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$$d_2$$-coloring of a Graph $$d_2$$-coloring of a Graph en en 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. 102 111 K. Selvakumar S. Nithya $$d_2$$-coloring $$d_2$$-chromatic number Article.1.pdf  G. Chartrand, L. Lesniak, Graphs and Digraphs, Wadsworth and Brooks/Cole, Monterey (1986) ## G. Fertin, E. Godard, A. Raspaud, Acyclic and k-distance coloring of the Grid, Information Processing Letters, 87 (2003), 51-58 ## J. Van den Heuvel, S. McGuinness, Colouring the square of a planar graph, J. Graph Theory, 42 (2005), 110-124
Nonexistence of Result for some p-Laplacian Systems Nonexistence of Result for some p-Laplacian Systems en en 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. 112 116 G. A. Afrouzi Z. Valinejad positive solutions p-Laplacian operator smooth bounded domain Article.2.pdf  G. A. Afrouzi, S. H. Rasouli, Population models involving the p-Laplacian with indefinite weight and constant yeild harvesting , Chaos Solitons Fractals, Vol. 31, 404--408 (2007) ## L. Boccardo, D. G. Figueiredo, Some remarks on a system of quasilinear elliptic equations, Nonl. Diff. Eqns. Appl., 9 (2002), 231-240 ## P. Clement, J. Fleckinger, E. Mitidieri, F. de Thelin, Existence of positive solutions for a nonvariational quasilinear elliptic systems, Journal of Differential Equations, 166 (2000), 455-477 ## R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Anal., 39 (2000), 559-568 ## A. Djellit, S. Tas, On some nonlinear elliptic systems, Nonlinear Anal., 59 (2004), 695-706 ## D. D. Hai, On a class of sublinear quasilinear elliptic problems, Proc. Amer. Math. Soc., 131 (2003), 2409-2414 ## G. A. Afrouzi, S. H. Rasouli, A remark on the Nonexistence of positive solutions for some p-Laplacian Systems., Global J. Pure. Appl. Math., 2005 (2005), 197-201
Multiple Solutions for a Two-point Boundary Value Problem Depending on Two Parameters Multiple Solutions for a Two-point Boundary Value Problem Depending on Two Parameters en en 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. 117 125 Shapour Heidarkhani Javad Vahidi Three solutions Critical point Multiplicity results Neumann problem. Article.3.pdf  G. A. Afrouzi, S. Heidarkhani, Three solutions for a Dirichlet boundary value problem involving the p-Laplacian, Nonlinear Anal., 66 (2007), 2281-2288 ## D. Averna, G. Bonanno, Three solutions for a Neumann boundary value problem involving the p-Laplacian, Le Matematiche, 60 (2005), 81-91 ## R. I. Avery, J. Henderson, Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. Lett., 13 (2000), 1-7 ## G. Bonanno, Existence of three solutions for a two point boundary value problem, Appl. Math. Lett., 13 (2000), 53-57 ## G. Bonanno, Multiple solutions for a Neumann boundary value problem, J. Nonlinear Convex Anal., 4 (2003), 287-290 ## G. Bonanno, P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math., 80 (2003), 424-429 ## A. R. Miciano, R. Shivaji, Multiple positive solutions for a class of semipositone Neumann two point boundary value problems, J. Math. Anal. Appl., 178 (1993), 102-115 ## M. Ramaswamy, R. Shivaji, Multiple positive solutions for classes of p-Laplacian equations, Differential and Integral Equations, 17 (2004), 1255-1261 ## B. Ricceri, A three critical points theorem revisited, Nonlinear Anal., 70 (2009), 3084-3089 ## B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problem, Math. Comput. Modelling, 32 (2000), 1485-1494 ## B. Ricceri, On a three critical points theorem, Arch. Math., 75 (2000), 220-226
A Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects A Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects en en 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. 126 134 S. H. Rasouli Z. Halimi Z. Mashhadban (p،q)- Laplacian system Sign-changing weight. Article.4.pdf  C. Atkinson, K. El-Ali, Some boundary value problems for the Bingham model, J. Non-Newtonian Fluid Mech., 41 (1992), 339-363 ## J. F. Escobar, Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate, Comm. Pure Appl. Math., 43 (1990), 857-883 ## P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150 ## G. S. Ladde, V. Lakshmikantham, A. S. Vatsale, Existence of coupled quase-solutions of systems of nonlinear elliptic boundary value problems, Nonlinear Anal., 8 (1984), 501-515 ## N. Dancer, Competing species systems with diffusion and large interaction, Rendiconti del Seminario Matematico e Fisico di Milano (Milan Journal of Mathematics), 65 (1995), 23-33 ## J. Ali, R. Shivaji, An existence result for a semipositone problem with a sign-changing weight, Abstr. Appl. Anal., 2006 (2006), 1-5 ## M. Chhetri, S. oruganti, R. Shivaji, Existence