Volume 126, Issue 2
May 2016, pages 143-287
pp 143-151 Research Article
Existence of non-abelian representations of the near hexagon 𝑄(5, 2) $\otimes$ 𝑄(5, 2)
In [5], a new combinatorial model with four types of points and nine types of lines of the slim dense near hexagon 𝑄(5, 2) $\otimes$ 𝑄(5, 2) was provided and it was then used to construct a non-abelain representation of 𝑄(5, 2) $\otimes$ 𝑄(5, 2) in the extraspecial 2-group 2$^{1+18}_{−}$ . In this paper, we give a direct proof for the existence of a non-abelian representation of 𝑄(5, 2) $\otimes$ 𝑄(5, 2) in 2$^{1+18}_{−}$ .
pp 153-157 Research Article
On trees with total domination number equal to edge-vertex domination number plus one
B Krishnakumari Y B Venkatakrishnan Marcin Krzywkowski
An edge $e \in E(G)$ dominates a vertex $v \in V(G)$ if $e$ is incident with $v$ or $e$ is incident with a vertex adjacent to $v$. An edge-vertex dominating set of a graph $G$ is a set $D$ of edges of $G$ such that every vertex of $G$ is edge-vertex dominated by an edge of $D$. The edge-vertex domination number of a graph $G$ is the minimum cardinality of an edge-vertex dominating set of $G$. A subset $D \subseteq V(G)$ is a total dominating set of $G$ if every vertex of $G$ has a neighbor in $D$. The total domination number of $G$ is the minimum cardinality of a total dominating set of $G$. We characterize all trees with total domination number equal to edge-vertex domination number plus one.
pp 159-165 Research Article
(3, 1)*-Choosability of graphs of nonnegative characteristic without intersecting short cycles
A graph 𝐺 is called (𝑘, 𝑑)*-choosable if for every list assignment 𝐿 satisfying $|L(v)|\geq k$ for all $v \in V (G)$, there is an 𝐿-coloring of 𝐺 such that each vertex of 𝐺 has at most 𝑑 neighbors colored with the same color as itself. In this paper, it is proved that every graph of nonnegative characteristic without intersecting 𝑖-cycles for all 𝑖 = 3, 4, 5 is (3, 1)*-choosable.
pp 167-177 Research Article
Quantitative metric theory of continued fractions
J Hančl A Haddley P Lertchoosakul R Nair
Quantitative versions of the central results of the metric theory of continued fractions were given primarily by C. De Vroedt. In this paper we give improvements of the bounds involved . For a real number 𝑥, let $$x=c_0+\dfrac{1}{c_1+\dfrac{1}{c_2+\dfrac{1}{c_3+\dfrac{1}{c_4+_\ddots}}}}.$$
A sample result we prove is that given $\epsilon > 0$, $$(c_1(x)\cdots c_n(x))^{\frac{1}{n}}=\prod^\infty_{k=1}\left( 1+\frac{1}{k(k+2)}\right)^{\frac{\log \, k}{\log \, 2}}+o\left(n^{-\frac{1}{2}}(\log \, n)^{\frac{3}{2}}(\log \, \log \, n)^{\frac{1}{2}+\epsilon}\right)$$
pp 179-186 Research Article
Non-properness of the functor of 𝐹-trivial bundles
We study the properness of the functor of 𝐹-trivial bundles by relating it to the base change question for the fundamental group scheme of Nori.
pp 187-197 Research Article
$\mathcal M^\ast$-supplemented subgroups of finite groups
Jia Zhang Long Miao Juping Tang
A subgroup $H$ of a group $G$ is said to be $\mathcal M^\ast$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G = HK$ and $H \cap K$ is $\mathcal M$-supplemented in $G$. In this paper, we prove as follows: Let $E$ be a normal subgroup of a group $G$. Suppose that every maximal subgroup of every non-cyclic Sylow subgroup $P$ of $F^\ast (E)$ is $\mathcal M^\ast$-supplemented in $G$, then $E \leq Z_{\mathcal U\Phi} (G)$.
pp 199-206 Research Article
Ray equations of a weak shock in a hyperbolic system of conservation laws in multi-dimensions
In this paper we give a complete proof of a theorem, which states that ‘for a weak shock, the shock ray velocity is equal to the mean of the ray velocities of nonlinear wavefronts just ahead and just behind the shock, provided we take the wavefronts ahead and behind to be instantaneously coincident with the shock front. Similarly, the rate of turning of the shock front is also equal to the mean of the rates of turning of such wavefronts just ahead and just behind the shock’. A particular case of this theorem for shock propagation in gasdynamics has been used extensively in applications. Since it is useful also in other physical systems, we present here the theorem in its most general form.
