Let Ω be a space with probability measure μ for which the notion of symmetry is defined. Given A ⊆ Ω, let ms(A) denote the supremum of μ(B) over symmetric B ⊆ A. An r-coloring of Ω is a measurable map χ : Ω → {1, . . . , r} possibly undefined on a set of measure 0. Given an r-coloring χ, let ms(Ω;χ) = max1≤i≤r ms(χ (i)). With each space Ω we associate a Ramsey type number ms(Ω, r) = infχ ms(Ω;χ...

Let G be a collection of graphs with n vertices. We present a simple description of [G]χ = {H ∈ G : χ(H) = χ(G)} where χ denotes the Randić index. We associate to G a Q-linear map ρ : Qm → Qk (for some integers k, m depending on G) such that the kernel of ρ contains the necessary information to describe [G]χ in terms of linear equations. These results provide precise tools for analyzing the beh...

Given a graph G we study the chromatic number of a random graph G1/2 obtained from G by removing each edge of G independently with probability 1/2. Studying χ(G1/2) has been suggested by Bukh [Buk], who asked whether E[χ(G1/2)] > Ω(χ(G)/ log(χ(G))) holds for all graphs G. In this paper we prove several results related this problem. Denoting the chromatic number of G by k = χ(G) we prove the fol...

If R-parity is not conserved, the SUSY-threshold production process e + e − → χ 0 1 χ 0 1 could be detectable with relatively low luminosity. Hence an interesting mass range for the lightest SUSY particle χ 0 1 could be explored at the CERN LEP collider during its intermediate energy development, even before the full LEP200 upgrade is completed. We present cross section formulas and discuss eve...

If G is a triangle-free graph, then two Gallai identities can be written as α(G)+ χ(L(G)) = |V (G)| = α(L(G))+ χ(G), where α and χ denote the stability number and the clique-partition number, and L(G) is the line graph of G. We show that, surprisingly, both equalities can be preserved for any graph G by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs, acc...

As a natural generalization of graph coloring, Vince introduced the star chromatic number of a graph G and denoted it by χ∗(G). Later, Zhu called it circular chromatic number and denoted it by χc(G). Let χ(G) be the chromatic number of G. In this paper, it is shown that if the complement of G is non-hamiltonian, then χc(G)=χ(G). Denote by M(G) the Mycielski graph of G. Recursively define Mm(G)=...

Inverse problems are typically ill-posed or ill-conditioned and require regularization. Tikhonov regularization is a popular approach and it requires an additional parameter called the regularization parameter that has to be estimated. The χ method introduced by Mead in [8] uses the χ distribution of the Tikhonov functional for linear inverse problems to estimate the regularization parameter. H...

PURPOSE This study was done to determine the risk factors for recurrent fallers (2+falls) compared to single fallers. METHODS Participants were 104 community-dwelling people 65 yr of age or older. The data were collected from June 1, 2008 to June 30, 2009 using the Residential Assessment Instrument-Home Care. RESULTS Over the past 90 days, 55.7% of the 104 participants fell once, and 44.2% ...

Given a link L ⊂ S 3 , a Seifert surface S for L is a compact, orientable surface with boundary L. The Euler characteristic χ(L) of the link L is dened to be the maximum over all Euler characteristics χ(S) of Seifert surfaces S for L. Seifert surfaces exist for all L, and this denition presents itself with the problem of calculating χ(L). An easily applicable method for producing Seifert surfac...

Supersymmetry (SUSY) uniquely opens the possibility to directly connect the Standard Model (SM) with an ultimate unification of the fundamental interactions. A primary goal for linear colliders is a precise measurement of SUSY parameters if it would be discovered at Tevatron or LHC. One of the most challenging signatures in SUSY will be events involving τ leptons. At large tanβ values, we could...