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abelian abelian group abstract algebra algorithm analysis applications asymptotic August 19 automorphism Banach space bifurcation boundary bounded C*-algebra coefficients combinatorial commutative ring compact complex Computer conjecture construction continuous functions convergence convex convex set curvature defined denote differential equations dimension dimensional discuss domain equivalent exists extended field finite formula functions geometry given graph harmonic Hilbert Hilbert space homeomorphism homotopy infinite integral invariant inverse isomorphic January 28 lattice Lie algebra Lie group linear manifold Math matrix matroids maximal method metric minimal modules nonlinear obtained October 17 operator paper parameters polynomial Preliminary report prime problem proof properties prove Received August 19 Received December Received January Received July Received October 17 Received September representation Riemann Riemannian Riemannian manifold ring semigroup sequence singular solution Sponsored structure subgroup subset subspace surfaces symmetric Theorem theory topological transformation variables vector