In this paper we present a numerical method to compute Diophantine rotation numbers of circle maps with high accuracy. We mainly focus on analytic circle diffeomorphisms, but the method also works in the case of (enough) finite differentiability. The keystone of the method is that, under these conditions, the map is conjugate to a rigid rotation of the circle. Moreover, albeit it is not fully j...

Gerotor pump is a special kind of internal rotary pump, which contains a trochoid profile (commonly called as cycloid). Generation of trochoid is normally realized by external rolling method, namely a circle rotating on a fixed circle without slipping. This paper proposes derivative process of the trochoid profile by means of internal rolling method, which is that internal surface of a circle c...

In this paper we have described in the Section 1 some equations and theorems concerning the Circle Method applied to the Goldbach’s Conjecture. In the Section 2, we have described some equations and theorems concerning the Circle Method to investigate Germain primes by the Major arcs. In the Section 3, we have described some equations concerning the equivalence between the Goldbach’s Conjecture...

Dense packing of equal circles on a sphere is investigated. A systematic algorithm, the Minimum-Zenith Method (MZM), is proposed in this report. Started from a proper initial configuration, a circle is sequentially packed one by one so that the zenith angle is as small as possible. It is necessary to fix the size of the circle and some initial configuration. Circle configurations we examined ha...

Among the many parameters affecting the design of hydraulic structures on fissured rocks are orientation of principal permeability axes and degree of anisotropy of fissured rocks. Geological explorations rarely provide this type of information. If at all the information is available, methods employed are computationally cumbersome and inefficient. A new concept of simple equations and geometric...

Published 15 We obtain a new proof of an asymptotic formula for the coefficients of the j-invariant 16 of elliptic curves. Our proof does not use the circle method. We use Laplace's method 17 of steepest descent and the Hardy–Ramanujan asymptotic formula for the partition 18 function. (The latter asymptotic formula can be derived without the circle method.)