Two-place Function Research Articles

Characterizations of monotone right continuous functions which generate associative functions

Associativity of a two-place function T:[0,1]2→[0,1] defined by T(x,y)=f(−1)(T⁎(f(x),f(y))) where T⁎:[0,1]2→[0,1] is an associative function with neutral element in [0,1], f:[0,1]→[0,1] is a monotone right continuous function and f(−1):[0,1]→[0,1] is the pseudo-inverse of f depends only on properties of the range of f. The necessary and sufficient conditions for the T to be associative are presented by applying the properties of the monotone right continuous function f.

Импликативно неявные базисы в импликативно замкнутых классах трехзначной логики

It is proved that for any k ≥ 3, any implicatively implicit extension in P<sub>k</sub> contains a class of H<sub>k</sub> homogeneous functions. In each of the 17 implicatively closed classes in P<sub>3</sub>, an implicatively implicitly generating system of functions is constructed. With the exception of the H<sub>3</sub> class, each of the systems consists of one or two no more than two-place functions. A one-place function is specified, the implicatively implicit extension of which is different from any implicatively closed class in P<sub>3</sub>.

A Simple Interpretation of Quantity Calculus

A simple interpretation of quantity calculus is given. Quantities are described as two-place functions from objects, states or processes (or some combination of them) into numbers that satisfy the mutual measurability property. Quantity calculus is based on a notational simplification of the concept of quantity. A key element of the simplification is that we consider units to be intentionally unspecified numbers that are measures of exactly specified objects, states or processes. This interpretation of quantity calculus combines all the advantages of calculating with numerical values (since the values of quantities are numbers, we can do with them everything we do with numbers) and all the advantages of calculating with standardly conceived quantities (calculus is invariant to the choice of units and has built-in dimensional analysis). This also shows that the standard metaphysics and mathematics of quantities and their magnitudes is not needed for quantity calculus. At the end of the article, arguments are given that the concept of quantity as defined here is a pivotal concept in understanding the quantitative approach to nature. As an application of this interpretation of quantity calculus, an easy proof of dimensional homogeneity of physical laws is given.

Generalization of the Harmonic Weighted Mean Via Pythagorean Invariance Identity and Application

Abstract Under some simple conditions on the real functions f and g defined on an interval I ⊂ (0, ∞), the two-place functions Af (x, y) = f (x) + y − f (y) and G g ( x , y ) = g ( x ) g ( y ) y {G_g}\left({x,y} \right) = {{g\left(x \right)} \over {g\left(y \right)}}y generalize, respectively, A and G, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity G ∘ (H, A) = G (equivalent to the Pythagorean harmony proportion), a suitable weighted extension Hf,g of the classical harmonic mean H is introduced. An open problem concerning the symmetry of Hf,g is proposed. As an application a method of effective solving of some functional equations involving means is presented.

On the completion of fuzzy metric spaces

This paper discusses the completion of a fuzzy metric space (in the sense of Kaleva and Seikkala) that is modeled by a pair of two-place functions, L and R . Previous work on this question is based on L = min , R = max . Under the condition that L and R are a generic class of two-place functions, we characterize the completable fuzzy metric space and show that the corresponding completion is uniquely determined up to isometry. Furthermore, we point out that with more conditions on R , including on R = max , the completion of each fuzzy metric space exists and is unique up to isometry. This improves the corresponding result of Kaleva [The completion of fuzzy metric spaces, J. Math. Anal. Appl. 109 (1985) 194–198].

A probabilistic theory of coherence

Desideratum (1) captures the qualitative features that a probabilistic generalization of logical coherence should satisfy — it requires C to respect the extreme deductive cases, and to be properly sensitive to probabilistic dependence (a general notion of probabilistic dependence will be defined precisely, and in a slightly non-standard way, below). I propose a probabilistic measure of coherence C based on a slight modification of Kemeny and Oppenheim’s (1952) measure of factual support F. The formulation of C is somewhat intricate. We begin with some preliminary definitions. First, we define the two-place function F(X,Y). F(X,Y) may be interpreted as the degree to which one proposition Y supports another proposition X (relative to a finitely additive, regular, Kolmogorov (1956) probability function Pr).

Different interpretations of triangular norms and related operations

Triangular norms (and triangular conorms and uni-norms) can be treated as semigroup, two-place function, also as commutative semiring multiplications. It can also be derived from the difference operation in a difference poset. We discuss here these different interpretations and their consequences.

