The classical family of Wishart distributions on a cone of positive definite matrices and its fun... more The classical family of Wishart distributions on a cone of positive definite matrices and its fundamental features are extended to a family of generalized Wishart distributions on a homogeneous cone using the theory of exponential families. The generalized Wishart ...
Manger tak Steen! You were 'Splendid'! (P-a.e.) " Life is more important than mathematics. " " Th... more Manger tak Steen! You were 'Splendid'! (P-a.e.) " Life is more important than mathematics. " " Think invariantly! " " Find the natural representations! " I must like to think that herein we have defined, created, orchestrated, & recorded (though perhaps not as nicely as Eva Legene would record) some measure of mathematical pattern & beauty. Likely neither as polished nor as complete as we would otherwise have liked to imagine, but enveloped in the confines of the vector space of life. Perhaps if I could have transcended, ascended, descended into an octonionic world... or borrowed passage through a Lorentzian space-time cone ... now let's just sit back & sip a cup of T-algebras ... or a glass of vvine while atop a beautiful berg ... perhaps Bernhard would even come to join us Thank you Dr. Mary Jo Wojnar Primosch! You have sporadically sustained me with your ever-constant sibling confidence and with your timely occasional wisdom. v x listening to it? Samuel Eilenberg, to whom the wide fields of algebraic topology, homological algebra, and category theory owe deep debt for their very existence, was sometimes described as investing inordinate efforts at reformulating or repackaging what had been already long-developed. I defend this dissertation for its significant volume of definitions, parallel definitions, footnotes, and abstractions. It is this mathematician's perspective that category-theoretic language is a natural language for dynamically evolving one's conceptions in the constructive & creative doing of mathematics. It even seems that one can loosely suggest that an analogy is but a functor between two worlds. Perhaps an argument can be made that mathematics is the science & fine art of analogies, little analogies being patterns. Moreover, the engagements of ascending & descending the ladders of abstraction and of exploring tangential byways are in large measure the foundation for recognizing threads of interconnections, and of recognizing in different levels of our conceptualizations some, if You would, scale-invariant or scale-almost-invariant self-similarity meta-mathematical fractals, even more wondrous than the concrete Madelbrot set & its many cousins. I seek to assemble the puzzle, or at least to lay some foundation work for others to assemble the puzzle, not just to pick out the more interesting and useful small clusters of pieces. Contributions to the ongoing development of mathematics are gained from mathematicians of a wide spectrum of philosophical dispositions and technical talents. This candidate believes strongly in the merits of notations, notions, definitions, abstractions, and categories, and it is in this spirit that I submit this dissertation. Lastly, paraphrasing Pierre, it disappoints me that several nice trinkets of mathematics remain begging to be included in this thesis, but the available temporal margins set by fidelity to my family are unsufficient to accomodate them.
The classical family of Wishart distributions on a cone of positive definite matrices and its fun... more The classical family of Wishart distributions on a cone of positive definite matrices and its fundamental features are extended to a family of generalized Wishart distributions on a homogeneous cone using the theory of exponential families. The generalized Wishart ...
Manger tak Steen! You were 'Splendid'! (P-a.e.) " Life is more important than mathematics. " " Th... more Manger tak Steen! You were 'Splendid'! (P-a.e.) " Life is more important than mathematics. " " Think invariantly! " " Find the natural representations! " I must like to think that herein we have defined, created, orchestrated, & recorded (though perhaps not as nicely as Eva Legene would record) some measure of mathematical pattern & beauty. Likely neither as polished nor as complete as we would otherwise have liked to imagine, but enveloped in the confines of the vector space of life. Perhaps if I could have transcended, ascended, descended into an octonionic world... or borrowed passage through a Lorentzian space-time cone ... now let's just sit back & sip a cup of T-algebras ... or a glass of vvine while atop a beautiful berg ... perhaps Bernhard would even come to join us Thank you Dr. Mary Jo Wojnar Primosch! You have sporadically sustained me with your ever-constant sibling confidence and with your timely occasional wisdom. v x listening to it? Samuel Eilenberg, to whom the wide fields of algebraic topology, homological algebra, and category theory owe deep debt for their very existence, was sometimes described as investing inordinate efforts at reformulating or repackaging what had been already long-developed. I defend this dissertation for its significant volume of definitions, parallel definitions, footnotes, and abstractions. It is this mathematician's perspective that category-theoretic language is a natural language for dynamically evolving one's conceptions in the constructive & creative doing of mathematics. It even seems that one can loosely suggest that an analogy is but a functor between two worlds. Perhaps an argument can be made that mathematics is the science & fine art of analogies, little analogies being patterns. Moreover, the engagements of ascending & descending the ladders of abstraction and of exploring tangential byways are in large measure the foundation for recognizing threads of interconnections, and of recognizing in different levels of our conceptualizations some, if You would, scale-invariant or scale-almost-invariant self-similarity meta-mathematical fractals, even more wondrous than the concrete Madelbrot set & its many cousins. I seek to assemble the puzzle, or at least to lay some foundation work for others to assemble the puzzle, not just to pick out the more interesting and useful small clusters of pieces. Contributions to the ongoing development of mathematics are gained from mathematicians of a wide spectrum of philosophical dispositions and technical talents. This candidate believes strongly in the merits of notations, notions, definitions, abstractions, and categories, and it is in this spirit that I submit this dissertation. Lastly, paraphrasing Pierre, it disappoints me that several nice trinkets of mathematics remain begging to be included in this thesis, but the available temporal margins set by fidelity to my family are unsufficient to accomodate them.
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