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# zbMATH — the first resource for mathematics

## Grubb, Gerd

Compute Distance To:
 Author ID: grubb.gerd Published as: Grubb, G.; Grubb, Gerd External Links: MGP · Wikidata
 Documents Indexed: 116 Publications since 1968, including 4 Books
all top 5

#### Co-Authors

 89 single-authored 7 Solonnikov, Vsevolod Alekseevich 3 Geymonat, Giuseppe 3 Seeley, Robert Thomas 2 Kokholm, Niels Jørgen 2 Schrohe, Elmar 2 Wood, Ian Geoffrey 1 Abels, Helmut 1 Atiyah, Michael Francis 1 Behrndt, Jussi 1 Boman, Jan 1 Bony, Jean-Michel 1 Booss-Bavnbek, Bernhelm 1 Broström, Sofia 1 Brown, B. Malcolm 1 Demailly, Jean-Pierre 1 Gaarde, Anders 1 Gilkey, Peter B. 1 Gimperlein, Heiko 1 Hansen, Lars H. 1 Helgason, Sigurdur 1 Hörmander, Lars Valter 1 Kiselman, Christer Oscar 1 Langer, Matthias 1 Lerner, Nicolas 1 Lotoreichik, Vladimir 1 Sigurðsson, Ragnar 1 Trèves, François 1 Wojciechowski, Krzysztof P.
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#### Serials

 9 Communications in Partial Differential Equations 7 Comptes Rendus de l’Académie des Sciences. Série I 7 Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Série A 6 Mathematica Scandinavica 4 Journal of Functional Analysis 3 Journal of Mathematical Analysis and Applications 3 Journal of Differential Equations 2 Communications in Mathematical Physics 2 Annales de l’Institut Fourier 2 Journal of Soviet Mathematics 2 Mathematische Annalen 2 Mathematische Nachrichten 2 Bollettino della Unione Matematica Italiana. Series V. B 2 Bulletin of the American Mathematical Society 2 Progress in Mathematics 2 Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta Imeni V. A. Steklova 2 Journal of Spectral Theory 1 Applicable Analysis 1 Israel Journal of Mathematics 1 Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV) 1 Arkiv för Matematik 1 Acta Mathematica 1 Advances in Mathematics 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1 Bulletin des Sciences Mathématiques. Deuxième Série 1 Duke Mathematical Journal 1 Inventiones Mathematicae 1 Journal für die Reine und Angewandte Mathematik 1 Mathematische Zeitschrift 1 Journal of Operator Theory 1 Annals of Global Analysis and Geometry 1 Differential and Integral Equations 1 The Journal of Geometric Analysis 1 Bulletin of the American Mathematical Society. New Series 1 Notices of the American Mathematical Society 1 Séminaire Équations aux Dérivées Partielles 1 Discrete and Continuous Dynamical Systems 1 Journal of Mathematical Fluid Mechanics 1 Journal of Evolution Equations 1 Annali della Scuola Normale Superiore di Pisa. Scienze Fisiche e Matematiche. III. Ser 1 Rendiconti del Seminario Matematico. Universitá e Politecnico di Torino 1 Contemporary Mathematics 1 Graduate Texts in Mathematics 1 Analysis & PDE
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#### Fields

 97 Partial differential equations (35-XX) 39 Global analysis, analysis on manifolds (58-XX) 37 Operator theory (47-XX) 11 Functional analysis (46-XX) 4 Fluid mechanics (76-XX) 3 Probability theory and stochastic processes (60-XX) 2 Approximations and expansions (41-XX) 2 Differential geometry (53-XX) 2 Quantum theory (81-XX) 1 General and overarching topics; collections (00-XX) 1 History and biography (01-XX) 1 Potential theory (31-XX)

