Abstract:
A graph G is said to have a perfect dominating set S if S is a set of vertices of G and for each vertex v of G,
either v is in S and v is adjacent to no other vertex in S, or v is not in S but is adjacent to precisely one vertex of S. A graph G may have
none, one or more than one perfect dominating sets. The problem of determining
if a graph has a perfect dominating set is NP-complete. The problem of
calculating the probability of an arbitrary graph having a perfect dominating
set seems also difficult. In 1994 Yue[1]

Abstract:
Khidr and El-Desouky [1] derived a symmetric sum involving the Stirling numbers of the first kind through the process of counting the number of paths along a rectangular array n*m denoted by ？A_{nm}. We investigate the generating function for the general case and hence some special cases as well. The probability function of the number of paths along is obtained. Moreover, the moment generating function of the random variable X and hence the mean and variance are obtained. Finally, some applications are introduced.

Abstract:
In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(X_{n}/Y_{n}), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (X^{0}_{n,k10},X^{1}_{n,k11};Y^{0}_{n,k20},Y^{1}_{n,k21}) where for i=0,1,X^{i}_{n,k1i} denotes the number of occurrences of i-runs of length k^{1}_{i} in the first component and Y^{i}_{n,k2i} denotes the number of occurrences of i-runs of length k^{2}_{i} in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ_{1} and Λ_{2} of lengths k_{1} and k_{2} respectively and obtain the pgf of joint distribution of (X_{n,Λ 1},Y_{n,Λ2} ) using method of conditional probability generating functions where X_{n,Λ1}(Y_{n,Λ2}) denotes the number of occurrences of pattern Λ_{1}(Λ_{2} ) of length k_{1} (k_{2}) in the

Abstract:
Statisticians are usually concerned with the proposition of new distributions. In this paper we point out that a unified and concise derivation procedure of the distribution of the minimum or maximum of a random number N of indepen-dent and identically distributed continuous random variables Y_{i},{i = 1,2,…,N} is obtained if one compounds the probability generating function of N with the survival or the distribution func-tion of Y_{i}. Expressions are then derived in closed form for the density, hazard and quantile func-tions of the minimum or maximum. The methodology is illustrated with examples of the distributions proposed by Adamidis and Loukas (1998), Kus (2007), Tahmasbi and Rezaei (2008), Barreto-Souza and Cribari-Neto (2009), Cancho, Louzada, and Barriga (2011) and Louzada, Roman and Cancho (2011).

Abstract:
In this paper, we study the k–Lucas numbers of arithmetic indexes of the form an+r , where n is a natural number and r is less than r. We prove a formula for the sum of these numbers and particularly the sums of the first k-Lucas numbers, and then for the even and the odd k-Lucas numbers. Later, we find the generating function of these numbers. Below we prove these same formulas for the alternated k-Lucas numbers. Then, we prove a relation between the k–Fibonacci numbers of indexes of the form 2rn and the k–Lucas numbers of indexes multiple of 4. Finally, we find a formula for the sum of the square of the k-Fibonacci even numbers by mean of the k–Lucas numbers.

Abstract:
This in virtue of the notion of likelihood ratio and the tool of moment generating function, the limit properties of the sequences of random discrete random variables are studied, and a class of strong deviation theorems which represented by inequalities between random variables and their expectation are obtained. As a result, we obtain some strong deviation theorems for Poisson distribution and binomial distribution.

Abstract:
Background We study the sparsification of dynamic programming based on folding algorithms of RNA structures. Sparsification is a method that improves significantly the computation of minimum free energy (mfe) RNA structures. Results We provide a quantitative analysis of the sparsification of a particular decomposition rule, Λ . This rule splits an interval of RNA secondary and pseudoknot structures of fixed topological genus. Key for quantifying sparsifications is the size of the so called candidate sets. Here we assume mfe-structures to be specifically distributed (see Assumption 1) within arbitrary and irreducible RNA secondary and pseudoknot structures of fixed topological genus. We then present a combinatorial framework which allows by means of probabilities of irreducible sub-structures to obtain the expectation of the Λ -candidate set w.r.t. a uniformly random input sequence. We compute these expectations for arc-based energy models via energy-filtered generating functions (GF) in case of RNA secondary structures as well as RNA pseudoknot structures. Furthermore, for RNA secondary structures we also analyze a simplified loop-based energy model. Our combinatorial analysis is then compared to the expected number of Λ -candidates obtained from the folding mfe-structures. In case of the mfe-folding of RNA secondary structures with a simplified loop-based energy model our results imply that sparsification provides a significant, constant improvement of 91% (theory) to be compared to an 96% (experimental, simplified arc-based model) reduction. However, we do not observe a linear factor improvement. Finally, in case of the “full” loop-energy model we can report a reduction of 98% (experiment). Conclusions Sparsification was initially attributed a linear factor improvement. This conclusion was based on the so called polymer-zeta property, which stems from interpreting polymer chains as self-avoiding walks. Subsequent findings however reveal that the O(n) improvement is not correct. The combinatorial analysis presented here shows that, assuming a specific distribution (see Assumption 1), of mfe-structures within irreducible and arbitrary structures, the expected number of Λ -candidates is Θ(n2). However, the constant reduction is quite significant, being in the range of 96%. We furthermore show an analogous result for the sparsification of the Λ -decomposition rule for RNA pseudoknotted structures of genus one. Finally we observe that the effect of sparsification is sensitive to the employed energy model.

Abstract:
In this note we have shown the existence of more general generating relation from the existence of a partial quasi-bilinear generating relation by using group theoretic method. Some particular cases of interest are also pointed out.

Abstract:
In the present paper, we obtain a theorem on bilateral generating functions of n +n (x) , a modification of n (x) by introducing a novel linear partial differential operator obtained by the suitable interpretations of the index n and the parameter of the polynomial under consideration.

Abstract:
n this note, we have obtained a novel result which is stated in the form of a theorem on mixed trilateral generating relations involving modified Laguerre polynomials. As a particular case, we obtain an interesting result, which is worthy of notice.