Volume 1, Issue 2

We establish an analog of the Morgan theorem for the Kontorovich–Lebedev operator of general complex index.

This paper is devoted to the study of the differential-difference Jacobi–Cherednik operator defined for f ∈ C 1 (𝕣) by T ( k,k′ ) f ( x ) = f′ ( x ) + ( k coth( x ) + k ′ tanh( x ))( f ( x ) – f (– x )) – ( k + k′ ) f (– x ), where k > 0 and k′ ≥ 0 are two parameters, and to the positivity of the operator which intertwines T ( k,k′ ) and the derivative operator .

In this paper we prove the absolute continuity of the representing measures of the Dunkl intertwining operator and of its dual. Next we present some applications of this result.

Given a directed graph G = ( V,A ), a subset X of V is a clan of G provided that for a, b ∈ X and x ∈ V \ X , ( a, x ) ∈ A if and only if ( b, x ) ∈ A , and similarly for ( x, a ) and ( x, b ). For instance, ∅, V and { x }, where x ∈ V , are clans of G , called trivial. A directed graph is clan-primitive if all its clans are trivial. Given a directed graph G = ( V,A ), a subset X of V is a cut of G if X and V \ X are clans of G . For example, ∅ and V are cuts of G , called trivial. A directed graph is cut-primitive if all its cuts are trivial. Ehrenfeucht and Rozenberg [Theoret. Comput. Sci. 70: 343–358, 1990] proved: Given a clan-primitive directed graph G = ( V,A ), if X is a subset of V such that | X | ≥ 3, | V \ X | ≥ 2 and G [ X ] is clan-primitive, then there are x ≠ y ∈ V \ X such that G [ X ∪ { x, y }] is clan-primitive. We show: Given a clan-primitive directed graph G = ( V,A ), if X is a proper subset of V such that | X | ≥ 3 and G [ X ] is cut-primitive, then there are x, y ∈ V \ X such that G [ X ∪ { x, y }] is cut-primitive and { x }, { y } are maximal proper clans of G [ X ∪ { x, y }].

Using the KKM technique, we establish some existence results for variational-hemivariational inequalities involving monotone set valued mappings on bounded, closed and convex subsets in reflexive Banach spaces. We also derive several sufficient conditions for the existence of solutions in the case of unbounded subsets.

Making use of the extremal points theory, we investigate some extremal problems for the class of analytic functions with negative coefficients defined by Wright's generalized hypergeometric function.

We study the range of validity of differentiation theorems and ergodic theorems for actions, for averages on “thick spheres” of Euclidean space.

This paper first generalizes the notion of basis graph of a finite matroid to that of a matroid of arbitrary cardinality. Then we follow by dealing with the relationship of a matroid of arbitrary cardinality between its connectivity and its basis graph.

A Just-Non-Compact group, or briefly a JNC group, is a Hausdorff topological group which is not a compact group but all of whose proper Hausdorff quotients are compact groups. Intuitively, it is clear that these groups are rich in compact quotients. Locally compact JNC groups are largely described in the present paper.