#### Citations contained in zbMATH

92 Publications have been cited 1,258 times in 676 Documents Cited by Year
Functional calculus of pseudodifferential boundary problems. 2nd ed. Zbl 0844.35002
Grubb, Gerd
1996
Distributions and operators. Zbl 1171.47001
Grubb, Gerd
2009
Fractional Laplacians on domains, a development of Hörmander’s theory of $$\mu$$-transmission pseudodifferential operators. Zbl 1318.47064
Grubb, Gerd
2015
A characterization of the non-local boundary value problems associated with an elliptic operator. Zbl 0182.14501
Grubb, G.
1968
Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems. Zbl 0851.58043
Grubb, Gerd; Seeley, Robert T.
1995
Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods. Zbl 0766.35034
Grubb, Gerd; Solonnikov, Vsevolod A.
1991
Functional calculus of pseudo-differential boundary problems. Zbl 0622.35001
Grubb, Gerd
1986
Pseudo-differential boundary problems in $$L_ p$$ spaces. Zbl 0723.35091
Grubb, Gerd
1990
$$M$$-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Zbl 1167.47057
Brown, B. M.; Grubb, G.; Wood, I. G.
2009
Local and nonlocal boundary conditions for $$\mu$$-transmission and fractional elliptic pseudodifferential operators. Zbl 1317.35310
Grubb, Gerd
2014
A global calculus of parameter-dependent pseudodifferential boundary problems in $$L_ p$$ Sobolev spaces. Zbl 0811.35176
Grubb, Gerd; Kokholm, Niels Jørgen
1993
Zeta and eta functions for Atiyah-Patodi-Singer operators. Zbl 0858.58050
Grubb, Gerd; Seeley, Robert T.
1996
The essential spectrum of elliptic systems of mixed order. Zbl 0361.35050
Grubb, Gerd; Geymonat, Giuseppe
1977
Singular Green operators and their spectral asymptotics. Zbl 0553.58034
Grubb, Gerd
1984
Regularity of spectral fractional Dirichlet and Neumann problems. Zbl 1448.47062
Grubb, Gerd
2016
Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems. Zbl 1021.35046
Grubb, Gerd
1999
Krein resolvent formulas for elliptic boundary problems in nonsmooth domains. Zbl 1206.35090
Grubb, G.
2008
Trace expansions and the noncommutative residue for manifolds with boundary. Zbl 0980.58017
Grubb, Gerd; Schrohe, Elmar
2001
Parameter-elliptic and parabolic pseudodifferential boundary problems in global $$L_ p$$ Sobolev spaces. Zbl 0877.35142
Grubb, Gerd
1995
Spectral asymptotics for the ”soft” selfadjoint extension of a symmetric elliptic differential operator. Zbl 0559.47035
Grubb, Gerd
1983
On coerciveness and semiboundedness of general boundary problems. Zbl 0231.35027
Grubb, Gerd
1971
Extension theory and Kreĭn-type resolvent formulas for nonsmooth boundary value problems. Zbl 1312.47019
Abels, Helmut; Grubb, Gerd; Wood, Ian Geoffrey
2014
Nonhomogeneous Dirichlet Navier-Stokes problems in low regularity $$L_p$$ Sobolev spaces. Zbl 0992.35065
Grubb, Gerd
2001
Solution of parabolic pseudo-differential initial-boundary value problems. Zbl 0713.35109
Grubb, Gerd; Solonnikov, Vsevolod A.
1990
Properties of normal boundary problems for elliptic even-order systems. Zbl 0309.35034
Grubb, Gerd
1975
Spectral asymptotics for Robin problems with a discontinuous coefficient. Zbl 1223.35146
Grubb, Gerd
2011
Heat operator trace expansions and index for general Atiyah-Patodi-Singer boundary problems. Zbl 0773.58025
Grubb, Gerd
1992
A resolvent approach to traces and zeta Laurent expansions. Zbl 1073.58021
Grubb, Gerd
2005
The transmission property. Zbl 0766.35088
Grubb, Gerd; Hörmander, Lars
1990
Boundary problems for systems of partial differential operators of mixed order. Zbl 0368.35030
Grubb, Gerd
1977
Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators. Zbl 1432.35107
Gimperlein, Heiko; Grubb, Gerd
2014
The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates. Zbl 1223.47050
Grubb, Gerd
2011
Les problèmes aux limites généraux d’un opérateur elliptique, provenant de la théorie variationnelle. Zbl 0201.46502
Grubb, G.
1970
Spectral results for mixed problems and fractional elliptic operators. Zbl 1302.35279
Grubb, Gerd
2015
Spectral boundary conditions for generalizations of Laplace and Dirac operators. Zbl 1039.58027
Grubb, Gerd
2003
Weakly semi-bounded boundary problems and sesquilinear forms. Zbl 0261.35011
Grubb, Gerd
1973
Perturbation of essential spectra of exterior elliptic problems. Zbl 1208.35094
Grubb, Gerd
2011
Traces and quasi-traces on the Boutet de Monvel algebra. Zbl 1078.58015
Grubb, Gerd; Schrohe, Elmar
2004
Spectral asymptotics for Douglis-Nirenberg elliptic and pseudo- differential boundary problems. Zbl 0381.35034
Grubb, Gerd
1978
Integration by parts and Pohozaev identities for space-dependent fractional-order operators. Zbl 1375.35600
Grubb, Gerd
2016
The local and global parts of the basic zeta coefficient for operators on manifolds with boundary. Zbl 1147.58028
Grubb, Gerd
2008
Spectral asymptotics for nonsmooth singular Green operators. Zbl 1290.35057
Grubb, Gerd
2014
Kreĭn-like extensions and the lower boundedness problem for elliptic operators. Zbl 1242.47041
Grubb, Gerd
2012
Known and unknown results on elliptic boundary problems. Zbl 1159.35004
Grubb, Gerd
2006
Complex powers of resolvents of pseudodifferential operators. Zbl 1129.58304
Grubb, Gerd; Hansen, Lars
2002
Poles of zeta and eta functions for perturbations of the Atiyah-Patodi-Singer problem. Zbl 0976.58018
Grubb, G.
2001
Logarithmic terms in asymptotic expansions of heat operator traces. Zbl 0911.35128
Gilkey, Peter B.; Grubb, Gerd
1998
Regularity in $$L_{p}$$ Sobolev spaces of solutions to fractional heat equations. Zbl 1396.35022
Grubb, Gerd
2018
Spectral asymptotics for resolvent differences of elliptic operators with $$\delta$$ and $$\delta^{'}$$-interactions on hypersurfaces. Zbl 1353.47090
Behrndt, Jussi; Grubb, Gerd; Langer, Matthias; Lotoreichik, Vladimir
2015
Nonhomogeneous time-dependent Navier-Stokes problems in $$L_ p$$ Sobolev spaces. Zbl 0831.35126
Grubb, Gerd
1995
Remainder estimates for eigenvalues and kernels of pseudo-differential elliptic systems. Zbl 0407.35075
Grubb, Gerd
1978
Remarks on trace estimates for exterior boundary problems. Zbl 0553.35022
Grubb, Gerd
1984
Fractional-order operators: boundary problems, heat equations. Zbl 1414.35099
Grubb, Gerd
2018
Extension theory for elliptic partial differential operators with pseudodifferential methods. Zbl 1331.47069
Grubb, Gerd
2012
The sectorial projection defined from logarithms. Zbl 1270.58016
Grubb, Gerd
2012
A weakly polyhomogeneous calculus for pseudodifferential boundary problems. Zbl 0998.58017
Grubb, Gerd
2001
Parabolic pseudo-differential boundary problems and applications. Zbl 0763.35116
Grubb, Gerd
1991
Estimation du reste dans l’étude des valeurs propres des problèmes aux limites pseudo-différentiels auto-adjoints. Zbl 0399.58019
Grubb, Gerd
1978
Green’s formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators. Zbl 1409.35216
Grubb, Gerd
2018
Logarithms and sectorial projections for elliptic boundary problems. Zbl 1189.35347
Gaarde, Anders; Grubb, Gerd
2008
On the logarithm component in trace defect formulas. Zbl 1087.35100
Grubb, Gerd
2005
Nonhomogeneous Navier-Stokes problems in $$L_p$$ Sobolev spaces over exterior and interior domains. Zbl 0934.35119
Grubb, G.
1998
Asymptotic expansions for the Atiyah-Patodi-Singer operator. Zbl 0813.58059
Grubb, Gerd; Seeley, Robert T.
1993
Initial value problems for the Navier-Stokes equations with Neumann conditions. Zbl 0781.35051
Grubb, Gerd
1992
Reduction of the basic initial-boundary value problems for the Navier- Stokes equations to initial-boundary value problems for nonlinear parabolic systems of pseudodifferential equations. Zbl 0729.35102
Grubb, G.; Solonnikov, V. A.
1991
Reduction of basic initial-boundary value problems for the Stokes equations to initial-boundary value problems for parabolic systems of pseudodifferential equations. Zbl 0655.35065
Grubb, Gerd; Solonnikov, V. A.
1987
Limited regularity of solutions to fractional heat and Schrödinger equations. Zbl 1415.35142
Grubb, Gerd
2019
Analysis of invariants associated with spectral boundary problems for elliptic operators. Zbl 1077.58014
Grubb, Gerd
2005
Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems. Zbl 1048.35095
Grubb, Gerd
2003
Parametrized pseudodifferential operators and geometric invariants. Zbl 0873.35116
Grubb, Gerd
1997
A pseudo-differential treatment of general inhomogeneous initial-boundary value problems for the Navier-Stokes equation. Zbl 0675.76028
Grubb, Gerd; Solonnikov, Vsevolod A.
1988
Complex powers of pseudo-differential boundary value problems with the transmission property. Zbl 0623.47068
Grubb, Gerd
1987
On coerciveness of Douglis-Nirenberg elliptic boundary value problems. Zbl 0456.35015
Grubb, Gerd
1979
Eigenvalue asymptotics for selfadjoint elliptic mixed order systems with nonempty essential spectrum. Zbl 0423.35071
Grubb, Gerd; Geymonat, Giuseppe
1979
Spectral theory for boundary value problems for elliptic systems of mixed order. Zbl 0301.35071
Geymonat, Giuseppe; Grubb, Gerd
1974
Inequalities for boundary value problems for systems of partial differential operators. Zbl 0267.35026
Grubb, Gerd
1973
Problèmes aux limites semi-bornes pour les systèmes elliptiques. Zbl 0243.35032
Grubb, Gerd
1972
Trace defect formulas and zeta values for boundary problems. Zbl 1154.35103
Grubb, Gerd
2008
Remarks on nonlocal trace expansion coefficients. Zbl 1119.58019
Grubb, Gerd
2006
Spectral boundary conditions and trace formulas. Zbl 1292.35348
Grubb, Gerd
2002
Pseudodifferential boundary problems and applications. Zbl 0873.35117
Grubb, G.
1997
Solution dans les espaces de Sobolev $$L_ p$$ anisotropes des problèmes aux limites pseudo-différentiels paraboliques et des problèmes de Stokes. (Solution in anisotropic $$L_ p$$ Sobolev spaces of parabolic pseudo-differential boundary value problems and Stokes problems). Zbl 0723.35093
Grubb, Gerd
1991
Reduction of basic initial-boundary value problems for Navier-Stokes equations to initial-boundary value problems for nonlinear parabolic systems of pseudo-differential equations. Zbl 0717.35069
Grubb, G.; Solonnikov, V. A.
1989
Une méthode pseudo-différentielle pour les perturbations singulières elliptiques. (A pseudo-differential method for elliptic singular perturbation problems). Zbl 0595.35045
Grubb, Gerd
1985
Comportement asymptotique du spectre des opérateurs de Green singuliers. Zbl 0553.35066
Grubb, Gerd
1983
Problèmes aux limites pseudodifférentiels dependant d’un paramètre. Zbl 0492.35079
Grubb, Gerd
1981
La resolvante d’un problème aux limites pseudo-différentiel elliptique. Zbl 0489.35083
Grubb, Gerd
1981
A resolvent construction for pseudo-differential boundary value problems, with applications. Zbl 0471.35064
Grubb, Gerd
1981
Caractérisation de quelques propriétés des problèmes aux limites pour les systèmes elliptiques: Problèmes aux limites variationnels. Zbl 0278.35035
Grubb, G.
1972
Le spectre négatif des problèmes aux limites auto-adjoints fortement elliptiques. (The negative spectrum of strongly elliptic selfadjoint boundary value problems). Zbl 0234.35075
Grubb, Gerd
1972
Une condition necessaire et suffisante pour la m-coercivite des problèmes aux limites elliptiques. Zbl 0192.49301
Grubb, G.
1970
Coerciveness of the normal boundary problems for an elliptic operator. Zbl 0188.16902
Grubb, G.
1970
Limited regularity of solutions to fractional heat and Schrödinger equations. Zbl 1415.35142
Grubb, Gerd
2019
Regularity in $$L_{p}$$ Sobolev spaces of solutions to fractional heat equations. Zbl 1396.35022
Grubb, Gerd
2018
Fractional-order operators: boundary problems, heat equations. Zbl 1414.35099
Grubb, Gerd
2018
Green’s formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators. Zbl 1409.35216
Grubb, Gerd
2018
Regularity of spectral fractional Dirichlet and Neumann problems. Zbl 1448.47062
Grubb, Gerd
2016
Integration by parts and Pohozaev identities for space-dependent fractional-order operators. Zbl 1375.35600
Grubb, Gerd
2016
Fractional Laplacians on domains, a development of Hörmander’s theory of $$\mu$$-transmission pseudodifferential operators. Zbl 1318.47064
Grubb, Gerd
2015
Spectral results for mixed problems and fractional elliptic operators. Zbl 1302.35279
Grubb, Gerd
2015
Spectral asymptotics for resolvent differences of elliptic operators with $$\delta$$ and $$\delta^{'}$$-interactions on hypersurfaces. Zbl 1353.47090
Behrndt, Jussi; Grubb, Gerd; Langer, Matthias; Lotoreichik, Vladimir
2015
Local and nonlocal boundary conditions for $$\mu$$-transmission and fractional elliptic pseudodifferential operators. Zbl 1317.35310
Grubb, Gerd
2014
Extension theory and Kreĭn-type resolvent formulas for nonsmooth boundary value problems. Zbl 1312.47019
Abels, Helmut; Grubb, Gerd; Wood, Ian Geoffrey
2014
Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators. Zbl 1432.35107
Gimperlein, Heiko; Grubb, Gerd
2014
Spectral asymptotics for nonsmooth singular Green operators. Zbl 1290.35057
Grubb, Gerd
2014
Kreĭn-like extensions and the lower boundedness problem for elliptic operators. Zbl 1242.47041
Grubb, Gerd
2012
Extension theory for elliptic partial differential operators with pseudodifferential methods. Zbl 1331.47069
Grubb, Gerd
2012
The sectorial projection defined from logarithms. Zbl 1270.58016
Grubb, Gerd
2012
Spectral asymptotics for Robin problems with a discontinuous coefficient. Zbl 1223.35146
Grubb, Gerd
2011
The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates. Zbl 1223.47050
Grubb, Gerd
2011
Perturbation of essential spectra of exterior elliptic problems. Zbl 1208.35094
Grubb, Gerd
2011
Distributions and operators. Zbl 1171.47001
Grubb, Gerd
2009
$$M$$-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Zbl 1167.47057
Brown, B. M.; Grubb, G.; Wood, I. G.
2009
Krein resolvent formulas for elliptic boundary problems in nonsmooth domains. Zbl 1206.35090
Grubb, G.
2008
The local and global parts of the basic zeta coefficient for operators on manifolds with boundary. Zbl 1147.58028
Grubb, Gerd
2008
Logarithms and sectorial projections for elliptic boundary problems. Zbl 1189.35347
Gaarde, Anders; Grubb, Gerd
2008
Trace defect formulas and zeta values for boundary problems. Zbl 1154.35103
Grubb, Gerd
2008
Known and unknown results on elliptic boundary problems. Zbl 1159.35004
Grubb, Gerd
2006
Remarks on nonlocal trace expansion coefficients. Zbl 1119.58019
Grubb, Gerd
2006
A resolvent approach to traces and zeta Laurent expansions. Zbl 1073.58021
Grubb, Gerd
2005
On the logarithm component in trace defect formulas. Zbl 1087.35100
Grubb, Gerd
2005
Analysis of invariants associated with spectral boundary problems for elliptic operators. Zbl 1077.58014
Grubb, Gerd
2005
Traces and quasi-traces on the Boutet de Monvel algebra. Zbl 1078.58015
Grubb, Gerd; Schrohe, Elmar
2004
Spectral boundary conditions for generalizations of Laplace and Dirac operators. Zbl 1039.58027
Grubb, Gerd
2003
Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems. Zbl 1048.35095
Grubb, Gerd
2003
Complex powers of resolvents of pseudodifferential operators. Zbl 1129.58304
Grubb, Gerd; Hansen, Lars
2002
Spectral boundary conditions and trace formulas. Zbl 1292.35348
Grubb, Gerd
2002
Trace expansions and the noncommutative residue for manifolds with boundary. Zbl 0980.58017
Grubb, Gerd; Schrohe, Elmar
2001
Nonhomogeneous Dirichlet Navier-Stokes problems in low regularity $$L_p$$ Sobolev spaces. Zbl 0992.35065
Grubb, Gerd
2001
Poles of zeta and eta functions for perturbations of the Atiyah-Patodi-Singer problem. Zbl 0976.58018
Grubb, G.
2001
A weakly polyhomogeneous calculus for pseudodifferential boundary problems. Zbl 0998.58017
Grubb, Gerd
2001
Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems. Zbl 1021.35046
Grubb, Gerd
1999
Logarithmic terms in asymptotic expansions of heat operator traces. Zbl 0911.35128
Gilkey, Peter B.; Grubb, Gerd
1998
Nonhomogeneous Navier-Stokes problems in $$L_p$$ Sobolev spaces over exterior and interior domains. Zbl 0934.35119
Grubb, G.
1998
Parametrized pseudodifferential operators and geometric invariants. Zbl 0873.35116
Grubb, Gerd
1997
Pseudodifferential boundary problems and applications. Zbl 0873.35117
Grubb, G.
1997
Functional calculus of pseudodifferential boundary problems. 2nd ed. Zbl 0844.35002
Grubb, Gerd
1996
Zeta and eta functions for Atiyah-Patodi-Singer operators. Zbl 0858.58050
Grubb, Gerd; Seeley, Robert T.
1996
Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems. Zbl 0851.58043
Grubb, Gerd; Seeley, Robert T.
1995
Parameter-elliptic and parabolic pseudodifferential boundary problems in global $$L_ p$$ Sobolev spaces. Zbl 0877.35142
Grubb, Gerd
1995
Nonhomogeneous time-dependent Navier-Stokes problems in $$L_ p$$ Sobolev spaces. Zbl 0831.35126
Grubb, Gerd
1995
A global calculus of parameter-dependent pseudodifferential boundary problems in $$L_ p$$ Sobolev spaces. Zbl 0811.35176
Grubb, Gerd; Kokholm, Niels Jørgen
1993
Asymptotic expansions for the Atiyah-Patodi-Singer operator. Zbl 0813.58059
Grubb, Gerd; Seeley, Robert T.
1993
Heat operator trace expansions and index for general Atiyah-Patodi-Singer boundary problems. Zbl 0773.58025
Grubb, Gerd
1992
Initial value problems for the Navier-Stokes equations with Neumann conditions. Zbl 0781.35051
Grubb, Gerd
1992
Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods. Zbl 0766.35034
Grubb, Gerd; Solonnikov, Vsevolod A.
1991
Parabolic pseudo-differential boundary problems and applications. Zbl 0763.35116
Grubb, Gerd
1991
Reduction of the basic initial-boundary value problems for the Navier- Stokes equations to initial-boundary value problems for nonlinear parabolic systems of pseudodifferential equations. Zbl 0729.35102
Grubb, G.; Solonnikov, V. A.
1991
Solution dans les espaces de Sobolev $$L_ p$$ anisotropes des problèmes aux limites pseudo-différentiels paraboliques et des problèmes de Stokes. (Solution in anisotropic $$L_ p$$ Sobolev spaces of parabolic pseudo-differential boundary value problems and Stokes problems). Zbl 0723.35093
Grubb, Gerd
1991
Pseudo-differential boundary problems in $$L_ p$$ spaces. Zbl 0723.35091
Grubb, Gerd
1990
Solution of parabolic pseudo-differential initial-boundary value problems. Zbl 0713.35109
Grubb, Gerd; Solonnikov, Vsevolod A.
1990
The transmission property. Zbl 0766.35088
Grubb, Gerd; Hörmander, Lars
1990
Reduction of basic initial-boundary value problems for Navier-Stokes equations to initial-boundary value problems for nonlinear parabolic systems of pseudo-differential equations. Zbl 0717.35069
Grubb, G.; Solonnikov, V. A.
1989
A pseudo-differential treatment of general inhomogeneous initial-boundary value problems for the Navier-Stokes equation. Zbl 0675.76028
Grubb, Gerd; Solonnikov, Vsevolod A.
1988
Reduction of basic initial-boundary value problems for the Stokes equations to initial-boundary value problems for parabolic systems of pseudodifferential equations. Zbl 0655.35065
Grubb, Gerd; Solonnikov, V. A.
1987
Complex powers of pseudo-differential boundary value problems with the transmission property. Zbl 0623.47068
Grubb, Gerd
1987
Functional calculus of pseudo-differential boundary problems. Zbl 0622.35001
Grubb, Gerd
1986
Une méthode pseudo-différentielle pour les perturbations singulières elliptiques. (A pseudo-differential method for elliptic singular perturbation problems). Zbl 0595.35045
Grubb, Gerd
1985
Singular Green operators and their spectral asymptotics. Zbl 0553.58034
Grubb, Gerd
1984
Remarks on trace estimates for exterior boundary problems. Zbl 0553.35022
Grubb, Gerd
1984
Spectral asymptotics for the ”soft” selfadjoint extension of a symmetric elliptic differential operator. Zbl 0559.47035
Grubb, Gerd
1983
Comportement asymptotique du spectre des opérateurs de Green singuliers. Zbl 0553.35066
Grubb, Gerd
1983
Problèmes aux limites pseudodifférentiels dependant d’un paramètre. Zbl 0492.35079
Grubb, Gerd
1981
La resolvante d’un problème aux limites pseudo-différentiel elliptique. Zbl 0489.35083
Grubb, Gerd
1981
A resolvent construction for pseudo-differential boundary value problems, with applications. Zbl 0471.35064
Grubb, Gerd
1981
On coerciveness of Douglis-Nirenberg elliptic boundary value problems. Zbl 0456.35015
Grubb, Gerd
1979
Eigenvalue asymptotics for selfadjoint elliptic mixed order systems with nonempty essential spectrum. Zbl 0423.35071
Grubb, Gerd; Geymonat, Giuseppe
1979
Spectral asymptotics for Douglis-Nirenberg elliptic and pseudo- differential boundary problems. Zbl 0381.35034
Grubb, Gerd
1978
Remainder estimates for eigenvalues and kernels of pseudo-differential elliptic systems. Zbl 0407.35075
Grubb, Gerd
1978
Estimation du reste dans l’étude des valeurs propres des problèmes aux limites pseudo-différentiels auto-adjoints. Zbl 0399.58019
Grubb, Gerd
1978
The essential spectrum of elliptic systems of mixed order. Zbl 0361.35050
Grubb, Gerd; Geymonat, Giuseppe
1977
Boundary problems for systems of partial differential operators of mixed order. Zbl 0368.35030
Grubb, Gerd
1977
Properties of normal boundary problems for elliptic even-order systems. Zbl 0309.35034
Grubb, Gerd
1975
Spectral theory for boundary value problems for elliptic systems of mixed order. Zbl 0301.35071
Geymonat, Giuseppe; Grubb, Gerd
1974
Weakly semi-bounded boundary problems and sesquilinear forms. Zbl 0261.35011
Grubb, Gerd
1973
Inequalities for boundary value problems for systems of partial differential operators. Zbl 0267.35026
Grubb, Gerd
1973
Problèmes aux limites semi-bornes pour les systèmes elliptiques. Zbl 0243.35032
Grubb, Gerd
1972
Caractérisation de quelques propriétés des problèmes aux limites pour les systèmes elliptiques: Problèmes aux limites variationnels. Zbl 0278.35035
Grubb, G.
1972
Le spectre négatif des problèmes aux limites auto-adjoints fortement elliptiques. (The negative spectrum of strongly elliptic selfadjoint boundary value problems). Zbl 0234.35075
Grubb, Gerd
1972
On coerciveness and semiboundedness of general boundary problems. Zbl 0231.35027
Grubb, Gerd
1971
Les problèmes aux limites généraux d’un opérateur elliptique, provenant de la théorie variationnelle. Zbl 0201.46502
Grubb, G.
1970
Une condition necessaire et suffisante pour la m-coercivite des problèmes aux limites elliptiques. Zbl 0192.49301
Grubb, G.
1970
Coerciveness of the normal boundary problems for an elliptic operator. Zbl 0188.16902
Grubb, G.
1970
A characterization of the non-local boundary value problems associated with an elliptic operator. Zbl 0182.14501
Grubb, G.
1968
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#### Cited by 725 Authors

