Let Ω be a topological space without isolated points, let E be a topological linear space which is continuously embedded into a product of countably boundedly generated topological linear spaces, and let X be a linear subspace of C(Ω, E). If a ∈ C(Ω) is not constant on any open subset of Ω and aX ⊂ X, then it is shown that there is at most one F-space topology on X that makes the multiplication by a continuous. Furthermore, if [Ufr ] is a subset of C(Ω) which separates strongly the points of Ω and [Ufr ]X ⊂ X, then it is proved that there is at most one F-space topology on X that makes the multiplication by a continuous for each a ∈ [Ufr ].

These results are applied to the study of the uniqueness of the F-space topology and the continuity of translation invariant operators on the Banach space L1(G, E) for a noncompact locally compact group G and a Banach space E. Furthermore, the problems of the uniqueness of the F-algebra topology and the continuity of epimorphisms and derivations on F-algebras and some algebras of vector-valued functions are considered.

A new invariant of (integrable) homotopy type for foliations is introduced: the tangential category of a foliated manifold.

The classical Lusternik–Schnirelmann theory is generalized to foliations and the relations of the tangential category with other known invariants such as the fibrewise and the equivariant category are studied. Cohomological lower bounds are provided in terms of foliated cohomology.

If the foliation is a product, the tangential category coincides with the ordinary category of the leaves. In general it is just bounded below. Estimates are given of the tangential category for compact-Hausdorff foliations and suspensions. Examples show that the lower and upper bounds are realized.

Let ω=(ωn)n[ges ]1 be a log concave sequence such that lim infn→+∞ ωn/nc>0 for some c>0 and ((log ωn)/nα)n[ges ]1 is nonincreasing for some α<1/2. We show that, if T is a contraction on the Hilbert space with spectrum a Carleson set, and if ∥T−n∥=O(ωn) as n tends to +∞ with [sum ]n[ges ]11/(n log ωn)=+∞, then T is unitary. On the other hand, if [sum ]n[ges ]11/(n log ωn)<+∞, then there exists a (non-unitary) contraction T on the Hilbert space such that the spectrum of T is a Carleson set, ∥T−n∥=O(ωn) as n tends to +∞, and lim supn→+∞ ∥T−n∥=+∞.

A diffeomorphism f of the disk has a periodic orbit [Oscr ], and the paper considers the minimal topological entropy l(f, [Oscr ]) in the isotopy class of f with respect to this orbit. The structure of the periodic orbits with l(f, [Oscr ]) = 0 has been studied lately and is now well understood. For l(f, [Oscr ]) non-zero, the paper obtains a lower bound of l(f, [Oscr ]) that depends only on the period of [Oscr ] and which improves previously known estimates. This result is a consequence of a stronger bound in the case where the isotopy class is pseudo-Anosov. It is obtained by studying the induced dynamics on a graph.

The paper first gives sufficient conditions on the critical points and the Schwarzian derivative of a real rational function $R$ such that the Julia set of $R$ is $\bar{{\bb C}}$ . Further, it is shown that under mild conditions on another real rational function $\tilde{R}$ with possibly non-empty Fatou set, the Julia set of $\tilde{R} \circ R$ is the whole Riemann sphere again. Then families of rational functions are given whose Julia set is $\bar{{\bb C}}$ and whose critical points are not necessarily preperiodic. Concrete examples were previously available only for the preperiodic case. Finally, it is demonstrated that the methods presented also apply to the construction of polynomials whose Julia sets are dendrites and whose critical points in the Julia set are not necessarily preperiodic.

We study boundedness and compactness properties of the Hardy integral operator Tf(x)=∫xAf from a weighted Banach function space X(v) into L∞ and BMO. We give a new simple characterization of compactness of T from X(v) into BMO. We construct examples of spaces X(v) such that T(X(v)) is (a) bounded in L∞ but not compact in BMO; (b) compact in BMO but not bounded in L∞; (c) bounded in BMO but neither bounded in L∞ nor compact in BMO; (d) bounded in L∞, compact in BMO and yet not compact in L∞. In order to obtain the last of the counterexamples we construct a new weighted Banach function space.

We investigate classes of uniform pseudodifferential operators on Riemannian manifolds with bounded geometry. We prove that operators belonging to the classes are bounded in function spaces of Hardy–Sobolev–Besov type defined on the manifold. The classes contain the powers of the Laplace–Beltrami operators so we get the fractional lift property for the function spaces.

It is shown that Alperin's weight conjecture is equivalent to the existence of contractible chain complexes whose entries have the right dimension coming from some of the alternating sum formulations. It is conjectured that for the other formulations and for Dade's ordinary conjecture, there also exist such contractible chain complexes.

It is proved that the simultaneous lattice packing and lattice covering constant of every three-dimensional centrally symmetric convex body is less than or equal to 7/4. Consequently, for any three-dimensional convex body K there is a corresponding lattice packing in which no extra translate of K can be added.

a new viewpoint is offered on some of the generic structures constructed using hrushovski's predimensions and it is shown that they are natural reducts of quite straightforward trivial, one-based stable structures.

We prove that if G is a non-trivial finite group, then the integral group ring $\mathbb{Z} G$ possesses a superdecomposable pure-injective module.

