Plasticity in metric spaces

In this paper we examine the properties of EC-plastic metric spaces, spaces which have the property that any noncontractive bijection from the space onto itself must be an isometry.     

Realization of constructive set theory into explicit mathematics: a lower bound for impredicative Mahlo universe

We define a realizability interpretation of Aczel's Constructive Set Theory CZF into Explicit Mathematics. The final results are that CZF extended by Mahlo principles is realizable in corresponding extensions of T₀, thus providing relative lower bounds for the proof-theoretic strength of the latter.     

Dirichlet空间上Toeplitz算子的紧性

本文给出了 Ｄｉｒｉｃｈｌｅｔ空间上 Ｔｏｅｌｐｉｔｚ算子为紧算子的充要条件,并证明具有 Ｃ１－符号的 Ｔｏｅｐｌｉｔｚ算子为紧算子当且仅当它为零算子,当且仅当符号的边值为零．     

Compactness of Product Operators on the Bergman Space

In this paper, we study the compactness of the product of a composition operator with another one＇s adjoint on the Bergman space. Some necessary and sufficient conditions for such operators to be compact are given.     

分形空间的局部紧性及网极限

对于完备度量空间 (X ,d) ,研究了X的局部紧性与相应分形空间 (H(X) ,h)的局部紧性之间的关系 ,得到结论 :(H(X) ,h)是局部紧的当且仅当X是局部紧的 .另一方面 ,给出了 (H(X) ,h)中收敛网的极限通过并、交及闭包运算的表示 .     

Finitary sequence spaces

This paper studies the metric structure of the space H_r of absolutely summable sequences of real numbers with at most r nonzero terms. H_r is complete, and is located and nowhere dense in the space of all absolutely summable sequences. Totally bounded and compact subspaces of H_r are characterized, and large classes of located, totally bounded, compact, and locally compact subspaces are constructed. The methods used are constructive in the strict sense. MSC: 03F65, 54E50.     

The swap of integral and limit in constructive mathematics

Integration within constructive, especially intuitionistic mathematics in the sense of L. E. J. Brouwer, slightly differs from formal integration theories: Some classical results, especially Lebesgue's dominated convergence theorem, have tobe substituted by appropriate alternatives. Although there exist sophisticated, but rather laborious proposals, e.g. by E. Bishop and D. S. Bridges (cf. [2]), the reference to partitions and the Riemann‐integral, also with regard to the results obtained by R. Henstock and J. Kurzweil (cf. [9], [12]), seems to give a better direction. Especially, convergence theorems can be proved by introducing the concept of “equi‐integrability”. The paper is strongly motivated by Brouwer's result that each function fully defined on a compact interval has necessarily to be uniformly continuous. Nevertheless, there are, with only one exception (a corollary of Theorem 4.2), no references to the fan‐theorem or to bar‐induction. Therefore, the whole paper can be read within the setting of Bishop's access to constructive mathematics. Nothing of genuine full‐fledged Brouwerian intuitionism is used for the main results in this note (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compact quantum metric spaces and ergodic actions of compact quantum groups

We show that for any co-amenable compact quantum group A=C(G) there exists a unique compact Hausdorff topology on the set EA(G) of isomorphism classes of ergodic actions of G such that the following holds: for any continuous field of ergodic actions of G over a locally compact Hausdorff space T the map T→EA(G) sending each t in T to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of γ in each fibre is continuous over T for every equivalence class γ of irreducible unitary representations of G. Generalizations for arbitrary compact quantum groups are also obtained. In the case G is a compact group, the restriction of this topology on the subset of isomorphism classes of ergodic actions of full multiplicity coincides with the topology coming from the work of Landstad and Wassermann. Podle? spheres are shown to be continuous in the natural parameter as ergodic actions of the quantum SU(2) group. We also introduce a notion of regularity for quantum metrics on G, and show how to construct a quantum metric from any ergodic action of G, starting from a regular quantum metric on G. Furthermore, we introduce a quantum Gromov-Hausdorff distance between ergodic actions of G when G is separable and show that it induces the above topology.     

Approach Theory in a Category: A Study of Compactness and Hausdorff Separation

In a topological construct endowed with a proper -factorization system and a concrete functor , we study -compactness and -Hausdorff separation, where is a class of “closed morphisms” in the sense of Clementino et al. (A functional approach to general topology. In: Categorical Foundations. Encyclopedia of Mathematics and Its Applications, vol. 97, pp. 103–163. Cambridge University Press, Cambridge, 2004), determined by Λ. In particular, we point out under which conditions on Λ, the notion of -compactness of an object of coincides with 0-compactness of the image in Prap. Our results will be illustrated by some examples: except for some well-known ones, like b-compactness of a topological space, we also capture some compactness notions that were not considered before in the literature. In particular, we obtain a generalization of b-compactness to the setting of approach spaces. This notion is shown to play an important role in the study of uniformizability. The author is research assistant at the Fund of Scientific Research Vlaanderen (FWO).     

