A subset S of V is called an i-set (\(i\geq 2\)) if no two vertices in S have the distance i. The 2-set number \(\alpha_2(G)\) of a graph is the maximum cardinality among all 2-sets of G. A \(d_2\)-coloring of a graph is an assign- ment of colors to its vertices so that no two vertices have the distance two get the same color. The \(d_2\)-chromatic number \(\chi_{d_2}(G)\) of a graph G is the minimum number of \(d_2\)-colors need to G. In this paper, we initiate a study of these two new parameters.

We study the nonexistence of positive solutions for the system \[ \begin{cases} -\Delta_{p}u=\lambda f(v),\,\,\,\,\, x\in \Omega,\\ -\Delta_{p}v=\mu g(u),\,\,\,\,\, x\in \Omega,\\ u=0=v,\,\,\,\,\, x\in \partial \Omega. \end{cases} \] where \(\Delta_p\) denotes the p-Laplacian operator defined by \(\Delta_pz=div(|\nabla z|^{p-2} \nabla z)\) for \(p >1\) and \(\Omega\) is a smooth bounded domain in \(N^R (N \geq 1)\) , with smooth boundary \(\partial \Omega\) , and \(\lambda\) , \({\mu}\) are positive parameters. Let \(f,g: [0,\infty)\rightarrow R\) be continuous and we assume that there exist positive numbers \(K_i\) and \(M_i ; i = 1;2\) such that \(f(v)\leq k_1v^{p-1}-M_1\) for all \(v\geq 0\) ; and \(g(u)\leq k_2u^{p-1}-M_2\) for all \(u\geq 0\); We establish the nonexistence of positive solutions when \(\lambda_{\mu}\) is large.

In this paper we deal with the existence of at least three weak solutions for a two-point boundary value problem with Neumann boundary condition. The approach is based on variational methods and critical point theory.

In this article, we study the existence of positive solution for a class of (p; q)- Laplacian system \[ \begin{cases} -\Delta_{p}u=\lambda a(x)f(u)h(v),\,\,\,\,\, x\in \Omega,\\ -\Delta_{p}v=\lambda b(x)g(u)k(v),\,\,\,\,\, x\in \Omega,\\ u=v=0,\,\,\,\,\, x\in \partial \Omega. \end{cases} \] where \(\Delta_p\) denotes the p-Laplacian operator defined by \(\Delta_pz=div(|\nabla z|^{p-2} \nabla z), p>1,\Omega>0\) is a parameter and \(\Omega\) is a bounded domain in \(R^N(N > 1)\) with smooth boundary \(\partial \Omega\). Here \(a(x)\) and \(b(x)\) are \(C^1\) sign-changing functions that maybe negative near the boundary and \(f, g, k, h\) are \(C^1\) nondecreasing functions such that \(f; g; h; k : [0,\infty)\rightarrow [0,\infty) ; f(s), k(s), h(s), g(s) > 0 ; s > 0\) and \[\lim_{x\rightarrow \infty}\frac{h(A(g(x))^{\frac{1}{q-1}})(f(x))^{p-1}}{x^{p-1}}=0\] for every \(A > 0\). We discuss the existence of positive solution when \(h, k, f, g, a(x)\) and \(b(x)\) satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.

In this article, we present a reliable combination of variational iterative method and Padé approximants to investigate two dimensional exponential stretching sheet problem. The proposed method is called variational iterative Pade´ method (VIPM). The method is capable of reducing the size of calculation and easily overcomes the difficulty of perturbation methods or Adomian polynomials. The results reveal that the VIPM is very effective and is easy to apply.

This paper derives a non linear optimal inventory policy involving instant deterioration of perishable items with allowing price discounts. This paper postulates that the inventory policy of perishable items very much resembles that of price discounts. Such a parallel policy suggests that improvements to production systems may be achievable by applying price discounts to increase demand rate of the perishable items. This paper shows how discounted approach reduces to perfect results, and how the post deteriorated discounted EOQ model is a generalization of optimization. The objective of this paper is to determine the optimal price discount, the cycle length and the replenishment quantity so that the net profit is maximized. The numerical analyses show that an appropriate discounted pricing policy can benefit the retailer and that discounted pricing policy is important, especially for deteriorating items. Furthermore the instant post deteriorated price discount crisp economic order quantity (CEOQ) model is shown to be superior in terms of profit maximization. The sensitivity analysis of parameters on the optimal solution is carried out.

In this paper we characterized the ( p - 3 )- regular graphs which have a 3−deletable and a 4−deletable set of vertices.

