Nonlinear Distributional Geometry
Nonlinear Distributional Geometry
Michael Kunzinger
(ORCID: 0000-0002-7113-0588)
Disciplines
Mathematics (80%); Physics, Astronomy (20%)
Keywords
-
Nonlinear Distributional Geometry,
Algebras of Generalized Functions,
Nonsmooth Differential Geometry,
Geometric Analysis of PDOs,
Nonsmooth General Relativity
The linear theory of generalized functions (distributions) has had a fundamental impact on diverse branches of mathematics and is today an indispensable tool for handling singular quantities both in mathematical analysis and in applications from physics (in particular in quantum mechanics and general relativity). One important limitation of distribution theory, however, is its inability to incorporate nonlinear operations into a comprehensive theory of nonlinear generalized functions adapted to the requirements of mathematical physics. The development of such a nonlinear extension of classical distribution theory was initiated in the early 1980ies by J.F. Colombeau, M. Oberguggenberger and others, utilizing the fundamental technique of regularizing singular quantities through approximations by smooth objects combined with asymptotic analysis in terms of a regularization parameter. Initially the resulting theory was mainly applied in problems of nonlinear partial differential equations involving singularities. However, an increasing number of applications to problems of a primarily geometric nature (general relativity, mathematical geophysics, nonsmooth differential geometry) brought about a fundamental reconstruction of the theory towards the development of a "nonlinear distributional geometry" capable of rigorously handling singular objects in a genuinely geometric context. It is the aim of this project to substantially advance nonlinear distributional geometry both as a mathematical theory and as a tool for applications. Development of the theory will therefore be carried out in close connection to relevant fields of applications, namely geometric analysis of partial differential operators, mathematical geophysics and Einstein`s theory of gravitation.
The linear theory of generalized functions (distributions) has had a fundamental impact on diverse branches of mathematics and is today an indispensable tool for handling singular quantities both in mathematical analysis and in applications from physics (in particular in quantum mechanics and general relativity). One important limitation of distribution theory, however, is its inability to incorporate nonlinear operations into a comprehensive theory of nonlinear generalized functions adapted to the requirements of mathematical physics. The development of such a nonlinear extension of classical distribution theory was initiated in the early 1980ies by J.F. Colombeau, M. Oberguggenberger and others, utilizing the fundamental technique of regularizing singular quantities through approximations by smooth objects combined with asymptotic analysis in terms of a regularization parameter. Initially the resulting theory was mainly applied in problems of nonlinear partial differential equations involving singularities. However, an increasing number of applications to problems of a primarily geometric nature (general relativity, mathematical geophysics, non-smooth differential geometry) brought about a fundamental reconstruction of the theory towards the development of a "nonlinear distributional geometry" capable of rigorously handling singular objects in a genuinely geometric context.The START-project Nonlinear Distributional Geometry has substantially advanced nonlinear distributional geometry both as a mathematical theory and as a tool for applications. The project was subdivided into three (interconnected) parts:Nonlinear Distributional GeometryThe main result here was the development of a global tensorial theory of nonlinear generalized functions containing the linear theory of distributions and suitable for applications in non-smooth differential geometry and mathematical physics. Nonsmooth Geometric Analysis of Partial Differential OperatorsHere, a new theory of Pseudodifferential- and Fourier-Integral Operators was developed and applied to problems of Mathematical Geophysics.Nonsmooth General RelativityNew methods for solving wave equations in General Relativity were created and the general problem of describing singularities in space-times of low regularity was addressed.The project has given strong momentum to research in several branches of nonlinear generalized functions and will have a lasting impact both on the mathematical field itself and on its applications in analysis, geometry, and mathematical physics.
Research institution(s)
- Universität Innsbruck - 1%
- Universität Wien - 99%
Project participants
- Michael Oberguggenberger, Universität Innsbruck , associated research partner
Research Output
- 1746 Citations
- 196 Publications
Publications
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2013
