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      Volume 129, Issue 3

      June 2019

    • Maps preserving $A^{\ast} A + AA^{\ast}$ on $C^{\ast}$-algebras

      ALI TAGHAVI

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      Let $\mathcal{A}$ be a $C^{*}$-algebra of real-rank zero and $\mathcal{B}$ be a $C^{*}$-algebra with unit $I$. It is shown that the mapping $\Phi: {\mathcal A}\longrightarrow {\mathcal B}$ which preserves arithmetic mean and satisfies$$\Phi(A^{*}A)=\frac{\Phi(A)^{*}\Phi(A)+\Phi(A)\Phi(A)^{*}}{2},$$for all normal elements $A\in \mathcal{A}$, is an $\mathbb R$-linear continuous Jordan $*$-homomorphism provided that $0\in {\rm Ran}\ \Phi$. Also, $\Phi$ is the sum of a linear Jordan $*$-homomorphism and a conjugate-linear Jordan $*$-homomorphism. This result also presents an application of maps which preserve the square absolute value.

    • Congruences for two restricted overpartitions

      SHANE CHERN LI-JUN HAO

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      Congruences for partitions have received a great deal of attention in literature. Recently, Bringmann et al. (Electron J. Combin. 22(3) (2015) Paper 3.17,16 pp.) studied overpartitions with restricted odd differences. In this paper, we present a number of Ramanujan-type congruences for these restricted overpartition functions.

    • Connectivity of the Julia sets of singularly perturbed rational maps

      JIANXUN FU YANHUA ZHANG

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      We consider a family of rational functions which is given by

      $$f_{\lambda}(z)=\frac{z^n(z^{2n}-\lambda^{n+1})}{z^{2n}-\lambda^{3n-1}},$$

      where $n\geq 2$ and $\lambda\in\mathbb{C}^{*}-\{\lambda:\lambda^{2n-2}=1\}$.When $\lambda\neq 0$ is small, $f_{\lambda}$ can be seen as a perturbation of the unicritical polynomial $z\mapsto z^{n}$. It was known that in this case the Julia set $J(f_\lambda)$ of $f_\lambda$ is a Cantor set of circles on which the dynamics of $f_\lambda$ is not topologically conjugate to that of any McMullen maps. In this paper, we prove that this is the unique case such that $J(f_\lambda)$ is disconnected.

    • Hardy’s inequality for the fractional powers of the Grushin operator

      RAKESH BALHARA

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      We prove Hardy’s inequality for the fractional powers of the generalized subLaplacian and the fractional powers of the Grushin operator. We also find an integral representation and a ground state representation for the fractional powers ofthe generalized subLaplacian.

    • The automorphism group of the bipartite Kneser graph

      S MORTEZA MIRAFZAL

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      Let $n$ and $k$ be integers with $n>2k$, $k\geq1$. We denote by $H(n, k)$ the bipartite Kneser graph, that is, a graph with the family of $k$-subsets and ($n-k$)-subsets of $[n] = \{1, 2,\ldots ,n\}$ as vertices, in which any two vertices are adjacent if and only if one of them is a subset of the other. In this paper, we determine the automorphism group of $H(n, k)$. We show that ${\rm Aut}(H(n, k))\cong {\rm Sym}([n]) \times \mathbb{Z}_2$, where $\mathbb{Z}_2$ is the cyclic group of order $2$. Then, as an application of the obtained result, we give a new proof for determining the automorphism group of the Kneser graph $K(n,k)$. In fact, we show how to determine the automorphism group of the Kneser graph $K(n,k)$ given the automorphism group of the Johnson graph $J(n,k)$. Note that the known proofs for determining the automorphism groups of Johnson graph $J(n,k)$ and Kneser graph $K(n,k)$ are independent of each other.

    • Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born–Infeld equation

      SHINTARO AKAMINE RAHUL KUMAR SINGH

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      In this paper, we investigate relations between solutions to the minimal surface equation in Euclidean 3-space $\mathbb{E}^{3}$, the zero mean curvature equation in the Lorentz–Minkowski 3-space $\mathbb{L}^{3}$ and the Born–Infeld equation under Wick rotations. We prove that the existence conditions of real solutions and imaginary solutions after Wick rotations are written by symmetries of solutions, and reveal how real and imaginary solutions are transformed under Wick rotations. We also give a transformation method for zero mean curvature surfaces containing light like lines with some symmetries. As an application, we give new correspondences among some solutions to the above equations by using the non-commutativity between Wick rotations and isometries in the ambient space.

