We present a curve approximation method which approximates each planar algebraic curve segment by discrete curve points at each of which the curve has its gradient from a set of uniformly distributed normals. This method, called Gaussian Approximation (GAP), provides eecient algorithms for various primitive geometric operations, especially for those related with gradients such as common tangent...

We show that in two dimensional flat torus the number of intersections between random eigenfunctions general eigenvalues and a given smooth curve is almost exponentially concentrated around its mean, even when randomness not gaussian.

This paper presents new geometric aspects of the behaviors of solutions to the porous medium equation (PME) and its associated equation. First we discuss thermostatistical structure with information geometry on a manifold of generalized exponential densities. A dualistic relation between the two existing formalisms by Naudts and Eguchi is elucidated. Next by equipping the manifold of what is ca...

For the inverse synthetic aperture radar (ISAR) imaging of a target at a long range, range alignment using the existing polynomial method brings about poor results because the flight trajectory changes depending on the initial position, and the motion parameters, meaning the polynomial cannot fit the trajectory. This paper proposes an improved range alignment method that models the trajectory u...

We report the first measurement of genus curves for the two-dimensional mass map in the neighborhood of rich, X-ray luminous galaxy cluster MS1054-03 at z = 0.83, reconstructed from weak lensing data obtained by Suprime-Cam on the prime focus of 8.2m Subaru telescope . We find that the genus curve measured in the whole survey field deviates from that expected from a random Gaussian field. We sh...

I show how one can estimate the shape of a thermal performance curve using information theory. This approach ranks plausible models by their Akaike information criterion (AIC), which is a measure of a model’s ability to describe the data discounted by the model’s complexity. I analyze previously published data to demonstrate how one applies this approach to describe a thermal performance curve....

Shi et al. (2006) proposed a Gaussian process functional regression (GPFR) model to model functional response curves with a set of functional covariates. Two main problems are addressed by this method: modelling nonlinear and nonparametric regression relationship and modelling covariance structure and mean structure simultaneously. The method gives very good results for curve fitting and predic...

A recent analysis of the productivity growth data shows, somewhat surprisingly, that the dependence of the 20-century productivity growth on time can be reasonably well described by a Gaussian formula. In this paper, we provide a possible theoretical explanation for this observation. 1 Formulation of the Problem An empirical fact. A recent book [2] shows that, when averaged over decades, the pr...

How do we perceive the predictability of functions? We derive a rational measure of a function’s predictability based on Gaussian process learning curves. Using this measure, we show that the smoothness of a function can be more important to predictability judgments than the variance of additive noise or the number of samples. These patterns can be captured well by the learning curve for Gaussi...

The effect of a weak source of noise on the chaotic scattering is relevant to situations of physical interest. We investigate how a weak source of additive uncorrelated Gaussian noise affects both the dynamics and the topology of a paradigmatic chaotic scattering problem as the one taking place in the open nonhyperbolic regime of the Hénon-Heiles Hamiltonian system. We have found long transient...