This paper provides examples of closed manifolds having a Morse number different from the stable Morse number.

In this paper, it is proved that if $B$ is a Brauer $p$-block of a $p$-solvable group, for some odd prime $p$, then the height of any ordinary character in $B$ is at most $2b$, where $p^b$ is the largest degree of the irreducible characters of the defect group of $B$. Some other results that relate the heights of characters with properties of the defect group are obtained.

Let $S$ be a compact Riemann surface of genus ${\rm g}\,{>}\,1$. By using Teichmüller theory, a new approach is obtained to the existence of a Jenkins–Strebel differential with prescribed type and heights on $S$. Moreover, the geometric structures of horizontal trajectories of certain classes of quadratic differentials are discussed.

Let f be a 1-periodic C1-function whose Fourier coefficients satisfy the condition [sum ]n[mid ]n[mid ]3[mid ] fˆ(n[mid ]2<∞. For every α∈R\Q and m∈Z\{0}, we consider the Anzai skew product T(x, y) = (x+α, y+mx+f(x)) acting on the 2-torus. It is shown that T has infinite Lebesgue spectrum on the orthocomplement L2(dx)⊥ of the space of functions depending only on the first variable. This extends some earlier results of Kushnirenko, Choe, Lemańczyk, Rudolph, and the author.

It is proved that if $P_G(s)$ has an Euler product expansion with all factors of the form $1-c_i/q_i^s$ where each $q_i$ is a prime power, then $G$ is soluble.

A class of operators is introduced on $L_1$ that is stable under taking sums of pointwise unconditionally convergent series, contains all compact operators and does not contain isomorphic embeddings. It follows that any operator from $L_1$ into a space with an unconditional basis belongs to this class.

If two operator algebras A and B are strongly Morita equivalent (in the sense of [5]), then their C*- envelopes C*(A) and C*(B) are strongly Morita equivalent (in the usual C*-algebraic sense due to Rieffel). Moreover, if Y is an equivalence bimodule for a (strong) Morita equivalence of A and B, then the operation, Y[otimes ]hA−, of tensoring with Y, gives a bijection between the boundary representations of C*(A) for A and the boundary representations of C*(B) for B. Thus the ‘noncommutative Choquet boundaries’ of Morita equivalent A and B are the same. Other important objects associated with an operator algebra are also shown to be preserved by Morita equivalence, such as boundary ideals, the Shilov boundary ideal, Arveson's property of admissability, and the lattice of C*-algebras generated by an operator algebra.

It is proved that the automorphism group of a countable structure cannot be a free uncountable group. The idea is that instead of proving that every countable set of equations of a certain form has a solution, it is proved that this holds for a co-meagre family of appropriate countable sets of equations.

Let F(z) = [sum ]∞n=0anzn be an analytic function in the unit disc satisfying

[formula here]

Then [formula here], which is familiar as Paley's inequality. In this paper, an analogue of this inequality with respect to the Jacobi expansions is established.

We describe an uncountable family of compact group automorphisms with entropy log 2. Each member of the family has a distinct dynamical zeta function, and the members are parametrised by a probability space. A positive proportion of the members have positive upper growth rate of periodic points, and almost all of them have an irrational dynamical zeta function.

If infinitely many Mersenne numbers have a bounded number of prime divisors, then a typical member of the family has upper growth rate of periodic points equal to log 2, and lower growth rate equal to zero.

An example is given of a random time $\rho$ associated with Brownian motion such that $\rho$ is not a stopping time but ${\bf E}M_{\rho}={\bf E}M_0$ for every uniformly integrable martingale $M$ .

A short proof is given that if $E$ is a super-reflexive Banach space, then $\mc B(E)$, the Banach algebra of operators on $E$ with composition product, is Arens regular. Some remarks are made on necessary conditions on $E$ for $\mc B(E)$ to be Arens regular.

