scholarly journals Spectral densities of singular values of products of Gaussian and truncated unitary random matrices

2019 ◽  
Vol 09 (04) ◽  
pp. 2050014
Author(s):  
Thorsten Neuschel
Keyword(s):  
Singular Values ◽  
Integral Representations ◽  
Algebraic Equations ◽  
Unitary Matrices ◽  
New Approach ◽  
Special Cases ◽  
Spectral Variable ◽  
Dimensional Growth ◽  
Complex Integral ◽  
Stieltjes Transforms

We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth. In the general case, we develop a new approach to obtain complex integral representations for densities of measures whose Stieltjes transforms satisfy algebraic equations of a certain type. In the special cases in which at most one factor of the product is a complex Gaussian, we derive elementary expressions for the limiting densities using suitable parameterizations for the spectral variable. Moreover, in all cases we study the behavior of the densities at the boundary of the spectrum.

scholarly journals An α-Model Parametrization Algorithm for Optimization with Differential-Algebraic Equations

2022 ◽  
Vol 12 (2) ◽  
pp. 890
Author(s):  
Paweł Dra̧g
Keyword(s):  
Optimization Algorithm ◽  
Numerical Optimization ◽  
Differential Algebraic Equations ◽  
Multiple Shooting ◽  
Small Range ◽  
Algebraic Equations ◽  
New Approach ◽  
Highly Nonlinear ◽  
Special Cases ◽  
Wide Range

An optimization task with nonlinear differential-algebraic equations (DAEs) was approached. In special cases in heat and mass transfer engineering, a classical direct shooting approach cannot provide a solution of the DAE system, even in a relatively small range. Moreover, available computational procedures for numerical optimization, as well as differential- algebraic systems solvers are characterized by their limitations, such as the problem scale, for which the algorithms can work efficiently, and requirements for appropriate initial conditions. Therefore, an αDAE model optimization algorithm based on an α-model parametrization approach was designed and implemented. The main steps of the proposed methodology are: (1) task discretization by a multiple-shooting approach, (2) the design of an α-parametrized system of the differential-algebraic model, and (3) the numerical optimization of the α-parametrized system. The computations can be performed by a chosen iterative optimization algorithm, which can cooperate with an outer numerical procedure for solving DAE systems. The implemented algorithm was applied to solve a counter-flow exchanger design task, which was modeled by the highly nonlinear differential-algebraic equations. Finally, the new approach enabled the numerical simulations for the higher values of parameters denoting the rate of changes in the state variables of the system. The new approach can carry out accurate simulation tests for systems operating in a wide range of configurations and created from new materials.

scholarly journals Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrey A. Pil’nik ◽  
Andrey A. Chernov ◽  
Damir R. Islamov
Keyword(s):  
Charge Transport ◽  
Thin Layers ◽  
Jump Processes ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Statistical Solution ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

AbstractIn this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

scholarly journals A New Approach for Analyzing the Reliability of the Repair Facility in a Series System with Vacations

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Renbin Liu ◽  
Yong Wu
Keyword(s):  
Renewal Process ◽  
Decomposition Method ◽  
Process Theory ◽  
Series System ◽  
New Approach ◽  
Numerical Examples ◽  
Special Cases ◽  
The Mean ◽  
Repair Facility

Based on the renewal process theory we develop a decomposition method to analyze the reliability of the repair facility in ann-unit series system with vacations. Using this approach, we study the unavailability and the mean replacement number during(0,t]of the repair facility. The method proposed in this work is novel and concise, which can make us see clearly the structures of the facility indices of a series system with an unreliable repair facility, two convolution relations. Special cases and numerical examples are given to show the validity of our method.

scholarly journals A (p,v)-Extension of Hurwitz-Lerch Zeta Function and its Properties

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
KS Nisar ◽  
Shahid Mubeen
Keyword(s):  
Zeta Function ◽  
Beta Function ◽  
Integral Representations ◽  
Basic Properties ◽  
Mellin Transformation ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
Generating Relations

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

On the Interaction at Anti-Flat Deformation of Stress Concentrators of the Type of Cracks and Stringers with Regard to the Layer Manufactured from Miscellaneous Materials