Title Wave Equations and Symmetric First-order Systems in Case of Low Regularity DOI 10.1007/978-3-0348-0585-8_15 Type Book Chapter Author Hanel C Publisher Springer Nature Pages 283-296 -
2013
Title Ordinary Differential Equations in Algebras of Generalized Functions DOI 10.1007/978-3-0348-0585-8_13 Type Book Chapter Author Erlacher E Publisher Springer Nature Pages 253-270 -
2013
Title Invariant meshless discretization schemes DOI 10.1088/1751-8113/46/6/062001 Type Journal Article Author Bihlo A Journal Journal of Physics A: Mathematical and Theoretical Pages 062001 Link Publication -
2013
Title An equivariant noncommutative residue DOI 10.4171/jncg/132 Type Journal Article Author Dave S Journal Journal of Noncommutative Geometry Pages 709-735 Link Publication -
2013
Title GENERALIZED SOLUTIONS FOR THE EULER–BERNOULLI MODEL WITH ZENER VISCOELASTIC FOUNDATIONS AND DISTRIBUTIONAL FORCES DOI 10.1142/s0219530513500176 Type Journal Article Author Hörmann G Journal Analysis and Applications Pages 1350017 Link Publication -
2016
Title Singular reduction modules of differential equations DOI 10.1063/1.4965227 Type Journal Article Author Boyko V Journal Journal of Mathematical Physics Pages 101503 Link Publication -
2009
Title Generalized oscillatory integrals and Fourier integral operators DOI 10.1017/s0013091506000915 Type Journal Article Author Garetto C Journal Proceedings of the Edinburgh Mathematical Society Pages 351-386 Link Publication -
2009
Title Seismic imaging with the generalized Radon transform: a curvelet transform perspective DOI 10.1088/0266-5611/25/2/025005 Type Journal Article Author De Hoop M Journal Inverse Problems Pages 025005 -
2009
Title Variational problems with fractional derivatives: Invariance conditions and Nöther’s theorem DOI 10.1016/j.na.2008.12.043 Type Journal Article Author Atanackovic T Journal Nonlinear Analysis: Theory, Methods & Applications Pages 1504-1517 Link Publication -
2009
Title Closed graph and open mapping theorems for topological $ \tilde \lc $-modules and applications DOI 10.1002/mana.200610793 Type Journal Article Author Garetto C Journal Mathematische Nachrichten Pages 1159-1188 Link Publication -
2009
Title A global approach to generalized function algebras DOI 10.1080/10652460802564522 Type Journal Article Author Dave S Journal Integral Transforms and Special Functions Pages 257-263 -
2009
Title Isomorphisms of algebras of generalized functions DOI 10.1007/s00605-009-0152-9 Type Journal Article Author Vernaeve H Journal Monatshefte für Mathematik Pages 225-237 -
2009
Title Conservation laws and hierarchies of potential symmetries for certain diffusion equations DOI 10.1016/j.physa.2008.10.018 Type Journal Article Author Ivanova N Journal Physica A: Statistical Mechanics and its Applications Pages 343-356 Link Publication -
2009
Title Generalized solutions for the Euler–Bernoulli model with distributional forces DOI 10.1016/j.jmaa.2009.03.069 Type Journal Article Author Hörmann G Journal Journal of Mathematical Analysis and Applications Pages 142-153 Link Publication -
2009
Title Is a nonclassical symmetry a symmetry? Type Conference Proceeding Abstract Author Kunzinger M Conference Proceedings of 4th Workshop 'Group Analysis of Differential Equations and Integrable Systems' -
2009
Title Block diagonalization of four-dimensional metrics DOI 10.1088/0264-9381/26/23/235014 Type Journal Article Author Grant J Journal Classical and Quantum Gravity Pages 235014 Link Publication -
2009
Title On the Geroch?Traschen class of metrics DOI 10.1088/0264-9381/26/6/065001 Type Journal Article Author Steinbauer R Journal Classical and Quantum Gravity Pages 065001 Link Publication -
2009
Title The Cauchy problem for Schrödinger-type partial differential operators with generalized functions in the principal part and as data DOI 10.48550/arxiv.0909.5672 Type Preprint Author Hörmann G -
2009
Title Symmetry justification of Lorenz’ maximum simplification DOI 10.1007/s11071-009-9634-5 Type Journal Article Author Bihlo A Journal Nonlinear Dynamics Pages 101-107 -
2008
Title Local conservation laws of second-order evolution equations DOI 10.1088/1751-8113/41/36/362002 Type Journal Article Author Popovych R Journal Journal of Physics A: Mathematical and Theoretical Pages 362002 Link Publication -
2008
Title On Lorentz geometry in algebras of generalized functions DOI 10.1017/s0308210506000898 Type Journal Article Author Mayerhofer E Journal Proceedings of the Royal Society of Edinburgh: Section A Mathematics Pages 843-871 Link Publication -
2008
Title Pointwise characterizations in generalized function algebras DOI 10.1007/s00605-008-0032-8 Type Journal Article Author Vernaeve H Journal Monatshefte für Mathematik Pages 195 -
2008
Title Foundations of the Calculus of Variations in Generalized Function Algebras DOI 10.1007/s10440-008-9228-0 Type Journal Article Author Konjik S Journal Acta Applicandae Mathematicae Pages 169-199 -
2008
Title Interactions of delta shock waves in a strictly hyperbolic system of conservation laws DOI 10.1016/j.jmaa.2008.03.040 Type Journal Article Author Nedeljkov M Journal Journal of Mathematical Analysis and Applications Pages 1143-1157 -
