For given finite (unordered) graphs G and H, we examine the existence of a Ramsey graph F for which the strong Ramsey arrow F −→ (G)r holds. We concentrate on the situation when H is not a complete graph. The set of graphs G for which there exists an F satisfying F −→ (G)2 2 (P2 is a path on 3 vertices) is found to be the union of the set of chordal comparability graphs together with the set of...

A fundamental property of convex functions in continuous space is that the convexity is preserved under affine transformations. A set function f on a finite set V is submodular if f(X) + f(Y ) ≥ f(X ∪ Y )− f(X ∩ Y ) for any pair X,Y ⊆ V . The symmetric difference transformation (SD-transformation) of f by a canonical set S ⊆ V is a set function g given by g(X) = f(X M S) for X ⊆ V , where X M S...

In this talk we are first going to state the so-called ’Representation Theorem’ which provides the representation of general f -divergences in terms of the class of elementary divergences. Then we consider the risk set of a (simple versus simple) testing problem and its characterization by the above mentioned class. These ingredients enable us to prove a sharpening of the ’Range of Values Theor...

Every F-probability (= coherent probability) F on a finite sample space Ωk with k elements defines a set of classical probabilities in accordance with the interval limits. This set, called “structure” of F , is a convex polytope having dimension ≤ k−1. We prove that the maximal number of vertices of structures is exactly k!.

For some given positive γ, a function f is called outer γ-convex if it satisfies the Jensen inequality f(zi) ≤ (1 − λi)f(x0) + λif(x1) for some z0 : = x0, z1, ..., zk : = x1 ∈ [x0, x1] satisfying ‖zi − zi+1‖ ≤ γ, where λi : = ‖x0 − zi‖/‖x0 − x1‖, i = 1, 2, ..., k − 1. Though the Jensen inequality is only required to hold true at some points (although the location of these points is uncertain) o...

where tn → 1 n → ∞ . We may assume that tn ≥ 1 for all n 1, 2, 3, . . . . Denote by F T the set of fixed points of T . Throughout this paper T and S : C → C are two commutative asymptotically nonexpansive mappings with asymptotical coefficients {tn} and {sn}, respectively. Suppose that F : F T ∩F S / ∅ 1, Goebel and Kirk’s theorem makes it possible . It is well known that F T and F S are convex...