On the Ordering Property and Law of Importation in Fuzzy Logic

In this paper, the authors investigate the ordering property (OP), , together with the general form of the law of importation(LI), i.e., , whereis a t-norm andis a fuzzy implication for the four main classes of fuzzy implications. The authors give necessary and sufficient conditions under which both (OP) and (LI) holds for S-, R-implications and some specific families of QL-, D-implications. Following this, the paper proposes the sufficient condition under which the equivalence between CRI and triple I method for FMP can be established. Moreover, this conclusion can be viewed as a unified triple I method, a generalized form of the known results proposed by Wang and Pei.

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On the law of the iterated logarithm in the infinite variance case

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700021868 ◽

1980 ◽

Vol 30 (1) ◽

pp. 5-14 ◽

Cited By ~ 2

Author(s):

R. A. Maller

Keyword(s):

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Infinite Variance ◽

Iterated Logarithm ◽

The Law ◽

Necessary And Sufficient ◽

Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

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Equipartitioning and balancing points of polygons

Pythagoras ◽

10.4102/pythagoras.v0i71.2 ◽

2010 ◽

Vol 0 (71) ◽

Author(s):

Shunmugam Pillay ◽

Poobhalan Pillay

Keyword(s):

General Theorem ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

The Other ◽

Sufficient Condition ◽

Centre Of Mass ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

The centre of mass G of a triangle has the property that the rays to the vertices from G sweep out triangles having equal areas. We show that such points, termed equipartitioning points in this paper, need not exist in other polygons. A necessary and sufficient condition for a quadrilateral to have an equipartitioning point is that one of its diagonals bisects the other. The general theorem, namely, necessary and sufficient conditions for equipartitioning points for arbitrary polygons to exist, is also stated and proved. When this happens, they are in general, distinct from the centre of mass. In parallelograms, and only in them, do the two points coincide.

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Asymptotic stability of heteroclinic cycles in systems with symmetry. II

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500003693 ◽

2004 ◽

Vol 134 (6) ◽

pp. 1177-1197 ◽

Cited By ~ 59

Author(s):

Martin Krupa ◽

Ian Melbourne

Keyword(s):

Asymptotic Stability ◽

Transition Matrix ◽

Sufficient Conditions ◽

Systematic Investigation ◽

Necessary And Sufficient Conditions ◽

Sufficient Condition ◽

Heteroclinic Cycles ◽

Higher Dimensional ◽

Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

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Necessary and sufficient conditions for asymptotic decay of oscillations in delayed functional equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500012695 ◽

1981 ◽

Vol 91 (1-2) ◽

pp. 135-145

Author(s):

Lu-San Chen ◽

Cheh-Chih Yeh

Keyword(s):

Differential Operator ◽

Functional Equations ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Oscillatory Solutions ◽

Asymptotic Decay ◽

Image Position ◽

Necessary And Sufficient

SynopsisThis paper studies the equationwhere the differential operator Ln is defined byand a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case wherewhere N is an integer with l≦N≦n–1.

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Some results on the complement of the annihilating ideal graph of a commutative ring

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500991 ◽

2015 ◽

Vol 14 (07) ◽

pp. 1550099 ◽

Cited By ~ 3

Author(s):

S. Visweswaran ◽

Hiren D. Patel

Keyword(s):

Commutative Ring ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

Rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. For such a ring R, we determine necessary and sufficient conditions in order that the complement of its annihilating ideal graph is connected and also find its diameter when it is connected. We discuss the girth of the complement of the annihilating ideal graph of R and prove that it is either equal to 3 or ∞. We also present a necessary and sufficient condition for the complement of the annihilating ideal graph to be complemented.

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The law of the iterated logarithm for exchangeable random variables

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171295000482 ◽

1995 ◽

Vol 18 (2) ◽

pp. 391-396

Author(s):

Hu-Ming Zhang ◽

Robert L. Taylor

Keyword(s):

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Exchangeable Random Variables ◽

Iterated Logarithm ◽

The Law ◽

Necessary And Sufficient

In this note, necessary and sufficient conditions for laws of the iterated logarithm are developed for exchangeable random variables.

