Keisuke Teramoto

We study pedal curves for planar curves, possibly with singularities, called frontal curves. In particular, we analyze the types of singularities on pedal curves for a given frontal in terms of the geometrical properties.

We deal with pseudo-spherical surfaces admitting certain singularities and their caustics. In particular, we give characterizations of cuspidal butterfly, cuspidal lip and cuspidal beak singularities on a pseudo-spherical surface. Moreover, characterizations of certain singularities on the caustics of a pseudo-spherical surface are given. Furthermore, when the caustic has a cuspidal edge singularity, we investigate geometric invariants defined at that point.

We show that only pseudo-spherical surface whose caustic becomes a minimal surface is Dini surface family. Moreover, we give the Weierstrass data for corresponding minimal surface to the caustic.

In exploratory factor analysis, model parameters are usually estimated by maximum likelihood method. The maximum likelihood estimate is obtained by solving a complicated multivariate algebraic equation. Since the solution to the equation is usually intractable, it is typically computed with continuous optimization methods, such as Newton-Raphson methods. With this procedure, however, the solution is inevitably dependent on the estimation algorithm and initial value since the log-likelihood function is highly non-concave. Particularly, the estimates of unique variances can result in zero or negative, referred to as improper solutions; in this case, the maximum likelihood estimate can be severely unstable. To delve into the issue of the instability of the maximum likelihood estimate, we compute exact solutions to the multivariate algebraic equation by using algebraic computations. We provide a computationally efficient algorithm based on the algebraic computations specifically optimized for maximum likelihood factor analysis. To be specific, Gröebner basis and cylindrical decomposition are employed, powerful tools for solving the multivariate algebraic equation. Our proposed procedure produces all exact solutions to the algebraic equation; therefore, these solutions are independent of the initial value and estimation algorithm. We conduct Monte Carlo simulations to investigate the characteristics of the maximum likelihood solutions.

We study singularities of Gauss maps of frontal surfaces. In particular, we discuss relations between types of singularities of Gauss maps and geometric invariants of initial frontal surfaces.