• Volume 129, Issue 2

April 2019

• Some notes on tetrahedrally closed spherical sets in Euclidean spaces

We describe a connection between a family of tetrahedrally closed spherical sets in Euclidean spaces and a family of point-line geometries called near polygons.

• An alternative approach to theory of weak factoriality

In this article, we define the notion of super quantum. We discuss several properties of super quanta and show an alternative approach to the theory of weak factoriality by replacing the notion of primary elements with super quanta.

• An essential representation for a product system over a finitely generated subsemigroup of $\mathbb{Z}^{d}$

Let $S \subset \mathbb{Z}^{d}$ be a finitely generated subsemigroup. Let $E$ be a product system over $S$. We show that there exists an infinite dimensional separable Hilbert space $\mathcal{H}$ and a semigroup $\alpha := \{\alpha_{x}\}_{x\in S}$ of unital normal $\ast$-endomorphisms of $B(\mathcal{H})$ such that $E$ is isomorphic to the product system associated to $\alpha$.

• Generalized skew-derivations annihilating and centralizing on multilinear polynomials in prime rings

Let $R$ be a prime ring of characteristic $\neq 2$, $Qr$ its right Martindale quotient ring, $C$ its extended centroid, $F \neq 0$ a generalized skew derivation of $R, f (x_{1}, . . . , x_{n})$ a multilinear polynomial over $C$ not central-valued on $R$ and $S$ the set of all evaluations of $f (x_{1}, . . . , x_{n})$ in $R$. If $a[F(x), x] \in C$ for all $x \in S$, then there exist $\lambda \in C$ and $b \in Qr$ such that $F(x) = bx + xb + \lambda x$, for all $x \in R$ and one of the following holds:(1) $b \in C$;(2) $f (x_{1}, . . . , x_{n})^{2}$ is central-valued on $R$;(3) $R$ satisfies $s_{4}$, the standered identity of degree 4.

• Regularity of certain diophantine equations

In Ramsey theory, there is a vast literature on regularity questions of linear diophantine equations. Some problems in higher degree have been considered recently. Here, we show that, for every pair of positive integers $r$ and $n$, there exists an integer $B = B(r)$ such that the diophantine equation$$\prod^{n}_{m=1} \left(\sum^{k_{m}}_{i=1} a_{m,i}x_{m,i} - \sum^{l_{m}}_{j=1} b_{m,j}y_{m,j}\right) = B$$with$$\sum^{k_{m}}_{i=1} a_{m,i} = \sum^{l_{m}}_{j=1} b_{m,j} \quad \forall m= 1,\cdots,n$$is $r$-regular, where $k_{m}, l_{m}$ are also positive integers and $a_{m,i}, b_{m,j}$ are non-zero integers.

• On arbitrarily graded rings

Let $\Re$ be a ring graded by an arbitrary set $A$. We show that $\Re$ decomposes as the sum of the well-described graded ideals plus (maybe) a certain subgroup.We also provide a context where the graded simplicity of $\Re$ is characterized and where a second Wedderburn-type theorem in the category of arbitrarily graded rings is stated.

• Balancing non-Wieferich primes in arithmetic progressions

A prime is called a balancing non-Wieferich prime if it satisfies $B_{p−(\frac{8}{p})} \not \equiv 0 (mod p^{2})$, where $\left(\frac{8}{p}\right)$ and $B_{n}$ denote the Jacobi symbol and the $n$-th balancing number respectively. For any positive integers $k$ > 2 and $n$ > 1, there are $\gg \rm{log x / \log \log x}$ balancing non-Wieferich primes $p \leq x$ such that $p \equiv 1 \pmod {k}$ under the assumption of the $abc$ conjecture for the number field $\mathbb{Q}(\sqrt{2})$ (Proc. Japan Acad. Ser. A 92 (2016) 112–116). In this paper, for any fixed $M$, the lower bound $\rm{\log x/ \log \log x}$ is improved to $\rm{(\log x/ \log \log x)(\log \log \log x)}^{M}$.

