We prove versions of de Leeuw's Theorems for the contraction of $S^{2p-1}$ to $H^{p-1}$.

We present congruences for Greene's ₃F₂ hypergeometric functions over finite fields, which relate values of these functions to a simple polynomial in the characteristic of the field.

In this note we prove two Positivstellensätze for definable functions of class $C^r$, $0\le r < \infty$, in any $o$-minimal structure $\mathcal{S}$ expanding a real closed field $R$. Namely, we characterize the definable functions that are nonnegative (resp. strictly positive) on basic definable sets of the form $F=\{f_1\ge 0,\dots, f_k\ge 0\}$.

In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomogeneity one. We characterize these submanifolds as the only minimal Lagrangian submanifolds in $\mathbb{C}\mathbb{H}^n$ that are foliated by umbilical hypersurfaces of Lagrangian subspaces $\mathbb{R}\mathbb{H}^n$ of $\mathbb{C}\mathbb{H}^n$. By suitably generalizing this construction, we obtain new families of minimal Lagrangian submanifolds in $\mathbb{C}\mathbb{H}^n$ from curves in $\mathbb{C}\mathbb{H}^1$ and $(n-1)$-dimensional minimal Lagrangian submanifolds of the complex space forms $\mathbb{C}\mathbb{P}^{n-1}$, $\mathbb{C}\mathbb{H}^{n-1}$ and $\mathbb{C}^{n-1}$. We give similar constructions in the complex projective space $\mathbb{C}\mathbb{P}^n$.

An effective description of the join of algebraic curves in the complex projective space $\mathbb{P}^{n}$ is given.

We prove that the category of precrossed modules is an algebraic category, and we develop a cotriple (co)homology theory for precrossed modules which generalizes the Eilenberg-MacLane theory of (co)homology groups. We study the relationship of this theory with the (co)homology of crossed modules.

Let $M$ be a (possibly non-orientable) compact 3-manifold with (possibly empty) boundary consisting of tori and Klein bottles. Let $X\subset\partial M$ be a trivalent graph such that $\partial M\setminus X$ is a union of one disc for each component of $\partial M$. Building on previous work of Matveev, we define for the pair $(M,X)$ a complexity $c(M,X)\in\mathbb{N}$ and show that, when $M$ is closed, irreducible and $\mathbb{P}^2$-irreducible, $c(M,\emptyset)$ is the minimal number of tetrahedra in a triangulation of $M$. Moreover $c$ is additive under connected sum, and, given any $n\geqslant 0$, there are only finitely many irreducible and $\mathbb{P}^2$-irreducible closed manifolds having complexity up to $n$. We prove that every irreducible and $\mathbb{P}^2$-irreducible pair $(M,X)$ has a finite splitting along tori and Klein bottles into pairs having the same properties, and complexity is additive on this splitting. As opposed to the JSJ decomposition, our splitting is not canonical, but it involves much easier blocks than all Seifert and simple manifolds. In particular, most Seifert and hyperbolic manifolds appear to have non-trivial splitting. In addition, a given set of blocks can be combined to give only a finite number of pairs $(M,X)$. Our splitting theorem provides the theoretical background for an algorithm which classifies 3-manifolds of any given complexity. This algorithm has been already implemented and proved effective in the orientable case for complexity up to 9.

A well-known formula for the topological entropy of a symbolic system is $h_{\operatorname{top}}(X)=\lim_{n\to\infty} \log N(\Lambda_n)/|\Lambda_n|$, where $\Lambda_n$ is the box of side $n$ in $\mathbb{Z}^d$ and $N(\Lambda)$ is the number of configurations of the system on the finite subset $\Lambda$ of $\mathbb{Z}^d$. We investigate the convergence of the above limit for sequences of regions other than $\Lambda_n$ and show in particular that if $\Xi_n$ is any sequence of finite `convex' sets in $\mathbb{Z}^d$ whose inradii tend to infinity, then the sequence $\log N(\Xi_n)/|\Xi_n|$ converges to $h_{\operatorname{top}}(X)$. We apply this to give a concrete proof of a `strong Variational Principle', that is, the result that for certain higher dimensional systems the topological entropy of the system is the supremum of the measure-theoretic entropies taken over the set of all invariant measures with the Bernoulli property.

Let $\mathcal{S}$ be a finite generalized quadrangle of order $(s,t)$, $s \ne 1 \ne t$. A spread is a set of $st + 1$ mutually non-concurrent lines of $\mathcal{S}$. A spread $\mathbf{T}$ of $\mathcal{S}$ is called a spread of symmetry if there is a group of automorphisms of $\mathcal{S}$ which fixes $\mathbf{T}$ elementwise and which acts transitively on the points of at least one (and hence every) line of $\mathbf{T}$. From spreads of symmetry of generalized quadrangles, there can be constructed near polygons, and new spreads of symmetry would yield new near polygons. In this paper, we focus on spreads of symmetry in generalized quadrangles of order $(s,s^2)$. Many new characterizations of the classical generalized quadrangle $\mathcal{Q}(5,q)$ which arises from the orthogonal group $\mathbf{O}^{-}(6,q)$ will be obtained. In particular, we show that a generalized quadrangle $\mathcal{S}$ of order $(s,t)$, $s \ne 1 \ne t$, containing a spread of symmetry {\bf T} is isomorphic to $\mathcal{Q}(5,s)$, under any of the following conditions:

(i) $\mathcal{S}$ contains a point which is incident with at least three axes of symmetry (Theorem 6.4);

(ii) $t = s^2$ with $s$ even and $\mathcal{S}$ has a center of transitivity (Theorem 6.6);

(iii) there exists a line $L \not\in \mathbf{T}$ such that $\mathcal{S}$ is an EGQ with base-line $L$ (Theorem 6.8).

