On equivariant maps related to the space of pairs of exceptional Jordan algebras

Let $\mathcal{J}$ be the exceptional Jordan algebra and $V=\mathcal{J}\oplus \mathcal{J}$. We construct an equivariant map from $V$ to $\mathrm{Hom}_k(\mathcal{J}\otimes \mathcal{J},\mathcal{J})$ defined by homogeneous polynomials of degree $8$ such that if $x\in V$ is a generic point, then the image of $x$ is the structure constant of the isotope of $\mathcal{J}$ corresponding to $x$. We also give an alternative way to define the isotope corresponding to a generic point of $\mathcal{J}$ by an equivariant map from $\mathcal{J}$ to the space of trilinear forms.Comment: 10 page

"A Semi-group Expansion for Pricing Barrier Options"

This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develops a semi-group expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs) arising in barrier option pricing. As an application, we propose a concrete approximation formula under a stochastic volatility model and demonstrate its validity by some numerical experiments.

Discontinuous elongation of nematic gels by a magnetic field

The effects of a magnetic field on the swelling of a nematic gel immersed in a low molecular weight liquid crystal solvent are examined by using a mean field theory. The nematic-isotropic (paranematic) phase transition temperature of the gel and the solvent is controlled by the external field. As a result of an anisotropic coupling between the gel and solvent, the shape of the gel is discontinuously (or continuously) elongated with increasing the strength of the external field. We examine the condition for a first- and second-order volume phase transition of the gel under a magnetic field

"Voice Matters in a Dictator Game"

We examine a dictator game with a "voice" option in the laboratory. In our experiment, the recipient has an opportunity to state a payoff-irrelevant request for the minimum acceptable offer before the dictator dictates his/her offer. In this game, it is predicted not only by the standard game theory, but by the behavioral game theory such as theories of other-regarding preferences, that the dictator's offer is independent of the recipient's request. Some findings based on our data are as follows: the above independence hypothesis is rejected; as the recipient's request increases, the dictator's offer increases when the requests are less than 50% of the pie; on the other hand, when the request goes beyond 50% of the pie, the offer decreases as the request increases. That is, "voice" matters in a dictator game. We also conduct a clustering analysis to classify dictators' behaviour into some notable patterns. As a result, we obtain the following three behavioural patterns: the other-disregarding, the punishing the greedy, and the lenient.