scholarly journals Strong Tolerance and Strong Universality of Interval Eigenvectors in a Max-Łukasiewicz Algebra

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1504
Author(s):  
Martin Gavalec ◽  
Zuzana Němcová ◽  
Ján Plavka
Keyword(s):  
Set Theory ◽  
Binary Operation ◽  
Sufficient Conditions ◽  
Discrete Events ◽  
Linear Combinations ◽  
Main Method ◽  
Max Algebra ◽  
Discrete Events Systems ◽  
Necessary And Sufficient ◽  
Theoretical Results

The Łukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the Łukasiewicz t-norm forms the basis for the so-called max-Łuk algebra, with applications to the investigation of systems working in discrete steps (discrete events systems; DES, in short). Similar algebras describing the work of DES’s are based on other pairs of operations, such as max-min algebra, max-plus algebra, or max-T algebra (with a given t-norm, T). The investigation of the steady states in a DES leads to the study of the eigenvectors of the transition matrix in the corresponding max-algebra. In real systems, the input values are usually taken to be in some interval. Various types of interval eigenvectors of interval matrices in max-min and max-plus algebras have been described. This paper is oriented to the investigation of strong, strongly tolerable, and strongly universal interval eigenvectors in a max-Łuk algebra. The main method used in this paper is based on max-Ł linear combinations of matrices and vectors. Necessary and sufficient conditions for the recognition of strong, strongly tolerable, and strongly universal eigenvectors have been found. The theoretical results are illustrated by numerical examples.

Using binary operations to construct a transitive set of block transformations

2020 ◽  
Vol 30 (6) ◽  
pp. 375-389
Author(s):  
Igor V. Cherednik
Keyword(s):  
Binary Operation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Binary Operations ◽  
The Family ◽  
Transitive Set ◽  
Necessary And Sufficient

AbstractWe study the set of transformations {ΣF : F∈ 𝓑∗(Ω)} implemented by a network Σ with a single binary operation F, where 𝓑∗(Ω) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family {ΣF : F∈ 𝓑∗(Ω)} in terms of the structure of the network Σ, identify necessary and sufficient conditions of transitivity of the set of transformations {ΣF : F∈ 𝓑∗(Ω)}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations {ΣF : F∈ 𝓑∗(Ω)}.

scholarly journals Clifford semigroups and monotonicity

1985 ◽  
Vol 32 (1) ◽  
pp. 83-92
Author(s):  
T.E. Hays
Keyword(s):  
Algebraic Structure ◽  
Binary Operation ◽  
Monotone Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Clifford Semigroups ◽  
Necessary And Sufficient

A semigroup S is said to be monotone if its binary operation is a monotone function from S × S into S. This paper utilizes some of the known algebraic structure of Clifford semigroups, semigroups which are unions of groups, to study topological Clifford semigroups which are monotone. It is shown that such semigroups are preserved under products, homomorphisms, and, under certain conditions, closures. Necessary and sufficient conditions for monotonicity of groups, paragroups, bands, compact orthodox Clifford semigroups, and compact bands of groups are developed.

Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization

2019 ◽  
Vol 20 (2) ◽  
pp. 125-136
Author(s):  
A. M. Yousef ◽  
S. Z. Rida ◽  
Y. Gh. Gouda ◽  
A. S. Zaki
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractIn this paper, we investigate the dynamical behaviors of a fractional-order predator–prey with Holling type IV functional response and its discretized counterpart. First, we seek the local stability of equilibria for the fractional-order model. Also, the necessary and sufficient conditions of the stability of the discretized model are achieved. Bifurcation types (include transcritical, flip and Neimark–Sacker) and chaos are discussed in the discretized system. Finally, numerical simulations are executed to assure the validity of the obtained theoretical results.

Multiconsensus of first-order multiagent systems with directed topologies

2020 ◽  
Vol 34 (23) ◽  
pp. 2050240
Author(s):  
Xiao-Wen Zhao ◽  
Guangsong Han ◽  
Qiang Lai ◽  
Dandan Yue
Keyword(s):  
Multiagent Systems ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Consensus Problem ◽  
Multiple Agents ◽  
First Order ◽  
The Matrix ◽  
Directed Topologies ◽  
Necessary And Sufficient ◽  
Theoretical Results

The multiconsensus problem of first-order multiagent systems with directed topologies is studied. A novel consensus problem is introduced in multiagent systems — multiconsensus. The states of multiple agents in each subnetwork asymptotically converge to an individual consistent value in the presence of information exchanges among subnetworks. Linear multiconsensus protocols are proposed to solve the multiconsensus problem, and the matrix corresponding to the protocol is designed. Necessary and sufficient conditions are derived based on matrix theory, under which the stationary multiconsensus and dynamic multiconsensus can be reached. Simulations are provided to demonstrate the effectiveness of the theoretical results.

