Article ID 0066 Published: 18 November 2020 Article
Connecting specific maps having two equal-sized faces and its genus
Graham Higman introduced the concept of januarial as a specific map having two equal sized faces under the action of $\langle x, y : x^2 = y^k = (xy)^l = 1\rangle$ on a finite set. In this paper we take up the question posed by Graham Higman that what is the maximum number of circuits in the subgraph of a simple januarial for any value of $k$? We describe conditions under which januarials are connected and larger januarials are obtained. In an effort to look at topological features of the connected januarial, we find out genus of the januarial, genera of the two faces and number of circuits.
Article ID 0065 Published: 13 November 2020 Article
Ovoidal fibrations in $PG(3, q)$, $q$ even
N S NARASIMHA SASTRY R P SHUKLA
Given a partition of a projective 3-space of odd cardinality by a set of ovoids,a line secant to one of the ovoids of the partition, and its polar relative to the symplectic polarity on the projective 3-space defined by this ovoid, are tangent to distinct ovoids of the partition (Theorem 2). The proof uses the fact that the radical of the linear code generated by the duals of the hyperbolic quadrics in a symplectic generalized quadrangle is of codimension one (Theorem 4).
Article ID 0064 Published: 13 November 2020 Article
On the automorphism group of doubled Grassmann graphs
In this paper, we study a class of regular graphs, which is related to the Grassmann graph. This class of graphs is called the doubled Grassman graph. The Grassmann graph is the class of graphs, which is defined similar to the Johnson graph. This time, however, we are concerned with the subspaces of a vector space, rather than with the subsets of a set. The doubled Grassmann graph is constructed from the Grassmann graph. This graph, was discovered by Biggs and Gardiner. In this paper, we determine the full automorphism group of the doubled Grassmann graph. For determining the automorphism group of the doubled Grassmann graph, we will use the methods used by Mirafzal (Proc. Indian Acad. Sci. (Math Sci.) 129(3) (2019), Art. 34, 8).
Article ID 0063 Published: 20 October 2020 Article
Proof of a supercongruence conjectured by Sun through a q-microscope
In 2011, Sun (Sci. China Math. 54 (2011) 2509--2535) made the following conjecture: for any odd prime $p$ and odd integer $m$,
$$\frac{1}{m^2\left({m-1}\atop{(m-1)/2}\right)}\left(\sum^{(mp-1)/2}_{k=0}\frac{{\left(2k\atop k\right) k}}{8^k}-\left(\frac{2}{p}\right)\sum^{(m-1)/2}_{k=0}\frac{\left({2k}\atop{k}\right)}{8^k}\right)\equiv 0 ({\rm mod}\, p^2).$$
By applying the creative microscoping method introduced by Guo and Zudilin (Adv. Math. 346 (2019) 329--35), we confirm the above conjecture of Sun.
Article ID 0062 Published: 7 October 2020 Article
In this paper, our purpose is to study the approximate controllability of abstract nonlocal neutral integro-differential equations with finite delay in a Hilbert space using the resolvent operator theory. We derive a variation of parameters formulafor representing a solution of the given neutral integro-differential system in the form of resolvent operators and then define a mild solution of the system. We also study the existence of a mild solution of the system with the help of resolvent operator theory. The fractional power theory, α-norm, resolvent operator theory, semigroup theory and Krasnoselskii’s fixed point theorem are used to prove the approximate controllability of the system. Finally, we illustrate our results with the help of an example.
Article ID 0061 Published: 7 October 2020 Article
On Euler–Ramanujan formula, Dirichlet series and minimal surfaces
RUKMINI DEY RISHABH SARMA RAHUL KUMAR SINGH
In this paper, we rewrite two forms of an Euler–Ramanujan identity in terms of certain Dirichlet series and derive functional equation of the latter.We also use the Weierstrass–Enneper representation of minimal surfaces to obtain some identitiesinvolving these Dirichlet series and one complex parameter.
Article ID 0060 Published: 30 September 2020 Article
In Rani et al. (Numerical inversion of Laplace transform based on Bernsteinoperational matrix, Mathematical Methods in the Applied Sciences (2018) pp. 1–13), a numerical method is developed to find the inverse Laplace transform of certain functions using Bernstein operational matrix. Here, we describe Bernstein operational matrix of integration and propose an algorithm to solve linear time-varying systems governing differential equations. Apart from discussing error estimate, the method is implemented to linear differential equations on Bessel equation of order zero, damped harmonic oscillator, some higher order differential equations, singular integral equation, Volterra integral and integro-differential equations and nonlinear Volterra integral equations of the first kind. A comparison with some existing methods like Haar operational matrix, block pulse operational matrix and others are discussed. The method is simple and easy to implement on a variety of problems. Relative errors estimate just for 5th or 6th approximation show high applicability of the method.
Article ID 0059 Published: 25 September 2020 Article
Supercongruences for sums involving fourth power of some rising factorials
In this paper, we give proof of certain recent conjectural supercongruenceson sums involving fourth power of some rising factorials.
Article ID 0058 Published: 23 September 2020 Article
The $x$-coordinates of Pell equations and sums of two Fibonacci numbers II
Let $\{Fn\}_{n≥0}$ be the sequence of Fibonacci numbers defined by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for all $n\geq 0$. In this paper, for an integer $d \geq 2$ which is square-free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^2 − dy^2 = \pm 4$ which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize.
