For a pythagorean field F with semiordering Q and associated preordering T, it is shown that the Witt ring WT (F) is isomorphic to the Witt ring W (K) whre K is a closure of F with respect to Q. For an arbitrary preordering T, it is shown how the covering number of T relates to the construction of WT (F).

We use a generalised Nevanlinna counting function to compute the Hilbert–Schmidt norm of a composition operator on the Bergman space $L_{a}^{2}(\mathbb{D})$ and weighted Bergman spaces $L_{a}^{1}(\text{d}A_{\unicode[STIX]{x1D6FC}})$ when $\unicode[STIX]{x1D6FC}$ is a nonnegative integer.

It is shown that if с is a finitely productive and hereditary class of uniformly pointed finite discrete universal algebras then any injective in с is an injective in Proс.

Given a closed set $C$ in a Banach space $(X,\Vert \cdot \Vert )$, a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_{C}(x)=\Vert x-z\Vert$, where $d_{C}$ is the distance of $x$ from $C$. We survey the problem of studying the size of the set of points in $X$ which have nearest points in $C$. We then turn to the topic of delta convex functions and indicate how it is related to finding nearest points.

Cordially dedicated to Professor Alexander Weinstein, on the occasion of his Seventyseventh birthday, January 21, 1974.

This note contains the proof of an extension of Alexander Weinstein's mean value theorem for Generalized Axially Symmetric Potential Theory.

Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i\leq ~k\} - \mathrm{ht} (I)$, for all $n\geq 1$. In addition we show that if $I$ is MHC and $w$ is an indeterminate which is not in the monomials generating $I$, then $\mathrm{reg} (S/ \mathop{(I+ {w}^{d} S)}\nolimits ^{n} )\leq \mathrm{reg} (S/ I)+ nd- 1$ for all $n\geq 1$ and $d$ large enough. Finally, we implement an algorithm for the computation of $m(I)$.

Any subset π* of the set of all planes through a line in a finite projective space PG(m, q) determines a subgeometry G(π*) of the combinatorial geometry associated with PG(m, q). In this paper the geometries G(π*) of rank greater than three in which every line contains at least four points, are characterized in terms of the existence of a certain set of automorphism groups Γ(C, X); where X is a copoint and C a point not in X, and each non-trivial element of Γ(C, X) fixes X and every copoint through C and fixes C and every point in X, but no other point; and where Γ(C, X) acts transitively on the points distinct from C and not in X of some line through C. As a corollary of the main theorem we obtain a characterization of the finite projective spaces PG(m, q) with m ≥ 3 and q ≥ 3.

Let G be a semiabelian variety defined over an algebraically closed field K of prime characteristic. We describe the intersection of a subvariety X of G with a finitely generated subgroup of $G(K)$.

The notion of relative normal-convexity is introduced and its connection to equations over groups is studied. We study the effect on relative normal-convexity by free products, amalgamations, and HNN extensions.

Let (Ω,ℱ,ℙ) be a probability space and Z=(ZK)k∈ℕ a Bernoulli noise on (Ω,ℱ,ℙ) which has the chaotic representation property. In this paper, we investigate a special family of functionals of Z, which we call the coherent states. First, with the help of Z, we construct a mapping ϕ from l2(ℕ) to ℒ2(Ω,ℱ,ℙ) which is called the coherent mapping. We prove that ϕ has the continuity property and other properties of operation. We then define functionals of the form ϕ(f) with f∈l2 (ℕ) as the coherent states and prove that all the coherent states are total in ℒ2 (Ω,ℱ,ℙ) . We also show that ϕ can be used to factorize ℒ2 (Ω,ℱ,ℙ) . Finally we give an application of the coherent states to calculus of quantum Bernoulli noise.

It is veil-known that every ring automorphism of the ring of all linear transformations of a real vector space into itself is inner. We shall show that, if this ring is regarded as a semigroup with respect to composition and the dimension of the vector space is not less than 2, every semigroup automorphism is inner.

Let Bn denote the unit ball in ℂn, n≥1. Given an α>0, let ℱα(n) denote the class of functions defined for z∈Bn by integrating the kernel (1−〈z,w〉)−α against a complex Borel measure dμ(w), w∈Bn. The family ℱ0(n) corresponds to the logarithmic kernel log (1/(1−〈z,w〉)). Various properties of the spaces ℱα(n), α≥0, are obtained. In particular, pointwise multiplies for ℱα(n) are investigated.

We prove the existence of bounded solution of the differential equation y′ = A(t)y + f(t, y) in a Banach space. The method used here is based on the concept of “admissibility” due to Massera and Schäffer when f satisfies the Caratheodory conditions and some regularity condition expressed in terms of the measure of noncompactness α.

The local connectivity, νk(G), of a graph G is the minimum of the connectivities of neighbourhoods of the vertices of G. G is minimally locally n-connected if νk(G) = n and for every edge x of G, νk(G−x) = n − 1. A necessary and sufficient condition for a locally connected graph to be minimally locally 1-connected is given, and it is shown that for n ≥ 7, is minimally locally 1-connected.