Multivariate Simultaneous Generalized ARCH

1995 ◽  
Vol 11 (1) ◽  
pp. 122-150 ◽  
Author(s):  
Robert F. Engle ◽  
Kenneth F. Kroner
Keyword(s):  
Sufficient Conditions ◽  
Likelihood Estimation ◽  
Positive Definiteness ◽  
Covariance Matrices ◽  
Simultaneous Equations ◽  
Equivalence Relations ◽  
Conditional Covariance ◽  
Arch Models ◽  
Necessary And Sufficient ◽  
Theoretical Results

This paper presents theoretical results on the formulation and estimation of multivariate generalized ARCH models within simultaneous equations systems. A new parameterization of the multivariate ARCH process is proposed, and equivalence relations are discussed for the various ARCH parameterizations. Constraints sufficient to guarantee the positive definiteness of the conditional covariance matrices are developed, and necessary and sufficient conditions for covariance stationarity are presented. Identification and maximum likelihood estimation of the parameters in the simultaneous equations context are also covered.

Embeddings of weighted Sobolev spaces into spaces of continuous functions

1992 ◽  
Vol 439 (1906) ◽  
pp. 279-296 ◽  
Keyword(s):  
Weighted Sobolev Spaces ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Compact Embeddings ◽  
Spaces Of Continuous Functions ◽  
Hölder Continuous Functions ◽  
Continuous And Compact Embeddings ◽  
Necessary And Sufficient ◽  
Theoretical Results

We give sufficient conditions and necessary conditions (which in some cases are both necessary and sufficient) for continuous and compact embeddings of the weighted Sobolev space W 1,p ( Ω ;v 0 v 1 )into spaces of weighted continuous and Holder continuous functions. The theoretical results are illustrated by several examples.

scholarly journals Computer-controlled variable-structure systems

1992 ◽  
Vol 34 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Xinghuo Yu ◽  
Renfrey B. Potts
Keyword(s):  
Computer Control ◽  
Sufficient Conditions ◽  
Variable Structure ◽  
Dimensional System ◽  
Sliding Modes ◽  
Variable Structure Systems ◽  
Computer Controlled ◽  
Two Dimensional System ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractA theory is developed for the computer control of variable-structure systems, using periodic zero-order-hold sampling. A simple two-dimensional system is first analysed, and necessary and sufficient conditions for the occurrence of pseudo-sliding modes are discussed. The method is then applied to a discrete model of a cylindrical robot. The theoretical results are illustrated by computer simulations.

scholarly journals Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

2021 ◽  
Vol 37 ◽  
pp. 359-369
Author(s):  
Marko Kostadinov
Keyword(s):  
Linear Combination ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

scholarly journals An Exposition of Fractional Spurs Resulting From Nonlinear Distortion

2021 ◽  
Author(s):  
Yann Donnelly ◽  
Michael Peter Kennedy
Keyword(s):  
Nonlinear Distortion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantization Noise ◽  
Linear Combinations ◽  
Upper Limit ◽  
Comprehensive Theory ◽  
Free Behavior ◽  
Necessary And Sufficient

The interaction between quantization noise intro-duced by the divider controller and memoryless nonlinearities in a fractional-N PLL causes fractional spurs to occur. This paper presents a comprehensive theory to explain why combinations of quantizers and memoryless nonlinearities produce fractional spurs. Necessary and sufficient conditions for spur-free behavior in the presence of an arbitrary memoryless nonlinearity or linear combinations of sets of arbitrary memoryless nonlinearities are derived. Finally, an upper limit on the number of nonlinearities for which a quantizer can exhibit spur-free performance is derived.

scholarly journals Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices

2015 ◽  
Vol 30 ◽  
pp. 843-870 ◽  
Author(s):  
Cheng-yi Zhang ◽  
Dan Ye ◽  
Cong-Lei Zhong ◽  
SHUANGHUA SHUANGHUA
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Sufficient Conditions ◽  
Numerical Linear Algebra ◽  
Convergence Results ◽  
Diagonally Dominant Matrices ◽  
Sufficient Conditions For Convergence ◽  
Diagonally Dominant ◽  
Necessary And Sufficient ◽  
Theoretical Results

It is well known that as a famous type of iterative methods in numerical linear algebra, Gauss-Seidel iterative methods are convergent for linear systems with strictly or irreducibly diagonally dominant matrices, invertible H−matrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with non-strictly diagonally dominant matrices and general H−matrices. This paper firstly proposes some necessary and sufficient conditions for convergence on Gauss-Seidel iterative methods to establish several new theoretical results on linear systems with nonstrictly diagonally dominant matrices and general H−matrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general H−matrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper.

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