A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF f is the value f(V (G)) = ∑ u∈V (G) f(u). A function f : V (G) → {0, 1, 2} with the ordered partition (V0, V1, V2) of V (G), where Vi = {v ∈ V (G) | f(v) = i} for i = 0...

a. More on Definition 0.1. In Definition 0.1 we think of the stochastic process {Yn : n ≥ 0} constituting the history or information. Then {Yk : 0 ≤ k ≤ n} is the history up to (and including) time n. The random variables Yk could be random vectors, as we illustrate below. We simply say that {Xn : n ≥ 0} is a martingale if {Xn : n ≥ 0} is a martingale with respect to {Xn : n ≥ 0}; i.e., if the ...

We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 (denoted by min TSP12 and max TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any 2 > 0, it is NP-hard to ap...

We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 (denoted by min TSP12 and max TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any 2 > 0, it is NP-hard to ap...

todeschini et al. have recently suggested to consider multiplicative variants of additive graphinvariants, which applied to the zagreb indices would lead to the multiplicative zagrebindices of a graph g, denoted by ( ) 1  g and ( ) 2  g , under the name first and secondmultiplicative zagreb index, respectively. these are define as  ( )21 ( ) ( )v v gg g d vand ( ) ( ) ( )( )2 g d v d v gu...

For a Markov chain X = {Xi, i = 1, 2, . . . , n} with the state space {0, 1}, the random variable S := ∑n i=1 Xi is said to follow a Markov binomial distribution. The exact distribution of S, denoted as LS, is very computationally intensive for large n [see Gabriel (1959) and Bhat and Lal (1988)] and this paper concerns suitable approximate distributions for LS when X is stationary. We conclude...

In this paper we investigate continuous and upper and lower semi-continuous real functions in the framework of TTE, Type-2 Theory of EEectivity. First some basic facts about TTE are summarized. For each of the function spaces, we introduce several natural representations based on diierent intiuitive concepts of \eeectivity" and prove their equivalence. Computability of several operations on the...

It is shown that a technicolor theory containing a color-octet technipion, usually denoted by P 0 ′ 8 , will give rise to an enhancement of tt̄ production at the Tevatron, LHC and SSC, via the process gg → P 0′ 8 → tt̄. The relevant cross-sections are computed taking into account the large lower bound on the top mass coming from the ”top search” experiments at LEP and CDF. At the LHC and SSC, the...

Consider a general polynomial system S in x1, . . . , xn of degree q and its corresponding vector of monomials of degree less than or equal to q. The system can be written as M0 · [xq1, xq−1 1 x2, . . . , xn, x1, . . . , xn, 1] = [0, 0, . . . , 0, 0, . . . , 0, 0] (1) in terms of its coefficient matrix M0. Here and hereafter, [...] T means the transposition. Further, [ξ1, ξ2, . . . , ξn] is one...

G. SZEKERES We shall deal with real functions which have continuous second derivatives in some open interval (a, b), — » ^a(x) is called convex (from below) if "(x)^0 in I(<p). If <p(x) is monotone increasing and convex, and ^i(x) is monotone increasing and concave such that the range of <p(x) is contained ...