Article ID 0057 Published: 19 September 2020 Article
On the automorphism groups of connected bipartite irreducible graphs
Let $G = (V, E)$ be a graph with the vertex-set $V$ and the edge-set $E$. Let $N(v)$ denote the set of neighbors of the vertex $v$ of $G$. The graph $G$ is called irreducible whenever for every $v,w\in V$ if $v\neq w$, then $N(v) \neq N(w)$. In this paper, we present a method for finding automorphism groups of connected bipartite irreducible graphs. Then, by our method, we determine automorphism groups of some classes of connected bipartite irreducible graphs, including a class of graphs which are derived from Grassmann graphs. Let $a_0$ be a fixed positive integer. We show that if $G$ is a connected non-bipartite irreducible graph such that $c(v,w) = \mid N(v)∩ N(w)\mid = a_0$ when $v, w$ areadjacent, whereas $c(v,w) \neq a_0$, when $v, w$ are not adjacent, then $G$ is a stable graph, that is, the automorphism group of the bipartite double cover of $G$ is isomorphic with the group ${\rm Aut}(G) \times \mathbb{Z}_2$. Finally, we show that the Johnson graph $J (n, k)$ is a stable graph.
Article ID 0056 Published: 19 September 2020 Article
On ramification index of composition of complete discrete valuation fields
For an extension $L/K$ of discrete valuation fields, let $e_{L/K}$ , $\mathfrak{O}_L$ denote the ramification index and valuation ring of $L/K$ respectively. Let $K$ be a complete discrete valuation field and $L_1/K$, $L_2/K$ be finite linearly disjoint extensions over $K$. We show that if $\mathfrak{O}_{L_1L_2}=\mathfrak{O}_{L_1}\mathfrak{O}_{L2}$ or gcd$(e_{L_1/K} , e_{L_2/K}) = 1$, and one of the residue fields $l_1/k$, $l_2/k$ is separable, then $e_{L_1L_2/L_1}= e_{L_2/K}$. The analogous results for the residue degrees are also true.
Article ID 0055 Published: 16 September 2020 Article
On Ricci solitons whose potential is convex
CHANDAN KUMAR MONDAL ABSOS ALI SHAIKH
In this paper, we consider theRicci curvature of a Ricci soliton. In particular,we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also it isometrically splits a line. We have also proved that a gradient Ricci soliton with non-constant concave potential function and bounded Ricci curvature is non-shrinking and hence the scalar curvaturehas at most one critical point.
Article ID 0054 Published: 11 September 2020 Article
The exact 2-domination number of generalized Petersen graphs
Let $G = (V, E)$ be a graph. A subset $S \subseteq V$ is a 2-dominating set of $G$ if each vertex in $V − S$ is adjacent to at least two vertices in $S$. The 2-domination number of $G$ is the cardinality of the smallest 2-dominating set of $G$. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs $P(5k+1, 3)$, $P(5k+2, 3)$ and $P(5k +3, 3)$ is $4k +2$, $4k +3$ and $4k +4$, respectively. This proves one conjecture due to Bakhshesh et al. (Proc. Indian Acad. Sci. (Math. Sci.) 128 (2018) 17).
Article ID 0053 Published: 8 September 2020 Article
Lifting to two-term relative maximal rigid subcategories in triangulated categories
Let $\mathscr{C}$ be a triangulated category with shift functor [1] and $\mathscr{R}$ a contravariantly rigid subcategory of $\mathscr{C}$ . We show that a tilting subcategory of $\mod\mathscr{ R}$ lifts to a two-term maximal $\mathscr{R}$[1]-rigid subcategory of $\mathscr{C}$ . As an application, our result generalizes a result by Xie and Liu (Proc. Amer. Math. Soc. 141(10) (2013) 3361–3367) for maximal rigid objects and a result by Fu and Liu (Comm. Algebra 37(7) (2009) 2410–2418) for cluster tilting objects.
Article ID 0052 Published: 18 August 2020 Article
Contact and isocontact embedding of $\pi$-manifolds
We prove some contact analogs of smooth embedding theorems for closed $\pi$-manifolds. We show that a closed, $k$-connected, $\pi$-manifold of dimension $(2n + 1)$ that bounds a $\pi$-manifold, contact embeds in the $(4n − 2k + 3)$-dimensional Euclideanspace with the standard contact structure. We also prove some isocontact embedding results for π-manifolds and parallelizable manifolds.
Article ID 0051 Published: 13 August 2020 Article
A note on bounds of the $p$-length of a $p$-soluble group
HUILONG GU JIAO LI HUAQUAN WEI LIYING YANG
Suppose that the finite group $G = AB$ is a mutually permutable product of two $p$-soluble subgroups $A$ and $B$. By using the Wielandt lengths of Sylow $p$-subgroups of $A$ and $B$, we present a new bound of the $p$-length of $G$. Some known results are improved.
Article ID 0050 Published: 12 August 2020 Article
A planar Tangle is a smooth simple closed curve piecewise defined by quadrants of circles with constant curvature. We can enumerate Tangles by counting their dual graphs, which consist of a certain family of polysticks. The number of Tangles with a given length or area grows exponentially, and we show the existence of their growth constants by comparing Tangles to two families of polyominoes.
Article ID 0049 Published: 12 August 2020 Article
Coloring of cozero-divisor graphs of commutative von Neumann regular rings
M BAKHTYIARI R NIKANDISH M J NIKMEHR
Let $R$ be a commutative ring with non-zero identity. The cozero-divisor graph of $R$, denoted by $\Gamma^{\prime }(R)$, is a graph with vertices in $W^*(R)$, which is the set of all non-zero and non-unit elements of $R$, and two distinct vertices $a$ and $b$ in $W^*(R)$ are adjacent if and only if $a\not\in Rb$ and $b\not\in Ra$. In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.
