The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as a continuous analogue of its diameter. We prove an analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytopes defined by 2d inequalities and for all d , then the order of the curvature is...

In this paper we give a new lower bound on the length of Snake-in-the-Box Codes, i.e., induced cycles in the d-dimensional cube. The bound is a linear function of the number of vertices of the cube and improves the bound c · 2/d, where c is a constant, proved by Danzer and Klee. AMS subject classification 1980: 05 C 35, 94 B 25

In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey. ∗This article appeared in Combinatorial and Graph-Theoretical Problems in Linear Algebra, R. A. Brualdi, S. Friedland, V. Klee, Eds., IMA Volumes in Mathema...

Since Walter Benjamin in “The Theses on the Philosophy of History” showcased Paul Klee’s angels, they remain mysterious figures that represent time, history or soul. The article focuses series drawings were created artist’s later period (1939−1940). can be regarded as final will and testament it expresses condensed philosophy mature views man’s place universe. It also reflects master’s cherishe...

A point-set embedding of a plane graph G with n vertices on a set S of n points is a straight-line drawing of G, where the vertices of G are mapped to distinct points of S. The problem of deciding whether a plane graph admits a point-set embedding on a given set of points is NPcomplete for 2-connected planar graphs, but polynomial-time solvable for outerplanar graphs and plane 3-trees. In this ...

Data-flow testing (DFT) checks the correctness of variable definitions by observing their corresponding uses. It has been empirically proved to be more effective than control-flow testing in fault detection, however, its complexities still overwhelm the testers in practice. To tackle this problem, we introduce a hybrid testing framework: (1) The core of our framework is symbolic execution, enha...

Let K be a nonempty closed convex nonexpansive retract of a uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be non-self asymptotically nonexpansive in the intermediate sense mapping with F (T ) 6= φ. Let {α n }, {β n } and {γ n } are sequences in [0, 1] with α (i) n + β (i) n + γ (i) n = 1 for all i = 1, 2, . . . , N . From arbitrary x1 ∈ K, define the sequenc...