Equality of two-variable functional means generated by different measures

We consider two-variable functional means of the form $$M_{f,g;\mu}(x,y) := \left(\frac{f}{g}\right)^{-1}\left(\frac{\int\nolimits_{[0,1]} f(tx+(1-t)y)\,d\mu(t)}{\int\nolimits_{[0,1]}g(tx+(1-t)y)\,d\mu(t)}\right),$$ where f, g are continuous functions on a real interval such that g is positive, f/g is strictly monotonic and??? is a measure over the Borel sets of [0,1]. The main results concern the functional equation M _f,g;?? ?=?M _f,g;?? for the unknown functions f, g, where??? and ?? are given measures. Depending on the symmetry properties of the measures, various necessary conditions and sufficient conditions are established.     

On the equality of quasi-arithmetic and Lagrangian means

The paper deals with the equality problem of quasi-arithmetic and Lagrangian means which is to determine all pairs of continuous strictly monotone functions φ,ψ:I→R such that, for all x,y∈I,

     

Computer aided solution of the invariance equation for two-variable Stolarsky means

We solve the so-called invariance equation in the class of two-variable Stolarsky means {S_p,q:p,q∈R}, i.e., we find necessary and sufficient conditions on the six parameters a, b, c, d, p, q such that the identity

     

Local monotonicity properties of two-variable Gini means and the comparison theorem revisited

In this paper we examine local monotonicity properties of the so-called Gini means, defined by

     

The equality problem in the class of conjugate means

Let ${I\subset\mathbb{R}}$ be a nonempty open interval and let ${L:I^2\to I}$ be a fixed strict mean. A function ${M:I^2\to I}$ is said to be an L-conjugate mean on I if there exist ${p,q\in{]}0,1]}$ and a strictly monotone and continuous function φ such that $$M(x,y):=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(L(x,y)))=:L_\varphi^{(p,q)}(x,y),$$ for all ${x,y\in I}$ . Here L(x, y) is a fixed quasi-arithmetic mean. We will solve the equality problem in this class of means.     

On the parseval equality

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 5, pp. 178–181, Spetember–October, 1990.     

On a general bilinear functional equation

Aequationes mathematicae - Let X, Y be linear spaces over a field $${\mathbb {K}}$$ . Assume that $$f :X^2\rightarrow Y$$ satisfies the general linear equation with respect to the first...     

On soft equality

Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we deal with the algebraic structure of soft sets. The lattice structures of soft sets are constructed. The concept of soft equality is introduced and some related properties are derived. It is proved that soft equality is a congruence relation with respect to some operations and the soft quotient algebra is established.     

On the sublinear functional associated to a family of invariant means

In this paper we give conditions under which we can obtain explicit analytic expressions of the fundamental sublinear functional P_K of the family of means K=[G]J where [G] is a subset of G-invariant means, G a semigroup of operators on 1 and J a saturated set of means. Such conditions allow us to assure that [G]J.We characterize the set of the almost convergent sequences related to the family K by means of the same functional p_K. We obtain also p_K in terms of p_J when p_J is known. This allow us to give different expressions of the same functional p_K when the family J changes and from which several examples are given.     

On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem

We present practical conditions under which the existence and uniqueness of a finite solution to a given equality quadratic program may be examined. Different sets of conditions allow this examination to take place when null-space, range-space or Lagrangian methods are used to find stationary points for the quadratic program.This research supported in part by the Natural Sciences and Engineering Research Council, Canada.