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## Zworski, Maciej

Compute Distance To:
 Author ID: zworski.maciej Published as: Zworski, Maciej; Zworski, M. External Links: MGP · Wikidata · GND · IdRef · theses.fr
 Documents Indexed: 123 Publications since 1985, including 3 Books 1 Contribution as Editor · 3 Further Contributions Co-Authors: 45 Co-Authors with 102 Joint Publications 1,018 Co-Co-Authors
all top 5

### Co-Authors

 25 single-authored 17 Sjöstrand, Johannes 10 Dyatlov, Semyon 9 Burq, Nicolas 8 Nonnenmacher, Stéphane 6 Christiansen, Tanya J. 6 Holmer, Justin 5 Guillopé, Laurent 5 Sá Barreto, Antônio 4 Melrose, Richard Burt 4 Petkov, Vesselin M. 3 Wunsch, Jared 3 Zelditch, Steve 2 Becker, Simon 2 Datchev, Kiril R. 2 Galkowski, Jeffrey 2 Hintz, Peter 2 Ji, Lizhen 2 Marzuola, Jeremy L. 2 Smith, Hart F. 2 Stefanov, Plamen D. 2 Tang, Siuhung 2 Vasy, András 1 Bindel, David S. 1 Bourgain, Jean 1 Dencker, Nils 1 Drouot, Alexis 1 Embree, Mark 1 Froese, Richard 1 Graham, C. Robin 1 Hassell, Andrew 1 Iantchenko, Alexai 1 Jin, Long 1 Klopp, Frédéric 1 Klukowski, Julian 1 Koch, Herbert 1 Lebeau, Gilles 1 Lee, Minjae 1 Lin, Kevin K. 1 Michel, Laurent 1 Nakamura, Shu 1 Perelman, Galina 1 Strain, John A. 1 Tataru, Daniel 1 Ward, Rachel A. 1 Wittsten, Jens
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### Serials

 19 Communications in Mathematical Physics 7 Journal of Functional Analysis 7 Séminaire Équations aux Dérivées Partielles 5 Inventiones Mathematicae 5 Mathematical Research Letters 4 Nonlinearity 4 Annales de l’Institut Fourier 4 Annales Henri Poincaré 3 Communications on Pure and Applied Mathematics 3 Duke Mathematical Journal 3 Proceedings of the American Mathematical Society 3 IMRN. International Mathematics Research Notices 3 Communications in Partial Differential Equations 3 SIAM Journal on Mathematical Analysis 2 Acta Mathematica 2 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 2 Journal of the American Mathematical Society 2 Geometric and Functional Analysis. GAFA 2 Annals of Mathematics. Second Series 2 Graduate Studies in Mathematics 2 Journal of Spectral Theory 1 Helvetica Physica Acta 1 Journal of Mathematical Physics 1 Letters in Mathematical Physics 1 Arkiv för Matematik 1 Canadian Journal of Mathematics 1 Demonstratio Mathematica 1 Journal of Differential Geometry 1 Osaka Journal of Mathematics 1 Proceedings of the London Mathematical Society. Third Series 1 Transactions of the American Mathematical Society 1 Ergodic Theory and Dynamical Systems 1 Asymptotic Analysis 1 Forum Mathematicum 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Journal of Physics A: Mathematical and General 1 SIAM Review 1 Bulletin of the American Mathematical Society. New Series 1 Notices of the American Mathematical Society 1 Journal of Nonlinear Science 1 Journées Équations aux Dérivées Partielles (Saint-Jean-de-Monts) 1 The Asian Journal of Mathematics 1 Mathematical Proceedings of the Royal Irish Academy 1 Journal of the European Mathematical Society (JEMS) 1 AMRX. Applied Mathematics Research eXpress 1 Astérisque 1 Contemporary Mathematics 1 Journal of Physics A: Mathematical and Theoretical 1 Journal of Modern Dynamics 1 Analysis and Mathematical Physics 1 Bulletin of Mathematical Sciences 1 Annals of PDE 1 Pure and Applied Analysis 1 Annales Henri Lebesgue 1 Probability and Mathematical Physics
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### Fields

 83 Partial differential equations (35-XX) 48 Global analysis, analysis on manifolds (58-XX) 39 Quantum theory (81-XX) 27 Operator theory (47-XX) 23 Dynamical systems and ergodic theory (37-XX) 7 Differential geometry (53-XX) 6 Number theory (11-XX) 6 Ordinary differential equations (34-XX) 4 Functions of a complex variable (30-XX) 4 Relativity and gravitational theory (83-XX) 3 Measure and integration (28-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Statistical mechanics, structure of matter (82-XX) 2 Combinatorics (05-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Numerical analysis (65-XX) 2 Systems theory; control (93-XX) 1 General and overarching topics; collections (00-XX) 1 History and biography (01-XX) 1 Mathematical logic and foundations (03-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Potential theory (31-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Approximations and expansions (41-XX) 1 Geometry (51-XX) 1 Optics, electromagnetic theory (78-XX) 1 Information and communication theory, circuits (94-XX)