 44 Grubb, Gerd 20 Schulze, Bert-Wolfgang 19 Behrndt, Jussi 15 Schrohe, Elmar 14 Loya, Paul A. 12 Malamud, Mark M. 10 Abels, Helmut 10 Lesch, Matthias 10 Mitrea, Marius 10 Ros-Oton, Xavier 10 Warma, Mahamadi 9 Gesztesy, Fritz 9 Natroshvili, David 9 Seiler, Jörg 8 Abatangelo, Nicola 8 Johnsen, Jon 8 Neidhardt, Hagen 8 Raymond, Jean-Pierre 8 Solonnikov, Vsevolod Alekseevich 7 Faierman, Melvin 7 Geymonat, Giuseppe 7 Kozhevnikov, Alexander 7 Lee, Yoonweon 7 Lotoreichik, Vladimir 7 Park, Jinsung 7 Rohleder, Jonathan 7 Scott, Simon G. 7 Uhlmann, Gunther Alberto 7 Wood, Ian Geoffrey 6 Borthagaray, Juan Pablo 6 Coriasco, Sandro 6 Denk, Robert 6 Dipierro, Serena 6 Jarohs, Sven 6 Kirsten, Klaus 6 Michelangeli, Alessandro 6 Rabinovich, Vladimir Samuilovich 6 Rüland, Angkana 6 Valdinoci, Enrico 6 Wang, Yong 5 Amann, Herbert 5 Booss-Bavnbek, Bernhelm 5 Fahrenwaldt, Matthias Albrecht 5 Gilkey, Peter B. 5 Gimperlein, Heiko 5 Krupchyk, Katsiaryna 5 Langer, Matthias 5 Posilicano, Andrea 5 Saldaña, Alberto 5 Seeley, Robert Thomas 5 Zakora, Dmitriĭ Aleksandrovich 4 Acosta, Gabriel 4 Antil, Harbir 4 Battisti, Ubertino 4 Brown, B. Malcolm 4 Chang, Tongkeun 4 Fall, Mouhamed Moustapha 4 Fischbacher, Christoph 4 Gallone, Matteo 4 Ghosh, Tuhin 4 Glusa, Christian 4 Goffeng, Magnus 4 Huang, Rung-Tzung 4 Iochum, Bruno 4 Jin, Bum Ja 4 Krainer, Thomas 4 Levendorskiĭ, Sergeĭ Zakharovich 4 Naboko, Serguei N. 4 Nistor, Victor 4 Salo, Mikko 4 Serra Montolí, Joaquim 4 Shibata, Yoshihiro 4 ter Elst, Antonius Frederik Maria 4 Wang, Jian 4 Zhu, Chaofeng 3 Ainsworth, Mark 3 Ashbaugh, Mark S. 3 Bohlen, Karsten 3 Bonforte, Matteo 3 Brüning, Jochen 3 Buchukuri, Tengiz 3 Chen, Guoyuan 3 Chkadua, O. O. 3 Dappiaggi, Claudio 3 Dias, Nuno Costa 3 Duduchava, Roland 3 Farwig, Reinhard 3 Hieber, Matthias 3 Kassmann, Moritz 3 Lassas, Matti 3 Latushkin, Yuri 3 Levy, Cyril 3 Liu, Genqian 3 Liu, Jie 3 Ludkovsky, Sergey Victor 3 Maniccia, Lidia 3 Melo, Severino Toscano 3 Mennicken, Reinhard 3 Moroianu, Sergiu 3 Pego, Robert L. ...and 625 more Authors
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#### Cited in 182 Serials

 39 Journal of Functional Analysis 30 Integral Equations and Operator Theory 27 Journal of Differential Equations 25 Communications in Partial Differential Equations 23 Journal of Mathematical Analysis and Applications 22 Mathematische Nachrichten 14 Journal of Geometry and Physics 14 Advances in Mathematics 14 Journal of Mathematical Sciences (New York) 13 Journal of Pseudo-Differential Operators and Applications 12 Journal of Mathematical Physics 11 Communications in Mathematical Physics 11 Transactions of the American Mathematical Society 9 Archive for Rational Mechanics and Analysis 8 Journal d’Analyse Mathématique 8 Mathematische Annalen 8 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 8 Proceedings of the American Mathematical Society 8 Annals of Global Analysis and Geometry 8 The Journal of Geometric Analysis 7 Applicable Analysis 7 Calculus of Variations and Partial Differential Equations 7 Discrete and Continuous Dynamical Systems 6 Mathematical Methods in the Applied Sciences 6 Duke Mathematical Journal 6 Journal of Soviet Mathematics 6 Russian Journal of Mathematical Physics 6 Journal of Evolution Equations 5 Journal of Computational Physics 5 Letters in Mathematical Physics 5 Reviews in Mathematical Physics 5 Annali di Matematica Pura ed Applicata. Serie Quarta 5 Journal für die Reine und Angewandte Mathematik 5 Mathematische Zeitschrift 5 SIAM Journal on Numerical Analysis 5 Journal de Mathématiques Pures et Appliquées. Neuvième Série 5 SIAM Journal on Mathematical Analysis 5 Annales Henri Poincaré 5 Complex Analysis and Operator Theory 4 Inverse Problems 4 Functional Analysis and its Applications 4 Monatshefte für Mathematik 4 Numerische Mathematik 4 SIAM Journal on Control and Optimization 4 International Journal of Mathematics 4 Differential Geometry and its Applications 4 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 4 Journal of Mathematical Fluid Mechanics 4 Communications on Pure and Applied Analysis 4 Journal of Noncommutative Geometry 4 Journal of Spectral Theory 3 Israel Journal of Mathematics 3 Mathematical Notes 3 Annales de l’Institut Fourier 3 Indiana University Mathematics Journal 3 Journal of Computational and Applied Mathematics 3 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 3 Journal of Integral Equations and Applications 3 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 3 Potential Analysis 3 Comptes Rendus. Mathématique. Académie des Sciences, Paris 3 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 3 Annali dell’Università di Ferrara. Sezione VII. Scienze Matematiche 2 International Journal of Modern Physics A 2 Computers & Mathematics with Applications 2 Computer Methods in Applied Mechanics and Engineering 2 Communications on Pure and Applied Mathematics 2 Nonlinearity 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 2 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 2 Applied Mathematics and Optimization 2 Manuscripta Mathematica 2 Memoirs of the American Mathematical Society 2 Rendiconti del Seminario Matematico della Università di Padova 2 Siberian Mathematical Journal 2 Transactions of the Moscow Mathematical Society 2 Journal of Scientific Computing 2 Computational Mathematics and Mathematical Physics 2 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2 St. Petersburg Mathematical Journal 2 Opuscula Mathematica 2 Electronic Research Announcements of the American Mathematical Society 2 The Journal of Fourier Analysis and Applications 2 Matematychni Metody ta Fizyko-Mekhanichni Polya 2 Fractional Calculus & Applied Analysis 2 Journal of High Energy Physics 2 Journal of Nonlinear Mathematical Physics 2 Advanced Nonlinear Studies 2 Bulletin of the American Mathematical Society 2 Annali della Scuola Normale Superiore di Pisa. Scienze Fisiche e Matematiche. III. Ser 2 Inverse Problems and Imaging 2 Science China. Mathematics 2 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 2 Mathematical Control and Related Fields 2 Transactions of A. Razmadze Mathematical Institute 1 Modern Physics Letters A 1 Bulletin of the Australian Mathematical Society 1 Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV) 1 Journal of Fluid Mechanics ...and 82 more Serials
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#### Cited in 44 Fields

 445 Partial differential equations (35-XX) 191 Global analysis, analysis on manifolds (58-XX) 190 Operator theory (47-XX) 65 Functional analysis (46-XX) 65 Fluid mechanics (76-XX) 54 Quantum theory (81-XX) 49 Numerical analysis (65-XX) 35 Mechanics of deformable solids (74-XX) 33 Differential geometry (53-XX) 21 Ordinary differential equations (34-XX) 21 Integral equations (45-XX) 19 Probability theory and stochastic processes (60-XX) 16 Potential theory (31-XX) 15 Calculus of variations and optimal control; optimization (49-XX) 15 Systems theory; control (93-XX) 13 Real functions (26-XX) 11 Statistical mechanics, structure of matter (82-XX) 10 Functions of a complex variable (30-XX) 9 Several complex variables and analytic spaces (32-XX) 9 Approximations and expansions (41-XX) 9 Harmonic analysis on Euclidean spaces (42-XX) 8 $$K$$-theory (19-XX) 6 Number theory (11-XX) 6 Dynamical systems and ergodic theory (37-XX) 6 Manifolds and cell complexes (57-XX) 6 Relativity and gravitational theory (83-XX) 6 Biology and other natural sciences (92-XX) 5 Algebraic geometry (14-XX) 5 Abstract harmonic analysis (43-XX) 5 Mechanics of particles and systems (70-XX) 4 Optics, electromagnetic theory (78-XX) 3 Linear and multilinear algebra; matrix theory (15-XX) 3 Topological groups, Lie groups (22-XX) 3 Integral transforms, operational calculus (44-XX) 3 Statistics (62-XX) 3 Classical thermodynamics, heat transfer (80-XX) 3 Geophysics (86-XX) 2 General and overarching topics; collections (00-XX) 2 Measure and integration (28-XX) 2 Special functions (33-XX) 1 Associative rings and algebras (16-XX) 1 Category theory; homological algebra (18-XX) 1 Convex and discrete geometry (52-XX) 1 Algebraic topology (55-XX)

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text:
Power Series - Differential Equation (check my answer)

1. Dec 7, 2009

vucollegeguy

Using the power series method to solve the differential equation
y'+xy=0 when y(0)=1

Write the solution in the form of a power series and then recognize what function it represents.
************************************

$$\sum$$(-1)k*[(x2k)/(2k)*k!]

Is my answer correct?
Is it written in the form of a power series?
What function does it represent?

Thanks to anyone that helps!

2. Dec 7, 2009

Dick

Yes, it's correct. And yes, it's a power series. It looks like exp(f(x)) to me. Can you guess what f(x) might be?

3. Dec 8, 2009

vucollegeguy

I'd guess and say that f(x)=cos(x).
So the function would be f(x)=x2cos(x).

Please correct me if I am wrong.
Thank you for helping thus far.

4. Dec 8, 2009

HallsofIvy

Staff Emeritus
Not quite. Your sum reduces to
$$\sum(-1)^k \frac{\left(\frac{x^2}{2}\right)^2}{k!}$$
which is a "cosine" but it is
$$cos(\frac{x}{2})$$

$$x^2cos(x)$$
would be
$$x^2\sum(-1)^k \frac{x^k}{k!}= \sum (-1)^k \frac{x^{k+2}}{k!}$$

5. Dec 8, 2009

vucollegeguy

Ok -got it.
A big "THANKKK YOUUU!" to all for all of your help!

format: plain


text:
# Homework Help: Checking My Method For Differentiation

1. Oct 22, 2013

### BOAS

1. The problem statement, all variables and given/known data

Differentiate the following with respect to x

y = $\frac{4}{x^{3}}$ + $\frac{x^{3}}{4}$

3. The attempt at a solution

So the problem here is really getting this into a form that is easy to differentiate and i'd just like to show what i'm doing before I go ahead and do the rest of my questions. The step i'm least confident with is applying chain rule to each term separately...

I want to use the chain rule here, so I would rearrange to this;

y = (x3)-4 + (4)-x3

dy/dx = -4(x3)-53x2 - x3(4)-x3-1.0

dy/dx = -4(x3)-53x2

dy/dx = -12x2(x3)-5

So, is this ok? I am concerned that my second term disappears, but that's what it looks like I have to do.

Last edited: Oct 22, 2013
2. Oct 22, 2013

### FeDeX_LaTeX

How did you get that?

3. Oct 22, 2013

### BOAS

$(x^{3})^{-4}$ = $4/x^{3}$ is it not?

4. Oct 22, 2013

### FeDeX_LaTeX

It is not -- $(x^3)^{-4} = x^{-12}$.

$\frac{4}{x^3} = 4x^{-3}$

5. Oct 22, 2013

### Staff: Mentor

No, it is not.

$\frac 4 {x^3} = 4x^{-3}$

For your problem there is no need for the chain rule.

6. Oct 22, 2013

### BOAS

I have made this mistake a few times before... Thanks for catching it.

7. Oct 22, 2013

### BOAS

That's become somewhat clearer to me after my mistake was pointed out.

so y = $4x^{-3} + x^{3}(4^{-1})$

dy/dx = $-12x^{-4} -4^{-2}3x^{2}$

dy/dx = $-12x^{-4} - (3/2) x^{2}$

8. Oct 22, 2013

### Staff: Mentor

y = 4x-3 + (1/4)x3
dy/dx = -12x-4 + (3/4)x2

You are using the power rule on something that is not a function. 4-1 is a constant, so its derivative is zero.

9. Oct 22, 2013

### BOAS

Ah, I understand.

