Non-strict don't care algebras and specifications

1996 ◽  
Vol 6 (1) ◽  
pp. 85-125 ◽  
Author(s):  
Egidio Astesiano ◽  
Maura Cerioli
Keyword(s):  
Computer Science ◽  
Essential Role ◽  
Sufficient Conditions ◽  
Appropriate Use ◽  
Partial Case ◽  
Basic Properties ◽  
Algebraic Specifications ◽  
Algebraic Framework ◽  
Necessary And Sufficient ◽  
The Relationship

Non-strict don't care functions, whose foremost representative is the ubiquitous if_then_else, play an essential role in computer science. As far as the semantics is concerned, they can be modelled by their totalizations with the appropriate use of elements representing undefinedness, as D. Scott has shown in his denotational approach. The situation is not so straightforward when we consider non-strict functions in the context of an algebraic framework; this point is discussed in the last section, where we explore the relationship between non-strict don't care and total algebras. The central part of this paper, after presenting the basic properties of the category of non-strict algebras, is an investigation of conditional algebraic specifications. It is shown that non-strict conditional specifications are equivalent to disjunctive specifications, and necessary and sufficient conditions for the existence of initial models are given. Since non-strict don't care specifications generalize both the total and the partial case, it is shown how the results for initiality can be obtained as specializations.

scholarly journals Property $(w)$ of upper triangular operator Matrices

2020 ◽  
Vol 51 (2) ◽  
pp. 81-99
Author(s):  
Mohammad M.H Rashid
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Extension Property ◽  
Operator Matrices ◽  
Single Valued Extension Property ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Number Of Authors

Let $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}\in\LB(\x,\y)$ be be an upper triangulate Banach spaceoperator. The relationship between the spectra of $M_C$ and $M_0,$ and theirvarious distinguished parts, has been studied by a large number of authors inthe recent past. This paper brings forth the important role played by SVEP,the {\it single-valued extension property,} in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type $M_0$ satisfies property $(w)$ $\Leftrightarrow$ $M_C$ satisfies property $(w)$ to hold. Moreover, we explore certain conditions on $T\in\LB(\hh)$ and $S\in\LB(\K)$ so that the direct sum $T\oplus S$ obeys property $(w)$, where $\hh$ and $\K$ are Hilbert spaces.

Insensitivity and reversed Markov processes

1983 ◽  
Vol 15 (4) ◽  
pp. 752-768 ◽  
Author(s):  
W. Henderson
Keyword(s):  
Markov Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
The Relationship

This paper is concerned with the relationship between insensitivity in a certain class of Markov processes and properties of that process when time is reversed. Necessary and sufficient conditions for insensitivity are established and linked to already proved results. A number of examples of insensitive systems are given.

Semicritical Rings and the Quotient Problem

1984 ◽  
Vol 27 (2) ◽  
pp. 160-170
Author(s):  
Karl A. Kosler
Keyword(s):  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Regular Elements ◽  
The Right ◽  
Minimal Prime ◽  
Necessary And Sufficient ◽  
Quotient Rings ◽  
The Relationship

AbstractThe purpose of this paper is to examine the relationship between the quotient problem for right noetherian nonsingular rings and the quotient problem for semicritical rings. It is shown that a right noetherian nonsingular ring R has an artinian classical quotient ring iff certain semicritical factor rings R/Ki, i = 1,…,n, possess artinian classical quotient rings and regular elements in R/Ki lift to regular elements of R for all i. If R is a two sided noetherian nonsingular ring, then the existence of an artinian classical quotient ring is equivalent to each R/Ki possessing an artinian classical quotient ring and the right Krull primes of R consisting of minimal prime ideals. If R is also weakly right ideal invariant, then the former condition is redundant. Necessary and sufficient conditions are found for a nonsingular semicritical ring to have an artinian classical quotient ring.

scholarly journals The Q0-matrix completion problem

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Kalyan Sinha
Keyword(s):  
Matrix Completion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Completion Problem ◽  
Matrix Completion Problem ◽  
Necessary And Sufficient ◽  
The Relationship

A matrix is a Q0-matrix if for every k∈{1,2,…,n}, the sum of all k×k principal minors is nonnegative. In this paper, we study some necessary and sufficient conditions for a digraph to have Q0-completion. Later on we discuss the relationship between Q and Q0-matrix completion problem. Finally, a classification of the digraphs of order up to four is done based on Q0-completion.

Insensitivity and reversed Markov processes

1983 ◽  
Vol 15 (04) ◽  
pp. 752-768 ◽  
Author(s):  
W. Henderson
Keyword(s):  
Markov Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
The Relationship

This paper is concerned with the relationship between insensitivity in a certain class of Markov processes and properties of that process when time is reversed. Necessary and sufficient conditions for insensitivity are established and linked to already proved results. A number of examples of insensitive systems are given.

