• Volume 126, Issue 2

      May 2016,   pages  143-287

    • Existence of non-abelian representations of the near hexagon 𝑄(5, 2) $\otimes$ 𝑄(5, 2)

      Binod Kumar Sahoo

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      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$^{1+18}_{−}$ .

    • On trees with total domination number equal to edge-vertex domination number plus one

      B Krishnakumari Y B Venkatakrishnan Marcin Krzywkowski

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      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.

    • (3, 1)*-Choosability of graphs of nonnegative characteristic without intersecting short cycles

      Haihui Zhang

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      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.

    • Quantitative metric theory of continued fractions

      J Hančl A Haddley P Lertchoosakul R Nair

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      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 $\epsilon > 0$, $$(c_1(x)\cdots c_n(x))^{\frac{1}{n}}=\prod^\infty_{k=1}\left( 1+\frac{1}{k(k+2)}\right)^{\frac{\log \, k}{\log \, 2}}+o\left(n^{-\frac{1}{2}}(\log \, n)^{\frac{3}{2}}(\log \, \log \, n)^{\frac{1}{2}+\epsilon}\right)$$

    • Non-properness of the functor of 𝐹-trivial bundles

      V B Mehta S Subramanian

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      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.

    • $\mathcal M^\ast$-supplemented subgroups of finite groups

      Jia Zhang Long Miao Juping Tang

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      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 $E \leq Z_{\mathcal U\Phi} (G)$.

    • Ray equations of a weak shock in a hyperbolic system of conservation laws in multi-dimensions

      Phoolan Prasad

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      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.

    • A summation due to Ramanujan revisited

      Dong Jun Jee Tuo Yeong Lee Nathaniel Zhi-Wei Leon

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      We employ Hardy’s regularly convergent double series to refine an argument of Nanjundiah [7]. In particular, we evaluate some alternating series.

    • Wavelet transform of generalized functions in $K^{\prime}\{ M_p\}$ spaces

      R S Pathak Abhishek Singh

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      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.

    • Approximation properties of fine hyperbolic graphs

      Benyin Fu

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      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.

    • Generators for finite depth subfactor planar algebras

      Vijay Kodiyalam Srikanth Tupurani

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      We show that a subfactor planar algebra of finite depth $k$ is generated by a single $s$-box, for $s\leq {\rm min}\{ k + 4, 2k\}$.

    • Structures of generalized 3-circular projections for symmetric norms

      A B Abubaker S Dutta

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      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 ${\mathbb M}_{m \times n}({\mathbb C})$.

    • Rigidity theorem forWillmore surfaces in a sphere

      Hongwei Xu Dengyun Yang

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      Let 𝑀2 be a compact Willmore surface in the (2 + 𝑝)-dimensional unit sphere 𝑆2+𝑝. Denote by 𝐻 and 𝑆 the mean curvature and the squared length of the second fundamental form of 𝑀2, 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 𝑀2 is a totally umbilical sphere.

    • Relative symplectic caps, 4-genus and fibered knots

      Siddhartha Gadgil Dheeraj Kulkarni

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      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.

    • Classification of smooth structures on a homotopy complex projective space

      Ramesh Kasilingam

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      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 $N^{10}$.

    • Characterizations of the power distribution by record values

      Min-Young Lee

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      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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