scholarly journals A Combinatorial Representation with Schröder Paths of Biorthogonality of Laurent Biorthogonal Polynomials

10.37236/955 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Shuhei Kamioka
Keyword(s):  
Linear Functional ◽  
Schröder Paths ◽  
Biorthogonal Polynomials ◽  
Combinatorial Representation ◽  
Weighted Plane

Combinatorial representation in terms of Schröder paths and other weighted plane paths are given of Laurent biorthogonal polynomials (LBPs) and a linear functional with which LBPs have orthogonality and biorthogonality. Particularly, it is clarified that quantities to which LBPs are mapped by the corresponding linear functional can be evaluated by enumerating certain kinds of Schröder paths, which imply orthogonality and biorthogonality of LBPs.

scholarly journals A Combinatorial Derivation with Schröder Paths of a Determinant Representation of Laurent Biorthogonal Polynomials

10.37236/800 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Shuhei Kamioka
Keyword(s):  
Recurrence Equation ◽  
Combinatorial Proof ◽  
Toeplitz Determinants ◽  
Determinant Representation ◽  
Combinatorial Derivation ◽  
Schröder Paths ◽  
Biorthogonal Polynomials ◽  
Weighted Plane

A combinatorial proof in terms of Schröder paths and other weighted plane paths is given for a determinant representation of Laurent biorthogonal polynomials (LBPs) and that of coefficients of their three-term recurrence equation. In this process, it is clarified that Toeplitz determinants of the moments of LBPs and their minors can be evaluated by enumerating certain kinds of configurations of Schröder paths in a plane.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

Fitting partially linear functional-coefficient panel-data models with Stata

2020 ◽  
Vol 20 (4) ◽  
pp. 976-998
Author(s):  
Kerui Du ◽  
Yonghui Zhang ◽  
Qiankun Zhou
Keyword(s):  
Monte Carlo ◽  
Panel Data ◽  
Fixed Effects ◽  
Linear Functional ◽  
Semiparametric Estimation ◽  
Data Models ◽  
Panel Data Models ◽  
Panel Models ◽  
Partially Linear ◽  
Functional Coefficient

In this article, we describe the implementation of fitting partially linear functional-coefficient panel models with fixed effects proposed by An, Hsiao, and Li [2016, Semiparametric estimation of partially linear varying coefficient panel data models in Essays in Honor of Aman Ullah ( Advances in Econometrics, Volume 36)] and Zhang and Zhou (Forthcoming, Econometric Reviews). Three new commands xtplfc, ivxtplfc, and xtdplfc are introduced and illustrated through Monte Carlo simulations to exemplify the effectiveness of these estimators.

scholarly journals Semi-normal operators on uniformly smooth Banach spaces

1990 ◽  
Vol 32 (3) ◽  
pp. 273-276 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Dual Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Uniformly Smooth ◽  
Normal Operators ◽  
Uniformly Smooth Banach Spaces ◽  
The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

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.

The weighted Cauchy problem for linear functional differential equations with strong singularities

2011 ◽  
Vol 18 (3) ◽  
pp. 577-586
Author(s):  
Zaza Sokhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Initial Point ◽  
Higher Order ◽  
Well Posedness ◽  
Deviating Arguments ◽  
Functional Differential

Abstract The sufficient conditions of well-posedness of the weighted Cauchy problem for higher order linear functional differential equations with deviating arguments, whose coefficients have nonintegrable singularities at the initial point, are found.

The Parameterization of All Disturbance Observers for Discrete-Time Plants with Input Disturbance

2011 ◽  
Vol 497 ◽  
pp. 197-209
Author(s):  
Kou Yamada ◽  
Iwanori Murakami ◽  
Yoshinori Ando ◽  
Takaaki Hagiwara ◽  
Da Zhi Gong ◽  
...  
Keyword(s):  
Discrete Time ◽  
Disturbance Observer ◽  
Linear Functional ◽  
Design Methods ◽  
Functional Disturbance ◽  
Input Disturbance

Disturbance observers have been used to estimate the disturbance in the plant. Several paperson design methods of disturbance observers have been published. Recently, the parameterizationof all disturbance observers for discrete-time plants with any output disturbance was clarified. However,no paper examines the parameterization of all disturbance observers for discrete-time plants withany input disturbance. In this paper, we clarify existance conditions of a disturbance observer and ofa linear functional disturbance observer for discrete-time plants with any input disturbance. Underthese conditions, we propose parameterizations of all disturbance observers and all linear functionaldisturbance observers for discrete-time plants with any input disturbance.

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