We prove that the bound on the Lp norms of the Kakeya type maximal functions studied by Cordoba [2] and Bourgain [1] are sharp for p>2. The proof is based on a construction originally due to Schoenberg [5], for which we provide an alternative derivation. We also show that r2 log (1/r) is the exact Minkowski dimension of the class of Kakeya sets in ℝ2, and prove that the exact Hausdorff dimension of these sets is between r2 log (1/r) and r2 log (1/r) [log log (1/r)]2+ε.

This paper proves that for every Lipschitz function $f:{\bb R}^n\longrightarrow {\bb R}^m,\;m < n$ , there exists at least one point of $\varepsilon$ -differentiability of $f$ which is in the union of all $m$ -dimensional affine subspaces of the form $q_0+{\rm span}\{q_1,q_2,\ldots,q_m\},\;{\rm where}\;q_j(j=0,1,\ldots,m)$ are points in ${\bb R}^n$ with rational coordinates.

We produce a family of algebraic curves (closed Riemann surfaces) Sλ admitting two cyclic groups H1 and H2 of conformal automorphisms, which are topologically (but not conformally) conjugate and such that S/Hi is the Riemann sphere [Copf ]ˆ. The relevance of this example is that it shows that the subvarieties of moduli space consisting of points parametrizing curves which occur as cyclic coverings (of a fixed topological type) of [Copf ]ˆ need not be normal.

Assuming that the integrated density of states of a Schrödinger operator admits a high-energy asymptotic expansion, the authors give explicit formulae for the coefficients of this expansion in terms of the heat invariants.

In this paper, we show that the set of all common fixed points of a one-parameter nonexpansive semigroup is that of some single nonexpansive mapping. We next compare our result with Bruck's famous fixed-point theorem. We finally prove very simple convergence theorems to a common fixed point. In our discussion, we assume neither the strict convexity of the underlying space, nor the weak compactness of the domain of a nonexpansive semigroup.

In this paper we consider connections between three classes of optimal (in different senses) convex lattice polygons. A classical result is that if G is a strictly convex curve of length s, then the maximal number of integer points lying on G is essentially

formula here

It is proved here that members of a class of digital convex polygons which have the maximal number of vertices with respect to their diameter are good approximations of these curves. We show that the number of vertices of these polygons is asymptotically

formula here

where s is the perimeter of such a polygon. This result implies that the area of these polygons is asymptotically less than 0.0191612·n3, where n is the number of vertices of the observed polygon. This result is very close to the result given by Colbourn and Simpson, which is 15/784·n3≈0.0191326·n3. The previous upper bound for the minimal area of a convex lattice n-gon is improved to 1/54·n3≈0.0185185·n3 as n→∞.

Let $M\subset \mathbb{R}^n$ be a connected component of an algebraic set $\varphi^{-1}(0)$, where $\varphi$ is a polynomial of degree $d$. Assume that $M$ is contained in a ball of radius $r$. We prove that the geodesic diameter of $M$ is bounded by $2r\nu(n)d(4d-5)^{n-2}$, where $\nu(n)=2{\Gamma({1}/{2})\Gamma(({n+1})/{2})}{\Gamma({n}/{2})}^{-1}$. This estimate is based on the bound $r\nu(n)d(4d-5)^{n-2}$ for the length of the gradient trajectories of a linear projection restricted to $M$.

The rank of a vector space $\mathcal{A}$ of $n\times n$-matrices is by definition the maximal rank of an element of $\mathcal{A}$. The space $\mathcal{A}$ is called rank-critical if any matrix space that properly contains $\mathcal{A}$ has a strictly higher rank. This paper exhibits a sufficient condition for rank-criticality, which is then used to prove that the images of certain Lie algebra representations are rank-critical. A rather counter-intuitive consequence, and the main novelty in this paper, is that for infinitely many $n$, there exists an eight-dimensional rank-critical space of $n \times n$-matrices having generic rank $n-1$, or, in other words: an eight-dimensional maximal space of non-invertible matrices. This settles the question, posed by Fillmore, Laurie, and Radjavi in 1985, of whether such a maximal space can have dimension smaller than $n$. Another consequence is that the image of the adjoint representation of any semisimple Lie algebra is rank-critical; in both results, the ground field is assumed to have characteristic zero.