### Citations contained in zbMATH Open

115 Publications have been cited 2,560 times in 1,284 Documents Cited by Year
Semiclassical analysis. Zbl 1252.58001
Zworski, Maciej
2012
Scattering matrix in conformal geometry. Zbl 1030.58022
Graham, C. Robin; Zworski, Maciej
2003
Complex scaling and the distribution of scattering poles. Zbl 0752.35046
Sjöstrand, Johannes; Zworski, Maciej
1991
Mathematical theory of scattering resonances. Zbl 1454.58001
Dyatlov, Semyon; Zworski, Maciej
2019
Geometric control in the presence of a black box. Zbl 1050.35058
Burq, Nicolas; Zworski, Maciej
2004
Distribution of poles for scattering on the real line. Zbl 0662.34033
Zworski, Maciej
1987
Quantum decay rates in chaotic scattering. Zbl 1226.35061
Nonnenmacher, Stéphane; Zworski, Maciej
2009
Scattering asymptotics for Riemann surfaces. Zbl 0898.58054
Guillopé, Laurent; Zworski, Maciej
1997
Dynamical zeta functions for Anosov flows via microlocal analysis. (Fonctions zêta dynamiques pour les flots d’Anosov en utilisant l’analyse microlocale.) Zbl 1369.37028
Dyatlov, Semyon; Zworski, Maciej
2016
Scattering metrics and geodesic flow at infinity. Zbl 0855.58058
Melrose, Richard; Zworski, Maciej
1996
Pseudospectra of semiclassical (pseudo-) differential operators. Zbl 1054.35035
Dencker, Nils; Sjöstrand, Johannes; Zworski, Maciej
2004
Fast soliton scattering by delta impurities. Zbl 1126.35068
Holmer, Justin; Marzuola, Jeremy; Zworski, Maciej
2007
Distribution of resonances for spherical black holes. Zbl 0883.35120
Sá Barreto, Antônio; Zworski, Maciej
1997
Ergodicity of eigenfunctions for ergodic billiards. Zbl 0840.58048
Zelditch, Steven; Zworski, Maciej
1996
From quasimodes to resonances. Zbl 0913.35101
Tang, Siu-Hung; Zworski, Maciej
1998
Sharp polynomial bounds on the number of scattering poles. Zbl 0705.35099
Zworski, Maciej
1989
Resonances in physics and geometry. Zbl 1177.58021
Zworski, Maciej
1999
Stochastic stability of Pollicott-Ruelle resonances. Zbl 1369.37037
Dyatlov, Semyon; Zworski, Maciej
2015
The Selberg zeta function for convex co-compact Schottky groups. Zbl 1075.11059
Guillopé, Laurent; Lin, Kevin K.; Zworski, Maciej
2004
Fractal upper bounds on the density of semiclassical resonances. Zbl 1201.35189
Sjöstrand, Johannes; Zworski, Maciej
2007
Semiclassical estimates in asymptotically Euclidean scattering. Zbl 0955.58023
Vasy, András; Zworski, Maciej
2000
Resonance expansions of scattered waves. Zbl 1032.35148
Tang, Siu-Hung; Zworski, Maciej
2000
Elementary linear algebra for advanced spectral problems. Zbl 1140.15009
Sjöstrand, Johannes; Zworski, Maciej
2007
Mathematical study of scattering resonances. Zbl 1368.35230
Zworski, Maciej
2017
Semiclassical $$L^p$$ estimates. Zbl 1133.58025
Koch, Herbert; Tataru, Daniel; Zworski, Maciej
2007
Upper bounds on the number of resonances for non-compact Riemann surfaces. Zbl 0841.58063
Guillopé, Laurent; Zworski, Maciej
1995
Sharp polynomial bounds on the number of scattering poles of radial potentials. Zbl 0681.47002
Zworski, Maciej
1989
Slow solution interaction with delta imperiums. Zbl 1137.35060
Holmer, Justin; Zworski, Maciej
2007
Resolvent estimates for normally hyperbolic trapped sets. Zbl 1228.81170
Wunsch, Jared; Zworski, Maciej
2011
Semi-classical estimates on the scattering determinant. Zbl 1041.81041
Petkov, Vesselin; Zworski, Maciej
2001
Soliton splitting by external delta potentials. Zbl 1128.35384
Holmer, J.; Marzuola, J.; Zworski, M.
2007
Lower bounds on the number of scattering poles. II. Zbl 0823.35137
Sjöstrand, Johannes; Zworski, Maciej
1994
Lower bounds on the number of scattering poles. Zbl 0784.35070
Sjöstrand, Johannes; Zworski, Maciej
1993
Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity. Zbl 0859.58028
Guillopé, Laurent; Zworski, Maciej
1995
Birkhoff normal forms in semi-classical inverse problems. Zbl 1258.35208
Iantchenko, Alexai; Sjöstrand, Johannes; Zworski, Maciej
2002
Quantum monodromy and semi-classical trace formulæ. Zbl 1038.58033
Sjöstrand, Johannes; Zworski, Maciej
2002
Decay of correlations for normally hyperbolic trapping. Zbl 1377.37049
Nonnenmacher, Stéphane; Zworski, Maciej
2015
Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces. Zbl 1016.58014
Zworski, Maciej
1999
Soliton interaction with slowly varying potentials. Zbl 1147.35084
Holmer, Justin; Zworski, Maciej
2008
Asymptotic distribution of resonances for convex obstacles. Zbl 0989.35099
Sjöstrand, Johannes; Zworski, Maciej
1999
A remark on a paper of E. B. Davies. Zbl 0981.35107
Zworski, Maciej
2001
Control for Schrödinger operators on tori. Zbl 1281.35011
Burq, Nicolas; Zworski, Maciej
2012
Ruelle zeta function at zero for surfaces. Zbl 1382.58022
Dyatlov, Semyon; Zworski, Maciej
2017
The wave trace for Riemann surfaces. Zbl 0947.58022
Guillopé, Laurent; Zworski, Maciej
1999
Existence of resonances in potential scattering. Zbl 0877.35087
Sá Barreto, Antônio; Zworski, Maciej
1996
Resonance expansions in semi-classical propagation. Zbl 1042.81582
Burq, Nicolas; Zworski, Maciej
2001
Bouncing ball modes and quantum chaos. Zbl 1072.81022
Burq, Nicolas; Zworski, Maciej
2005
Breit-Wigner approximation and the distribution of resonances. Zbl 0936.47004
Petkov, Vesselin; Zworski, Maciej
1999
Quantum ergodicity for restrictions to hypersurfaces. Zbl 1278.81100
Dyatlov, Semyon; Zworski, Maciej
2013
The inverse spectral problem. Zbl 1061.58029
Zelditch, Steve
2004
Resonances for asymptotically hyperbolic manifolds: Vasy’s method revisited. Zbl 1365.58012
Zworski, Maciej
2016
Counting scattering poles. Zbl 0823.35139
Zworski, M.
1994
Numerical linear algebra and solvability of partial differential equations. Zbl 1021.35077
Zworski, Maciej
2002
Distribution of resonances for open quantum maps. Zbl 1114.81043
Nonnenmacher, Stéphane; Zworski, Maciej
2007
Poisson formula for resonances in even dimensions. Zbl 0932.47004
Zworski, Maciej
1998
Spectral asymptotics for manifolds with cylindrical ends. Zbl 0818.58046
Christiansen, Tanya; Zworski, Maciej
1995
Distribution of resonances for asymptotically Euclidean manifolds. Zbl 1030.58024
Wunsch, Jared; Zworski, Maciej
2000
Resonance expansions of propagators in the presence of potential barriers. Zbl 1037.35064
Nakamura, Shu; Stefanov, Plamen; Zworski, Maciej
2003
Control for Schrödinger operators on 2-tori: rough potentials. Zbl 1279.35016
Bourgain, Jean; Burq, Nicolas; Zworski, Maciej
2013
Existence of resonances in three dimensions. Zbl 0835.35099
Sá Barreto, Antônio; Zworski, Maciej
1995
Semiclassical resolvent estimates in chaotic scattering. Zbl 1181.81055
Nonnenmacher, Stéphane; Zworski, Maciej
2009
Density of resonances for strictly convex analytic obstacles. With an appendix by M. Zworski. Zbl 0863.35072
Sjöstrand, Johannes
1996
The complex scaling method for scattering by strictly convex obstacles. Zbl 0839.35095
Sjöstrand, Johannes; Zworski, Maciej
1995
Effective dynamics of double solitons for perturbed mKdV. Zbl 1300.37046
Holmer, Justin; Perelman, Galina; Zworski, Maciej
2011
Resonances and lower resolvent bounds. Zbl 1326.35299
Datchev, Kiril; Dyatlov, Semyon; Zworski, Maciej
2015
Sharp polynomial bounds on the number of Pollicott-Ruelle resonances. Zbl 1332.37006
Datchev, Kiril; Dyatlov, Semyon; Zworski, Maciej
2014
The FBI transform on compact $${\mathcal{C}^\infty}$$ manifolds. Zbl 0974.35005
Wunsch, Jared; Zworski, Maciej
2001
Generic simplicity of resonances. Zbl 0844.47040
Klopp, Frédéric; Zworski, Maciej
1995
High frequency scattering by a convex obstacle. Zbl 0732.35060
Zworski, Maciej
1990
Resonance wave expansions: Two hyperbolic examples. Zbl 0955.58024
Christiansen, T.; Zworski, M.
2000
A remark on isopolar potentials. Zbl 0988.34067
Zworski, Maciej
2001
Symmetry of bound and antibound states in the semiclassical limit. Zbl 1161.81430
Bindel, David; Zworski, Maciej
2007
Fractal Weyl laws in discrete models of chaotic scattering. Zbl 1082.81079
Nonnenmacher, Stéphane; Zworski, Maciej
2005
Distribution of scattering poles near the real axis. Zbl 0766.35031
Sjöstrand, Johannes; Zworski, Maciej
1992
Fractal Weyl law for open quantum chaotic maps. Zbl 1293.81022
Nonnenmacher, Stéphane; Sjöstrand, Johannes; Zworski, Maciej
2014
Poisson formulae for resonances. Zbl 1255.35084
Zworski, Maciej
1997
Semi-linear diffraction of conormal waves. Zbl 0902.35004
Melrose, Richard B.; Sá Barreto, Antônio; Zworski, Maciej
1996
Breathing patterns in nonlinear relaxation. Zbl 1173.35685
Holmer, Justin; Zworski, Maciej
2009
From open quantum systems to open quantum maps. Zbl 1223.81127
Nonnenmacher, Stéphane; Sjöstrand, Johannes; Zworski, Maciej
2011
Microlocal analysis of forced waves. Zbl 1426.35013
Dyatlov, Semyon; Zworski, Maciej
2019
Estimates on the number of scattering poles near the real axis for strictly convex obstacles. Zbl 0784.35073
Sjöstrand, J.; Zworski, M.
1993
Instability for the semiclassical nonlinear Schrödinger equation. Zbl 1133.35089
Burq, Nicolas; Zworski, Maciej
2005
Heat traces and existence of scattering resonances for bounded potentials. (La trace du noyau de la chaleur et l’existence de résonances de diffusion pour les potentiels bornés.) Zbl 1358.35077
Smith, Hart F.; Zworski, Maciej
2016
Scattering matrices and scattering geodesics of locally symmetric spaces. Zbl 1026.53026
Ji, Lizhen; Zworski, Maciej
2001
Quantum resonances and partial differential equations. Zbl 1136.35432
Zworski, M.
2002
The remainder estimate in spectral accumulation for degenerating hyperbolic surfaces. Zbl 0783.58078
Ji, Lizhen; Zworski, Maciej
1993
Weighted eigenfunction estimates with applications to compressed sensing. Zbl 1262.41004
Burq, Nicolas; Dyatlov, Semyon; Ward, Rachel; Zworski, Maciej
2012
A Fermi golden rule for quantum graphs. Zbl 1347.81046
Lee, Minjae; Zworski, Maciej
2016
Growth of the zeta function for a quadratic map and the dimension of the Julia set. Zbl 1066.37031
Strain, John; Zworski, Maciej
2004
Spacing between phase shifts in a simple scattering problem. Zbl 0946.47006
Zelditch, Steve; Zworski, Maciej
1999
Probabilistic Weyl laws for quantized tori. Zbl 1205.53092
Christiansen, T. J.; Zworski, M.
2010
Magnetic oscillations in a model of graphene. Zbl 1417.82035
Becker, Simon; Zworski, Maciej
2019
A quantitative version of Catlin-D’Angelo-Quillen theorem. Zbl 1270.32001
Drouot, Alexis; Zworski, Maciej
2013
Scattering resonances as viscosity limits. Zbl 1420.35081
Zworski, Maciej
2018
Resonances for obstacles in hyperbolic space. Zbl 1393.35138
Hintz, Peter; Zworski, Maciej
2018
Control theory and high energy eigenfunctions. Zbl 1064.35120
Burq, Nicolas; Zworski, Maciej
2004
Rough controls for Schrödinger operators on 2-tori. (Contrôles peu réguliers pour l’equation de Schrödinger sur les tores de dimension $$2$$.) Zbl 1421.35306
Burq, Nicolas; Zworski, Maciej
2019
A semiclassical approach to the Kramers-Smoluchowski equation. Zbl 1403.35299
Michel, Laurent; Zworski, Maciej
2018
Finite volume surfaces with resonances far from the unitarity axis. Zbl 0797.58016
Froese, Richard; Zworski, Maciej
1993
Decomposition of normal currents. Zbl 0658.49019
Zworski, Maciej
1988
Analytic hypoellipticity of Keldysh operators. Zbl 1481.35100
Galkowski, Jeffrey; Zworski, Maciej
2021
Fractal uncertainty for transfer operators. Zbl 1442.42059
Dyatlov, Semyon; Zworski, Maciej
2020
Mathematical theory of scattering resonances. Zbl 1454.58001
Dyatlov, Semyon; Zworski, Maciej
2019
Microlocal analysis of forced waves. Zbl 1426.35013
Dyatlov, Semyon; Zworski, Maciej
2019
Magnetic oscillations in a model of graphene. Zbl 1417.82035
Becker, Simon; Zworski, Maciej
2019
Rough controls for Schrödinger operators on 2-tori. (Contrôles peu réguliers pour l’equation de Schrödinger sur les tores de dimension $$2$$.) Zbl 1421.35306
Burq, Nicolas; Zworski, Maciej
2019
Remarks on Vasy’s operator with analytic coefficients. Zbl 1412.35207
Lebeau, Gilles; Zworski, Maciej
2019
Scattering resonances as viscosity limits. Zbl 1420.35081
Zworski, Maciej
2018
Resonances for obstacles in hyperbolic space. Zbl 1393.35138
Hintz, Peter; Zworski, Maciej
2018
A semiclassical approach to the Kramers-Smoluchowski equation. Zbl 1403.35299
Michel, Laurent; Zworski, Maciej
2018
Commentary on “Differentiable dynamical systems” by Stephen Smale. Zbl 1390.37004
Zworski, Maciej
2018
Mathematical study of scattering resonances. Zbl 1368.35230
Zworski, Maciej
2017
Ruelle zeta function at zero for surfaces. Zbl 1382.58022
Dyatlov, Semyon; Zworski, Maciej
2017
Wave decay for star-shaped obstacles in $${\mathbb R}^3$$: papers of Morawetz and Ralston revisited. Zbl 1393.35137
Hintz, Peter; Zworski, Maciej
2017
A local trace formula for Anosov flows. Zbl 1373.37088
Jin, Long; Zworski, Maciej
2017
Dynamical zeta functions for Anosov flows via microlocal analysis. (Fonctions zêta dynamiques pour les flots d’Anosov en utilisant l’analyse microlocale.) Zbl 1369.37028
Dyatlov, Semyon; Zworski, Maciej
2016
Resonances for asymptotically hyperbolic manifolds: Vasy’s method revisited. Zbl 1365.58012
Zworski, Maciej
2016
Heat traces and existence of scattering resonances for bounded potentials. (La trace du noyau de la chaleur et l’existence de résonances de diffusion pour les potentiels bornés.) Zbl 1358.35077
Smith, Hart F.; Zworski, Maciej
2016
A Fermi golden rule for quantum graphs. Zbl 1347.81046
Lee, Minjae; Zworski, Maciej
2016
Stochastic stability of Pollicott-Ruelle resonances. Zbl 1369.37037
Dyatlov, Semyon; Zworski, Maciej
2015
Decay of correlations for normally hyperbolic trapping. Zbl 1377.37049
Nonnenmacher, Stéphane; Zworski, Maciej
2015
Resonances and lower resolvent bounds. Zbl 1326.35299
Datchev, Kiril; Dyatlov, Semyon; Zworski, Maciej
2015
Sharp polynomial bounds on the number of Pollicott-Ruelle resonances. Zbl 1332.37006
Datchev, Kiril; Dyatlov, Semyon; Zworski, Maciej
2014
Fractal Weyl law for open quantum chaotic maps. Zbl 1293.81022
Nonnenmacher, Stéphane; Sjöstrand, Johannes; Zworski, Maciej
2014
Inverse problems and applications. Proceedings of two conferences in honour of Gunther Uhlmann’s 60th birthday, Irvine, CA, USA, June 18–22, 2012 and Hangzhou, China, September 17–21, 2012. Zbl 1290.35005
2014
Quantum ergodicity for restrictions to hypersurfaces. Zbl 1278.81100
Dyatlov, Semyon; Zworski, Maciej
2013
Control for Schrödinger operators on 2-tori: rough potentials. Zbl 1279.35016
Bourgain, Jean; Burq, Nicolas; Zworski, Maciej
2013
A quantitative version of Catlin-D’Angelo-Quillen theorem. Zbl 1270.32001
Drouot, Alexis; Zworski, Maciej
2013
Pointwise bounds on quasimodes of semiclassical Schrödinger operators in dimension two. Zbl 1284.58009
Smith, Hart F.; Zworski, Maciej
2013
Semiclassical analysis. Zbl 1252.58001
Zworski, Maciej
2012
Control for Schrödinger operators on tori. Zbl 1281.35011
Burq, Nicolas; Zworski, Maciej
2012
Weighted eigenfunction estimates with applications to compressed sensing. Zbl 1262.41004
Burq, Nicolas; Dyatlov, Semyon; Ward, Rachel; Zworski, Maciej
2012
Resolvent estimates for normally hyperbolic trapped sets. Zbl 1228.81170
Wunsch, Jared; Zworski, Maciej
2011
Effective dynamics of double solitons for perturbed mKdV. Zbl 1300.37046
Holmer, Justin; Perelman, Galina; Zworski, Maciej
2011
From open quantum systems to open quantum maps. Zbl 1223.81127
Nonnenmacher, Stéphane; Sjöstrand, Johannes; Zworski, Maciej
2011
Probabilistic Weyl laws for quantized tori. Zbl 1205.53092
Christiansen, T. J.; Zworski, M.
2010
Quantum decay rates in chaotic scattering. Zbl 1226.35061
Nonnenmacher, Stéphane; Zworski, Maciej
2009
Semiclassical resolvent estimates in chaotic scattering. Zbl 1181.81055
Nonnenmacher, Stéphane; Zworski, Maciej
2009
Breathing patterns in nonlinear relaxation. Zbl 1173.35685
Holmer, Justin; Zworski, Maciej
2009
A mathematical formulation of the Mahaux-Weidenmüller formula for the scattering matrix. Zbl 1179.81157
Christiansen, T. J.; Zworski, M.
2009
Soliton interaction with slowly varying potentials. Zbl 1147.35084
Holmer, Justin; Zworski, Maciej
2008
Fast soliton scattering by delta impurities. Zbl 1126.35068
Holmer, Justin; Marzuola, Jeremy; Zworski, Maciej
2007
Fractal upper bounds on the density of semiclassical resonances. Zbl 1201.35189
Sjöstrand, Johannes; Zworski, Maciej
2007
Elementary linear algebra for advanced spectral problems. Zbl 1140.15009
Sjöstrand, Johannes; Zworski, Maciej
2007
Semiclassical $$L^p$$ estimates. Zbl 1133.58025
Koch, Herbert; Tataru, Daniel; Zworski, Maciej
2007
Slow solution interaction with delta imperiums. Zbl 1137.35060
Holmer, Justin; Zworski, Maciej
2007
Soliton splitting by external delta potentials. Zbl 1128.35384
Holmer, J.; Marzuola, J.; Zworski, M.
2007
Distribution of resonances for open quantum maps. Zbl 1114.81043
Nonnenmacher, Stéphane; Zworski, Maciej
2007
Symmetry of bound and antibound states in the semiclassical limit. Zbl 1161.81430
Bindel, David; Zworski, Maciej
2007
Bouncing ball modes and quantum chaos. Zbl 1072.81022
Burq, Nicolas; Zworski, Maciej
2005
Fractal Weyl laws in discrete models of chaotic scattering. Zbl 1082.81079
Nonnenmacher, Stéphane; Zworski, Maciej
2005
Instability for the semiclassical nonlinear Schrödinger equation. Zbl 1133.35089
Burq, Nicolas; Zworski, Maciej
2005
Geometric control in the presence of a black box. Zbl 1050.35058
Burq, Nicolas; Zworski, Maciej
2004
Pseudospectra of semiclassical (pseudo-) differential operators. Zbl 1054.35035
Dencker, Nils; Sjöstrand, Johannes; Zworski, Maciej
2004
The Selberg zeta function for convex co-compact Schottky groups. Zbl 1075.11059
Guillopé, Laurent; Lin, Kevin K.; Zworski, Maciej
2004
The inverse spectral problem. Zbl 1061.58029
Zelditch, Steve
2004
Growth of the zeta function for a quadratic map and the dimension of the Julia set. Zbl 1066.37031
Strain, John; Zworski, Maciej
2004
Control theory and high energy eigenfunctions. Zbl 1064.35120
Burq, Nicolas; Zworski, Maciej
2004
Scattering matrix in conformal geometry. Zbl 1030.58022
Graham, C. Robin; Zworski, Maciej
2003
Resonance expansions of propagators in the presence of potential barriers. Zbl 1037.35064
Nakamura, Shu; Stefanov, Plamen; Zworski, Maciej
2003
Birkhoff normal forms in semi-classical inverse problems. Zbl 1258.35208
Iantchenko, Alexai; Sjöstrand, Johannes; Zworski, Maciej
2002
Quantum monodromy and semi-classical trace formulæ. Zbl 1038.58033
Sjöstrand, Johannes; Zworski, Maciej
2002
Numerical linear algebra and solvability of partial differential equations. Zbl 1021.35077
Zworski, Maciej
2002
Quantum resonances and partial differential equations. Zbl 1136.35432
Zworski, M.
2002
Semi-classical estimates on the scattering determinant. Zbl 1041.81041
Petkov, Vesselin; Zworski, Maciej
2001
A remark on a paper of E. B. Davies. Zbl 0981.35107
Zworski, Maciej
2001
Resonance expansions in semi-classical propagation. Zbl 1042.81582
Burq, Nicolas; Zworski, Maciej
2001
The FBI transform on compact $${\mathcal{C}^\infty}$$ manifolds. Zbl 0974.35005
Wunsch, Jared; Zworski, Maciej
2001
A remark on isopolar potentials. Zbl 0988.34067
Zworski, Maciej
2001
Scattering matrices and scattering geodesics of locally symmetric spaces. Zbl 1026.53026
Ji, Lizhen; Zworski, Maciej
2001
Singular part of the scattering matrix determines the obstacle. Zbl 1181.35348
Zworski, Maciej
2001
Semiclassical estimates in asymptotically Euclidean scattering. Zbl 0955.58023
Vasy, András; Zworski, Maciej
2000
Resonance expansions of scattered waves. Zbl 1032.35148
Tang, Siu-Hung; Zworski, Maciej
2000
Distribution of resonances for asymptotically Euclidean manifolds. Zbl 1030.58024
Wunsch, Jared; Zworski, Maciej
2000
Resonance wave expansions: Two hyperbolic examples. Zbl 0955.58024
Christiansen, T.; Zworski, M.
2000
Erratum: “Breit-Wigner approximation and the distribution of resonances”. Zbl 1046.81551
Petkov, Vesselin; Zworski, Maciej
2000
Resonances in physics and geometry. Zbl 1177.58021
Zworski, Maciej
1999
Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces. Zbl 1016.58014
Zworski, Maciej
1999
Asymptotic distribution of resonances for convex obstacles. Zbl 0989.35099
Sjöstrand, Johannes; Zworski, Maciej
1999
The wave trace for Riemann surfaces. Zbl 0947.58022
Guillopé, Laurent; Zworski, Maciej
1999
Breit-Wigner approximation and the distribution of resonances. Zbl 0936.47004
Petkov, Vesselin; Zworski, Maciej
1999
Spacing between phase shifts in a simple scattering problem. Zbl 0946.47006
Zelditch, Steve; Zworski, Maciej
1999
Resonant rigidity of $$S^2$$. Zbl 0954.35125
Hassell, Andrew; Zworski, Maciej
1999
From quasimodes to resonances. Zbl 0913.35101
Tang, Siu-Hung; Zworski, Maciej
1998
Poisson formula for resonances in even dimensions. Zbl 0932.47004
Zworski, Maciej
1998
Scattering asymptotics for Riemann surfaces. Zbl 0898.58054
Guillopé, Laurent; Zworski, Maciej
1997
Distribution of resonances for spherical black holes. Zbl 0883.35120
Sá Barreto, Antônio; Zworski, Maciej
1997
Poisson formulae for resonances. Zbl 1255.35084
Zworski, Maciej
1997
Scattering metrics and geodesic flow at infinity. Zbl 0855.58058
Melrose, Richard; Zworski, Maciej
1996
Ergodicity of eigenfunctions for ergodic billiards. Zbl 0840.58048
Zelditch, Steven; Zworski, Maciej
1996
Existence of resonances in potential scattering. Zbl 0877.35087
Sá Barreto, Antônio; Zworski, Maciej
1996
Density of resonances for strictly convex analytic obstacles. With an appendix by M. Zworski. Zbl 0863.35072
Sjöstrand, Johannes
1996
Semi-linear diffraction of conormal waves. Zbl 0902.35004
Melrose, Richard B.; Sá Barreto, Antônio; Zworski, Maciej
1996
Harmonic functions of polynomial growth on certain complete manifolds. Zbl 0881.58018
Christiansen, T.; Zworski, M.
1996
Upper bounds on the number of resonances for non-compact Riemann surfaces. Zbl 0841.58063
Guillopé, Laurent; Zworski, Maciej
1995
Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity. Zbl 0859.58028
Guillopé, Laurent; Zworski, Maciej
1995
Spectral asymptotics for manifolds with cylindrical ends. Zbl 0818.58046
Christiansen, Tanya; Zworski, Maciej
1995
Existence of resonances in three dimensions. Zbl 0835.35099
Sá Barreto, Antônio; Zworski, Maciej
1995
The complex scaling method for scattering by strictly convex obstacles. Zbl 0839.35095
Sjöstrand, Johannes; Zworski, Maciej
1995
Generic simplicity of resonances. Zbl 0844.47040
Klopp, Frédéric; Zworski, Maciej
1995
...and 15 more Documents
all top 5

