Journal of Prime Research in Mathematics

Abstract: In this paper we try to connect the Grassmannian subcomplex defined over the projective differential map \(\acute{d}\) and the variant of Cathelineau’s complex. To do this we define some morphisms over the configuration space for both weight 2 and 3. we also prove the commutativity of corresponding diagrams.

Abstract: Let \(G\) be a finite non-abelian group and \(g\) a fixed element of \(G\). In 2014, Tolue et al. introduced the g-noncommuting graph of \(G\), which was denoted by \(Γ^{g}_G\) with vertex set \(G\) and two distinct vertices \(x\) and \(y\) join by an edge if \([x, y] \neq g\) and \(g^{−1}\). In this paper, we consider induced subgraph of \(Γ^{g}_{G}\) on \(G /Z(G)\) and survey some graph theoretical properties like connectivity, the chromatic and independence numbers of this graph associated to symmetric, alternating and dihedral groups.

Abstract: In this paper, we enumerate self-dual AG-groupoids up to order 6, and classify them on the basis of commutativity and associativity. A self-dual AG-groupoid-test is introduced to check an arbitrary AG-groupoid for a self-dual AG-groupoid. We also respond to an open problem regarding cancellativity of an element in an AG-groupoid. Some features of ideals in self-dual AG-groupoids are explored. Some desired algebraic structures are constructed from the known ones subject to certain conditions and some subclasses of self-dual AG-groupoids are introduced.

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.

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.

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.