2008
Title A Note on Distribution Spaces on Manifolds. Type Journal Article Author Grosser M -
2008
Title Spectral asymptotic Analysis in Algebras of Generalized Functions. Type Journal Article Author Delcroix A -
2008
Title Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source DOI 10.1007/s10440-008-9280-9 Type Journal Article Author Vaneeva O Journal Acta Applicandae Mathematicae Pages 1-46 -
2008
Title Generalized solutions for the Euler-Bernoulli model with distributional forces DOI 10.48550/arxiv.0812.1958 Type Preprint Author Hörmann G -
2008
Title A Note on Distribution Spaces on Manifolds DOI 10.48550/arxiv.0812.5099 Type Preprint Author Grosser M -
2008
Title Conservation laws and potential symmetries of systems of diffusion equations DOI 10.1088/1751-8113/41/23/235201 Type Journal Article Author Ivanova N Journal Journal of Physics A: Mathematical and Theoretical Pages 235201 -
2008
Title Generalized Fourier Integral Operators on spaces of Colombeau type DOI 10.48550/arxiv.0803.0284 Type Preprint Author Garetto C -
2008
Title Microlocal Analysis of Generalized Pullbacks of Colombeau Functions DOI 10.1007/s10440-008-9266-7 Type Journal Article Author Haller S Journal Acta Applicandae Mathematicae Pages 83 -
2008
Title Generalized analytic functions on generalized domains DOI 10.48550/arxiv.0811.1521 Type Preprint Author Vernaeve H -
2008
Title Banach $\widetilde{\mathbb C}$-algebras DOI 10.48550/arxiv.0811.1742 Type Preprint Author Vernaeve H -
2008
Title Generalized regularity and solution concepts for differential equations DOI 10.48550/arxiv.0806.1451 Type Preprint Author Haller S -
2008
Title Variational problems with fractional derivatives: Euler–Lagrange equations DOI 10.1088/1751-8113/41/9/095201 Type Journal Article Author Atanackovic T Journal Journal of Physics A: Mathematical and Theoretical Pages 095201 Link Publication -
2008
Title A Note on Distributional Semi-Riemannian Geometry. Type Journal Article Author Steinbauer R -
2008
Title Full algebra of generalized functions and non-standard asymptotic analysis DOI 10.1007/s11813-008-0008-y Type Journal Article Author Todorov T Journal Logic and Analysis Pages 205 -
2008
Title A Note on Distributional Semi-Riemannian Geometry DOI 10.48550/arxiv.0812.0173 Type Preprint Author Steinbauer R -
2008
Title Singular reduction operators in two dimensions DOI 10.48550/arxiv.0808.3577 Type Preprint Author Kunzinger M -
2008
Title Potential Conservation Laws DOI 10.48550/arxiv.0803.1156 Type Preprint Author Kunzinger M -
2008
Title Symmetry justification of Lorenz' maximum simplification DOI 10.48550/arxiv.0805.4061 Type Preprint Author Bihlo A -
2008
Title Tensor Valued Colombeau Functions on Manifolds DOI 10.48550/arxiv.0812.3275 Type Preprint Author Grosser M -
2008
Title The ADHM construction and non-local symmetries of the self-dual Yang-Mills equations DOI 10.48550/arxiv.0812.2005 Type Preprint Author Grant J -
2008
Title Reducible connections and non-local symmetries of the self-dual Yang-Mills equations DOI 10.48550/arxiv.0812.2147 Type Preprint Author Grant J -
2008
Title Conservation laws and hierarchies of potential symmetries for certain diffusion equations DOI 10.48550/arxiv.0806.1698 Type Preprint Author Ivanova N -
2008
Title Block diagonalisation of four-dimensional metrics DOI 10.48550/arxiv.0809.3327 Type Preprint Author Grant J -
2008
Title Potential conservation laws DOI 10.1063/1.2993117 Type Journal Article Author Kunzinger M Journal Journal of Mathematical Physics Pages 103506 Link Publication -
2008
Title Comparison of some solution concepts for linear first-order hyperbolic differential equations with non-smooth coefficients DOI 10.2298/pim0898123h Type Journal Article Author Haller S Journal Publications de l'Institut Mathematique Pages 123-157 Link Publication -
2008
Title Singular reduction operators in two dimensions DOI 10.1088/1751-8113/41/50/505201 Type Journal Article Author Kunzinger M Journal Journal of Physics A: Mathematical and Theoretical Pages 505201 Link Publication -
2008
Title The Wave Equation on Singular Space-Times DOI 10.1007/s00220-008-0549-7 Type Journal Article Author Grant J Journal Communications in Mathematical Physics Pages 399-420 -
2008
Title Internal sets and internal functions in Colombeau theory DOI 10.1016/j.jmaa.2007.10.030 Type Journal Article Author Oberguggenberger M Journal Journal of Mathematical Analysis and Applications Pages 649-659 Link Publication -
2008
Title Generalized Fourier Integral Operators on Spaces of Colombeau Type DOI 10.1007/978-3-7643-8969-7_8 Type Book Chapter Author Garetto C Publisher Springer Nature Pages 137-184 Link Publication -
2008
Title Reduction operators of linear second-order parabolic equations DOI 10.1088/1751-8113/41/18/185202 Type Journal Article Author Popovych R Journal Journal of Physics A: Mathematical and Theoretical Pages 185202 Link Publication -
2008
Title Spectral asymptotic analysis in algebras of generalized functions DOI 10.3233/asy-2008-0885 Type Journal Article Author Delcroix A Journal Asymptotic Analysis Pages 83-107 Link Publication -