Article ID 0048 Published: 6 August 2020 Article
On Betti numbers in the linear strand and regularity of triangular graphs
SHAHNAWAZ AHMAD RATHER PAVINDER SINGH ROHIT VERMA
To every finite simple graph $G$, we associate the so-called edge ideal $I (G)$, which is a square-free quadratic monomial ideal generated by the monomials corresponding to the edges of $G$. In this paper, we determine all the initial graded Betti numbers of edge ideals of triangular graphs and alternate triangular graphs in terms of the underlying graph. We also compute the regularity of edge ideals of triangular and alternate triangular graphs.
Article ID 0047 Published: 25 July 2020 Article Special Issue in Honour of Professor C R Rao on His Birth Centenary
Completely mixed bimatrix games
T Parthasarathty Vasudha Sharma A R Sricharan
Kaplansky (Ann. Math. 46(3) (1945) 474–479) introduced the notion of completely mixed games. Fifty years later in 1995, he wrote another beautiful paper where he gave a set of necessary and sufficient conditions for a skew symmetric matrix game to be completely mixed. In this work, we attempt to answer when bimatrix games will be completely mixed. In particular, we give a set of necessary and sufficient condition for a bimatrix game $(A, B)$ to be completely mixed when $A$ and $B$ are skew symmetric matrices of order 3. We give several examples to show the sharpness of our result. We also formulate a conjecture when $A$ and $B$ are skew symmetric matrices of order 5.
Article ID 0046 Published: 25 July 2020 Article Special Issue in Honour of Professor C R Rao on His Birth Centenary
C R Rao and Mahalanobis’ distance
C R Rao has made seminal contributions in many areas in statistics. A review of some of his fundamental contributions towards the development of the theory and application of Mahalanobis distance in classification problems is presented here. The review is based on information provided in some of Rao’s famous research and autobiographical papers.
Article ID 0045 Published: 25 July 2020 Article Special Issue in Honour of Professor C R Rao on His Birth Centenary
Dr C R Rao’s contributions to the advancement of economic science
T Krishna Kumar H D Vinod Suresh Deman
In this paper, the authors review Dr C R Rao’s contributions to statistical foundations in economic science, and the importance of his work in advancing econometric modelling and statistical inference in celebration of his birth centenary.
Article ID 0044 Published: 25 July 2020 Article Special Issue in Honour of Professor C R Rao on His Birth Centenary
This paper summarizes a talk given by the author at the Indian Academy of Sciences, Bengaluru on December 12, 2019. It outlines the multiplicity of situations where measurements on directions in two, three, or more dimensions are observed, and form the basis for answering various scientific questions. After briefly outlining the novelty in dealing with such data and the need for an entirely different set of analytical tools, one of the basic questions that arises before any further inference viz. testing isotropy of circular data, is addressed. The author was initiated into this topic of directional statistics by Professor C R Rao during the 1960s and is pertinent to this occasion celebrating him.
Article ID 0043 Published: 25 July 2020 Article Special Issue in Honour of Professor C R Rao on His Birth Centenary
Professor C R Rao’s contributions in statistical signal processing and its long-term implications
Professor C R Rao has made significant contributions in different areas of statistics and in related fields particularly in inference, biometrics, design of experiments, linear models, variance components, econometrics, and most of his contributions are well known to the statistical community. But it may not be known to many statisticians that Professor Rao has worked quite extensively, for about six to seven years, on statistical signal processing and has made some fundamental contributions which has generated significant interest along that direction. He along with his collaborators have worked mainly on three classical signal processing problems, and provided theoretical foundations and efficient estimation procedures. In my opinion, the main contribution of Professor Rao is that he has provided new insights in all these problems, which has helped to bring new way of solving these and some related problems. He has guided two Ph.D. students in the area of Statistical Signal Processing. The main aim of this article istwo-fold. First, we would like to introduce to the statistical community the contributionof Professor Rao in this area and our second aim is to provide a class of different relatedopen problems which are of interest and which require sophisticated statistical tools toprovide efficient solutions.
Article ID 0042 Published: 25 July 2020 Article Special Issue in Honour of Professor C R Rao on His Birth Centenary
On foundation of statistical inference by C R Rao relating to information inequality
This is the birth centenary year of the living legend and giant in the world of statistics, Prof. Calyampudi Radhakrishna (C R) Rao. This article is a partial reflection of Dr. Rao’s contributions to statistical theory and methodology, including unbiased estimation, variance reduction by sufficiency, efficiency of estimation, information geometry, as well as the application of matrix theory in linear statistical inference. His contributions are so massive, it will be impossible to review all of them. This article will comprise mostly his major discovery, which is Cramér–Rao lower bound and related applications and extensions.
Article ID 0041 Published: 24 July 2020 Article Special Issue in Honour of Professor C R Rao on His Birth Centenary
Cramér–Rao inequality revisited
Among C R Rao’s many contributions to statistical inference, one which has been and still is considered to be of extreme importance in the areas of statistics, physics and of signal processing in electrical engineering beside other sciences is an inequality which is now known as the Cramér–Rao inequality. This result was studied in recent years by several other scientists to relax the conditions under which it holds and to generalize it in different directions. Contributions by Bhattacharya (Sankhya 8 (1946) 1–14, 201–218, 315–328), Barankin (Ann. Math. Statist. 20 (1949) 477–501) and Fabian and Hannan (Ann. Statist. 5 (1977) 197–205) are significant in this area. We do not propose to give an extensive survey of results connected with the inequality. Our aim in this communication is to highlight some recent advances.
