scholarly journals AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET INTEGRALS

2011 ◽  
Vol 19 (02) ◽  
pp. 193-199 ◽  
Author(s):  
MARTA CARDIN ◽  
MIGUEL COUCEIRO ◽  
SILVIO GIOVE ◽  
JEAN-LUC MARICHAL
Keyword(s):  
Boolean Function ◽  
Functional Equations ◽  
Choquet Integral ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lovász Extension ◽  
Choquet Integrals ◽  
Necessary And Sufficient

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lovász extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory.

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

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.

scholarly journals An Extension of a Ger’s Result

2018 ◽  
Vol 32 (1) ◽  
pp. 263-274
Author(s):  
Dan Ştefan Marinescu ◽  
Mihai Monea
Keyword(s):  
Continuous Function ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bregman Distance ◽  
International Conference ◽  
Necessary And Sufficient

Abstract The aim of this paper is to extend a result presented by Roman Ger during the 15th International Conference on Functional Equations and Inequalities. First, we present some necessary and sufficient conditions for a continuous function to be convex. We will use these to extend Ger’s result. Finally, we make some connections with other mathematical notions, as g-convex dominated function or Bregman distance.

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

2012 ◽  
Vol 20 (01) ◽  
pp. 1-20 ◽  
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.

scholarly journals On linear functional equations modulo $${\mathbb {Z}}$$

2021 ◽  
Author(s):  
Attila Gilányi ◽  
Agata Lewicka
Keyword(s):  
Linear Space ◽  
Functional Equations ◽  
Linear Functional ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Functional Equations ◽  
Necessary And Sufficient

AbstractIn this paper, we consider the condition $$\sum _{i=0}^{n+1}\varphi _i(r_ix+q_iy)\in {\mathbb {Z}}$$ ∑ i = 0 n + 1 φ i ( r i x + q i y ) ∈ Z for real valued functions defined on a linear space V. We derive necessary and sufficient conditions for functions satisfying this condition to be decent in the following sense: there exist functions $$f_i:V\rightarrow {\mathbb {R}}$$ f i : V → R , $$g_i:V\rightarrow {\mathbb {Z}}$$ g i : V → Z such that $$\varphi _i=f_i+g_i$$ φ i = f i + g i , $$(i=0,\dots ,n+1)$$ ( i = 0 , ⋯ , n + 1 ) and $$\sum _{i=0}^{n+1}f_i(r_ix+q_iy)=0$$ ∑ i = 0 n + 1 f i ( r i x + q i y ) = 0 for all $$x, y\in V$$ x , y ∈ V .

scholarly journals Necessary and sufficient conditions for eventually vanishing oscillatory solutions of functional equations with small delays

1978 ◽  
Vol 1 (3) ◽  
pp. 269-283 ◽  
Author(s):  
Bhagat Singh
Keyword(s):  
Functional Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Oscillatory Solutions ◽  
All Solutions ◽  
Approach Zero ◽  
Necessary And Sufficient

Necessary and sufficient conditions are found for all oscillatory solutions of the equation(rn−1(t)(rn−2(t)(−−−(r2(t)(r1(t)y′(t)))−−−)))+a(t)h(y(g(t)))=b(t)to approach zero. Sufficient conditions are also given to ensure that all solutions of this equation are unbounded.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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