Several authors have proved Lefschetz type formulas for the local Euler obstruction. In particular, a result of this type has been proved that turns out to be equivalent to saying that the local Euler obstruction, as a constructible function, satisfies the local Euler condition (in bivariant theory) with respect to general linear forms. The purpose of the paper is to determine what prevents the local Euler obstruction from satisfying the local Euler condition with respect to functions which are singular at the considered point. This is measured by an invariant (or ‘defect’) of such functions. An interpretation of this defect is given in terms of vanishing cycles, which allows it to be calculated algebraically. When the function has an isolated singularity, the invariant can be defined geometrically, via obstruction theory. This invariant unifies the usual concepts of the Milnor number of a function and the local Euler obstruction of an analytic set.

The problem of studying the largest rearrangement invariant space of functions with almost everywhere convergent Fourier series is considered. A quasi-Banach space QA of functions is defined with almost everywhere convergent Fourier series that strictly contains Antonov and Soria's spaces ([Lscr ] log [Lscr ] log log log [Lscr ] and B*ϕ1([ ])).

The paper considers n-dimensional hypersurfaces with constant scalar curvature of a unit sphere Sn−1(1). The hypersurface Sk(c1)×Sn−k(c2) in a unit sphere Sn+1(1) is characterized, and it is shown that there exist many compact hypersurfaces with constant scalar curvature in a unit sphere Sn+1(1) which are not congruent to each other in it. In particular, it is proved that if M is an n-dimensional (n > 3) complete locally conformally flat hypersurface with constant scalar curvature n(n−1)r in a unit sphere Sn+1(1), then r > 1−2/n, and

(1) when r ≠ (n−2)/(n−1), if

then M is isometric to S1(√1−c2)×Sn−1(c), where S is the squared norm of the second fundamental form of M;

(2) there are no complete hypersurfaces in Sn+1(1) with constant scalar curvature n(n−1)r and with two distinct principal curvatures, one of which is simple, such that r = (n−2)/(n−1) and

We develop a theory of modules of type FP∞ over group algebras of hierarchically decomposable groups. This class of groups is denoted H[Fscr ] and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an h[Fscr ]-group and Mis a ℤG-module of type FP∞ then M has finite projective dimension over ℤH for all torsion-free subgroups H of G. We also show that if G is an h[Fscr ]-group of type FP∞ and M is a ℤG-module which is ℤF-projective for all finite subgroups F of G, then M has finite projective dimension over ℤG. Both of these results have as a special case the striking fact that if G is an h[Fscr ]-group of type FP∞ then the torsion-free subgroups of G have finite cohomological dimension. A further result in this spirit states that every residually finite h[Fscr ]-group of type FP∞ has finite virtual cohomological dimension.

We investigate the location and nature of the spectrum of the fourth-order self-adjoint equation

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subject to certain asymptotic assumptions on the coefficients. The main tools are the theory of asymptotic integration and the Titchmarsh–Weyl M-matrix. Asymptotic integration yields asymptotic formulae for the solutions of the differential equation which are then used to derive properties of the M-matrix. The characterisation of spectral properties in terms of the boundary behaviour of M leads to the desired results.

Four methods for constructing anti-Pasch Steiner triple systems are developed. The first generalises a construction of Stinson and Wei to obtain a general singular direct product construction. The second generalises the Bose construction. The third employs a construction due to Lu. The fourth employs Wilson-type inflation techniques using Latin squares having no subsquares of order 2. As a consequence of these constructions we are able to produce anti-Pasch systems of order v for v 31 or 7 (mod 18), for v ≡ 49 (mod 72), and for many other values of v.

The weak type 1 for the Mehler maximal function is studied via a precise estimate for the ‘maximal kernel’. This, in turn, allows the geometry involved in this setting to be described.

The existing results are surveyed and several new results are proved for the cardinality of the restricted doubling 2ˆA = {a′ + a″:a′,a″ ∈ A,a′ ≠ a″}, where A ⊆ G is a subset of the set of elements of an (additively written) group G. In particular, known estimates for G = ℤ and G = ℤ/pℤ are improved and a first-of-a-kind general estimate valid for arbitrary G is given.

Consider a parabolic $N\times N$-system of order $m$ on $\mathbb{R}^n$ with top-order coefficients $a_\alpha \in \mathrm{VMO} \cap L^\infty$. Let $1<p,q < \infty$ and let $\omega$ be a Muckenhoupt weight. It is proved that systems of this kind possess a unique solution $u$ satisfying

$$\|u'\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)} + \|\mathcal{A} u\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)} \le C \|f\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)},$$

where $\mathcal{A} u = \sum_{|\alpha| \le m}a_\alpha D^\alpha u$ and $J=[0,\infty)$. In particular, choosing $\omega =1$, the realization of $\mathcal{A}$ in $L^p({\mathbb{R}}^n)^N$ has maximal $L^p-L^q$ regularity.

A new technique is presented which gives conditions under which perturbations of certain base potentials are uniquely determined from the location of eigenvalues and resonances in the context of a Schrödinger operator on a half line. The method extends to complex-valued potentials and certain potentials whose first moment is not integrable.Research supported in part by the US National Science Foundation under grants DMS-9970299 and DMS-0107492 and by the UK EPSRC.

For each $p$ in $[1, \infty)$ let ${\bf E}_p$ denote the closure of the region of holomorphy of the Ornstein–Uhlenbeck semigroup $\{{\cal H}_t : t >0\}$ on $L^p$ with respect to the Gaussian measure. Sharp weak type and strong type estimates are proved for the maximal operator $f \mapsto {{\cal H}^*}_pf=\sup\{\vert {\cal H}_zf\vert :z\in {\bf E}_p\}$ and for a class of related operators. As a consequence, a new and simpler proof of the weak type 1 estimate is given for the maximal operator associated to the Mehler kernel.