Thank you.


text:
# Convergence rates for variational regularization of inverse problems with stochastic data

Yusufu Simayi (Georg-August-Universität Göttingen)
Torsdag 16. maj 2019 13:15–14:00 Koll. D (1531-211)
Stochastics-seminar
We consider variational regularization methods for mildly ill-posed inverse problems described by operator equations $F(f)=g$ in Hilbert spaces. One focus of this talk will be on statistical data noise models: We will present a general framework which allows to treat many noise models and data fidelity terms in a unified setting, including Gaussian and Poisson and empirical processes and show (nearly) optimal rates of convergence as the noise level tends to zero. Such inverse problems for example occur in parameter identification problems in stochastic differential equations and Photonic imaging etc. The main tool of our analysis are large deviation inequalities for stochastic processes in negative Besov norms.
Organiseret af: Department of Mathematics
Kontakt: Jevgenijs Ivanovs Revideret: 24.06.2021


text:
# Features and the power of representations

🕑 13 min • 👤 Thomas Graf • 📆 June 06, 2019 in Discussions

As you might have gleaned from my previous post, I’m not too fond of features, but I haven’t really given you a reason for that. It is actually straight-forward: features lower complexity. By itself, that is actually a useful property. Trees lower the complexity of syntax, and nobody (or barely anybody) uses that as an argument that we should use strings. Distributing the workload between representations and operations/constraints over these representations is considered a good thing. Rightfully so, because factorization is generally a good idea.

But there is a crucial difference between trees and features. We actually have models of how trees are constructed from strings — you might have heard of them, they’re called parsers. And we have some ways of measuring the complexity of this process, e.g. asymptotic worst-case complexity. We lack a comparable theory for features. We’re using an enriched representation without paying attention to the computational cost of carrying out this enrichment. That’s no good, we’re just cheating ourselves in this case. Fortunately, listening to people talk about features for 48h at the workshop gave me an epiphany, and I’m here to share it with you.

## Recap: Fear the Feature

Let’s first make clear where my concern about features is coming from. Gillian Ramchand has a beautifully written recap of my workshop talk that might make for a more accessible starter. I can’t resist including my favorite passage as a sample here:

When you transform your landscape in Feature Land into Constraint Land, lakes turn into rain and volcanoes emerge from the mist, and vice versa.

I’m afraid I can’t offer a comparable level of prose, but I’ll try to make up for it by bringing in equally beautiful math.

Like most linguists, I want our theories to be highly restrictive. Some overgeneration is always unavoidable because of substantive universals, e.g. that we can’t have the same phonetic exponent for each lexical root. But on a formal level we don’t want to see widely unnatural patterns like the following:

Absolute counting

There must be at least three C-heads in the tree.

Modulo counting

The number of lexical items must be a multiple of 3.

Symmetric opposites

An NPI must c-command its licensor.

Boolean constraint conjunction

Satisfy either NPI-licensing or V2 iff Principle B is satisfied.

No locality

The last word of the first TP must be the same as the first word of the last TP.

Domain mixing

If the first word is downward entailing, then the last word must not contain an onset cluster.

Constraint flutter

If A is an adjunct containing an adjunct C, then A and C are both islands only if there is some adjunct B contained by A and containing C that is not an island.

I don’t even…

If the Levenshtein distance between omniscience and at least one lexical item in the tree is a prime number, then the tree is well-formed iff the number of instances of God in it exceeds the greatest Levenshtein distance between all lexical items in the tree.

All of this crazy stuff can be done with constraints that are definable in monadic second-order logic (MSO). MSO is a relatively simple extension of first-order logic (FO) where we can also quantify over sets. While MSO isn’t nearly as well-studied as FO, it is quite popular in theoretical computer science because of its connections to automata theory. An MSO formula over strings can be translated into an equivalent finite-state automaton, and the other way round. Over trees, MSO corresponds to bottom-up tree automata, which are the natural tree counterpart of finite-state automata. All of this has been known for a long time (Thatcher and Wright 1968; Doner 1970), making MSO the logic counterpart to computer scientists’ favorite classes of automata.

Cute history lesson, but let’s bring this back to linguistics before you close the tab in favor of some kitten pics. Here’s the rub: Every syntactic theory with a subcategorization mechanism can actually express any arbitrary MSO constraint. The pipeline for this is fairly simple:

1. Express the constraint as an MSO formula over trees.
2. Translate the MSO formula into an equivalent bottom-up tree automaton. The automaton may be much larger than the constraint as every quantifier alternation in the MSO formula induces an exponential blow-up. In general, the resulting automaton will be a confusing mess, and once you combine multiple constraints into a single automaton you should forfeit all hope of ever making sense of the automaton. It’s like translating Prolog into machine code.
3. Compile the automaton directly into the feature calculus by refining category and subcategorziation features (Graf 2011; Kobele 2011). Intuitively, an automaton that enforces NPI licensing would split every category XP into XP[+/- licensor, +/- needs licensing] and then use subcategorization requirements to ensure that we never end up with [+ needs licensing] at the root. In practice, you get things like XP357 selects YP28 and ZP39, with no idea what the heck any of this does.

The details of the procedure aren’t all the important actually. If you’re curious, check Graf (2017) for a relatively newbie-friendly discussion of this translation mechanism. The bottom line is that features allow us to express arbitrary MSO constraints, and that’s something we simply do not want.

And the problem actually goes much deeper than the translation suggests. You might think that it’s all a problem with subcategorization, but that’s just one specific instantiation of a much more general principle: every recognizable set is a projection of a local set, or equivalently, every local set is a cylindrification of a recognizable set. Yikes, that’s two of the most jargony jargon sentences that ever jargoned. Let’s run them through the DeJargonator 2000:

1. A recognizable set is a set of structures that is definable in MSO. For instance a regular string language (recognized by finite state-automata), or a regular tree language (recognized by a bottom-up tree automaton). Remember, MSO is very powerful, so recognizable sets can be very complex.
2. A local set is a set of structures that can be described in terms of a finite list of licit, locally bounded contexts. The string language $$(ab)^+$$ab, abab, ababab, and so on — is a local string set because we can describe it in terms of finitely many forbidden substructures:
1. Start with a: $$a \mid \ \_$$
2. Only have b after a: $$b \mid a \_$$
3. You may also have a after b: $$a \mid b \_$$
4. End with b: $$b \mid \_ \$$
Local sets are much simpler than recognizable sets. They’re pretty much bottom-tier when it comes to complexity.
3. A projection is a relabeling that conflates distinctions, for instance mapping both a and b to c. A cylindrification is the opposite, it adds new distinctions, e.g. replacing a by either c or d depending on context (which may be arbitrarily complex).

In plain terms, we can take a very complex recognizable set and push it to the bottom-tier of local sets by choosing a smart relabeling. The prototypical example for this is how modulo counting can be made strictly local. Consider the set of all strings that only contain a and have an even length greater than 0. That’s aa, aaaa, aaaaaa, and so on. This is a recognizable set, and it involves one of the most powerful mechanism of MSO, i.e. modulo counting. But now suppose that we give each a an extra feature that tells us whether it is an odd or an even position. Now we can describe the well-formed strings in terms of four forbidden substructures:

2. Only have a[-odd] after a[+odd].
3. You may also have a[+odd] after a[-odd].
4. End with a[-odd].

Yup, that’s exactly the same constraints as for the ab-language above. So this is clearly a local set. Through the addition of a single feature, complexity has plummeted from recognizable to local. And this isn’t just a mathematical curiosity, this stuff actually happens in linguistics. A concrete instance is GPSG, which takes the complex movement dependencies and reduces them to local dependencies by adding slash-features to the representation. That’s why every feature is by default suspicious — even if it is linguistically motivated and seems innocuous, it might have accidentally reduced complexity.

You might wonder now: Isn’t reducing complexity a good thing? Isn’t this why we factor problems into subproblems and always look for the simplest explanation? Those are valid points, but the feature coding above has very little to do with that. It doesn’t furnish new insights or bring to light important parallels. All it does is artificially lower the complexity of the problem through feature coding. It obscures the problem rather than illuminating it. Once we allow ourselves this kind of feature coding, every MSO-definable constraint can be expressed this way. Even if we limit ourselves to linguistically natural feature uses, that might open the flood gates just enough to let in all kinds of ludicrous overgeneration. We simply do not want feature coding.

Unfortunately, it is very difficult to distinguish feature coding from innocent uses of features. I’ve tried many different routes and didn’t find anything that works. So that’s why I’ve taken a very strict methodological stance: fear the feature. During the workshop, though, a question by Gillian prompted me to think about the problem in a new way, and now I might have a solution so that linguists can have their cake and eat it, too.

## Feature use without feature abuse

From a complexity perspective, the central problem of features is that they hide complexity. Throw in some features, and a complex computation becomes much simpler. We’re shifting the workload from the computation into the representation. As I said at the beginning of this post, this can be a very fruitful route. Syntax is the prime example where things become much more manageable once one moves from strings to trees. The problem isn’t so much with having features in the representation, it is that we have no good handle of how specific features affect the complexity of representations.

At least that’s what I thought, until I realized we do have a way of measuring this: transductions! Transductions are the computational counterpart to rewrite rules. We can have string-to-string transductions, tree-to-tree transductions, tree-to-string transductions, tree-to-graph transductions, whatever. Transductions are simply a way of manipulating structures. One of the first papers on tree-to-tree transductions, Rounds (1970), even cites transformational grammar as an inspiration. As with string languages and tree languages, there’s a veritable zoo of complexity classes for transductions. And that’s exactly the leverage we need to measure the complexity of features. Basically, how hard is it to add the features to the representation? That’s the computational work hidden by making features explicit in the representation!

Let’s dive right into an example. Remember the recognizable set of even-length strings over a and how we made it local by adding a feature [+/- odd]? How hard is it to add this feature? Arguably the simplest kind of string-to-string transductions are the input strictly local (ISL) functions that Jane Chandlee has been doing lots of work on (Chandlee 2014; Chandlee and Heinz 2018). ISL functions are basically SPE rewrite rules where the context is finitely bounded and there’s no optionality. In addition, all the rewrite rules apply simultaneously, so there’s no feeding or bleeding either. Whether a rewrite rule applies to some segment depends only on the segments itself and its local context in the underlying representation. So can an ISL function add the [+/- odd] features? No. The starting point is easy enough. The first a should get [+odd], which is captured by the rule $a \Rightarrow a[+\text{odd}] \mid \ \_$ Crucially, though, we cannot use the fact that the first a is [+odd] in any other context specifications. So the only way to make sure that the second a gets [-odd] is to look at a larger chunk of the input: $a \Rightarrow a[-\text{odd}] \mid \ a \_$ We can continue like this for a while. \begin{align*} a \Rightarrow a[+\text{odd}] \mid & \ aa \_\\ a \Rightarrow a[-\text{odd}] \mid & \ aaa \_\\ a \Rightarrow a[+\text{odd}] \mid & \ aaaa \_\\ a \Rightarrow a[-\text{odd}] \mid & \ aaaaa \_\\ a \Rightarrow a[+\text{odd}] \mid & \ aaaaaa \_\\ a \Rightarrow a[-\text{odd}] \mid & \ aaaaaaa \_\\ \end{align*} But as we can only have a finite number of contexts, we cannot keep counting from the left edge. At some point, we have to say something like “Okay, this a is preceded by at least 17 other instances of a, but I don’t know how many. Should it get [+odd] or [-odd]?” And there’s no safe answer given the available information. Therefore, an ISL function cannot correctly assign [+/- odd].

