A double triangle subspace lattice in a Hilbert space H is a 5-element set of subspaces of H, containing (0) and H, with each pair of non-trivial elements intersecting in (0) and spanning H. It is shown that if any pair of non-trivial elements has a closed vector sum the double triangle is both non-reflexive and non-transitive. A double triangle in H⊕H is an operator double triangle if each non-trivial elements is the graph of an operator acting on H. A sufficient condition is given for any operator double triangle to be non-reflexive.

Let G denote any locally compact abelian group with the dual group Γ. We construct a new kind of subalgebra L1(G) ⊗ΓS of L1(G) from given Banach ideal S of L1(G). We show that L1(G) ⊗гS is the larger amoung all strongly character invariant homogeneous Banach algebras in S. when S contains a strongly character invariant Segal algebra on G, it is show that L1(G) ⊗гS is also the largest among all strongly character invariant Segal algebras in S. We give applications to characterizations of two kinds of subalgebras of L1(G)-strongly character invariant Segal algebras on G and Banach ideal in L1(G) which contain a strongly character invariant Segal algebra on G.

We prove a conjecture of Lennox and Wiegold that a finitely generated soluble group, in which every infinite subset contains two elements generating a supersoluble group, is finite-by-supersoluble.

We have defined a mini-injective module and given some structures of self mini-injective rings and certain relationships between such rings and QF-rings in [8] and [9].

In this short note we shall study the modules dual to mini-injective modules, which we call maxi-quasiprojective modules. We shall give a characterization and some structures, in terms of the above modules, of those rings whose every injective modules has the lifting property of direct decompositions modulo the Jacobian radical (see [5], [6] and [7]). Furthermore, we shall show that the above rings are closely related to QF-rings (see [8] and [9]).

A maximal tolerance of a lattice L without infinite chains is either a congruence or a central relation. A finite lattice L is order-polynomially complete if and only if L is simple and has no central relation.

The main goal of this paper is to describe radical classes closed under essential extensions. It turns out that such classes are precisely the homomorphically closed semisimple classes, and hence a radical class is essentially closed if and only if it is subdirectly closed. Moreover, a class is closed under homomorphic images, direct sums and essential extensions if and only if it is an essentially closed radical class. Also radical classes are investigated which are closed under Dorroh essentially extensions only, such a radical class R consists of idempotent rings provided that R does not contain the ring of integers, meanwhile all the other radicals satisfy this requirement. A description of (hereditary and) Dorroh essentially closed radicals is given in Theorem 4.

Stampfli and Embry have shown that a point of the numerical range of an operator is extreme if and only if a set of vectors corresponding to it is linear. This is generalized here to show that a point of the closure of the numerical range is extreme if and only if a corresponding set of sequences forms a linear space. A more geometric alternative proof is given for a theorem of Das and Garske concerning weak convergence to zero at the unattained extreme points of the closure of the numerical range.The result is shown to hold also for lone extreme points of the numerical range which lie on line segments on its boundary. Further, a bound is obtained on the norm of the weak limit of the weakly convergent sequences corresponding to points on a line segment on the boundary of numerical range.

Several morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).

All inverse semigroups with idempotents dually well-ordered may be constructed inductively. The techniques involved are the constructions of ordinal sums, direct limits and Bruck-Reilly extensions.

If two functions of a real variable are integrable over two intervals, say of t, τ, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals of t and τ. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Çesàro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof.

This paper defines descriptive, Riemann, and constructive integrals equivalent to the approximately continuous integral of Burkill.

It is proved that an arbitrary pair of positive integers can be simultaneously represented by products of the values at integer points of certain rational functions. Linear recurrences in Z-modules and elliptic power sums are applied.

This paper is concerned with a characterization of the optimal order of convergence of Tikhonov regularization for first kind operator equations in terms of the “smoothness” of the data.

The structure and geometry of Banach spaces with the property that E(4) = Ê** + E⊥ ⊥ are investigated: such spaces are called quotient reflexive spaces here. For these spaces, if E is very smooth, Ê is also very smooth, and if E* is weakly locally uniformly rotund (WLUR), E(4) is smooth on a certain (relatively) norm dense subset of Ê**. Consequently, for quotient reflexive spaces, WLUR and very-WLUR are equivalent in E*.

We introduce the class of p-adic F-functions which contains both the p-adic E-function and p-adic G-functions, as well as other functions. In this paper we obtain lower bounds for polynomials in the values at algebraic points of a class of p-adic F-functions defined over the completion of the algebraic closure of a p-adic field.

For a cardinal m with ω ≤ m ≤ 2≤m is embeddable in it as a closed subspace. We derive this result from the corresponding statement for metric spaces.