pp 207-212 Research Article
A summation due to Ramanujan revisited
Dong Jun Jee Tuo Yeong Lee Nathaniel Zhi-Wei Leon
We employ Hardy’s regularly convergent double series to refine an argument of Nanjundiah [7]. In particular, we evaluate some alternating series.
pp 213-226 Research Article
Wavelet transform of generalized functions in $K^{\prime}\{ M_p\}$ spaces
Using convolution theory in $K\{ M_p\}$ space we obtain bounded results for the wavelet transform. Calderón-type reproducing formula is derived in distribution sense as an application of the same. An inversion formula for the wavelet transform of generalized functions is established.
pp 227-234 Research Article
Approximation properties of fine hyperbolic graphs
In this paper, we propose a definition of approximation property which is called the metric invariant translation approximation property for a countable discrete metric space. Moreover, we use the techniques of Ozawa’s to prove that a fine hyperbolic graph has the metric invariant translation approximation property.
pp 235-240 Research Article
Generators for finite depth subfactor planar algebras
Vijay Kodiyalam Srikanth Tupurani
We show that a subfactor planar algebra of finite depth $k$ is generated by a single $s$-box, for $s\leq {\rm min}\{ k + 4, 2k\}$.
pp 241-252 Research Article
Structures of generalized 3-circular projections for symmetric norms
Recently several authors investigated structures of generalized bi-circular projections in spaces where the descriptions of the group of surjective isometries are known. Following the same idea in this paper we give complete descriptions of generalized 3-circular projections for symmetric norms on ${\mathbb C}^n$ and ${\mathbb M}_{m \times n}({\mathbb C})$.
pp 253-260 Research Article
Rigidity theorem forWillmore surfaces in a sphere
Let 𝑀^{2} be a compact Willmore surface in the (2 + 𝑝)-dimensional unit sphere 𝑆^{2+𝑝}. Denote by 𝐻 and 𝑆 the mean curvature and the squared length of the second fundamental form of 𝑀^{2}, respectively. Set $\rho^2 = S − 2H^2$. In this note, we proved that there exists a universal positive constant 𝐶, such that if $\parallel \rho^2\parallel_2 \lt C$, then $\rho^2 = 0$ and 𝑀^{2} is a totally umbilical sphere.
pp 261-275 Research Article
Relative symplectic caps, 4-genus and fibered knots
Siddhartha Gadgil Dheeraj Kulkarni
We prove relative versions of the symplectic capping theorem and sufficiency of Giroux’s criterion for Stein fillability and use these to study the 4-genus of knots. More precisely, suppose we have a symplectic 4-manifold 𝑋 with convex boundary and a symplectic surface 𝛴 in 𝑋 such that 𝛿𝛴 is a transverse knot in 𝛿𝑋. In this paper, we prove that there is a closed symplectic 4-manifold 𝑌 with a closed symplectic surface 𝑆 such that (𝑋, 𝛴) embeds into (𝑌, 𝑆) symplectically. As a consequence we obtain a relative version of the symplectic Thom conjecture. We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in $\mathbb S^3$. Further, we give a criterion for quasipositive fibered knots to be strongly quasipositive.
pp 277-281 Research Article
Classification of smooth structures on a homotopy complex projective space
We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective $n$-space ${\mathbb C}{\bf P}^n$, where $n = 3$ and 4. Let $M^{2n}$ be a closed smooth $2n$-manifold homotopy equivalent to ${\mathbb C}{\bf P}^n$. We show that, up to diffeomorphism, $M^6$ has a unique differentiable structure and $M^8$ has at most two distinct differentiable structures. We also show that, up to concordance, there exist at least two distinct differentiable structures on a finite sheeted cover $N^2n$ of ${\mathbb C}{\bf P}^n$ for 𝑛 = 4, 7 or 8 and six distinct differentiable structures on $N^{10}$.
pp 283-287 Research Article
Characterizations of the power distribution by record values
Let $\{ X_n, n \geq 1\}$ be a sequence of i.i.d. random variables which has absolutely continuous distribution function $F(x)$ with probability density function $f(x)$ and $F(0) = 0$. Assume that $X_n$ belongs to the class $C^\ast_1$ or $C_2$. Then $X_k$ has the power distribution if and only if $X_k$ and $\frac{X_{L(n+1)}}{X_{L(n)}}$ or $\frac{X_{L(n+1)}}{X_{L(n)}}$ and $\frac{X_{L(n)}}{X_{L(n--1)}}$ are identically distributed, respectively. Suppose that $X_n$ belongs to the class $C_3$. Also, $X_k$ has the power distribution if and only if $X_{L(n+1)}$ and $X_{L(n)}\cdot V$ are identically distributed, where $V$ is independent of $X_{L(n)}$ and $X_{L(n+1)}$ and is distributed as $X_n$’s.
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