Article ID 0040 Published: 24 July 2020 Article Special Issue in Honour of Professor C R Rao on His Birth Centenary
Professor Calyampudi Radhakrishna Rao - Dr.Rao to most of us - was in thefirst batch of Master’s students in Statistics of the Calcutta University. He graduated with marks that remain unsurpassed. His name is etched in the annals of statistical science through Cramér–Rao bound, Rao–Blackwell theorem and Rao’s Score Test. He has been elected to the Fellowship of The Royal Society, awarded the U.S. National Medal of Science and Padma Vibhushan by the Government of India. We celebrate his birth centenary because through his contributions he has elevated statistics as an indispensable applied tool in all walks of life, with firm theoretical foundations.
Article ID 0039 Published: 20 July 2020 A message from C. R. Rao Special Issue in Honour of Professor C R Rao on His Birth Centenary
Article ID 0038 Published: 20 July 2020 Preface Special Issue in Honour of Professor C R Rao on His Birth Centenary
Article ID 0037 Published: 20 June 2020 Research Article
For two natural numbers $1$ < $p1$ < $p2$, with $\alpha = \frac{log(p1)} {log(p2)}$ irrational, we describe in the Main Theorem $\Omega$ and in Note 1.5, the factorization of two adjacent numbers in the multiplicatively closed subset $S = \{p^{i}_{1} p^{j}_{2}\,|\, i, j \in \mathbb{N}\cup\{0\}\}$ using primary and secondary convergents of $\alpha$. This suggests the general Question 1.2 for more than two generators which is still open.
Article ID 0036 Published: 15 June 2020 Research Article
A limit set intersection theorem for graphs of relatively hyperbolic groups
Let $G$ be a relatively hyperbolic group that admits a decomposition intoa finite graph of relatively hyperbolic groups structure with quasi-isometrically (qi)embedded condition. We prove that the set of conjugates of all the vertex and edge groups satisfy the limit set intersection property for conical limit points (refer to Definition 3 and Definition 23 for the definitions of conical limit points and limit set intersection property respectively). This result is motivated by the work of Sardar for graph of hyperbolic groups [16].
Article ID 0035 Published: 15 June 2020 Research Article
A submultiplicative property of the Carathéodory metric on planar domains
AMAR DEEP SARKAR KAUSHAL VERMA
Given a pair of smoothly bounded domains $D_{1}$, $D_{2} \subset \mathbb{C}$, the purpose of this paper is to obtain an inequality that relates the Carathéodory metrics on $D_{1}$, $D_{2}$, $D_{1}\cap D_{2}$ and $D_{1} \cup D_{2}$.
Article ID 0034 Published: 11 June 2020 RESEARCH ARTICLE
Picard bundle on the moduli space of torsionfree sheaves
Let $Y$ be an integral nodal projective curve of arithmetic genus $g\geq2$ with $m$ nodes defined over an algebraically closed field. Let $n$ and $d$ be mutually coprime integers with $n\geq2$ and $d$ > $n(2g −2)$. Fix a line bundle $L$ of degree $d$ on $Y$ .We prove that the Picard bundle $E_{L}$ over the ‘fixed determinant moduli space’ $U_{L}(n, d)$ is stable with respect to the polarisation $\theta_{L}$ and its restriction to the moduli space $U'_{L}(n, d)$, of vector bundles of rank $n$ and determinant $L$, is stable with respect to any polarisation. There is an embedding of the compactified Jacobian $\bar{J}(Y)$ in the moduli space $U_{Y}(n, d)$ of rank $n$ and degree $d$. We show that the restriction of the Picard bundle of rank $ng$ (over $U_{Y}(n, n(2g − 1)))$ to $\bar{J}(Y)$ is stable with respect to any theta divisor $\theta_{\bar{J}(Y)}$.
Article ID 0033 Published: 7 May 2020 Research Article
Blow up property for viscoelastic evolution equations on manifolds with conical degeneration
MOHSEN ALIMOHAMMADY MORTEZA KOOZEHGAR KALLEJI
This paper is concerned with the study of nonlinear viscoelastic evolutionequation with strong damping and source terms, described by $$u_{tt} − \Delta_{\mathbb{B}}u +\int^{t}_{0}\,g(t − \tau)\Delta_{\mathbb{B}}u(\tau)d\tau + f (x)u_{t} |u_{t}|^{m−2}\\ = h(x)|u|^{p−2}u, \,\,\,\,x \in int\,\mathbb{B}, t > 0,$$
where $\mathbb{B}$ is a stretched manifold. First, we prove the solutions of problem (1.1) in the cone Sobolev space $\mathcal{H}^{1,\frac{n}{2}}_{2,0} (\mathbb{B})$, which admit a blow up in finite time for $p$ > $m$ and positive initial energy. Then, we construct a lower bound for obtaining blow up time under appropriate assumptions on data.
