For a holomorphic function $f$ in the open unit disc $D$, the $N$th partial sum of its Taylor series with center $\zeta \in D$ is denoted by $S_N(f,\zeta)(z)=$${\sum\nolimits^N_{n=0}}({{f^{(n)}(\zeta)}/n!})(z-\zeta)^n$. Generically, all functions $f$ in $H(D)$ satisfy the following. For every compact set $K\subset\Bbb C$ with $K{\cap}\,D=\varnothing$ and $K^c$ connected and every polynomial $h$, there exists a sequence of positive integers $\{\lambda_n\}^{\infty}_{n=1}$ such that, for every $l \in \{ 0,1,2, \ldots \}$, \[ \sup_{z \in K}\Big\vert {{\partial^l}\over {\partial z^l}} S_{\lambda_n}(f,0)(z)-h^{(l)}(z)\Big\vert \,{\to}\,0 \quad {\rm as} \; n\,{\to}\,{+}\,\,\infty.\]

Let $l$ be an oriented link of $d$ components in a homology $3$-sphere. For any nonnegative integer $q$, let $l(q)$ be the link of $d-1$ components obtained from $l$ by performing $1/q$ surgery on its $d$th component $l_d$. The Mahler measure of the multivariable Alexander polynomial $\Delta_{l(q)}$ converges to the Mahler measure of $\Delta_l$ as $q$ goes to infinity, provided that $l_d$ has nonzero linking number with some other component. If $l_d$ has zero linking number with each of the other components, then the Mahler measure of $\Delta_{l(q)}$ has a well defined but different limiting behavior. Examples are given of links $l$ such that the Mahler measure of $\Delta_l$ is small. Possible connections with hyperbolic volume are discussed.

A new large class of rearrangement-invariant (symmetric) spaces is introduced which contains most classical spaces used in applications. It consists of spaces with arbitrary fundamental function $\f(t)$ which are interpolation spaces with respect to the corresponding extreme spaces $\Lambda_\varphi$ and $M_\varphi$ (the quasinorm case is also permitted). The (quasi)norms for these new spaces are of the form $\|\varphi(t)f^*(t)\|_{\widetilde E}$, where $\widetilde E$ is an arbitrary rearrangement-invariant function space with respect to the measure $dt/t$. Thus the considered spaces include and generalize Lorentz, Lorentz–Zygmund and other similar spaces as well as the spaces $L_{p,\alpha, E}$, previously studied by Pustylnik. In spite of their generality, these spaces can be investigated deeply and in detail with rather sharp results that open a simple way to various applications, extending results about their classical examples. For instance, their dual spaces, sharp conditions for separability and mutual embeddings, dilation functions, Boyd indices and so on are found. A theorem on optimal weak type interpolation in these spaces is also proved.

It is important and interesting to study harmonic functions on a Riemannian manifold. In an earlier work of Li and Tam [21] it was demonstrated that the dimensions of various spaces of bounded and positive harmonic functions are closely related to the number of ends of a manifold. For the linear space consisting of all harmonic functions of polynomial growth of degree at most d on a complete Riemannian manifold Mn of dimension n, denoted by [Hscr ]d(Mn), it was proved by Li and Tam [20] that the dimension of the space [Hscr ]1(M) always satisfies dim[Hscr ]1(M) [les ] dim[Hscr ]1(ℝn) when M has non-negative Ricci curvature. They went on to ask as a refinement of a conjecture of Yau [32] whether in general dim [Hscr ]d(Mn) [les ] dim[Hscr ]d(ℝn) for all d. Colding and Minicozzi made an important contribution to this question in a sequence of papers [5–11] by showing among other things that dim[Hscr ]d(M) is finite when M has non-negative Ricci curvature. On the other hand, in a very remarkable paper [16], Li produced an elegant and powerful argument to prove the following. Recall that M satisfies a weak volume growth condition if, for some constant A and ν,

formula here

for all x ∈ M and r [les ] R, where Vx(r) is the volume of the geodesic ball Bx(r) in M; M has mean value property if there exists a constant B such that, for any non- negative subharmonic function f on M,

formula here

for all p ∈ M and r > 0.

It is shown that every strongly-cyclic branched covering of a (1,1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds coincides with the class of strongly-cyclic branched coverings of (1,1)-knots. As a consequence, a parametrization of (1,1)-knots by 4-tuples of integers is obtained. Moreover, using a representation of (1,1)-knots by the mapping class group of the twice-punctured torus, an algorithm is provided which gives the parametrization of all torus knots in $\mathbf{S}^3$.