Let α > 0. The operator of the form

formula here

is considered, where the real weight function v(x) is locally integrable on R+ := (0, ∞). In case v(x) = 1 the operator coincides with the Riemann–Liouville fractional integral, Lp → Lq estimates of which with power weights are well known. This work gives Lp → Lq boundedness and compactness criteria for the operator Tα in the case 0 < p, q < ∞, p > max(1/α, 1).

We are interested in how small a root of multiplicity k can be for a power series of the form f(z:= 1+[sum ]∞n=1aizi with coefficients ai in [−1, 1]. Let r(k) denote the size of the smallest root of multiplicity k possible for such a power series. We show that

formula here

We describe the form that the extremal power series must take and develop an algorithm that lets us compute the optimal root (which proves to be an algebraic number). The computations, for k[les ]27, suggest that the upper bound is close to optimal and that r(k)∼1−c/(k+1), where c=1.230….

Topologists say that a space is sequential if every sequentially closed set is closed. Directly from the definitions, metrizable ⇒ Fréchet–Urysohn ⇒ sequential ⇒ $k$ -space. Kąkol showed that for an (LM)-space (the inductive limit of a sequence of locally convex metrizable spaces), metrizable , Fréchet–Urysohn. The Cascales and Orihuela result that every (LM)-space is angelic proved that for an (LM)-space, sequential [hArr ] $k$ -space. Within the class of (LM)-spaces, then, the four notions become only two distinct ones bearing the relation metrizable ⇒ sequential. Webb proved that every infinite-dimensional Montel (DF)-space is sequential but not Fréchet–Urysohn, and equivalently, not metrizable, since Montel (DF)-spaces are (LB)-spaces and, a fortiori, (LM)-spaces. Does the converse hold in the (LB)-space, (DF)-space or (LM)-space settings? Yes, in all cases. If a (DF)-space or (LM)-space is sequential, then it is either metrizable or it is a Montel (DF)-space. Pfister's result that every (DF)-space is angelic is needed, and the paper provides elementary proofs for this and the similar theorem by Cascales and Orihuela. The strongest locally convex topology plays a vital role throughout.

Szemerédi's theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi's theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation.

The paper studies the permutation representations of a finite general linear group, first on finite projective space and then on the set of vectors of its standard module. In both cases the submodule lattices of the permutation modules are determined. In the case of projective space, the result leads to the solution of certain incidence problems in finite projective geometry, generalizing the rank formula of Hamada. In the other case, the results yield as a corollary the submodule structure of certain symmetric powers of the standard module for the finite general linear group, from which one obtains the submodule structure of all symmetric powers of the standard module of the ambient algebraic group.

Certain discrete-time Markov processes on locally compact metric spaces which arise from graph-directed constructions of fractal sets with place-dependent probabilities are studied. Such systems naturally extend finite Markov chains and inherit some of their properties. It is shown that the Markov operator defined by such a system has a unique invariant probability measure in the irreducible case and an attractive probability measure in the aperiodic case if the vertex sets form an open partition of the state space, the restrictions of the probability functions on their vertex sets are Dini-continuous and bounded away from zero, and the system satisfies a condition of contractiveness on average.

Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field ${\bb Q}$ , are contained in the complex field ${\bb C}$ . Let $P$ be the set of all prime numbers. For any algebraic number field $F$ , let $C_F$ denote the ideal class group of $F$ and, writing $F^+$ for the maximal real subfield of $F$ , let $C^-_F$ denote the kernel of the norm map from $C_F$ to the ideal class group of $F^+$ ; for each $l \in P$ , let $C_F(l)$ denote the $l$ -class group of $F$ , that is, the $l$ -primary component of $C_F$ , and let $C^-_F(l)$ denote the $l$ -primary component of $C^-_F$ . Furthermore, for each $l \in P$ , we denote by ${\bb Z}_l$ the ring of $l$ -adic integers.

Let $X$ be a subset in $[-1,1]^{n_0}\,{\subset}\,\Real^{n_0}$ defined by the formula \[ X=\{ {\bf x}_0\,{\mid}\,Q_1{\bf x}_1 Q_2{\bf x}_2 \cdots Q_{\nu}{\bf x}_{\nu} (({\bf x}_0,{\bf x}_1,\,{\ldots}\,,{\bf x}_{\nu}) \in X_{\nu})\}, \] where $Q_i \in \{ \exists, \forall \}$, $Q_i \neq Q_{i{+}1}$, ${\bf x}_i \in \Real^{n_i}$, and $X_{\nu}$ may be either an open or a closed set in $[-1,1]^{n_0+ \cdots +n_{\nu}}{\!}$, being the difference between a finite CW-complex and its subcomplex. An upper bound on each Betti number of $X$ is expressed via a sum of Betti numbers of some sets defined by quantifier-free formulae involving $X_{\nu}$.

In important particular cases of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae with polynomials and Pfaffian functions respectively, upper bounds on Betti numbers of $X_{\nu}$ are well known. The results allow to extend the bounds to sets defined with quantifiers, in particular to sub-Pfaffian sets.

Through the use of the so-called variational Lyapunov method, which is developed by combining the method of variation of parameters and the Lyapunov second method, stability and instability properties in terms of two measures for impulsive differential equations with variable moments of impulsive effects are discussed. Some stability and instability criteria are established. These theorems, together with an example, show that perturbation and impulsive effects may make a stable system uniformly asymptotically stable or unstable. These results significantly generalize the known ones.