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For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the following*monotonic integral transform*

where the integral is assumed to exist for*T* a positive operator on a complex Hilbert space*H*. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)^{2} ≤ Δ for some constants α, β, δ, Δ, then

and

where

Applications for power function and logarithm are also provided.

Let ƒ be analytic in the unit disk B and normalized so that ƒ (z) = z + a_{2}z^{2} + a_{3}z^{3} +܁܁܁. In this paper, we give upper bounds of the Hankel determinant of second order for the classes of starlike functions of order α, Ozaki close-to-convex functions and two other classes of analytic functions. Some of the estimates are sharp.

The authors have studied the curvature of the focal conic in the isotropic plane and the form of the circle of curvature at its points has been obtained. Hereby, we discuss several properties of such circles of curvature at the points of a parabola in the isotropic plane.

Let *k* ≥ 1. A *Sperner k-family* is a maximum-sized subset of a finite poset that contains no chain with *k* + 1 elements. In 1976 Greene and Kleitman defined a lattice-ordering on the set *S _{k}
*(

*P*) of Sperner

*k*-families of a fifinite poset

*P*and posed the problem: “Characterize and interpret the join- and meet-irreducible elements of

*S*(

_{k}*P*),” adding, “This has apparently not been done even for the case

*k*= 1.”

In this article, the case *k* = 1 is done.

A linear operator on a Hilbert space *S* is shown to be densely defined and closed if and only if

In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.

Let *X* be a topological space. For any positive integer *n*, we consider the *n*-fold symmetric product of *X*, ℱ*
_{n}
*(

*X*), consisting of all nonempty subsets of

*X*with at most

*n*points; and for a given function

*ƒ*:

*X*→

*X*, we consider the induced functions ℱ

*(*

_{n}*ƒ*): ℱ

*(*

_{n}*X*) → ℱ

*(*

_{n}*X*). Let

*M*be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ

_{+}-transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal,

*I N, T T*

_{++}, semi-open and irreducible. In this paper we study the relationship between the following statements:

*ƒ*∈

*M*and ℱ

*(*

_{n}*ƒ*) ∈

*M*.

Infinite matroids have been defined by Reinhard Diestel and coauthors in such a way that this class is (together with the finite matroids) closed under dualization and taking minors. On the other hand, Andreas Dress introduced a theory of matroids with coefficients in a fuzzy ring which is – from a combinatorial point of view – less general, because within this theory every circuit has a finite intersection with every cocircuit. Within the present paper, we extend the theory of matroids with coefficients to more general classes of matroids, if the underlying fuzzy ring has certain properties to be specified.

In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.

This paper solves an enumerative problem which arises naturally in the context of Pascal’s hexagram. We prove that a general Desargues configuration in the plane is associated to *six* conical sextuples via the theorems of Pascal and Kirkman. Moreover, the Galois group associated to this problem is isomorphic to the symmetric group on six letters.

The purpose of this paper is to study the principal fibre bundle (*P*, *M*, *G*, *π*
_{
p
} ) with Lie group *G*, where M admits Lorentzian almost paracontact structure (*Ø*, *ξ*
_{
p
}, η_{
p
}, *g*) satisfying certain condtions on (1, 1) tensor field *J*, indeed possesses an almost product structure on the principal fibre bundle. In the later sections, we have defined trilinear frame bundle and have proved that the trilinear frame bundle is the principal bundle and have proved in Theorem 5.1 that the Jacobian map *π*
^{
*
} is the isomorphism.