Let $\mathbb K$ be an algebraically closed field, let $X$ be a $\mathbb K$-variety, and let $X(\mathbb K)$ be the set of closed points in $X$. A constructible set$C$ in $X(\mathbb K)$ is a finite union of subsets $Y(\mathbb K)$ for subvarieties $Y$ in $X$. A constructible function$f:X(\mathbb K)\rightarrow\mathbb Q$ has $f(X(\mathbb K))$ finite and $f^{-1}(c)$ constructible for all $c\ne 0$. Write CF$(X)$ for the vector space of such $f$. Let $\phi:X\rightarrow Y$ and $\psi: Y\rightarrow Z$ be morphisms of ${\mathbb C}$-varieties. MacPherson defined a linear pushforward CF$(\phi):{\rm CF}(X)\rightarrow{\rm CF}(Y)$ by ‘integration’ with respect to the topological Euler characteristic. It is functorial, that is, CF$(\psi\circ\phi)={\rm CF}(\psi)\circ{\rm CF}(\phi)$. This was extended to $\mathbb K$ of characteristic zero by Kennedy.

This paper generalizes these results to $\mathbb K$-schemes and Artin$\mathbb K$-stacks with affine stabilizer groups. We define the notions of Euler characteristic for constructible sets in $\mathbb K$-schemes and $\mathbb K$-stacks, and pushforwards and pullbacks of constructible functions, with functorial behaviour. Pushforwards and pullbacks commute in Cartesian squares. We also define pseudomorphisms, a generalization of morphisms well suited to constructible functions problems.

Let G be a permutation group on a finite set Ω. A sequence B=(ω1, …, ωb) of points in Ω is called a base if its pointwise stabilizer in G is the identity. Bases are of fundamental importance in computational algorithms for permutation groups. For both practical and theoretical reasons, one is interested in the minimal base size for (G, Ω), For a nonredundant base B, the elementary inequality 2[mid ]B[mid ][les ][mid ]G[mid ][les ][mid ]Ω[mid ][mid ]B[mid ] holds; in particular, [mid ]B[mid ][ges ]log[mid ]G[mid ]/log[mid ]Ω[mid ]. In the case when G is primitive on Ω, Pyber [8, p. 207] has conjectured that the minimal base size is less than Clog[mid ]G[mid ]/log[mid ]Ω[mid ] for some (large) universal constant C.

It appears that the hardest case of Pyber's conjecture is that of primitive affine groups. Let H=GV be a primitive affine group; here the point stabilizer G acts faithfully and irreducibly on the elementary abelian regular normal subgroup V of H, and we may assume that Ω=V. For positive integers m, let mV denote the direct sum of m copies of V. If (v1, …, vm)∈mV belongs to a regular G-orbit, then (0, v1, …, vm) is a base for the primitive affine group H. Conversely, a base (ω1, …, ωb) for H which contains 0∈V=Ω gives rise to a regular G-orbit on (b−1) V. Thus Pyber's conjecture for affine groups can be viewed as a regular orbit problem for G-modules, and it is therefore a special case of an important problem in group representation theory. For a related result on regular orbits for quasisimple groups, see [4, Theorem 6].

An RC-group is a unital group G with a distinguished compression base with respect to which G satisfies the Rickart projection and general comparison properties. We prove that a monotone $\sigma$-complete RC-group is a union of subgroups each of which is a lattice-ordered Dedekind $\sigma$-complete RC-group.

A formula of Walczak is applied to two situations in differential geometry. Holomorphic distributions on Kähler manifolds are studied, and it is shown how the formula simplifies to a Bochner type formula, which is particularly useful in the study of integrable distributions. Then the Walczak formula is applied in the context of harmonic morphisms, where it provides a means of investigating the vertical Laplacian of the dilation. It is shown that, under some additional conditions on the map and the domain, the $p$-energy is infinite for $p$ sufficiently large.

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), we shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [8].

A group is accessible if there is a G-tree T such that every edge stabilizer is finite and every vertex stabilizer has at most one end. A group is inaccessible if it is not accessible. In [6] I began the study of group actions on protrees. These actions arise naturally when considering inaccessible groups. This study is continued in this paper.

It is proved that a Banach space $X$ is a subspace of a weakly compactly generated Banach space if and only if, for every $\varepsilon\,{>}\,0$, $X$ can be covered by a countable collection of bounded closed convex symmetric sets where the weak$^*$ closure in $X^{**}$ of each of them lies within the distance $\varepsilon$ from $X$. A new short functional-analytic proof of the known result that a continuous image of an Eberlein compact is Eberlein is given as a corollary.

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 $l \in \{ 0,1,2, \ldots \}$, \[ \sup_{z \in K}\Big\vert {{\partial^l}\over {\partial z^l}} S_{\lambda_n}(f,0)(z)-h^{(l)}(z)\Big\vert \,{\to}\,0 \quad {\rm as} \; n\,{\to}\,{+}\,\,\infty.\]

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.

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.

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.

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 $g$.

The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian \[ \left\{ \begin{array}{@{}r@{\,}c@{\,}l@{\qquad}l} u_{t}+( -\Delta ) ^{{\alpha}/{2}}u+u^{1+\sigma } &=&0, & x\in {\mathbf{R}}^{n},\text{ }t>0, \\[4pt] u( 0,x) &=& u_{0} ( x), &x\in {\mathbf{R}}^{n}, \end{array} \right. \label{A} \] where $\alpha \in ( 0,2)$, with critical $\sigma ={\alpha }/{ n}$ and sub-critical $\sigma \in ( 0,{\alpha }/{n}) $ powers of the nonlinearity. Let $u_{0}\,{\in}\, \mathbf{L}^{1,a}\cap \mathbf{L}^{\infty }\cap \mathbf{C}, $$u_{0}( x) \,{\geq}\, 0$ in $\mathbf{R}^{n},$$\theta \,{=}\,\int_{ \mathbf{R}^{n}}u_{0}( x) \,dx\,{>}\,0.$ The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution $u \in \mathbf{C}( [0,\infty); \mathbf{L}^{\infty }\cap \mathbf{L}^{1,a}\cap \mathbf{C})$ and the large time asymptotics are obtained.

Necessary and sufficient conditions are given for linearly recurrent Cantor dynamical systems to have measurable and continuous eigenfunctions. Also an example of a linearly recurrent system with a nontrivial Kronecker factor and a trivial maximal equicontinuous factor is constructed explicitly.

For a sequence (cn) of complex numbers, the quadratic polynomials fcn(z) := z2 + cn and the sequence (Fn) of iterates Fn := fcn∘…∘fc1 are considered. The Fatou set [Fscr ](cn) is by definition the set of all z ∈ [Copf ]ˆ such that (Fn) is normal in some neighbourhood of z, while the complement of [Fscr ](cn) is called the Julia set [Jscr ](cn). The aim of this article is to study the connectedness and stability of the Julia set [Jscr ](cn) provided that the sequence (cn) is bounded.

A point λ in the complex plane is said to be free if the group generated by the matrices

formula here

is free. The paper gives an infinite family of polynomials whose roots are the non-free points. The main idea in the paper is to employ a symmetry relation.

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.

We prove some general results about quasi-actions on trees and define Property (QFA), which is analogous to Serre's Property (FA), but in the coarse setting. This property is shown to hold for a class of groups, including SL$(n,\mathbb{Z})$ for $n\geq 3$. We also give a way of thinking about Property (QFA) by breaking it down into statements about particular classes of trees.