Infinite Log-Concavity and r-Factor

D. Uminsky and K. Yeats [6] studied the properties of the log- operator L on the subset of the finite symmetric sequences and prove the existence of an infinite region R, bounded by parametrically de- fined hypersurfaces such that any sequence corresponding a point of R is infinitely log concave. We study the properties of a new operator L_r and redefine the hypersurfaces which generalizes the one defined by Uminsky and Yeats [6]. We show that any sequence corresponding a point of the region R, bounded by the new generalized parametrically defined r-factor hypersurfaces, is Generalized r-factor infinitely log concave. We also give an improved value of r_0 found by McNamara and Sagan [4] as the log-concavity criterion using the new log-operator

(1−2uk)(1-2u^k)-constacyclic codes over Fp+uFp+u2F+u3Fp+⋯+ukFp\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_+u^{3}\mathbb{F}_{p}+\dots+u^{k}\mathbb{F}_{p}

Let $\mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $\mathcal{R}=\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_p+u^{3}\mathbb{F}_{p}+\cdots+u^{k}\mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem

Gravitational Dust Collapse in f(R)f(R) Gravity

This paper is devoted to investigate gravitational collapse of dust in metric $f(R)$ gravity. We take FRW metric for the interior region while the Schwarzchild spacetime is considered for the exterior region of a star. The junction conditions have been derived to match interior and exterior spacetimes. The assumption of constant scalar curvature is used to find a solution of field equations. Gravitational mass is found by using the junction conditions. It is concluded that the constant curvature term $f(R_0)$ plays the role of the cosmological constant involved in the field equations of general relativity.Comment: 17 Page

Admissible local systems for a class of line arrangements

A rank one local system \LL on a smooth complex algebraic variety $M$ is admissible roughly speaking if the dimension of the cohomology groups H^m(M,\LL) can be computed directly from the cohomology algebra H^*(M,\C). We say that a line arrangement \A is of type \CC_k if $k \ge 0$ is the minimal number of lines in \A containing all the points of multiplicity at least 3. We show that if \A is a line arrangement in the classes \CC_k for $k\leq 2$, then any rank one local system \LL on the line arrangement complement $M$ is admissible. Partial results are obtained for the class \CC_3.Comment: 9 pages, 2figure

On the Structure of Involutions and Symmetric Spaces of Quasi Dihedral Group

Let $G=QD_{8k}~$ be the quasi-dihedral group of order $8n$ and $\theta$ be an automorphism of $QD_{8k}$ of finite order. The fixed-point set $H$ of $\theta$ is defined as $H_{\theta}=G^{\theta}=\{x\in G \mid \theta(x)=x\}$ and generalized symmetric space $Q$ of $\theta$ given by Q_{\theta}=\{g\in G \mid g=x\theta(x)^{-1}~\mbox{for some}~x\in G\}. The characteristics of the sets $H$ and $Q$ have been calculated. It is shown that for any $H$ and $Q,~~H.Q\neq QD_{8k}.$ the $H$-orbits on $Q$ are obtained under different conditions. Moreover, the formula to find the order of $v$-th root of unity in $\mathbb{Z}_{2k}$ for $QD_{8k}$ has been calculated. The criteria to find the number of equivalence classes denoted by $C_{4k}$ of the involution automorphism has also been constructed. Finally, the set of twisted involutions $R=R_{\theta}=\{~x\in G~\mid~\theta(x)=x^{-1}\}$ has been explored.Comment: major revisio

Solution of Certain Pell Equations

Let $a,b,c$ be any positive integers such that $c\mid ab$ and $d_i^\pm$ is a square free positive integer of the form $d_i^\pm=a^{2k} b^{2l}\pm i c^m$ where $k,l \geq m$ and $i=1,2.$ The main focus of this paper to find the fundamental solution of the equation $x^2-d_i^\pm y^2=1$ with the help of the continued fraction of $\sqrt{d_i^\pm}.$ We also obtain all the positive solutions of the equations $x^2-d_i^\pm y^2=\pm 1$ and $x^2-d_i^\pm y^2=\pm 4$ by means of the Fibonacci and Lucas sequences. Furthermore, in this work, we derive some algebraic relations on the Pell form $F_{d_i^\pm}(x, y) = x^2-d_i^\pm y^2$ including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation $F_{\Delta_{d_i^\pm}} (x, y) = 1$ in terms of $d_i^\pm. We generalized all the results of the papers [2], [9], [26], and [37].Comment: 16 page

Cylindrically Symmetric Solutions in f(R,T)f(R,T) Gravity

The main purpose of this paper is to investigate the exact solutions of cylindrically symmetric spacetime in the context of $f(R,T)$ gravity [1], where $f(R,T)$ is an arbitrary function of Ricci scalar $R$ and trace of the energy momentum tensor $T$. We explore the exact solutions for two different classes of $f(R,T)$ models. The first class $f(R,T)=R+2f(T)$ yields a solution which corresponds to an exterior metric of cosmic string while the second class $f(R,T)=f_1(R)+f_2(T)$ provides an additional solution representing a non-null electromagnetic field. The energy densities and corresponding functions for $f(R,T)$ models are evaluated in each case.Comment: 18 Pages. arXiv admin note: text overlap with arXiv:1506.0869

Algebraic characterization of the SSC Δs(Gn,r1)\Delta_s(\mathcal{G}_{n,r}^{1})

In this paper, we characterize the set of spanning trees of $\mathcal{G}_{n,r}^1$ (a simple connected graph consisting of $n$ edges, containing exactly one $1$-edge-connected chain of $r$ cycles $\mathbb{C}_r^1$ and $\mathcal{G}_{n,r}^{1}\setminus\mathbb{C}_r^1$ is a forest). We compute the Hilbert series of the face ring $k[\Delta_s (\mathcal{G}_{n,r}^1)]$ for the spanning simplicial complex $\Delta_s (\mathcal{G}_{n,r}^1)$. Also, we characterize associated primes of the facet ideal $I_{\mathcal{F}} (\Delta_s (\mathcal{G}_{n,r}^1))$. Furthermore, we prove that the face ring $k[\Delta_s(\mathcal{G}_{n,r}^{1})]$ is Cohen-Macaulay.Comment: 12 page

On Algebraic Characterization of SSC of the Jahangir's Graph Jn,m\mathcal{J}_{n,m}

In this paper, some algebraic and combinatorial characterizations of the spanning simplicial complex $\Delta_s(\mathcal{J}_{n,m})$ of the Jahangir's graph $\mathcal{J}_{n,m}$ are explored. We show that $\Delta_s(\mathcal{J}_{n,m})$ is pure, present the formula for $f$-vectors associated to it and hence deduce a recipe for computing the Hilbert series of the Face ring $k[\Delta_s(\mathcal{J}_{n,m})]$. Finaly, we show that the face ring of $\Delta_s(\mathcal{J}_{n,m})$ is Cohen-Macaulay and give some open scopes of the current work.Comment: arXiv admin note: text overlap with arXiv:1509.0430

Counting Of Binary Matrices Avoiding Some 2 × 2 Matrices

The number of matrices avoiding certain types of matrices is NP-hard in general. In this paper the binary matrices are considered. In particular, the problem of finding the total number of special binary matrices avoiding some types of 2 × 2 matrices is the main objective of this paper. The solution of the problem is given under some constraints as well as under general situation. The formula for the special binary matrices is obtained for total count of matrices of order n × k and also obtained the formula for special binary matrices avoiding some matrices of order 2 × 2. The formula is obtained in terms of the Catalan numbers