We show that the region lit by a point light source inside a simple n-gon after at most k reeections oo the boundary has combinatorial complexity O(n 2k), for any k 1. A lower bound of ((n=k ? (1)) 2k) is also established which matches the upper bound for any xed k. A simple near-optimal algorithm for computing the illuminated region is presented, which runs in O(n 2k log n) time and O(n 2k) sp...

Let F2k,k2 consist of all simple graphs on 2k vertices and k2 edges. For a simple graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G in at most λ colors, and let f(2k, k2, λ) = max{PG(λ) : G ∈ F2k,k2}. We prove that f(2k, k2, 3) = PKk,k(3) and Kk,k is the only extremal graph. We also prove that f(2k, k2, 4) = (6 + o(1))4k as k →∞.

Matrix descriptions of higher dimensional spherical branes are investigated. It is known that a fuzzy 2k-sphere is described by the coset space SO(2k+1)/U(k) and has some extra dimensions. It is shown that a fuzzy 2k-sphere is comprised of n k(k−1) 2 spherical D(2k − 1)-branes and has a fuzzy 2(k − 1)-sphere at each point. We can understand the relationship between these two viewpoints by the d...

We prove that any cycle Cn, n ≥ 4, with list assignment L, has a k -fold list coloring from the given lists if (i) each list contains at least 2k colors and (ii) Cn and L satisfy Hall’s condition for k -fold list colorings. Further, 2k in (i) cannot be replaced by 2k − 1 if either n is odd, or n is even and n ≥ k + 2. In other words, if n ≥ 4, the k -fold Hall number of a cycle Cn satifies h (C...

Let m,n, and k be integers satisfying 0 < k 6 n < 2k 6 m. A family of sets F is called an (m,n, k)-intersecting family if ([n] k ) ⊆ F ⊆ ([m] k ) and any pair of members of F have nonempty intersection. Maximum (m, k, k)and (m, k + 1, k)-intersecting families are determined by the theorems of Erdős-KoRado and Hilton-Milner, respectively. We determine the maximum families for the cases n = 2k − ...

It is shown that the existence of a BIB design with parameters v = 2k − 2k+1, b = 2v, r = 2k, k, λ = 1 implies the existence of Hadamard matrices of orders 4v and 8vt, where t is an integer for which an orthogonal design of order 4t and type (t, t, t, t) exists.

An explicit definition of a 1-factorization of B k (the bipartite graph defined by the kand (k+l)-element subsets of [ 2k+ l ] ) , whose constituent matchings are defined using addition modulo k + 1, is introduced. We show that the matchings are invariant under rotation (mapping under ~r = (1, 2, 3 ..... 2k + 1 )), describe the effect of reflection (mapping under p = (1, 2k + 1)(2, 2k).. . (k, ...

We conjecture that every planar graph of odd-girth 2k + 1 admits a homomorphism to Cayley graph C(Z 2 , S2k+1), with S2k+1 being the set of (2k + 1)vectors with exactly two consecutive 1’s in a cyclic order. This is an strengthening of a conjecture of T. Marshall, J. Nešetřil and the author. Our main result is to show that this conjecture is equivalent to the corresponding case of a conjecture ...

For integers a and b, 0 ≤ a ≤ b, an [a, b]-graph G satisfies a ≤ deg(x,G) ≤ b for every vertex x of G, and an [a, b]-factor is a spanning subgraph F such that a ≤ deg(x, F ) ≤ b for every vertex x of F . An [a, b]-factor is almost-regular if b = a+1. A graph is [a, b]-factorable if its edges can be decomposed into [a, b]-factors. When both k and t are positive integers and s is a nonnegative in...

eEF-2K is a potential target for treating cancer. However, potent specific inhibitors for this enzyme are lacking. Previously, we identified 2,6-diamino-4-(2-fluorophenyl)-4H-thiopyran-3,5-dicarbonitrile (DFTD) as an inhibitor of eEF-2K. Here we describe its mechanism of action against eEF-2K, on the basis of kinetic, mutational, and docking studies, and use chemoinformatic approaches to identi...