2008
Title Evolution systems for paraxial wave equations of Schrödinger-type with non-smooth coefficients DOI 10.1016/j.jde.2008.06.022 Type Journal Article Author De Hoop M Journal Journal of Differential Equations Pages 1413-1432 Link Publication -
2008
Title Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations DOI 10.1007/s10440-008-9321-4 Type Journal Article Author Popovych R Journal Acta Applicandae Mathematicae Pages 315-359 Link Publication -
2008
Title Homogeneity in generalized function algebras DOI 10.1016/j.jmaa.2007.07.049 Type Journal Article Author Hanel C Journal Journal of Mathematical Analysis and Applications Pages 889-904 Link Publication -
2008
Title Real analytic generalized functions DOI 10.1007/s00605-008-0524-6 Type Journal Article Author Pilipovic S Journal Monatshefte für Mathematik Pages 85 Link Publication -
2008
Title Fundamental Solutions in the Colombeau Framework: Applications to Solvability and Regularity Theory DOI 10.1007/s10440-008-9220-8 Type Journal Article Author Garetto C Journal Acta Applicandae Mathematicae Pages 281-318 -
2008
Title Generalized solutions of the Vlasov–Poisson system with singular data DOI 10.1016/j.jmaa.2007.08.051 Type Journal Article Author Kmit I Journal Journal of Mathematical Analysis and Applications Pages 575-587 Link Publication -
2007
Title Real Analytic Generalized Functions DOI 10.48550/arxiv.math/0702794 Type Preprint Author Pilipovic S -
2007
Title Fundamental solutions in the Colombeau framework: applications to solvability and regularity theory DOI 10.48550/arxiv.math/0701344 Type Preprint Author Garetto C -
2007
Title Multidimensional integrable systems and deformations of Lie algebra homomorphisms DOI 10.1063/1.2777008 Type Journal Article Author Dunajski M Journal Journal of Mathematical Physics Pages 093502 Link Publication -
2007
Title Multidimensional integrable systems and deformations of Lie algebra homomorphisms DOI 10.48550/arxiv.nlin/0702040 Type Preprint Author Dunajski M -
2007
Title Microlocal Asymptotic Analysis in Algebras of Generalized Functions DOI 10.48550/arxiv.0704.1077 Type Preprint Author Delcroix A -
2007
Title Weak homogeneity in generalized function algebras DOI 10.48550/arxiv.0711.2495 Type Preprint Author Vernaeve H -
2007
Title Group invariant Colombeau generalized functions DOI 10.1007/s00605-007-0485-1 Type Journal Article Author Vernaeve H Journal Monatshefte für Mathematik Pages 165-175 -
2007
Title Geometrical embeddings for distributions into algebras of generalized functions DOI 10.48550/arxiv.0709.2039 Type Preprint Author Dave S -
2007
Title Internal sets and internal functions in Colombeau theory DOI 10.48550/arxiv.0707.0994 Type Preprint Author Oberguggenberger M -
2007
Title Pointwise characterizations in generalized function algebras DOI 10.48550/arxiv.0712.3464 Type Preprint Author Vernaeve H -
2007
Title Comparison of some solution concepts for linear first-order hyperbolic differential equations with non-smooth coefficients DOI 10.48550/arxiv.0712.1669 Type Preprint Author Haller S -
2007
Title Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis DOI 10.48550/arxiv.0712.2603 Type Preprint Author Todorov T -
2007
Title Ideals in the ring of Colombeau generalized numbers DOI 10.48550/arxiv.0707.0698 Type Preprint Author Vernaeve H -
2007
Title Hilbert $\widetilde{\C}$-modules: structural properties and applications to variational problems DOI 10.48550/arxiv.0707.1104 Type Preprint Author Garetto C -
2007
Title Distributional solution concepts for the Euler–Bernoulli beam equation with discontinuous coefficients DOI 10.1080/00036810701595944 Type Journal Article Author Hörmann G Journal Applicable Analysis Pages 1347-1363 Link Publication -
2007
Title Wave equations on space-times of low regularity: Existence results and regularity theory in the framework of generalized function algebras. Type Conference Proceeding Abstract Author Grant J Conference Proceedings of GF2007, Bedlewo -
2007
Title Spherical completeness of the non-archemedian ring of Colombeau generalized numbers. Type Journal Article Author Mayerehofer E Journal Bulletin of the Institute of Mathematics, Academia Sinica -
2007
Title On the characterization of p -adic Colombeau–Egorov generalized functions by their point values DOI 10.1002/mana.200510554 Type Journal Article Author Mayerhofer E Journal Mathematische Nachrichten Pages 1297-1301 -
2007
Title Conservation Laws and Potential Symmetries of Linear Parabolic Equations DOI 10.1007/s10440-007-9178-y Type Journal Article Author Popovych R Journal Acta Applicandae Mathematicae Pages 113-185 -
2007
Title Sequence spaces with exponent weights. Realizations of Colombeau type algebras DOI 10.4064/dm447-0-1 Type Journal Article Author Delcroix A Journal Dissertationes Mathematicae Pages 1-56 Link Publication -
2007
Title Wave equations on space-times of low regularity: Existence results and regularity theory in the framework of generalized function algebras DOI 10.48550/arxiv.0711.1972 Type Preprint Author Grant J -