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ON A NEW CLASS OF IMPLICATIONS: (g,min)-IMPLICATIONS AND SEVERAL CLASSICAL TAUTOLOGIES

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488512500018 ◽

2012 ◽

Vol 20 (01) ◽

pp. 1-20 ◽

Cited By ~ 14

Author(s):

HUA-WEN LIU

Keyword(s):

Classical Logic ◽

Functional Equations ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Triangular Norms ◽

Basic Properties ◽

New Class ◽

Fuzzy Implications ◽

Necessary And Sufficient

A new class of fuzzy implications, called (g, min )-implications, is introduced by means of the additive generators of continuous Archimedean t-conorms, called g-generators. Basic properties of these implications are discussed. It is shown that the (g, min )-implications are really a new class different from the known ( S , N )-, R -, QL - and Yager's f- and g-implications. Generalizations of three classical logic tautologies with implications, viz. law of importation, contraction law and distributivity over triangular norms ( t -norms) and triangular conorms ( t -conorms) are investigated. A series of necessary and sufficient conditions are proposed, under which the corresponding functional equations are satisfied.

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Hendricks libraries and Tsetlin piles

Advances in Applied Probability ◽

10.1017/s0001867800036697 ◽

1982 ◽

Vol 14 (01) ◽

pp. 37-55 ◽

Cited By ~ 1

Author(s):

Jacques-Edouard Dies

Keyword(s):

Markov Chains ◽

Large Class ◽

Special Class ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Stationary Measure ◽

Sufficient Condition ◽

Positive Recurrence ◽

Necessary And Sufficient

In order to study the transience of Hendricks libraries, we introduce and study a special class of Markov chains, the Tsetlin d-piles, generalizing Tsetlin libraries and briefly defined as follows: a 1-pile is a Tsetlin library and a d-pile is a Tsetlin library where each book is replaced by a (d − 1)-pile. We give a stationary measure of these chains and establish the necessary and sufficient conditions for positive recurrence and transience. Finally, the study of d-piles allows us to determine a sufficient condition for transience of quite a large class of Hendricks libraries.

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Normal and Quasi-Normal Modes in Damped Linear Dynamic Systems

Journal of Applied Mechanics ◽

10.1115/1.3625058 ◽

1966 ◽

Vol 33 (2) ◽

pp. 413-416 ◽

Cited By ~ 3

Author(s):

J. S. Maybee

Keyword(s):

Linear Systems ◽

Dynamic Systems ◽

Normal Modes ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Linear Dynamic Systems ◽

Linear Dynamic ◽

Necessary And Sufficient

A generalization of the concept of classical normal modes in damped linear systems is presented. It is then shown that a necessary and sufficient condition that such quasi-normal modes exist is that certain matrices associated with the system commute. Necessary and sufficient conditions of the same type are also obtained for the classical normal modes, but under more restrictive conditions.

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On a certain kind of reducible rational fractions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700006183 ◽

1980 ◽

Vol 21 (3) ◽

pp. 321-328

Author(s):

Mordechai Lewin

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Rational Fraction ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Image Position ◽

Necessary And Sufficient ◽

Positive Coefficients ◽

Positive Integers ◽

Exact Positions

The rational fractiona, c, p, q positive integers, reduces to a polynomial under conditions specified in a result of Grosswald who also stated necessary and sufficient conditions for all the coefficients to tie nonnegative.This last result is given a different proof using lemmas interesting in themselves.The method of proof is used in order to give necessary and sufficient conditions for the positive coefficients to be equal to one. For a < 2pq, a = αp + βq, α, β nonnegative integers, c > 1, the exact positions of the nonzero coefficients are established. Also a necessary and sufficient condition for the number of vanishing coefficients to be minimal is given.

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