As the first step of research on functional equations for multiple zeta-functions, we present a candidate of the functional equation for a class of two variable double zeta-functions of the Hurwitz–Lerch type, which includes the classical Euler sum as a special case.

We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves.

Let ${\frak M}_{g}$ be the moduli space of smooth, integral curves of genus $g$ over the complex field ${\mathbb C}$ and let $\overline{{\frak M}}_{g}$ be its Mumford–Deligne compactification. If $C$ is a smooth, integral curve of genus $g$ we denote by [$C$] its isomorphism class in ${\frak M}_{g}$.

We prove a conjecture made by Frank Peterson on the global structure of the Dickson algebras arising as odd primary general linear group invariants. The Dickson algebra $W_{n}$ of invariants in a rank $n$ polynomial algebra over $\mathbb{F}_{p}$ is an unstable algebra over the mod $p$ Steenrod algebra. We prove that $W_{n}$ is a free unstable algebra on a certain cyclic module, modulo just one additional relation. The result is both similar to and different from the corresponding result we previously obtained with Frank Peterson at the prime 2. We also extend our characterization to the algebras of invariants under the special linear groups.

We show that projective unitary groups, or more generally, central quotients of products of unitary groups are totally $N$-determined when considered as $p$-compact groups at odd primes $p$.

Existence varieties (briefly e-varieties) of regular semigroups were introduced in [1] as classes of regular semigroups closed under the formation of homomorphic images, regular subsemigroups and direct products. Existence varieties of orthodox semigroups, which are simply the sub-e-varieties of the e-variety $\hbox{\sp O}$ of all orthodox semigroups, were independently introduced in [4] under the name of bivarieties. Moreover, in [4], the notions of bifree objects, biidentities and biinvariant congruences were introduced within $\hbox{\sp O}$ in such a way that a theory properly generalizing the theory of varieties of inverse semigroups arose. Existence varieties of orthodox semigroups include all varieties of orthodox completely regular semigroups, and hence, in particular, all varieties of idempotent semigroups, that is, all subvarieties of the variety $\hbox{\sp B}$ of all bands. These varieties form a countable lattice whose full description is well known.

Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra $S$ in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adjoin inverses to some of the arrows in the quiver, we show that the path algebra of the new quiver with relations is $M_n(S)$ where $n$ is the number of vertices in our quiver. The slogan would be that every finitely presented algebra is Morita equivalent to a universal localization of a finite dimensional algebra.

We discuss several methods which yield spanning sets for the irreducible polynomial $GL(n,K)$-modules, where $K$ is an infinite field. We show that the spanning set which arises from our first method coincides, up to sign, with the spanning set produced by a method due to Pittaluga and Strickland. We prove that these spanning sets are related to the spanning set produced by a third method via the Désarménien matrix.

In this paper we apply the techniques and results from the theory of multifractal divergence points to give a systematic and detailed account of the Hausdorff dimensions of sets of numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their $N$-adic expansion. Using earlier methods and results we investigate and compute the Hausdorff dimension of several new sets of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of numbers for which the limiting frequencies of the digits in their $N$-adic expansion do not exist. Such sets have only very rarely been studied. In addition, our techniques provide simple proofs of (substantial generalizations of) known results, by Cajar and Drobot and Turner and others, on the Hausdorff dimension of sets of normal and non-normal numbers.

We examine the multifractal spectra of one-sided local dimensions of Ahlfors regular measures on ${\bf R}$. This brings into a natural context a curious property that has been observed in a number of instances, namely that the Hausdorff dimension of the set of points of non-differentiability of a self-affine ‘devil's staircase’ function is the square of the dimension of the set of points of increase.

We extend a result of M. Heins by showing that for any sequence of points $(z_n)$ in the unit disk ${\Bbb D}$ tending to the boundary, there is a Blaschke product $B$ which is universal for noneuclidian translates in the sense that the set $\{B((z\,{+}\,z_n)/(1\,{+}\,\overline{z}_nz))\,{:} n\,{\in}\,{\Bbb N}\}$ is locally uniformly dense in the set of all holomorphic functions bounded by one on ${\Bbb D}$. From this, we conclude that for every countable set ${\sp L}$ of hyperbolic/parabolic automorphisms of the unit disk there exists a Blaschke product which is a common cyclic vector in $H^2$ for the composition operators associated with the elements in ${\sp L}$. These results are obtained by transferring the associated approximation problems to interpolation problems on the corona of $H^\infty$.

In this paper, we are concerned with the boundedness of all the solutions of the two-dimensional differential system $$ \left.\begin{array}{@{}l@{}}x'\,{=}\,{-}|y|^{p-2} y+f(t)\\ y'\,{=}\,|x|^{q-2} x+g(t) \end{array}\right\} $$ where $1\,{<}\,p$, $1\,{<}\,q\,{<}\,2$ and $5pq\,{>}\,3p+4q$, $pq/(p+q)\,{<}\,1$, $f, g$ are $C^5$ 1-periodic functions.

The multifractal formalism for functions has been proved to be valid for a large class of selfsimilar functions. All the functions that have been studied are (or turned out to be) associated with a family of contractions which satisfies some separation conditions. In this paper, we extend the validity in the presence of overlaps involved by the well-known $n$-scale dilation family. Our method of proof is based on wavelet analysis and some interesting properties of this family.

The annihilator $J^{\perp}$ of a weak*-closed inner ideal $J$ in the JBW*-triple $A$ consists of elements $b$ of $A$ for which $\{J\,b\,A\}$ is equal to zero, and the kernel $\mathrm{Ker}(J)$ of $J$ consists of those elements $b$ in $A$ for which $\{J\,b\,J\}$ is equal to zero. The annihilator $J^{\perp}$ is also a weak*-closed inner ideal in $A$, and $A$ enjoys the Peirce decomposition \begin{eqnarray*} A &=& J \oplus_M J^{\perp} \oplus J_1\\ &=& J_2 \oplus_M J_0 \oplus J_1, \end{eqnarray*} where $J_1$ is the intersection of the kernels of $J$ and $J^{\perp}$. When all of the usual Peirce arithmetical relations hold, $J$ is said to be a Peirce inner ideal. A pair $(J,K)$ of weak*-closed inner ideals is said to be compatible when \[A = \bigoplus_{j,k = 0}^2 J_j \cap K_k.\] By analysing the biannihilator $J^{\perp\perp}$ of a Peirce inner ideal $J$ it is shown that, if $J$ is a Peirce inner ideal, then $J^{\perp}$ is a Peirce inner ideal. Furthermore, if $K$ is a weak*-closed inner ideal in $A$ such that $(J,K)$ is a compatible pair, then $(J^{\perp},K)$ is a compatible pair.

We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus in the complex projective plane and of the Whitney spheres in the complex projective, complex Euclidean and complex hyperbolic planes.

Let ${\cal F}$ be an oriented conformal foliation of connected, totally geodesic and 1-dimensional leaves in $\mathbb{R}^{n+1}$. We prove that if $n\geq 3$ then the Gauss map $\phi{:}\,\,U\,{\to}\,S^n$ of ${\cal F}$ is a non-constant $n$-harmonic morphism if and only if it is a radial projection.