Symplectic Modules Over Colombeau-Generalized Numbers

REFERENCES

• Abraham , R. , Marsden , J. E. ( 1978 ). Foundations of Mechanics. , Second ed. Cambridge , MA : Perseus Books .
   Google Scholar
• Aragona , J. , Fernandez , R. , Juriaans , S. O. ( 2009 ). Natural topologies on Colombeau algebras . Topol. Methods Nonlinear Anal. 34 ( 1 ): 161 – 180 .
   Web of Science ®Google Scholar
• Aragona , J. , Juriaans , S. O. ( 2001 ). Some structural properties of the topological ring of Colombeau's generalized numbers . Comm. Algebra 29 ( 5 ): 2201 – 2230 .
   Web of Science ®Google Scholar
• Aragona , J. , Juriaans , S. O. , Oliveira , O. R. B. , Scarpalezos , D. ( 2008 ). Algebraic and geometric theory of the topological ring of Colombeau generalized functions . Proc. Edinb. Math. Soc. (2) 51 ( 3 ): 545 – 564 .
   Web of Science ®Google Scholar
• Bhatia , R. ( 1987 ). Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics 162, Longman, Harlow .
   Google Scholar
• Bourbaki , N. ( 1974 ). Algebra I, Chapters 1–3 . Elements of Mathematics. Paris , Massachusetts : Hermann, Addison-Wesley .
   Google Scholar
• Burtscher , A. , Kunzinger , M. ( 2012 ). Algebras of generalized functions with smooth parameter dependence . Proc. Edinb. Math. Soc. (2) 55 ( 1 ): 105 – 124 .
   Google Scholar
• Colombeau , J.-F. ( 1984 ). New Generalized Functions and Multiplication of Distributions. North-Holland, Amsterdam .
   Google Scholar
• Colombeau , J.-F. (1985). Elementary Introduction to New Generalized Functions. North-Holland, Amsterdam.
   Google Scholar
• Garetto , C. , Hörmann , G. ( 2005 ). Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities . Proc. Edinb. Math. Soc. (2) 48 ( 3 ): 603 – 629 .
   Web of Science ®Google Scholar
• Garetto , C. , Hörmann , G. , Oberguggenberger , M. ( 2009 ). Generalized oscillatory integrals and Fourier integral operators . Proc. Edinb. Math. Soc. (2) 52 ( 2 ): 351 – 386 .
   Google Scholar
• Garetto , C. , Oberguggenberger , M. ( 2012 ). Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients. Math. Nachrichten, to appear .
   Google Scholar
• Grosser , M. , Kunzinger , M. , Oberguggenberger , M. , Steinbauer , R. ( 2001 ). Geometric Theory of Generalized Functions, volume 537 of Mathematics and its Applications . Dordrecht : Kluwer Academic Publishers .
   Google Scholar
• Hörmander , L. ( 1985 ). The Analysis of Linear Partial Differential Operators III . Grundlehren der Mathematischen Wissenschaften Vol. 274. Berlin : Springer-Verlag .
   Google Scholar
• Oberguggenberger , M. , Pilipović , S. , Scarpalézos , D. ( 2007 ). Positivity and positive definiteness in generalized function algebras . J. Math. Anal. Appl. 328 ( 2 ): 1321 – 1335 .
   Web of Science ®Google Scholar
• Mayerhofer , E. ( 2008 ). On Lorentz geometry in algebras of generalized functions . Proc. R. Soc. Edinb. Sect. A, Math. 138 ( 4 ): 843 – 871 .
   Google Scholar
• Nedeljkov , M. , Pilipović , S. , Scarpalézos , D. ( 1998 ). The Linear Theory of Colombeau Generalized Functions. Pitman Research Notes in Mathematics Series, 385. Longman, Harlow .
   Google Scholar
• Oberguggenberger , M. ( 1992 ). Multiplication of Distributions and Applications to Partial Differential Equations. Pitman Research Notes in Mathematics 259. Longman .
   Google Scholar
• Oberguggenberger , M. , Vernaeve , H. ( 2008 ). Internal sets and internal functions in Colombeau theory . J. Math. Anal. Appl. 341 ( 1 ): 649 – 659 .
   Web of Science ®Google Scholar
• Oberguggenberger , M. ( 2009 ). Hyperbolic systems with discontinuous coefficients: generalized wavefront sets. In: New Developments in Pseudo-differential Operators, Oper. Theory Adv. Appl. Vol. 189. Basel: Birkhäuser, pp. 117–136 .
   Google Scholar
• Oeljeklaus , E. , Remmert , R. ( 1974 ). Lineare Algebra. I . Berlin : Springer-Verlag. Heidelberger Taschenbücher, Band 150 .
   Google Scholar
• Scheja , G. , Storch , U. ( 1980 ). Lehrbuch der Algebra. Teil 1 . Stuttgart : B. G. Teubner .
   Google Scholar
• Scheja , G. , Storch , U. ( 1988 ). Lehrbuch der Algebra. Teil 2 . Stuttgart : B. G. Teubner .
   Google Scholar
• Vernaeve , H. Banach -algebras. Preprint, arxiv:0811.1742
   Google Scholar
• Vernaeve , H. ( 2010 ). Ideals in the ring of Colombeau generalized numbers . Comm. Algebra 38 ( 6 ): 2199 – 2228 .
   Web of Science ®Google Scholar
• Communicated by I. Shestakov.
   Google Scholar