Homomorphic actions of ℍ on a compact groupon a compact group

The way the additive semigroup [0, ∞) carrying the usual topology can act as a semigroup of homomorphisms on a compact group is studied in this paper. It is shown that such actions of [0, ∞) on a compact group G can be completely described in terms of continuous homomorphisms from [0, ∞) into G. Furthermore, it is proven that each such action of [0, ∞) on a compact group G has a unique extension to an action of the additive group of all real numbers endowed with the usual topology on G so that a similar characterization is also valid for actions of the real numbers on a compact group.     

Square roots and continuity in strictly linearly ordered semigroups on real intervals

In this article we show that the semigroup operation of a strictly linearly ordered semigroup on a real interval is automatically continuous if each element of the semigroup admits a square root. Hence, by a result of Aczél, such a semigroup is isomorphic to an additive subsemigroup of the real numbers.     

RESULTS ON SUBGROUPS OF THE GROUP OF HOMEOMORPHISMS OF THE RATIONAL NUMBERS

Abstract

If G is the group of homeomorphisms of the rationals with usual topology and H ? G is such that |G:H| < 2^x0 then there exists a finite, non-empty subset Y of the rational numbers such that G(Y) ? H ? G^{Y} where G(Y) is the group o all homeomorphisms in G fixing a neighbourhood of Y and G^{Y} is the set-wise stabilizer of Y in G.     

Gruppentopologien auf nichtabelschen abzählbaren Gruppen

Group topologies of countable groups are constructed. They form a dense subset of the lattice of all group topologies. Every such group topology induces an injective mapping from the set of all filters of the natural numbers, which are finer than the filter of those sets of natural numbers having finite complements, into the lattice of group topologies. A sufficient condition is given, which groups can be topologized. It is shown, that every group can be topologized by a non-linear group topology, if it can be topologized by a group topology.     

One-variable and multi-mariable calculus on a non-Archimedean field extension of the real numbers

New elements of calculus on a complete real closed non-Archimedean field extension F of the real numbers ? will be presented. It is known that the total disconnectedness of F in the topology induced by the order makes the usual (topological) notions of continuity and differentiability too weak to extend real calculus results to F. In this paper, we introduce new stronger concepts of continuity and differentiability which we call derivate continuity and derivate differentiability [2, 12]; andwe show that derivate continuous and differentiable functions satisfy the usual addition, product and composition rules and that n-times derivate differentiable functions satisfy a Taylor formula with remainder similar to that of the real case. Then we generalize the definitions of derivate continuity and derivate differentiability to multivariable F-valued functions and we prove related results that are useful for doing analysis on F and F ⁿ in general.     

The continuum as a formal space

A constructive definition of the continuum based on formal topology is given and its basic properties studied. A natural notion of Cauchy sequence is introduced and Cauchy completeness is proved. Other results include elementary proofs of the Baire and Cantor theorems. From a classical standpoint, formal reals are seen to be equivalent to the usual reals. Lastly, the relation of real numbers as a formal space to other approaches to constructive real numbers is determined. Received: 11 November 1996     

THE NUMBER AND SIZE OF ORBITS OF SOME PERMUTATION GROUPS

Abstract

It is shown that Aut ?, the group of homeomorphisms of the rational numbers with the usual topology, has 2 ^N_o orbits on the power set P(?). We call S ? ? a moiety if S and its complement in ? are infinite. It is shown that the orbit of any moiety S under Aut ? has cardinality 2^N_o while the orbit of S under Aut(?, ≤), the group of order preserving automorphisms of ?, has cardinality N_o if and only if S is a finite union of disjoint rational intervals with rational endpoints.     

Pseudo‐c‐archimedean and pseudo‐finite cyclically ordered groups

Robinson and Zakon gave necessary and sufficient conditions for an abelian ordered group to satisfy the same first‐order sentences as an archimedean abelian ordered group (i.e., which embeds in the group of real numbers). The present paper generalizes their work to obtain similar results for infinite subgroups of the group of unimodular complex numbers. Furthermore, the groups which satisfy the same first‐order sentences as ultraproducts of finite cyclic groups are characterized.     

Une application des nombres de Pisot à l'algorithme de Jacobi-Perron

SinceO. Perron introduced in 1907 Jacobi-Perron algorithm, which is the simplest generalization of continued fractions to finite sets of real numbers, the main question of characterising the periodicity is still open. The usual conjecture is that the development of any basis of a real number field by this algorithm is periodic. But we only know some infinite families for which this is true. In this paper we prove that for any real number field there exists a basis for which we have periodicity.     

