For a graph G the symbol G v → (a1, . . . , ar) means that in every r-coloring of the vertices of G for some i ∈ {1, . . . , r} there exists a monochromatic ai-clique of color i. The vertex Folkman numbers Fv(a1, . . . , ar; q) = min{|V(G)| : G v → (a1, . . . , ar) and Kq " G} are considered. In this paper we shall compute the Folkman numbers Fv(2, . . . , 2 } {{ } r ; r− k + 1) when k ≤ 12 and...

For every integer r > 3 and every ε > 0 we construct a graph with maximum degree r − 1 whose circular total chromatic number is in the interval (r, r + ε). This proves that (i) every integer r > 3 is an accumulation point of the set of circular total chromatic numbers of graphs, and (ii) for every ∆ > 2, the set of circular total chromatic numbers of graphs with maximum degree ∆ is infinite. Al...

the zarankiewicz number z(b; s) is the maximum size of a subgraph of kb,b which does not contain ks,s as a subgraph. the two-color bipartite ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of kb,b with two colors contains a ks,s in the rst color or a kt,t in the second color.in this work, we design and exploit a computational method for bounding and computin...

Rotylenchulus reniformis is rapidly becoming the most economically important pest associated with cotton in the southeastern United States. Incentive programs have been implemented to support sampling of production fields to determine the presence and abundance of R. reniformis. These sampling programs have dramatically increased the number of soils samples submitted to nematology laboratories ...

data obtained in association with many real-life random experiments from different fields cannot be perfectly/exactly quantified.hspace{.1cm}often the underlying imprecision can be suitably described in terms of fuzzy numbers/values. for these random experiments, the scale of fuzzy numbers/values enables to capture more variability and subjectivity than that of categorical data, and more accura...

New perturbation analyses are presented for the block Cholesky downdating problem U T U = R T R ? X T X. These show how changes in R and X alter the Cholesky factor U. There are two main cases for the perturbation matrix R in R: (1) R is a general matrix; (2))R is an upper triangular matrix. For both cases, rst order perturbation bounds for the downdated Cholesky factor U are given using two ap...

3 ABSTRACT. Let c > 0 be a fixed constant. Let 0 ≤ r < s be an arbitrary pair of real numbers. Let a, b be any pair of real numbers such that | b − a | ≤ c(s − r). Define C s r to be the set of continuous real-valued functions on [r, s], and define C r to be the set of continuous real-valued functions on [ r, +∞). Finally, consider the following sets of Lipschitz functions:

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets K ⊂ H of a Hilbert space H by the metric entropy of the set K where the covering numbers N(K, ") of K by "-balls of H satisfy the Lorentz condition ∫ ∞ 0 ( log2N(K, ") )r/s d" <∞ for some fixed 0 < r, s ≤ ∞ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 <...