It is known that any locally compact group that is approximable by finite groups must be unimodular. However, this condition is not sufficient. For example, simple Lie groups are not approximable by finite ones. In this paper we consider the approximation of locally compact groups by more general finite algebraic systems. We prove that a locally compact group is approximable by finite semigroups iff it is approximable by finite groups. Thus, there exist some locally compact groups and even some compact groups that are not approximable by finite semigroups. We prove also that whenever a locally compact group is approximable by finite quasigroups (latin squares) it is unimodular. The converse theorem is also true: any unimodular group is approximable by finite quasigroups and even by finite loops. In this paper we prove this theorem only for discrete groups. For the case of non-discrete groups the proof is rather long and complicated and is given in a separate paper.

Recall that a locally compact group $G$ is called unimodular if the left Haar measure on $G$ is equal to the right one. It is shown that $G$ is unimodular iff it is approximable by finite quasigroups (Latin squares).

We demonstrate that a certain class of problems in linear arithmetic combinatorics is equivalent to a problem in the representation theory of finite groups which relates the sizes of various stabilizer groups.

This paper is concerned with linear partial differential operators with constant coefficients in $H^p(\mathbf{R} ^n)$. In the case $0 \lt p\le1$, we establish some basic properties and the spectral mapping property, and determine completely the essential spectrum, point spectrum, approximate point spectrum, continuous spectrum, and residual spectrum of such differential operators. In the case $p \gt 2$, we show that the point spectrum of such differential operators in $L^p(\mathbf{R} ^n)$ is the empty set for $p\in(2,{2n\over n-1})$, but not for $p \gt {2n\over n-1}$ in general. Moreover, we make some remarks on the case $p \gt 1$ and give several examples.

It is shown that spacelike surfaces of constant mean curvature $\pm 1$ (abbreviated as CMC $\pm 1$) in de Sitter $3$-space ${\mathbb S}^3_1(1)$ can be constructed from holomorphic curves in ${\mathbb P}\mathrm{SL} (2;{\mathbb C})=\mathrm{SL} (2;{\mathbb C})/\{\pm\mathrm{id} \}$ via a Bryant type representation formula. This Bryant type representation formula is used to investigate an explicit one-to-one correspondence, the so-called {Lawson correspondence}, between spacelike CMC $\pm 1$ surfaces in de Sitter $3$-space ${\mathbb S}^3_1(1)$ and spacelike maximal surfaces in Lorentz $3$-space ${\mathbb E}^3_1$. The hyperbolic Gauss map of spacelike surfaces in ${\mathbb S}^3_1(1)$, which is a close analogue of the classical Gauss map, is considered. It is shown that the hyperbolic Gauss map plays an important role in the study of spacelike CMC $\pm 1$ surfaces in ${\mathbb S}^3_1(1)$. In particular, the relationship between the holomorphicity of the hyperbolic Gauss map and spacelike CMC $\pm 1$ surfaces in ${\mathbb S}^3_1(1)$ is studied.

In this paper we develop the theory of covers for locally definable groups in o-minimal structures.

Absolute-valuable Banach spaces are introduced as those Banach spaces which underlie complete absolute-valued algebras. Examples and counterexamples are given. It is proved that every Banach space can be isometrically enlarged to an absolute-valuable Banach space, which has the same density character as the given Banach space, and whose dual space is also absolute-valuable. It is also shown that every weakly countably determined Banach space different from $\mathbb{R}$ can be renormed in such a way that neither it nor its dual are absolute-valuable. Hereditarily indecomposable Banach spaces are examples of Banach spaces which cannot be renormed as absolute-valuable Banach spaces.

We give a characterisation of the maximal Abelian sub-algebras of the bounded operators on a Hilbert space that are normalised by the unitary representations of the group of unimodular two by two matrices.

Let $(X_i,{\mathscr B}_i,m_i)$ $(i\in{\mathbb N})$ be a sequence of regular Borel measure spaces. There is a Borel measure $\mu$ on $\prod_{i\in{\mathbb N}}{X_i}$ such that if $K_i\subseteq X_i$ is compact for all $i\in{\mathbb N}$ and $\prod_{i\in{\mathbb N}}{m_i(K_i)}$ converges, then $\mu({\prod_{i\in{\mathbb N}}{K_i}})=\prod_{i\in{\mathbb N}}{m_i(K_i)}$.

