On the stability of some classical operators from approximation theory

We obtain some results on Hyers–Ulam stability for some classical operators from approximation theory. For Bernstein operators we determine the Hyers–Ulam constant using a result concerning coefficient bounds of Lorentz representation for a polynomial.     

On the moduli spaces of multipolygonal linkages in the plane

One can form a polygonal linkage by identifying initial and terminal points of two free linkages. Likewise, one can form a multipolygonal linkage by identifying initial and terminal points of three free linkages. The geometric and topological properties of moduli spaces of multipolygonal linkages in the plane are studied. These spaces are compact algebraic varieties. Some conditions under which these spaces are smooth manifolds, cross products or disjoint unions of moduli spaces of polygonal linkages, or connected, are determined. Dimensions in smooth manifold cases and some Euler characteristics are computed. A classification of generic multiquadrilateral linkages is also made.     

The embeddability ordering of topological spaces

For K a set of topological spaces and X,Y∈K, the notation Xh_⊆Y means that X embeds homeomorphically into Y; and X∼Y means Xh_⊆Yh_⊆X. With , the equivalence relation ∼ on K induces a partial order h_? well-defined on K/∼ as follows: if Xh_⊆Y.For posets (P,P_?) and (Q,Q_?), the notation (P,P_?)?(Q,Q_?) means: there is an injection such that p₀P_?p₁ in P if and only if h(p₀)Q_?h(p₁) in Q. For κ an infinite cardinal, a poset (Q,Q_?) is a κ-universal poset if every poset (P,P_?) with |P|?κ satisfies (P,P_?)?(Q,Q_?).The authors prove two theorems which improve and extend results from the extensive relevant literature.

Theorem 2.2. There is a zero-dimensional Hausdorff space S with|S|=κsuch that(P(S)/∼,h_?)is a κ-universal poset.     

Compact factors in finally compact products of topological spaces

We present instances of the following phenomenon: if a product of topological spaces satisfies some given compactness property then the factors satisfy a stronger compactness property, except possibly for a small number of factors.The first known result of this kind, a consequence of a theorem by A.H. Stone, asserts that if a product is regular and Lindelöf then all but at most countably many factors are compact. We generalize this result to various forms of final compactness, and extend it to two-cardinal compactness. In addition, our results need no separation axiom.     

On topological spaces of singular density and minimal weight

We introduce a weakening of the generalized continuum hypothesis, which we will refer to as the prevalent singular cardinals hypothesis, and show it implies that every topological space of density and weight ℵω₁ is not hereditarily Lindelöf.The assumption PSH is very weak, and in fact holds in all currently known models of ZFC.     

Note on initial topologies on rational vector spaces induced by realvalued linear mappings

LetG be a vector space over the field of rational numbers andf, g:G -linear mappings. equipped with the usual norm topology. Denote by _f , _g the initial topologies onG induced byf respectivelyg.Then the following result holds: If there is a nonvoid open setU whose complement contains at least one inner point such thatf ^–1 U _g , then there is ac withf=cg. In particular, iff0, the topologies coincide.Furthermore, a -linear mappingh: (G, _f )(G, _g ) is continuous if and only if there is a real constantc withg ^o h=cf.Dedicated to Professor János Aczél on his 60th birthday     

On some estimates and Carleson type measure for multifunctional holomorphic spaces in the unit ball

The aim of this paper is to obtain some new estimates for multifunctional holomorphic expressions by using properties of Bergman metric ball. Also we obtain some characterizations of Carleson type measure for some multifunctional holomorphic spaces defined with Bergman metric ball.     

On the stability of the translation equation and dynamical systems

In this paper we prove the stability of the functional equation F(s,F(t,x))=F(s+t,x) in the class of functions F:R×I→I, which are continuous with respect to each variable, and where I⊂R is a real interval. We also discuss the stability in the sense of Hyers-Ulam of dynamical systems on I. We show some properties of δ-approximate solutions of the translation equation on a real interval.     

Preorderable topologies and order-representability of topological spaces

We furnish characterizations of topologies that coincide with the lower topology or with the order topology of some total preorder defined on a set. Leaning on these characterizations we introduce some applications to the study of continuous and semicontinuous order-representability properties of topological spaces.     

Linear recurrence relations for binomial coefficients modulo a prime

We investigate when the sequence of binomial coefficients modulo a prime p, for a fixed positive integer k, satisfies a linear recurrence relation of (positive) degree h in the finite range 0?i?k. In particular, we prove that this cannot occur if 2h?k<p−h. This hypothesis can be weakened to 2h?k<p if we assume, in addition, that the characteristic polynomial of the relation does not have −1 as a root. We apply our results to recover a known bound for the number of points of a Fermat curve over a finite field.     