Approximations of Two-Place Functions by Finite Sums of Terms with Separable Variables

It is known that if a smooth function h in two real variables x and y belongs to the class Σ n of all sums of the form Σ n k=1 ƒ k ( x) g k ( y), then its ( n + 1)th order "Wronskian" det[ h x i y j ] n i, j=0 is identically equal to zero. The present paper deals with the approximation problem h( x, y) ≍ Σ n k=1 ƒ k ( x) g k ( y) with a prescribed n, for general smooth functions h not lying in Σ n . Two natural approximation sums T= T( h) ∈ Σ n , S= S( h) ∈ Σ n are introduced and the errors | h- T|, | h- S| are estimated by means of the above mentioned Wronskian of the function h. The proofs utilize the technique of ordinary linear differential equations.

Subadditive functions and partial converses of Minkowski's and Mulholland's inequalities

Let be an arbitrary bijection of R+. We prove that if the two-place function 1 ( (s) + (t)) is subadditive in R 2 then must be a convex homeomorphism of R+. This is a partial converse of Mulholland's inequality. Some new properties of subadditive bijections of R+ are also given. We apply the above results to obtain several converses of Minkowski's inequality.

On generalization of Nguyen's theorem

The goal of this paper is to generalize certain results of Nguyen [1] (concerning the α-cuts of two-place functions defined by the Zadeh's extension principle) to the case of extended two-place functions defined via a sup-t-norm convolution.

Alternative axiomatizations of elementary probability theory.

1 Outline In this paper I offer some new axiomatizations of elementary (finite) probability theory. The new axiomatizations have the following distinctive features. First, both conditional and unconditional probabilities are simultaneously axiomatized. Second, the axiomatizations are extensions of Kolmogorov probability theory, and have the usual definitions linking conditional and unconditional probabilities as theorems. Third, while both probabilities are simultaneously axiomatized, the resultant axioms are about as simple as the usual axiomatizations of conditional probability alone. Fourth, the axioms adopted have strong and direct justifications independently of the interpretation of the probability function (axiomatization L however is only a permissible extension of intuitions). The reader who wishes to review the axiomatizations before their rationale can at this point go to section 5 (the main axiomatization is system KK). 2 What is unsatisfactor y about usual axiomatizations? I offer alternative axiomatizations because I am dissatisfied with the customary ones. But this dissatisfaction has a special nature. In improving existing axiomatizations, one might reduce the number of axioms, or simplify them, or adopt axioms that make proofs easier and more elegant. Although in my own axiomatizations I am concerned with these factors, none of these traditional formal motivations is the source of my dissatisfaction. Instead, I am basically concerned about how the relationship between conditional and unconditional probabilities is developed (for some related misgivings, see de Finetti [2], pp. 81-83). There are two probability functions about which we have intuitions. The basic formal difference is that unconditional probability is a one-place function, and conditional probability is a twoplace function. As a result, there are two probabilistic concepts that can be axiomatized, and the two can be axiomatized independently. What makes a function a one-place probability function, and what makes a function a two-place probability function, are separate questions, and the correct answer for each involves a separate characterizing theory.

Representation for measures of information with the branching property

The representation for measures of information which are symmetric, expansible, and have the branching property in the form of a sum is provided. This class of measures includes, in particular, Shannon's entropy, entropies of degree β , Kullback's directed divergence, and Kerridge's inaccuracy. Rényi's entropy and information gain of order β are, however, excluded from this class. The proof is based on an algebraic theorem concerning the representation of a two-place function by the superposition of a one-place function.

On the degrees of index sets. II

A class of recursively enumerable sets may be classified either as an object in itself -the range of a two-place function in the obvious way or by means of the corresponding set of indices. The latter approach is not only more precise but also, as we show below, provides an alternative method for solving certain problems on recursively enumerable sets and their degrees of unsolvability. The main result of the present paper is the computation, for every recursively enumerable degree a, of the degree (in fact, isomorphism-type) of the index-set corresponding to the recursively enumerable sets of degree a: its degree is a(. It follows from a theorem of Sacks [10] that the degrees of such index-sets are exactly those which are > 0(3) and recursively enumerable in 0(3). In particular, this proves Rogers' conjecture [9] that the index-set corresponding to Ol) is of degree 0(4); partial results on this problem have been obtained by Rogers [9] and by Lacombe (unpublished). The most interesting immediate consequence of our result is a different proof of Sacks' theorem [11] that the recursively enumerable degrees are dense. We refer the reader to Kleene [5] and Sacks [10] for our basic terminology and notation. A useful summary of many results which connect degrees with the arithmetical hierarchy is presented in [9], which is a good background to the present paper since without it the latter would not exist. For an assortment of results on classes of recursively enumerable sets the reader is referred to [2]. If e is a number and A is a set, then we define the partial function OA by setting: