• Volume 128, Issue 5

      November 2018

    • Homotopy type of neighborhood complexes of Kneser graphs, $K G_{2,k}$


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      Schrijver (Nieuw Archief voor Wiskunde, 26(3) (1978) 454–461) identified a family of vertex critical subgraphs of the Kneser graphs called the stable Kneser graphs $SG_{n,k}$ . Björner and de Longueville (Combinatorica 23(1) (2003) 23–34) proved that the neighborhood complex of the stable Kneser graph $SG_{n,k}$ is homotopy equivalent to a$k$-sphere. In this article, we prove that the homotopy type of the neighborhood complex of the Kneser graph $K G_{2,k}$ is a wedge of $(k + 4)(k + 1) + 1$ spheres of dimension $k$. We construct a maximal subgraph $S_{2,k}$ of $K G_{2,k}$ , whose neighborhood complex is homotopy equivalent to the neighborhood complex of $SG_{2,k}$ . Further, we prove that the neighborhood complex of $S_{2,k}$ deformation retracts onto the neighborhood complex of $SG_{2,k}$ .

    • On the family of elliptic curves $y^2 = x^3 − m^2x + p^2$


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      In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of $E_{m,p} : y^{2} = x^{3} − m^{2}x + p^{2}$, where $m$ is a positive integer and $p$ is a prime. We prove that for any prime $p$, the torsion subgroup of $E_{m,p}(\mathbb{Q})$ is trivial for both the cases $\{m \geq 1, m \not\equiv 0 (mod 3)\}$ and $\{m \geq 1, m \equiv 0 (mod 3)$, with $gcd(m, p) = 1\}$. We also show that given any odd prime $p$ and for any positive integer $m$ with $m \not\equiv 0 (mod 3)$ and $m \equiv 2 (mod 32)$, the lower bound for the rank of $E_{m,p}(\mathbb{Q})$ is 2. Finally, we find curves of rank 9 in this family.

    • A generalization of sumset and its applications


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      Let $A$ be a nonempty finite subset of an additive abelian group $G$ and let $r$ and $h$ be positive integers. The generalized h-fold sumset of $A$, denoted by $h^{(r)} A$, is the set of all sums of $h$ elements of $A$, where each element appears in a sum at most $r$ times. The direct problem for $h^{(r)} A$ is to find a lower bound for $|h^{(r)} A|$ in terms of $|A|$. The inverse problem for $h^{(r)} A$ is to determine the structure of the finite set $A$ for which $|h^{(r)} A|$ is minimal with respect to some fixed value of $|A|$. If $G = \mathbb{Z}$, the direct and inverse problems are well studied. In case of $G = \mathbb{Z}/p\mathbb{Z}$, $p$ a prime, the direct problem has been studied very recently by Monopoli (J. Number Theory, 157 (2015) 271–279). In this paper, we express the generalized sumset $h^{(r)} A$ in terms of the regular and restrictedsumsets. As an application of this result, we give a new proof of the theorem of Monopoli and as the second application, we present new proofs of direct and inverse theorems for the case $G = \mathbb{Z}$.

    • Comparison of graphs associated to a commutative Artinian ring


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      Let $R$ be a commutative ringwith $1 \neq 0$ and the additive group $R^{+}$. Several graphs on $R$ have been introduced by many authors, among zero-divisor graph $\Gamma_{1}(R)$, co-maximal graph $\Gamma_{2}(R)$, annihilator graph $AG(R)$, total graph $T (\Gamma(R))$, cozero-divisors graph $\Gamma_{c}(R)$, equivalence classes graph $\Gamma_{E}(R)$ and the Cayley graph Cay$(R^{+}, Z^{\ast}(R))$.Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to Cay$(R^{+}, Z^{\ast}(R))$. Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when $R$ is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if $R$ has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results,we prove that for a commutative finite ring $R$ with $|Max(R)| = n \geq 3$, $\Gamma_{1}(R) \simeq \Gamma_{2}(R)$ if and only if $R \simeq \mathbb{Z}^{n}_{2}$; if and only if $\Gamma_{1}(R) \simeq \Gamma_{E}(R)$. Also then annihilator graph is identical to the cozero-divisor graph if and only if $R$ is a Frobenius ring.

    • Moduli space of parabolic vector bundles over hyperelliptic curves


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      Let $X$ be a smooth projective hyperelliptic curve of arbitrary genus $g$. In this article, we will classify the rank 2 stable vector bundles with parabolic structure along a reduced divisor of degree 4.

    • Recognition of PSL$(2, 2^{a})$ by the orders of vanishing elements


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      Here, we show that the simple groups PSL$(2, 2^{a}), a \geq 2$, are characterized by the orders of vanishing elements.

    • Coleman automorphisms of holomorphs of completely reducible and almost simple groups


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      Let $H$ be the holomorph of a finite group $G$. It is proved that every Coleman automorphism of $H$ is inner whenever $G$ is either completely reducible or almost simple; in particular, this is the case when $G$ is either characteristically simple or simple. As an application, we obtain the normalizer the conjecture holds for integral group rings of holomorphs of such groups in question.