Article ID 0032 Published: 7 May 2020 Research Article
New criteria for Vandiver’s conjecture using Gauss sums – Heuristics and numerical experiments
The link between Vandiver’s conjecture and Gauss sums is well known sincethe papers of Iwasawa (Symposia Mathematica, vol 15, Academic Press, pp 447–459,1975), Thaine (Mich Math J 42(2):311–344, 1995; Trans Am Math Soc 351(12):4769–4790, 1999) and Anglès and Nuccio (Acta Arith 142(3):199–218, 2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver’s conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of the $p$-th cyclotomic field (Sur la $p$-ramification abélienne (1984) vol. 20, pp. 1–26). Then we provide a specific use of Gauss sums of characters of order $p$ of $\mathbb{F}^{\times}_{ell}$ and prove new criteria for Vandiver’s conjecture to hold (Theorem 2 (a) using both the sets of exponents of $p$ irregularity and of $p$-primarity of suitable twists of the Gauss sums, and Theorem 2 (b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in $\S$5.2 new heuristics showing that any counter example to the conjecture leads to excessive constraints modulo $p$ on the above twists as $\ell$ varies and suggests analytical approaches to evidence. We perform numerical experiments to strengthen our arguments in the direction of the very probable truth of Vandiver’s conjecture and to inspire future research. The calculations with their PARI/GP programs are given in appendices.
Article ID 0031 Published: 1 May 2020 Research Article
Computing $n$-th roots in $\rm{SL_{2}}$ and Fibonacci polynomials
Let $k$ be a field of characteristic $\neq$2. In this paper, we study squares, cubesand their products in split and anisotropic groups of type $A_{1}$. In the split case,we show that computing $n$-th roots is equivalent to finding solutions of certain polynomial equations in at most two variables over the base field $k$. The description of these polynomials involves generalised Fibonacci polynomials. Using this we obtain asymptotic proportions of $n$-th powers, and conjugacy classes which are $n$-th powers, in $SL_{2}(\mathbb{F}_{q})$ when $n$ is a prime or $n = 4$. We also extend the already known Waring type result for $SL_{2}(\mathbb{F}_{q})$, that every element of $SL_{2}(\mathbb{F}_{q})$ is a product of two squares, to $SL_{2}(k)$ for an arbitrary $k$. For anisotropic groups of type $A_{1}$, namely $SL_{1}(Q)$ where $Q$ is a quaternion division algebra, we prove that when 2 is a square in $k$, every element of $SL_{1}(Q)$ is a product of two squares if and only if $−1$ is a square in $SL_{1}(Q)$.
Article ID 0030 Published: 22 April 2020 Research Article
Correction to: Sparse subsets of the natural numbers and Euler’s totient function
MITHUN KUMAR DAS PRAMOD EYYUNNI BHUWANESH RAO PATIL
Correction to: Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:84https://doi.org/10.1007/s12044-019-0512-x
Article ID 0029 Published: 15 March 2020 Research Article
Berger’s formulas and their applications in symplectic mean curvature flow
In this paper, we recall some well known Berger’s formulas. As their applications, we prove that if the local holomorphic pinching constant is $\gamma$ < 2, then there exists a positive constant $\delta$ > $\frac{29(\lambda−1)} {\sqrt{(48−24\lambda)^{2}+(29\lambda−29)^{2}}}$ such that cos $\alpha \geq \delta$ is preserved along the mean curvature flow, improving Li–Yang’s main theorem in Li and Yang (Geom. Dedicata 170 (2014) 63–69). We also prove that when cos $\alpha$ is close enough to 1, then the symplectic mean curvature flow exists globally and converges to a holomorphic curve.
Article ID 0028 Published: 15 March 2020 Research Article
Repdigits as products of two Fibonacci or Lucas numbers
In this study, we show that if $2 \leq m \leq n$ and $F_{m} F_{n}$ represents a repdigit, then $(m, n)$ belongs to the set $$\{(2, 2), (2, 3), (3, 3), (2, 4), (3, 4), (4, 4), (2, 5), (2, 6), (2, 10)\}.$$ Also, we show that if $0 \leq m \leq n$ and $L_{m} L_{n}$ represents a repdigit, then $(m, n)$ belongs to the set $$\left\{ \begin{array}{l}(0, 0), (0, 1), (1, 1), (0, 2), (1, 2), (2, 2), (0, 3),\\ (1, 3), (1, 4), (1, 5), (2, 5), (3, 5), (4, 5) \end{array}\right\}$$
Article ID 0027 Published: 12 March 2020 Research Article
On the number of distinct exponents in the prime factorization of an integer
Let $f (n)$ be the number of distinct exponents in the prime factorization ofthe natural number n.We prove some results about the distribution of $f (n)$. In particular, for any positive integer $k$, we obtain that
$$\{n \leq x : f (n) = k\} \sim A_{k} x$$
and
$$\{n \leq x : f (n) = \omega(n) − k\} \sim \frac{Bx(log log x)^{k}}{k! log x},$$
as $x \rightarrow +\infty$, where $\omega(n)$ is the number of prime factors of $n$ and $A_{k}$, $B$ > 0 are some explicit constants. The latter asymptotic extends a result of Aktas and Ram Murty (Proc. Indian Acad. Sci. (Math. Sci.) 127(3) (2017) 423–430) about numbers having mutually distinct exponents in their prime factorization.
Article ID 0026 Published: 12 March 2020 Research Article
A set of generators for the Picardmodular group $SU(2,1,O_{2})$
MAHBOUBEH GHOSHOUNI MAJID HEYDARPOUR
In this work, we find a system of generators for the Picard modular group $SU(2, 1,O_{2})$. This system contains five transformations, three translations a rotation and an involution.
