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This book provides a very readable description of a technique, developed by the author years ago but as current as ever, for proving that solutions to certain non-elliptic partial differential equations only have real analytic solutions when the data are real analytic locally. The resulting vector spaces, in turn, form the algebraic setting for the standard model of Euclidean geometry. Chapter 14 Treves Approach. Newtonian velocity addition is the common vector addition, which is both commutative and associative. In full analogy, Einsteinian velocity addition is a gyrovector addition, which is both gyrocommutative and gyroassociative. The novel relativistic resultant mass of the system, concentrated at the relativistic center of mass, dictates the validity of the dark matter and the dark energy that were introduced by cosmologists as ad hoc postulates to explain cosmological observations about missing gravitational force and late-time cosmic accelerated expansion. Chapter 3 Overview of Proofs. In full analogy with classical results, the book presents a novel relativistic interpretation of stellar aberration in terms of relativistic gyrotrigonometry and gyrovector addition. The technique is completely Chapter 4 Full Proof for the Heisenberg Group. The novel relativistic resultant mass of the system, concentrated at the relativistic center of mass, dictates the validity of the dark matter and the dark energy that were introduced by cosmologists as ad hoc postulates to explain cosmological observations about missing gravitational force and late-time cosmic accelerated expansion. The discovery of the relativistic center of mass in this book thus demonstrates once again the usefulness of the study of Einstein's special theory of relativity in terms of its underlying analytic hyperbolic geometry. Chapter 6 Pseudodifferential Problems. Chapter 3 Overview of Proofs.