    • Regularity of binomial edge ideals of certain block graphs

      A V JAYANTHAN N NARAYANAN B V RAGHAVENDRA RAO

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      We prove that the regularity of binomial edge ideals of graphs obtained by gluing two graphs at a free vertex is the sum of the regularity of individual graphs. As a consequence, we generalize certain results of Zafar and Zahid (Electron J Comb 20(4), 2013). We obtain an improved lower bound for the regularity of trees. Further, we characterize trees which attain the lower bound. We prove an upper bound for the regularity of certain subclass of block-graphs. As a consequence, we obtain sharp upper and lower bounds for a class of trees called lobsters.

    • $C^{\ast}$-algebra-valued partial metric space and fixed point theorems

      SUMIT CHANDOK DEEPAK KUMAR CHOONKIL PARK

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      In this paper, we introduce the notion of $C^{\ast}$-algebra-valued partial metric space which is more general than partial metric space. Some fixed point results using C-class functions on such spaces are obtained. Moreover, some illustrated examples are also provided.

    • Ramification theory and formal orbifolds in arbitrary dimension

      MANISH KUMAR

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      Formal orbifolds are defined in higher dimension to study wild ramification. Their étale fundamental groups are also defined. It is shown that the fundamental groups of formal orbifolds have certain finiteness property and it is also shown that they can be used to approximate the étale fundamental groups of normal varieties. Étale site on formal orbifolds are also defined.This framework allows one to study wild ramification in an organized way. Brylinski–Kato filtration, Lefschetz theorem for fundamental groups and $l$-adic sheaves in these contexts are also studied.

    • Combinatorial identities for tenth order mock theta functions

      MEGHA GOYAL M RANA

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      In this paper, the open problem posed by Sareen and Rana (Proc. Indian Acad. Sci. (Math. Sci.) 126 (2016) 549–556) is addressed. Here, we interpret two tenth order mock theta functions combinatorially in terms of lattice paths. Then we extend enumeration of one of these with Bender–Knuth matrices; the other by using Frobenius partitions. The combinatorial interpretation of one of these mock theta functions in terms of Frobenius partitions gives an answer to the open problem. Finally, we establish bijections between different classes of combinatorial objects which lead us to one 4-way and one 3-way combinatorial identity.

    • An identity on generalized derivations involving multilinear polynomials in prime rings

      B DHARA C GARG R K SHARMA

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      Let $R$ be a prime ring of characteristic different from $2$ with its Utumi ring of quotients $U$, extended centroid $C$, $f(x_{1},\ldots,x_{n})$ a multilinear polynomial over $C$, which is not central-valued on $R$ and $d$ a nonzero derivation of $R$. By $f(R)$, we mean the set of all evaluations of the polynomial $f(x_{1},\ldots,x_{n})$ in $R$. In the present paper, we study $b[d(u),u]+p[d(u),u]q+[d(u),u]c=0$ for all $u\in f(R)$, which includes left-sided, right-sided as well as two-sided annihilating conditions of the set $\{[d(u),u] : u\in f(R)\}$.We also examine some consequences of this result related to generalized derivations and we prove that if $F$ is a generalized derivation of $R$ and $d$ is a nonzero derivation of $R$ such that $$F^{2}([d(u), u])=0$$ for all $u\in f(R)$, then there exists $a\in U$ with $a^{2}=0$ such that $F(x)=xa$ for all $x\in R$ or $F(x)=ax$ for all $x\in R$.

    • Euler’s criterion for eleventh power nonresidues

      S A KATRE JAGMOHAN TANTI

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      Let $p$ be a prime $\equiv 1\pmod{p}$. If an integer $D$ with $(p,D)=1$ is an eleventh power nonresidue $\pmod{p}$, then $D^{(p-1)/11} \equiv \alpha\, \pmod{p}$ for some eleventh root of unity $\alpha(\not \equiv 1)\,\pmod{p}$. In this paper, we establish an explicit expression for $\alpha$ in terms of a particular solution of certain quadratic partition of $p$. Euler's criterion for eleventh power residues and nonresidues is given with explicit results for $D=2, 7,11$.

    • Onthe gaps inmultiplicatively closed sets generated by atmost two elements

      C P ANIL KUMAR

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      We prove in the the main theorem, Theorem 3.2, that the multiplicatively closed subset of natural numbers, generated by two elements $1$ < $p_{1}$ < $p_{2}$ with $\alpha=\frac{\log\ p_{1}}{\log\ p_{2}}$ irrational, has arbitrarily large gaps by explicitly constructing large integer intervals, with known factorization for the endpoints in terms of generators $p_{1},p_{2}$ obtained from the stabilization sequence of the irrational $\alpha$ (Definition 3.1). Example 5.6 is also illustrated. In the Appendix, for a finitely generated multiplicatively closed subset of natural numbers, we mention another constructive proof (refer to Theorem A.1}) for arbitrarily large gap intervals, where the factorization of the right endpoint is not known in terms of generators unlike in the constructive proof of the main result. The suggested general Question 1.1 remains still open

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