On the special spherical triangles for physical and cosmological applications

It is well known that a spherical triangle of 270 degree triangle is constructible on the surface of a sphere; a globe is a good example. Take a point (A) on the equator, draw a line 1/4 the way around (90 degrees of longitude) on the equator to a new point (B).

Black Holes as Markers of Dimensionality

Black hole temperature TBH = TP/2πd as a function of its Planck length real diameter multiplier d is derived from black hole surface gravity and Hawking temperature w.l.o.g. It is conjectured d = 1/2π describes primordial Big Bang singularity as in this case TBH = TP. A black hole interacts with the environment and observable black holes have uniquely defined Delaunay triangulations with a natural number of spherical triangles having Planck areas (bits), where a Planck triangle is active and has gravitational potential of -c2 if all its vertices have black hole gravitational potential of -c2/2 and is inactive otherwise. As temporary distribution of active triangles on an event horizon tends to maximize Shannon entropy a black hole is a fundamental, one-sided thermodynamic equilibrium limit for a dissipative structure. Black hole blackbody radiation, informational capacity fluctuations, and quantum statistics are discussed. On the basis of the latter, wavelength bounds for BE, MB, and FD statistics are derived as a function of the diameter multiplier d. It is shown that black holes feature wave-particle duality only if d ≤ 8π, which also sets the maximum diameter of a totally collapsible black hole. This outlines the program for research of other nature phenomena that emit perfect blackbody radiation, such as neutron stars and white dwarfs.

Black Holes as Markers of Dimensionality

Black hole temperature as a function of its Planck length diameter multiple is derived from black hole surface gravity and Hawking temperature. It is conjectured that this multiple corresponds to dimensionality of the graph of nature with d = 1/2pi describing primordial Big Bang singularity. A black hole interacts with the environment and observable black holes have uniquely defined Delaunay triangulations with a natural number of spherical triangles having Planck areas (bits), where a Planck triangle is active and has gravitational potential of -c^2 if all its vertices have black hole gravitational potential of -c^2/2 and is inactive otherwise. Temporary distribution of active triangles on an event horizon tends to maximize Shannon entropy. Black hole blackbody radiation, informational capacity fluctuations, and quantum statistics are discussed. On the basis of the latter, wavelength bounds for BE, MB, and FD statistics are derived as a function of a diameter. A similarity of the logistic function and black hole FD statistics leads to the BE logistic function and map. This outlines the program for research of other nature phenomena that emit perfect blackbody radiation, such as neutron stars and white dwarfs.

Numerical Analysis of Steel Geodesic Dome under Seismic Excitations

The paper presents the response of two geodesic domes under seismic excitations. The structures subjected to seismic analysis were created by two different methods of subdividing spherical triangles (the original octahedron face), as proposed by Fuliński. These structures are characterised by the similar number of elements. The structures are made of steel, which is a material that undoubtedly gives lightness to structures and allows large spans. Designing steel domes is currently a challenge for constructors, as well as architects, who take into account their aesthetic considerations. The analysis was carried out using the finite element method of the numerical program. The two designed domes were analysed using four different seismic excitations. The analysis shows what influence particular earthquakes have on the geodesic dome structures by two different methods. The study analysed the maximum displacements, axial forces, velocities, and accelerations of the designed domes. In addition, the Time History method was used for the analysis, which enabled the analysis of the structure in the time domain. The study will be helpful in designing new structures in seismic areas and in assessing the strength of various geodesic dome structures under seismic excitation.

The method of images in the pricing of barrier derivatives in three dimensions

The thesis describes the joint distributions of minima, maxima and endpoint values for a three dimensional Wiener process. In particular, the results provide the point cumulative distributions for the maxima and/or minima of the components of the process. The densities are obtained explicitly for special type of correlations by the method of images; the analysis requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used to obtain explicit expressions for price of options in financial mathematics. We provide closed-form expressions for the price of several barrier type derivatives with a three dimensional geometric Wiener process as underlying. These solutions are found for special correlation matrices and are given by linear combinations of three dimensional Gaussian cumulative distributions. In order to extend the results to a wider set of correlation matrices the method of random correlations is outlined.

The method of images in the pricing of barrier derivatives in three dimensions

The thesis describes the joint distributions of minima, maxima and endpoint values for a three dimensional Wiener process. In particular, the results provide the point cumulative distributions for the maxima and/or minima of the components of the process. The densities are obtained explicitly for special type of correlations by the method of images; the analysis requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used to obtain explicit expressions for price of options in financial mathematics. We provide closed-form expressions for the price of several barrier type derivatives with a three dimensional geometric Wiener process as underlying. These solutions are found for special correlation matrices and are given by linear combinations of three dimensional Gaussian cumulative distributions. In order to extend the results to a wider set of correlation matrices the method of random correlations is outlined.

An analytic solution of full-sky spherical geometry for satellite relative motions

Herein, an exact and efficient analytic solution for an unperturbed satellite relative motion was developed using a direct geometrical approach. The derivation of the relative motion geometrically interpreted the projected Keplerian orbits of the satellites on a sphere (Earth and celestial spheres) using the solutions of full-sky spherical triangles. The results were basic and computationally faster than the vector and plane geometry solutions owing to the advantages of the full-sky spherical geometry. Accordingly, the validity of the proposed solution was evaluated by comparing it with other analytic relative motion theories in terms of modeling accuracy and efficiency. The modeling accuracy showed an equivalent performance with Vadali’s nonlinear unit sphere approach, which is essentially equal to the Yan–Alfriend nonlinear theory. Moreover, the efficiency was demonstrated by the lowest computational cost compared with other models. In conclusion, the proposed modeling approach illustrates a compact and efficient closed-form solution for satellite relative motion dynamics.