### Cited by 1,132 Authors

 49 Zworski, Maciej 34 Vasy, András 29 Sjöstrand, Johannes 28 Guillarmou, Colin 25 Hassell, Andrew 23 Dyatlov, Semyon 22 Galkowski, Jeffrey 21 Korotyaev, Evgeny L. 20 Gover, Ashwin Rod 19 Wunsch, Jared 18 Vodev, Georgi 17 Zelditch, Steve 16 Burq, Nicolas 16 Rivière, Gabriel 15 Sá Barreto, Antônio 14 Bony, Jean-François 14 Datchev, Kiril R. 13 Christiansen, Tanya J. 13 Toth, John Andrew 12 Christianson, Hans 11 Iantchenko, Alexei 11 Stefanov, Plamen D. 11 Weich, Tobias 10 Macià, Fabricio 10 Marzuola, Jeremy L. 10 Nonnenmacher, Stéphane 10 Sire, Yannick 10 Spence, Euan A. 10 Vögel, Martin 9 Drouot, Alexis 9 Fujiié, Setsuro 9 González, María del Mar 9 Jin, Long 9 Tacy, Melissa 8 Borthwick, David 8 Hitrik, Michael 8 Naud, Frédéric 8 Petkov, Vesselin M. 8 Ramond, Thierry 8 Royer, Julien 8 Vũ Ngọc, San 7 Bruneau, Vincent 7 Case, Jeffrey S. 7 Chang, Sun-Yung Alice 7 Chtioui, Hichem 7 Hezari, Hamid 7 Hintz, Peter 7 Jakobson, Dmitry 7 Kim, Seunghyeok 7 Perry, Peter A. 7 Pravda-Starov, Karel 7 Sogge, Christopher D. 7 Xiong, Jingang 6 Canzani, Yaiza 6 Dimassi, Mouez 6 Faure, Frédéric 6 Gannot, Oran 6 Häfner, Dietrich 6 Ingremeau, Maxime 6 Isozaki, Hiroshi 6 Juhl, Andreas 6 Laurent, Camille 6 Léautaud, Matthieu 6 Muñoz, Claudio 6 Nier, Francis 6 Uhlmann, Gunther Alberto 6 Zerzeri, Maher 6 Zuazua, Enrique 5 Angulo Pava, Jaime 5 Bonthonneau, Yannick Guedes 5 Bouclet, Jean-Marc 5 Carles, Rémi 5 Cuccagna, Scipio 5 Eswarathasan, Suresh 5 Fermanian-Kammerer, Clotilde 5 Gell-Redman, Jesse 5 Hager, Mildred 5 Helffer, Bernard 5 Hilgert, Joachim 5 Hillairet, Luc 5 Holmer, Justin 5 Jin, Tianling 5 Joshi, Mark S. 5 Lassas, Matti J. 5 Maeda, Masaya 5 Michel, Laurent 5 Pohl, Anke D. 5 Salo, Mikko 5 Shen, Shu 5 Simon, Barry 5 Smith, Hart F. 5 Thomann, Laurent 5 Tohaneanu, Mihai 5 Trélat, Emmanuel 5 Weinstein, Michael I. 5 Zhang, Junyong 4 Abdelhedi, Wael 4 Alexandrova, Ivana 4 Anantharaman, Nalini 4 Bandtlow, Oscar F. ...and 1,032 more Authors
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### Cited in 209 Serials

 112 Communications in Mathematical Physics 75 Journal of Functional Analysis 70 Communications in Partial Differential Equations 60 Annales Henri Poincaré 43 Journal of Mathematical Physics 38 Journal of Differential Equations 34 Annales de l’Institut Fourier 33 Advances in Mathematics 30 Duke Mathematical Journal 29 Journal of Spectral Theory 26 The Journal of Geometric Analysis 25 Transactions of the American Mathematical Society 24 Proceedings of the American Mathematical Society 21 SIAM Journal on Mathematical Analysis 17 Letters in Mathematical Physics 17 Comptes Rendus. Mathématique. Académie des Sciences, Paris 16 Journal de Mathématiques Pures et Appliquées. Neuvième Série 15 Inventiones Mathematicae 14 Journal of Mathematical Analysis and Applications 14 Mathematische Annalen 13 Reviews in Mathematical Physics 12 Journal of the European Mathematical Society (JEMS) 11 Nonlinearity 11 Calculus of Variations and Partial Differential Equations 10 Archive for Rational Mechanics and Analysis 10 Geometric and Functional Analysis. GAFA 10 Journal of High Energy Physics 9 Journal d’Analyse Mathématique 9 Journal of Computational Physics 9 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 9 Journal of the American Mathematical Society 9 Annales de l’Institut Henri Poincaré. Physique Théorique 8 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 8 Annals of Mathematics. Second Series 8 Communications in Contemporary Mathematics 8 Journal of the Institute of Mathematics of Jussieu 8 Analysis & PDE 8 Séminaire Laurent Schwartz. EDP et Applications 8 Pure and Applied Analysis 7 Journal für die Reine und Angewandte Mathematik 7 Annales Henri Lebesgue 6 Journal of Statistical Physics 6 Mathematische Zeitschrift 6 Ergodic Theory and Dynamical Systems 6 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 6 Physica D 6 Bulletin of the American Mathematical Society. New Series 6 Discrete and Continuous Dynamical Systems 6 Journal of Hyperbolic Differential Equations 5 Inverse Problems 5 Acta Mathematica 5 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 5 Mathematische Nachrichten 5 Numerische Mathematik 5 Asymptotic Analysis 5 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 5 The Journal of Fourier Analysis and Applications 5 Complex Variables and Elliptic Equations 4 Communications on Pure and Applied Mathematics 4 ZAMP. Zeitschrift für angewandte Mathematik und Physik 4 Integral Equations and Operator Theory 4 SIAM Journal on Control and Optimization 4 Chinese Annals of Mathematics. Series B 4 Annals of Global Analysis and Geometry 4 Revista Matemática Iberoamericana 4 Differential Geometry and its Applications 4 Indagationes Mathematicae. New Series 4 Russian Journal of Mathematical Physics 4 Applied and Computational Harmonic Analysis 4 Inverse Problems and Imaging 4 Bulletin of Mathematical Sciences 4 Journal de l’École Polytechnique – Mathématiques 3 Israel Journal of Mathematics 3 Arkiv för Matematik 3 Journal of Geometry and Physics 3 Annali di Matematica Pura ed Applicata. Serie Quarta 3 Functional Analysis and its Applications 3 Journal of Computational and Applied Mathematics 3 Memoirs of the American Mathematical Society 3 Proceedings of the London Mathematical Society. Third Series 3 Acta Applicandae Mathematicae 3 SIAM Journal on Applied Mathematics 3 St. Petersburg Mathematical Journal 3 Selecta Mathematica. New Series 3 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 3 Conformal Geometry and Dynamics 3 Chaos 3 Communications on Pure and Applied Analysis 3 Journal of Pseudo-Differential Operators and Applications 3 Analysis and Mathematical Physics 3 Annales Mathématiques du Québec 3 Annals of PDE 3 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 2 Applicable Analysis 2 Classical and Quantum Gravity 2 Reports on Mathematical Physics 2 Mathematics of Computation 2 American Journal of Mathematics 2 Indiana University Mathematics Journal 2 Publications Mathématiques ...and 109 more Serials
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### Cited in 56 Fields

 840 Partial differential equations (35-XX) 347 Global analysis, analysis on manifolds (58-XX) 347 Quantum theory (81-XX) 245 Operator theory (47-XX) 167 Differential geometry (53-XX) 156 Dynamical systems and ergodic theory (37-XX) 72 Relativity and gravitational theory (83-XX) 65 Ordinary differential equations (34-XX) 46 Numerical analysis (65-XX) 46 Systems theory; control (93-XX) 44 Number theory (11-XX) 35 Optics, electromagnetic theory (78-XX) 34 Functional analysis (46-XX) 33 Statistical mechanics, structure of matter (82-XX) 28 Several complex variables and analytic spaces (32-XX) 26 Harmonic analysis on Euclidean spaces (42-XX) 25 Probability theory and stochastic processes (60-XX) 21 Fluid mechanics (76-XX) 19 Mechanics of deformable solids (74-XX) 16 Functions of a complex variable (30-XX) 16 Mechanics of particles and systems (70-XX) 13 Special functions (33-XX) 13 Calculus of variations and optimal control; optimization (49-XX) 12 Topological groups, Lie groups (22-XX) 12 Manifolds and cell complexes (57-XX) 11 Algebraic geometry (14-XX) 11 Measure and integration (28-XX) 10 Linear and multilinear algebra; matrix theory (15-XX) 9 Combinatorics (05-XX) 9 Integral transforms, operational calculus (44-XX) 8 Potential theory (31-XX) 8 Abstract harmonic analysis (43-XX) 6 Information and communication theory, circuits (94-XX) 5 Approximations and expansions (41-XX) 5 Geometry (51-XX) 4 History and biography (01-XX) 4 Group theory and generalizations (20-XX) 4 Difference and functional equations (39-XX) 4 Astronomy and astrophysics (85-XX) 3 Commutative algebra (13-XX) 3 Real functions (26-XX) 3 Integral equations (45-XX) 3 General topology (54-XX) 3 Geophysics (86-XX) 2 Field theory and polynomials (12-XX) 2 Associative rings and algebras (16-XX) 2 Nonassociative rings and algebras (17-XX) 2 $$K$$-theory (19-XX) 2 Statistics (62-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Operations research, mathematical programming (90-XX) 2 Biology and other natural sciences (92-XX) 1 General and overarching topics; collections (00-XX) 1 Category theory; homological algebra (18-XX) 1 Algebraic topology (55-XX) 1 Computer science (68-XX)

### Wikidata Timeline

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text:
# page numbering

This post is a wiki. Anyone with karma >75 is welcome to improve it.

Greetings, I am working in Libre 5.2.2 and I have tried everything I have read on page numbering. The problem is trying to put page 1 at chapter 1 which is 5 pages down. I can get a 1 there but the upper pages have numbers attached to them and if I delete one of those numbers, then all numbers disappear. I do see that 5.2.2 has moved around things in the tool bars. Is there anyone or anywhere where I can see how to replace a numbered page with another number or, how to make a template for a manuscript for a novel in Libre? or how to start format a manuscript from the beginning. Thanks Hawk

edit retag close merge delete

You should get informed about headers and footers and how text in them behaves. The best thing to do is to make a kind of sandbox document, where you can play with the options, so you get comfortable with page styles, headers/footers, numbering and the lot. Just reading about it won't help you much, you have to get it all inside your head.

( 2016-10-31 21:03:29 +0200 )edit

Sort by » oldest newest most voted

Here's a tutorial on page numbering: https://forum.openoffice.org/en/forum... - hope that helps. It covers a lot of the issues.

more

( 2016-11-01 10:03:11 +0200 )edit

This post is a wiki. Anyone with karma >75 is welcome to improve it.

Thanks for all the help but I have spent two days on page numbering and styles and I really have to get to work on other stuff. What is needed is for someone to make a slow, complete video on page numbering. Numbering that will start on a different page than the first page and once you start with the number 1, then how to rid the previous pages with their numbers. And, of course I would think Styles would fall into that.

more

It looks like your document consists of several "independent" parts, each with specific properties (page number sequence and style, title, ...).

The basic tool for that is the page style. A page style defines a common appearance (margins, header/footer, page number kind, background, ...) for a set of pages which are sequentially numbered. A change of these characteristics requires a transition to another page style.

Transition is performed with a break: either manual Insert->Manual Break..., or automatic with a configured paragraph style (Text Flow tab). In both cases, you can force the page number to restart at any value.

The principles are quite simple, but it requires practice to get quickly what you need. Follow @floris v's advice: read the tutorial and make trials to see how the parameters interact.

If this helps, please tick the check mark and, optionally, upvote the answer.

more

In this particular case, what you need is:

a) a document with at least two page styles: one, say, Default, with no header/footer (or with no page numbers there) and another with a header/footer and page numbers there;

b) the first paragraph of the first page where you need numbering containing a break in text flow.

See the attached file: sample_numbering.odt.

more

format: plain


text:
# 选择可以从战略上降低通勤成本的住宅

B住房的总费用+交通费用

## 夫妻通勤费用

B住房的总费用+交通费用

When presented with this scenario, many couples seemingly gravitate towards what they deem to be a 妥协. In other words, “somewhere in 中间”。让我们在此后面添加视觉效果。

• 1号工人在密歇根州荷兰的某个地点体彩官网（红色箭头）
• 2号工人在密歇根州大急流城的某个地点体彩官网，大约在24英里外（绿色箭头）
• “Compromise commute”住所一个位置（黄色箭头）位于密歇根州的哈德逊维尔，距离大急流城和荷兰约12英里
• “Long-short commute”住所B（紫色箭头）距离1号工人1英里’的体彩官网地点在大急流城，距离＃2号工人23英里’在荷兰的体彩官网地点

At first glance, the 妥协 residence (yellow arrow) seems like a win. The couple shares the commute load evenly. With residence B (purple arrow), one half of the couple would see their commute dip to a mile, while the other would have a longer commute of 23 miles versus the residence A commute. Total costs would be the same because total commuted miles are the same, right?

1. 蜜雪儿
• 杰夫
2. TC
3. 柯比


text:
# Regularity of reflection coefficients (or more generally the scattering transform)

Consider the Schrodinger operator $L(q) = -\partial_x^2 + q(x)$ where the potential $q$ is a real-valued function of a real variable which decays sufficiently rapidly at $\pm \infty$.

We define the scattering data in the usual way, as follows:

The essential spectrum of $L(q)$ is the positive real axis $[0,\infty)$ and it has multiplicity two. The Jost functions $f_\pm(\cdot,k)$ corresponding to $L(q)$ solve $L(q)f_\pm = k^2f_\pm$ with $f_\pm(x,k) \sim e^{ikx}$ as $x \to \pm \infty$.

The reflection coefficients $R_\pm(k)$ are defined so that $f_\pm = \bar{f_\mp} + R_\pm(k)f_\mp$ where here overbar denotes complex conjugate. The intuition is that $R$ measures the amount of energy which is reflected back to spatial $\infty$ when a wave with spatial frequency $k$ that originates at spatial $\infty$ interacts with the potential $q$.

The scattering transform (the map from $q$ to the scattering data, of which $R_+$ is a part) and its inverse are important in the theory of integrable PDE.

My question is the following: What is the regularity of the map $q \mapsto R_+$? Is it continuously differentiable?

To answer this question, we first must specify the spaces that $q$ and $R_+$ live in. I don't really care so long as they are reasonable spaces, for example take $q$ in weighted $H^1$ where the weight enforces a rapid decay at $\pm \infty$.

-


text:
# Graphing Waves

Standard

In these two videos we learn how to graph waves using trigonometric functions.

# Graphs of Sine and Cosine

Standard

In these videos we learn about the graphs of sin(x) and cos(x).

# Reciprocal Trig Ratios

Standard

In this video we learn three new trig ratios called the reciprocal ratios. (secant,cosecant,and cotangent)

# Unit Circle Part 2

Standard

More about why the unit circle is useful.

# The Unit Circle

Standard

In this video I introduce the unit circle.

Standard

A radian is a unit of measuring angles just like a degree is. A radian is a distance on the circle’s circumference which is equal to 1 radius.

This means there are $2\pi$ radians in one circle or that $2\pi$ radians equals 360 degrees.

If $2\pi$ radians equals 360 degrees that means that 1 degree equals $2\pi$ divided by 360 radians or $\pi$ divided by 180 radians.

1 radian = $\frac{180}{\pi}$ degrees

1 degree = $\frac{\pi}{180}$ radians

This is a degrees and radians chart. If 1 degree = $\frac{\pi}{180}$radians then 30 degrees equals 30 times that or $\frac{\pi}{6}$ radians. The same thing can be done to find the radian measure for other angles.


text:
## Calculus, 10th Edition (Anton)

Just because the average value of $g(x)$ is less than the average value of $f(x)$ doesn't mean that $g(x)$ can't be greater than $f(x)$ at a single point. For example, the average value of $f(x)=2$ on the interval $[0,3]$ is greater than the average value of $g(x)=x$ on $[0,3]$, but $g(3)$ $(y=3)$ is greater than $f(3)$ $(y=2).$


text:
# Integrals of Limits and the Calculation of surface area

1. Jan 25, 2010

### Yaridovich

I recently had a very interesting idea. Would it be possible to calculate the surface area of a parametric surface by looking at the equation for the parametric line that wraps around that surface? If there is a way to define that surrounding line by using sines and cosines as the x and y parameters, then it is possible to wrap that line around the parametric surface as tightly as you want (that is, for each point in the sine and cosine functions, the distance between consecutive layers of the functions can be made as tight as possible). This is most easily seen with an example. Consider the following parametric function:

x = Cos[2 $$\pi$$ t] (1 - t^2/a^2)
y = Sin[2 $$\pi$$ t](1 - t^2/a^2)
z = t^2/a^2

This parametric line wraps around a cone of height 1 and radius 1. As a gets larger, the line wraps more and more times around the cone. Consider the distance between two points on this function for a values of sine and cosine, so maybe Cos[t] = 1 and sin[t] = 0 ). As a increases, that distance between "rungs" of the line decreases. In this case, the distance between rungs would be the square root of (x(t) -x(t-1))^2+(y(t) - y(t-1)) + (z(t) - z(t-1)). Now, if we multiply that function by the equation that needs to be integrated in order to find arc length (which is easily found by taking derivatives), we have an equation that can help us find the surface area of that cone. All we need to do is take the integral with respect to t from 0 to infinity and then take the limit as a goes to infinite. Unfortunately, that integral is quite difficult, which takes me to my second point:

I know (or at least I think I know) that the limit of an integral is equal to the integral of limit. For example, the integral of (limit(a->infinite)(1+x^2/a)^a) with respect to x is just the integral of e^x^2. However, if the expression (1+x^2/a)^a by itself is integrated with respect to x and then the limit is taken, we get the same result. If we attempt to take the limit as a goes to infinity of the parametric equation described above, we find that the limit evaluates to 0. The limit only produces a non zero result when t is taken to infinity as well, which is what is done once the integral is evaluated. My question is this: is there some way to evaluate this limit so that it does not collapse to zero before taking its integral from zero to infinity?


text:
Susan Nall

2022-01-14

For $\alpha \in \left({0}^{\circ };{90}^{\circ }\right)$ simplify ${\mathrm{sin}}^{2}\alpha +{\mathrm{tan}}^{2}\alpha +{\mathrm{sin}}^{2}\alpha \cdot {\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{4}\alpha$
My try: ${\mathrm{sin}}^{2}\alpha +{\mathrm{tan}}^{2}\alpha +{\mathrm{sin}}^{2}\alpha \cdot {\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{4}\alpha ={\mathrm{sin}}^{2}\alpha +\frac{{\mathrm{sin}}^{2}\alpha }{{\mathrm{cos}}^{2}\alpha }+{\mathrm{sin}}^{2}\alpha \cdot {\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{4}\alpha$
$=\frac{{\mathrm{sin}}^{2}\alpha \cdot {\mathrm{cos}}^{2}\alpha +{\mathrm{sin}}^{2}+{\mathrm{sin}}^{2}\alpha \cdot {\mathrm{cos}}^{4}\alpha +{\mathrm{cos}}^{6}\alpha }{{\mathrm{cos}}^{2}\alpha }$

poleglit3

Express everything in terms of $\mathrm{cos}\alpha =c$
$\left(1-{c}^{2}\right)+\frac{\left(1-{c}^{2}\right)}{{c}^{2}}+\left(1-{c}^{2}\right){c}^{2}+{c}^{4}$
$=\frac{{c}^{2}-{c}^{4}+1-{c}^{2}+{c}^{4}-{c}^{6}+{c}^{6}}{{c}^{2}}=\frac{1}{{c}^{2}}$

RizerMix

Youre

user_27qwe

Instead of dividing by ${\mathrm{cos}}^{2}\alpha$, you could do the following, ${\mathrm{sin}}^{2}\alpha +{\mathrm{tan}}^{2}\alpha +{\mathrm{sin}}^{2}\alpha +{\mathrm{sin}}^{2}\alpha {\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{4}\alpha$ $={\mathrm{sin}}^{2}\alpha +{\mathrm{tan}}^{2}\alpha +{\mathrm{sin}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha \left({\mathrm{sin}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha \right)$ $={\mathrm{sin}}^{2}\alpha +{\mathrm{tan}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha$ $=1+{\mathrm{tan}}^{2}\alpha$ $={\mathrm{sec}}^{2}\alpha$


text:
# Finite Element Method Formulation for the Poisson's Equation

(Difference between revisions)
 Revision as of 12:01, 15 August 2008 (view source)JMora (Talk | contribs) (New page: === Suggested Prerequisites === * Linear algebra * Partial differential equations === Our Partial Differential Equation: Poisson's equation === As our first example, we will use the [htt...) Latest revision as of 09:07, 4 November 2009 (view source)JMora (Talk | contribs) (→Finite Element Method Formulation) (3 intermediate revisions by one user not shown) Line 68: Line 68: :$x=[x,y]^T = \sum N_i[x_i,y_i]^T$ :$x=[x,y]^T = \sum N_i[x_i,y_i]^T$ − with N_i the shape forms of the element. + with '''Ni''' the shape forms of the element. :$\varphi = \sum N_i \varphi_i = N a^{(e)}$ :$\varphi = \sum N_i \varphi_i = N a^{(e)}$ Line 112: Line 112: A very symple example (to be prepared). A very symple example (to be prepared). + + + [[Category:Poisson's Equation]]

## Contents

### Suggested Prerequisites

• Linear algebra
• Partial differential equations

### Our Partial Differential Equation: Poisson's equation

As our first example, we will use the Poisson's equation in two-dimensional Cartesian coordinates, that it takes the form:

$\left[ \frac{\partial}{\partial x}\cdot \left( K_{x} \cdot \frac{\partial}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(K_{y} \cdot \frac{\partial}{\partial y} \right) \right]\varphi(x,y) = f(x,y)$

for a domain Ω, equivalent to write:

$\nabla \cdot (K \nabla \varphi )= f$

where K is the material property, $\varphi$ is the unknown to obtain and f is a source term.

