Let G be a finite group, K a field, and V a finite-dimensional K G-module. Write L(V) for the free Lie algebra on V; similarly, let M ( V) be the free metabelian Lie algebra. The action of G extends naturally to these algebras, so they become KG-modules, which are direct sums of finite-dimensional submodules. This paper explores whether indecomposable direct summands of such a KG-module (for some specific choices of G, K and V) must fall into finitely many isomorphism classes. Of course this is not a question unless there exist infinitely many isomorphism classes of indecomposable KG-modules (that is, K has positive characteristic p and the Sylow p-subgroups of G are non-cyclic) and dim V > 1.

The first two results show that the answer is positive for M(V) when K is finite and dim V = 2, but negative when G is the Klein four-group, the characteristic of K is 2, and V is the unique 3-dimnsional submodule of the regular module D. In the third result, G is again the Klein four-group, K is any field of charateristic 2 with more than 2 elements, V is any faithfull module of dimension 2, and B is the unique 3-dimensional quotient of D; the answer is positive for L(V) if and only if it is positive for each of L(B), L(D), and L(V ⊗ V).

In this paper, the variety of three-valued closure algebras, that is, closure algebras with the property that the open elements from a three-valued Heyting algebra, is investigated. Particularly, the structure of the finitely generated free objects in this variety is determined.

This paper presents a sharp boundary growth estimate for all positive superharmonic functions u in a smooth domain Ω in ℝ2 satisfying the nonlinear inequality where c>0, α∈ℝ and p>0, and δΩ(x) stands for the distance from a point x to the boundary of Ω. A result is applied to show the existence of nontangential limits of such superharmonic functions.

A module M is said to satisfy the condition (℘*) if M is a direct sum of a projective module and a quasi-continuous module. In an earlier paper, we described the structure of rings over which every (countably generated) right module satisfies (℘*), and it was shown that such a ring is right artinian. In this note some additional properties of these rings are obtained. Among other results, we show that a ring over which all right modules satisfy (℘*) is also left artinian, but the property (℘*) is not left-right symmetric.

In this paper we study the asymptotic behavior of cylindrical ends in compact foliated 3-manifolds and give a sufficient condition for these ends to spiral onto a toral leaf.

By a topological semiring we mean a Hausdorff space S together with two continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, we insist that for all x, y and z in S. Note that, in contrast to the purely algebraic situation [1,2], we do not postulate the existence of an additive identity which is a multiplicative zero.

In this paper the following notation will be used: for any group G, positive integer c and non-negative integer n, Gc is the cth term of the lower central series of G and δ nGc is the nth term of the derived series of Gc.

An algebra A is homogeneous if the automorphism group of A acts transitively on the one dimensional subspaces of A. Suppose A is a homogeneous algebra over an infinite field k. Let La denote left multiplication by any nonzero element a ∈ A. Several results are proved concerning the structure of A in terms of La. In particular, it is shown that A decomposes as the direct sum A = ker La Im La. These results are then successfully applied to the problem of classifying the infinite homogeneous algebras of small dimension.

We consider the interior and Dirichiet problems and problems with first order boundary conditions, for a second order homogeneous elliptic partial differential operator with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. From there we obtain a priori estimates and regularity results for boundary value problems in Sobolev spaces.

This paper gives a complete classification of essentially commutative C*-algebras whose essential spectrum is homeomorphic to S2n−1 by their characteristic numbers. Let 1, 2 be such two C*-algebras; then they are C*-isomorphic if and only if they have the same n-th characteristic number. Furthermore, let γn() = m then is C*-isomorphic to C*(Mzl, …, Mzn) if m = 0, is C*-isomorphic C*(Tz1, …, Tzn−1, Tznm) if m ≠ 0. Some examples are given to show applications of the classfication theorem. We finally remark that the proof of the theorem depends on a construction of a complete system of representatives of Ext(S2n−1).

A semilattice tree T with 0 is slim if there is a chain C with 0 so that the lattices θ (T) and θ(C) of semilattice congruences are isomorphic. This paper establishes elementary consequences of slimness and uses simple constructive techniques to show certain small trees slim. If T is the union of at most countably many branches, each of which has a maximum or a countable cotinal subset, then T is slim. For trees with enough maximals slimness is equivalent with not having any uncountable anti-chains. If a tree T has a countable cofinal subset then T is slim. Thus finitary trees are slim.

We study the variational problem for $N$-parallel curves on a Finsler surface by means of exterior differential systems using Griffiths’ method. We obtain the conditions when these curves are extremals of a length functional and write the explicit form of Euler–Lagrange equations for this type of variational problem.

We classify semigroups in the title according to whether they have a finite or an infinite number ofℒ-classes or ℛ-classes. For each case, we provide a concrete construction using Rees matrix semigroups and their translational hulls. An appropriate relatively free semigroup is used to complete the classification. All this is achieved by first treating the special case in which one of the generators is idempotent. We conclude by a discussion of a possible classification of 2-generator completely regular semigroups.

Let F(z) be a continuous complex-valued function defined on the closed upper half plane H whose generalized derivative ∂F(z) is unbounded. In this paper, we discuss the relationship between the increasing order of ]∂F(x + iy)] when y → 0 and that of λf(x, t) ](F(x + t) − 2F(x) + F(x − t))/t], (x, t ∈ R), when t → 0.

We prove the boundedness of Bergman-type operators on mixed norm spaces Lp·q (φ) for 0 < q < 1 and 0 < p ≤ ∞ of functions on the unit ball of ” with an application to Gleason's problem.

The aim of this note is to investigate the structure of general surjectivity problem for a continuous linear map between locally convex spaces. We shall do so by using the method introduced in Yamamuro (1980). Its basic notion is that of calibrations which has been introduced in Yamamuro (1975), studied in detail in Yamamuro (1979) and appliced to several problems in Yamamuro (1978) and Yamamuro (1979a).

We give an upper bound for the minimum s with the property that every sufficiently large integer can be represented as the sum of s positive kth powers of integers, each of which is represented as the sum of three positive cubes for the cases $2\leq k\leq 4.$

For a doubly commuting n-tuple of unbounded normal operators in a Hilbert space the joint spectral measure can be constructed and its closed support described as the joint spectrum of the given n-tuple. The same is here shown for larger, possibly uncountable, families of operators.

For an odd prime $p$, a $p$-transposition group is a group generated by a set of involutions such that the product of any two has order 2 or $p$. We first classify a family of $(G,2)$-geodesic transitive Cayley graphs ${\rm\Gamma}:=\text{Cay}(T,S)$ where $S$ is a set of involutions and $T:\text{Inn}(T)\leq G\leq T:\text{Aut}(T,S)$. In this case, $T$ is either an elementary abelian 2-group or a $p$-transposition group. Then under the further assumption that $G$ acts quasiprimitively on the vertex set of ${\rm\Gamma}$, we prove that: (1) if ${\rm\Gamma}$ is not $(G,2)$-arc transitive, then this quasiprimitive action is the holomorph affine type; (2) if $T$ is a $p$-transposition group and $S$ is a conjugacy class, then $p=3$ and ${\rm\Gamma}$ is $(G,2)$-arc transitive.

The most important open problem in monotone operator theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar’s constraint qualification holds. In this paper, we establish the maximal monotonicity of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}A+B$ provided that $A$ and $B$ are maximally monotone operators such that ${\rm star}({\rm dom}\ A)\cap {\rm int}\, {\rm dom}\, B\neq \varnothing $, and $A$ is of type (FPV). We show that when also ${\rm dom}\ A$ is convex, the sum operator $A+B$ is also of type (FPV). Our result generalizes and unifies several recent sum theorems.