2007
Title Conservation Laws and Potential Symmetries of Linear Parabolic Equations DOI 10.48550/arxiv.0706.0443 Type Preprint Author Popovych R -
2007
Title Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source DOI 10.48550/arxiv.0708.3457 Type Preprint Author Vaneeva O -
2007
Title The wave equation on singular space-times DOI 10.48550/arxiv.0710.2007 Type Preprint Author Grant J -
2007
Title Positivity and positive definiteness in generalized function algebras DOI 10.1016/j.jmaa.2006.06.034 Type Journal Article Author Oberguggenberger M Journal Journal of Mathematical Analysis and Applications Pages 1321-1335 -
2007
Title Reduction Operators of Linear Second-Order Parabolic Equations DOI 10.48550/arxiv.0712.2764 Type Preprint Author Popovych R -
2006
Title Distributional solution concepts for the Euler-Bernoulli beam equation with discontinuous coefficients DOI 10.48550/arxiv.math/0606058 Type Preprint Author Hoermann G -
2006
Title Generalized Oscillatory Integrals and Fourier Integral Operators DOI 10.48550/arxiv.math/0607706 Type Preprint Author Garetto C -
2006
Title Closed graph and open mapping theorems for topological $\wt{\C}$-modules and applications DOI 10.48550/arxiv.math/0608087 Type Preprint Author Garetto C -
2006
Title Colombeau solutions to nonlinear wave equations DOI 10.48550/arxiv.math/0612445 Type Preprint Author Oberguggenberger M -
2006
Title CHARACTERIZATION OF WAVE FRONT SETS BY WAVELET TRANSFORMS DOI 10.2748/tmj/1163775136 Type Journal Article Author Pilipovic S Journal Tohoku Mathematical Journal, Second Series Pages 369-391 Link Publication -
2006
Title Spherical completeness of the non-archimedean ring of Colombeau generalized numbers DOI 10.48550/arxiv.math/0604495 Type Preprint Author Mayerhofer E -
2006
Title Regularity theory in Colombeau algebras DOI 10.2298/bmat0631148o Type Journal Article Author Oberguggenberger M Journal Bulletin: Classe des sciences mathematiques et natturalles Pages 147-162 Link Publication -
2006
Title On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets DOI 10.2298/bmat0631115g Type Journal Article Author Garetto C Journal Bulletin: Classe des sciences mathematiques et natturalles Pages 115-136 Link Publication -
2006
Title Homogeneity in generalized function algebras DOI 10.48550/arxiv.math/0611377 Type Preprint Author Hanel C -
2006
Title An equivariant noncommutative residue DOI 10.48550/arxiv.math/0610371 Type Preprint Author Dave S -
2006
Title Microlocal hypoellipticity of linear partial differential equations with generalized functions as coefficients. Type Journal Article Author Hörmann G -
2006
Title Isomorphisms of algebras of Colombeau generalized functions DOI 10.48550/arxiv.math/0612646 Type Preprint Author Vernaeve H -
2006
Title On Lorentz geometry in algebras of generalized functions DOI 10.48550/arxiv.math-ph/0604052 Type Preprint Author Mayerhofer E -
2006
Title Admissible Transformations and Normalized Classes of Nonlinear Schroedinger Equations DOI 10.48550/arxiv.math-ph/0611061 Type Preprint Author Popovych R -
2006
Title The use of Generalised Functions and Distributions in General Relativity DOI 10.48550/arxiv.gr-qc/0603078 Type Preprint Author Steinbauer R -
2006
Title Generalized group actions in a global setting DOI 10.1016/j.jmaa.2005.08.065 Type Journal Article Author Konjik S Journal Journal of Mathematical Analysis and Applications Pages 420-436 Link Publication -
2006
Title Global Gronwall Estimates for Integral Curves on Riemannian Manifolds DOI 10.5209/rev_rema.2006.v19.n1.16639 Type Journal Article Author Kunzinger M Journal Revista Matemática Complutense Link Publication -
2006
Title Equalities in algebras of generalized functions DOI 10.1515/forum.2006.039 Type Journal Article Author Pilipovic S Journal Forum Mathematicum Pages 789-801 -
2006
Title Divergent Type Quasilinear Dirichlet Problem with Singularities DOI 10.1007/s10440-006-9067-9 Type Journal Article Author Pilipovic S Journal Acta Applicandae Mathematica Pages 67-82 -
2006
Title Fourier-integral-operator approximation of solutions to first-order hyperbolic pseudodifferential equations II: Microlocal analysis DOI 10.1016/j.matpur.2006.08.004 Type Journal Article Author Le Rousseau J Journal Journal de Mathématiques Pures et Appliquées Pages 403-426 Link Publication -
2006
Title The use of generalized functions and distributions in general relativity DOI 10.1088/0264-9381/23/10/r01 Type Journal Article Author Steinbauer R Journal Classical and Quantum Gravity Link Publication -
2006
Title Global Representatives of Colombeau Holomorphic Generalized Functions DOI 10.1007/s00605-006-0416-6 Type Journal Article Author Oberguggenberger M Journal Monatshefte für Mathematik Pages 67-74 -
2006
Title Group invariants in algebras of generalized functions DOI 10.1080/10652460500436536 Type Journal Article Author Konjik S Journal Integral Transforms and Special Functions Pages 77-84 Link Publication -
2006