### Boundary Conditions

Two different boundary conditions will be considered:

#### Dirichlet boundary condition

The first type boundary condition specifies the values the solution needs to take $(\varphi_{0})$ on a boundary $({\Gamma_{\varphi}})$ of the domain:

$\varphi(x,y)\mid_{\Gamma_{\varphi}} = \varphi_{0}$

#### Neumann boundary condition

The second type boundary condition specifies the values that the derivative of a solution (q0) is to take on the boundary (Γq) of the domain:

$K \nabla \varphi(x,y)\mid_{\Gamma_{q}} = q_{0}$

### Finite Element Method Formulation

The Weak formulation using the Weighted Residual Form (Galerkin Method) of Poisson equation can be written as follows:

${ \int_{\Omega} W [\nabla^T D \nabla \varphi + \rho_v]\partial \Omega + \oint_{\Gamma_q}\overline{W}[n^T D \nabla \varphi + \overline{q}]\partial \Gamma_q=0 }$

with $W, \overline{W}$ the weights for each differential equation. If $\overline{W}=W$ and using the integration by parts, this equation becomes:

${ \int_{\Omega} \nabla^T W^T D \nabla \varphi \partial \Omega = \int_{\Omega} W^T \rho_v \partial \Omega - \oint_{\Gamma_q} W^T \overline{q_n} \partial \Gamma_q - \oint_{\Gamma_{\varphi}} W^T q_n \partial \Gamma_{\varphi} }$

for the two-dimensional Cartesian coordinates, each term can be written:

$D = \begin{bmatrix} K_x & 0 \\ 0 & K_y \end{bmatrix}$
$\partial \Omega = \partial x \partial y$
$x=[x,y]^T = \sum N_i[x_i,y_i]^T$

with Ni the shape forms of the element.

$\varphi = \sum N_i \varphi_i = N a^{(e)}$
$q=- D \nabla \varphi = - D \nabla N a^{(e)} = - D B a^{(e)}$

with B defined as:

$B = \begin{bmatrix} B_1... & B_n \end{bmatrix}$
$B_i = \begin{bmatrix} \frac{\partial N_i}{\partial x} \\ \frac{\partial N_i}{\partial y} \end{bmatrix}$

These system of equations can be written as the following matricial form:

$K \cdot a = f$

with K the stiffness matrix of the system, a the vector of nodal values to obatin (unknown vector), and f the source vector (independent forces or charges).

The superscript (e) refers to the equation applied for each element.

$K^{(e)} = { \int_{\Omega^{(e)}} B^T D B \partial \Omega^{(e)} }$
$f^{(e)} = { \int_{\Omega^{(e)}} N^T \rho_v \partial \Omega^{(e)} - \oint_{\Gamma_q^{(e)}} N^T \overline{q} \partial \Gamma_q^{(e)} - \oint_{\Gamma_{\varphi}^{(e)}} n^T N^T q_n \partial \Gamma_{\varphi}^({(e)} }$

A very symple example (to be prepared).


text:
# Agent Failover - use always DEFAULT EM, if it's available

Question asked by pivst01 on Apr 5, 2016
Latest reply on Apr 7, 2016 by pivst01

Hi All,

is there please any possibility to force the Agent to always (re)connect to the DEFAULT EM, when it's available

even in case it was unavailable, BACKUP EM took it's place, but after some time the DEFAULT EM comes up again? so it should be use..

1.

introscope.agent.enterprisemanager.connectionorder=DEFAULT,BACKUP

introscope.agent.enterprisemanager.failbackRetryIntervalInSeconds=30

DEFAULT & BACKUP EM are in different cluster, so i think loafbalancing.xml shouldnt be used as the agent shouldnt be loadbalanced, the agent should use the BACKUP EM just in some exceptional case.

2.

DEFAULT EM goes down and the agent connects to the BACKUP EM

3.

DEFAULT EM comes up and it should be used

i tried it, but the agent kept connecting to the BACKUP EM even the DEFAULT EM came up and were available again.

any idea?

thank you,

stefan


text:
# zbMATH — the first resource for mathematics

## Lafont, Yves

Compute Distance To:
 Author ID: lafont.yves Published as: Lafont; Lafont, Y.; Lafont, Yves External Links: MGP
 Documents Indexed: 26 Publications since 1987, including 3 Books
all top 5

#### Co-Authors

 15 single-authored 3 Girard, Jean-Yves 2 Métayer, François 2 Regnier, Laurent 1 Dehornoy, Patrick 1 Ehrhard, Thomas 1 Prouté, Alain 1 Rannou, Pierre 1 Scedrov, Andre 1 Taylor, Paul H. 1 Worytkiewicz, Krzysztof
all top 5

#### Serials

 3 Journal of Pure and Applied Algebra 3 Theoretical Computer Science 2 The Journal of Symbolic Logic 2 Information and Computation 2 MSCS. Mathematical Structures in Computer Science 1 Advances in Mathematics 1 Annales de l’Institut Fourier 1 Cahiers de Topologie et Géométrie Différentielle Catégoriques 1 Gazette des Mathématiciens 1 Applied Categorical Structures 1 Diagrammes 1 Cambridge Tracts in Theoretical Computer Science 1 London Mathematical Society Lecture Note Series
all top 5

#### Fields

 17 Computer science (68-XX) 11 Mathematical logic and foundations (03-XX) 8 Category theory; homological algebra (18-XX) 4 Group theory and generalizations (20-XX) 2 General and overarching topics; collections (00-XX) 1 General algebraic systems (08-XX) 1 Associative rings and algebras (16-XX) 1 Information and communication theory, circuits (94-XX)

#### Citations contained in zbMATH Open

24 Publications have been cited 598 times in 529 Documents Cited by Year
Proofs and types. Zbl 0671.68002
Girard, Jean-Yves; Taylor, Paul; Lafont, Yves
1989
Linear logic and lazy computation. Zbl 0647.03016
Girard, J. Y.; Lafont, Y.
1987
From proof-nets to interaction nets. Zbl 0830.03028
Lafont, Yves
1995
Soft linear logic and polynomial time. Zbl 1079.03057
Lafont, Yves
2004
Towards an algebraic theory of Boolean circuits. Zbl 1037.94015
Lafont, Yves
2003
A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier). Zbl 0832.20080
Lafont, Yves
1995
The linear abstract machine. Zbl 0648.68016
Lafont, Y.
1988
The finite model property for various fragments of linear logic. Zbl 0897.03010
Lafont, Yves
1997
Interaction combinators. Zbl 0882.68058
Lafont, Yves
1997
Homology of Gaussian groups. Zbl 1100.20036
Dehornoy, Patrick; Lafont, Yves
2003
A folk model structure on omega-cat. Zbl 1236.18017
Lafont, Yves; Métayer, François; Worytkiewicz, Krzysztof
2010
Church-Rosser property and homology of monoids. Zbl 0748.68035
Lafont, Yves; Prouté, Alain
1991
The undecidability of second order multiplicative linear logic. Zbl 0852.03003
Lafont, Yves; Scedrov, Andre
1996
Algebra and geometry of rewriting. Zbl 1132.68036
Lafont, Yves
2007
Penrose diagrams and 2-dimensional rewriting. Zbl 0789.18004
Lafont, Yves
1992
Penrose diagrams and rewriting in dimension 2. Zbl 0769.68060
Lafont, Y.
1992
The undecidability of second order linear logic without exponentials. Zbl 0866.03005
Lafont, Yves
1996
Polygraphic resolutions and homology of monoids. Zbl 1169.18002
Lafont, Yves; Métayer, François
2009
Corrigenda to: “The linear abstract machine”. Zbl 0659.68016
Lafont, Y.
1988
Diagram rewriting and operads. Zbl 1277.18004
Lafont, Yves
2011
A boundary version of Cartan-Hadamard and applications to rigidity. Zbl 1190.53031
Jean-François; Lafont
2009
Rewriting and the word problem. Zbl 1231.20031
Lafont, Yves
2009
Diagram rewriting for orthogonal matrices: A study of critical peaks. Zbl 1145.68451
Lafont, Yves; Rannou, Pierre
2008
Advances in linear logic. Based on the linear logic workshop held June 14-18, 1993 at the Mathematical Sciences Institute, Cornell University, Ithaca, New York, USA. Zbl 0816.00018
Girard, Jean-Yves (ed.); Lafont, Yves (ed.); Regnier, Laurent (ed.)
1995
Diagram rewriting and operads. Zbl 1277.18004
Lafont, Yves
2011
A folk model structure on omega-cat. Zbl 1236.18017
Lafont, Yves; Métayer, François; Worytkiewicz, Krzysztof
2010
Polygraphic resolutions and homology of monoids. Zbl 1169.18002
Lafont, Yves; Métayer, François
2009
A boundary version of Cartan-Hadamard and applications to rigidity. Zbl 1190.53031
Jean-François; Lafont
2009
Rewriting and the word problem. Zbl 1231.20031
Lafont, Yves
2009
Diagram rewriting for orthogonal matrices: A study of critical peaks. Zbl 1145.68451
Lafont, Yves; Rannou, Pierre
2008
Algebra and geometry of rewriting. Zbl 1132.68036
Lafont, Yves
2007
Soft linear logic and polynomial time. Zbl 1079.03057
Lafont, Yves
2004
Towards an algebraic theory of Boolean circuits. Zbl 1037.94015
Lafont, Yves
2003
Homology of Gaussian groups. Zbl 1100.20036
Dehornoy, Patrick; Lafont, Yves
2003
The finite model property for various fragments of linear logic. Zbl 0897.03010
Lafont, Yves
1997
Interaction combinators. Zbl 0882.68058
Lafont, Yves
1997
The undecidability of second order multiplicative linear logic. Zbl 0852.03003
Lafont, Yves; Scedrov, Andre
1996
The undecidability of second order linear logic without exponentials. Zbl 0866.03005
Lafont, Yves
1996
From proof-nets to interaction nets. Zbl 0830.03028
Lafont, Yves
1995
A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier). Zbl 0832.20080
Lafont, Yves
1995
Advances in linear logic. Based on the linear logic workshop held June 14-18, 1993 at the Mathematical Sciences Institute, Cornell University, Ithaca, New York, USA. Zbl 0816.00018
Girard, Jean-Yves (ed.); Lafont, Yves (ed.); Regnier, Laurent (ed.)
1995
Penrose diagrams and 2-dimensional rewriting. Zbl 0789.18004
Lafont, Yves
1992
Penrose diagrams and rewriting in dimension 2. Zbl 0769.68060
Lafont, Y.
1992
Church-Rosser property and homology of monoids. Zbl 0748.68035
Lafont, Yves; Prouté, Alain
1991
Proofs and types. Zbl 0671.68002
Girard, Jean-Yves; Taylor, Paul; Lafont, Yves
1989
The linear abstract machine. Zbl 0648.68016
Lafont, Y.
1988
Corrigenda to: “The linear abstract machine”. Zbl 0659.68016
Lafont, Y.
1988
Linear logic and lazy computation. Zbl 0647.03016
Girard, J. Y.; Lafont, Y.
1987
all top 5

#### Cited by 581 Authors

 14 Mackie, Ian 13 Dal Lago, Ugo 11 Lafont, Yves 10 Fernández, Maribel 8 Guiraud, Yves 7 Kanovich, Max Iosifovich 7 Kobayashi, Yuji 6 Abramsky, Samson 6 Kesner, Delia 6 Nour, Karim 6 Pym, David J. 5 Baillot, Patrick 5 Blute, Richard F. 5 Ehrhard, Thomas 5 Ferreira, Fernando 5 Ferreira, Gilda 5 Masini, Andrea 5 Okada, Mitsuhiro 5 Otto, Friedrich 5 Roversi, Luca 5 Scedrov, Andre 5 Sobociński, Paweł 5 Solov’ëv, Sergeĭ Vladimirovich 5 Yoshida, Nobuko 4 Abel, Andreas M. 4 Ara, Dimitri 4 Aschieri, Federico 4 de Paiva, Valeria 4 Gaboardi, Marco 4 Hofmann, Martin 4 Mazza, Damiano 4 Ronchi Della Rocca, Simona 4 Sato, Shinya 4 Scott, Philip J. 4 Terui, Kazushige 4 Tortora de Falco, Lorenzo 4 van Bakel, Steffen 3 Alves, Sandra 3 Asperti, Andrea 3 Avron, Arnon 3 Bickford, Mark 3 Bonchi, Filippo 3 Braüner, Torben 3 Bucciarelli, Antonio 3 Buszkowski, Wojciech 3 Colson, Loïc 3 Constable, Robert Lee 3 Curien, Pierre-Louis 3 Došen, Kosta 3 Dowek, Gilles 3 Florido, Mário 3 Fu, Peng 3 Fu, Yuxi 3 Girard, Jean-Yves 3 Harper, Robert 3 Kamide, Norihiro 3 Kirchner, Claude 3 Komendantskaya, Ekaterina 3 Laurent, Olivier 3 Lippi, Sylvain 3 Lipton, James B. 3 Longo, Giuseppe 3 Malbos, Philippe 3 Malheiro, António 3 Martini, Simone 3 Miquel, Alexandre 3 Mogbil, Virgile 3 Montanari, Ugo G. 3 O’Hearn, Peter W. 3 Paolini, Luca 3 Piccolo, Mauro 3 Pientka, Brigitte 3 Regnier, Laurent 3 Retoré, Christian 3 Ritter, Eike 3 Salo, Ville O. 3 Salvati, Sylvain 3 Sinot, François-Régis 3 Straßburger, Lutz 3 Stump, Aaron 3 Tranquilli, Paolo 3 Wadler, Philip Lee 3 Zorzi, Margherita 2 Ageron, Pierre 2 Akama, Yohji 2 Ariola, Zena M. 2 Aubert, Clément 2 Baaz, Matthias 2 Banach, Richard 2 Béchet, Denis 2 Bethke, Inge 2 Blanqui, Frédéric 2 Boykett, Tim 2 Brown, Carolyn 2 Bruni, Roberto 2 Burel, Guillaume 2 Callegaro, Filippo 2 Coquand, Thierry 2 Čubrić, Djordje 2 Danos, Vincent ...and 481 more Authors
all top 5

