This introductory text explores and develops the basic notions of geometry on the hyperbolic plane. Topics covered include the upper half-space model of the hyperbolic plane, Möbius transformations, the general Möbius group and the subgroup preserving path length in the upper half-space model, arc-length and distance, the Poincar disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications. Coverage provides readers with a firm grasp of the concepts and techniques of this beautiful area of mathematics.
From birth through age fifty-seven, I considered myself a staunch believer in my creator, but failed to embrace the reality that Jesus was truly the son of God in human form. Those struggles lead me to contrast my life to other organized groups of believers and wonder what I might have been capable of doing if my conceptual belief system had a Jihadist foundation instead of my reality. I describe two brushes with death in which the second event took my life in a totally unexpected direction. I describe that profound shift as if I were writing an op-ed, and the wild-goose chase that ensued.
Included is an appendix for struggling and skeptical Christians everywhere. It is a contemporary guide to help the reader become comfortable reading the Bible, so that he or she might discover the answer to each and every issue that faces humankind today.
In this unfiltered look at living the American dream and great loss, Jim shares his candid assessment of struggles with his conceptual belief system and the faith required to keep his faith alive for the next generation. His unique early-life encounters provided Jim with a sound foundation, a foundation that will be critically tested from unsuspecting angles as he navigates through life.
--Shan Rutherford, Senior Pastor, Greenwood Christian Church, Greenwood, Indiana.
From birth through age fifty-seven, I considered myself a staunch believer in my creator, but failed to embrace the reality that Jesus was truly the son of God in human form. Those struggles lead me to contrast my life to other organized groups of believers and wonder what I might have been capable of doing if my conceptual belief system had a Jihadist foundation instead of my reality. I describe two brushes with death in which the second event took my life in a totally unexpected direction. I describe that profound shift as if I were writing an op-ed, and the wild-goose chase that ensued.
Included is an appendix for struggling and skeptical Christians everywhere. It is a contemporary guide to help the reader become comfortable reading the Bible, so that he or she might discover the answer to each and every issue that faces humankind today.
In this unfiltered look at living the American dream and great loss, Jim shares his candid assessment of struggles with his conceptual belief system and the faith required to keep his faith alive for the next generation. His unique early-life encounters provided Jim with a sound foundation, a foundation that will be critically tested from unsuspecting angles as he navigates through life.
--Shan Rutherford, Senior Pastor, Greenwood Christian Church, Greenwood, Indiana.
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