scholarly journals A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems

Mathematics ◽  
2018 ◽  
Vol 6 (9) ◽  
pp. 166
Author(s):  
Xiaojie Dou  ◽  
Jin-San Cheng 
Keyword(s):  
Real Zero ◽  
Heuristic Method ◽  
Polynomial Systems ◽  
Sum Of Squares ◽  
Interval Methods ◽  
Necessary And Sufficient Condition ◽  
Real Zeros ◽  
Input System ◽  
Necessary And Sufficient ◽  
The Given

In this paper, by transforming the given over-determined system into a square system, we prove a necessary and sufficient condition to certify the simple real zeros of the over-determined system by certifying the simple real zeros of the square system. After certifying a simple real zero of the related square system with the interval methods, we assert that the certified zero is a local minimum of sum of squares of the input polynomials. If the value of sum of squares of the input polynomials at the certified zero is equal to zero, it is a zero of the input system. As an application, we also consider the heuristic verification of isolated zeros of polynomial systems and their multiplicity structures.

scholarly journals A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems

2018 ◽  
Author(s):  
Jin-San Cheng ◽  
Xiaojie Dou
Keyword(s):  
Local Minimum ◽  
Heuristic Method ◽  
Polynomial Systems ◽  
Sum Of Squares ◽  
Real System ◽  
Interval Methods ◽  
The Real ◽  
Real Zeros ◽  
Input System ◽  
Complex Zeros

We construct a real square system related to a given over-determined real system. We prove that the simple real zeros of the over-determined system are the simple real zeros of the related square system and the real zeros of the two systems are one-to-one correspondence with the constraint that the value of the sum of squares of the polynomials in the over-determined system at the real zeros is identically zero. After certifying the simple real zeros of the related square system with the interval methods, we assert that the certified zero is a local minimum of the sum of squares of the input polynomials. If the value of the sum of the squares of the input polynomials at the certified zero is equal to zero, it is a zero of the input system. As an application, we also consider the heuristic verification of the isolated zeros of polynomial systems and their multiplicity structures. Notice that a complex system with complex zeros can be transformed into a real system with real zeros.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

scholarly journals An application of bivariate polynomial factorization on decoding of Reed-Solomon based codes

2018 ◽  
Vol 12 (1) ◽  
pp. 166-177
Author(s):  
Ivan Pavkov ◽  
Nebojsa Ralevic ◽  
Ljubo Nedovic
Keyword(s):  
Efficient Algorithm ◽  
Sufficient Condition ◽  
Polynomial Factorization ◽  
Necessary And Sufficient Condition ◽  
Bivariate Polynomials ◽  
Bivariate Polynomial ◽  
Integer Coefficients ◽  
Necessary And Sufficient ◽  
The Given

A necessary and sufficient condition for the existence of a non-trivial factorization of an arbitrary bivariate polynomial with integer coefficients was presented in [2]. In this paper we develop an efficient algorithm for factoring bivariate polynomials with integer coefficients. Also, we shall give a proof of the optimality of the algorithm. For a given codeword, formed by mixing up two codewords, the algorithm recovers those codewords directly by factoring corresponding bivariate polynomial. Our algorithm determines uniquely the given polynomials which are used in forming the mixture of two codewords.

Controllability of nonlinear stochastic fractional higher order dynamical systems

2019 ◽  
Vol 22 (4) ◽  
pp. 1063-1085
Author(s):  
R. Mabel Lizzy ◽  
K. Balachandran ◽  
Yong-Ki Ma
Keyword(s):  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Banach Fixed Point Theorem ◽  
Necessary And Sufficient Condition ◽  
Bounded Operators ◽  
Semilinear Systems ◽  
Operator Functions ◽  
Fractional System ◽  
Necessary And Sufficient ◽  
The Given

Abstract This paper deals with the study of controllability of stochastic fractional dynamical systems with 1 < α ≤ 2. Necessary and sufficient condition for controllability of linear stochastic fractional system is obtained. Sufficient conditions for controllability of stochastic fractional semilinear systems, integrodifferential systems, systems with neutral term, systems with delays in control and systems with Lévy noise is formulated and established. The solution is obtained in terms of Mittag-Leffler operator functions by considering bounded operators. The Banach fixed point theorem is used to obtain the desired results from an equivalent nonlinear integral equation of the given system.

