• Volume 127, Issue 3

June 2017,   pages  393-549

• New modular relations involving cubes of the Göllnitz–Gordon functions

Chen and Huang established some elegant modular relations for the Göllnitz–Gordon functions analogous to Ramanujan’s list of forty identities for the Rogers–Ramanujan functions. In this paper, we derive some new modular relations involving cubes of the Göllnitz–Gordon functions. Furthermore, we also provide newproofs of some modular relations for the Göllnitz–Gordon functions due to Gugg.

• On a problem of Pillai with Fibonacci numbers and powers of 2

In this paper, we find all integers $c$ having at least two representations as a difference between a Fibonacci number and a power of 2.

• On the number of special numbers

For lack of a better word, a number is called special if it has mutually distinct exponents in its canonical prime factorizaton for all exponents. Let $V (x)$ be the number of special numbers $\leq x$. We will prove that there is a constant $c$ > 1 such that $V (x) \sim \frac{cx}{log x}$. We will make some remarks on determining the error term at the end. Using the explicit abc conjecture, we will study the existence of 23 consecutive special integers.

• On a conjecture on linear systems

In a remark to Green’s conjecture, Paranjape and Ramanan analysed the vector bundle $E$ which is the pullback by the canonical map of the universal quotient bundle $T_\mathbb{P}^{g−1}(−1)$ on $\mathbb{P}^{g−1}$ and stated a more general conjecture and proved it for the curveswith Clifford Index 1 (trigonal and plane quintics). In this paper, we state the conjecturefor general linear systems and obtain results for the case of hyper-elliptic curves.

• Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

For −1 $\leq B < A \leq 1$, let $S^{\ast}(A,B)$ denote the class of normalized analytic functions $f(z) = z+\sum^{\infty}_{n=2}a_{n}z^{n}$ in $\mid z\mid <1$ which satisfy the subordination relation $zf'(z)/f(z)\prec(1+Az)/(1+Bz)$ and $\sum^{\ast}(A,B)$ be the corresponding class of meromorphic functions in $\mid z\mid > 1$. For $f \in S^{\ast}(A,B)$ and $\lambda > 0$, we shall estimate the absolute value of the Taylor coefficients $a_{n}(−\lambda,f )$ of the analytic function $(f(z)/z)^{−\lambda}$. Using this we shall determine the coefficient estimate for inverses of functions in the classes $S^{\ast}(A,B)$ and $\sum^{\ast}(A,B)$.

• Analysing the Wu metric on a class of eggs in $\mathbb{C}^{n} – \rm{II}$

We study the Wu metric for the non-convex domains of the form $E_{2m}= \{z \in \mathbb{C}^n : \mid z_1\mid ^{2m} + \mid z_2\mid ^2 +\cdots +\mid z_{n−1}\mid^2 + \mid z_n \mid^2$ < $1\}$, where 0 < m < 1/2. We give explicit expressions for the Kobayashi metric and the Wu metric on such pseudo-eggs $E_{2m}$. We verify that the Wu metric is a continuous Hermitian metric on $E_{2m}$, real analytic everywhere except along the complex hypersurface $Z = \{(0, z_{2}, . . . , z_{n}) \in E_{2m}\}$. We also show that the holomorphic curvature of the Wu metric for this noncompactfamily of pseudoconvex domains is bounded above in the sense of currents by a negative constant independent of $m$. This verifies a conjecture of S.Kobayashi and H.Wu for such $E_{2m}$.

• Schrödinger operators on a periodically broken zigzag carbon nanotube

In this paper, we study the spectra of Schrödinger operators on zigzag carbon nanotubes, which are broken by abrasion or during refining process. Throughout this paper, we assume that the carbon nanotubes are broken periodically and we deal with one of those models. Making use of the Floquet–Bloch theory, we examine the spectra of the Schrödinger operators and compare the spectra of the broken case and the pure unbroken case.

• On certain geodesic conjugacies of flat cylinders

We prove $C^0$-conjugacy rigidity of any flat cylinder among two different classes of metrics on the cylinder, namely among the class of rotationally symmetric metrics and among the class of metrics without conjugate points.

• Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

In this paper, we study the noncommutative Orlicz space $L_{\varphi}( \tilde{\cal M}, \tau)$,which generalizes the concept of noncommutative $L^p$ space, where $\cal M$ is a von Neumann algebra, and $\varphi$ is an Orlicz function. As a modular space, the space $L_{\varphi}( \tilde{\cal M}, \tau)$ possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace $E_{\varphi}( \tilde{\cal M}, \tau)$ =$\overline{\cal {M}\bigcap L_{\varphi}( \tilde{\cal M}, \tau)}$ in $L_{\varphi}( \tilde{\cal M}, \tau)$, which is closed under the norm topology and dense under the measure topology, is given. Moreover, if the Orlicz function $\varphi$ satisfies the $\Delta_2$-condition, then $L_{\varphi}( \tilde{\cal M}, \tau)$ is uniformly monotone, and convergence in the norm topology and measure topology coincide onthe unit sphere. Hence, $E_{\varphi}( \tilde{\cal M}, \tau)$ = $L_{\varphi}( \tilde{\cal M}, \tau)$ if $\varphi$ satisfies the $\Delta_2$-condition.

• Lie $n$-derivations on $\cal J$-subspace lattice algebras

Let $\cal L$ be a $\cal J$ -subspace lattice on a Banach space $X$ over the real or complex field $\mathbb F$ with dim $X \geq 3$ and let $n \geq 2$ be an integer. Suppose thatdim $K \neq 2$ for every $K \in \cal {J (L)}$ and $L : Alg \,\cal L → Alg \,\cal L$ is a linear map. Itis shown that $L$ satisfies $\sum^{n}_{i=1} p_{n}(A_{1}, \cdots, A_{i−1}, L(A_{i}), A_{i+1}, \cdots, A_{n}) = 0$ whenever $p_{n}(A_{1}, A_{2},\cdots, A_{n}) = 0$ for $A_{1}, A_{2},\cdots, A_{n} \in Alg \,\cal L$ if and only if for each $K \in \cal {J (L)}$, there exists a bounded linear operator $T_{K} \in \cal{B}\rm(K)$, a scalar $\lambda_K$ and a linearfunctional $h_{K} : Alg \,\cal {L} → \mathbb F$ such that $L(A)x = (T_{K}A−AT_{K} + \lambda_{K}A + h_{K}(A)I )x$ for all $x \in K$ and all $A \in Alg \,\cal L$. Based on this result, a complete characterization of linear $n$-Lie derivations on $Alg \,\cal L$ is obtained.

• Line bundles and flat connections

We prove that there are cocompact lattices $\Gamma$ in $\rm SL(2,\mathbb C)$ with the property that there are holomorphic line bundles $L$ on $\rm SL(2,\mathbb C)/ \Gamma$ with $c_{1}(L) = 0$ such that $L$ does not admit any unitary flat connection.

• # Proceedings – Mathematical Sciences

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