On the mixed hodge structure associated hypersurface singularities

JPRM-Vol. 1 (2015), Issue 1, pp. 137 – 161 Open Access Full-Text PDF
Mohammad Reza Rahmati
Abstract: Let $$f : \mathbb{C}^{n+1} → \mathbb{C}$$ be a germ of hypersurface with isolated singularity. One can associate to f a polarized variation of mixed Hodge structure $$H$$ over the punctured disc, where the Hodge filtration is the limit Hodge filtration of W. Schmid and J. Steenbrink. By the work of M. Saito and P. Deligne the VMHS associated to cohomologies of the fibers of $$f$$ can be extended over the degenerate point $$0$$ of disc. The new fiber obtained in this way is isomorphic to the module of relative differentials of $$f$$ denoted $$Ω_f$$ . A mixed Hodge structure can be defined on $$Ω_f$$ in this way. The polarization on \mathcal{H} deforms to Grothendieck residue pairing modified by a varying sign on the Hodge graded pieces in this process. This also proves the existence of a Riemann-Hodge bilinear relation for Grothendieck pairing and allow to calculate the Hodge signature of Grothendieck pairing.
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Existence and non existence of mean cordial labeling of certain graphs

JPRM-Vol. 1 (2015), Issue 1, pp. 123 – 136 Open Access Full-Text PDF
R. Ponraj, S. Sathish Narayanan
Abstract: Let f be a function from the vertex set $$V (G)$$ to $${0, 1, 2}$$. For each edge $$uv$$ assign the label $$\frac{f(u)+f(v)}{2}$$. $$f$$ is called a mean cordial labeling if $$|v_f (i) − v_f (j)| ≤ 1$$ and $$|e_f (i) − e_f (j)| ≤ 1$$, $$i, j ∈ {0, 1, 2}$$, where $$v_f (x)$$ and $$e_f (x)$$ respectively denote the number of vertices and edges labeled with $$x$$ $$(x = 0, 1, 2)$$. A graph with a mean cordial labeling is called a mean cordial graph. In this paper we investigate mean cordial labeling behavior of prism, $$K_2 +\overline{K_m}$$, $$K_n + \overline{2K_2}$$, book $$B_m$$ and some snake graphs.
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$$A_19/B_6$$: A new lanczos-type algorithm and its implementation

JPRM-Vol. 1 (2015), Issue 1, pp. 106 – 122 Open Access Full-Text PDF
Zakir Ullah, Muhammad Farooq, Abdellah Salhi
Abstract: Lanczos-type algorithms are mostly derived using recurrence relationships between formal orthogonal polynomials. Various recurrence relations between these polynomials can be used for this purpose. In this paper, we discuss recurrence relations A19 and B6 for the choice $$U_i(x) = P_{i}^{(1)}$$, where $$U_i$$ is an auxiliary family of polynomials of exact degree $$i$$. This leads to new Lanczos-type algorithm $$A_19/B_6$$ that shows superior stability when compared to existing algorithms of the same type. This new algorithm is derived and described here. Computational results obtained with it are compared to those of the most robust algorithms of this type namely $$A_12$$,  (A^{new}_12\) $$A_5/B_{10}$$ and $$A_8/B_{10}$$ on the same test problems. These results are included.
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A study of the flow of non-newtonian fluid between heated parallel plates by HAM

JPRM-Vol. 1 (2015), Issue 1, pp. 93 – 105 Open Access Full-Text PDF
H. A. Wahab, Saira Bhatti, Saifullah Khan, Muhammad Naeem ,Sajjad Hussain
Abstract: This paper presents the heat transfer of a third grade fluid between two heated parallel plates for two models: constant viscosity model and Reynold’s model. In both cases the nonlinear energy and momentum equations have been solved by HAM. The graphs for the velocity and temperature profiles are plotted and discussed for various values of the emerging parameters in the problem. The main effect is governed by whether or not the fluid is non-Newtonian and the temperature effects are being referred to have a less dominant role.
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$$(λ, µ)$$-Fuzzy ideals in ternary semirings

JPRM-Vol. 1 (2015), Issue 1, pp. 85 – 90 Open Access Full-Text PDF
T. Anitha, D. Krishaswamy.
Abstract: In this paper we introduce the notion of $$(λ, µ)$$-Fuzzy ternary subsemirings and $$(λ, µ)$$-Fuzzy ideals in ternary semirings which can be regarded as the generalization of fuzzy ternary subsemirings and fuzzy ideals in ternary semirings.
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On multiplication group of an AG-group

JPRM-Vol. 1 (2015), Issue 1, pp. 77 – 84 Open Access Full-Text PDF
M. Shah, A. Ali, I. Ahmad, V. Sorge
Abstract: We are investigating the multiplication group of a special class of quasigroup called AG-group. We prove some interesting results such as: The multiplication group of an AG-group of order n is a non-abelian group of order 2n and its left section is an abelian group of order n. The inner mapping group of an AG-group of any order is a cyclic group of order 2.
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The $$t$$-pebbling number of squares of cycles

