Gama and Nguyen [‘Finding short lattice vectors within Mordell’s inequality’, in: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, New York, 2008, 257–278] have presented slide reduction which is currently the best SVP approximation algorithm in theory. In this paper, we prove the upper and lower bounds for the ratios $\Vert { \mathbf{b} }_{i}^{\ast } \Vert / {\lambda }_{i} (\mathbf{L} )$ and $\Vert {\mathbf{b} }_{i} \Vert / {\lambda }_{i} (\mathbf{L} )$, where ${\mathbf{b} }_{1} , \ldots , {\mathbf{b} }_{n} $ is a slide reduced basis and ${\lambda }_{1} (\mathbf{L} ), \ldots , {\lambda }_{n} (\mathbf{L} )$ denote the successive minima of the lattice $\mathbf{L} $. We define generalised slide reduction and use slide reduction to approximate $i$-SIVP, SMP and CVP. We also present a critical slide reduced basis for blocksize 2.

An algebra A is said to be finitely related if the clone Clo(A) of its term operations is determined by a finite set of finitary relations. We prove that each finite idempotent semigroup satisfying the identity xyxzx≈xyzx is finitely related.

A geometrically aberrant Banach space with uniformly normal structure, a modification of Brown's example, is given.

In recent years fixed point theorems have been proved for non-expansive and similar mappings on uniformly convex Banach spaces. The only role the linear structure plays in the statement of these results occurs in the definition of uniform convexity. It is therefore natural to ask whether the results depend essentially on the linear structure, or whether an extension of the notion of uniform convexity to metric spaces would allow the hypothesis of linear structure on the underlying space to be removed.

For a locally compact group G, let LUC(G) denote the space of all left uniformly continuous functions on G. Here, we investigate projectivity, injectivity and flatness of LUC(G) and its dual space LUC(G)* as Banach left modules over the group algebra as well as the measure algebra of G.

Using Moore's ergodicity theorem, S.G. Dani and S. Raghavan proved that the linear action of SL(n, ℤ) on ℝn is topologically (n − l)-transitive; that is, topologically transitive on the Cartesian product of n − 1 copies of ℝn. In this paper, we give a more direct proof, using the prime number theorem. Further, using the congruence subgroup theorem, we generalise the result to arbitrary finite index subgroups of SL(n, ℤ).

If A is a complex Banach algebra which is also a Bezout domain, it is shown that for any prime p and a non-negative integer n, pn is not a topological divisor of zero. Using the above result it is shown that a complex Banach algebra which is a principal ideal domain is isomorphic to the complex field.

In this paper, class operators are used to give a complete listing of distinct base radical and semisimple classes for universal classes of finite associative rings. General relations between operators reveal that the maximum order of the semigroup formed is 46. In this setting, the homomorphically closed semisimple classes are precisely the hereditary radical classes and hence radical–semisimple classes, and the largest homomorphically closed subclass of a semisimple class is a radical–semisimple class.

In [2], it was asserted in Theorem 3 that the measure M(x0 + … + xx) is asymptotically , where c was an explicit constant. The value of c given was incorrect, and should be e-½γ where γ is Euler's constant. This was pointed out by the first author. In fact

where we have tried to make amends by improving the error term.

A general process for the desingularization of smooth actions of compact Lie groups is described. If G is a compact Lie group, it is shown that there is naturally associated to any compact G manifold M a compact G × (Z/2)p manifold on which G acts principally. Here Z/2 denotes the cyclic group of order two and p + 1 is the number of orbit types of the G action on M.

In this article we use real methods to study the Dirichlet and the boundary value problems of an annulus. Then we establish various properties of real Hardy spaces on the annulus, and give some applications.

We prove some existence theorems for solutions of a semilinear two point boundary value problem at resonance in which the nonlinear terms can have arbitrary polynomial growth in one of the directions ∞ or –∞, and are bounded in the other. The results are based on degree theoretic arguments and the Borsuk odd mapping theorem.

van Mill et al. posed in ‘Classes defined by stars and neighborhood assignments’, Topology Appl.154 (2007), 2127–2134 the following question: Is a star-compact space metrizable if it has a Gδ-diagonal? In this paper, we give a negative answer to this question.

Let (ωi, σi, μi.) be two positive finite measure spaces, V a non-zero Hilbert space, and 1 ≤ p < ∞, p # 2. In this article it is shown that each surjective linear isometry between the Bochner spaces induces a Boolean isomorphism between the measure algebras , thus generalizing a result of Cambern's for separable Hilbert spaces.

This Banach–Stone type theorem is achieved via a description of the Lp-structure of .

We establish a separation principle for a class of fractional order time-delay nonlinear differential systems. We show that a nonlinear time-delay observer is globally convergent and give sufficient conditions under which the observer-based controller stabilises the system.

Let { Xn, n ≥ 1 } be a sequence of independent Banach valued random variables and { an, n, ≥ 1 } a sequence of real numbers such that 0 < an ↑ ∞. It is shown that, under the assumption with some restrictions on φ, Sn/an → 0 a.s. if and only if Sn/an → 0 in probability if and only if Sn/an → 0 in L1. From this result several known strong laws of large numbers in Banach spaces are easily derived.

We study bounded linear regularity of finite sets of closed subspaces in a Hilbert space. In particular, we construct for each natural number $n\geq 3$ a set of $n$ closed subspaces of ${\ell }^{2} $ which has the bounded linear regularity property, while the bounded linear regularity property does not hold for each one of its nonempty, proper nonsingleton subsets. We also establish a related theorem regarding the bounded regularity property in metric spaces.

We identify the multiplier space of continuous vector-valued function spaces as the bounded continuous multiplier-valued space in the strong topology.

Let n be a positive integer, Bn the closed unit ball in Euclidean n-space, and X any countable CW-complex of dimension at most n. It is shown that the free Abelian topological group on Bn, F(Bn), has F(X) as a closed subgroup. It is also shown that for every differentiable manifold Y of dimension at most n, F(Y) is a closed subgroup of F(Bn).

Roth’s theorem on the consistency of the generalized Sylvester equation AX−YB=C is a special case of a rank minimization theorem.