Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

Continuous weak selections for products

A weak selection on an infinite set X   is a function σ:[X]²→X$\sigma :{[X]}^2\to X$ such that σ({x,y})∈{x,y}$\sigma (\{x,y\})\in \{x,y\}$ for each {x,y}∈[X]²$\{x,y\}\in {[X]}^2$. A weak selection on a space is said to be continuous if it is a continuous function with respect to the Vietoris topology on [X]²${[X]}^2$ and the topology on X  . We study some topological consequences from the existence of a continuous weak selection on the product X×Y$X\times Y$ for the following particular cases:

(i)

Both X and Y are spaces with one non-isolated point.     

Certain strict topologies on the space of regular Borel measures on locally compact groups

Let G   denote a locally compact Hausdorff group and M(G)$M(G)$ be the space of all bounded complex-valued regular Borel measures on G  . In this paper, we define two strict topologies on M(G)$M(G)$ and study various properties of these topologies such as metrizability, barrelledness and completeness. We also determine the dual space of M(G)$M(G)$ and consider various continuity properties for the convolution product on M(G)$M(G)$ under these topologies.     

Structured total least norm and approximate GCDs of inexact polynomials

The determination of an approximate greatest common divisor (GCD) of two inexact polynomials f=f(y)$f=f(y)$ and g=g(y)$g=g(y)$ arises in several applications, including signal processing and control. This approximate GCD can be obtained by computing a structured low rank approximation S^*(f,g)$S^{*}(f,g)$ of the Sylvester resultant matrix S(f,g)$S(f,g)$. In this paper, the method of structured total least norm (STLN) is used to compute a low rank approximation of S(f,g)$S(f,g)$, and it is shown that important issues that have a considerable effect on the approximate GCD have not been considered. For example, the established works only yield one matrix S^*(f,g)$S^{*}(f,g)$, and therefore one approximate GCD, but it is shown in this paper that a family of structured low rank approximations can be computed, each member of which yields a different approximate GCD. Examples that illustrate the importance of these and other issues are presented.     

On diagonal resolvable spheres

In this paper we give the complete answer to a question posed by A. Arhangel?skii and prove that the sphere Sⁿ$S^n$ is diagonal resolvable if and only if Sⁿ$S^n$ is an H  -space if and only if n∈{0,1,3,7}$n\in \{0,1,3,7\}$. Moreover, we prove that any upper half even dimensional Q$\mathbb{Q}$-sphere cannot be diagonal resolvable.     

The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n  -dimensional Euclidean space Rⁿ${\mathbb{R}}^n$, let F(Y)$\mathbb{F}(Y)$ be the family of all fuzzy sets of Rⁿ${\mathbb{R}}^n$ which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)$\mathbb{F}(Y)$ with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?²${?}^2$ if Y   is compact; and the space F(Rⁿ)$\mathbb{F}({\mathbb{R}}^n)$ is also homeomorphic to ?²${?}^2$.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.     

Productivity of paracompactness in the class of GO-spaces

The main result of the paper says that if X is a paracompact GO-space, meaning a subspace of a linearly ordered space and M a paracompact space satisfying the first axiom of countability such that X   can be embedded in M^_ω1$M^{{\omega }_1}$ then the product X×Y$X\times Y$ is paracompact for every paracompact space Y   if and only if the first player of the G(DC,X)$G(DC,X)$ game, introduced by Telgarsky has a winning strategy. In particular we obtain that if X   is paracompact GO-space of weight not greater than _ω1${\omega }_1$ then the product X×Y$X\times Y$ is paracompact for every paracompact space Y   if and only if the first player of the G(DC,X)$G(DC,X)$ game has a winning strategy.     

A note on irreducible,infinite Coxeter groups

A new proof is given for the statement: For an irreducible, infinite Coxeter group (W,S)$(W,S)$ and w∈W$w\in W$, if wSw^-1=S${\mathit{wSw}}^{-1}=S$, then w=1$w=1$ (the identity element of W).     

Completeness,precompactness and compactness in finite-dimensional asymmetrically normed lattices

If (X,‖⋅‖)$(X,\Vert \cdot \Vert )$ is a real normed lattice, then p(x)=‖x⁺‖$p(x)=\Vert x^{+}\Vert$ defines an asymmetric norm on X. We characterise the left-K   sequentially complete, precompact and compact subsets of (R^m,p)$({\mathbb{R}}^m,p)$.     

Generic global rigidity of body–bar frameworks

A basic geometric question is to determine when a given framework G(p)$G(\mathbf{p})$ is globally rigid in Euclidean space R^d${\mathbb{R}}^d$, where G is a finite graph and p is a configuration of points corresponding to the vertices of G  . G(p)$G(\mathbf{p})$ is globally rigid in  R^d${\mathbb{R}}^d$ if for any other configuration q for G   such that the edge lengths of G(q)$G(\mathbf{q})$ are the same as the corresponding edge lengths of G(p)$G(\mathbf{p})$, then p is congruent to q. A framework G(p)$G(\mathbf{p})$ is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.     

Simple geometry facilitates iterative solution of a nonlinear equation via a special transformation to accelerate convergence to third order

Direct substitution _xk+1=g(_xk)$x_{k+1}=g(x_k)$ generally represents iterative techniques for locating a root z   of a nonlinear equation f(x)$f(x)$. At the solution, f(z)=0$f(z)=0$ and g(z)=z$g(z)=z$. Efforts continue worldwide both to improve old iterators and create new ones. This is a study of convergence acceleration by generating secondary solvers through the transformation _gm(x)=(g(x)-m(x)x)/(1-m(x))$g_m(x)=(g(x)-m(x)x)/(1-m(x))$ or, equivalently, through partial substitution _gmps(x)=x+G(x)(g-x)$g_{m\mathrm{ps}}(x)=x+G(x)(g-x)$, G(x)=1/(1-m(x))$G(x)=1/(1-m(x))$. As a matter of fact, _gm(x)≡_gmps(x)$g_m(x)\equiv g_{m\mathrm{ps}}(x)$ is the point of intersection of a linearised g   with the g=x$g=x$ line. Aitken's and Wegstein's accelerators are special cases of _gm$g_m$. Simple geometry suggests that m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ is a good approximation for the ideal slope of the linearised g  . Indeed, this renders a third-order _gm$g_m$. The pertinent asymptotic error constant has been determined. The theoretical background covers a critical review of several partial substitution variants of the well-known Newton's method, including third-order Halley's and Chebyshev's solvers. The new technique is illustrated using first-, second-, and third-order primaries. A flexible algorithm is added to facilitate applications to any solver. The transformed Newton's method is identical to Halley's. The use of m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ thus obviates the requirement for the second derivative of f(x)$f(x)$. Comparison and combination with Halley's and Chebyshev's solvers are provided. Numerical results are from the square root and cube root examples.