We exhibit a large class K* of real 2 × 2 matrices of determinant ±1 such that, for nearly all A and B in K*, the group generated by A and B1 (the transpose of B) is the free product of the cyclic groups It is shown that K* contains all matrices of determinant ±1 with integer entries satisfying |b| > |a|, |c|, |d|. This gives a generalization of a theorem of Goldberg and Newman [2]. We also prove related results concerning the dominance of b and the discreteness of the free products .

§ 1. Introductory. The formula to be proved is

where n is zero or a positive integer, R(m)>0, R(αs-l)>0, s = 1, 2, …, p, p≧q.

A function φ(p) is operationally related to h(t) when they satisfy the integral equation

provided that the integral is convergent and R(p)> 0.

As usual, we shall denote (1) by the symbolic expression

φ(p) ≑ h(t).

The object of this paper is to evaluate some infinite integrals involving E-functions by the methods of the operational calculus. Most of the results obtained are believed to be new.

A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m: Σ → Χ is a countably additive vector measure, then the range of m has not only the Banach–Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p > 1, the p-Banach–Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p < ∞, no weakly-p-summable sub-sequences.

An element a in a ring R is left annihilator-stable (or left AS) if, whenever $Ra+{\rm l}(b)=R$ with $b\in R$ , $a-u\in {\rm l}(b)$ for a unit u in R, and the ring R is a left AS ring if each of its elements is left AS. In this paper, we show that the left AS elements in a ring form a multiplicatively closed set, giving an affirmative answer to a question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.]. This result is used to obtain a necessary and sufficient condition for a formal triangular matrix ring to be left AS. As an application, we provide examples of left AS rings R over which the triangular matrix rings ${\mathbb T}_n(R)$ are not left AS for all $n\ge 2$ . These examples give a negative answer to another question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.] whether R/J(R) being left AS implies that R is left AS.

For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗minA is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.

We show that the order of an automorphism of an arithmetic Riemann surface of genus $g$ is not greater than $2g\,{-}\,2$, provided $g$ is large enough. This bound is an arithmetic analog of the classical Wiman bound. We prove that it is sharp and attained for any genus but in contrast to the general case the automorphisms of maximal order act without fixed points. This allows us to consider the automorphisms which act on arithmetic Riemann surfaces and have a given number of fixed points. For these automorphisms we describe the asymptotic behavior of their orders.

We present a ready to compute trace formula for Hecke operators on vector-valued modular forms of integral weight for SL2(ℤ) transforming under the Weil representation. As a corollary, we obtain a ready to compute dimension formula for the corresponding space of vector-valued cusp forms, which is more general than the dimension formulae previously published in the vector-valued setting.

For the Kronecker algebra, Zwara found in [14] an example of a module whose orbit closure is neither unibranch nor Cohen-Macaulay. In this paper, we explain how to extend this example to all representation-infinite algebras with a preprojective component.

We prove some inequalities for the spectral radius of positive operators on Banach function spaces. In particular, we prove the following extension of Levinger's theorem. Let $K$ be a positive compact kernel operator on $L^2(X, \mu)$ with the spectral radius $r(K)$. Then the function $\phi$ defined by $\phi(t) = r(t K + (1-t) K^*)$ is non-decreasing on $[0, \frac{1}{2}]$. We also prove that $\| A + B^* \| \ge 2 \cdot \sqrt{r(A B)}$ for any positive operators $A$ and $B$ on $L^2(X, \mu)$.

If A(z) belongs to the Bergman space , then the differential equation f″+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ .

We compute the precise bounds, for every positive integer n, for the nilpotency class of nilpotent subgroups of {\text GL}(n,{\tf="P7DC6"Q}) and {\text GL}(n,{\tf="P7DC6"Z}).

Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in manifold stratified spaces, and to spaces of manifold stratified approximate fibrations.

Let $p$ be a prime number, $\Q_p$ the field of $p$-adic numbers, $K$ a finite field extension of $\Q_p$, $\skew4\bar K$ a fixed algebraic closure of $K$, and $\C_p$ the completion of $\skew4\bar K$ with respect to the $p$-adic valuation. We discuss some properties of Lipschitzian elements, which are elements $T$ of $\C_p$ defined by a certain metric condition that allows one to integrate Lipschitzian functions along the Galois orbit of $T$ over $K$ with respect to the Haar distribution.

We calculate the moments of the characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, at a position t in the bulk of the spectrum, as a series expansion in powers of t. We focus in particular on the Gaussian Unitary Ensemble. We employ a novel approach to calculate the coefficients in this series expansion of the moments, appropriately scaled. These coefficients are polynomials in N. They therefore grow as $N\to\infty$ , meaning that in this limit the radius of convergence of the series expansion tends to zero. This is related to oscillations as t varies that are increasingly rapid as N grows. We show that the $N\to\infty$ asymptotics of the moments can be derived from this expansion when $t=0$ . When $t\ne 0$ we observe a surprising cancellation when the expansion coefficients for N and $N+1$ are formally averaged: this procedure removes all of the N-dependent terms leading to values that coincide with those expected on the basis of previously established asymptotic formulae for the moments. We obtain as well formulae for the expectation values of products of the secular coefficients.

We introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the exponential map.

We characterize norm closed subspaces $B$ of $\linf (\partial D)$ such that $C(\partial D) B \subset B$ and maximal ones in the family of proper closed subspaces $B$ of $L^\infty(\partial D)$ such that $A(D) B \subset B$, where $A(D)$ is the disk algebra. Analogously, we characterize closed subspaces of $H^\infty$ that are simultaneously invariant under $S$ and $S^\ast$, the forward and the backward shift operators, and maximal invariant subspaces of $H^\infty$.

We consider a class of operators that contains the strictly singular operators and it is contained in the perturbation class of the upper semi-Fredholm operators PΦ+. We show that this class is strictly contained in PΦ+, solving a question of Friedman. We obtain similar results for the strictly cosingular operators and the perturbation class of the lower semi-Fredholm operators PΦ−. We also characterize in terms of PΦ+ and in terms of PΦ−. As a consequence, we show that and are the biggest operator ideals contained in PΦ+ and PΦ−, respectively.

We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.

A maximal symmetry group is a group of isomorphisms of a three-dimensional hyperbolic manifold of maximal order in relation to the volume of the manifold. In this paper we determine all maximal symmetry groups of the types PSL(2, q) and PGL(2, q). Depending on the prime p there are one or two such groups with q=pk and k always equals 1, 2 or 4.