Title Notes on regularity results in Colombeau algebras DOI 10.1080/10652460500436601 Type Journal Article Author Oberguggenberger M Journal Integral Transforms and Special Functions Pages 101-107 -
2005
Title On Local Solutions of the Relativistic Vlasov-Klein-Gordon system Type Journal Article Author Kunzinger M Journal Electronic Journal of Differential Equations Pages 1-17 -
2005
Title On Classical Solutions of the Relativistic Vlasov-Klein-Gordon System. Type Journal Article Author Kunzinger M -
2005
Title SYMMETRIES AND MODULI SPACES OF THE SELF-DUAL YANG-MILLS EQUATIONS DOI 10.1142/9789812775061_0010 Type Conference Proceeding Abstract Author Grant J Pages 159-173 -
2005
Title Pseudodifferential operators with generalized symbols and regularity theory. Type Journal Article Author Garetto C -
2005
Title Generalised connections and curvature DOI 10.1017/s0305004105008649 Type Journal Article Author Kunzinger M Journal Mathematical Proceedings of the Cambridge Philosophical Society Pages 497-521 Link Publication -
2005
Title On the characterization of p-adic Colombeau-Egorov generalized functions by their point values DOI 10.48550/arxiv.math/0508027 Type Preprint Author Mayerhofer E -
2005
Title Group invariant Colombeau generalized functions DOI 10.48550/arxiv.math/0512219 Type Preprint Author Vernaeve H -
2005
Title Generalized Group Actions in a Global Setting DOI 10.48550/arxiv.math/0504061 Type Preprint Author Konjik S -
2005
Title Evolution systems for paraxial wave equations of Schroedinger-type with non-smooth coefficients DOI 10.48550/arxiv.math/0506410 Type Preprint Author De Hoop M -
2004
Title Colombeau solutions to nonlinear wave equations. Type Journal Article Author Oberguggenberger M Journal Matemática Contemporânea -
2001
Title Microlocal Analysis and Global Solutions of Some Hyperbolic Equations with Discontinuous Coefficients DOI 10.1023/a:1010614332739 Type Journal Article Author Hörmann G Journal Acta Applicandae Mathematica Pages 173-224 -
2012
Title Factorization of Second-order strictly hyperbolic operators with non-smooth coefficients and microlocal diagonalization. Type Journal Article Author Glogowatz M -
2012
Title Topological and algebraic structures on the ring of Fermat reals DOI 10.1007/s11856-012-0079-z Type Journal Article Author Giordano P Journal Israel Journal of Mathematics Pages 459-505 Link Publication -
2011
Title Notes on generalized pseudo-differential operators. Type Journal Article Author Dave S -
2014
Title Singularity structures for noncommutative spaces DOI 10.1090/s0002-9947-2014-06024-8 Type Journal Article Author Dave S Journal Transactions of the American Mathematical Society Pages 251-273 Link Publication -
2014
Title Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients DOI 10.1002/mana.201400192 Type Journal Article Author Garetto C Journal Mathematische Nachrichten Pages 185-205 Link Publication -
2011
Title New energy inequalities for tensorial wave equations on spacetimes that satisfy a one-sided bound DOI 10.48550/arxiv.1105.0168 Type Preprint Author Burtscher A -
2011
Title An axiomatic approach to the nonlinear theory of generalized functions and consistency of Laplace transforms DOI 10.1080/10652469.2011.604960 Type Journal Article Author Todorov T Journal Integral Transforms and Special Functions Pages 695-708 Link Publication -
2011
Title Synthetic geometry and generalised functions. Type Journal Article Author Grant J -
2011
Title Topological and algebraic structures on the ring of Fermat reals DOI 10.48550/arxiv.1104.1492 Type Preprint Author Giordano P -
2011
Title Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients DOI 10.48550/arxiv.1104.2281 Type Preprint Author Garetto C -
2011
Title Manifold-valued generalized functions in full Colombeau spaces DOI 10.48550/arxiv.1103.5845 Type Preprint Author Kunzinger M -
2011
Title Inversion of a ‘discontinuous coordinate transformation’ in general relativity DOI 10.1080/00036811.2010.490526 Type Journal Article Author Erlacher E Journal Applicable Analysis Pages 1707-1728 Link Publication -
2011
Title An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms DOI 10.1080/10652469.2010.541057 Type Journal Article Author Todorov T Journal Integral Transforms and Special Functions Pages 367-374 Link Publication -
2011
Title Generalised Fourier integral operator methods for hyperbolic equations with singularities DOI 10.48550/arxiv.1104.3035 Type Preprint Author Garetto C -
2011
Title Generalized solutions for the Euler-Bernoulli model with Zener viscoelastic foundations and distributional forces DOI 10.48550/arxiv.1102.2148 Type Preprint Author Hörmann G -
2011
Title Factorization of Second-order strictly hyperbolic operators with non-smooth coefficients and microlocal diagonalization DOI 10.48550/arxiv.1112.5615 Type Preprint Author Glogowatz M -
2011
Title An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms DOI 10.48550/arxiv.1101.5740 Type Preprint Author Todorov T -
2011
Title Areas and Volumes for Null Cones DOI 10.1007/s00023-011-0090-7 Type Journal Article Author Grant J Journal Annales Henri Poincaré Pages 965-985 Link Publication -
