A vertex coloring of a graph G=(V,E) is called an exact square G if any pair vertices at distance 2 receive distinct colors. The minimum number colors required by the chromatic and denoted χ[#2](G). set clique are 2. G, ω[#2](G), maximum cardinality clearly, ω[#2](G)≤χ[#2](G). In this article, we give tight upper bounds various standard operations, including Cartesian product, strong lexicograp...

(1) (c, χ) = (7n, n), (2) (c, χ) = (5n, n), (3) (c, χ) = (4n, n), (4) (c, χ) = (2n, n), (5) (c, χ) = ((6 + 8g)n, (1 + g)n (for g ≥ 0), (6) (c, χ) = (7n+ (6 + 8g)m,n+ (1 + g)m), (7) (c, χ) = (7n+ 5m,n+m), (8) (c, χ) = (7n+ 4m,n+m), (9) (c, χ) = (7n+ 2m,n+m), (10) (c, χ) = ((6 + 8g)n+ 5m, (1 + g)n+m) (for g ≥ 0), (11) (c, χ) = ((6 + 8g)n+ 4m, (1 + g)n+m) (for g ≥ 0), (12) (c, χ) = ((6 + 8g)n+ 2m,...

Calculi are used to reason about functional languages from a theoretical perspective. They allows us to make bold statements about programs based on sound and correct deductive methods. Methods of visualization allow us to gain a better understanding of the communication and interaction involved between processes. Often, graphs are used because of the level of intuition they can bring to a syst...

Suppose that T is an acyclic r-uniform hypergraph, with r ≥ 2. We define the (t-color) chromatic Ramsey number χ(T, t) as the smallest m with the following property: if the edges of any m-chromatic r-uniform hypergraph are colored with t colors in any manner, there is a monochromatic copy of T . We observe that χ(T, t) is well defined and ⌈ R(T, t)− 1 r − 1 ⌉ + 1 ≤ χ(T, t) ≤ |E(T )| + 1 where R...

The general quantum χ(2)-divergence has recently been introduced by Temme et al. [Temme K, Kastoryano M, Ruskai M, Wolf M, Verstrate F (2010) J Math Phys 51:122201] and applied to quantum channels (quantum Markov processes). The quantum χ(2)-divergence is not unique, as opposed to the classical χ(2)-divergence, but depends on the choice of quantum statistics. It was noticed that the elements in...

1. χtrivial(x) = 1 for every x ∈ G is a character. It is called the trivial character. 2. If χ1 and χ2 are characters of G then so is χ1 · χ2 (where χ1 · χ2(x) = χ1(x)χ2(x)). 3. For every character χ of G and x ∈ G, |χ(x)| = 1 (the absolute norm is of course in C). In particular, if we define χ1(x) = χ1(x), then χ1 is also a character and χ · χ = χtrivial. 4. This implies that Ĝ = {χ ∈ C[G] | χ...

BACKGROUND We review our single institution experience to determine predictive factors for early and delayed distant brain failure (DBF) after radiosurgery without whole brain radiotherapy (WBRT) for brain metastases. MATERIALS AND METHODS Between January 2000 and December 2010, a total of 464 patients were treated with Gamma Knife stereotactic radiosurgery (SRS) without WBRT for primary mana...

The notations F p, ν1,ν2 and χ 2 p ,ν in Eqs. (1), (2) (3)

as χ varies over all multiplicative characters of k. For each χ, S(χ) is real, and (by Weil) has absolute value at most 2. Evans found empirically that, for large q = #k, these q − 1 sums were approximately equidistributed for the “Sato-Tate measure” (1/2π) √ 4− x2dx on the closed interval [−2, 2], and asked if this equidistribution was provably true. Rudnick had done numerics on the sums T (χ)...

Let L(G) denote the set of all odd cycle lengths of a graph G. Gyárfás gave an upper bound for χ(G) depending on the size of this set: if |L(G)| = k ≥ 1, then χ(G) ≤ 2k+1 unless some block of G is a K2k+2, in which case χ(G) = 2k+2. This bound is generally tight, but when investigating L(G) of special forms, better results can be obtained. Wang completely analyzed the case L(G) = {3, 5}; Camach...