In this paper, by the method of heat flow and the
method of exhaustion, we prove an existence theorem of Hermitian-Yang-Mills-Higgs metrics on holomorphic line bundle over a class of non-compact Gauduchon manifold.
Authors: James A. Smith
Comments: 54 Pages.
Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting point, we then employ GA's capacities for handling rotations to determine the orientation of a gnomon at any given latitude and longitude during the period between the December solstices of 2016 and 2017. Subsequently, we derive equations for two angles: that between the Sun's rays and the gnomon's shaft, and that between the gnomon's shadow and the direction ``north" as traced on the ground at the gnomon's location. To validate our equations, we convert those angles to Solar azimuths and altitudes for comparison with simulations made by the program Stellarium. As further validation, we analyze our equations algebraically to predict (for example) the precise timings and locations of sunrises, sunsets, and Solar zeniths on the solstices and equinoxes. We emphasize that the accuracy of the results is only to be expected, given the high accuracy of the heliocentric model itself, and that the relevance of this work is the efficiency with which that model can be implemented via GA for teaching at the introductory level. On that point, comments and debate are encouraged and welcome.
Authors: Hiroshi Okumura
Comments: 2 Pages. This paper will be submitted to Sangaku Journal of Mathematics.
A problem involving an isosceles triangle with a square and three congruent circles is generalized.
The first definition (prior to the well-known five postulates) of Euclid describes the point as “that of which there is no part”. Here we show how the Euclidean account of manifolds is untenable in our physical realm and that the concepts of points, lines, surfaces, volumes need to be revisited, in order to allow us to be able to describe the real world. Here we show that the basic object in a physical context is a traversal of spacetime via tiny subregions of spatial regions, rather than the Euclidean point. We also elucidate the psychological issues that lead our mind to think to points and lines as really existing in our surrounding environment.
Authors: Ryan Haddad
Comments: 1 Page.
This conjecture may be a tool in defining the indefinite tangent of 90 degrees, and is a (new) mathematical coincidence that is indeed strange; why would the tangent of angles near 90 degrees be equal to the angle of the radian multiplied by powers of 10? In fact, if there is no geometrical explanation in current mathematics, it may resides in metamathematics.