Working over an algebraically closed, complete, non-Archimedean, nontrivially valued field of characteristic zero, it is shown that (i) any subanalytic subset of a one-dimensional semialgebraic set is semialgebraic, (ii) any one-dimensional subanalytic set is semianalytic, and (iii) any subanalytic subset of a two-dimensional semianalytic set is semianalytic.

On page 45 of his lost notebook, Ramanujan recorded two asymptotic formulas for two continued fractions involving the Riemann zeta-function and Dirichlet $L$ -functions. The paper proves a more general theorem and derives Ramanujan's claims as a corollary of the theorem.

Assume that $G$ is a group covered by countably many sets $X_n,\,n\,{<}\,\omega$. It is proved that $G$ is generated in two or three steps by a small number of the sets $X_n$. These results are generalized for countable coverings of types and countable colourings of graphs.

Let $\tau$ be a conjugation, alias a conjugate linear isometry of order 2, on a complex Banach space $X$ and let $X^{\tau}$ be the real form of $X$ of $\tau$-fixed points. In contrast to the Dunford–Pettis property, the alternative Dunford–Pettis property need not lift from $X^{\tau}$ to $X$. If $X$ is a C*-algebra it is shown that $X^{\tau}$ has the alternative Dunford–Pettis property if and only if $X$ does and an analogous result is shown when $X$ is the dual space of a C*-algebra. One consequence is that both Dunford–Pettis properties coincide on all real forms of C*-algebras.

Two classes of vector-valued BMOA spaces are defined, in the complex ball and on the complex sphere, respectively. In the case of the complex sphere, vector measures are involved, since the argument in the scalar setting is not appropriate. Several properties (the Lp-equivalent norm theorem, exponential decay, the Baernstein theorem, and so on) of BMOA in the complex ball are extended to the Banach space setting. The two classes of BMOA spaces are proved to be isomorphic; in particular, the corresponding John–Nirenberg exponential decay is shown. Finally, the vector-valued H1-BMOA duality theorem is proved.

For each permutation f of the set of positive integers, all triples s, t, u are determined such that t and u are the lower and upper limits of the sequence of partial sums of the ‘f-rearrangement’ [sum ]af(n) of some real series [sum ]an with sum s.

Cameron–Praeger designs with parameters $t-(v,k,\lambda)$ are studied. Cameron and Praeger showed that in such designs, $t=2$ or 3. In 1989, Delandtsheer and Doyen proved that if $t=2$ then \[ v \leqslant \left(\left({k \atop 2}\right)-1\right)^2. \] In 2000, Mann and Tuan improved this equality and showed that if $t=3$ then \[ v \leqslant \left({k \atop 2}\right)+1. \] Three infinite families of Cameron–Praeger 3-designs for which this bound is met have been constructed by Mann and Tuan and by Sebille. The paper constructs infinitely many infinite families of such Cameron–Praeger 3-designs via a study of a divisibility problem for polynomials. Further, the construction generalizes the previous constructions.

It is shown that the composition operator w[map ]w∘u generated by an inner function u is a self-map of the Muckenhoupt class A2, but that it need not preserve the Muckenhoupt classes Ap for p∈(1, 2)∪(2, ∞).

The spectral gaps and thus the exponential rates of convergence to equilibrium are compared for ergodic one-dimensional diffusions on an interval. One of the results may be thought of as the diffusion analogue of a recent result for the spectral gap of one-dimensional Schrödinger operators. The similarities and differences between spectral gap results for diffusions and for Schrödinger operators are also discussed.

It is proved that every solution of the Neumann initial-boundary problem

[formula here]

converges to some equilibrium, if the system satisfies (i) ∂Fi/∂uj [ges ] 0 for all 1 [les ] i ≠ j [les ] n, (ii) F(u * g(s)) [ges ] h(s) [midast ] F(u) whenever u ∈ ℝn+ and 0 [les ] s [les ] 1, where x * y = (x1y1, …, xnyn) and g, h : [0, 1] → [0, 1]n are continuous functions satisfying gi(0) = hi(0) = 0, gi(1) = hi(1) = 1, 0 < gi(s); hi(s) < 1 for all s ∈ (0, 1) and i = 1, 2, …, n, and (iii) the solution of the corresponding ordinary differential equation system is bounded in ℝn+. We also study the convergence of the solution of the Lotka–Volterra system

[formula here]

where ri > 0, α [ges ] 0, and aij [ges ] 0 for i ≠ j.

The theory of order $p$ linear differential equations with univalued solutions can be presented in terms of the Schubert calculus in the Grassmannian of $p$-dimensional subspaces of the vector space of complex polynomials in one variable. The regular singular points and the exponents of an equation determine an intersection of Schubert varieties.

Heine and Stieltjes in their studies of order $p = 2$ linear differential equations with polynomial coefficients having a polynomial solution of a prescribed degree discovered that such a solution was given by a critical point of a certain remarkable symmetric function. In terms of the Schubert calculus, the critical points determine the elements of the intersection of Schubert varieties.

The case $p > 2$ is studied. A function whose critical points determine the non-degenerate elements in the intersection of Schubert varieties is presented. The number of Fuchsian equations with univalued solutions and prescribed exponents at singular points is estimated from above by the intersection number of the corresponding Schubert classes. Conjecturally, for generic disposition of singular points, these numbers coincide, that is, the intersection of Schubert varieties is transversal and consists of non-degenerate elements only. For $p = 2$ the conjecture was proved earlier by the author.

