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In this paper, we introduce the notion of a Gel’fand Γ-semiring and discuss the various characterization of simple, *k*-ideal, strong ideal, *t*-small elements and additively cancellative elements of a Gel’fand Γ-semiring *R*, and prove that the set of additively cancellative elements, set of all *t*-small elements of *R* and set of all maximal ideal of *R* are strong ideals. Further, let *R* be a simple Gel’fand Γ-semiring and 1 ≠ *t* ∈ *R*. Let *M* be the set of all maximal left (right) ideals of *R*. Then an element *x* of *R* is *t*-small if and only if it belongs to every maximal one sided left (right)ideal of *R* containing *t*.

For a continuous and positive function *w*(λ), *λ >
* 0 and

*μ*a positive measure on (0, ∞) we consider the following

*integral transform*

where the integral is assumed to exist for *t* > 0.

We show among others that *D*(*w, μ*) is operator convex on (0, ∞). From this we derive that, if *f* : [0, ∞) → **R** is an operator monotone function on [0, ∞), then the function [*f*(0) -*f*(*t*)] *t*
^{-1} is operator convex on (0, ∞). Also, if *f* : [0, ∞) → **R** is an operator convex function on [0, ∞), then the function

under some natural assumptions for the positive operators A and B are given. Examples for power, exponential and logarithmic functions are also provided.

Problem 2 of Welsh’s 1976 text *Matroid Theory*, asking for criteria telling when two families of sets have a common transversal, is solved.

Another unsolved problem in the text *Matroid Theory*, on whether the “join” of two non-decreasing submodular functions is submodular, is answered in the negative. This resolves an issue first raised by Pym and Perfect in 1970.

This manuscript deals with the global existence and asymptotic behavior of solutions for a Kirchhoff beam equation with internal damping. The existence of solutions is obtained by using the Faedo-Galerkin method. Exponential stability is proved by applying Nakao’s theorem.

We consider hypersphere x = x(*u, v, w*) in the four dimensional Euclidean space. We calculate the Gauss map, and the curvatures of it. Moreover, we compute the second Laplace-Beltrami operator the hypersphere satisfying Δ^{II}x = *A*x, where *A* ϵ *Mat* (4,4).

In this paper, we show a Marcinkiewicz type interpolation theorem for Orlicz spaces. As an application, we obtain an existence result for a parabolic equation in divergence form.

Let *E, G* be Fréchet spaces and *F* be a complete locally convex space. It is observed that the existence of a continuous linear not almost bounded operator *T* on *E* into *F* factoring through *G* causes the existence of a common nuclear Köthe subspace of the triple (*E, G, F*). If, in addition, *F* has the property (*y*), then (*E, G, F*) has a common nuclear Köthe quotient.

In this paper we study the sum *n*, and {*n _{p}
*} is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is

In this paper we derive new inequalities involving the generalized Hardy operator. The obtained results generalized known inequalities involving the Hardy operator. We also get new inequalities involving the classical Hardy–Hilbert inequality.

The bipartite domination number of a graph is the minimum size of a dominating set that induces a bipartite subgraph. In this paper we initiate the study of this parameter, especially bounds involving the order, the ordinary domination number, and the chromatic number. For example, we show for an isolate-free graph that the bipartite domination number equals the domination number if the graph has maximum degree at most 3; and is at most half the order if the graph is regular, 4-colorable, or has maximum degree at most 5.