§1. A series

has been defined by J. M. Whittaker to be absolutely summable (A), if

is convergent in (0 ≤ x < 1) and f (x) is of bounded variation in (0, 1), i.e.

for all subdivisions 0 = x0 < x1 < x2 < . … < xm < 1.

Characterization and uniqueness results are given for uniform approximation which extend to suns previously known results for convex sets.

This paper deals with a reaction–diffusion model with inner absorptions and coupled nonlinear boundary conditions of exponential type. The critical exponents are described via a pair of parameters that satisfy a certain matrix equation containing all the six nonlinear exponents of the system. Whether the solutions blow up or not is determined by the signs of the two parameters. A more precise analysis, depending on the geometry of $\varOmega$ and the absorption coefficients, is proposed for the critical sign of the parameters.

AMS 2000 Mathematics subject classification: Primary 35K55; 35B33

In this paper we extend results due to Vogt on line bundles over Cousin groups to the case of domains stable by the maximal compact subgroup. This is used to show that the algebraic dimension of Oeljeklaus—Toma manifolds (OT-manifolds) is 0. In the last part we establish that certain Cousin groups, in particular those arising from the construction of OT-manifolds, have finite-dimensional irregularity.

Let $S$ and $T$ be symmetric unbounded operators. Denote by $\overline{S+T}$ the closure of the symmetric operator $S+T$. In general, the deficiency indices of $\overline{S+T}$ are not determined by the deficiency indices of $S$ and $T$. The paper studies some sufficient conditions for the stability of the deficiency indices of a symmetric operator $S$ under self-adjoint perturbations $T$. One can associate with $S$ the largest closed $^*$-derivation $\delta_{S}$ implemented by $S$. We prove that if the unitary operators $\exp(\ri tT)$, for $t\in\mathbb{R}$, belong to the domain of $\delta_{S}$ and $\delta_{S}(\exp(\ri tT))\rightarrow0$ in the strong operator topology as $t\rightarrow0$, then the deficiency indices of $S$ and $\overline{S+T}$ coincide. In particular, this holds if $S$ and $\exp(\ri tT)$ commute or satisfy the infinitesimal Weyl relation.

We also study the case when $S$ and $T$ anticommute: $\exp(-\ri tT)S\subseteq S\exp(\ri tT)$, for $t\in\mathbb{R}$. We show that if the deficiency indices of $S$ are equal, or if the group $\{\exp(\ri tT):t\in\mathbb{R}\}$ of unitary operators has no stationary points in the deficiency space of $S$, then $S$ has a self-adjoint extension which anticommutes with $T$, the operator $S+T$ is closed and the deficiency indices of $S$ and $S+T$ coincide.

AMS 2000 Mathematics subject classification: Primary 47B25

In this paper quasilinear elliptic boundary value equations without an Ambrosetti and Rabinowitz growth condition are considered. Existence of a non-trivial solution result is established. For this, we show the existence of a Cerami sequence by using a variant of the mountain-pass theorem due to Schechter. The novelty here is that we may consider nonlinearities that satisfy a local p-superlinear condition and may change sign.

Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

We study two different heat-type equations. First, global well-posedness in the energy space of some high-order semilinear heat-type equation with exponential nonlinearity is obtained for even space dimensions. Second, a finite-time blow-up result for the critical monomial focusing heat equation with the p-Laplacian is proved.

In this paper, we prove that if is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that then the Fréchet space C(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence satisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.

We consider Sturm–Liouville boundary-value problems on the interval $[0,1]$ of the form $-y''+qy=\lambda y$ with boundary conditions $y'(0)\sin\alpha=y(0)\cos\alpha$ and $y'(1)=(a\lambda+b)y(1)$, where $a\lt0$. We show that via multiple Crum–Darboux transformations, this boundary-value problem can be transformed ‘almost’ isospectrally to a boundary-value problem of the same form, but with the boundary condition at $x=1$ replaced by $y'(1)\sin\beta=y(1)\cos\beta$, for some $\beta$.

