Geometric continuity and compatibility conditions for 4-patch surfaces

When considering regularity of surfaces, it is its geometry that is of interest. Thus, the concept of geometric regularity or geometric continuity of a specific order is a relevant concept. In this paper we discuss necessary and sufficient conditions for a 4-patch surface to be geometrically continuous of order one and two or, in other words, being tangent plane continuous and curvature continuous respectively. The focus is on the regularity at the point where the four patches meet and the compatibility conditions that must appear in this case. In this article the compatibility conditions are proved to be independent of the patch parametrization, i.e., the compatibility conditions are universal. In the end of the paper these results are applied to a specific parametrization such as Bezier representation in order to generalize a 4-patch surface result by Sarraga.Comment: 25 pages, 6 figure

Origin of Mass. Mass and Mass-Energy Equation from Classical-Mechanics Solution

We establish the classical wave equation for a particle formed of a massless oscillatory elementary charge generally also traveling, and the resulting electromagnetic waves, of a generally Doppler-effected angular frequency \w, in the vacuum in three dimensions. We obtain from the solutions the total energy of the particle wave to be \eng=\hbarc\w, 2\pi \hbarc being a function expressed in wave-medium parameters and identifiable as the Planck constant. In respect to the train of the waves as a whole traveling at the finite velocity of light $c$, \eng=mc^2 represents thereby the translational kinetic energy of the wavetrain, m=\hbarc\w/c^2 being its inertial mass and thereby the inertial mass of the particle. Based on the solutions we also write down a set of semi-empirical equations for the particle's de Broglie wave parameters. From the standpoint of overall modern experimental indications we comment on the origin of mass implied by the solution.Comment: 13 pages, no figure. Augmented introductio

Structural and electronic properties of Li intercalated graphene on SiC(0001)

We investigate the structural and electronic properties of Li-intercalated monolayer graphene on SiC(0001) using combined angle-resolved photoemission spectroscopy and first-principles density functional theory. Li intercalates at room temperature both at the interface between the buffer layer and SiC and between the two carbon layers. The graphene is strongly $n$-doped due to charge transfer from the Li atoms and two $\pi$-bands are visible at the $\bar{K}$-point. After heating the sample to 300$^\circ$C, these $\pi$-bands become sharp and have a distinctly different dispersion to that of Bernal-stacked bilayer graphene. We suggest that the Li atoms intercalate between the two carbon layers with an ordered structure, similar to that of bulk LiC$_6$. An AA-stacking of these two layers becomes energetically favourable. The $\pi$-bands around the $\bar{K}$-point closely resemble the calculated band structure of a C$_6$LiC$_6$ system, where the intercalated Li atoms impose a super-potential on the graphene electronic structure that opens pseudo-gaps at the Dirac points of the two $\pi$-cones.Comment: 9 pages, 7 figure