Now if the transduction were output strictly local (OSL), then things would be easy. An OSL function gets to look at the segment and the context in the output. It’s comparable to an SPE-rule that applies left-to-right or right-to-left. This makes it fairly easy to enforce the alternation between [+odd] and [-odd]. \begin{align*} a \Rightarrow a[+\text{odd}] \mid & \ \_\\ a \Rightarrow a[-\text{odd}] \mid & a[+\text{odd}] \_\\ a \Rightarrow a[+\text{odd}] \mid & a[-\text{odd}] \_\\ \end{align*} It seems, then, that OSL would be a bad model of feature inference because it allows us to do all kinds of undesirable things via feature coding. ISL, on the other hand, is a reasonable baseline, as long as we don’t allow ridiculously large contexts.

I gave a string-based example here for simplicity, but rest assured that we can do the very same thing for trees. For instance, we can require category and subcategorization features to be inferable by an ISL function. This greatly reduces what subcategorization is allowed to do, effectively shutting down the translation pipeline of Graf (2011) and Kobele (2011). I think this also makes interesting empirical predictions about subcategorization, but this would lead us too far here. ISL functions won’t be able to correctly infer movement features, for two reasons. The more obvious problem is that movement is not locally bounded in the sense that ISL requires. But movement also has the property of being partially non-deterministic. The only difference between the derivation trees for this book, John gave to his daughter and to his daughter, John gave his book lies in which D-head carries a topicalization feature. ISL functions, by virtue of being functions, struggle with such non-determinism. Neither challenge seems insurmountable, though, so I might have something workable in the near future. It sure would be nice to finally be able to distinguish feature use from feature abuse.

## A dessert of conceptual remarks

Let’s take a step back to take a broader view of what this post is about. The primary issue is that features in the technical sense of generative grammar aren’t just properties or building blocks, they are reified units of computation. This computation could be inferring class membership — the feature pl conveys that we have determined that the carrier of the feature belongs to a class of elements that we call plural. But it could also represent more complex computations, e.g. modulo counting in the case of [+/- odd]. Putting features into the computation is a loose analog of dynamic programming: we have already carried out a specific computation, and now we’re storing the result of this computation in the form of a feature.

In and of itself, there’s nothing wrong with that. But the way features are used, we completely skip the actual computation step and go immediately to storing the result. Without a principled linking theory between features and computation, features can hide incredibly complex computations. I think this is a point that’s not sufficiently appreciated by linguists. In a comment on my previous post, Peter Svenonius says

If features are real, then it follows that they are “an explicit part of the representation”.

I hope that this post has shown that things aren’t as clear-cut. Features represent computations that may well be real, yet this does not entail that features are an explicit part of the representation. They may be, but methodologically we cannot take this as carte blanche to freely put features into our representations, no matter how plausible or natural they may seem. Two weeks ago, my stance would have been “Sorry, you can’t do that without opening Pandora’s box”. I think I can now offer a more palatable compromise: Sorry, features aren’t as innocent as you think, but by measuring the transduction complexity of feature assignment we might soon be able to separate appropriate feature use from inappropriate feature abuse.

In hindsight, it’s actually baffling to me that I didn’t think of this any sooner. The trade-off between representations and operations/constraints is basically folklore in mathematical linguistics. Jeff has been insisting on the importance of complexity classes for representations for ages now and has had some ideas for getting at this through transductions. And together with my student Aniello De Santo I have recently argued that TSL, a very basic approximation of autosegmental phonology, should be studied in terms of constraints on the representation and a transduction for inferring the representation (De Santo and Graf 2019). It was all there in plain sight, but it took a nudge from Gillian for the pieces to assemble in the right way. So, thank you Gillian! You guys can expect a paper on feature inference within the next few months, with some nifty applications to subcategorization.

## References

Chandlee, Jane. 2014. Strictly local phonological processes. University of Delaware phd thesis. http://udspace.udel.edu/handle/19716/13374.

Chandlee, Jane., and Jeffrey Heinz. 2018. Strict locality and phonological maps. Linguistic Inquiry 49.23–60.

De Santo, Aniello., and Thomas Graf. 2019. Structure sensitive tier projection: Applications and formal properties. Proceedings of formal grammar.

Doner, John. 1970. Tree acceptors and some of their applications. Journal of Computer and System Sciences 4.406–451. doi:10.1016/S0022-0000(70)80041-1.

Graf, Thomas. 2011. Closure properties of Minimalist derivation tree languages. (Ed. by.) Sylvain Pogodalla and Jean-Philippe Prost. LACL 2011. Lecture notes in artificial intelligence. Heidelberg: Springer. doi:10.1007/978-3-642-22221-4_7. https://dx.doi.org/10.1007/978-3-642-22221-4_7.

Graf, Thomas. 2017. A computational guide to the dichotomy of features and constraints. Glossa 2.1–36. doi:10.5334/gjgl.212. https://dx.doi.org/10.5334/gjgl.212.

Kobele, Gregory M. 2011. Minimalist tree languages are closed under intersection with recognizable tree languages. (Ed. by.) Sylvain Pogodalla and Jean-Philippe Prost. LACL 2011. Lecture notes in artificial intelligence. doi:10.1007/978-3-642-22221-4_9. https://doi.org/10.1007/978-3-642-22221-4_9.

Rounds, William C. 1970. Mappings on grammars and trees. Mathematical Systems Theory 4.257–287.

Thatcher, James W., and J. B. Wright. 1968. Generalized finite automata theory with an application to a decision problem of second-order logic. Mathematical Systems Theory 2.57–81. doi:10.1007/BF01691346.


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# Sergei Yakovenko's blog: on Math and Teaching

## Topology of the plane

1. Functions of two variables as maps $f\colon \mathbb R^2\to\mathbb R$. Domains in the plane. Open and closed domains.
2. Convergence of planar points: $\lim_{n\to\infty}(x_n,y_n)=(X,Y)\in\mathbb R^2\iff\lim x_n=X\ \&\ \lim y_n=Y$. Alternative description: any square $Q_\varepsilon(X,Y)=\{|x-X|<\varepsilon,\ |y-Y|<\varepsilon\}$ contains almost all elements of the sequence.
3. Limits of functions: $f\colon U\to\mathbb R$, $Z=(X,Y)\in\mathbb R$. We say that $A=\lim_{(x,y)\to Z}f(x,y)$, if for any sequence of points $\{(x_n,y_n)\}$ converging to $Z$, the sequence $f(x_n,y_n)$ converges to $A$ as $n\to\infty$. We say that $f$ is continuous at $Z$, if $A=f(Z)$.
4. Exercise: $f$ is continuous at $Z$, if the preimage $f^{-1}(J)$, of any interval $J=(A-\delta,A+\delta),\ \delta>0$, contains the intersection $Q_\varepsilon\cap U$ for some sufficiently small $\varepsilon>0$.
5. Exercise: if $f$ is continuous at all points of the rectangle $\{x\in I,\ y\in J\}\subseteq\mathbb R^2$, then for any $y\in J$ the function $f_y\colon I\to\mathbb R$, defined by the formula $f_y(x)=f(x,y)$ is continuous on $I$. Can one exchange the role of $x$ and $y$? Formulate and think about the inverse statement.
6. Exercise: Check that the functions $f(x,y)=x\pm y$ and $g(x,y)=xy$ are continuous. What can be said about the continuity of the function $f(x,y)=y/x$?
7. Exercise: formulate and prove a theorem on continuity of the composite functions.
8. Exercise: Give the definition of a continuous function $f\colon I\to\mathbb R^2$ for $I\subseteq\mathbb R$. Planar curves.  Simple curves. Closed curves.
9. Intermediate value theorem for curves. Connected sets, connected components. Jordan lemma.
10. Rotation of a closed curve around a point. Continuity of the rotation number. Yet another “Intermediate value theorem” for functions of two variables.

1. I have a question about the difference between the term function and map, in which cases it is used? 😦

Also on the terms variety and manifold 😦
Regards. 🙂

Comment by Goethe — Thursday, August 12, 2010 @ 9:11

2. Dear Johann Wolfgang 😉

The terms “function” and “map” formally are completely interchangeable (another terms for the same is “application”, “mapping” etc.) However, in conjunction with some adjectives the usage becomes more restricted: you may say “vector-valued function” but never “vector-valued map”. Very loosely speaking, the word “function” is used when the source and the target sets have completely different nature (like function of matrix, function of several arguments), while “map” often assumes that these two sets have some common features, – like coordinate maps on a manifold etc. But the thumb rule should always be “read the definition first”…

Variety and manifold are also almost interchangeable. In more advanced parts of Algebraic Geometry the term “variety” may denote “manifold with singularities”, while “manifold” usually associates with the “smooth manifold”, “topological manifold” e.a. Here the above thumb rule also applies completely 😉

Comment by Sergei Yakovenko — Thursday, August 12, 2010 @ 9:27

3. oh! thank you very much for the reply 🙂

Courses I’ve taken (in spanish), at least I do not remember that one teacher has made known the difference between the definition of function and map, in fact (at the risk of being wrong) in spanish, the management of both terms often used without distinction of any kind (in spanish).

I am self reading a book on algebraic geometry, Shafarevich – BAG. In which the author handles both terms, function and map. So to translate this into spanish, it becomes a somewhat confusing, e.g. Regular map and regular function, clearly looking at the definitions, the picture is clarifying, but I was a somewhat confusing at first instance.

Again, thank you very much for the reply.

Comment by goethe — Friday, August 13, 2010 @ 7:52

4. In algebraic geometry one has to be especially careful, using the above thumb rule on every step. Sometimes the terms can be misleading: for instance, a “rational function”, the central notion of AG, is NOT A FUNCTION in the standard sense!! Even if you extend the set of values by the “infinite point” (i.e., replacing the affine complex line by the projective line a.k.a. the Riemann sphere), there will remain the “indeterminacy points” like the origin on the (x,y)-plane for the “rational function” y/x. Beware 😦 !