Given a codimension-one foliation $\mathcal F$ of a closed manifold $M$ and a vector field $X$ on $M$, we show that if $X$ is transverse to $\mathcal F$, then there are many functions $f$ on $M$ so that $fX$ is the mean curvature vector of $\mathcal F$ with respect to some Riemannian metric on $M$. Further we give a necessary and sufficient condition for $X$ to become the mean curvature vector of $\mathcal F$ with respect to some Riemannian metric on $M$.

Let $K$ be a non-archimedean field with residue field $k$, and suppose that $k$ is not an algebraic extension of a finite field. We prove two results concerning wandering domains of rational functions $\phi\in K(z)$ and Rivera-Letelier's notion of nontrivial reduction. First, if $\phi$ has nontrivial reduction, then assuming some simple hypotheses, we show that the Fatou set of $\phi$ has wandering components by any of the usual definitions of ``components of the Fatou set''. Second, we show that if $k$ has characteristic zero and $K$ is discretely valued, then the existence of a wandering domain implies that some iterate has nontrivial reduction in some coordinate.

Under some regularity assumptions, we compute the modulus of continuity of the generalized Mazur map between unit balls of Orlicz spaces. Our estimate coincides with the known estimates in the setting of $L_p(\mu)$-spaces. We apply this estimate to approximate uniformly continuous mappings between balls of reflexive Orlicz spaces by $\alpha$-Hölder maps, with $\alpha$ as large as possible. We also relate this optimal value of $\alpha$ to the Boyd indices of the spaces and to the problem of isomorphic extension of Hölder maps.

We study isometries between spaces of weighted holomorphic functions defined on bounded domains in $\mathbf{C}^n$. Using the Bergman kernel we see that it is possible to define a `natural' weight on bounded domains in $\mathbf{C}^n$. We calculate the isometries of weighted spaces of holomorphic functions on the unit ball, the Thullen domains, the generalised Thullen domains and the domain with minimal complex norm.

This paper establishes a defect relation for linearly nondegenerate meromorphic mappings from parabolic manifolds into the projective space intersecting hypersurfaces, extending a result of H. Cartan result and an earlier result of Min Ru.

We give a proof of the fact that the upper and the lower sectional curvature bounds of a complete manifold vary at a bounded rate under the Ricci flow.

Let $I = (F_1,\ldots,F_r)$ be a homogeneous ideal of the ring $R = k[x_0,\ldots,x_n]$ generated by a regular sequence of type $(d_1,\ldots,d_r)$. We give an elementary proof for an explicit description of the graded Betti numbers of $I^s$ for any $s \geq 1$. These numbers depend only upon the type and $s$. We then use this description to: (1) write $H_{R/I^s}$, the Hilbert function of $R/I^s$, in terms of $H_{R/I}$; (2) verify that the $k$-algebra $R/I^s$ satisfies a conjecture of Herzog-Huneke-Srinivasan; and (3) obtain information about the numerical invariants associated to sets of fat points in $\mathbb{P}^n$ whose support is a complete intersection or a complete intersection minus a point.

We answer a question of Pillay and Ziegler and construct a type of $U$-rank 1 (of the theory of separably closed fields) which is thin but not very thin.

Using the theory of infinite iterated function systems, we show that the Julia set of any function of the type $G=\lambda\exp \circ F$, $\lambda \in \mathbb{C} \setminus \{0\}$, with $F:\mathbb{C} \to \hat{\mathbb{C}}$ a non-constant elliptic function, has Hausdorff dimension two. However, there exist elliptic functions $F$ such that the Julia sets of the maps $G=\exp \circ F$ are nowhere dense in $\mathbb{C}$.

Although previous research on CR extension has emphasized the concept of wedge extendability, wedges do not have some of the properties we expect of the regions described by a general theory. In particular, the regions we describe should be of roughly the same size and shape as the full regions of extendability, they should vary smoothly as one varies the base point or the size of the open neighborhood on the manifold, and they should satisfy a natural containment condition. We illustrate through an example the failure of wedges to satisfy these conditions. We then develop an alternative description of the sort outlined above for the class of tube-like CR submanifolds of $\mathbb{C}^n$ of CR dimension 1.