#### Cited in 80 Serials

 106 Theoretical Computer Science 46 Information and Computation 39 Annals of Pure and Applied Logic 31 MSCS. Mathematical Structures in Computer Science 21 Journal of Pure and Applied Algebra 18 The Journal of Symbolic Logic 10 Journal of Automated Reasoning 9 Journal of Logic, Language and Information 7 Journal of Functional Programming 7 Logical Methods in Computer Science 6 Advances in Mathematics 6 Studia Logica 6 International Journal of Algebra and Computation 6 The Bulletin of Symbolic Logic 5 Journal of Algebra 5 Journal of Symbolic Computation 5 Applied Categorical Structures 5 RAIRO. Theoretical Informatics and Applications 5 Journal of Applied Logic 4 RAIRO. Informatique Théorique et Applications 4 Annals of Mathematics and Artificial Intelligence 3 Journal of Computer and System Sciences 3 Journal of Philosophical Logic 3 Formal Aspects of Computing 3 Mathematical Logic Quarterly (MLQ) 3 Theory and Applications of Categories 3 Higher-Order and Symbolic Computation 3 Comptes Rendus. Mathématique. Académie des Sciences, Paris 3 ACM Transactions on Computational Logic 3 Logica Universalis 2 Discrete Mathematics 2 International Journal of Theoretical Physics 2 Information Processing Letters 2 Mathematische Zeitschrift 2 Notre Dame Journal of Formal Logic 2 Transactions of the American Mathematical Society 2 Journal of Computer Science and Technology 2 Archive for Mathematical Logic 2 Applicable Algebra in Engineering, Communication and Computing 2 Journal of Applied Non-Classical Logics 2 Journal of Mathematical Sciences (New York) 2 Diagrammes 2 International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 1 Acta Informatica 1 Artificial Intelligence 1 Communications in Algebra 1 Communications in Mathematical Physics 1 Israel Journal of Mathematics 1 Journal of Statistical Physics 1 Algebra Universalis 1 Annales de l’Institut Fourier 1 Compositio Mathematica 1 Fuzzy Sets and Systems 1 Geometriae Dedicata 1 Kybernetika 1 Publications of the Research Institute for Mathematical Sciences, Kyoto University 1 Semigroup Forum 1 Synthese 1 Bulletin of the Section of Logic 1 Topology and its Applications 1 Ergodic Theory and Dynamical Systems 1 Linear Algebra and its Applications 1 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 1 Journal of Algebraic Combinatorics 1 Acta Mathematica et Informatica Universitatis Ostraviensis 1 Theory of Computing Systems 1 Journal of the ACM 1 Journal of the European Mathematical Society (JEMS) 1 The Journal of Logic and Algebraic Programming 1 Central European Journal of Mathematics 1 Journal of Homotopy and Related Structures 1 Journal of Topology and Analysis 1 Journal of $$K$$-Theory 1 The Review of Symbolic Logic 1 Science China. Mathematics 1 European Journal for Philosophy of Science 1 Journal of Theoretical Biology 1 Mathematics 1 Nonlinear Analysis. Theory, Methods & Applications 1 Higher Structures
all top 5

#### Cited in 32 Fields

 336 Mathematical logic and foundations (03-XX) 305 Computer science (68-XX) 73 Category theory; homological algebra (18-XX) 40 Group theory and generalizations (20-XX) 16 Order, lattices, ordered algebraic structures (06-XX) 14 Algebraic topology (55-XX) 11 Manifolds and cell complexes (57-XX) 10 Information and communication theory, circuits (94-XX) 8 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 7 Combinatorics (05-XX) 7 Quantum theory (81-XX) 5 General algebraic systems (08-XX) 5 Associative rings and algebras (16-XX) 4 General topology (54-XX) 3 Algebraic geometry (14-XX) 3 Biology and other natural sciences (92-XX) 2 General and overarching topics; collections (00-XX) 1 History and biography (01-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Nonassociative rings and algebras (17-XX) 1 $$K$$-theory (19-XX) 1 Topological groups, Lie groups (22-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Functional analysis (46-XX) 1 Geometry (51-XX) 1 Convex and discrete geometry (52-XX) 1 Differential geometry (53-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Mechanics of particles and systems (70-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Operations research, mathematical programming (90-XX)


text:
# Python Vs R : The Eternal Question for Data Scientists

Python and R are the two most commonly used languages for data science today. They are both fully open source products and completely free to use and modify as required under the GNU public license.

But which one is better? And, more importantly, which one should you learn?
Both are widely used and are standard tools in the hands of every data scientist.

The answer may surprise you – because as a professional data scientist, you should be ready to deal with both.

Python has certain use cases and so does R. The scenarios in which they are used vary. It is more often the environment and the needs of the client or your employer which dictates the choice between Python and R. Many things are easier in Python. But R also has its place in your development toolkit.

## Python

Python is a general-purpose programming language released by Guido Van Rossum in 1991.

Since then, Python has been used in multiple environments for multiple purposes, including, but not limited to:
● Web Development (Django)
● Zappa Serverless Framework for Python
● TensorFlow (Deep Learning Machine Learning Models)
● Keras (High-Level Abstractions to Simplify TensorFlow Development)
● Popular apps built in Python include Dropbox, BitTorrent, Morpheus, Calibre, Blender, and Mercurial – among many, many others.

Python has more appeal for software engineers. This is mainly because industry use, production-ready code can be usually written in Python. If you have the background of a software engineer or already know programming, Data Science in Python is a better choice for you (especially if you’re a beginner).

Another situation where Python shines is the sheer number pre-existing libraries that are readily available and open sourced to use. The large number of packages available in the PyPI (short for Python Package Index) repository with over 121k packages that automate many programming tasks at various levels of abstraction, making life easy for the programmer. At least 6k out of the packages on PyPI are focused on data science. Python also excels in readability. Compared to R, Python is much easier to read and to understand. Python is faster than R, in some cases dramatically faster.

## R

R is a statisticians programming language designed for statisticians by statisticians. It originated in the ‘90s through George Ross Ihaka and Robert Gentleman. R excels in academic use and in the hands of a statistician. People who have formal training in Statistics, such as a Statistics degree, find working with R extremely simple. The repository for R packages or libraries, called CRAN (Comprehensive R Archive Network) contains nearly 12k packages, roughly half of which are for data science with R. R also excels at data visualization. Analyzing data on a one-time basis is often simpler and more easily expressible in R.

Also, once upon a time, using Python meant linking many libraries together, some of which would become incompatible after feature revisions and library updates. That is no longer true because of Anaconda – see below. For a short time, deep learning was strictly a Python feature – which shifted the balance of the machine learning world towards Python, for a short time. However, with the release of Keras for TensorFlow in R, that factor changed as well, and deep learning models could now be used in R.

So, what is the answer? Which one should you use?

### Jupyter Notebook – Integrating Python and R

The Anaconda distribution from Continuum Analytics has completely disrupted the machine learning picture. Anaconda supports the standard libraries required for Python and machine learning – NumPy, SciPy, Pandas, SymPy, Seaborn, Matplotlib – as well as full support for R with an outstanding IDE called R Studio.
For Deep Learning it supports TensorFlow, Theano, Caffe, Scikit-Learn, and Torch. One of its most remarkable features is the introduction of the Jupyter Notebook, an integrated platform which supported the use of Python and R in the same environment while keeping everything open source.

Another option is the Hydrogen plug-in for the Atom text editor. It allows you to enter any code that you can use in a Jupyter Notebook and returns the result in the editor. However, it is still in alpha and crashed with an error on my local machine. The Jupyter Lab application allows Python and R notebook editing in the same environment, using the concept of separate and even remote kernels.

As the machine learning field progresses, one can expect more plugins like Hydrogen (which I can’t wait to test once it’s out of alpha) produced in the very near future. So, Python excels in machine learning, while R excels in statistics.
But why should you learn both?

Because a professional data scientist needs to understand statistics and the mathematics behind the machine learning algorithms in great detail.

We shall examine two SVM machine learning models, one through Python code, and then another through R code. This will give us a good picture of how both languages work.

### Python Code

This code performs binary classification using non-linear support vector machine using a Gaussian kernel. The target to predict is a XOR of the inputs.
The color map illustrates the decision function learned by the Support Vector Machine Classifier. (SVC)

print(__doc__)

import numpy as np
import matplotlib.pyplot as plt
from sklearn import svm

xx, yy = np.meshgrid(np.linspace(-3, 3, 500),
np.linspace(-3, 3, 500))
np.random.seed(0)
X = np.random.randn(300, 2)
Y = np.logical_xor(X[:, 0] > 0, X[:, 1] > 0)

# fit the model
clf = svm.NuSVC()
clf.fit(X, Y)

# plot the decision function for each datapoint on the grid
Z = clf.decision_function(np.c_[xx.ravel(), yy.ravel()])
Z = Z.reshape(xx.shape)

plt.imshow(Z, interpolation=’nearest’,
extent=(xx.min(), xx.max(), yy.min(), yy.max()), aspect=’auto’,
origin=’lower’, cmap=plt.cm.PuOr_r)
contours = plt.contour(xx, yy, Z, levels=[0], linewidths=2,
linetypes=’–‘)
plt.scatter(X[:, 0], X[:, 1], s=30, c=Y, cmap=plt.cm.Paired,
edgecolors=’k’)
plt.xticks(())
plt.yticks(())
plt.axis([-3, 3, -3, 3])
plt.show()

.Output:

### R Code

This program uses the iris dataset to illustrate the use of a non-linear SVM classifier. This code is deliberately a little more complex since it applies ML techniques to a full-fledged built in dataset – the iris dataset – one of the canonical data sets used to illustrate the capacities of the ML techniques traditionally. THis code also illustrates the usage of the built-in statistical functions of R.
You will need to install R package e1071 and add it to the compile list by calling library(e1071) before executing the code below. But don’t worry – installing new packages in R Studio is ridiculously simple.

# NOT RUN {
data(iris)
attach(iris)

## classification mode
# default with factor response:

library(e1071)

#loads the svm library into the compile path

model <- svm(Species ~ ., data = iris)

x <- subset(iris, select = -Species)
y <- Species
model <- svm(x, y)

print(model)
summary(model)

# test with train data
pred <- predict(model, x)
# (same as:)
pred <- fitted(model)

# Check accuracy:
table(pred, y)

.Output:

y

pred         setosa versicolor virginica

setosa         50          0         0

versicolor      0         48         2

virginica       0          2        48

<code>

# visualize (classes by color, SV by crosses):
plot(cmdscale(dist(iris[,-
5])),
col = as.integer(iris[,5]),
pch = c(
“o”,“+”)[1:150 %in% model$index + 1]) ## try regression mode on two dimensions # create data x <- seq( 0.1, 5, by = 0.05) y <- log(x) + rnorm(x, sd = 0.2) # estimate model and predict input values m <- svm(x, y) new <- predict(m, x) # visualize plot(x, y) points(x, log(x), col = 2) points(x, new, col = 4) ## density-estimation # create 2-dim. normal with rho=0: X <- data.frame(a = rnorm( 1000), b = rnorm(1000)) attach(X) # traditional way: m <- svm(X, gamma = 0.1) # formula interface: m <- svm(~., data = X, gamma = 0.1) # or: m <- svm(~ a + b, gamma = 0.1) # test: newdata <- data.frame(a = c( 0, 4), b = c(0, 4)) predict (m, newdata) # visualize: plot(X, col = 1:1000 %in% m$index + 1, xlim = c(-5,5), ylim=c(-5,5))
points(newdata, pch =
“+”, col = 2, cex = 5)

i2 <- iris
levels(i2$Species)[3] <-“versicolor” summary(i2$Species)
wts <-
100 / table(i2\$Species)
wts
m <- svm(Species ~ ., data = i2, class.weights = wts)
# }

Output:

As you can see, the R code is fundamentally more powerful in its graphing and statistical abilities than Python. Being a language of statisticians by statisticians, if you have a statistics background, using R will be the best launchpad for your new career in data science.

## Conclusion

Thus, when it comes to choosing between Python and R, any data scientist who is worth his money will know that he is supposed to know both.
And in the end, all the most advanced software engineering skills won’t get you anywhere unless you have a firm foundation in Statistics – or a professional statistician in your team. The main reason that we use analytics is to make business decisions. And we can utilize it best when we have an iron-clad grasp of the entire picture.
So, on Python versus R, to sum up:
Both perform similar tasks in data science but are optimized toward different domains. If you are a software engineer, choose Python. If you are an academic researcher, choose R.
And if you are a data scientist – choose both.

Original Article Published here


text:
## Pat(A) 1095. Cars on Campus (30)

### 1095. Cars on Campus (30)

Zhejiang University has 6 campuses and a lot of gates. From each gate we can collect the in/out times and the plate numbers of the cars crossing the gate. Now with all the information available, you are supposed to tell, at any specific time point, the number of cars parking on campus, and at the end of the day find the cars that have parked for the longest time period.

Input Specification:

Each input file contains one test case. Each case starts with two positive integers N (<= 10000), the number of records, and K (<= 80000) the number of queries. Then N lines follow, each gives a record in the format

plate_number hh:mm:ss status

where plate_number is a string of 7 English capital letters or 1-digit numbers; hh:mm:ss represents the time point in a day by hour:minute:second, with the earliest time being 00:00:00 and the latest 23:59:59; and status is either in or out.

