A module of a ring of differential operators $\mathcal{D}$ over a smooth surface has order $1$ if it is isomorphic to a factor module of $\mathcal{D}$ by a cyclic ideal generated by an operator of order $1$. Let $k$ be a positive integer. We give conditions under which an indecomposable $\mathcal{D}$-module of order $1$ is GK-critical of length $k$. We also give examples of indecomposable, non-critical, $\mathcal{D}$-modules whose subfactors have order $1$.

AMS 2000 Mathematics subject classification: Primary 16S32. Secondary 37F75

We consider symplectic difference systems, which contain as special cases linear Hamiltonian difference systems and Sturm–Liouville difference equations of any even order. An associated discrete quadratic functional is important in discrete variational analysis, and while its positive definiteness has been characterized and is well understood, a characterization of its positive semidefiniteness remained an open problem. In this paper we present the solution to this problem and offer necessary and sufficient conditions for such discrete quadratic functionals to be non-negative definite.

AMS 2000 Mathematics subject classification: Primary 39A12; 39A13. Secondary 34B24; 49K99

We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz–Krieger algebras. Wedescribe a variant of the Cuntz–Krieger relations which applies to graphs with sources, and describe a local convexitycondition which characterizes the higher-rank graphs that admit a non-trivial Cuntz–Krieger family. We then proveversions of the uniqueness theorems and classifications of ideals for the $C^*$-algebras generated by Cuntz–Krieger families.

AMS 2000 Mathematics subject classification: Primary 46L05

We establish two characterizations of local Laplace transforms in Banach spaces. The first result follows the classic approach of Widder, while the second is in terms of vector-valued moment sequences. As a consequence, we derive characterizations of nilpotent semigroups.

AMS 2000 Mathematics subject classification: Primary 44A10; 47D03

Given an embedded hypersurface $M$ in $\mathbb{R}^4$ we consider families of projections $H$ of $M$ to lines and families of projections $P$ of $M$ to 3-spaces. We characterize generically the singularities of these projections. We also show that there is a duality relation between some strata of the bifurcation sets of $H$ and $P$, and deduce geometric properties about these sets.

AMS 2000 Mathematics subject classification: Primary 58K05. Secondary 58K40

The best-constant problem for Nash and Sobolev inequalities on Riemannian manifolds has been intensively studied in the last few decades, especially in the compact case. We treat this problem here for a more general family of Gagliardo–Nirenberg inequalities including the Nash inequality and the limiting case of a particular logarithmic Sobolev inequality. From the latter, we deduce a sharp heat-kernel upper bound.

AMS 2000 Mathematics subject classification: Primary 58J05

Let $\varOmega$ be a convex, open subset of $\mathbb{R}^n$ and let $\mathcal{D}'(\varOmega)$ be the space of distributions on $\varOmega$. It is shown that there exist linear embeddings of $\mathcal{D}'(\varOmega)$ into a differential algebra that commute with partial derivatives and that embed $\mathcal{C}^{\infty}(\varOmega)$ as a subalgebra. This embedding appears to be the first one after Colombeau’s to possess these properties. We show that many nonlinear operations on distributions can be defined that are not definable in the Colombeau setting.

AMS 2000 Mathematics subject classification: Primary 46F30. Secondary 13C11

We prove that the separating space of an epimorphism from a Lie–Banach algebra onto the (continuous) derivation algebra $\mathfrak{Der}(A)$ of a Banach algebra $A$ consists of derivations which map into the radical of $A$.

AMS 2000 Mathematics subject classification: Primary 46H40; 46H70. Secondary 47B47; 47B48

The notion of $B$-convexity for operator spaces, which a priori depends on a set of parameters indexed by $\sSi$, is defined. Some of the classical characterizations of this geometric notion for Banach spaces are studied in this new context. For instance, an operator space is $B_{\sSi}$-convex if and only if it has $\sSi$-subtype. The class of uniformly non-$\mathcal{L}^1(\sSi)$ operator spaces, which is also the class of $B_{\sSi}$-convex operator spaces, is introduced. Moreover, an operator space having non-trivial $\sSi$-type is $B_{\sSi}$-convex. However, the converse is false. The row and column operator spaces are nice counterexamples of this fact, since both are Hilbertian. In particular, this result shows that a version of the Maurey–Pisier Theorem does not hold in our context. Some other examples of Hilbertian operator spaces will be considered. In the last part of this paper, the independence of $B_{\sSi}$-convexity with respect to $\sSi$ is studied. This provides some interesting problems, which will be posed.

AMS 2000 Mathematics subject classification: Primary 46L07. Secondary 42C15

In this paper finite groups factorized as products of pairwise totally permutable subgroups are studied in theframework of Fitting classes.

AMS 2000 Mathematics subject classification: Primary 20D10; 20D40

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

We deal with a class of $p$-Laplacian Dirichlet boundary-value problems where the combined effects of ‘sublinear’ and‘superlinear’ growths allow us to establish the existence of at least two positive solutions.