results for a class of p-Laplacian problems with sign-changing weiht, Diff. Int. Equs., 18 (2005), 991-996 ## R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Analysis: Theory, Methods & Applications, 39 (2000), 559-568 ## J. Ali, R. Shivaji, M. Ramaswamy, Multiple positive solutions for a class of elliptic systems with combined nonlinear effects, Differential and Integral Equations, 19 (2006), 669-680 ## D. D. Hai, R. Shivaji, An existence result on positive solutions for a class of semilinear elliptic systems, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 134 (2004), 137-141 ## D. D. Hai, R. Shivaji, An existence result on positive solutions for a class of p-Laplacian systems, Nonlinear Analysis: Theory, Methods & Applications, 56 (2004), 1007-1010 ## A. Canada, P. Drabek, J. L. Gamez, Existence of positive solutions for some problems with nonlinear diffusion, Trans. Amer. Math. Soc., 349 (1997), 4231-4249 ## P. Drabek, J. Hernandez, Existence and uniqueness of positive solutions for some quasilinear elliptic problem, Nonlinear Anal., 44 (2001), 189-204 ## A. Ambrosetti, J. G. Azorero, I. Peral, Existence and multiplicity results for some nonlinear elliptic equations: a survey, Rend. Mat. Appl., 20 (2000), 167-198 ## C. O. Alves, D. G. De Figueiredo, Nonvariational elliptic systems, Discr. Contin. Dyn. Systems-A, 8 (2002), 289-302 ## G. A. Afrouzi, S. H. Rasouli, A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 445-455 ## G. A. Afrouzi, S. H. Rasouli, A remark on the linearized stability of positive solutions for systems involving the p-Laplacian, Positivity, 11 (2007), 351-356 ## A. Djellit, S. Tas, On some nonlinear elliptic systems, Nonlinear Analysis: Theory, Methods & Applications, 59 (2004), 695-706 ## D. D. Hai, Uniqueness of positive solutions for a class of semilinear elliptic systems, Nonlinear Analysis: Theory, Methods & Applications, 52 (2003), 596-603
Quasi-Permutation Representations for the Borel and Maximal Parabolic Subgroups of $$Sp(4,2^n)$$ Quasi-Permutation Representations for the Borel and Maximal Parabolic Subgroups of $$Sp(4,2^n)$$ en en 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 $$SP(4,2^f)$$ . 165 175 M. Ghorbany General linear group Quasi-permutation. Article.8.pdf  H. Behravesh , Quasi-permutation representations of p-groups of class 2 , J. London Math. Soc., 55 (1997), 251-260 ## J. M. Burns, B. Goldsmith, B. Hartley, R. Sandling, On quasi-permutation representations of finite groups, Glasgow Math. J., 36 (1994), 301-308 ## M. R. Darafsheh, M. Ghorbany, A. Daneshkhah, H. Behravesh, Quasi-permutation representation of the group $$GL(2,q)$$, Journal of Algebra , 243 (2001), 142-167 ## M. R. Darafsheh, M. Ghorbany, Quasi-permutation representations of the groups $$SU (3, q^2)$$ and $$PSU(3,q^2 )$$, Southest Asian Bulletin of Mathemetics, 26 (2003), 395-406 ## M. R. Darafsheh, M. Ghorbany, Quasi-permutation representations of the groups $$SL(3,q)$$ and $$PSL(3,q)$$, Iranian Journal of Science and Technology Trans. A -Sci., 26 (2002), 145-154 ## H. Enomoto, The characters of the finite symplectic group $$SP(4, q), q = 2^f$$, Osaka J. Math., 9 (1972), 75-94 ## M. Ghorbany, Special representations of the group $$G_2(2^n)$$ with minimal degrees, Southest Asian Bulletin of Mathemetics , 30 (2006), 663-670 ## W. J. Wong, Linear groups analogous to permutation groups, J. Austral. Math. Soc. (Sec. A), 3 (1963), 180-184
Estimating the Average Worth of a Subset Selected from Binomial Populations Estimating the Average Worth of a Subset Selected from Binomial Populations en en 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. 236 245 Riyadh Al-Mosawi binomial populations selected subset average worth estimation Article.15.pdf  R. Al-Mosawi, P. Vellaisamy, A. Shanubhogue, Risk-Unbiased estimation of the selected subset of Poisson populations, Journal of Indian Statistical Association, Vol. 49, (2011) ## R. R. Al-Mosawi, A. Shanubhogue, P. Vellaisamy, Average worth estimation of the selected subset of Poisson populations, Statistitcs, 46 (2012), 813-831 ## J. D. Gibbons, I. Olkin, M. Sobel, Selecting and ordering populations: a new statistical methodology.Society for Industrial and Applied Mathematics (SIAM), SIAM, Philadelphia (1999) ## S. S. Gupta, On some multiple decision (selection and ranking) rules, Technometrics, 7 (1965), 225-245 ## S. S. Gupta, S. Panchapakesan, Multiple decision procedures: theory and methodology of selection and ranking populations. Society for Industrial and Applied Mathematics (SIAM), SIAM, Philadelphia (2002) ## S. Jeyarathnam, S. Panchapakesan, An estimation problem relating to subset selection