    • Nilpotent groups related to an automorphism


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      The aim of this paper is to state some results on an $\alpha$-nilpotent group, which was recently introduced by Barzegar and Erfanian (Caspian J. Math. Sci. 4(2)(2015) 271–283), for any fixed automorphism $\alpha$ of a group $G$. We define an identity nilpotent group and classify all finitely generated identity nilpotent groups. Moreover, we prove a theorem on a generalization of the converse of the known Schur’s theorem. In the last section of the paper, we study absolute normal subgroups of a finite group.

    • On some results for a class of meromorphic functions having quasiconformal extension


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      We consider the class $\Sigma(p)$ of univalent meromorphic functions $f$ on $\mathbb{D}$ having a simple pole at $z = p \in [0, 1)$ with residue 1. Let $\Sigma_{k}(p)$ be the class of functions in $\Sigma(p)$ which have $k$-quasiconformal extension to the extended complex plane $\hat{\mathbb{C}}$ , where $0 \leq k$ < $1$. We first give a representation formula for functions in this class and using this formula, we derive an asymptotic estimate of the Laurent coefficients for the functions in the class $\Sigma_{k}(p)$. Thereafter, we give a sufficient condition for functions in $\Sigma(p)$ to belong to the class $\Sigma_{k}(p)$. Finally, we obtain a sharp distortion result for functions in $\Sigma(p)$ and as a consequence, we obtain a distortion estimate for functions in $\Sigma_{k}(p)$.

    • Existence of positive solutions to semilinear elliptic problems with nonlinear boundary condition


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      In this paper, a semilinear elliptic equation with a nonlinear boundary condition and a perturbation in the reaction term is studied. The existence of a positive solution and another non-zero solution to the problem is proved when $|\lambda|$ is small enough without any specific assumptions on the perturbation term. Moreover, it is shown that thenon-zero solution becomes a positive one for small $\lambda$ > 0 under suitable assumptions on the perturbation term.

    • Set of periods of a subshift


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      In this article, subsets of $\mathbb{N}$ that can arise as sets of periods of the following subshifts are characterized: (i) subshifts of finite type, (ii) transitive subshifts of finite type, (iii) sofic shifts, (iv) transitive sofic shifts, and (v) arbitrary subshifts.

    • Iteration of certain exponential-like meromorphic functions


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      The dynamics of functions $f_{\lambda}(z) = \lambda\frac{e^{z}} {z+1}$ for $z \in \mathbb{C}$, $\lambda$ > 0 is studied showing that there exists $\lambda^{\ast}$ > 0 such that the Julia set of $f_{\lambda}$ is disconnected for 0 < $\lambda$ < $\lambda^{\ast}$ whereas it is the whole Riemann sphere for $\lambda$ > $\lambda^{\ast}$. Further, for 0 < $\lambda$ < $\lambda^{\ast}$, the Julia set is a disjoint union of two topologically and dynamically distinct completely invariant subsets, one of which is totally disconnected. The union of the escaping set and the backward orbit of $\infty$ is shown to be disconnected for 0 < $\lambda$ < $\lambda^{\ast}$ whereas it is connected for $\lambda$ > $\lambda^{\ast}$. For complex $\lambda$, it is proved that either all multiply connected Fatou components ultimately land on an attracting or parabolic domain containing the omitted value of the function or the Julia set is connected. In the latter case, the Fatou set can be empty or consists of Siegel disks. All these possibilities are shown to occur for suitable parameters. Meromorphic functions $E_{n}(z) = e^{z} (1+z + \frac{z^{2}} {2!} + · · · + \frac{z^{n}} {n!})^{−1}$, which we call exponential-like, are studied as a generalization of $f(z) = \frac{e^{z}} {z+1}$ which is nothing but $E_{1}(z)$. This name is justified by showing that $E_{n}$ has an omitted value 0 and there are no other finite singular value. In fact, it is shown that there is only onesingularity over 0 as well as over $\infty$ and both are direct. Non-existence of Herman rings are proved for $\lambda E_{n}$.

    • Polynomially peripheral range-preserving maps between Banach algebras


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      Let $A$ and $B$ be two Banach function algebras and $p$ a two variable polynomial $p(z,w) = zw + az + bw + c, (a, b, c \in \mathbb{C})$. We characterize the general form of a surjection $T : A \rightarrow B$ which satisfies $\rm{Ran}_{\pi}(p(T f, Tg)) \cap \rm{Ran}_{\pi}(p( f, g)) \neq \emptyset$, ($f, g \in A$ and $c \neq ab$), where $\rm{Ran}_{\pi}( f )$ is the peripheral range of $f$ .

    • Quasi hyperrigidity and weak peak points for non-commutative operator systems


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      In this article, we introduce the notions of weak boundary representation, quasi hyperrigidity and weak peak points in the non-commutative setting for operator systems in $C^{\ast}$-algebras. An analogue of Saskin’s theorem relating quasi hyperrigidity and weak Choquet boundary for particular classes of $C^{\ast}$-algebras is proved. We also show that, if an irreducible representation is a weak boundary representation and weak peak then it is a boundary representation. Several examples are provided to illustrate these notions.

    • Homotopy classification of contact foliations on open contact manifolds


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      We give a homotopy classification of foliations on open contact manifolds whose leaves are contact submanifolds of the ambient space. The result is an extension of Haefliger’s classification of foliations on open manifold in the contact setting. While proving the main theorem, we also prove a result on equidimensional isocontactimmersions on open contact manifolds.

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