Volume 130, All articles
Continuous Article Publishing mode
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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