On a manifold with polynomial volume growth satisfying Gaussian upper bounds of the heat kernel, a simple characterization of the matching lower bounds is given in terms of a certain Sobolev inequality. The method also works in the case of so-called sub-Gaussian or sub-diffusive heat kernels estimates, which are typical of fractals. Together with previously known results, this yields a new characterization of the full upper and lower Gaussian or sub-Gaussian heat kernel estimates.

The orientation preserving topological symmetry group of a graph embedded in the 3-sphere is the subgroup of the automorphism group of the graph consisting of those automorphisms which can be induced by an orientation preserving homeomorphism of the ambient space. We characterize all possible orientation preserving topological symmetry groups of embeddings of complete graphs in the 3-sphere.

Let $X,Y$ be Banach spaces. Of concern are the higher order abstract Cauchy problem $({\rm ACP}_n)$ in $X$ and its inhomogeneous version $({\rm IACP}_n)$ . A new operator family of bounded linear operators from $Y$ to $X$ is introduced, called an existence family for $({\rm ACP}_n)$ , so that the existence and continuous dependence on initial data of the solutions of $({\rm ACP}_n)$ and $({\rm IACP}_n)$ can be studied, and some basic results in a quite general setting can be obtained. A sufficient and necessary condition ensuring that $({\rm ACP}_n)$ possesses an exponentially bounded existence family, in terms of Laplace transforms, is presented. As a partner of the existence family, for $({\rm ACP}_n)$ , a uniqueness family of bounded linear operators on $X$ is defined to guarantee the uniqueness of solutions. These two operator families for $({\rm ACP}_n)$ are generalizations of the classical strongly continuous semigroups and sine operator functions, the $C$ -regularized semigroups and sine operator functions, the existence and uniqueness families for $(ACP_1)$ , and the $C$ -propagation families for $({\rm ACP}_n)$ . They have a special function in treating those ill-posed $({\rm ACP}_n)$ and $({\rm IACP}_n)$ whose coefficient operators lack commutativity.

We study the following question: given an open set $\Omega$, symmetric about 0, and a continuous, integrable, positive definite function $f$, supported in $\Omega$ and with $f(0)=1$, how large can $\int f$ be? This problem has been studied so far mostly for convex domains $\Omega$ in Euclidean space. In this paper we study the question in arbitrary locally compact abelian groups and for more general domains. Our emphasis is on finite groups as well as Euclidean spaces and ${\mathbb Z}^d$. We exhibit upper bounds for $\int f$ assuming geometric properties of $\Omega$ of two types: (a) packing properties of $\Omega$ and (b) spectral properties of $\Omega$. Several examples and applications of the main theorems are shown. In particular, we recover and extend several known results concerning convex domains in Euclidean space. Also, we investigate the question of estimating $\int_{\Omega}f$ over possibly dispersed sets solely in dependence of the given measure $m:=|\Omega|$ of $\Omega$. In this respect we show that in ${\mathbb R}$ and ${\mathbb Z}$ the integral is maximal for intervals.

It is proved that the finite soluble groups can be characterized among finite groups by a first-order sentence, namely, the sentence that asserts that no non-trivial element $g$ is a product of 56 commutators $[x,y]$ with entries $x$, $y$ conjugate to $g$.

A univalent harmonic map of the unit disk Δ[ratio ]={z∈[Copf ][ratio ][mid ]z[mid ]<1} is a complex-valued function f(z) on Δ that satisfies Laplace's equation fzz[bar]=0 and is injective. The Jacobian J[ratio ]=[mid ]fz[mid ]2 −[mid ]fz[bar][mid ]2 of a univalent harmonic map can never vanish [18], and so we might as well assume that J>0 throughout Δ. Then [mid ]fz[mid ]>0 and a short computation verifies that the analytic dilatation ω[ratio ] =f[bar]z[bar]/fz is indeed an analytic function, with [mid ]ω[mid ]<1 since J>0. Clearly ω≡0 when f is a conformal map, and in general the dilatation ω measures how far f is from being conformal. Also, if ω happens to be the square of an analytic function, then f ‘lifts’ to give an isothermal coordinate map for a minimal surface, and in that case i/√ω equals the stereographic projection of the Gauss map of the surface.

The classical transmission problem for the scattering of time-harmonic electromagnetic waves by a dielectric object is solved using layer potential techniques. The surface of the object is only assumed to be a Lipschitz manifold. Unlike in the case of smoother surfaces, neither the calculus of pseudodifferential operators nor the compactness of certain boundary operators are available tools. Nevertheless, the particular representation of the solutions and the choice of the space of boundary data used in the paper make the problem tractable via harmonic analysis techniques for elliptical partial differential equations in nonsmooth domains.

