location:  Publications → journals
Search results

Search: All articles in the CMB digital archive with keyword convexity

 Expand all        Collapse all Results 1 - 5 of 5

1. CMB 2012 (vol 57 pp. 61)

Geschke, Stefan
 2-dimensional Convexity Numbers and $P_4$-free Graphs For $S\subseteq\mathbb R^n$ a set $C\subseteq S$ is an $m$-clique if the convex hull of no $m$-element subset of $C$ is contained in $S$. We show that there is essentially just one way to construct a closed set $S\subseteq\mathbb R^2$ without an uncountable $3$-clique that is not the union of countably many convex sets. In particular, all such sets have the same convexity number; that is, they require the same number of convex subsets to cover them. The main result follows from an analysis of the convex structure of closed sets in $\mathbb R^2$ without uncountable 3-cliques in terms of clopen, $P_4$-free graphs on Polish spaces. Keywords:convex cover, convexity number, continuous coloring, perfect graph, cographCategories:52A10, 03E17, 03E75

2. CMB 2011 (vol 54 pp. 217)

Chen, William Y. C.; Wang, Larry X. W.; Yang, Arthur L. B.
 Recurrence Relations for Strongly $q$-Log-Convex Polynomials We consider a class of strongly $q$-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly $q$-log-convex. We also prove that the Bessel transformation preserves log-convexity. Keywords:log-concavity, $q$-log-convexity, strong $q$-log-convexity, Bell polynomials, Bessel polynomials, Ramanujan polynomials, Dowling polynomialsCategories:05A20, 05E99

3. CMB 2009 (vol 52 pp. 39)

Cimpri\v{c}, Jakob
 A Representation Theorem for Archimedean Quadratic Modules on $*$-Rings We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand--Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution. Keywords:Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometryCategories:16W80, 46L05, 46L89, 14P99

4. CMB 2003 (vol 46 pp. 242)

Litvak, A. E.; Milman, V. D.
 Euclidean Sections of Direct Sums of Normed Spaces We study the dimension of random'' Euclidean sections of direct sums of normed spaces. We compare the obtained results with results from \cite{LMS}, to show that for the direct sums the standard randomness with respect to the Haar measure on Grassmanian coincides with a much weaker'' randomness of diagonal'' subspaces (Corollary~\ref{sle} and explanation after). We also add some relative information on phase transition''. Keywords:Dvoretzky theorem, random'' Euclidean section, phase transition in asymptotic convexityCategories:46B07, 46B09, 46B20, 52A21

5. CMB 2000 (vol 43 pp. 448)

Li, Chi-Kwong; Zaharia, Alexandru
 Nonconvexity of the Generalized Numerical Range Associated with the Principal Character Suppose $m$ and $n$ are integers such that $1 \le m \le n$. For a subgroup $H$ of the symmetric group $S_m$ of degree $m$, consider the {\it generalized matrix function} on $m\times m$ matrices $B = (b_{ij})$ defined by $d^H(B) = \sum_{\sigma \in H} \prod_{j=1}^m b_{j\sigma(j)}$ and the {\it generalized numerical range} of an $n\times n$ complex matrix $A$ associated with $d^H$ defined by $$\wmp(A) = \{d^H (X^*AX): X \text{ is } n \times m \text{ such that } X^*X = I_m\}.$$ It is known that $\wmp(A)$ is convex if $m = 1$ or if $m = n = 2$. We show that there exist normal matrices $A$ for which $\wmp(A)$ is not convex if $3 \le m \le n$. Moreover, for $m = 2 < n$, we prove that a normal matrix $A$ with eigenvalues lying on a straight line has convex $\wmp(A)$ if and only if $\nu A$ is Hermitian for some nonzero $\nu \in \IC$. These results extend those of Hu, Hurley and Tam, who studied the special case when $2 \le m \le 3 \le n$ and $H = S_m$. Keywords:convexity, generalized numerical range, matricesCategory:15A60
 top of page | contact us | privacy | site map |