THE INDEPENDENCE OF FUZZY VARIABLES WITH APPLICATIONS TO FUZZY RANDOM OPTIMIZATION

2007 ◽  
Vol 15 (supp02) ◽  
pp. 1-20 ◽  
Author(s):  
YIAN-KUI LIU ◽  
JINWU GAO
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Possibility Distribution ◽  
Fuzzy Variables ◽  
Fundamental Properties ◽  
Random Optimization ◽  
Fuzzy Random Programming ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Fuzzy Random

This paper presents the independence of fuzzy variables as well as its applications in fuzzy random optimization. First, the independence of fuzzy variables is defined based on the concept of marginal possibility distribution function, and a discussion about the relationship between the independent fuzzy variables and the noninteractive (unrelated) fuzzy variables is included. Second, we discuss some properties of the independent fuzzy variables, and establish the necessary and sufficient conditions for the independent fuzzy variables. Third, we propose the independence of fuzzy events, and deal with its fundamental properties. Finally, we apply the properties of the independent fuzzy variables to a class of fuzzy random programming problems to study their convexity.

scholarly journals A Note on the⊤-Stein Matrix Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Chun-Yueh Chiang
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Comprehensive Treatment ◽  
Backward Error ◽  
Linear Matrix Equation ◽  
Perturbation Bounds ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Linear Matrix Equations

This note is concerned with the linear matrix equationX=AX⊤B + C, where the operator(·)⊤denotes the transpose (⊤) of a matrix. The first part of this paper sets forth the necessary and sufficient conditions for the unique solvability of the solutionX. The second part of this paper aims to provide a comprehensive treatment of the relationship between the theory of the generalized eigenvalue problem and the theory of the linear matrix equation. The final part of this paper starts with a brief review of numerical methods for solving the linear matrix equation. In relation to the computed methods, knowledge of the residual is discussed. An expression related to the backward error of an approximate solution is obtained; it shows that a small backward error implies a small residual. Just like the discussion of linear matrix equations, perturbation bounds for solving the linear matrix equation are also proposed in this work.Erratum to “A Note on the⊤-Stein Matrix Equation”

scholarly journals k-Kernel Symmetric Matrices

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
A. R. Meenakshi ◽  
D. Jaya Shree
Keyword(s):  
Scalar Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Symmetric Matrices ◽  
Fuzzy Matrix ◽  
Basic Properties ◽  
Necessary And Sufficient

In this paper we present equivalent characterizations ofk-Kernel symmetric Matrices. Necessary and sufficient conditions are determined for a matrix to bek-Kernel Symmetric. We give some basic results of kernel symmetric matrices. It is shown that k-symmetric impliesk-Kernel symmetric but the converse need not be true. We derive some basic properties ofk-Kernel symmetric fuzzy matrices. We obtain k-similar and scalar product of a fuzzy matrix.

scholarly journals Process tree-based analysis method of DECLARE relation constraints in acyclic bridge-less well-structured workflow nets

2019 ◽  
Vol 13 (2) ◽  
pp. 104-111
Author(s):  
Shingo Yamaguchi ◽  
Mohd Anuaruddin Bin Ahmadon
Keyword(s):  
Polynomial Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Analysis Method ◽  
Declarative Language ◽  
Tree Representation ◽  
Workflow Nets ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Process Tree

In this paper, we proposed a method to analyze workflows’ constraints whose templates are defined in a declarative language called DECLARE. Checking such constraints is important but known to be intractable in general. Our results show three things. First, utilizing a tree representation of workflow process called {\it process tree}, we provided necessary and sufficient conditions on the constraints. Second, those conditions enable us to not only check a given constraint in polynomial time but also find a clue for improving the net if it violates the constraint. Third, we revealed the relationship among the constraint templates.

Core for Game with Fuzzy Generalized Triangular Payoff Value

2019 ◽  
Vol 27 (05) ◽  
pp. 789-813 ◽  
Author(s):  
Xiaohui Yu ◽  
Qiang Zhang
Keyword(s):  
Cooperative Game ◽  
Sufficient Conditions ◽  
Triangular Fuzzy Number ◽  
Existence Condition ◽  
Stable Conditions ◽  
Fuzzy Core ◽  
Fuzzy Payoff ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Sufficient Existence Condition

In this paper, we investigate cooperative game with fuzzy payoff value in the generalized triangular fuzzy number directly. Based on the fuzzy max order, we define three kinds of fuzzy cores, i.e., fuzzy strong core, fuzzy non-dominated core and fuzzy weak core. All three kinds of fuzzy cores can be regarded as the generalization of crisp core. Convexity is one of the sufficient conditions for the existence of fuzzy core. By the balanced cooperative game, a necessary and sufficient existence condition of fuzzy strong core is also given. Further, the fuzzy strong core is represented by crisp core, and the relationship between fuzzy strong core and crisp one is shown. For the fuzzy non-dominated core and fuzzy weak core, we show their necessary and sufficient existence condition, and their properties to construct the fuzzy imputations. Hence, the verification of fuzzy non-dominated core and fuzzy weak core become easier. The above three fuzzy cores are all the extensions of crisp core, but their stable conditions are not different. The weak core is least restricted, but is least stable. Hence, we could choose the fuzzy core according the stable neediness of fuzzy cooperative game.

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