Necessary and sufficient conditions for accurate and efficient rational function evaluation and factorizations of rational matrices

2001 ◽  
pp. 117-143 ◽  
Author(s):  
James Demmel ◽  
Plamen Koev
Keyword(s):  
Rational Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Function Evaluation ◽  
Necessary And Sufficient

scholarly journals Existence conditions for a class of modular subgroups of genus zero

2002 ◽  
Vol 66 (3) ◽  
pp. 517-525
Author(s):  
Joachim A. Hempel
Keyword(s):  
Rational Function ◽  
Finite Index ◽  
Modular Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Genus Zero ◽  
Existence Conditions ◽  
Necessary And Sufficient

Every subgroup of finite index of the modular groupPSL(2, ℤ) has asignatureconsisting of conjugacy-invariant integer parameters satisfying certain conditions. In the case of genus zero, these parameters also constitute a prescription for the degree and the orders of the poles of a rational functionFwith the property:Functions correspond to subgroups, and we use this to establish necessary and sufficient conditions for existence of subgroups with a certain subclass of allowable signatures.

scholarly journals Waring’s Problem for Rational Functions in One Variable

2020 ◽  
Vol 71 (2) ◽  
pp. 439-449
Author(s):  
Bo-Hae Im ◽  
Michael Larsen
Keyword(s):  
Rational Function ◽  
Sufficient Conditions ◽  
Rational Functions ◽  
Laurent Polynomial ◽  
Necessary And Sufficient Conditions ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Odd Degree ◽  
Necessary And Sufficient ◽  
Polynomial Of Odd Degree

Abstract Let $f\in{\mathbb{Q}}(x)$ be a non-constant rational function. We consider ‘Waring’s problem for $f(x)$,’ i.e., whether every element of ${\mathbb{Q}}$ can be written as a bounded sum of elements of $\{f(a)\mid a\in{\mathbb{Q}}\}$. For rational functions of degree $2$, we give necessary and sufficient conditions. For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring’s problem. We also consider the ‘easier Waring’s problem’: whether every element of ${\mathbb{Q}}$ can be represented as a bounded sum of elements of $\{\pm f(a)\mid a\in{\mathbb{Q}}\}$.

A complete characterization of the existence of rational functional solutions with infinite support bases

2016 ◽  
Vol 15 (09) ◽  
pp. 1650178 ◽  
Author(s):  
Lan Nguyen
Keyword(s):  
Rational Function ◽  
Characteristic Zero ◽  
Polynomial Solution ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Solutions ◽  
Function Solution ◽  
Necessary And Sufficient

It is known that there exist polynomial solutions [Formula: see text] with infinite support base [Formula: see text], of certain functional equations arising from quantum arithmetics, which cannot be constructed from quantum integers. A description of the necessary and sufficient conditions on a set of primes [Formula: see text] for the existence of a polynomial solution, with field of coefficients of characteristic zero and support base [Formula: see text], which cannot be constructed from quantum integers is also known, leading to the classification of the set of polynomial solutions. In his papers on quantum arithmetics, Melvyn Nathanson raises a question concerning the classification of the possibly non-trivially broader set of solutions, namely the set of rational function solutions. It is not known at the time that the set of rational function solutions is more than just the set of ratio of polynomial solutions. However, it is now known that there are infinitely many rational function solutions [Formula: see text], with support base [Formula: see text] and field of coefficients of characteristic zero, which are not ratios of polynomial solutions with the same support base, even in the purely cyclotomic case. Thus, a natural question that should be asked in order to classify the set of rational function solutions, is: If polynomial solutions are replaced by merely rational function solutions, what would the necessary and sufficient conditions be on the support base [Formula: see text]? In this paper, we give a complete description of the necessary and sufficient conditions on the set of primes [Formula: see text] for the existence of a rational function solution, with field of coefficients of characteristic zero and support base [Formula: see text], which cannot be constructed from quantum integers.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

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