2014
Title Null injectivity estimate under an upper bound on the curvature DOI 10.4310/cag.2014.v22.n5.a8 Type Journal Article Author Grant J Journal Communications in Analysis and Geometry Pages 965-996 Link Publication -
2013
Title Reduction operators of Burgers equation DOI 10.1016/j.jmaa.2012.08.062 Type Journal Article Author Pocheketa O Journal Journal of Mathematical Analysis and Applications Pages 270-277 Link Publication -
2010
Title The ADHM Construction and Non-local Symmetries of the Self-dual Yang–Mills Equations DOI 10.1007/s00220-010-1024-9 Type Journal Article Author Grant J Journal Communications in Mathematical Physics Pages 405-428 -
2010
Title Internal sets and internal functions in Colombeau theory. Type Journal Article Author Oberguggenberger M -
2010
Title Generalized Hamilton's principle with fractional derivatives DOI 10.1088/1751-8113/43/25/255203 Type Journal Article Author Atanackovi? T Journal Journal of Physics A: Mathematical and Theoretical Pages 255203 Link Publication -
2010
Title Tensor Valued Colombeau Functions on Manifolds. Type Journal Article Author Grosser M -
2010
Title Wave-type equations of low regularity DOI 10.1080/00036811.2010.495340 Type Journal Article Author Hanel C Journal Applicable Analysis Pages 1691-1705 Link Publication -
2010
Title Reducible Connections and Non-local Symmetries of the Self-dual Yang–Mills Equations DOI 10.1007/s00220-010-1025-8 Type Journal Article Author Grant J Journal Communications in Mathematical Physics Pages 429-446 Link Publication -
2010
Title Hilbert C ~ \widetilde {\mathbb {C}} -modules: Structural properties and applications to variational problems DOI 10.1090/s0002-9947-2010-05143-8 Type Journal Article Author Garetto C Journal Transactions of the American Mathematical Society Pages 2047-2090 Link Publication -
2010
Title Lecture Notes: Non-Standard Approach to J.F. Colombeau's Theory of Generalized Function: University of Vienna, Austria, May 2006 DOI 10.48550/arxiv.1010.3482 Type Preprint Author Todorov T -
2010
Title Equivariant homology for pseudo-differential operators DOI 10.48550/arxiv.1005.2282 Type Preprint Author Dave S -
2010
Title Notes on generalized pseudo-differential operators DOI 10.48550/arxiv.1005.2279 Type Preprint Author Dave S -
2010
Title Algebras of generalized functions with smooth parameter dependence DOI 10.48550/arxiv.1010.5752 Type Preprint Author Burtscher A -
2010
Title The Cauchy problem for Schrödinger-type partial differential operators with generalized functions in the principal part and as data DOI 10.1007/s00605-010-0232-x Type Journal Article Author Hörmann G Journal Monatshefte für Mathematik Pages 445-460 -
2010
Title Optimal regularization processes on complete Riemannian manifolds DOI 10.48550/arxiv.1003.3341 Type Preprint Author Dave S -
2010
Title Null injectivity estimate under an upper bound on the curvature DOI 10.48550/arxiv.1008.5167 Type Preprint Author Grant J -
2010
Title Generalized conditional symmetries of evolution equations DOI 10.48550/arxiv.1011.0277 Type Preprint Author Kunzinger M -
2010
Title Wave-type equations of low regularity DOI 10.48550/arxiv.1011.4108 Type Preprint Author Hanel C -
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Title Areas and volumes for null cones DOI 10.48550/arxiv.1008.0524 Type Preprint Author Grant J -
2010
Title Weak homogeneity in generalized function algebras DOI 10.1002/mana.200810007 Type Journal Article Author Vernaeve H Journal Mathematische Nachrichten Pages 1506-1522 Link Publication -
2009
Title Rapidly converging approximations and regularity theory DOI 10.48550/arxiv.0906.1374 Type Preprint Author Dave S -
2009
Title Harmonic generalized functions in generalized function algebras DOI 10.1007/s00605-009-0180-5 Type Journal Article Author Pilipovic S Journal Monatshefte für Mathematik Pages 81-106 -
2009
Title Is a nonclassical symmetry a symmetry? DOI 10.48550/arxiv.0903.0821 Type Preprint Author Kunzinger M -
2009
Title Sheaves of nonlinear generalized functions and manifold-valued distributions DOI 10.1090/s0002-9947-09-04621-2 Type Journal Article Author Kunzinger M Journal Transactions of the American Mathematical Society Pages 5177-5192 Link Publication -
2013
Title Optimal Regularization Processes on Complete Riemannian Manifolds DOI 10.3836/tjm/1374497511 Type Journal Article Author Shantanu D Journal Tokyo Journal of Mathematics Pages 25-47 Link Publication -
2013
Title Inversion of Colombeau generalized functions DOI 10.1017/s0013091512000429 Type Journal Article Author Erlacher E Journal Proceedings of the Edinburgh Mathematical Society Pages 469-500 -
2013
Title New topologies on Colombeau generalized numbers and the Fermat–Reyes theorem DOI 10.1016/j.jmaa.2012.10.005 Type Journal Article Author Giordano P Journal Journal of Mathematical Analysis and Applications Pages 229-238 Link Publication -
2013
Title Generalized Fourier Integral Operator Methods for Hyperbolic Equations with Singularities DOI 10.1017/s0013091513000424 Type Journal Article Author Garetto C Journal Proceedings of the Edinburgh Mathematical Society Pages 426-463 -