Article ID 0025 Published: 12 March 2020 Research Article
Repdigits in Euler functions of associated pell numbers
In this paper, we discuss some properties of Euler totient function ofassociated Pell numbers which are repdigits in base 10.
Article ID 0024 Published: 12 March 2020 Research Article
Exponential sums of squares of Fourier coefficients of cusp forms
We prove nontrivial estimates for linear sums of squares of Fourier coefficients of holomorphic and Maass cusp forms twisted by additive characters. For holomorphic forms $f$ , we show that if $|\alpha − a/q| \leq 1/q^{2}$ with $(a, q) = 1$, then for any $\varepsilon$ > 0,
$$\sum_{n\leqslant x}\lambda_{f}(n)^{2}e(n\alpha)\, \ll _{f,\varepsilon} X^{\frac{4}{5}+\varepsilon}\,\, for X^\frac{1}{5}\, \ll \,q \, \ll \, X^\frac{4}{5}.$$
Moreover, for any $\varepsilon$ > 0, there exists a set $S \subset (0, 1)$ with $\mu(S) = 1$ such that for every $\alpha \in S$, there exists $X_{0} = X_{0}(\alpha)$ such that the above inequality holds true for any $\alpha \in S$ and $X \geqslant X_{0}(\alpha)$. A weaker bound for Maass cusp forms is also established.
Article ID 0023 Published: 28 February 2020 Research Article
A ternary diophantine inequality with prime numbers of a special type
We consider the inequality $$|p^{c}_{1}+ p^{c}_{2}+ p^{c}_{3}− N| < (log N)^{−E},$$ where $1$ < $c$ < $\frac{281}{250}$ , $N$ is a sufficiently large real number and $E$ > $0$ is an arbitrarily large constant. We prove that the above inequality has a solution in primes $p_{1}, p_{2}, p_{3}$ such that each of the numbers $p_{1} + 2, p_{2} + 2, p_{3} + 2$ has at most [$\frac{1475}{562−500c}$] prime factors, counted with the multiplicity. This result constitutes an improvement upon that of Tolev.
Article ID 0022 Published: 20 February 2020 Research Article
Palindromic width of graph of groups
KRISHNENDU GONGOPADHYAY SWATHI KRISHNA
In this paper, we answer questions raised by Bardakov and Gongopadhyay (Commun. Algebra 43(11) (2015) 4809–4824). We prove that the palindromic width of HNN extension of a group by proper associated subgroups is infinite. We also prove that the palindromic width of the amalgamated free product of two groups via a proper subgroup is infinite (except when the amalgamated subgroup has index two in each of the factors). Combining these results it follows that the palindromic width of the fundamental group of a graph of groups is mostly infinite.
Article ID 0021 Published: 13 February 2020 Research Article
Upper bound for the first nonzero eigenvalue related to the $p$-Laplacian
Let $M$ be a closed hypersurface in $\mathbb{R}^{n}$ and $\Omega$ be a bounded domain such that $M = \partial\Omega$. In this article, we obtain an upper bound for the first nonzero eigenvalue of the following problems:
(1) Closed eigenvalue problem:
$$\Delta_{p}u = \lambda_{p} |u|^{p−2} u \quad {\rm on}\; M.$$
(2) Steklov eigenvalue problem:
$$\begin{array}{ll} \Delta_{p}u =0 \quad \;\;\;\;\;\;\rm{in} \;\Omega,\\ |\nabla u|^{p−2} \frac{\partial u}{\partial v} = \mu_{p}|u|^{p−2} u \quad {\rm on} \ M. \end{array}$$
These bounds are given in terms of the first nonzero eigenvalue of the usual Laplacianon the geodesic ball of the same volume as of $\Omega$.
Article ID 0020 Published: 30 January 2020 Research Article
Gerstenhaber algebra structure on the cohomology of a hom-associative algebra
A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we define a cup product on the cohomology of a hom-associative algebra. A direct verification shows that this cup product together with the degree−1 graded Lie bracket (which controls the deformation of the hom-associative algebra structure) on the cohomology makes it a Gerstenhaber algebra.
Article ID 0019 Published: 30 January 2020 Research Article
Rigidity of Bott–Samelson–Demazure–Hansen variety for $F_{4}$ and $G_{2}$
S SENTHAMARAI KANNAN PINAKINATH SAHA
Let $G$ be a simple algebraic group of adjoint type over $\mathbb{C}$, whose root system is of type $F_{4}$. Let $T$ be a maximal torus of $G$ and $B$ be a Borel subgroup of $G$ containing $T$. Let $w$ be an element of the Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w$. Let $Z(w, \underline{i})$ be the Bott–Samelson–Demazure–Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w$. In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i})$, where $w_{0}$ is the longest element of the Weyl group $W$. We describe all the reduced expressions of $w_{0}$ in terms of a Coxeter element such that $Z(w_{0}, \underline{i})$ is rigid (see Theorem 7.1). Further, if $G$ is of type $G_{2}$, there is no reduced expression $\underline{i}$ of $w_{0}$ for which $Z(w_{0}, \underline{i})$ is rigid (see Theorem 8.2).
Article ID 0018 Published: 30 January 2020 Research Article
The fundamental group and extensions of motives of Jacobians of curves
SUBHAM SARKAR RAMESH SREEKANTAN
In this paper, we construct extensions of mixed Hodge structure coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth projective pointed curve which correspond to the regulators of certain motivic cohomology cycles on the Jacobian of the curve essentially constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. This is a generalisation of a theorem of Colombo (J. Algebr. Geom. 11(4) (2002) 761–790) where she constructed the extension corresponding to Collino’s cycles in the Jacobian of a hyperelliptic curve.