AMS 2000 Mathematics subject classification: Primary 34B07; 47E05; 34L05

We study conical density properties of general Borel measures on Euclidean spaces. Our results are analogous to the previously known result on the upper density properties of Hausdorff and packing-type measures.

Clay (3), Johnson (5) and Krimmel (6) have each considered the near-rings with identity on dihedral groups. Krimmel actually generalised the class of dihedral groups and investigated the class of finite non-abelian groups with a cyclic normal subgroup of prime index: we shall call this class . Krimmel considered the near-rings with identity that might be denned on members of and he determined the subclass of groups in which support near-rings of this kind. He also managed to calculate the number of non-isomorphic near-rings involved for certain cases. His methods were essentially combinatorial, and his results were expressed in terms of various integers which characterised the individual members of . Certain features of this work led us to investigate the structure of near-rings on members of from a more algebraic point of view and thereby to complete and extend Krimmel's programme. Part of the work in this paper formed the basis of the second author's thesis (7). We should like to thank Dr. J. Krimmel for permission to include some of his results, and Dr. J. Meldrum who detected an error in our original formulation of Theorem 7.1.

This paper contains some basic relations between Ganea strong category and Lusternik Schnirelmann category in proper homotopy theory. We focus our interest on the case of category 2 in order to show that ℚn is the unique open n-manifold with proper Lusternik-Schnirelmann category 2 (n ≠ 3).

A subalgebra U of a Lie algebra L over a field F is called modular* in L if U satisfies the dual of the modular identities in the lattice of subalgebras of L. Our aim is the study of the influence of the modular* identities in the structure of the algebra. First we prove that if the modular* conditions are imposed on an ideal of L then every element of L acts as an scalar on this ideal and if they are imposed on a non-ideal subalgebra U of L then the largest ideal of L contained in U also satisfies the modular* identities. We determine Lie algebras having a subalgebra which satisfies both the modular and modular* identities, provided that either L is solvable or char(F)≠ 2,3. As immediate consequences of this result we obtain that the existence of a co-atom satisfying the modular* identities in the lattice L(L) forces that the lattice L(L) is modular and that the modular* identities on any subalgebra U forces that U is quasi-abelian. In the case when L is supersolvable we obtain that the modular* conditions on any non-ideal of L are enough to guarantee that L(L) is modular. For arbitrary fields and any Lie algebra L, we prove that the modular* conditions on every co-atom of the lattice L(L) guarantee that L(L) is modular.

Equations which may be regarded as extensions of the dual series equations discussed by Noble (1) and the present author (2) are the triple series equations of the first kind

and the triple series equations of the second kind

where f, f1, g, g1h and h1 are all known functions,

is the Jacobi polynomial (3).

The following theorems in ring theory are well-known:

1. Let R be a ring. If e is a unique left identity, then e is also a right identity.

2. If R is a ring with more than one element such that aR = R for every nonzero element a ε R, then R is a division ring.

3. A ring R with identity e ≠ 0 is a division ring if and only if it has no proper right ideals.

A correspondence is established between a class of coverings of an inverse semigroup S and a class of embeddings of S, generalising results of McAlister and Reilly on E-unitary covers of inverse semigroups.

The present paper is mainly concerned with decomposition theorems of the Jordan, Yosida-Hewitt, and Lebesgue type for vector measures of bounded variation in a Banach lattice having property (P). The central result is the Jordan decomposition theorem due to which these vector measures may alternately be regarded as order bounded vector measures in an order complete Riesz space or as vector measures of bounded variation in a Banach space. For both classes of vector measures, properties like countable additivity, purely finite additivity, absolute continuity, and singularity can be defined in a natural way and lead to decomposition theorems of the Yosida-Hewitt and Lebesgue type. In the Banach lattice case, these lattice theoretical and topological decomposition theorems can be compared and combined.