2013
Title Rapidly converging approximations and regularity theory DOI 10.1007/s00605-013-0480-7 Type Journal Article Author Dave S Journal Monatshefte für Mathematik Pages 121-145 -
2012
Title A global theory of algebras of generalized functions II: tensor distributions. Type Journal Article Author Grosser M -
2012
Title Volume Comparison for Hypersurfaces in Lorentzian Manifolds and Singularity Theorems DOI 10.48550/arxiv.1201.4249 Type Preprint Author Treude J -
2012
Title Singular reduction modules of differential equations DOI 10.48550/arxiv.1201.3223 Type Preprint Author Boyko V -
2012
Title Invariant discretization schemes for the shallow-water equations DOI 10.48550/arxiv.1201.0498 Type Preprint Author Bihlo A -
2012
Title Wave Equations on Non-smooth Space-times DOI 10.1007/978-3-0348-0454-7_9 Type Book Chapter Author Hörmann G Publisher Springer Nature Pages 163-186 -
2012
Title Complete group classification of a class of nonlinear wave equations DOI 10.1063/1.4765296 Type Journal Article Author Bihlo A Journal Journal of Mathematical Physics Pages 123515 Link Publication -
2012
Title Symmetric hyperbolic systems in algebras of generalized functions and distributional limits DOI 10.1016/j.jmaa.2011.11.014 Type Journal Article Author Hörmann G Journal Journal of Mathematical Analysis and Applications Pages 1166-1179 Link Publication -
2012
Title Geometric Regularization on Riemannian and Lorentzian Manifolds DOI 10.1007/978-3-0348-0454-7_5 Type Book Chapter Author Dave S Publisher Springer Nature Pages 87-102 -
2012
Title Wave equations and symmetric first-order systems in case of low regularity DOI 10.48550/arxiv.1202.0406 Type Preprint Author Hanel C -
2012
Title New topologies on Colombeau generalized numbers and the Fermat-Reyes theorem DOI 10.48550/arxiv.1201.3887 Type Preprint Author Giordano P -
2012
Title Reduction operators of Burgers equation DOI 10.48550/arxiv.1208.0232 Type Preprint Author Pocheketa O -
2012
Title Algebras of generalized functions with smooth parameter dependence DOI 10.1017/s0013091510001410 Type Journal Article Author Burtscher A Journal Proceedings of the Edinburgh Mathematical Society Pages 105-124 Link Publication -
2012
Title New Energy Inequalities for Tensorial Wave Equations on Spacetimes that Satisfy a One-Sided Bound DOI 10.1080/03605302.2011.647199 Type Journal Article Author Burtscher A Journal Communications in Partial Differential Equations Pages 1596-1619 Link Publication -
2012
Title Volume comparison for hypersurfaces in Lorentzian manifolds and singularity theorems DOI 10.5283/epub.23578 Type Other Author Grant J Link Publication -
2012
Title Volume comparison for hypersurfaces in Lorentzian manifolds and singularity theorems DOI 10.1007/s10455-012-9343-z Type Journal Article Author Treude J Journal Annals of Global Analysis and Geometry Pages 233-251 -
2012
Title Invariant Discretization Schemes for the Shallow-Water Equations DOI 10.1137/120861187 Type Journal Article Author Bihlo A Journal SIAM Journal on Scientific Computing Link Publication -
2011
Title Singularity structures for noncommutative spaces DOI 10.48550/arxiv.1111.6570 Type Preprint Author Dave S -
2011
Title Symmetric hyperbolic systems in algebras of generalized functions and distributional limits DOI 10.48550/arxiv.1107.5967 Type Preprint Author Hoermann G -
2011
Title Generalized conditional symmetries of evolution equations DOI 10.1016/j.jmaa.2011.01.027 Type Journal Article Author Kunzinger M Journal Journal of Mathematical Analysis and Applications Pages 444-460 Link Publication -
2011
Title Wave equations on non-smooth space-times DOI 10.48550/arxiv.1107.0014 Type Preprint Author Hoermann G -
2011
Title Geometric regularization on Riemannian and Lorentzian manifolds DOI 10.48550/arxiv.1107.0010 Type Preprint Author Dave S -
2010
Title Ideals in the Ring of Colombeau Generalized Numbers DOI 10.1080/00927870903055222 Type Journal Article Author Vernaeve H Journal Communications in Algebra Pages 2199-2228 Link Publication -
2010
Title A geometric approach to full Colombeau algebras DOI 10.4064/bc88-0-21 Type Conference Proceeding Abstract Author Steinbauer R Pages 267-272 Link Publication -
2010
Title Recent progress in special Colombeau algebras: geometry, topology, and algebra DOI 10.4064/bc88-0-14 Type Conference Proceeding Abstract Author Kunzinger M Pages 175-184 Link Publication -
2010
Title Tensor valued Colombeau functions on manifolds DOI 10.4064/bc88-0-11 Type Conference Proceeding Abstract Author Grosser M Pages 145-152 Link Publication -
2010
Title Geometrical embeddings of distributions into algebras of generalized functions DOI 10.1002/mana.200710147 Type Journal Article Author Dave S Journal Mathematische Nachrichten Pages 1575-1588 Link Publication -
2010
Title Waves in fractional Zener type viscoelastic media DOI 10.1016/j.jmaa.2009.10.043 Type Journal Article Author Konjik S Journal Journal of Mathematical Analysis and Applications Pages 259-268 Link Publication