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)$F_n(X)$.     

The inverse inertia problem for the complements of partial k-trees

Let F$\mathbb{F}$ be an infinite field with characteristic not equal to two. For a graph G=(V,E)$G=(V,E)$ with V={1,…,n}$V=\{1,\dots ,n\}$, let S(G;F)$S(G;\mathbb{F})$ be the set of all symmetric n×n$n\times n$ matrices A=[_ai,j]$A=[a_{i,j}]$ over F$\mathbb{F}$ with _ai,j≠0$a_{i,j}\ne 0$, i≠j$i\ne j$ if and only if ij∈E$ij\in E$. We show that if G is the complement of a partial k  -tree and m?k+2$m?k+2$, then for all nonsingular symmetric m×m$m\times m$ matrices K   over F$\mathbb{F}$, there exists an m×n$m\times n$ matrix U   such that U^TKU∈S(G;F)$U^{T}KU\in S(G;\mathbb{F})$. As a corollary we obtain that, if k+2?m?n$k+2?m?n$ and G is the complement of a partial k-tree, then for any two nonnegative integers p and q   with p+q=m$p+q=m$, there exists a matrix in S(G;R)$S(G;\mathbb{R})$ with p positive and q negative eigenvalues.     

A new approach to fixed point theorems on G-metric spaces

In this paper, we introduce the metric d^G$d^G$ on a G  -metric space (X,G)$(X,G)$ and use this notion to show that many contraction conditions for maps on the G  -metric space (X,G)$(X,G)$ reduce to certain contraction conditions for maps on the metric space (X,d^G)$(X,d^G)$. As applications, the proofs of many fixed point theorems for maps on the G  -metric space (X,G)$(X,G)$ may be simplified, and many fixed point theorems for maps on the G  -metric space (X,G)$(X,G)$ are direct consequences of preceding results for maps on the metric space (X,d^G)$(X,d^G)$.     

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.     

Triangularizing matrix polynomials

For an algebraically closed field F$\mathbb{F}$, we show that any matrix polynomial P(λ)∈F[λ]^n×m$P(\lambda )\in \mathbb{F}{[\lambda ]}^{n\times m}$, n?m$n?m$, can be reduced to triangular form, preserving the degree and the finite and infinite elementary divisors. We also characterize the real matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal blocks of sizes 1×1$1\times 1$ and 2×2$2\times 2$. The proofs we present solve the structured inverse problem of building up triangular matrix polynomials starting from lists of elementary divisors.     

Linear systems on a class of anticanonical rational threefolds

Let X be the blow-up of the three dimensional complex projective space along r   points in very general position on a smooth elliptic quartic curve B⊂P³$B\subset {\mathbb{P}}^3$ and let L∈Pic(X)$L\in \mathrm{Pic}(X)$ be any line bundle. The aim of this paper is to provide an explicit algorithm for determining the dimension of H⁰(X,L)$H^0(X,L)$.     

On characterized subgroups of compact abelian groups

Let X be a compact abelian group. A subgroup H of X   is called characterized if there exists a sequence u=(_un)$\mathbf{u}=(u_n)$ of characters of X   such that H=_su(X)$H=s_{\mathbf{u}}(X)$, where _su(X):={x∈X:(_un,x)→0 in T}$s_{\mathbf{u}}(X):=\{x\in X:(u_n,x)\to 0\text{in}\mathbb{T}\}$. Every characterized subgroup is an _Fσδ$F_{\sigma \delta }$-subgroup of X  . We show that every _Gδ$G_{\delta }$-subgroup of X is characterized. On the other hand, X   has non-characterized _Fσ$F_{\sigma }$-subgroups.     

Representation theorems for Banach algebras

For a space X   denote by _Cb(X)$C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on X   and denote by _C0(X)$C_0(X)$ the set of all elements in _Cb(X)$C_b(X)$ which vanish at infinity.     

Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

A quasiplane f(V)$f(V)$ is the image of an n-dimensional Euclidean subspace V   of R^N${\mathbb{R}}^N$ (1≤n≤N−1$1\le n\le N-1$) under a quasiconformal map f:R^N→R^N$f:{\mathbb{R}}^N\to {\mathbb{R}}^N$. We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz n  -manifold and for a quasiplane to have big pieces of bi-Lipschitz images of Rⁿ${\mathbb{R}}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension N−n$N-n$. To establish the big pieces criterion, we prove new extension theorems for “almost affine” maps, which are of independent interest. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion.     

On affine motions and universal rigidity of tensegrity frameworks

Recently, Alfakih and Ye (2013) [4] proved that if an r  -dimensional bar framework (G,p)$(G,p)$ on n?r+2$n?r+2$ nodes in general position in R^r${\mathbb{R}}^r$ admits a positive semidefinite stress matrix with rank n−r−1$n-r-1$, then (G,p)$(G,p)$ is universally rigid. In this paper, we generalize this result in two directions. First, we extend this result to tensegrity frameworks. Second, we replace the general position assumption by the weaker assumption that in configuration p, each point and its neighbors in G   affinely span R^r${\mathbb{R}}^r$.     

Best quadrature formula on Sobolev class with Chebyshev weight

Using best interpolation function based on a given function information, we present a best quadrature rule of function on Sobolev class KW^r[-1,1]${\mathit{KW}}^r[-1,1]$ with Chebyshev weight. The given function information means that the values of a function f∈KW^r[-1,1]$f\in {\mathit{KW}}^r[-1,1]$ and its derivatives up to r-1$r-1$ order at a set of nodes x$\mathbf{x}$ are given. Error bounds are obtained, and the method is illustrated by some examples.     

Re-nnd generalized inverses

In this paper, we give some necessary and sufficient conditions for the existence of Re-nnd and nonnegative definite {1,3}$\{1,3\}$- and {1,4}$\{1,4\}$-inverses of a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ and completely described these sets. Moreover, we prove that the existence of nonnegative definite {1,3}$\{1,3\}$-inverse of a matrix A   is equivalent with the existence of its nonnegative definite {1,2,3}$\{1,2,3\}$-inverse and present the necessary and sufficient conditions for the existence of Re-nnd {1,3,4}$\{1,3,4\}$-inverse of A.