We establish upper bounds for multiplicative character sums and exponential sums over sets of integers that are described by various properties of their digits in a fixed base $g\ge 2$. Our main tools are the Weil and Vinogradov bounds for character sums and exponential sums. Our results can be applied to study the distribution of quadratic non-residues and primitive roots among these sets of integers.

The notions of an LEF group and of an LEF algebra, introduced by Gordon and Vershik in \cite{GV}, have been formulated in terms of local embeddability into finite groups or finite dimensional algebras, respectively. We will prove that the group algebra $\mC G$ of a group $G$ is LEF if and only if $G$ is. This solves a question raised in \cite{GV}.

Let $f:\mathcal{X}\rightarrow \mathcal{Y}$ be a morphism of flows, $y$ an almost periodic point of $\mathcal{Y}$, and $x\in f^{-1}(y)$. In general $x$ is not ne\-cessa\-rily almost periodic, but several conditions are known under which that happens. They fall into either ``compact" or ``noncompact" conditions, depending on whether $\mathcal{X}$ and $\mathcal{Y}$ are assumed to be compact or not. In ``noncompact" conditions other assumptions are restrictive. We find a criterion for almost periodicity of $x$, which generalizes both ``compact" and ``noncompact" statements at the same time. We deduce theorems of Ellis, Markley, Kutaibi-Rhodes and Pestov as corollaries.

We study nonlinear potential theory on a metric measure space equipped with a doubling measure and supporting a Poincaré inequality. Minimizers, superminimizers and the obstacle problem for the $p$-Dirichlet integral play an important role in the theory. We prove lower semicontinuity of superminimizers and continuity of the solution to the obstacle problem with a continuous obstacle. We also show that the limit of an increasing sequence of superminimizers is a superminimizer provided it is bounded above. Moreover, we consider superharmonic functions and study their relations to superminimizers. Our proofs are based on the direct methods of the calculus of variations and on De Giorgi type estimates. In particular, we do not use the Euler-Lagrange equation and our arguments are based on the variational integral only.

A group is called a $CP$-group if every element of the group has prime power order. The complete classification of locally finite $CP$-groups is given in this article.

This paper examines viscosity solutions to a class of fully nonlinear equations on Grushin-type planes. First, viscosity solutions are defined, using subelliptic second order superjets and subjets. Then, a Grushin maximum principle is proved, and as an application, comparison principles for certain types of nonlinear functions follow. This is accomplished by establishing a natural relationship between Euclidean and subelliptic jets, in order to use the viscosity solution technology of Crandall, Ishii, and Lions (1992). The particular example of infinite harmonic functions on certain Grushin-type planes is examined in further detail.

A well-known theorem of K. Zhu \cite{6} asserts that, for $2 \leq p <\infty $, the Hankel operators $H_f$ and $H_{\bar f}$ on the Bergman space $L^2_a(B_n,dV)$ of the unit ball belong to the Schatten class ${\mathcal{C}}_p$ if and only if the mean oscillation $\MO(f)(z) = \{\widetilde{|f|^2}(z) - |\tilde f(z)|^2\}^{1/2}$ belongs to $L^p(B_n,(1-|z|^2)^{-n-1}dV(z))$. It is well known that, for trivial reasons, this theorem cannot be extended to the case $p \leq 2n/(n+1)$. This paper fills the gap between $2n/(n+1)$ and 2. More precisely, we prove that, when $2n/(n+1) < p < 2$, the same theorem holds true.

We show that the interpolating sequences for the algebra of holomorphic functions in the unit disk of order at most $\alpha > 0$ are characterized by a hyperbolic density condition. We also give conditions along the same lines for the analogous problem in the unit ball of $\mathbb{C}^n$.

We give a short proof of the fact that there are no measurable subsets of Euclidean space (in dimension $d\ge 3$) which, no matter how translated and rotated, always contain exactly one integer lattice point. In dimension $d=2$ (the original Steinhaus problem) the question remains open.

For a CW-complex $X$ and for $0\leq j\leq 2$, we construct natural homomorphisms $\beta_{j}^{X}\colon H_{j}(X;\,\mathbb{Z}) \longrightarrow K_{j}(X)$ that are rationally right-inverses of the Chern character. We show that $\beta_{j}^{X}$ is injective for $j=0$ and $j=1$. The case $j=3$ is treated using $\mathbb{Z}[\frac12]$-coefficients. The study of these maps is motivated by the connection with the Baum-Connes conjecture on the $K$-theory of group $C^{*}$-algebras.