JPRM-Vol. 1 (2015), Issue 1, pp. 61 – 76 Open Access Full-Text PDF
Lourdusamy Arockiam, Mathivanan Thanaraj
Abstract: Let $$C$$ be a configuration of pebbles on a graph $$G$$. A pebbling move (step) consists of removing two pebbles from one vertex, throwing one pebble away, and moving the other pebble to an adjacent vertex. The $$t$$-pebbling number, $$f_t(G)$$, of a connected graph $$G$$, is the smallest positive integer such that from every configuration of $$f_t(G)$$ pebbles, t pebbles can be moved to any specified target vertex by a sequence of pebbling moves. In this paper, we determine the t-pebbling number for squares of cycles.
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The complement of subgroup graph of a group

JPRM-Vol. 1 (2015), Issue 1, pp. 55 – 60 Open Access Full-Text PDF
F. Kakeri, A. Erfanian
Abstract: Let $$G$$ be a finite group and $$H$$ a subgroup of $$G$$. In 2012, David F. Anderson et al. introduced the subgroup graph of $$H$$ in $$G$$ as a simple graph with vertex set consisting all elements of G and two distinct vertices $$x$$ and $$y$$ are adjacent if and only if $$xy ∈ H$$. They denoted this graphby $$Γ_H(G)$$. In this paper, we consider the complement of $$Γ_H(G)$$, denoted by $$\overline{Γ_H(G)}$$ and will give some graph properties of this graph, for instance diameter, girth, independent and dominating sets, regularity. Moreover, the structure of this graph, planerity and 1-planerity are also investigated in the paper.
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Clique-to-vertex detour distance in graphs

JPRM-Vol. 1 (2015), Issue 1, pp. 42 – 54 Open Access Full-Text PDF
I. Keerthi Asir, S. Athisayanathan
Abstract: Let $$C$$ be a clique and $$v$$ a vertex in a connected graph $$G$$. A clique-to-vertex $$C − v$$ path $$P$$ is a $$u − v$$ path, where u is a vertex in $$C$$ such that $$P$$ contains no vertices of $$C$$ other than $$u$$. The clique-to-vertex distance, $$d(C, v)$$ is the length of a smallest $$C − v$$ path in $$G$$. A $$C − v$$ path of length $$d(C, v)$$ is called a $$C − v$$ geodesic. The clique-to-vertex eccentricity $$e_2(C)$$ of a clique $$C$$ in G is the maximum clique-to-vertex distance from $$C$$ to a vertex $$v ∈ V$$ in $$G$$. The clique-to-vertex radius $$r_2$$ of $$G$$ is the minimum clique-to-vertex eccentricity among the cliques of $$G$$, while the clique-to-vertex diameter $$d-2$$ of $$G$$ is the maximum cliqueto-vertex eccentricity among the cliques of $$G$$. Also The clique-to-vertex detour distance, $$D(C, v)$$ is the length of a longest $$C − v$$ path in $$G$$. A $$C −v$$ path of length $$D(C, v)$$ is called a  (C −v\) detour. The clique-to-vertex detour eccentricity $$e_{D2}(C)$$ of a clique $$C$$ in $$G$$ is the maximum clique-tovertex detour distance from $$C$$ to a vertex $$v ∈ V$$ in  (G\). The clique-to-vertex detour radius $$R_2$$ of $$G$$ is the minimum clique-to-vertex detour eccentricity among the cliques of $$G$$, while the clique-to-vertex detour diameter $$D_2$$ of $$G$$ is the maximum clique-to-vertex detour eccentricity among the cliques of $$G$$. It is shown that $$R_2 ≤ D_2$$ for every connected graph $$G$$ and that every two positive integers a and b with $$2 ≤ a ≤ b$$ are realizable as the clique-tovertex detour radius and the clique-to-vertex detour diameter respectively of some connected graph. Also it is shown that for any two positive integers a and b with $$2 ≤ a ≤ b$$, there exists a connected graph $$G$$ such that $$r_2 = a$$, $$R_2 = b$$ and it is shown that for any two positive integers a and b with $$2 ≤ a ≤ b$$, there exists a connected graph $$G$$ such that $$d_2 = a$$, $$D_2 = b$$.
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On extensions of PVMHS and mixed hodge modules

JPRM-Vol. 1 (2015), Issue 1, pp. 01 – 41 Open Access Full-Text PDF
Mohammad Reza Rahmati
Abstract: We employ the techniques of mixed Hodge modules in order to answer some questions on extension of mixed Hodge structures. Specifically a theorem of M. Saito tells that, the mixed Hodge modules on a complex algebraic manifold X, correspond to polarized variation of mixed Hodge structures on Zariski open dense subsets of X. In this article we concern with the minimal extension of MHM or PVMHS related to this criteria. In [26] we studied the extension of VMHS associated to isolated hypersurface singularities. This article generalizes some of the results there to the admissible VMHS on open dense submanifolds. Some applications to the Neron models of Hodge structures are also given. A short discussion on abelian positivity in the positive characteristic and of height pairing on arithmetic varieties have been included.
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