Article ID 0017 Published: 30 January 2020 Research Article
Prime intersection graph of ideals of a ring
KUKIL KALPA RAJKHOWA HELEN K SAIKIA
Let $R$ be a ring. The prime intersection graph of ideals of $R$, denoted by $G_{P}(R)$, is the graph whose vertex set is the collection of all non-trivial (left) ideals of $R$ with two distinct vertices $I$ and $J$ are adjacent if and only if $I \cap J \neq 0$ and either one of $I$ or $J$ is a prime ideal of $R$. We discuss connectedness in $G_{P}(R)$. The concepts of bipartition, planarity and colorability are interpreted. Finally, we introduce the idea of traversability in $G_{P}(\mathbb{Z}_{n})$. The core part of this paper is observed in the ring $\mathbb{Z}_{n}$.
Article ID 0016 Published: 30 January 2020 Research Article
For positive integers $d$ and $n$, the vector space $H_{d} (x_{1}, x_{2}, . . . , x_{n})$ of homogeneous polynomials of degree $d$ is a representation of the symmetric group $S_{n}$ acting by permutation of variables. Regarding this as a representation for the dihedral subgroup $D_{n}$, we prove a formula for the dimension of all the isotypical subrepresentations. Our formula is simpler than the existing one found by Zamani and Babaei (Bull. Iranian Math. Soc. 40(4) (2014) 863–874). By varying the degrees $d$ we compute the generating functions for these dimensions. Further, our formula leads us naturally to a specific supercharacter theory of $D_{n}$. It turns out to be a $\ast$-product of a specific supercharacter theory studied in depth by Fowler et al. (The Ramanujan Journal (2014)), with the unique supercharacter theory of a group of order 2.
Article ID 0015 Published: 30 January 2020 Research Article
Examples of blown up varieties having projective bundle structures
We give some examples of blow up of projective space along someprojective subvariety, such that these blown up spaces are isomorphic to a projective bundle over some projective space.
Article ID 0014 Published: 30 January 2020 Research Article
In this paper, we introduce mixed coloured permutations, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and combinatorial identities for the counting sequence, for mixed Stirling numbers of the first kind. In this comprehensive study, we consider further the conditions on the length of the cycles, $r$-mixed Stirling numbers and the connection to Bell polynomials.
Article ID 0013 Published: 30 January 2020 Research Article
Markov approximation and the generalized entropy ergodic theorem for non-null stationary process
In an earlier work, we proved a generalized entropy ergodic theorem forfinite nonhomogeneous Markov chains (NMC). In this paper, we establish a generalized strong law of large numbers for finite $m$-th order NMC. Then we deduce a generalized entropy ergodic theorem for finite $m$-th order NMC, under some assumptions on the continuity rate and of non-nullness. Explicit upper and lower bounds relating the generalized relative entropy density of the original finite non-null stationary sequence and its canonical $m$-order Markov approximation is obtained.
Article ID 0012 Published: 10 January 2020 Research Article
Ring endomorphisms satisfying the central reversible property
ARNAB BHATTACHARJEE UDAY SHANKAR CHAKRABORTY
A ring $R$ is called reversible if for $a, b \in R$, $ab = 0$ implies $ba = 0$. These rings play an important role in the study of noncommutative ring theory. Kafkas et al.(Algebra Discrete Math. 12 (2011) 72–84) generalized the notion of reversible rings to central reversible rings. In this paper, we extend the notion of central reversibility of rings to ring endomorphisms. We investigate various properties of these rings and answer relevant questions that arise naturally in the process of development of these rings, and as a consequence many new results related to central reversible rings are also obtained as corollaries to our results.
Article ID 0011 Published: 10 January 2020 Research Article
The group of invertible ideals of a Prüfer ring
Let $R$ be a commutative ring and $\mathcal{I}(R)$ denote the multiplicative group of all invertible fractional ideals of $R$, ordered by $A \leq B$ if and only if $B \subseteq A$. We investigate when there is an order homomorphism from $\mathcal{I}(R)$ into the cardinal direct sum $\coprod_{i\in I} G_{i}$, where $G_{i}$ ’s are value groups, if $R$ is a Marot Prüfer ring of finite character. Furthermore, over Prüfer rings with zero-divisors, we investigate the conditions that make this monomorphism onto.
Article ID 0010 Published: 10 January 2020 Research Article
On weighted signed color partitions
In this paper, we provide combinatorial interpretations of certain provedRogers–Ramanujan type identities using signed color partitions with attached weights. The approach of using the signed color partitions is interesting since negative exponents do not make an explicit appearance in these identities.