According to a 1904 conjecture of Dickson [2] unless prevented by congruence conditions, any finite collection of linear forms in $\Z[x]$ with positive leading coefficients infinitely often simultaneously represent primes. For the forms $x,x+2$ this includes the conjectured infinitude of prime-pairs.

In this paper we study sequence spaces that arise from the concept of strong weighted mean summability. Let q = (qn) be a sequence of positive terms and set Qn = [sum ]nk=1qk. Then the weighted mean matrix Mq = (ank) is defined by

formula here

It is well known that Mq defines a regular summability method if and only if Qn→∞. Passing to strong summability, we let 0<p<∞. Then

formula here

are the spaces of all sequences that are strongly Mq-summable with index p to 0, strongly Mq-summable with index p and strongly Mq-bounded with index p, respectively. The most important special case is obtained by taking Mq = C1, the Cesàro matrix, which leads to the familiar sequence spaces

formula here

respectively, see [4, 21]. We remark that strong summability was first studied by Hardy and Littlewood [8] in 1913 when they applied strong Cesàro summability of index 1 and 2 to Fourier series; orthogonal series have remained the main area of application for strong summability. See [32, §6] for further references.

When we abstract from the needs of summability theory certain features of the above sequence spaces become irrelevant; for instance, the qk simply constitute a diagonal transform. Hence, from a sequence space theoretic point of view we are led to study the spaces

The paper gives new examples of [Ascr ]-structures on certain SO(3)-principal fibre bundles.

Let $\textbf{F}\,{=}\,(F_1, \ldots, F_m)$ be an $m$-tuple of primitive positive binary quadratic forms and let $U_{\textbf{F}}(x)$ be the number of integers not exceeding $x$ that can be represented simultaneously by all the forms $F_j$, $j = 1, \ldots, m$. Sharp upper and lower bounds for $U_{\textbf{F}}(x)$ are given uniformly in the discriminants of the quadratic forms.

As an application, a problem of Erdős is considered. Let $V(x)$ be the number of integers not exceeding $x$ that are representable as a sum of two squareful numbers. Then $V(x) = x(\log x)^{-\alpha+o(1)}$ with $\alpha\,{=}\,1-2^{-1/3}\,{=}\,0.206\ldots$.

The asymptotic behaviour of entropy numbers of Trudinger–Strichartz embeddings of radial Besov spaces on ${\mathbb R}^n$ into exponential Orlicz spaces is calculated. Estimates of the entropy numbers as well as estimates of entropy numbers of Sobolev embeddings of radial Besov spaces are applied to spectral theory of certain pseudo-differential operators.

Remez-type inequalities provide upper bounds for the uniform norms of polynomials $p$ on given compact sets $K$, provided that $|p(x)|\leq1$ for every $x\in K\setminus E$, where $E$ is a subset of $K$ of small measure. In this paper we prove sharp Remez-type inequalities for homogeneous polynomials on star-like surfaces in $\mathbb{R}^d$. In particular, this covers the case of spherical polynomials (when $d=2$ we deduce a result of Erdélyi for univariate trigonometric polynomials).

Let ϕ be a hyperbolic map. Cocycle equations of the form f = uϕ·g·αu−1 are considered, with f, g, u taking values in a compact connected Lie group G, α being an automorphism of G and f, g being Hölder continuous. When the eigenvalues of the derivative of α have modulus 1, it is proved that any measurable solution u has a Hölder continuous version. This condition on α is optimal. When f, g are Ck then u may be taken to be Ck−1+ε for any ε ∈ (0, 1).

Consider a standard braid diagram as a three-dimensional figure viewed from the top; what happens when this figure is looked at from the side? Then a new braid can be obtained, and studying the connection between the initial braid and the derived braid so obtained provides both a new simple proof for the existence of the right greedy normal form of positive braids and a geometrical interpretation for the automatic structure of the braid groups.

The motivation for the theory of Euler characteristics of groups, which was introduced by C. T. C. Wall [21], was topology, but it has interesting connections to other branches of mathematics such as group theory and number theory. This paper investigates Euler characteristics of Coxeter groups and their applications. In his paper [20], J.-P. Serre obtained several fundamental results concerning the Euler characteristics of Coxeter groups. In particular, he obtained a recursive formula for the Euler characteristic of a Coxeter group, as well as its relation to the Poincaré series (see §3). Later, I. M. Chiswell obtained in [10] a formula expressing the Euler characteristic of a Coxeter group in terms of orders of finite parabolic subgroups (Theorem 1). These formulae enable us to compute Euler characteristics of arbitrary Coxeter groups.

On the other hand, the Euler characteristics of Coxeter groups W happen to be intimately related to their associated complexes [Fscr ]W, which are defined by means of the posets of nontrivial parabolic subgroups of finite order (see §2.1 for the precise definition). In particular, it follows from the recent result of M. W. Davis [13] that if [Fscr ]W is a product of a simplex and a generalized homology 2n-sphere, then the Euler characteristic of W is zero (Corollary 3.1). The first objective of this paper is to generalize the previously mentioned result to the case when [Fscr ]W is a PL-triangulation of a closed 2n-manifold which is not necessarily a homology 2n-sphere. In other words (as given below in Theorem 3), if W is a Coxeter group such that [Fscr ]W is a PL- triangulation of a closed 2n-manifold, then the Euler characteristic of W is equal to 1−χ([Fscr ]W)/2.