AMS 2000 Mathematics subject classification: Primary 35J25; 35J60. Secondary 35J65; 35J70

In this paper the authors establish the $L^p$ boundedness for several classes of Marcinkiewicz integral operators with kernels satisfying a condition introduced by Grafakos and Stefanov in Indiana Univ. Math. J.47 (1998), 455–469.

AMS 2000 Mathematics subject classification: Primary 42B25; 42B99

Montel introduced the concept of quasi-normal families $f:\varOmega\to\mathbb{C}$ in 1922: $\mathcal{F}$ is quasi-normal of order $N$ if every sequence $\{f_n\}$ from $\mathcal{F}$ has a subsequence which converges uniformly on compact subsets of $\varOmega\setminus Z^\dagger$, where $Z^\dagger\subset\varOmega$ contains at most $N\in\mathbb{N}$ elements. ($\mathcal{F}$ is of order $N:=\infty$ if every such exceptional set $Z^\dagger$ is finite.) The problem is that $Z^\dagger$ normally depends on the subsequence. So even if every sequence has a subsequence which converges to a given function $f$ in $\varOmega$ except at $N$ points, the sequence itself may not converge in any domain $D\subseteq\varOmega$.

In this paper we introduce the concept of general convergence. Indeed, $\{f_n\}$ above converges generally to $f$. We also introduce a related concept, restrained sequences, and study some of their properties. The definitions extend earlier concepts introduced for sequences of linear fractional transformations.

AMS 2000 Mathematics subject classification: Primary 20H10; 30B70; 30D45; 30F35; 58F08

The characterization theorem for the Banach-space-valued local Laplace transform established by Keyantuo, Müller and Vieten is used to obtain a real variable characterization of generators of local convoluted semigroups. The concept of local convoluted semigroups extends that of distribution as well as ultradistribution semigroups. Complete characterizations existed only for exponentially bounded semigroups integrated $\alpha$ times, whereas for the non-exponential case generation results had been obtained in terms of complex conditions only.

AMS 2000 Mathematics subject classification: Primary 47D03; 47D06; 44A10

It is well known that every non-reflexive $M$-ideal is weakly compactly generated (in short, WCG). We present a family of Banach spaces $\{V_{s}:0 \lt s \lt 1\}$ which are not WCG and such that every $V_{s}$ satisfies the inequality

$$ \|\f\|\geq\|\pi\f\|+s\|\f-\pi\f\|\quad\forall\f\in V_{s}^{\ast\ast\ast}, $$

where $\pi$ is the canonical projection from $V_{s}^{\ast\ast\ast}$ onto $V_{s}^{\ast}$. In particular, no $V_{s}$ can be renormed to be an $M$-ideal.

AMS 2000 Mathematics subject classification: Primary 46B20

Let $q$ be a positive integer, let $\mathcal{I}=\mathcal{I}(q)$ and $\mathcal{J}=\mathcal{J}(q)$ be subintervals of integers in $[1,q]$ and let $\mathcal{M}$ be the set of elements of $\mathcal{I}$ that are invertible modulo $q$ and whose inverses lie in $\mathcal{J}$. We show that when $q$ approaches infinity through a sequence of values such that $\varphi(q)/q\rightarrow0$, the $r$-spacing distribution between consecutive elements of $\mathcal{M}$ becomes exponential.

AMS 2000 Mathematics subject classification: Primary 11K06; 11B05; 11N69

We prove that the space $L_\infty(\mu,X)$ has the same numerical index as the Banach space $X$ for every $\sigma$-finite measure $\mu$. We also show that $L_\infty(\mu,X)$ has the Daugavet property if and only if $X$ has or $\mu$ is atomless.

AMS 2000 Mathematics subject classification: Primary 46B20; 47A12

The Equation (1) $(r(x)y')'=q(x)y(x)$ is regarded as a perturbation of (2) $(r(x)z'(x))'=q_1(x)z(x)$. The functions $r(x)$, $q_1(x)$ are assumed to be continuous real valued, $r(x)>0$, $q_1(x)\ge0$, whereas $q(x)$ is continuous complex valued. A problem of Hartman and Wintner regarding the asymptotic integration of (1) for large $x$ by means of solutions of (2) is studied. Sufficiency conditions for solvability of this problem expressed by means of coefficients $r(x)$, $q(x)$, $q_1(x)$ of Equations (1) and (2) are obtained.

AMS 2000 Mathematics subject classification: Primary 34E20

Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate forpredator–prey interactions where predation involves serious searching processes. Due to the fact that every populationgoes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled.The presence of time delay often greatly complicates the analytical study of such models. In this paper, thequalitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in theinterior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable andthere exist non-trivial periodic solutions for the model. Moreover, by choosing delay $\tau$ as the bifurcationparameter we study the Hopf bifurcation and the stability of the periodic solutions.

AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14