Bisection hardly ever converges linearly

Summary. For real functions that cross the unit interval, the method of bisection converges linearly if, but only if, the point of crossing is a diadic number where the function does not vanish, or, except for finitely many digits, its binary expansion coincides with that of one third or two thirds. Otherwise, the order of convergence remains undefined. If the point of crossing is one of Borel's normal real numbers (Lebesgue's measure of all of which equals one), then the sequence of ratios of two consecutive errors accumulates simultaneously at zero, one half, and negative infinity. Thus, in every finite sequence of estimates from the bisection, the last estimate need not be more accurate than the first one.     

The Structure of Extended Real-valued Metric Spaces

An extended metric on a set X is a distance function that satisfies the usual properties of a metric except that it can assume values of infinity, in addition to nonnegative real values. Given a metrizable space we exhibit a universal space for all extended metric spaces compatible with the topology. Defining a set in an extended metric space to be bounded if it is contained in a finite union of open balls, we characterize those bornologies on X that can be realized as bornologies of metrically bounded sets. We also consider a second possible definition of bounded set in this setting.     

From image processing to topological modelling with <Emphasis Type="Italic">p</Emphasis>-adic numbers

Encoding the hierarchical structure of images by p-adic numbers allows for image processing and computer vision methods motivated from arithmetic physics. The p-adic Polyakov action leads to the p-adic diffusion equation in low level vision. Hierarchical segmentation provides another way of p-adic encoding. Then a topology on that finite set of p-adic numbers yields a hierarchy of topological models underlying the image. In the case of chain complexes, the chain maps yield conditions for the existence of a hierarchy, and these can be expressed in terms of p-adic integrals. Such a chain complex hierarchy is a special case of a persistence complex from computational topology, where it is used for computing persistence barcodes for shapes. The approach is motivated by the observation that using p-adic numbers often leads to more efficient algorithms than their real or complex counterparts.     

Compactness of the product topology and the solvability of infinite systems of infinite linear equations

If every finite subsystem of an infinite system of linear equations (say, over the field of real numbers) each with finitely many unknowns has a solution then the entire system has a solution. The situation is not so if the equations contain infinitely many unknowns. In this case, as shown below, the solvability of every finite subsystem implies the solva. bility of the entire system provided finite subsystems have solution with common upper and lower bounds and the coefficients of ever equation satisfy some boundedness or convergence conditions. The passage from the solvability of finite subsystem to the solvability of the entire system is achieved based on Tychnoff’s theorem stating that any product of compact topological spaces is compact in their product topology.     

On the variety of Riesz spaces

Finitely generated linearly ordered Riesz spaces are described, leading to a proof that the variety of Riesz spaces is generated as a quasivariety by the Riesz space ? of real numbers. The finitely generated Riesz spaces are also described: they are the subalgebras of real-valued function spaces on root systems of finite height.     

Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation

For each real number , let denote the set of real numbers with exact order . A theorem of Güting states that for the Hausdorff dimension of is equal to . In this note we introduce the notion of exact t–logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes Güting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem. Received: 12 December 2000 / Published online: 25 June 2001     

Certain density theorems applied to the embeddability of iteration semigroups

Aequationes mathematicae - Let A be an additive semigroup of real numbers the additive group generated by which is non-cyclic. Let $$I=(a,b)$$ be an open interval and $$\mathcal {A}=\left\{...     

Generalized power series on a non-Archimedean field

Power series with rational exponents on the real numbers field and the Levi-Civita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we generalize that result and study power series with rational exponents on the Levi-Civita field. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asserts strong (order) convergence for points whose distance from the center is infinitely smaller than the radius of convergence. Then we study a class of functions that are given locally by power series with rational exponents, which are shown to form a commutative algebra over the Levi-Civita field; and we study the differentiability properties of such functions within their domain of convergence.     

Group topologies on the real line

It is shown that there exist 2^2c non-isomorphic group topologies on the real line, among them 2^2c are non-locally compact, ?₀ are compact (connected), ?₀ are locally compact and compactly generated. We also give some characterizations of the usual topology among group topologies on the real line.     

On Haar measurable additive maps on Β(N)

Let Β(N) be the power set of the set of natural numbers endowed with the usual structure of a compact abelian group and let Μ be the normed Haar measure on it. The paper studies the properties of Μ-measurable additive maps of Β(N) into Hausdorff abelian groups.