The volume conjecture,perturbative knot invariants,and recursion relations for topological strings

We study the relation between perturbative knot invariants and the free energies defined by topological string theory on the character variety of the knot. Such a correspondence between SL(2;C)$\mathit{SL}(2;\mathbb{C})$ Chern–Simons gauge theory and the topological open string theory was proposed earlier on the basis of the volume conjecture and AJ conjecture. In this paper we discuss this correspondence beyond the subleading order in the perturbative expansion on both sides. In the computation of the perturbative invariants for the hyperbolic 3-manifold, we adopt the state integral model for the hyperbolic knots, and the factorized AJ conjecture for the torus knots. On the other hand, we iteratively compute the free energies on the character variety using the Eynard–Orantin topological recursion relation. We discuss the correspondence for the figure eight knot complement and the once punctured torus bundle over S¹${\mathbb{S}}^1$ with the monodromy L²R$L^{2}R$ up to the fifth order. For the torus knots, we find trivial the recursion relations on both sides.     

Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter

We provide existence results of multiple solutions for quasilinear elliptic equations depending on a parameter under the Neumann and Dirichlet boundary condition. Our main result shows the existence of two opposite constant sign solutions and a sign changing solution in the case where we do not impose the subcritical growth condition to the nonlinear term not including derivatives of the solution. The studied equations contain the \(p\) -Laplacian problems as a special case. Our approach is based on variational methods combining super- and sub-solution and the existence of critical points via descending flow.     

On the frame of the unit ball of Banach spaces

The notion of the frame of the unit ball of Banach spaces was introduced to construct a new calculation method for the Dunkl-Williams constant. In this paper, we characterize the frame of the unit ball by using k-extreme points and extreme points of the unit ball of two-dimensional subspaces. Furthermore, we show that the frame of the unit ball is always closed, and is connected if the dimension of the space is not less than three. As infinite dimensional examples, the frame of the unit balls of c ₀ and ? _p are determined.     

In-situ observations of dissolution process of GaSb into InSb melt by X-ray penetration method

Dissolution process of GaSb into InSb melt was observed by an X-ray penetration method. The intensity of X-rays penetrated through the rectangular shaped GaSb (seed)/InSb/GaSb (feed) sandwich sample was recorded by the CdTe line sensor detector. The penetrated X-ray intensities and images of the sample were obtained as a function of time and temperature. The gallium (Ga) composition profile of the sample was calculated as a function of time by making the calibration line with the penetrated X-ray intensities of GaSb and InSb standard samples. The calculated Ga composition profile of the grown sample agreed well with the data measured by energy dispersive X-ray spectroscopy analysis. The result suggested that lower GaSb seed dissolved faster than upper GaSb feed despite of the low temperature at the lower GaSb seed. It clearly indicates that the solutal transport induced by gravity strongly affects the dissolution process.     

Glycerophosphocholines of the earthworm, Pheretima asiatica.

The lipid composition of the earthworm, Pheretima asiatica (Annelida), was analyzed. Four glycerophospholipids, together with four known glycosphingolipids, were isolated in pure form. The former four were 1-alkyl 2-acyl glycerophosphocholines possessing a C17:0 and/or C18:1 fatty acid residue. Their structures, including the position and geometry of the double bond, were determined on the bases of chemical and spectral data.     

Enantioselective total synthesis of anti HIV-1 active (+)-calanolide A through a quinine-catalyzed asymmetric intramolecular oxo-Michael addition

Enantioselective total synthesis of anti HIV-1 active (+)-calanolide A was achieved by a quinine-catalyzed asymmetric intramolecular oxo-Michael addition as a key step.