Henselian valued fields: a constructive point of view

This article is a logical continuation of the Henri Lombardi and Franz‐Viktor Kuhlmann article [9]. We address some classical points of the theory of valued fields with an elementary and constructive point of view. We deal with Krull valuations, and not simply discrete valuations. First of all, we show how (in the spirit of [9]) to construct the Henselization of a valued field; we restrict to fields in which one has at one's disposal algorithmic tools to test the nullity or the valuation ring membership. It is therefore a work that differs as much in spirit as in field of application from that of Mines, Richman and Bridges (cf. [10]), who address the framework of Heyting fields and discrete valuation. We show then in a constructive way a batch of classical results in Henselian fields, notably factorization criteria and Krasner's Lemma. We conclude by a construction of the inertia field of a valued field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A characterization of compactly generated metric groups

Recall that a topological group is: (a) -compact if where each is compact, and (b) compactly generated if is algebraically generated by some compact subset of . Compactly generated groups are -compact, but the converse is not true: every countable non-finitely generated discrete group (for example, the group of rational numbers or the free (Abelian) group with a countable infinite set of generators) is a counterexample. We prove that a metric group is compactly generated if and only if is -compact and for every open subgroup of there exists a finite set such that algebraically generates . As a corollary, we obtain that a -compact metric group is compactly generated provided that one of the following conditions holds: (i) has no proper open subgroups, (ii) is dense in some connected group (in particular, if is connected itself), (iii) is totally bounded (= subgroup of a compact group). Our second major result states that a countable metric group is compactly generated if and only if it can be generated by a sequence converging to its identity element (eventually constant sequences are not excluded here). Therefore, a countable metric group can be generated by a (possibly eventually constant) sequence converging to its identity element in each of the cases (i), (ii) and (iii) above. Examples demonstrating that various conditions cannot be omitted or relaxed are constructed. In particular, we exhibit a countable totally bounded group which is not compactly generated.

     

ON THE LEAST DOMINANT CONTINUOUS MODULUS AND ITS APPLICATION

o.IntroductionInthispaperwedealwithqualltitativeKorovkintypetheoremsfortheaPproalmationbyboundedlinearoperatorsdefinedonC(X),andinparticularbypositiveones.HereC(X)=CR(X,d)denotestheBanachlatticeofreal-valuedcontinuousfunctionsdefinedonthecompactmetricspace(X,d)withnormgivenbylIfIlx=ma-xlf(x)I,xEX.WealsoassumethatXhasdiameterd(X)>o.ThefirstsuchtheoremforgeneralpositivelinearoperatorsandX=[a,b]equippedwiththeeuclidiandistanceisduetoR.Mamedovl4].Forspaces(X,d)beingmetricallyconvexinthes…     

Generic properties of compact metric spaces

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose elements enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.     

On a problem of iteration invariants for distributional chaos

We show that if f is a DC3 continuous map of a compact metric space then also f^N is DC3, for every N > 0. This solves a problem given by [Li R. A note on the three versions of distributional chaos. Commun Nonlinear Sci Numer Simulat 2011;16:1993-1997].     

Metric aspects of noncommutative homogeneous spaces

For a closed cocompact subgroup Γ of a locally compact group G, given a compact abelian subgroup K of G and a homomorphism satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations of the homogeneous space G/Γ, generalizing Rieffel's construction of quantum Heisenberg manifolds. We show that when G is a Lie group and G/Γ is connected, given any norm on the Lie algebra of G, the seminorm on induced by the derivation map of the canonical G-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on ρ continuously, with respect to quantum Gromov-Hausdorff distances.     

Étude constructive de problèmes de topologie pour les réels irrationnels

We study in a constructive manner some problems of topology related to the set Irr of irrational reals. The constructive approach requires a strong notion of an irrational number; constructively, a real number is irrational if it is clearly different from any rational number. We show that the set Irr is one-to-one with the set Dfc of infinite developments in continued fraction (dfc). We define two extensions of Irr, one, called Dfc₁, is the set of dfc of rationals and irrationals preserving for each rational one dfc, the other, called Dfc₂, is the set of dfc of rationals and irrationals preserving for each rational its two dfc. We introduce six natural distances over Irr wich we denote by d_fc0, d_fc1, d_fc2, d, d_mir and d_cut. We show that only the four distances d_fco, d_fc1, d and d_mir among the six make Irr a complete metric space. The last distances define in Irr the same topology in a constructive sens. We study further the set Dfc₁ in which we show that the irrationals constitute a closed subset. Finally, we make a particular study of the completion Dfc₂ of Dfc for the two equivalent metrics d_fc2 and d_cut.     

Lazy bases: a minimalist constructive theory of Noetherian rings

We give a constructive treatment of the theory of Noetherian rings. We avoid the usual restriction to coherent rings; we can even deal with non‐discrete rings. We introduce the concept of rings with certifiable equality which covers discrete rings and much more. A ring R with certifiable equality can be fitted with a partial ideal membership test for ideals of R. Lazy bases of ideals of R [X ] are introduced in order to derive a partial ideal membership test for ideals of R [X ]. It is then proved that if R is Noetherian, then so is R [X ]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compact Metric Spaces and Weak Forms of the Axiom of Choice

It is shown that for compact metric spaces (X, d) the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{G_n : n ∈ ω}, ∣G_n∣ < ω, with lim_n→∞ diam (G _n) = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF⁰ , the Zermelo‐Fraenkel set theory without the axiom of regularity, and that the countable axiom of choice for families of finite sets CAC_fin does not imply the statement “Compact metric spaces are separable”.