Note that all times will be within a single day. Each "in" record is paired with the chronologically next record for the same car provided it is an "out" record. Any "in" records that are not paired with an "out" record are ignored, as are "out" records not paired with an "in" record. It is guaranteed that at least one car is well paired in the input, and no car is both "in" and "out" at the same moment. Times are recorded using a 24-hour clock.

Then K lines of queries follow, each gives a time point in the format hh:mm:ss. Note: the queries are given in ascending order of the times.

Output Specification:

For each query, output in a line the total number of cars parking on campus. The last line of output is supposed to give the plate number of the car that has parked for the longest time period, and the corresponding time length. If such a car is not unique, then output all of their plate numbers in a line in alphabetical order, separated by a space.

Sample Input:
16 7
JH007BD 18:00:01 in
ZD00001 11:30:08 out
DB8888A 13:00:00 out
ZA3Q625 23:59:50 out
ZA133CH 10:23:00 in
ZD00001 04:09:59 in
JH007BD 05:09:59 in
ZA3Q625 11:42:01 out
JH007BD 05:10:33 in
ZA3Q625 06:30:50 in
JH007BD 12:23:42 out
ZA3Q625 23:55:00 in
JH007BD 12:24:23 out
ZA133CH 17:11:22 out
JH007BD 18:07:01 out
DB8888A 06:30:50 in
05:10:00
06:30:50
11:00:00
12:23:42
14:00:00
18:00:00
23:59:00

Sample Output:
1
4
5
2
1
0
1
JH007BD ZD00001 07:20:09

### 代码

#include "iostream"
#include "algorithm"
#include "vector"
#include "string"
#include "map"
using namespace std;

struct node{
string car;
int time;
string status;
};

bool cmp(struct node a,struct node b){
return a.time < b.time;
}
bool cmpWithCarNum(struct node a,struct node b){
if(a.car != b.car)
return a.car < b.car;
return a.time < b.time;
}

int N,K;
int maxTime = 0;
map<string,int> parkInTime;
map<string,bool> parkStatus;
vector<struct node> records;
vector<struct node> realRecords;
map<string,int> parkTime;

void init(){
scanf("%d %d",&N,&K);
int i;
string car,status;
int h,m,s;
char c1[10];
char c2[10];
for(i = 0; i < N; i ++){
scanf("%s %d:%d:%d %s",c1,&h,&m,&s,c2);
car = string(c1);
status = string(c2);
struct node temp;
temp.car = car;
temp.time = h * 3600 + m * 60 + s;
temp.status = status;
records.push_back(temp);
}
}

void cal(){
parkStatus.clear();
int i,j;
int h,m,s,time;
char c;
int carNum = 0;
for(i = 0,j = 0; i < K; i++){
scanf("%d:%d:%d",&h,&m,&s);
time = h * 3600 + m * 60 + s;
while(j < realRecords.size() && realRecords[j].time <= time){
if(realRecords[j].status == "in"){
parkStatus[realRecords[j].car] = true;
parkInTime[realRecords[j].car] = realRecords[j].time;
carNum ++;
}else if(parkStatus[realRecords[j].car]){
parkStatus[realRecords[j].car] = false;
parkTime[realRecords[j].car] += realRecords[j].time - parkInTime[realRecords[j].car];
maxTime = max(maxTime,parkTime[realRecords[j].car]);
carNum --;
}
j ++;
}
printf("%d\n",carNum);
}
while(j < realRecords.size()){
if(realRecords[j].status == "in"){
parkStatus[realRecords[j].car] = true;
parkInTime[realRecords[j].car] = realRecords[j].time;
}else if(parkStatus[realRecords[j].car]){
parkStatus[realRecords[j].car] = false;
parkTime[realRecords[j].car] += realRecords[j].time - parkInTime[realRecords[j].car];
maxTime = max(maxTime,parkTime[realRecords[j].car]);
}
j ++;
}
}

void fil(){
for (int i = 0; i < records.size()-1; i++){
if(records[i].car != records[i+1].car){
continue;
}
if(records[i].status == "in" && records[i+1].status == "out"){
realRecords.push_back(records[i ++]);
realRecords.push_back(records[i]);
}
continue;
}
}

int main(){
init();
sort(records.begin(),records.end(),cmpWithCarNum);
fil();
sort(realRecords.begin(),realRecords.end(),cmp);
cal();
for(auto it = parkTime.begin();it != parkTime.end();it++){
if(maxTime == it->second)
printf("%s ",it->first.c_str());
}
printf("%02d:%02d:%02d\n",maxTime/3600,maxTime%3600/60,maxTime%60);
system("pause");

}

还没有人评论...


text:
# Highly Nonlinear Approximations for Sparse Signal Representation

## Construction of Oblique Projectors for signal splitting

Oblique projectors in the context of signal reconstruction were introduced in [2] and further analyzed in [3]. The application to signal splitting, also termed structured noise filtering, amongst a number of other applications, is discussed in [4]. For advanced theoretical studies of oblique projector operators in infinite dimensional spaces see [5,6]. We restrict our consideration to numerical constructions in finite dimension, with the aim of addressing the problem of signal splitting when the problem is ill posed.

Given and disjoint, i.e., such that in order to provide a prescription for constructing we proceed as follows. Firstly we define as the direct sum of and , which we express as

Let be the orthogonal complement of in . Thus we have

The operations and are termed direct and orthogonal sum, respectively.

Considering that is a spanning set for a spanning set for is obtained as

Denoting as the standard orthonormal basis in , i.e., the Euclidean inner product , we define the operators and as

Thus the adjoint operators and are

It follows that and hence, defined as:

is self-adjoint and its matrix representation, , has elements

From now on we restrict our signal space to be , since we would like to build the oblique projector onto and along having the form

 (3)

Clearly for the operator to map to zero every vector in the dual vectors must span . This entails that for each there exists a set of coefficients such that

 (4)

which guarantees that every is orthogonal to all vectors in and therefore is included in the null space of . Moreover, since every signal, say, in can be written as with and , the fact that implies . Hence, , which implies that the null space of restricted to is .

In order for to be a projector it is necessary that . As will be shown in the next proposition, if the coefficients are the matrix elements of a generalised inverse of the matrix this condition is fulfilled.

Proposition 1   If the coefficients in (4) are the matrix elements of a generalised inverse of the matrix , which has elements , the operator in (3) is a projector.

Proof. For the measurement vectors in (4) to yield a projector of the form (3), the corresponding operator should be idempotent, i.e.,

 (5)

Defining

 (6)

and using the operators and , as given above, the left hand side in (5) can be expressed as

 (7)

and the right hand side as

 (8)

Assuming that is a generalised inverse of indicated as it satisfies (c.f. Section )

 (9)

and therefore, from (7), the right hand side of (5) follows. Since and , we have . Hence, if the elements determining in (6) are the matrix elements of a generalised inverse of the matrix representation of , the corresponding vectors obtained by (4) yield an operator of the form (3), which is an oblique projector.

Property 1   Let be the oblique projector onto and along and the orthogonal projector onto . Then the following relation holds

Proof. given in (3) can be recast, in terms of operator and , as:

Applying both sides of the equation we obtain:

which is a well known form for the orthogonal projector onto .

Remark 2   Notice that the operative steps for constructing an oblique projector are equivalent to those for constructing an orthogonal one. The difference being that in general the spaces are different. For the special case , , both sets of vectors span and we have an orthogonal projector onto along .

Subsections


text:
# EM scattering

Would I be right in saying the difference between Rayleigh and Tomson scattering is that Tomson involves scattering off a free electron, whereas in Rayleigh the incident wave is scattered off a bound nucleus?

Alos are there any other differences in the two scattering processes?

Thanks

## Answers and Replies

One final thing. Are both elastic scattering?

In the classical limit, Thomson scattering is treated as elastic scattering off a single "free" electron; If the electron atomic binding energy is B, and electron rest mass is m0c2, then B << hv << m0c2 for Thomson scattering. It is the classical limit of Compton scattering, which is inelastic. The energy of the scattered photon is given by
(1/hv') -(1/hv) = 1/[m0c2(1-cos{theta})]
which shows that the scattered photon hv' at theta=90 degrees is nearly exactly the same energy as the incident photon hv for UV and soft x-rays (hv ~ 1000 eV).


text:
# SGPR notes¶

James Hensman, March 2016 Small corrections Alex Matthews, December 2016

The GPflow method SGPR performs Sparse Gaussian Process Regression. It is based mainly on Titsias (2009), though other works (Hensman et al. (2013), Matthews et. al. 2016) are useful for clarifying the prediction density.

This notebook contains a derivation of the form of the equations used in GPflow for the marginal likelihood bound and for the prediction equations.

## Marginal Likelihood bound¶

The bound on the marginal likelihod is (Titsias 2009):

\begin{align}\begin{aligned}\log p(\mathbf y) \geq \log \mathcal N(\mathbf y\,|\,\mathbf 0,\, \mathbf Q_{ff} + \sigma^2 \mathbf I) - \tfrac{1}{2} \sigma^{-2}\textrm{tr}(\mathbf K_{ff} - \mathbf Q_{ff}) \triangleq \mathcal L\\with\end{aligned}\end{align}
$\mathbf Q_{ff} = \mathbf K_{fu}\mathbf K_{uu}^{-1}\mathbf K_{uf}$

.

The kernel matrices $$\mathbf K_{ff}$$, $$\mathbf K_{uu}$$, $$\mathbf K_{fu}$$ represent the kernel evaulated at the data points $$\mathbf X$$, the inducing input points $$\mathbf Z$$ and between the data and inducing points, respectively. We refer to the value of the GP at the data points $$\mathbf X$$ as $$\mathbf f$$, at the inducing points $$\mathbf Z$$ as $$\mathbf u$$ and at the remainder of the function as $$f^\star$$.

To obtain an efficient and stable evaluation os the bound $$\mathcal L$$, we first apply the Woodbury identity to the effective covariance matrix:

$[\mathbf Q_{ff} + \sigma^2 \mathbf I ]^{-1} = \sigma^{-2} \mathbf I - \sigma^{-4} \mathbf K_{fu}[\mathbf K_{uu} + \mathbf K_{uf}\mathbf K_{fu}\sigma^{-2}]^{-1}\mathbf K_{uf}$

Now, to obtain a better conditioned matrix for inversion, we rotate by $$\mathbf L$$, where $$\mathbf L\mathbf L^\top = \mathbf K_{uu}$$:

$[\mathbf Q_{ff} + \sigma^2 \mathbf I ]^{-1} = \sigma^{-2} \mathbf I - \sigma^{-4} \mathbf K_{fu}\color{red}{\mathbf L^{-\top} \mathbf L^\top}[\mathbf K_{uu} + \mathbf K_{uf}K_{fu}\sigma^{-2}]^{-1}\color{red}{\mathbf L \mathbf L^{-1}}\mathbf K_{uf}$

This matrix is better conditioned because for many kernels it has eigenvalues which are bounded above and below. For more detail see GPML section 3.4.3 .

$\phantom{[\mathbf Q_{ff} + \sigma^2 \mathbf I ]^{-1}} = \sigma^{-2} \mathbf I - \sigma^{-4} \mathbf K_{fu}\color{red}{\mathbf L^{-\top}} [\color{red}{\mathbf L^{-1}}\mathbf (K_{uu} + \mathbf K_{uf}K_{fu}\sigma^{-2})\color{red}{\mathbf L^{-\top}}]^{-1}\color{red}{ \mathbf L^{-1}}\mathbf K_{uf}$
$\phantom{[\mathbf Q_{ff} + \sigma^2 \mathbf I ]^{-1} }= \sigma^{-2} \mathbf I - \sigma^{-4} \mathbf K_{fu}\color{red}{\mathbf L^{-\top}} [\mathbf I + \color{red}{\mathbf L^{-1}}\mathbf (\mathbf K_{uf}K_{fu})\color{red}{\mathbf L^{-\top}}\sigma^{-2}]^{-1}\color{red}{ \mathbf L^{-1}}\mathbf K_{uf}$

For notational convenience, we’ll define $$\mathbf L^{-1}\mathbf K_{uf}\sigma^{-1} \triangleq \mathbf A$$, and $$[\mathbf I + \mathbf A\mathbf A^\top]\triangleq \mathbf B$$.