Comment by Sergei Yakovenko — Friday, August 13, 2010 @ 11:14

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# Biblioteca

Música » Stick to Your Guns »

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# Ideal Gases

DEAL GASES AND THE IDEAL GAS LAW

Kinetic Theory assumptions about ideal gases

There is no such thing as an ideal gas, of course, but many gases behave approximately as if they were ideal at ordinary working temperatures and pressures. Real gases are dealt with in more detail on another page.

The assumptions are:

• Gases are made up of molecules which are in constant random motion in straight lines.
• The molecules behave as rigid spheres.
• Pressure is due to collisions between the molecules and the walls of the container.
• All collisions, both between the molecules themselves, and between the molecules and the walls of the container, are perfectly elastic. (That means that there is no loss of kinetic energy during the collision.)
• The temperature of the gas is proportional to the average kinetic energy of the molecules.

And then two absolutely key assumptions, because these are the two most important ways in which real gases differ from ideal gases:

• There are no (or entirely negligible) intermolecular forces between the gas molecules.
• The volume occupied by the molecules themselves is entirely negligible relative to the volume of the container.

The Ideal Gas Equation The ideal gas equation is:

pV = nRT

On the whole, this is an easy equation to remember and use. The problems lie almost entirely in the units. Exploring the various terms Pressure, p Pressure is measured in pascals, Pa – sometimes expressed as newtons per square metre, N m-2. These mean exactly the same thing. Be careful if you are given pressures in kPa (kilopascals). For example, 150 kPa is 150,000 Pa. You must make that conversion before you use the ideal gas equation. Should you want to convert from other pressure measurements:

• 1 atmosphere = 101,325 Pa
• 1 bar = 100 kPa = 100,000 Pa

Volume, V This is the most likely place for you to go wrong when you use this equation. That’s because the SI unit of volume is the cubic metre, m3not cm3 or dm3. 1 m3 = 1000 dm3 = 1,000,000 cm3 So if you are inserting values of volume into the equation, you first have to convert them into cubic metres. You would have to divide a volume in dm3 by 1000, or in cm3 by a million. Similarly, if you are working out a volume using the equation, remember to convert the answer in cubic metres into dm3 or cmif you need to – this time by multiplying by a 1000 or a million. If you get this wrong, you are going to end up with a silly answer, out by a factor of a thousand or a million. So it is usually fairly obvious if you have done something wrong, and you can check back again.

The gas constant, R

A value for R will be given you if you need it, or you can look it up in a data source. The SI value for R is 8.31441 J K-1 mol-1.

The temperature, T

The temperature has to be in kelvin. Don’t forget to add 273 if you are given a temperature in degrees Celsius.


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# Polar Form of Complex Numbers with Examples

Here you will learn what is the polar form of complex numbers with examples.

Let’s begin –

## Polar Form of Complex Numbers

Let z = x + iy be a complex number represented by a point P(x, y) in the argand plane.

Then by the geometrical representation of z = x + iy, we have

OP = | z |  and angle POX = $$\theta$$ = arg (z)

In triangle POM, we have

$$cos\theta$$ = $$OM\over OP$$ = $$x\over | z |$$ $$\implies$$ x = | z | $$cos\theta$$

and, $$sin\theta$$ = $$PM\over OP$$ = $$y\over | z |$$ $$\implies$$ x = | z | $$sin\theta$$

So,  z = x + iy

z = | z | $$cos\theta$$ + i | z | $$sin\theta$$

$$\implies$$  z = | z | $$(cos\theta + i sin\theta)$$

$$\implies$$ z = r$$(cos\theta + i sin\theta)$$, where r = | z | and $$\theta$$ = arg (z)

This form of z is called a polar form of z. If we use the general value of the argument of $$\theta$$, then the polar form of z is given by

z = r$$(cos(2n\pi + \theta) + i sin(2n\pi + \theta))$$, where r = | z | and $$\theta$$ = arg (z) and n is an integer.

Example : Write the following complex numbers in the polar form :

(i) $$-3\sqrt{2} + 3\sqrt{2}i$$

(ii) 1 + i

Solution :

(i) Let z = $$-3\sqrt{2} + 3\sqrt{2}i$$. Then,

r = | z | = $$\sqrt{(-3\sqrt{2})^2 + (3\sqrt{2})^2}$$ = 6

Let $$tan\alpha$$ = |$$Im(z)\over Re(z)$$| = 1 $$\implies$$ $$\alpha$$ = $$\pi\over 4$$

Since the point representing z lies in the second quadrant. Therefore, the argument of z is given by

$$\theta$$ = $$\pi – \alpha$$ = $$\pi$$ – $$\pi\over 4$$ = $$3\pi\over 4$$.

So, the polar form of z is

z = r$$(cos\theta + i sin\theta)$$ = 6$$cos{3\pi\over 4} + isin{3\pi\over 4}$$

(ii) Let z = 1 + i. Then,

r = | z | = $$\sqrt{1^2 + 1^2}$$ = $$\sqrt{2}$$

Let $$tan\alpha$$ = |$$Im(z)\over Re(z)$$| = 1 $$\implies$$ $$\alpha$$ = $$\pi\over 4$$

Since the point (1, 1) representing z lies in the first quadrant. Therefore, the argument of z is given by

$$\theta$$ = $$\alpha$$ = $$\pi\over 4$$

So, the polar form of z is

z = r$$(cos\theta + i sin\theta)$$ = $$\sqrt{2}$$$$cos{\pi\over 4} + isin{\pi\over 4}$$


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On the complexity of sets of free lines and line segments among balls in three dimensions - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year :

## On the complexity of sets of free lines and line segments among balls in three dimensions

Marc Glisse
Sylvain Lazard

#### Abstract

We present two new fundamental lower bounds on the worst-case combinatorial complexity of sets of free lines and sets of maximal free line segments in the presence of balls in three dimensions. We first prove that the set of maximal non-occluded line segments among n disjoint unit balls has complexity Omega(n^4), which matches the trivial O(n^4) upper bound. This improves the trivial Omega(n^2) bound and also the Omega(n^3) lower bound for the restricted setting of arbitrary-size balls [Devillers and Ramos, 2001]. This result settles, negatively, the natural conjecture that this set of line segments, or, equivalently, the visibility complex, has smaller worst-case complexity for disjoint fat objects than for skinny triangles. We also prove an Omega(n^3) lower bound on the complexity of the set of non-occluded lines among n balls of arbitrary radii, improving on the trivial Omega(n^2) bound. This new bound is however not known to be tight as the only known upper bound is the trivial O(n^4) bound.

### Dates and versions

inria-00442751 , version 1 (22-12-2009)
inria-00442751 , version 2 (17-03-2010)

### Identifiers

• HAL Id : inria-00442751 , version 1

### Cite

Marc Glisse, Sylvain Lazard. On the complexity of sets of free lines and line segments among balls in three dimensions. 2009. ⟨inria-00442751v1⟩

### Export

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# Pls answer me this question asap, with working.The average monthly salary of "m" male employees and "f" female employees of a company is $2000. If the average monthly salary of the male employees is$(b+200),find the average monthly salary of the female employees.

###### Question:

Pls answer me this question asap, with working.
The average monthly salary of "m" male employees and "f" female employees of a company is $2000. If the average monthly salary of the male employees is$(b+200),
find the average monthly salary of the female employees.

### Glass is an example of a ____ material. a. translucent c. opaque b. medium d. transparent

Glass is an example of a ____ material. a. translucent c. opaque b. medium d. transparent...

### Which choice correctly identifies the legs and hypotenuse of a right triangle

which choice correctly identifies the legs and hypotenuse of a right triangle...

### Find the measure of DF

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### Le Which two structures produce energy that cells can use? A and B, B and C, C and D, D and A

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### PLZZ HELP MEE WORTH 15 POINTZZ

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### If 10 x + 3 = 20, then x = ?

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### Explain which trig functions and identities produce circles, ellipses and hyperbolas. Will give Brainliest for correct answer.

Explain which trig functions and identities produce circles, ellipses and hyperbolas. Will give Brainliest for correct answer....

### What strategies can aid in leading a substance-free life? Check all that apply. building a strong relationship with your parents modeling behaviors seen on reality TV shows choosing friends who do not use drugs being a positive role model for others finding activities you can enjoy without drugs answers: A,C,D,E

What strategies can aid in leading a substance-free life? Check all that apply. building a strong relationship with your parents modeling behaviors seen on reality TV shows choosing friends who do not use drugs being a positive role model for others finding activities you can enjoy without drugs ans...

### List two ways in which an invasive species can be introduced to a new environment.

List two ways in which an invasive species can be introduced to a new environment....

### Based on the sequence of events in the excerpt, which sentence best states a cause-and-effect relationship? Because Antonio Meucci was a trained engineer, he was able to invent a “talking telegraph.” Because electricity was used to cure sickness, Antonio Meucci invented the “talking telegraph.” Because the device was so useful, Antonio Meucci invented the “talking telegraph.” Because he heard a patient’s yelp through a copper wire, Antonio Meucci invented the “talking telegraph.”

Based on the sequence of events in the excerpt, which sentence best states a cause-and-effect relationship? Because Antonio Meucci was a trained engineer, he was able to invent a “talking telegraph.” Because electricity was used to cure sickness, Antonio Meucci invented the “talking telegraph...

### A cell-phone company has noticed that the probability of a customer experiencing a dropped call decreases as the customer approaches a cell-site base station. A company representative approached a cell site at a constant speed and calculated the probability of a dropped call at regular intervals, and the probabilities formed the recursive sequence 0.6, 0.4, 0.24, 0.096, 0.02304. If the company representative continues calculating the probability of a dropped call while approaching the cell site

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### Expository writing is fact based so why does it sometimes include more than just facts

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### Meter 1 can measure voltage to within 0.1 volts. Meter 2 can measure voltage to within 0.01 volts. A.) Which meter has the greatest precision? B.) Which meter is more accurate? Why?

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According to the explained model, for optimization of transient protection devices against pump station power failure in this pipeline, all connection points of pipes have been considered as candidate points for locating protective devices. Seven types of air-chamber with different volumes and three types of air-inlet valves with different orifices were considered as possible selections for candidate points. Table 3 shows the list of protective device specifications that are usable in this optimization along with the cost of each one. The optimization model selects decision variables for each candidate point from these two lists. Although in the model there is the possibility of a lack of protective device for each candidate point, it is the state that nothing is selected from the options presented in a candidate point.

