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.

A theorem of Kac on quiver representations states that the dimension vectors of indecomposable representations are precisely the positive roots of the associated symmetric Kac–Moody Lie algebra. Here this result is generalised to representations respecting an admissible quiver automorphism, and the positive roots of an associated symmetrisable Kac–Moody Lie algebra are obtained.

Also the relationship with species of valued quivers over finite fields is discussed. It is known that the number of isomorphism classes of indecomposable representations of a given dimension vector for a species is a polynomial in the size of the base field. It is shown that these polynomials are non-zero if and only if the dimension vector is a positive root of the corresponding symmetrisable Kac–Moody Lie algebra.

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.

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 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….

The Skolem–Mahler–Lech theorem states that if $f(n)$ is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that $f(m)$ is equal to 0 is the union of a finite number of arithmetic progressions in $m\ge 0$ and a finite set. We prove that if X is a subvariety of an affine variety Y over a field of characteristic 0 and q is a point in Y, and $\sigma$ is an automorphism of Y, then the set of m such that $\sigma^m({\bf q})$ lies in X is a union of a finite number of complete doubly-infinite arithmetic progressions and a finite set. We show that this is a generalisation of the Skolem–Mahler–Lech theorem.

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.

It is shown that every special Lagrangian cone in $\C^3$ determines, and is determined by, a primitive harmonic surface in the 6-symmetric space ${\rm SU}_3/{\rm SO}_2$. For cones over tori, this allows the classification theory of harmonic tori to be used to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some of the examples found recently by Joyce are put into this context.

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.

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([ ])).

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.

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.

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.

The paper studies projective stationary sets. The Projective Stationary Reflection Principle is the statement that every projective stationary set contains an increasing continuous ∈-chain of length ω1. It is shown that, if Martin's Maximum holds, then the Projective Stationary Reflection Principle holds. Also, this principle is equivalent to the Strong Reflection Principle. The paper shows that the saturation of the nonstationary ideal on ω1 is equivalent to a certain kind of reflection.

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.

It is proved that all modules over an integral domain $R$ have strongly flat cover if and only if every flat $R$ -module is strongly flat. The domains satisfying this property are characterized by the property that all their proper quotients are perfect rings, and are called almost perfect. They are exactly the $h$ -local domains which are locally almost perfect. Various relevant classes of modules admitting or not admitting strongly flat cover are exhibited.

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.

Let T be a d×d matrix with integral coefficients. Then T determines a self-map of the d-dimensional torus X = ℝd/ℤd. Choose for each natural number n a ball B(n) in X and suppose that B(n+1) has smaller radius than B(n) for all n. Now let W be the set of points x ∈ X such that Tn(x) ∈ B(n) for infinitely many n ∈ ℕ. The Hausdorff dimension of W is studied by analogy with the Jarník–Besicovitch theorem on the dimension of the set of well-approximable real numbers. The dimension depends on the quantity

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A complete description is given only when the matrix is diagonalizable over ℚ. In other cases a result is obtained for sufficiently large τ. The results, in as far as they go, show that the Hausdorff dimension of W is a strictly decreasing, continuous function of τ which is piecewise of the form (Aτ+B)/(Cτ+D). The numbers A, B, C and D which arise in this way are typically sums of logarithms of the absolute values of eigenvalues of T.

A 4n-dimensional Riemannian manifold (M, g) is hyperkähler if it possesses three anti-commuting complex structures I, J, K such that the metric g is Kähler with respect to each of them. The reduced holonomy group of such a manifold is necessarily a subgroup of Sp(n) so the Ricci tensor of g vanishes and (M, g) can be regarded as a positive definite solution to Einstein's equations in vacuum.