On rigidity of Roe algebras

Roe algebras are C^?$C^{?}$-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d)$(X,d)$. In the special case that X=Γ$X=\Gamma$ is a finitely generated group and d   is a word metric, the simplest Roe algebra associated to (Γ,d)$(\Gamma ,d)$ is isomorphic to the crossed product C^?$C^{?}$-algebra l^∞(Γ)_?rΓ$l^{\infty }(\Gamma ){?}_r\Gamma$.     

Capturing two elements in unavoidable minors of 3-connected binary matroids

Let M be a 3-connected binary matroid and let n   be an integer exceeding 2. Ding, Oporowski, Oxley, and Vertigan proved that there is an integer f(n)$f(n)$ so that if |E(M)|>f(n)$|E(M)|>f(n)$, then M has a minor isomorphic to one of the rank-n wheel, the rank-n   tipless binary spike, or the cycle or bond matroid of _K3,n$K_{3,n}$. This result was recently extended by Chun, Oxley, and Whittle to show that there is an integer g(n)$g(n)$ so that if |E(M)|>g(n)$|E(M)|>g(n)$ and x∈E(M)$x\in E(M)$, then x is an element of a minor of M isomorphic to one of the rank-n wheel, the rank-n   binary spike with a tip and a cotip, or the cycle or bond matroid of _K1,1,1,n$K_{1,1,1,n}$. In this paper, we prove that, for each i   in {2,3}$\{2,3\}$, there is an integer _hi(n)$h_i(n)$ so that if |E(M)|>_hi(n)$|E(M)|>h_i(n)$ and Z is an i-element rank-2 subset of M, then M has a minor from the last list whose ground set contains Z.     

Sharp upper bounds on the distance spectral radius of a graph

Let M=(_mij)$M=(m_{ij})$ be a nonnegative irreducible n×n$n\times n$ matrix with diagonal entries 0. The largest eigenvalue of M is called the spectral radius of the matrix M  , denoted by ρ(M)$\rho (M)$. In this paper, we give two sharp upper bounds of the spectral radius of matrix M. As corollaries, we give two sharp upper bounds of the distance matrix of a graph.     

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${\mathbb{R}}^r$.     

Inequivalent representations of matroids over prime fields

It is proved that for each prime field GF(p)$\mathit{GF}(p)$, there is an integer _np$n_p$ such that a 4-connected matroid has at most _np$n_p$ inequivalent representations over GF(p)$\mathit{GF}(p)$. We also prove a stronger theorem that obtains the same conclusion for matroids satisfying a connectivity condition, intermediate between 3-connectivity and 4-connectivity that we term “k-coherence”.     

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.     

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.     

A characterization of quintic helices

A polynomial curve of degree 5, α,$\alpha ,$ is a helix if and only if both ∥α^′∥$\parallel {\alpha }^{\prime }\parallel$ and ∥α^′∧α^″∥$\parallel {\alpha }^{\prime }\wedge {\alpha }^{″}\parallel$ are polynomial functions.     

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)$(X,d^G)$.     

Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs

For a countable ultrahomogeneous graph G=〈G,ρ〉$\mathbb{G}=〈G,\rho 〉$ let P(G)$\mathbb{P}(\mathbb{G})$ denote the collection of sets A⊂G$A\subset G$ such that 〈A,ρ∩[A]²〉≅G$〈A,\rho \cap {[A]}^2〉\cong \mathbb{G}$. The order types of maximal chains in the poset 〈P(G)∪{∅},⊂〉$〈\mathbb{P}(\mathbb{G})\cup \{\varnothing \},\subset 〉$ are characterized as:     

Test ideals of non-principal ideals: Computations,jumping numbers,alterations and division theorems

Given an ideal a⊆R$\mathfrak{a}\subseteq R$ in a (log) Q$\mathbb{Q}$-Gorenstein F  -finite ring of characteristic p>0$p>0$, we study and provide a new perspective on the test ideal τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ for a real number t>0$t>0$. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F  -jumping numbers of τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.