Hyperbolic forms admit reversible weighted shift determinantal representation

2019 ◽  
Vol 362 ◽  
pp. 124595
Author(s):  
M.T. Chien ◽  
H. Nakazato
Keyword(s):  
Weighted Shift ◽  
Determinantal Representation

scholarly journals On the Implicit Equation of Conics and Quadrics Offsets

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1784
Author(s):  
Jorge Caravantes ◽  
Gema M. Diaz-Toca ◽  
Mario Fioravanti ◽  
Laureano Gonzalez-Vega
Keyword(s):  
Three Dimensions ◽  
Analytic Geometry ◽  
Implicit Equation ◽  
Determinantal Representation ◽  
Point Positioning ◽  
Geometric Queries ◽  
Expanded Form

A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in “A Treatise on the Analytic Geometry of Three Dimensions” by G. Salmon for offsets to non-degenerate conics and central quadrics.

Stability of Weighted Darma Filters

1998 ◽  
Vol 41 (1) ◽  
pp. 49-64
Author(s):  
K. J. Harrison ◽  
J. A. Ward ◽  
L-J. Eaton
Keyword(s):  
Difference Equations ◽  
Shift Operator ◽  
Variable Coefficients ◽  
Weighted Shift ◽  
Linear Difference Equations ◽  
Linear Filters ◽  
Weighted Shift Operator ◽  
Rational Transfer ◽  
The Stability ◽  
Special Case

AbstractWe study the stability of linear filters associated with certain types of linear difference equations with variable coefficients. We show that stability is determined by the locations of the poles of a rational transfer function relative to the spectrum of an associated weighted shift operator. The known theory for filters associated with constant-coefficient difference equations is a special case.

scholarly journals Determinantal Representations of the Core Inverse and Its Generalizations with Applications

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Ivan I. Kyrchei
Keyword(s):  
Direct Method ◽  
Linear Equations ◽  
Close Relation ◽  
Numerical Example ◽  
Determinantal Representation ◽  
The Core ◽  
Core Inverse ◽  
Cramer’S Rule ◽  
The Right

In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.

CHAOS IN A CLASS OF NONCONSTANT WEIGHTED SHIFT OPERATORS

2013 ◽  
Vol 23 (01) ◽  
pp. 1350010 ◽  
Author(s):  
XINXING WU ◽  
PEIYONG ZHU
Keyword(s):  
Positive Integer ◽  
Linear Space ◽  
Normed Linear Space ◽  
Shift Operator ◽  
Weighted Shift ◽  
Constructive Proof ◽  
Sufficient Condition ◽  
Weighted Shift Operator ◽  
Shift Operators ◽  
Devaney Chaos

In this paper, chaos generated by a class of nonconstant weighted shift operators is studied. First, we prove that for the weighted shift operator Bμ : Σ(X) → Σ(X) defined by Bμ(x0, x1, …) = (μ(0)x1, μ(1)x2, …), where X is a normed linear space (not necessarily complete), weak mix, transitivity (hypercyclity) and Devaney chaos are all equivalent to separability of X and this property is preserved under iterations. Then we get that [Formula: see text] is distributionally chaotic and Li–Yorke sensitive for each positive integer N. Meanwhile, a sufficient condition ensuring that a point is k-scrambled for all integers k > 0 is obtained. By using these results, a simple example is given to show that Corollary 3.3 in [Fu & You, 2009] does not hold. Besides, it is proved that the constructive proof of Theorem 4.3 in [Fu & You, 2009] is not correct.

scholarly journals Definite determinantal representations of multivariate polynomials

2019 ◽  
Vol 19 (07) ◽  
pp. 2050129 ◽  
Author(s):  
Papri Dey
Keyword(s):  
Semidefinite Programming ◽  
Matrix Polynomial ◽  
Heuristic Method ◽  
Linear Matrix ◽  
Multivariate Polynomials ◽  
Multivariate Polynomial ◽  
Bivariate Polynomial ◽  
Determinantal Representation ◽  
Representation Problem ◽  
Feasible Sets

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming (SDP) problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant (GMD) of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree [Formula: see text]. Then we propose a heuristic method to obtain a monic symmetric determinantal representation (MSDR) of a multivariate polynomial of degree [Formula: see text].

scholarly journals Weighted shift and composition operators on ℓ which are (m,q)-isometries

2016 ◽  
Vol 505 ◽  
pp. 152-173 ◽  
Author(s):  
Teresa Bermúdez ◽  
Antonio Martinón ◽  
Juan Agustín Noda
Keyword(s):  
Composition Operators ◽  
Weighted Shift
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