Let $G$ be a finite group scheme operating on an algebraic variety $X$ , both defined over an algebraically closed field $k$ . The paper first investigates the properties of the quotient morphism $X\longrightarrow X/G$ over the open subset of $X$ consisting of points whose stabilizers have maximal index in $G$ . Given a $G$ -linearized coherent sheaf on $X$ , it describes similarly an open subset of $X$ over which the invariants in the sheaf behave nicely in some way. The points in $X$ with linearly reductive stabilizers are characterized in representation theoretic terms. It is shown that the set of such points is nonempty if and only if the field of rational functions $k(X)$ is an injective $G$ -module. Applications of these results to the invariants of a restricted Lie algebra ${\frak g}$ operating on the function ring $k[X]$ by derivations are considered in the final section. Furthermore, conditions are found ensuring that the ring $k[X]^{\frak g}$ is generated over the subring of $p$ th powers in $k[X]$ , where $p={\rm char}\,k>0$ , by a given system of invariant functions and is a locally complete intersection.

A variation on the unique product property of groups is described which seems natural from a geometric point of view. It is stronger than the unique product property, and hence implies, for example, that the group rings have no zero divisors. Some of its closure properties are described. It is shown that most surface groups satisfy this condition, and various other examples are given. It is explained how these ideas can give a more geometric interpretation of Promislow's example of a non-unique-product group.

Asymptotic cones were first used by Gromov in [6], where he constructed limit spaces of nilpotent groups in order to prove that groups with polynomial growth are virtually nilpotent. Gromov does not use the term asymptotic cone, which was introduced later by Van den Dries and Wilkie in [12], when they gave a nonstandard interpretation of Gromov's results. Ultrafilters appear in [12] for the first time in this context. Later, Gromov gave an extensive treatment of asymptotic cones in [7]. Since then several authors have used asymptotic cones to obtain interesting results, for instance, in identifying quasi-isometry classes of 3-manifolds [8] or relating asymptotic cones with Dehn functions of finitely presented groups (see [2] and [11], the results of which are stated in Section 2).

The purpose of this paper is to develop some of the results stated in [7], in particular those describing the asymptotic cone of the Baumslag–Solitar groups and of Sol. According to [2], these spaces are not simply connected, since their Dehn functions are exponential, so our primary goal is to study their fundamental groups. It will be proved that these fundamental groups are uncountable and nonfree (Section 9), by constructing subgroups isomorphic to the fundamental group of the Hawaiian earring. These subgroups are constructed by finding subspaces in the asymptotic cones which are homotopically equivalent to the Hawaiian earring, and which induce injections in the fundamental group level (Section 8). Crucial to the proof of these facts is the computation of the covering dimension of these asymptotic cones (Section 7), which is done using a more general theorem on dimensions of spaces which admit certain maps into well-known spaces (Section 6).

A study is made of groups with finitely many derived groups of subgroups or of infinite subgroups. These groups are classified completely in the locally graded case. In the general case, detailed structural information about groups in each class is found.

Some elementary results are given on restricted sums $2^{\wedge}{\cal A}$ and $3^{\wedge}{\cal A}$ of a subset ${\cal A}\subset {\bb Z}/n{\bb Z}$ in the case when the cardinality of ${\cal A}$ , is close to $n/2$ . In the last section, some of these results are used to derive some new values of a function related to the Erdős–Ginzburg–Ziv problem.

The norm problem is considered for elementary operators of the form $U_{a,b}\,{:}\,{\cal A}\,{\longrightarrow}\,{\cal A},\;x\longmapsto axb\,{+}\,bxa (a,\,b\,{\in}\,{\cal A})$ in the special case when ${\cal A}$ is a subalgebra of the algebra of bounded operators on a Banach space. In particular, the lower estimate $\|U_{a,b}\|\geq\|a\|\|b\|$ is established when the Banach space is a Hilbert space and ${\cal A}$ is the algebra of all bounded linear operators.

An interpretation is given of the Shintani descent from the points of a connected algebraic group over a finite field to its group of points over a subfield in terms of a Shintani descent between two different rational structures over the same finite field, using restriction of scalars. From this a generalization is deduced of a result of Gyoja's on the Shintani descent of Deligne–Lusztig characters of a finite group of Lie type.

For an irrational rotation α of the circle group T=R/Z and a piecewise absolutely continuous function f@[ratio ]T→R, the unitary operator Vh(x)=e2πif(x)h(x+α) on L2(T) is studied. It is shown that if f has a single discontinuity with non-integer jump then V is κ-weakly mixing for some κ with 0<[mid ]κ[mid ]<1. In particular V has continuous singular spectrum. The property of κ-weak mixing (with possible change of the value of κ, 0<[mid ]κ[mid ]<1) holds for all irrational rotations and, given α, is stable under perturbations of f by functions with sufficiently small O(1/n)-norm. On the other hand, there exists a piecewise linear function f with two non-integer jumps such that the spectrum of V is continuous singular for one value of α and Lebesgue for another.