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix.Thus every permutation matrix over C is a quasi-permutation matrix . The minimal degree of a faithful representation of G by quasi-permutation matrices over the complex numbers is denoted by c(G), and r(G) denotes the minimal degree of a faithful rational valued complex character of G . In this paper c(G) and r(G) are calculated for the Borel or maximal parabolic subgroups of \( SP(4,2^f)\) .

In this work, we solve the inverse nodal problem for the diffusion operator on a finite interval with separated boundary conditions. We investigation the oscillation of the eigenfunctions and derive an asymptotic formula for the nodal points. Uniqueness theorem is proved, and a constructive procedure for the solution is provided.

This paper compare modified homotopy perturbation method with the exact solution for solving Fourth order Volterra integro-differential equations. From the computational viewpoint, the modified homotopy perturbation method is more efficient and easy to use.

Vehicle Routing Problem with Time windows (VRPTW) is an example of scheduling in constrained environment. It is a well known NP hard combinatorial scheduling optimization problem in which minimum number of routes have to be determined to serve all the customers within their specified time windows. So far different analytic and heuristic approaches have been tried to solve such problems. In this paper we proposed algorithms which incorporate new local search techniques with genetic algorithm approach to solve VRPTW scheduling problems in various scenarios.

The annual cost of corrosion worldwide is over 3% of the world’s GDP. There are hundreds of thousands of kilometers of pipelines in various sectors of industry, which include many uncoated pipelines in chemical manufacturing plants, interstate natural gas transmission lines, and offshore oil-and-gas production pipelines. Mathematical modeling is richly endowed with many analytic computational techniques for analyzing real life situations. This paper reviewed that the predictive models on corrosion rate for natural gas pipeline. These models were selected based on the thermodynamic properties of the fluid and the developed rate is plotted against various operating conditions.

In this paper, the modification of the homotopy perturbation method (MHPM) Zaid M. Odibat (Appl. Math. Comput. 2007 ) is extended to derive approximate solutions of the nonlinear coupled wave equations. This work will present a numerical comparison between the modification and the homotopy perturbation method (HPM). In order to show the ability and reliability of the method some examples are provided. The results reveal that the method is very effective and simple. The modified method accelerates the rapid convergence of the series solution and reduces the size of work.

In this paper, the Homotopy analysis Method (HAM) is applied to the Maxwell system. The HAM yields an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms.

Suppose \(\overline{X}=(\overline{X}_1,...\overline{X}_p), (p\geq 2)\); where \(\overline{X}_i\)represents the mean of a random sample of size ni drawn from binomial \(bin(1,\theta_i)\) population. Assume the parameters \(\theta_1,...,\theta_p\) are unknown and the populations \(bin(1,\theta_1),...,bin(1,\theta_p)\) are independent. A subset of random size is selected using Gupta's (Gupta, S. S. (1965). On some multiple decision(selection and ranking) rules. Technometrics 7,225-245) subset selection procedure. In this paper, we estimate of the average worth of the parameters for the selected subset under squared error loss and normalized squared error loss functions. First, we show that neither the unbiased estimator nor the risk- unbiased estimator of the average worth (corresponding to the normalized squared error loss function) exist based on a single-stage sample. Second, when additional observations are available from the selected populations, we derive an unbiased and risk-unbiased estimators of the average worth and also prove that the natural estimator of the average worth is positively biased. Finally, the bias and risk of the natural, unbiased and risk-unbiased estimators are computed and compared using Monti Carlo simulation method.

We present a coupled fixed point theorems for mixed monotone operators in partially ordered metric spaces.

Using the technique of Brown and Wu [11]; we present a note on the paper [22] by Wu. Indeed, we extend the multiplicity results for a class of semilinear problems to the quasilinear elliptic problems with singular weights of the form: \[ \begin{cases} -div(|x|^{-ap}|\nabla u|^{p-2}\nabla u)\lambda|x|^{-(a+1)p+c}f(x)|u|^{q-2}u,\,\,\,\,\, x\in \Omega,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial n}=|x|^{-(a+1)p+c}g(x)|u|^{r-2}u, \,\,\,\,\, x\in \partial \Omega. \end{cases} \] Here \(0\leq a<\frac{N-p}{p}, c\) is a positive parameter, \(1 < q < p < r < p*(p* = \frac{pN}{N-p}\) if \(N > p, p* =\infty\) if \(N \leq p), \Omega\subset R^N\) is a bounded domain with smooth boundary, \(\frac{\partial }{\partial n}\) is the outer normal derivative, \(\lambda\in R-{0}\); and \(f(x); g(x)\) are continuous functions which change sign in \(\overline{\Omega}\).