For any nonnegative self-adjoint operators $A$ and $B$ in a separable Hilbert space, the Trotter-type formula $[;(e^{i2tA/n}+e^{i2tB/n})/2]^n$ is shown to converge strongly in the norm closure of $\dom(A^{1/2})\cap\dom(B^{1/2})$ for some subsequence and for almost every $t\in\mathbb{R}$. This result extends to the degenerate case, and to Kato-functions following the method of T. Kato (see ‘Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroup’, Topics in functional analysis (ed. M. Kac, Academic Press, New York, 1978) 185–195). Moreover, the restrictions on the convergence can be removed by considering functions other than the exponential.

It is shown that a hypercentral group that has all subgroups subnormal and every non-nilpotent subgroup of bounded defect is nilpotent. As a consequence, a hypercentral group of length at most ω in which every subgroup is subnormal is nilpotent.

Keedwell has shown that none of the groups of order less than 5 has a weak uniquely completable set. We prove that a weak uniquely completable set exists in a latin square based on a finite group if and only if the group is of order greater than 5.

It is shown that if Ω is an infinite, metrizable, compact space such that some finite order iterated derived set of Ω is empty, then $\dg C(\om)=\db C(\om)=2$.

Let F be a field of characteristic other than 2. Let F(2) denote the compositum over F of all quadratic extensions of F, let F(3) denote the compositum over F(2) of all quadratic extensions of F(2) that are Galois over F, and let F{3} denote the compositum over F(2) of all quadratic extensions of F(2). This paper shows that F(3) = F{3} if and only if F is a rigid field, and that F(3) = K(3) for some extension K of F if and only if F is Pythagorean and K = F(√−1). The proofs depend mainly on the behavior of quadratic forms over quadratic extensions, and the corresponding norm maps.

a proof is given to show that for an inner form of $\gl_n$ over a global field of zero characteristic, there exist only a finite number of automorphic representations with fixed local factor (up to equivalence) at almost every place. what is new in comparison to earlier work (see a. i. badulescu and p. broussous, ‘un théorème de finitude’, compositio math. 132 (2002) 177–190) is the case when the local factors are not fixed at the infinite places, as well as the statement of the result for the automorphic spectrum, rather than the cuspidal one.

In this paper, a geometric interpretation is provided of a new rational Landen transformation. The convergence of its iterates is also established.

It is a well-known problem of Von Neumann to discover whether the countable chain condition and weak distributivity of a complete Boolean algebra imply that it carries a strictly positive probability measure. It was shown recently by Balcar, Jech and Pazák, and by Veličković, that it is consistent with ZFC, modulo the consistency of a supercompact cardinal, that every ccc weakly distributive complete Boolean algebra carries a contiuous strictly positive submeasure – that is, it is a Maharam algebra. We use some ideas of Gitik and Shelah and implications from the inner model theory to show that some large cardinal assumptions are necessary for this result.

The aim of this paper is to prove that for certain generalized Weyl algebras A (including the first Weyl algebra A1 over a field of characteristic zero) and for every simple left (right) A-module M, there are infinitely many non-isomorphic simple left (right) A-modules {Ni} such that Ext1A (M, Ni) ≠ 0 (respectively Ext1A(Ni, M) ≠ 0.

We give a formula for the Betti numbers of 3-Sasakian manifolds or orbifolds which can be obtained as 3-Sasakian quotients of a sphere by a torus. This answers a question of Galicki and Salamon about the topology of 3-Sasakian manifolds.

This paper provides a method of generating infinitely many integer solutions of the simultaneous equations $x_1^r+x_2^r+x_3 ^r=y_1^r+y_2^r+y_3^r$ where $r$ is equal to $2$, $3$ or $4$. Three numerical solutions of this system of equations have also been given in this paper.

Let G be a residually-finite virtually soluble minimax group. We give upper and lower linear bounds for the degree of subgroup growth of G in terms of the Hirsch length of G.

Let $f \in \Z[x]$ be a polynomial of degree d. The paucity of non-trivial positive integer solutions to the equation $f(x_1)+f(x_2)=f(x_{3})+f(x_{4})$ is established, provided that $d \geq 7$. Also the corresponding situation is investigated for equal sums of three like polynomials.

A group is said to be (2,3)-generated if it can be generated by an element of order 2 and an element of order 3. The class of (2,3)-generated groups seems to be rather extensive. From results of Schupp [7] and Mason and Pride [3], it is known that there are 2ℵ0 isomorphism classes of (2,3)-generated groups, and moreover that every countable group can be embedded in a (2,3)-generated group. All finite non-abelian simple groups are (2,3)-generated, with the exception of some groups of low rank in characteristics 2 and 3 (see Wilson [13] or Sanchini and Tamburini [6]).

We deal with weighted inequalities of the type

formula here

where:

T is either the Hardy–Littlewood maximal operator or a singular integral operator; G is any measurable subset of ℝn; ƒ is any measurable function, vanishing outside G, such that Tƒ is well-defined; v and w are weights, that is, nonnegative locally integrable functions on G;

p, q ∈(1, ∞).

In the paper, three lower bounds are given for the Morse index of a constant mean curvature torus in Euclidean $3$ -space, in terms of its spectral genus $g$ . The first two lower bounds grow linearly in $g$ and are stronger for smaller values of $g$ , while the third grows quadratically in $g$ but is weaker for smaller values of $g$ .

We define a Henstock-type integral on measure spaces with metric topologies, and give an example of a function which is integrable but whose absolute value is not.

If p is a prime, then a finite group is p-soluble if each of its composition factors is either a p-group or has order coprime to p. For example, soluble groups are p-soluble. However, there are many insoluble groups that are p-soluble. We shall prove the following result.