long time simulations, stability, linear system

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Authors: BenedictI Ita, Etido P. Inyang

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In this work, the resolutions of the N-dimensional Schrödinger equation with the screened modified Kratzerplus inversely quadratic Yukawa potential (SMKIQYP) have been obtained with the Greene-Aldrich approximation scheme using the Nikiforov-Uvarov method. The eigenvalues and the normalized eigenfunctions are obtained. We then apply the energy spectrum to study four (HCl, N₂, NO, and CO) diatomic molecules. The results show that the energy spectra of these diatomic molecules increase as quantum numbers increase. The energy equation was also used to calculate the partition function and other thermodynamic properties. We predicted the partition function of CO and NO. To check the accuracy of our work, the special case (Modified Kratzer and screened Modified Kratzer potentials) of the collective potential energy eigenvalues agrees excellently with the existing literature.

Keywords: Schrödinger equation, Nikiforov-Uvarov method, modified screened Kratzer, inversely quadratic Yukawa potential, diatomic molecules

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Authors: Feng Wang, Shencheng Ni, Shuying Han, Shanhong You

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With the development of optical communication, optical performance monitoring (OPM) has received more and more attentions. Since optical signal-to-noise ratio (OSNR) is directly related to bit error rate (BER), it is one of the important parameters in optical networks. Recently, artificial neural network (ANN) has been greatly developed. ANN has strong learning and generalization ability. In this paper, a method of OSNR monitoring based on delay-tap sampling (DTS) and ANN has been proposed. DTS technique is used to extract the eigenvalues of the signal. Then, the eigenvalues are input into the ANN to realize the OSNR monitoring. The experiments of 10 Gb/s non-return-to-zero (NRZ) on–off keying (OOK), 20 Gb/s pulse amplitude modulation (PAM4) and 20 Gb/s return-to-zero (RZ) differential phase-shift keying (DPSK) systems are demonstrated for the OSNR monitoring based on the proposed method. The experimental results show that the range of OSNR monitoring is from 15 to 30 dB and the root-mean-square errors (RMSEs) for 10 Gb/s NRZ-OOK, 20 Gb/s PAM4 and 20 Gb/s RZ-DPSK systems are 0.36 dB, 0.45 dB and 0.48 dB respectively. The impact of chromatic dispersion (CD) on the accuracy of OSNR monitoring is also investigated in the three experimental systems mentioned above.

Keywords: artificial neural network (ANN), chromatic dispersion (CD), delay-tap sampling (DTS), optical signal-to-noise ratio (OSNR)

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