Table 3

The specification of protective devices used in optimization model

Air-inlet valve
Air-chamber
TypeOrifice diameter Diameter (mm)Cost ($)TypeVolume (m3)Cost ($)
300 7,000 10 55,000
450 9,000 20 100,000
600 12,000 25 250,000
30 300,000
35 340,000
40 367,000
60 530,000
Air-inlet valve
Air-chamber
TypeOrifice diameter Diameter (mm)Cost ($)TypeVolume (m3)Cost ($)
300 7,000 10 55,000
450 9,000 20 100,000
600 12,000 25 250,000
30 300,000
35 340,000
40 367,000
60 530,000

Close Modal


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### Theory:

The arithmetic mean is a fancy term for the average. The simplest formula for mean or average adds up the numbers you want to average and divided by the number of items.

To write a simple mathematical expression for arithmetic mean or average or mean is,

$\mathit{Arithmetic}\phantom{\rule{0.147em}{0ex}}\mathit{mean}\phantom{\rule{0.147em}{0ex}}=\frac{\sum _{n\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}1}^{k}{X}_{n}}{k}\phantom{\rule{0.147em}{0ex}}$

The sum of $$k$$ different numbers divided by $$k$$.
Example:
A batsman scored the following number of runs in six innings:
$$36, 35, 50, 46, 60, 55$$
Calculate the mean runs scored by him in an inning.

Solution:
Total runs $$= 36 + 35 + 50 + 46 + 60 + 55 = 282$$.
To find the mean, we do the sum of all the observations and divide it by the number of observations.
Therefore, in this case, mean $$=$$ $\frac{282}{6}$ $$= 47$$.
Thus, the mean runs scored in an inning is $$47$$.


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# Tag Info

6

$\ln\left(\int_0^1 x\,dx\right)=\ln\left(\frac{1}{2}\right),$ but $\int_0^1 \ln x\,dx=-1$. So no, it's not that simple.

5

The intuition is that in an optimal strategy, the picker should be indifferent to what the guesser chooses. Suppose we just take $n=3$ for simplicity. Suppose the picker chooses $1$ with probability $p_1$, chooses $2$ with probability $p_2$, and $3$ with probability $p_3$. The selection of $p_1, p_2, p_3$ constitutes the picker's strategy. The ...

4

Some quick calculations, to showcase (presumably) what the OP has done. For the first game, the probability that the game ends on the $k$th turn is $\frac{(k-1)!}{(k+1)!}=\frac{1}{k(k+1)}$. Thus when $X$ is the number of turns, $$E(X)=\sum_{i=1}^\infty\frac{1}{i+1}=\infty$$ which is known to diverge. For game two, the probability that the game ends on the $... 4 If$E[\phi(X)]=E[\phi(Y)]$for every$\phi$, the most you can conclude is that$X$and$Y$have the same distribution. This follows from taking$\phi(x) = I(x\in B)$as$B$ranges over an appropriately large collection of sets. To see why$X=Y$need not be true, even if$X$and$Y$are defined on the same space, consider$X$uniformly distributed on$[0,1]$... 3 For example, take$X_n$s.t.$P(X_n = 2^n) = \frac{1}{2^n}$,$P(X_n = 0) = 1 - \frac{1}{2^n}$. Then a.s. all but finitely many$X_n$are zeroes, so$\sum X_n$converges a.s., but$\mathbb E (X_n) = 1$, so series of mean diverges. 3 If no such$\omega$exists then$Y$prescribed by$\omega\mapsto X(\omega)-\mathbb EX$is a negative random variable. Then on base of$P(\bigcup_{n=1}^{\infty}\{Y<-\frac1n\})=P(\Omega)=1$it can be shown that some$n\in\mathbb N$exist such that$P(Y<-\frac1n)>0$which leads in combination with the fact that$Y$is negative to the conclusion that$...

3

$W(1)$ is a random variable and you cannot just suppose that it is equal to $a$. In fact, you know its distribution. If $W$ is a standard Brownian motion then $W(t) \sim \mathcal{N}(0,t)$ for every $t$. In particular, $W(1) \sim \mathcal{N}(0,1)$ which is why $\mathbb{E}[W(1)] = 0$.

2

Simple mistake you made: $Var(Z)=E[Z^2]-(E[Z])^2$ (and not with a $+$). So, $E[Z^2]=Var(Z)+(E[Z])^2$.

2

The second term in the second step has a plus sign.

2

You can do this with any math software (such as Mathematica) or any reasonable language. I used Python. I attach a picture of the code, which you should try translating into your language / software of choice. There are three cells. The first loads the required required mathematical module ("numpy") and a plotting module. It also defines your transition ...

2

Since all the terms being summed are positive, Fubini's theorem allows you to shuffle them, so that $$\sum_{y = - \infty}^{\infty} \sum_{x = -\infty}^{\infty} xP(X = x| Y = y)P(Y=y) = \sum_{x = - \infty}^{\infty} \sum_{y = -\infty}^{\infty} xP(X = x| Y = y)P(Y=y)$$ Then the innermost sum doesn't depend on $x$ and you can pull it out without worry. It's ...

2

Note that $$X_{n-1} Y_{n-1} = (X_{n-1} - X_n) Y_{n-1} + X_n Y_{n-1}$$ and $$X_{n-1} Y_{n-1} = X_{n-1} (Y_{n-1} - Y_n) + X_{n-1} Y_n.$$ For the right-hand side of the first equation we can use \begin{align*} E[(X_{n-1} - X_n) Y_{n-1}] &= E[E[(X_{n-1} - X_n) Y_{n-1}| F_{n-1}]] \\ &= E[Y_{n-1}\ E[(X_{n-1} - X_n) | F_{n-1}]] \\ &= 0, \end{align*}...

2

Just as you did it: \begin{align} \sum^\infty_{n=1}(2n-1)P(X\geq n) &=\sum^\infty_{n=1}(2n-1)\Big(\sum^\infty_{j=n}P(X=j)\Big)\\ &=\sum^\infty_{j=1}\sum^j_{n=1}P(X=j)(2n-1)\\ &=\sum^\infty_{j=1}P(X=j)\Big(\sum^j_{n=1}(2n-1)\Big)\\ &=\sum^\infty_{j=1}P(X=j)j^2 \end{align} The last line follows from $\sum^j_{n=1}(2n-1)=2\frac{j(j+1)}{2}-j$

2

You are dealing with a probability space $(\Omega, \mathcal{F}, \mathbb{P})$; and your random variable $X$ is a measurable function $X\colon \Omega\to S$, from the sample space to the state space. Then, $\omega$ is just an element of $\Omega$. $X(\omega)$ is the realization of the random variable, $\mathbb{P}(\omega)$ is the probability of $\omega$.

2

$\omega\$ stands for an outcome, i. e. for an element from the sample space $\Omega$.

2

If $(X,Y,Z)$ is uniformly distributed in $T=\{(x,y,z)\in[0,1]^3 : x+y+z=1\}$, then $(X,Y)$ is uniformly distributed in $T'=\{(x,y)\in[0,1]^2 : x+y\leqslant 1\}$, and $Z=1-X-Y$ a.s., thus $$\mathbb{E}f(X,Y,Z)=2\iint_{T'}f(x,y,1-x-y)\,dx\,dy.$$ A simplification by symmetry, $\mathbb{E}\max\{X,Y,Z\}=6\,\mathbb{E} Z[X\leqslant Y\leqslant Z]$, gives $$\mathbb{E}\... 2 Either you get a 1 the first time with probability \frac1{100}, taking just one attempt, or you get a 0 the first time with probability 1-\frac1{100} and then start over again, taking one attempt plus the number of attempts you would expect when restarting (this latter value being the same as the expected number when originally starting) So to find ... 2 First of all, note that Erfc does not denote the Error function but its "complement", i.e.$$\DeclareMathOperator{\erfc}{Erfc}\DeclareMathOperator{\erf}{Erf} \erfc(x) := \frac{2}{\sqrt{\pi}} \int_x^{\infty} \exp (-y^2) \, dy = 1- \erf(x)$$where$$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp (-y^2) \, dy$$is the Error function. Since$$|2(-k)+1| = |...

2

Hint: When faced with a random variable $X$ and an event $A$, the following may help: $$E[X|A] = \frac{E[X1_{A}]}{P(A)}$$ where $1_{()}$ is the indicator function. In your case replace $X$ with $f(X,Y)$ and $A$ with $\{y \in \mathcal{Y}\}$. Let me know if you need more help.

2

No. As a counterexample, take $f(x)=e^x$.

1

Using a bit more machinery. First note that $$X^2=\sum_{n=1}^\infty(2n-1)I(X\geq n)\tag{0}$$ with probability one where $I$ is the indicator function. To see that this identity is true note that when $X^2=k^2$ (so $X=k$) for an integer $k\geq 1$ the RHS is $$\sum_{n=1}^k(2n-1)=\sum_{n=1}^k [n^2-(n-1)^2]=k^2$$ as desired. Take expectations of $(0)$ (use ...

1

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Matthew and Michael - you made a lot of progress on Puzzle 168, but I have the feeling I didn't clearly explain the point of the puzzle. The main point of Puzzle 168 was not to show that this square commutes:

$\begin{matrix} & & \alpha_{a,b} & & \\\\ & \mathcal{B}(F(a),b) & \rightarrow & \mathcal{A}(a,G(b)) &\\\\ \mathcal{B}(F(f),g) & \downarrow & & \downarrow & \mathcal{A}(f,G(g)) \\\\ & \mathcal{B}(F(a'),b') & \rightarrow & \mathcal{A}(a',G(b')) &\\\\ & & \alpha_{a',b'} & & \\\\ \end{matrix}$

As Michael notes, it obviously commutes, because when we have an adjunction we get a _natural_ isomorphism

$\alpha : \mathcal{B}(F(-), -) \to \mathcal{A}(-, G(-))$

Naturality is all about commutative squares.

The main point of the puzzle was to get a better feeling for what this naturality "really means". I wrote:

> To see what this means concretely, take a guy \$$h \in \mathcal{B}(F(a),b) \$$, meaning a morphism \$$h : F(a) \to b\$$, and send it around the square both ways: down and then across, or across and then down. You get two morphisms from \$$a' \$$ to \$$G(b')\$$; **what are they?** Since the square commutes these must be equal.

As Michael sort of noted, these two morphisms are

$$\alpha_{a',b'}(g \circ h \circ F(f))$$

and

$$G(g) \circ \alpha_{a,b}(h) \circ f$$

so what the naturality "really means" is that

$$\alpha_{a',b'}(g \circ h \circ F(f)) = G(g) \circ \alpha_{a,b}(h) \circ f$$

However, now we have to figure what _this_ equation "really means"! At first glance it looks pretty intimidating.

Perhaps the best approach would be to look at an example - some example of an adjunction that we understand very well. Then this equation will wind up saying something really nice and obvious, which make us feel "_of course_ we want this equation to be true."