scholarly journals Sharp Z-eigenvalue inclusion set-based method for testing the positive definiteness of multivariate homogeneous forms

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3131-3139
Author(s):  
Gang Wang ◽  
Linxuan Sun ◽  
Yiju Wang
Keyword(s):  
Convergence Rate ◽  
Numerical Experiments ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Upper Bounds ◽  
Rank One ◽  
Inclusion Set ◽  
Even Order ◽  
Weakly Symmetric ◽  
The Given

In this paper, we establish a sharp Z-eigenvalue inclusion set for even-order real tensors by Z-identity tensor and prove that new Z-eigenvalue inclusion set is sharper than existing results. We propose some sufficient conditions for testing the positive definiteness of multivariate homogeneous forms via new Z-eigenvalue inclusion set. Further, we establish upper bounds on the Z-spectral radius of weakly symmetric nonnegative tensors and estimate the convergence rate of the greedy rank-one algorithms. The given numerical experiments show the validity of our results.

scholarly journals Identifying the Positive Definiteness of Even-Order Weakly Symmetric Tensors via Z-Eigenvalue Inclusion Sets

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1239
Author(s):  
Feichao Shen ◽  
Ying Zhang ◽  
Gang Wang
Keyword(s):  
Asymptotic Stability ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Polynomial Systems ◽  
Invariant Polynomial ◽  
Symmetric Tensors ◽  
Even Order ◽  
Weakly Symmetric ◽  
Time Invariant ◽  
The Given

The positive definiteness of even-order weakly symmetric tensors plays important roles in asymptotic stability of time-invariant polynomial systems. In this paper, we establish two Brauer-type Z-eigenvalue inclusion sets with parameters by Z-identity tensors, and show that these inclusion sets are sharper than existing results. Based on the new Z-eigenvalue inclusion sets, we propose some sufficient conditions for testing the positive definiteness of even-order weakly symmetric tensors, as well as the asymptotic stability of time-invariant polynomial systems. The given numerical experiments are reported to show the efficiency of our results.

scholarly journals POSITIVE SEMIDEFINITENESS OF DISCRETE QUADRATIC FUNCTIONALS

2003 ◽  
Vol 46 (3) ◽  
pp. 627-636 ◽  
Author(s):  
Martin Bohner ◽  
Ondřej Došlý ◽  
Werner Kratz
Keyword(s):  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Quadratic Functional ◽  
Quadratic Functionals ◽  
Positive Semidefiniteness ◽  
Difference Systems ◽  
Special Cases ◽  
Even Order ◽  
Sturm Liouville ◽  
Necessary And Sufficient

AbstractWe consider symplectic difference systems, which contain as special cases linear Hamiltonian difference systems and Sturm–Liouville difference equations of any even order. An associated discrete quadratic functional is important in discrete variational analysis, and while its positive definiteness has been characterized and is well understood, a characterization of its positive semidefiniteness remained an open problem. In this paper we present the solution to this problem and offer necessary and sufficient conditions for such discrete quadratic functionals to be non-negative definite.AMS 2000 Mathematics subject classification: Primary 39A12; 39A13. Secondary 34B24; 49K99

scholarly journals New iterative codes for 𝓗-tensors and an application

2016 ◽  
Vol 14 (1) ◽  
pp. 212-220 ◽  
Author(s):  
Feng Wang ◽  
Deshu Sun
Keyword(s):  
Homogeneous Polynomial ◽  
Sufficient Conditions ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Polynomial Form ◽  
Numerical Examples ◽  
Even Order

AbstractNew iterative codes for identifying 𝓗 -tensor are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor, i.e., an even-degree homogeneous polynomial form are given. Advantages of results obtained are illustrated by numerical examples.

scholarly journals New criteria for H-tensors and an application

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Feng Wang ◽  
Deshu Sun
Keyword(s):  
Sufficient Conditions ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Numerical Examples ◽  
Even Order ◽  
New Criteria

AbstractSome new criteria for identifying H-tensors are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given. Advantages of results obtained are illustrated by numerical examples.

scholarly journals New S-Type Bounds of M-Eigenvalues for Elasticity Tensors with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Zhuanzhou Zhang ◽  
Jun He ◽  
Yanmin Liu ◽  
Zerong Ren
Keyword(s):  
Sufficient Conditions ◽  
Upper Bounds ◽  
Elasticity Tensor ◽  
Strong Ellipticity ◽  
Extreme Eigenvalues ◽  
Elasticity Tensors ◽  
The Given

In this paper, based on the extreme eigenvalues of the matrices arisen from the given elasticity tensor, S-type upper bounds for the M-eigenvalues of elasticity tensors are established. Finally, S-type sufficient conditions are introduced for the strong ellipticity of elasticity tensors based on the S-type M-eigenvalue inclusion sets.

scholarly journals Practical Criteria for H-Tensors and Their Application

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 155
Author(s):  
Min Li ◽  
Haifeng Sang ◽  
Panpan Liu ◽  
Guorui Huang
Keyword(s):  
Sufficient Conditions ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Tensor Analysis ◽  
Symmetric Tensors ◽  
Numerical Examples ◽  
Even Order

Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method.

scholarly journals Lower Estimate of Clique Size via Edge Coloring

2021 ◽  
Vol 27_NS1 (1) ◽  
pp. 8-15
Author(s):  
Balázs Király ◽  
Sándor Szabó
Keyword(s):  
Lower Bounds ◽  
Numerical Experiments ◽  
Lower Estimate ◽  
Edge Coloring ◽  
Clique Number ◽  
Upper Bounds ◽  
Search Algorithms ◽  
The One ◽  
The Given ◽  
Better Than

In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.

scholarly journals Computer Modeling of Aerosol Emissions Spread in the Atmosphere

2019 ◽  
Vol 97 ◽  
pp. 05023 ◽  
Author(s):  
Daler Sharipov ◽  
Sharofiddin Aynakulov ◽  
Otabek Khafizov
Keyword(s):  
Boundary Layer ◽  
Mathematical Model ◽  
Atmospheric Boundary Layer ◽  
Computer Modeling ◽  
Numerical Experiments ◽  
Climatic Factors ◽  
Numerical Algorithms ◽  
Aerosol Emissions ◽  
The Given ◽  
And Diffusion

The paper deals with the development of mathematical model and numerical algorithms for solving the problem of transfer and diffusion of aerosol emissions in the atmospheric boundary layer. The model takes into account several significant parameters such as terrain relief, characteristics of underlying surface and weather-climatic factors. A series of numerical experiments were conducted based on the given model. The obtained results presented here show how these factors affect aerosol emissions spread in the atmosphere.

scholarly journals Convergence rate analysis of an iterative algorithm for solving the multiple-sets split equality problem

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shijie Sun ◽  
Meiling Feng ◽  
Luoyi Shi
Keyword(s):  
Convergence Rate ◽  
Iterative Algorithm ◽  
Sufficient Conditions ◽  
Linear Convergence ◽  
Regularity Property ◽  
Step Size ◽  
Bounded Linear ◽  
Split Equality Problem ◽  
Linear Convergence Rate ◽  
Equality Problem

Abstract This paper considers an iterative algorithm of solving the multiple-sets split equality problem (MSSEP) whose step size is independent of the norm of the related operators, and investigates its sublinear and linear convergence rate. In particular, we present a notion of bounded Hölder regularity property for the MSSEP, which is a generalization of the well-known concept of bounded linear regularity property, and give several sufficient conditions to ensure it. Then we use this property to conclude the sublinear and linear convergence rate of the algorithm. In the end, some numerical experiments are provided to verify the validity of our consequences.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

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