Pl. Kannappan

QUADRATIC FUNCTIONAL EQUATION AND INNER PRODUCT SPACES

Quadratic functional equation was used to characterize inner product spaces. Several other functional equations were also used to characterize inner product spaces. In this paper we solve five funtional equations (1), (2), (3), (4), and (5) connected to quadratic functional equation and inner product spaces.

Measures of distance between probability distributions

Abstract In statistical estimation problems measures between probability distributions play significant roles. Hellinger coefficient, Jeffreys distance, Chernoff coefficient, directed divergence, and its symmetrization J -divergence are examples of such measures. Here these and like measures are characterized through a composition law and the sum form they possess. The functional equations f ( pr , qs ) + f ( ps , qr ) = ( r + s ) f ( p , q ) + ( p + q ) f ( r , s ) and f ( pr , qs ) + f ( ps , qr ) = f ( p , q ) f ( r , s ) are instrumental in their deduction.

A generalization of the cosine-sine functional equation on groups

Abstract The functional equation f ( xy )= f ( x ) g ( y )+ g ( x ) f ( y )+ h ( x ) h ( y ) is solved where f , g , h are complex functions defined on a group.

Functional Equations and Inequalities in ‘Rational Group Decision Making’

This paper consists of a reformulation and generalization of results in [2–4]. A problem of ‘rational group decision making’ is the following: A fixed amount s is to be allocated to a fixed number m of competing projects. Each member of a group of n decision makers makes recommendations, the jth allocating, say, xij to the ith project, in order to establish the ‘consensus’ allocation fi(x i) [we write x i = (xil,..., xin)]. We suppose only that fi(0) = 0 [‘consensus of rejection’; 0 = (0,...,0)] and that the allocations are non-negative. In the case m > 2, we prove that each fi is the same weighted arithmetic mean. There are other solutions for m = 2, even if f1 = f2 is supposed. We determine all of them. The solutions are also established for variable s.

A mixed theory of information — IV: Inset-inaccuracy and directed divergence

The general form of recursive, symmetric and regular (measurable) deviation of randomized systems of events is derived in the frame work of the mixed theory of information, which includes as special cases the purely probabilistic measures the inaccuracy and the directed divergence.

Recursive inset entropies of multiplicative type on open domains

SummaryLet ℬ be a ring of sets, and letI be thek-dimensional open unit interval. The functional equation

General two-place information functions

\begin{gathered} \varphi (E \cup F,G;p) + \mu (1 - p)\varphi \left( {E,F;\frac{q}{{1 - p}}} \right) \hfill \\ = \varphi (E \cup G,F;q) + \mu (1 - q)\varphi \left( {E,G;\frac{p}{{1 - q}}} \right), \hfill \\ \end{gathered}

On a functional equation connected with generalized directed-divergence

for all disjoint triplesE, F, G of nonvoid sets in ℬ and all pairsp, q inI withp + q ∈ I, is solved for ϕ and multiplicative μ. This problem was posed by Aczél in Aequationes Math.26 (1984), 255–260. Our solution to this problem leads to an axiomatic characterization of measures of inset informationIn(E1,⋯,En;