for normal populations, Design of Experiments: Ranking and Selection (Technical rept.), New York (1983) ## S. Jeyarathnam, S. Panchapakesan, Estimation after subset selection from exponential populations, Communications in Statistics-Theory and Methods, 15 (1986), 3459-3473 ## S. Kumar, A. K. Mahapatra, P. Vellaisamy, Relaibility estimation of the selected exponential populations, Statistics & Probability Letters, 52 (2009), 305-318 ## E. L. Lehmann, G. Casella, Theory of point estimation, Springer-Verlag, New York (1998) ## P. Vellaisamy, Average worth and simulatneous estimation of the selected subset, Ann. Inst. Statist. Math., 44 (1992), 551-562 ## P. Vellaisamy, On UMVUE estimation following selection, Comm. Statist--Theory Methods, 22 (1993), 1031-1043 ## P. Vellaisamy, Simultaneous estimation of the selected subset of uniform populations, J. Appl. Statist.Sci., 5 (1996), 39-46 ## P. Vellaisamy, R. R. Al-Mosawi, Simultaneous estimation of Poisson means of the selected subset, J. Statist. Plann. Infer., 140 (2010), 3355-3364
The Nehari Manifold for a Quasilinear Elliptic Equation with Singular Weights and Nonlinear Boundary Conditions The Nehari Manifold for a Quasilinear Elliptic Equation with Singular Weights and Nonlinear Boundary Conditions en en Using the technique of Brown and Wu ; we present a note on the paper  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 $$\overline{\Omega}$$. 262 277 S. H. Rasouli K. Fallah Quasilinear elliptic problem Singular weights Nehari manifold Nonlinear boundary condition. Article.17.pdf  C. O. Alves, A. El Hamidi, Nehari manifold and existence of positive solutions tob a class of quasilinear problems, Nonlinear Analysis: Theory, Methods & Applications, 60 (2005), 611-624 ## A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543 ## H. Amman, J. Lopez-Gomez, A priori bounds and multiple solution for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374 ## D. Arcoya, J. I. Diaz, S-shaped bifurcation branch in a quasilinear multivalued model arising in climatology, J. Differential Equations, 150 (1998), 215-225 ## C. Atkinson, K. El-Ali, Some boundary value problems for the Bingham model, J. Non-Newtonian Fluid Mech., 41 (1992), 339-363 ## C. Atkinson, C. R. Champion, On some boundary value problems for the equation $$\nabla(F(|\nabla w|)\nabla w)=0$$, Proc. R. Soc. Lond. A, 448 (1995), 269-279 ## P. A. Binding, Y. X. Huang, P. Drábek, Existence of multiple solutions of critical quasilinear elliptic Neuman problems, Nonlinear Analysis: Theory, Methods & Applications, 42 (2000), 613-629 ## P. A. Binding, P. Drabek, Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations, 1997 (1997), 1-11 ## K. J. Brown, The Nehari manifold for a semilinear elliptic equation involving a sublinear term, Calculus of variations and partial differential equations, 22 (2004), 483-494 ## K. J. Brown, T. F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations, 69 (2007), 1-9 ## K. J. Brown, T. F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function, J. Math. Anal. Appl., 337 (2008), 1326-1336 ## K. J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499 ## M. del Pino, C. Flores, Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains, Comm. Partial Differential Equations, 26 (2001), 2189-2210 ## J. I. Diaz, Nonlinear partial differential equations and free boundaries, Elliptic Equations, Boston (1985) ## P. Drabek, S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Royal Soc. Edinburgh Soc. A, 127 (1997), 721-747 ## J. F. Escobar, Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate, Comm. Pure Appl. Math., 43 (1990), 857-883 ## J. F. Bonder, J. D. Rossi, Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains, Comm. Pure Appl. Anal., 1 (2002), 359-378 ## J. F. Bonder, E. Lami-Dozo, J. D. Rossi , Symmetry properties for the extremals of the Sobolev trace embedding, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 795-805 ## J. F. Bonder, S. Martínez, J. D. Rossi, The behavior of the best Sobolev trace constant and extremals in thin domains, J. Differential Equations, 198 (2004), 129-148 ## P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150 ## T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270 ## T. F. Wu, A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential, Electron. J. Differential Equations, 131 (2006), 1-15 ## T. F. Wu, Multiplicity of positive solution of p-Laplacian problems with sign-changing weight function, Int. J. Math. Anal., 1 (2007), 557-563 ## T. F. Wu, Multiplicity results for a semilinear elliptic equation involving sign-changing weight function, Rocky Mountain J. Math., 2009 (2009), 995-1011