• Volume 130, All articles

      Continuous Article Publishing mode

    • Onthe factorization of two adjacent numbers in multiplicatively closed sets generated by two elements

      C P ANIL KUMAR

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      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.

    • A limit set intersection theorem for graphs of relatively hyperbolic groups

      SWATHI KRISHNA

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      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].

    • A submultiplicative property of the Carathéodory metric on planar domains

      AMAR DEEP SARKAR KAUSHAL VERMA

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      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 $D_{1} \cup D_{2}$.

    • Picard bundle on the moduli space of torsionfree sheaves

      USHA N BHOSLE

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      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 $\theta_{\bar{J}(Y)}$.

    • Blow up property for viscoelastic evolution equations on manifolds with conical degeneration

      MOHSEN ALIMOHAMMADY MORTEZA KOOZEHGAR KALLEJI

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      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.

    • New criteria for Vandiver’s conjecture using Gauss sums – Heuristics and numerical experiments

      GEORGES GRAS

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      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.

    • Computing $n$-th roots in $\rm{SL_{2}}$ and Fibonacci polynomials

      AMIT KULSHRESTHA ANUPAM SINGH

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      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 $SL_{1}(Q)$.

    • Correction to: Sparse subsets of the natural numbers and Euler’s totient function

      MITHUN KUMAR DAS PRAMOD EYYUNNI BHUWANESH RAO PATIL

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      Correction to: Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:84https://doi.org/10.1007/s12044-019-0512-x

    • Berger’s formulas and their applications in symplectic mean curvature flow

      SHIJIN ZHANG

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      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.

    • Repdigits as products of two Fibonacci or Lucas numbers

      FATIH ERDUVAN REFIK KESKIN

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      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 $$\left\{ \begin{array}{l}(0, 0), (0, 1), (1, 1), (0, 2), (1, 2), (2, 2), (0, 3),\\ (1, 3), (1, 4), (1, 5), (2, 5), (3, 5), (4, 5) \end{array}\right\}$$

    • On the number of distinct exponents in the prime factorization of an integer

      CARLO SANNA

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      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.

    • A set of generators for the Picardmodular group $SU(2,1,O_{2})$

      MAHBOUBEH GHOSHOUNI MAJID HEYDARPOUR

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      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.

    • Repdigits in Euler functions of associated pell numbers

      G K PANDA M K SAHUKAR

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      In this paper, we discuss some properties of Euler totient function ofassociated Pell numbers which are repdigits in base 10.

    • Exponential sums of squares of Fourier coefficients of cusp forms

      RATNADEEP ACHARYA

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      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.

    • A ternary diophantine inequality with prime numbers of a special type

      LI ZHU

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      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.

    • Palindromic width of graph of groups

      KRISHNENDU GONGOPADHYAY SWATHI KRISHNA

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      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.

    • Upper bound for the first nonzero eigenvalue related to the $p$-Laplacian

      SHEELA VERMA

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      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 $\Omega$.

    • Gerstenhaber algebra structure on the cohomology of a hom-associative algebra

      APURBA DAS

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      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.

    • Rigidity of Bott–Samelson–Demazure–Hansen variety for $F_{4}$ and $G_{2}$

      S SENTHAMARAI KANNAN PINAKINATH SAHA

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      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).

    • The fundamental group and extensions of motives of Jacobians of curves

      SUBHAM SARKAR RAMESH SREEKANTAN

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      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.

    • Prime intersection graph of ideals of a ring

      KUKIL KALPA RAJKHOWA HELEN K SAIKIA

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      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 $\mathbb{Z}_{n}$.

    • Dimension formula for the space of relative symmetric polynomials of $D_{n}$ with respect to any irreducible representation

      S RADHA P VANCHINATHAN

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      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.

    • Examples of blown up varieties having projective bundle structures

      NABANITA RAY

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      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.

    • Mixed coloured permutations

      BEÁTA BÉNYI DANIEL YAQUBI

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      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.

    • Markov approximation and the generalized entropy ergodic theorem for non-null stationary process

      ZHONGZHI WANG WEIGUO YANG

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      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.

    • Ring endomorphisms satisfying the central reversible property

      ARNAB BHATTACHARJEE UDAY SHANKAR CHAKRABORTY

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      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.

    • The group of invertible ideals of a Prüfer ring

      BASAK AY SAYLAM

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      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.

    • On weighted signed color partitions

      V GUPTA M RANA S SHARMA

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      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.

    • 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

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      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.

    • Exceptional set in Waring–Goldbach problem: Two squares, two cubes and two sixth powers

      YUHUI LIU

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      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.

    • Normalized unit groups and their conjugacy classes

      S KAUR M KHAN

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      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 $V_{\ast}(FH)$.

    • A simple method to extract zeros of certain Eisenstein series of small level

      ARADHITA CHATTOPADHYAYA

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      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 $\Gamma_{0}(N)$.

    • Finite groups with exactly one composite conjugacy class size

      QINHUI JIANG CHANGGUO SHAO YAN ZHAO

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      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.

    • A note on the weak law of large numbers of Kolmogorov and Feller

      FUQIANG MA YU MIAO JIANYONG MU

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      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.

    • Class group of the ring of invariants of an exponential map on an affine normal domain

      S M BHATWADEKAR J T MAJITHIA

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      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)$.

    • Signs of Fourier coefficients of cusp form at sumof two squares

      SOUMYARUP BANERJEE MANISH KUMAR PANDEY

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      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.

    • On $\mathcal{D}$-closed submodules

      YILMAZ DURGUN SALAHATTIN ÖZDEMIR

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      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.

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