scholarly journals On the notion ofL 1-completeness of a stochastic flow on a manifold

2002 ◽  
Vol 7 (12) ◽  
pp. 627-635 ◽  
Author(s):  
Yu. E. Gliklikh ◽  
L. A. Morozova
Keyword(s):  
Functional Space ◽  
Riemannian Metric ◽  
Stochastic Flow ◽  
Time Interval ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
The Given

We introduce the notion ofL 1-completeness for a stochastic flow on manifold and prove a necessary and sufficient condition for a flow to beL 1-complete.L 1-completeness means that the flow is complete (i.e., exists on the given time interval) and that it belongs to some sort ofL 1-functional space, natural for manifolds where no Riemannian metric is specified.

A new method for inextensible flows of timelike curves in 4-dimensional LP-Sasakian manifolds

2015 ◽  
Vol 08 (04) ◽  
pp. 1550073 ◽  
Author(s):  
Talat Körpinar
Keyword(s):  
Tensor Field ◽  
Sasakian Manifold ◽  
New Method ◽  
Conformally Flat ◽  
Necessary And Sufficient Condition ◽  
Sasakian Manifolds ◽  
Practical Applications ◽  
Inextensible Flows ◽  
Necessary And Sufficient ◽  
The Given

Inextensible flows of timelike curves plays an important role in practical applications. In this paper, we construct a new method for inextensible flows of timelike curves in a conformally flat, quasi conformally flat and conformally symmetric 4-dimensional LP-Sasakian manifold. With this new representation, we derive the necessary and sufficient condition for the given curve to be the inextensible flow. By using curvature tensor field, we give some characterizations for curvatures of a timelike curve in a conformally flat, quasi conformally flat and conformally symmetric 4-dimensional LP-Sasakian manifold. Finally, we obtain flows of some associated curves of timelike curves.

scholarly journals Polynomials with Real Zeros and Compatible Sequences

10.37236/2674 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Li Liu
Keyword(s):  
Simple Proof ◽  
Opposite Sign ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Zeros ◽  
Independence Polynomial ◽  
Necessary And Sufficient ◽  
Interlacing Property

In this paper, we study polynomials with only real zeros based on the method of compatible zeros. We obtain a necessary and sufficient condition for the compatible property of two polynomials whose leading coefficients have opposite sign. As applications, we partially answer a question proposed by M. Chudnovsky and P. Seymour in the recent publication [M. Chudnovsky, P. Seymour, The roots of the independence polynomial of a clawfree graph, J. Combin. Theory Ser. B 97 (2007) 350--357]. We also establish the connection between the interlacing property and the compatible property of two polynomials and give a simple proof of some known results.

scholarly journals On the extension of orders in ordered modules

1970 ◽  
Vol 2 (1) ◽  
pp. 81-88 ◽  
Author(s):  
P. Ribenboim
Keyword(s):  
Abelian Groups ◽  
Independent Set ◽  
Total Order ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Ordered Ring ◽  
The Given

We introduce the notion of a positively independent set of elements in an ordered module. With this concept we determine a necessary and sufficient condition which insures that on a strictly ordered module over a strictly ordered ring there exists a strict total order refining the given order. This generalizes a previous result of Fuchs, concerning the case of ordered abelian groups.As an application, let R be a strictly ordered totally ordered ring and let M be the R-module of all mappings from a set I into R, with pointwise order; then this order on M may be refined to a strict total order.

scholarly journals On smallest radical and semi-simple classes

1971 ◽  
Vol 12 (2) ◽  
pp. 98-104 ◽  
Author(s):  
W. G. Leavitt ◽  
J. F. Watters
Keyword(s):  
Admissible Function ◽  
Radical Class ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Universal Class ◽  
Necessary And Sufficient ◽  
The Given

In a recent paper [5] one of us has given a sufficient condition to be satisfied by a given property of radical classes within a universal class w in order that, for any subclass ℳ of w, there should be a smallest radical class having the given property and containing ℳ. The sufficient condition is that the classof all radical classes with the given property can be characterised as the class of all radical classes fixed by an admissible function F (see Section 1 below). In this paper a necessary and sufficient condition is derived and the corresponding result for semi-simpleclasses is also presented. These results are given in Section 2.

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