Let $\zeta(s, \alpha)$ be the Hurwitz zeta function with parameter $\alpha$ . Power mean values of the form $\sum^q_{a=1}\zeta(s,a/q)^h$ or $\sum^q_{a=1}|\zeta(s,a/q)|^{2h}$ are studied, where $q$ and $h$ are positive integers. These mean values can be written as linear combinations of $\sum^q_{a=1}\zeta_r(s_1,\ldots,s_r;a/q)$ , where $\zeta_r(s_1,\ldots,s_r;\alpha)$ is a generalization of Euler–Zagier multiple zeta sums. The Mellin–Barnes integral formula is used to prove an asymptotic expansion of $\sum^q_{a=1}\zeta_r(s_1,\ldots,s_r;a/q)$ , with respect to $q$ . Hence a general way of deducing asymptotic expansion formulas for $\sum^q_{a=1}\zeta(s,a/q)^h$ and $\sum^q_{a=1}|\zeta(s,a/q)|^{2h}$ is obtained. In particular, the asymptotic expansion of $\sum^q_{a=1}\zeta(1/2,a/q)^3$ with respect to $q$ is written down.

Consider the image of the Monge–Ampère operator acting on bounded functions, defined on a compact Kähler manifold, whose sum with the local Kähler potential is plurisubharmonic. It is shown that a nonnegative Borel measure belongs to this image if and only if it belongs to the image locally. In particular, those measures form a convex set.

The aim of this paper is to determine which (finite) 3-dimensional classical groups are completions of the Goldschmidt G3-amalgam. We recall, first, that an amalgam (of rank 2) consists of three groups P1, P2, B and two group monomorphisms ϕ1, ϕ2 such that

formula here

Usually, when ϕ1 and ϕ2 are understood, this amalgam is denoted by [Ascr ](P1, P2, B). Now a group G is a completion of [Ascr ](P1, P2, B) if there exist group homomorphisms ψi[ratio ]Pi → G (i = 1, 2) satisfying G = 〈im ψ1, im ψ2〉 and ψ1ϕ1 = ψ2ϕ2[ratio ]B→G. The Goldschmidt G3-amalgam, which appears in [6], is defined as follows: P1 ≅ Sym(4) ≅ P2, B ≅ Dih(8) (Dih(n) denotes the dihedral group of order n) with ϕ−11(O2(P1)) ≠ ϕ−12(O2(P2)).

An example is given which shows that the Denef–Loeser zeta function (usually called the topological zeta function) associated to a germ of a complex hypersurface singularity is not a topological invariant of the singularity. The idea is the following. Consider two germs of plane curves singularities with the same integral Seifert form but with different topological type and which have different topological zeta functions. Make a double suspension of these singularities (consider them in a 4-dimensional complex space). A theorem of M. Kervaire and J. Levine states that the topological type of these new hypersurface singularities is characterized by their integral Seifert form. Moreover the Seifert form of a suspension is equal (up to sign) to the original Seifert form. Hence these new singularities have the same topological type. By means of a double suspension formula the Denef–Loeser zeta functions are computed for the two 3-dimensional singularities and it is verified that they are not equal.

In 1868 Zolotarev determined the polynomial which deviates least from zero with respect to the maximum norm on $[-1,1]$ among all polynomials of the form $x^n + \sigma x^{n-1} + a_{n-2}x^{n-2} + \ldots +a_1x + a_0$, where $\sigma \in {\mathbb R}$ is given. The polynomial was given explicitly in terms of elliptic functions by Zolotarev. It is now called the Zolotarev polynomial. Zolotarev also gave an explicit expression for the minimum deviation. In the sequel attempts have been made to replace the elliptic functions and to express the Zolotarev polynomial and the minimum deviation in terms of elementary functions, at least asymptotically. In 1913 Bernstein succeeded in finding an asymptotic formula for the minimum deviation, which has been improved several times since then. Here we give the first asymptotic representation of the Zolotarev polynomials. For the asymptotic representation we use the rational functions introduced by Bernstein. Furthermore, we obtain asymptotic representations of minimal polynomials with interpolation constraints which are of interest in the theory of the iterative solution of inconsistent linear systems of equations.

A partition of the class of all Hermite–Biehler entire functions of finite order into subclasses ${\cal P}_\kappa$ is introduced. A given function $E(z)$ belongs to ${\cal P}_\kappa$ if and only if $-z^{-1}\log E(z)\in{\mc N}_\kappa$, where the class ${\mc N}_\kappa$ is a well studied family of meromorphic functions on the upper half plane, which originates from operator theoretic problems. It is also proved that the subclasses ${\cal P}_\kappa$ are stable under bounded type perturbation.

We prove a complex analytic analogue of the classical Rolle theorem asserting that the number of zeros of a real smooth function can exceed that of its derivative by at most 1. This result is used then to obtain upper bounds for the number of complex isolated zeros of:

(1) functions defined by linear ordinary differential equations (in terms of the magnitude of the coefficients of the equations);

(2) elements from the polynomial envelope of a linear differential equation with an irreducible monodromy group (in terms of the degree of the envelope);

(3) successive derivatives of a function defined by a linear irreducible equation (in terms of the order of the derivative).

These results generalize the bounds from [2, 5, 6] that were previously obtained for the number of real isolated zeros.

The purpose of this paper is to give characterizations of weights for which reverse inequalities of Hölder type for quasi-monotone functions are satisfied. Our inequalities with general weights and with sharp constants complement the results of [2, 8, 9] and [16, 17] for the values of parameters p and q with 1[les ]p[les ]q<∞.

Let $G$ be a metacyclic group of order $pq$ , where $p$ and $q$ are distinct odd prime numbers, let $N| k$ be a Galois extension whose Galois group $G(N| k)$ is isomorphic to $G$ . Let $R_N, R_k$ be the rings of integers of $N$ and $k$ . As $R_k$ -module $R_N$ is completely determined by $[N:k]$ and by its class in the class group of $R_k$ . The paper determines the classes realized by tame Galois extensions $N|k$ with $G(N|k)\cong G$ and proves that they form a group.