$\phantom{[\mathbf Q_{ff} + \sigma^2 \mathbf I ]^{-1} }= \sigma^{-2} \mathbf I - \sigma^{-2} \mathbf A^{\top} [\mathbf I + \mathbf A\mathbf A^\top]^{-1}\mathbf A$
$$\phantom{[\mathbf Q_{ff} + \sigma^2 \mathbf I ]^{-1}}= \sigma^{-2} \mathbf I - \sigma^{-2} \mathbf A^{\top} \mathbf B^{-1}\mathbf A$$

We also apply the matrix determinant lemma to the same:

$|{\mathbf Q_{ff}} + \sigma^2 {\mathbf I}| = |{\mathbf K_{uu}} + \mathbf K_{uf}\mathbf K_{fu}\sigma^{-2}| \, |\mathbf K_{uu}^{-1}| \, |\sigma^{2}\mathbf I|$

Substituting $$\mathbf K_{uu} = {\mathbf {L L}^\top}$$:

$|{\mathbf Q_{ff}} + \sigma^2 {\mathbf I}| = |{\mathbf {L L}^\top} + \mathbf K_{uf}\mathbf K_{fu}\sigma^{-2}| \, |\mathbf L^{-\top}|\,| \mathbf L^{-1}| \, |\sigma^{2}\mathbf I|$
$|{\mathbf Q_{ff}} + \sigma^2 {\mathbf I}| = |\mathbf I + \mathbf L^{-1}\mathbf K_{uf}\mathbf K_{fu} \mathbf L^{-\top}\sigma^{-2}| \, |\sigma^{2}\mathbf I|$
$|{\mathbf Q_{ff}} + \sigma^2 {\mathbf I}| = |\mathbf B| \, |\sigma^{2}\mathbf I|$

With these two definitions, we’re ready to expand the bound:

$\mathcal L = \log \mathcal N(\mathbf y\,|\,\mathbf 0,\, \mathbf Q_{ff} + \sigma^2 \mathbf I) - \tfrac{1}{2} \sigma^{-2}\textrm{tr}(\mathbf K_{ff} - \mathbf Q_{ff})$
$= -\tfrac{N}{2}\log{2\pi} -\tfrac{1}{2}\log|\mathbf Q_{ff}+\sigma^2\mathbf I| - \tfrac{1}{2}\mathbf y^\top [ \mathbf Q_{ff} + \sigma^2 \mathbf I]^{-1}\mathbf y - \tfrac{1}{2} \sigma^{-2}\textrm{tr}(\mathbf K_{ff} - \mathbf Q_{ff})$
$= -\tfrac{N}{2}\log{2\pi} -\tfrac{1}{2}\log(|\mathbf B||\sigma^{2}\mathbf I|) - \tfrac{1}{2}\sigma^{-2}\mathbf y^\top (\mathbf I - \sigma^{-2} \mathbf A^{\top} \mathbf B^{-1}\mathbf A)\mathbf y - \tfrac{1}{2} \sigma^{-2}\textrm{tr}(\mathbf K_{ff} - \mathbf Q_{ff})$
$= -\tfrac{N}{2}\log{2\pi} -\tfrac{1}{2}\log|\mathbf B| -\tfrac{N}{2}\log\sigma^{2} -\tfrac{1}{2}\sigma^{-2}\mathbf y^\top\mathbf y +\tfrac{1}{2}\sigma^{-2}\mathbf y^\top\mathbf A^{\top} \mathbf B^{-1}\mathbf A\mathbf y -\tfrac{1}{2}\sigma^{-2}\textrm{tr}(\mathbf K_{ff}) + \tfrac{1}{2}\textrm{tr}(\mathbf {AA}^\top)$

where we have used $$\sigma^{-2}\textrm{tr}(\mathbf Q) = \textrm{tr}(\mathbf {AA}^\top)$$.

Finally, we define $$\mathbf c \triangleq \mathbf L_{\mathbf B}^{-1}\mathbf A\mathbf y \sigma^{-1}$$, with $$\mathbf {L_BL_B}^\top = \mathbf B$$, so that:

$\sigma^{-2}\mathbf y^\top\mathbf A^{\top} \mathbf B^{-1}\mathbf A\mathbf y = \mathbf c^\top \mathbf c$

The SGPR code implements this equation with small changes for multiple concurrent outputs (columns of the data matrix Y) and also a prior mean function.

## Prediction¶

At precition time, we need to compute the mean and variance of the variational approximation at some new points $$\mathbf X^\star$$. Following Hensman et al. (2013), we know that all the information in the posterior approximation is contained in the Gaussian distribution $$q(\mathbf u)$$, which represents the distribution on function values at the inducing points $$\mathbf Z$$. Recall:

$q(\mathbf u) = \mathcal N(\mathbf u\,|\, \mathbf m, \mathbf \Lambda)$

with

$\mathbf \Lambda = \mathbf K_{uu}^{-1} + \mathbf K_{uu}^{-1}\mathbf K_{uf}\mathbf K_{fu}\mathbf K_{uu}^{-1} \sigma^{-2}$
$\mathbf m = \mathbf \Lambda^{-1} \mathbf K_{uu}^{-1}\mathbf K_{uf}\mathbf y\sigma^{-2}$

To make a prediction, we need to integrate:

\begin{align}\begin{aligned} p(\mathbf f^\star) = \int p(\mathbf f^\star \,|\, \mathbf u) q (\mathbf u) \textrm d \mathbf u\\ with\end{aligned}\end{align}
$p(\mathbf f^\star \,|\, \mathbf u) = \mathcal N(\mathbf f^\star\,|\, \mathbf K_{\star u}\mathbf K_{uu}^{-1}\mathbf u, \,\mathbf K_{\star\star} - \mathbf K_{\star u}\mathbf K_{uu}^{-1}\mathbf K_{u\star})$

The integral results in

$p(\mathbf f^\star) = \mathcal N(\mathbf f^\star\,|\, \mathbf K_{\star u}\mathbf K_{uu}^{-1}\mathbf m, \,\mathbf K_{\star\star} - \mathbf K_{\star u}\mathbf K_{uu}^{-1}\mathbf K_{u\star} + \mathbf K_{\star u}\mathbf K_{uu}^{-1}\mathbf \Lambda^{-1}\mathbf K_{uu}^{-1}\mathbf K_{u\star})$

Note from our above definitions that we have

$\mathbf K_{uu}^{-1}\mathbf \Lambda^{-1}\mathbf K_{uu}^{-1} = \mathbf L^{-\top}\mathbf B^{-1}\mathbf L^{-1}$

and further

$\mathbf K_{uu}^{-1}\mathbf m = \mathbf L^{-\top}\mathbf L_{\mathbf B}^{-\top}\mathbf c$

.

substituting:

$p(\mathbf f^\star) = \mathcal N(\mathbf f^\star\,|\, \mathbf K_{\star u}\mathbf L^{-\top}\mathbf L_{\mathbf B}^{-\top}\mathbf c, \,\mathbf K_{\star\star} - \mathbf K_{\star u}\mathbf L^{-1}(\mathbf I - \mathbf B^{-1})\mathbf L^{-1}\mathbf K_{u\star})$

.

The code in sgpr.py implements this equation, with an additional switch depending on whether the full covariance matrix is required.


text:
# When to use lorentz transformations vs time dilation?

1. Oct 20, 2013

### channel1

I don't understand when I should use the Lorentz transformation versus time dilation or length contraction.

I found this: http://www.phas.ubc.ca/~mav/p200/lttips.html

but it's still unclear to me...

"Length contraction applies when you are talking about a distance that is independent of time (e.g. the distance between two objects that are fixed relative to one another or the distance between two ends of a single object)."
I *think* I understand that...
- length contraction would be the distance between two planets to a moving observer
- lorentz transformation would be the distance between a planet and a moving observer after some time t
Is that correct?

"Time dilation applies when you are talking about two events that are at the same place in some frame (e.g. two ticks on the same clock)."
this one I'm just utterly lost on. can someone give me an example for time dilation and then a slightly tweaked example for a lorentz transformation? (like the ones I made for length contraction above)

2. Oct 20, 2013

### UltrafastPED

The Lorentz transformation allows you to shift the point of view between two (or more) coordinate systems - moving observer vs laboratory, etc.

Or if you are only interested in a particular time or length you can just directly use the Lorentz factor to extract that information ... based on one set of coordinates you obtain that single result as seen by the other coordinate system - but without converting everything.

3. Oct 20, 2013

### Staff: Mentor

If you have a single observer with a clock in one frame of reference sweeping past a sequence of clocks in another frame of reference, he will observe that the time exhibited on his clock is changing more slowly than the times he observes on the clocks that he sweeps past. If there are also a sequence of observers at the locations of the clocks in the other frame of reference, and they note down the time on his clock and on their own clocks as he sweeps past them, when they get together later and compare notes, they will conclude that the time on his clock was changing more slowly than the time on their clocks. Both these sets of observations describe time dilation.

In many calculations, if you have some experience and really know what you are doing, then you can confidently take time dilation into consideration and get the right answer. However, if you are less confident and experienced, you can always use the Lorentz Transformation and have a better chance of getting the right answer. I recommend doing problems both ways until you have more experience and confidence.

4. Oct 20, 2013

### channel1

...wouldn't the observer in this case see the OTHER clock moving more slowly, since in his frame it's the other frame that's moving, not him?

5. Oct 20, 2013

### channel1

ultrafast - That didn't help me, tweaked examples as referred to please.

6. Oct 21, 2013

### vela

Staff Emeritus
You can derive time dilation from the Lorentz transformations. Say you have two events separated by $\Delta t$ and $\Delta x$ in one frame and $\Delta t'$ and $\Delta x'$ in another. The Lorentz transformations tell us that
\begin{align*}
c\Delta t' &= \gamma(c\Delta t - \beta \Delta x) \\
\Delta x' &= \gamma(\Delta x - \beta c\Delta t')
\end{align*} You recover the time dilation formula $\Delta t' = \gamma \Delta t$ when $\Delta x=0$.

Say there's a moving clock. In its rest frame, two ticks occur at the same place in space. In any other frame, the time interval $\Delta t'$ will be greater than the time $\Delta t$ because $\gamma\ge 1$. In other words, observers will see the moving clock ticking more slowly than their own clocks.

7. Oct 21, 2013

### Staff: Mentor

No. He's not looking at ONE OTHER clock. He is looking at the times shown on a sequence of clocks in the other frame as he sweeps past each of them. As he goes by, if he writes down the time on his own clock and the time on the clock directly opposite him in the other frame, and does this for each of the clocks he goes past, he will find that the times displayed on the clocks in the other frame (as captured on his data sheet) will show intervals that are greater than the intervals displayed on his own clock.

Chet


text:
# Linear Algebra (Odd Distance )

I need assistance in understanding the proof of this theorem :

"There are no four points in a plane such that the distance between each pair is an odd integer".

We proof this in contradiction and assume that one of the point is zero and the remaining ones $a, b, c$ and $\| a- b \| , \|b - c\| , \|c - a\|$ are odd integers. considering euclidean length of a vector $\|x\|$. We observe that if $m$ is an odd integer, then $m ^2 = 1 \pmod 8$.

We use the cosine theorem, asserting that every two vectors $a, b \in \mathbb R^n$ satisfy $2(a, b) = \|a\|^2 + \|b\|^2 - \|a-b\|^2 = 1 \pmod 8$ and the same holds for $2(a, c)$ and $2(b,c)$ but $(a, a ) , (b , b), (c, c)$ are all equal to $1$.

My question is why is the latter so? If I can get a better explanation, that will be great for me.

-
(a,a),(b,b),(c,c) are all equal to 1 (mod 8) –  Hassan Feb 11 '12 at 17:41
yes Asaf Karagila –  Heat Feb 11 '12 at 18:46
Say "yes" to Asaf Karagila! Four more years! –  Asaf Karagila Feb 11 '12 at 19:00

Since $\|a-b\|$ is an odd integer, $\|a-b\|^2$ is congruent to 1 mod 8.

We also have that $\|a-0\|=\|a\|$ is an odd integer (remember $0$ is one of the points). So $\|a\|^2$ is also congruent to 1 mod 8 (likewise $\|b\|^2$ and $\|c\|^2$ are congruent to 1 mod 8).

Thus mod 8 we have $\|a-b\|^2-\|a\|^2-\|b\|^2=1+1-1=1$.

$\|a-b\|^2=(a-b,a-b)=(a,a-b)-(b,a-b)$ $=(a,a)-(a,b)-(b,a)-(b,b)$ $=\|a\|^2-2(a,b)+\|b\|^2$. Therefore, $2(a,b)=\|a\|^2+\|b\|^2-\|a-b\|^2$ is congruent to 1 mod 8.

Likewise, $2(a,c)=\|a\|^2+\|c\|^2-\|a-c\|^2$ and $2(b,c)=\|b\|^2+\|c\|^2-\|b-c\|^2$ are both congruent to 1 mod 8.

Then as Joriki points out there's a good bit more of the proof left.

Notice if $a=(a_1,a_2)$, $b=(b_1,b_2)$, and $c=(c_1,c_2)$ (these are points in the plane). Then we can form the matrix $A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{bmatrix}$.

$$B = A^TA = \begin{bmatrix} a_1 & a_2 \\ b_1 & b_2 \\ c_1 & c_2 \end{bmatrix} \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{bmatrix} = \begin{bmatrix} a_1^2+a_2^2 & a_1b_1+a_2b_2 & a_1c_1+a_2c_2 \\ a_1b_1+a_2b_2 & b_1^2+b_2^2 & b_1c_1+b_2c_2 \\ a_1c_1+a_2c_2 & b_1c_1+b_2c_2 & c_1^2+c_2^2 \end{bmatrix}$$

Therefore, $B = \begin{bmatrix} \|a\|^2 & (a,b) & (a,c) \\ (a,b) & \|b\|^2 & (b,c) \\ (a,c) & (b,c) & \|c\|^2 \end{bmatrix}$. Thus $2B$ is congruent to $\begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}$ mod $8$. But then $\mathrm{det}(2B)$ is congruent to $4$ mod $8$. So $\mathrm{det}(B) \not=0$ [otherwise $\mathrm{det}(B)=0$ mod $8$ and so would $\mathrm{det}(2B)$]. Therefore, $B$ is invertible. But this is impossible since $B=A^TA$ which has rank at most 2 (since rank of $B$ is no more than rank $A$ which is at most $2$).

-
perfect... Bill. That explain my worries. –  Heat Feb 11 '12 at 19:50

It seems you somehow lost the second part of the proof, which you can find here. There's a lot more to it than what you reproduced. As Asaf remarked, $(a,a)=(b,b)=(c,c)=1$ holds only $\bmod8$.

-
I made no such remark! I just made $\LaTeX$ improvements. –  Asaf Karagila Feb 11 '12 at 19:04
@ Joriki, yes, I'm aware of that but all the same , thank you.. –  Heat Feb 11 '12 at 19:52
@Asaf: Sorry -- I saw "yes Asaf" and wrongly inferred that the comment above it was yours :-) –  joriki Feb 11 '12 at 20:36