On the Steinhaus property in topological groups

Let G be a locally compact Abelian group and μ a Haar measure on G. We prove: (a) If G is connected, then the complement of a union of finitely many translates of subgroups of G with infinite index is μ-thick and everywhere of second category. (b) Under a simple (and fairly general) assumption on G, for every cardinal number m such that ℵ₀?m?|G| there is a subgroup of G of index m that is μ-thick and everywhere of second category. These results extend theorems by Muthuvel and Erd?s-Marcus, respectively. (b) also implies a recent theorem by Comfort-Raczkowski-Trigos stating that every nondiscrete compact Abelian group G admits ²|G|-many μ-nonmeasurable dense subgroups.     

CLP-compactness for topological spaces and groups

We study CLP-compact spaces (every cover consisting of clopen sets has a finite subcover) and CLP-compact topological groups. In particular, we extend a theorem on CLP-compactness of products from [J. Steprāns, A. Šostak, Restricted compactness properties and their preservation under products, Topology Appl. 101 (3) (2000) 213-229] and we offer various criteria for CLP-compactness for spaces and topological groups, that work particularly well for precompact groups. This allows us to show that arbitrary products of CLP-compact pseudocompact groups are CLP-compact. For every natural n we construct:

(i)

a totally disconnected, n-dimensional, pseudocompact CLP-compact group; and

(ii)

a hereditarily disconnected, n-dimensional, totally minimal, CLP-compact group that can be chosen to be either separable metrizable or pseudocompact (a Hausdorff group G is totally minimal when all continuous surjective homomorphisms G→H, with a Hausdorff group H, are open).

     

On topological entropy of transitive triangular maps

In the paper of Alsedà, Kolyada, Llibre and Snoha [L. Alsedà, S.F. Kolyada, J. Llibre, L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999) 1551-1573] there was—among others—proved that a nonminimal continuous transitive map f of a compact metric space (X,ρ) can be extended to a triangular map F on X×I (i.e., f is the base for F) in such a way that F is transitive and has the same entropy as f. The presented paper shows that under certain conditions the extension of minimal maps is guaranteed, too: Let (X,f) be a solenoidal dynamical system. Then there exist a transitive triangular map F such that h(F)=h(f).     

Separable subspaces of affine function spaces on convex compact sets

Let K be a compact convex subset of a separated locally convex space (over R) and let Ap(K) denote the space of all continuous real-valued affine mappings defined on K, endowed with the topology of pointwise convergence on the extreme points of K. In this paper we shall examine some topological properties of Ap(K). For example, we shall consider when Ap(K) is monolithic and when separable compact subsets of Ap(K) are metrizable.     

Very I-favorable spaces

We prove that a Hausdorff space X is very I-favorable if and only if X is the almost limit space of a σ-complete inverse system consisting of (not necessarily Hausdorff) second countable spaces and surjective d-open bonding maps. It is also shown that the class of Tychonoff very I-favorable spaces with respect to the co-zero sets coincides with the d-openly generated spaces.     

On fundamental groups of quotient spaces

In classical covering space theory, a covering map induces an injection of fundamental groups. This paper reveals a dual property for certain quotient maps having connected fibers, with applications to orbit spaces of vector fields and leaf spaces in general.     

On the resolvent condition in the Kreiss Matrix Theorem

The Kreiss Matrix Theorem asserts the uniform equivalence over allN ×N matrices of power boundedness and a certain resolvent estimate. We show that the ratio of the constants in these two conditions grows linearly withN, and we obtain the optimal proportionality factor up to a factor of 2. Analogous results are also given for the related problem involving matrix exponentialse ^At . The proofs make use of a lemma that may be of independent interest, which bounds the arc length of the image of a circle in the complex plane under a rational function.Dedicated to Germund Dahlquist on the occasion of his sixtieth birthday.Research supported by NSF Mathematical Sciences Postdoctoral Fellowships, by the Courant Institute of Mathematical Sciences, and by the National Aeronautics and Space Administration under Contract No. NAS1-17070 while the authors were in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23665.     

Compact Hausdorff fuzzy topological spaces are topological

Many examples of compact fuzzy topological spaces which are highly non topological are known [5, 6]. Equally many examples of Hausdorff fuzzy topological spaces which are highly non topological can be given. In this paper we show that the two properties - compact and Hausdorff - combined however necessarily imply that the fuzzy topological space is topological. This at once solves some open questions with regard to the compactification of fuzzy topological spaces [8]. It also emphasizes once more the particular role played by compact Hausdorff topological spaces not only in the category of topological spaces but even in the category of fuzzy topological spaces.