2019 ◽  
Vol 828 ◽  
pp. 81-88
Author(s):  
Nune Grigoryan ◽  
Mher Mkrtchyan
Keyword(s):  
Integral Transform ◽  
Composite Layer ◽  
Singular Integral Equations ◽  
Original System ◽  
Fourier Integral ◽  
Quadrature Formulas ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Tangential Forces

In this paper, we consider the problem of determining the basic characteristics of the stress state of a composite in the form of a piecewise homogeneous elastic layer reinforced along its extreme edges by stringers of finite lengths and containing a collinear system of an arbitrary number of cracks at the junction line of heterogeneous materials. It is assumed that stringers along their longitudinal edges are loaded with tangential forces, and along their vertical edges - with horizontal concentrated forces. In addition, the cracks are laden with distributed tangential forces of different intensities. The case is also considered when the lower edge of the composite layer is free from the stringer and rigidly clamped. It is believed that under the action of these loads, the composite layer in the direction of one of the coordinate axes is in conditions of anti-flat deformation (longitudinal shift). Using the Fourier integral transform, the solution of the problem is reduced to solving a system of singular integral equations (SIE) of three equations. The solution of this system is obtained by a well-known numerical-analytical method for solving the SIE using Gauss quadrature formulas by the use of the Chebyshev nodes. As a result, the solution of the original system of SIE is reduced to the solution of the system of systems of linear algebraic equations (SLAE). Various special cases are considered, when the defining SIE and the SLAE of the task are greatly simplified, which will make it possible to carry out a detailed numerical analysis and identify patterns of change in the characteristics of the tasks.

scholarly journals Chebyshev Wavelet Finite Difference Method: A New Approach for Solving Initial and Boundary Value Problems of Fractional Order

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
A. Kazemi Nasab ◽  
A. Kılıçman ◽  
Z. Pashazadeh Atabakan ◽  
S. Abbasbandy
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Finite Difference Methods ◽  
Algebraic Equations ◽  
Chebyshev Wavelets ◽  
New Approach ◽  
Nonlinear Algebraic Equations ◽  
Chebyshev Wavelet ◽  
Fractional Differential

A new method based on a hybrid of Chebyshev wavelets and finite difference methods is introduced for solving linear and nonlinear fractional differential equations. The useful properties of the Chebyshev wavelets and finite difference method are utilized to reduce the computation of the problem to a set of linear or nonlinear algebraic equations. This method can be considered as a nonuniform finite difference method. Some examples are given to verify and illustrate the efficiency and simplicity of the proposed method.

scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

Dynamic Modeling and Simulation of Parallel Mechanisms Using the Virtual Spring Approach

2000 ◽  
Author(s):  
Jiegao Wang ◽  
Clément M. Gosselin ◽  
Li Cheng
Keyword(s):  
Closed Loop ◽  
Differential Algebraic Equations ◽  
Parallel Mechanisms ◽  
Algebraic Equations ◽  
New Approach ◽  
Flexible Links ◽  
Simulation Results ◽  
Loop Constraints ◽  
Virtual Springs ◽  
Good Agreement

Abstract A new approach for the dynamic simulation of parallel mechanisms or mechanical systems is presented in this paper. This approach uses virtual springs and dampers to include the closed-loop constraints thereby avoiding the solution of differential-algebraic equations. Examples illustrating the approach are given and include the four-bar mechanism with both rigid and flexible links as well as the 6-dof Gough-Stewart platform. Simulation results are given for the four-bar linkages and the 6-dof manipulator. The results achieve a good agreement with the results obtained from other conventional approaches.

scholarly journals Further Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 48
Author(s):  
Kottakkaran Sooppy Nisar
Keyword(s):  
Zeta Function ◽  
Hypergeometric Function ◽  
Summation Formula ◽  
Integral Representations ◽  
Generalized Hypergeometric Function ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Function Of Two Variables

The main aim of this paper is to provide a new generalization of Hurwitz-Lerch Zeta function of two variables. We also investigate several interesting properties such as integral representations, summation formula, and a connection with the generalized hypergeometric function. To strengthen the main results we also consider some important special cases.

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