• Three solutions for a semilinear elliptic boundary value problem

The purpose of this work is to study the following elliptic problem:

$${\rm (P_\lambda)} \left\{\begin{array}{ll} -\Delta u =f(x)|u(x)|^{p-2}u(x) +\la g(x)|u|^{q-2}u \;\;\mbox{ in }\,\Omega ;\\ u = 0 \;\;\mbox{ in }\,\partial\Omega,$$

\end{array}

\right.

where $\Omega\subset\mathbb{R}^N\;(N\geq 3)$ be a bounded smooth domain, $f,\,g\in L^\infty(\Omega),$ $\lambda$ is a positive parameter. Under adequate assumptions on the sources terms $f$ and $g$, we establish the existence of three solutions: one is positive, one is negative and the other changes sign.

• Hybrid mean value of 2$k$-th power inversion of $L$-functions and general quartic Gauss sums

In this paper, we find the 2$k$-th power mean of the inversion of $L$-functions with the weight of the general quartic Gauss sums. We establish the results with the help of Dirichlet characters and properties of classical Gauss sums. We also describe asymptotic behaviour for it.

• McKay correspondence in quasi-SL quasitoric orbifoldsMcKay correspondence in quasi-SL quasitoric orbifolds

We show McKay correspondence for Betti numbers of Chen–Ruan cohomology of omnioriented quasi-$SL$ quasitoric orbifolds. This generalizes a correspondence for $SL$ projective toric orbifolds due to Batyrev and Dias (Topology 35(4) (1996) 901–929) to a setting that does not require a complex or even an almost complex structure. In previous works with Ganguli and Poddar (Osaka J. Math. 50(2) (2013) 397–415; 50(4) (2013) 977–1005), we have proved the correspondence in dimensions four and six. Here we deal with the general case.

• GIT quotient of a Bott–Samelson–Demazure–Hansen variety by a maximal torus

Let $G$ be an almost simple, simply connected algebraic group over the field $\mathbb{C}$ of complex numbers. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G$, and let $W$ be the Weyl group defined by $T$ . The Borel group $B$ determines a subset of simple reflections in $W$. For $w$ in $W$, we let $Z(w, \underline{i})$ be the Bott–Samelson–Demazure–Hansen variety corresponding to a reduced expression $\underline{i}$ of $w$ as a product of these simple reflections. In this article, we study the geometric invariant theoretic quotient of $Z(w, \underline{i})$ for the $T$ -linearized ample line bundles.

• On some ternary pure exponential diophantine equations with three consecutive positive integers bases

By using the lower bound of linear forms in two logarithms of Laurent(Acta Arith. 133(4) (2008) 325–348), we give here a new solution that the ternary pure exponential diophantine equation $(n + 1)^{x} + (n + 2)^{y} = n^{z}$ has no positive integer solutions except for $(n, x, y, z) = (3, 1, 1, 2)$. This proof is very different from Le (J. Yulin Teachers College 28(3) (2007) 1–2), in which he used the classification method of solutions of exponential decomposition form equation. Furthermore, we solved completely another similar ternary pure exponential diophantine equation $n^{x} +(n+2)^{y} = (n+1)^{z}$ by using $m$-adic estimation of linear forms due to Bugeaud (Compos. Math. 132(2) (2002) 137–158).

• On a presentation of the spin planar algebra

We define a certain abstract planar algebra by generators and relations, study the various aspects of its structure, and then identify it with Jones’ spin planar algebra.

• Cross-sections of the multicorns

For each integer $d \geq 2$, we identify the intersections of the connectedness locus of $\bar{z}^{d} + c$ with the rays $\omega\mathbb{R}^{+}$, where $\omega^{d+1} = \pm{1}$.

• Comparison between two differential graded algebras in noncommutative geometry

Starting with a spectral triple, one can associate two canonical differential graded algebras (DGA) defined by Connes (Noncommutative geometry (1994) Academic Press Inc., San Diego) and Fröhlich et al. (Comm. Math. Phys. 203(1) (1999)119–184). For the classical spectral triples associated with compact Riemannian spinmanifolds, both these DGAs coincide with the de-Rham DGA. Therefore, both are candidates for the noncommutative space of differential forms. Here we compare these two DGAs in a very precise sense.

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