It's well known (see Endo [1]) that for a commutative ring A, if A is semihereditary then w.gl. dim. A ≤ 1. It seems worth recording the noncommutative version of this.

We study boundedness properties of the $q$-mean-square operator {{S}^{(q)}} on $E$-valued analytic martingales, where $E$ is a complex quasi-Banach space and $2\,\le \,q\,<\,\infty $. We establish that a.s. finiteness of ${{S}^{(q)}}$ for every bounded $E$-valued analytic martingale implies strong $(p,\,p)$-type estimates for ${{S}^{(q)}}$ and all $p\,\in \,(0,\,\infty )$. Our results yield new characterizations (in terms of analytic and stochastic properties of the function ${{S}^{(q)}}$) of the complex spaces $E$ that admit an equivalent $q$-uniformly $\text{PL}$-convex quasi-norm. We also obtain a vector-valued extension (and a characterization) of part of an observation due to Bourgain and Davis concerning the ${{L}^{p}}$-boundedness of the usual square-function on scalar-valued analytic martingales.

In this note we give a very elementary proof of a result of Meyer's [1]. Let r(A) be the rank of the matrix A.

A sequence (xi) in a Banach space X is a Schauder basis for X provided for each x∊X there is a unique sequence of scalars (ai) such that

1.1

convergence in the norm topology. It is well known [1] that if (xi) is a (Schauder) basis for X and (fi) is defined by

1.2

where then fi(xj) = δij and fi∊X* for each positive integer i.

A sequence (xi) is a éasic sequence in X if (xi) is a basis for [xi], where the bracketed expression denotes the closed linear span of (xi).

We consider the problem: If K is a compact normal operator on a Hilbert module E, and f ∈ C0(SpK) is a function which is zero in a neighbourhood of the origin, is f(K) of finite rank? We show that this is the case if the underlying C*-algebra is abelian, and that the range of f(K) is contained in a finitely generated projective submodule of E.

Let $r=[a_1(r), a_2(r),\ldots ]$ be the continued fraction expansion of a real number $r\in \mathbb R$ . The growth properties of the products of consecutive partial quotients are tied up with the set admitting improvements to Dirichlet’s theorem. Let $(t_1, \ldots , t_m)\in \mathbb R_+^m$ , and let $\Psi :\mathbb {N}\rightarrow (1,\infty )$ be a function such that $\Psi (n)\to \infty $ as $n\to \infty $ . We calculate the Hausdorff dimension of the set of all $ (x, y)\in [0,1)^2$ such that

$$ \begin{align*} \max\left\{\prod_{i=1}^ma_{n+i}^{t_i}(x), \prod_{i=1}^ma_{n+i}^{t_i}(y)\right\} \geq \Psi(n) \end{align*} $$

$n\geq 1$

We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.

In earlier papers [3] and [4] some Vandermonde type convolution identities with several summations were established which were the extensions of Gould's work that involved one summation. Furthermore, these identities were utilized to obtain some inverse series relations for functions of several variables. As a continuation, this paper deals with some further generalizations of Gould's work in [1] and [2].

In [1] Lambek calls the injective R-module I nice if every torsionfree factor module of the ring of quotients Q of R with respect to lis divisible. If lis nice then g is a dense subring of the bicommutator BicRI of I with respect to the finite topology (see [1, Proposition 2]). We now give an example of an injective R-module over an Artinian ring R which is not nice. Since R is Artinian, Q=BicRI, by Proposition B of [1].

Before we give the example, we state the following, which depends on [2] for terminology.

Let p be a rational prime and n a positive integer ≥2. We denote by an(p) the least positive integral value of a for which the polynomial xn+x+a is irreducible (mod p), and set

(1)

Let N be an ideal of a ring A. We say that idempotents modulo N can be lifted provided that for every a of A such that a2-a ∈ N there exists an element e2=e ∈ A such that e-a ∈ N. The technique of lifting idempotents is considered to be a fundamental tool in the classical theory of nonsemiprimitive Artinian rings (refer [2; p. 72]).

Previous results by the author on the connection between three measures of noncompactness obtained for ${{\mathcal{L}}_{p}}$ are extended to regular spaces of measurable functions. An example is given of the advantages of some cases in comparison with others. Geometric characteristics of regular spaces are determined. New theorems for $\left( k,\,\beta \right)$-boundedness of partially additive operators are proved.

Let $X$ be a smooth projective curve of arbitrary genus $g\,>\,3$ over the complex numbers. In this short note we will show that the moduli space of rank $2$ stable vector bundles with determinant isomorphic to ${{L}_{x}}$ , where ${{L}_{x}}$ denotes the line bundle corresponding to a point $x\,\in \,X$, is isomorphic to a certain variety of lines in the moduli space of $S$-equivalence classes of semistable bundles of rank $2$ with trivial determinant.

We show that every element in the Brauer-Wall group of a field with characteristic different from 2 is represented uniquely by a division graded algebra, (i.e. homogeneous elements are invertible) but, of course, not necessarily by a graded (division algebra). This is a fairly direct consequence of Wall's structure theory for central simple Z/2-graded algebras.

We build on the recent techniques of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894), which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of $\mathrm {K}$ -semistable Fano varieties. Here, we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of the Harder and Narasimhan polygons. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894). In doing so, we expand upon the slope stability theory, for filtered vector spaces, that was initiated by Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138). One source of inspiration for our abstract study of Harder and Narasimhan data, which is a concept that we define here, is the lattice reduction methods of Grayson (1984, Commentarii Mathematici Helvetic 59, 600–634). Another is the work of Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138), and Evertse and Ferretti (2013, Annals of Mathematics 177, 513–590), which is within the context of Diophantine approximation for projective varieties.

In this work, we study the existence of solutions of nonlinear fractional coupled system of $\varphi $-Hilfer type in the frame of Banach spaces. We improve a property of a measure of noncompactness in a suitably selected Banach space. Darbo’s fixed point theorem is applied to obtain a new existence result. Finally, the validity of our result is illustrated through an example.

Free binary systems are shown not to admit idempotent means. This refutes a conjecture of the author. It is also shown that the extension of Hindman’s theorem to nonassociative binary systems formulated and conjectured by the author is false.

In this paper we study some aspects of the behaviour of p-lattice summing operators. We prove first that an operator T from a Banach space E to a Banach lattice X is p-lattice summing if and only if its bitranspose is. Using this theorem we prove a characterization for 1 -lattice summing operators defined on a C(K) space by means of the representing measure, which shows that in this case 1 -lattice and ∞-lattice summing operators coincide. We present also some results for the case 1 ≤ p < ∞ on C(K,E).