Article ID 0030 June 2019 Research Article
Maps preserving $A^{\ast} A + AA^{\ast}$ on $C^{\ast}$-algebras
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.
Article ID 0031 June 2019 Research Article
Congruences for two restricted overpartitions
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.
Article ID 0032 June 2019 Research Article
Connectivity of the Julia sets of singularly perturbed rational maps
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.
Article ID 0033 June 2019 Research Article
Hardy’s inequality for the fractional powers of the Grushin operator
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.
Article ID 0034 June 2019 Research Article
The automorphism group of the bipartite Kneser graph
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.
Article ID 0035 June 2019 Research Article
SHINTARO AKAMINE RAHUL KUMAR SINGH
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.
Article ID 0036 June 2019 Research Article
Regularity of binomial edge ideals of certain block graphs
A V JAYANTHAN N NARAYANAN B V RAGHAVENDRA RAO
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.
Article ID 0037 June 2019 Research Article
$C^{\ast}$-algebra-valued partial metric space and fixed point theorems
SUMIT CHANDOK DEEPAK KUMAR CHOONKIL PARK
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.
Article ID 0038 June 2019 Research Article
Ramification theory and formal orbifolds in arbitrary dimension
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.
Article ID 0039 June 2019 Research Article
Combinatorial identities for tenth order mock theta functions
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.
Article ID 0040 June 2019 Research Article
An identity on generalized derivations involving multilinear polynomials in prime rings
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 $x\in R$.
Article ID 0041 June 2019 Research Article
Euler’s criterion for eleventh power nonresidues
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 $D=2, 7,11$.
Article ID 0042 June 2019 Research Article
Onthe gaps inmultiplicatively closed sets generated by atmost two elements
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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Volume 129 | Issue 5
November 2019
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