scholarly journals The Tensor Pade´-Type Approximant with Application in Computing Tensor Exponential Function

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Chuanqing Gu ◽  
Yong Liu
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Exponential Function ◽  
Efficient Algorithm ◽  
Linear Functional ◽  
Function Form ◽  
Numerical Examples ◽  
First Time ◽  
Formal Orthogonal Polynomials

Tensor exponential function is an important function that is widely used. In this paper, tensor Pade´-type approximant (TPTA) is defined by introducing a generalized linear functional for the first time. The expression of TPTA is provided with the generating function form. Moreover, by means of formal orthogonal polynomials, we propose an efficient algorithm for computing TPTA. As an application, the TPTA for computing the tensor exponential function is presented. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.

scholarly journals Second-order neutrosophic boundary-value problem

2021 ◽  
Author(s):  
Sandip Moi ◽  
Suvankar Biswas ◽  
Smita Pal(Sarkar)
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Differential Calculus ◽  
Boundary Value ◽  
Second Order ◽  
Numerical Examples ◽  
Different Types ◽  
First Time

AbstractIn this article, some properties of neutrosophic derivative and neutrosophic numbers have been presented. This properties have been used to develop the neutrosophic differential calculus. By considering different types of first- and second-order derivatives, different kind of systems of derivatives have been developed. This is the first time where a second-order neutrosophic boundary-value problem has been introduced with different types of first- and second-order derivatives. Some numerical examples have been examined to explain different systems of neutrosophic differential equation.

scholarly journals Period-Life of a Branching Process with Migration and Continuous Time

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 868
Author(s):  
Khrystyna Prysyazhnyk ◽  
Iryna Bazylevych ◽  
Ludmila Mitkova ◽  
Iryna Ivanochko
Keyword(s):  
Random Process ◽  
Generating Function ◽  
Continuous Time ◽  
Branching Process ◽  
Probability Generating Function ◽  
Time Interval ◽  
The Moment ◽  
First Time ◽  
Critical Branching Process ◽  
Number Of Particles

The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found.

scholarly journals Spectral Transformations and Associated Linear Functionals of the First Kind

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 107
Author(s):  
Juan Carlos García-Ardila ◽  
Francisco Marcellán
Keyword(s):  
Orthogonal Polynomials ◽  
Linear Space ◽  
Linear Functional ◽  
Linear Functionals ◽  
Spectral Transformations ◽  
Christoffel Transformation ◽  
Complex Coefficients

Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations.

A NEW LAYOUT DESIGN SYSTEM FOR MULTICHIP MODULES

1995 ◽  
Vol 06 (03) ◽  
pp. 509-538 ◽  
Author(s):  
BERNHARD M. RIESS ◽  
ANDREAS A. SCHOENE
Keyword(s):  
Efficient Algorithm ◽  
State Of The Art ◽  
Layout Design ◽  
The State ◽  
Design System ◽  
Multichip Modules ◽  
Analytical Technique ◽  
Cut Method ◽  
First Time ◽  
Three Components

A new layout design system for multichip modules (MCMs) consisting of three components is described. It includes a k-way partitioning approach, an algorithm for pin assignment, and a placement package. For partitioning, we propose an analytical technique combined with a problem-specific multi-way ratio cut method. This method considers fixed module-level pad positions and assigns the cells to regularly arranged chips on the MCM substrate. In the subsequent pin assignment step the chip-level pads resulting from cut nets are positioned on the chip borders. Pin assignment is performed by an efficient algorithm, which profits from the cell coordinates generated by the analytical technique. Global and final placement for each chip is computed by the state-of-the-art placement tools GORDIANL and DOMINO. For the first time, results for MCM layout designs of benchmark circuits with up to 100,000 cells are presented. They show a small number of required chip-level pads, which is the most restricted resource in MCM design, and short total wire lengths.

Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory

2009 ◽  
Vol 52 (1) ◽  
pp. 95-104 ◽  
Author(s):  
L. Miranian
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Classical Theory ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
The Matrix ◽  
Classical Subject ◽  
First Time

AbstractIn the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel–Darboux formulas are presented for the first time.

A Comparative Study of Integer and Noninteger Order Wavelets for Fractional Nonlinear Fredholm Integro-Differential Equations

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
F. Mohammadi ◽  
J. A. Tenreiro Machado
Keyword(s):  
Differential Equations ◽  
Comparative Study ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Efficient Algorithm ◽  
Operational Matrix ◽  
Tau Method ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
Integer Order

This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

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