Article ID 0009 Published: 10 January 2020 Research Article
Rainbow 2-connectivity of edge-comb product of a cycle and a Hamiltonian graph
MARTIN BACA A N M SALMAN RINOVIA SIMANJUNTAK BETY HAYAT SUSANTI
An edge-colored graph $G$ is rainbow $k$-connected, if for every two verticesof $G$, there are $k$ internally disjoint rainbow paths, i.e., if no two edges of each path are colored the same. The minimum number of colors needed for which there exists a rainbow $k$-connected coloring of $G$, $rc_{k} (G)$, is the rainbow $k$-connection number of $G$. Let $G$ and $H$ be two connected graphs, where $O$ is an orientation of $G$. Let $\vec{e}$ be an oriented edge of $H$. The edge-comb product of $G$ (under the orientation $O$) and $H$ on $\vec{e}$, $G^{o} \vartriangleleft_{\vec{e}} H$, is a graph obtained by taking one copy of $G$ and $|E(G)|$ copies of $H$ and identifying the $i$-th copy of $H$ at the edge $\vec{e}$ to the $i$-th edge of $G$, where the two edges have the same orientation. In this paper, we provide sharp lower and upper bounds for rainbow 2-connection numbers of edge-comb product of a cycle and a Hamiltonian graph. We also determine the rainbow 2-connection numbers of edge-comb product of a cycle with some graphs, i.e. complete graph, fan graph, cycle graph, and wheel graph.
Article ID 0008 Published: 10 January 2020 Research Article
Exceptional set in Waring–Goldbach problem: Two squares, two cubes and two sixth powers
Let $R(n)$ denote the number of representations of an even integer $n$ as thesum of two squares, two cubes and two sixth powers of primes, and by $\mathcal{E}(N)$ we denote the number of even integers $n \leq N$ such that the expected asymptotic formula for $R(n)$ fails to hold. In this paper, it is proved that $\mathcal{E}(N) \ll N^{\frac{127}{288}+\varepsilon}$ for any $\varepsilon$ > 0.
Article ID 0007 Published: 10 January 2020 Research Article
Normalized unit groups and their conjugacy classes
Let $G = H \times A$ be a finite 2-group, where $H$ is a non-abelian group of order 8 and $A$ is an elementary abelian 2-group. We obtain a normal complement of $G$ in the normalized unit group $V(FG)$ and in the unitary subgroup $V_{\ast}(FG)$ over the field $F$ with 2 elements. Further, for a finite field $F$ of characteristic 2, we derive class size of elements of $V(FG)$. Moreover, we provide class representatives of $V_{\ast}(FH)$.
Article ID 0006 Published: 10 January 2020 Research Article
A simple method to extract zeros of certain Eisenstein series of small level
This paper provides a simple method to extract the zeros of some weight two Eisenstein series of level $N$ where $N$ = 2, 3, 5 and 7. The method is based on the observation that these Eisenstein series are integral over the graded algebra of modular forms on $SL(2, Z)$ and their zeros are ‘controlled’ by those of $E_{4}$ and $E_{6}$ in the fundamental domain of $\Gamma_{0}(N)$.
Article ID 0005 Published: 10 January 2020 Research Article
Finite groups with exactly one composite conjugacy class size
QINHUI JIANG CHANGGUO SHAO YAN ZHAO
A composite number is a positive integer that has at least one divisor integerother than 1 and itself. In this paper, we give a detailed structural description of a group if it has a unique composite conjugacy class size.
Article ID 0004 Published: 10 January 2020 Research Article
A note on the weak law of large numbers of Kolmogorov and Feller
FUQIANG MA YU MIAO JIANYONG MU
In this paper, we establish the weak laws of large numbers for the negative quadrant-dependent random sequences which extend the classic Kolmogorov–Feller weak law of large numbers. In addition, the moment convergence for the negative quadrant-dependent random sequences are also given.
Article ID 0003 Published: 10 January 2020 Research Article
Class group of the ring of invariants of an exponential map on an affine normal domain
Let $k$ be a field and let $B$ be an affine normal domain over $k$. Let $\phi$ be a non-trivial exponential map on $B$ and let $A = B^{\phi}$ be the ring of $\phi$-invariants. Since $A$ is factorially closed in $B$, $A = K \cap B$ where $K$ denotes the field of fractions of $A$. Hence $A$ is a Krull domain. We investigate here a relation between the class group $\rm{Cl}$$(A)$ of $A$ and the class group $\rm{Cl}$$(B)$ of $B$. In this direction, we give a sufficient condition for an injective group homomorphism from $\rm{Cl}$$(A)$ to $\rm{Cl}$$(B)$. We also give an example to show that $\rm{Cl}$$(A)$ may not be realized as a subgroup of $\rm{Cl}$$(B)$.
Article ID 0002 Published: 10 January 2020 Research Article
Signs of Fourier coefficients of cusp form at sumof two squares
SOUMYARUP BANERJEE MANISH KUMAR PANDEY
In this article,we investigate the sign changes of the sequence of coefficientsat sum of two squareswhere the coefficients are the Fourier coefficients of the normalized Hecke eigen cusp form for the full modular group.We provide the quantitative result for the number of sign changes of the sequence in a small interval.
Article ID 0001 Published: 10 January 2020 Research Article
On $\mathcal{D}$-closed submodules
YILMAZ DURGUN SALAHATTIN ÖZDEMIR
A submodule $N$ of a module $M$ is called $\mathcal{D}$-closed if the socle of $M/N$ is zero. $\mathcal{D}$-closed submodules are similar to $\mathcal{S}$-closed submodules (a generalization of closed submodules) defined through nonsingular modules. First, we describe the smallest proper class (due to Buchsbaum) containing the class of short exact sequences determined by $\mathcal{D}$-closed submodules in terms of that submodule, and showthat it coincides with other classes of modules under certain conditions. Second, we study coprojective modules of this class, called edc-flat modules. We give some equivalent conditions for injective modules to be edc-flat for special rings, and for edc-flat modules to be projective (flat) for any ring.
Volume 130, 2020
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