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The main aim of this paper is to prove that the nonnegativity of the Riesz’s logarithmic kernels with respect to the Walsh– Kaczmarz system fails to hold.

Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group *V* is a *P*
_{0}(*V*) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order *p*
^{n} (*p* an odd prime and *n* ≥ 1 an integer ≤ 7).

Column-row products have zero determinant over any commutative ring. In this paper we discuss the converse. For domains, we show that this yields a characterization of pre-Schreier rings, and for rings with zero divisors we show that reduced pre-Schreier rings have this property.

Finally, for the rings of integers modulo *n*, we determine the 2x2 matrices which are (or not) full and their numbers.

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.