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## Feireisl, Eduard

Compute Distance To:
 Author ID: feireisl.eduard Published as: Feireisl, Eduard; Feireisl, E.; Feireisel, Eduard Homepage: http://www.math.cas.cz/homepage/main_page.php?id_membre=37 External Links: MGP · ORCID · Wikidata · Math-Net.Ru · dblp · GND · IdRef · theses.fr
 Documents Indexed: 331 Publications since 1987, including 8 Books 11 Contributions as Editor · 1 Further Contribution Reviewing Activity: 82 Reviews Co-Authors: 109 Co-Authors with 241 Joint Publications 3,055 Co-Co-Authors
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### Co-Authors

 100 single-authored 57 Novotný, Antonín 41 Petzeltová, Hana 18 Nečasová, Šárka 13 Hofmanová, Martina 12 Rocca, Elisabetta 11 Breit, Dominic 11 Málek, Josef 10 Schimperna, Giulio 9 Kreml, Ondřej 9 Lukáčová-Medvid’ová, Mária 8 Laurençot, Philippe 8 Sun, Yongzhong 7 Simondon, Frédérique 6 Ducomet, Bernard 6 Lu, Yong 6 Mizerová, Hana 6 She, Bangwei 5 Březina, Jan 5 Bucur, Dorin 5 Donatelli, Donatella 5 Takáč, Peter 4 Bella, Peter 4 Bulíček, Miroslav 4 Cimatti, Giovanni 4 Gwiazda, Piotr 4 Hilhorst, Danielle 4 Karper, Trygve G. 4 Petcu, Madalina 4 Pražák, Dalibor 4 Stebel, Jan 4 Straškraba, Ivan 4 Świerczewska-Gwiazda, Agnieszka 4 Zarnescu, Arghir Dani 3 Abbatiello, Anna 3 Aizicovici, Sergiu 3 Chiodaroli, Elisabetta 3 Ciuperca, Ionel Sorin 3 Dafermos, Constantine M. 3 Fanelli, Francesco 3 Hošek, Radim 3 Issard-Roch, Françoise 3 Jai, Mohammed 3 Jin, Bumja 3 Mácha, Václav 3 Markfelder, Simon 3 Neustupa, Jiří 3 O’Dowd, Geoffrey 3 Pokorný, Milan 3 Rokyta, Mirko 3 Schonbek, Maria Elena 3 Wiedemann, Emil 2 Dai, Mimi 2 Gallagher, Isabelle 2 Herrmann, Leopold 2 Maltese, David 2 Maslowski, Bohdan 2 Michálek, Martin 2 Mimura, Masayasu 2 Norbury, John W. 2 Petrov, Aleksandr Georgievich 2 Vasseur, Alexis F. 2 Weidenfeld, Rémi 2 Zuazua, Enrique 1 Abels, Helmut 1 Ammari, Kais 1 Ball, John MacLeod 1 Bruell, Gabriele 1 Carrillo de la Plata, José Antonio 1 De Andrade, Nirzi G. 1 Dell’Oro, Filippo 1 Diening, Lars 1 Ersoy, Mehmet 1 Escobedo Martínez, Miguel 1 Fašangová, Eva 1 Flandoli, Franco 1 Frémond, Michel 1 Friedmann, Elfriede 1 Gérard-Varet, David 1 Ghoshal, Shyam Sundar 1 Hillairet, Matthieu 1 Jana, Animesh 1 Kaplický, Petr 1 Klein, Rupert 1 Kwon, Young-Sam 1 Lewicka, Marta 1 Li, Yang 1 Marcati, Pierangelo 1 Mikelić, Andro 1 Mikyška, jiří 1 Mucha, Piotr Bogusław 1 Namlyeyeva, Yuliya V. 1 Nicaise, Serge 1 Otto, Felix 1 Petrov, Adrien 1 Poláčik, Peter 1 Poul, Lukas 1 Rautmann, Raimund 1 Růžička, Michael 1 Shvydkoy, Roman V. 1 Süli, Endre E. ...and 11 more Co-Authors
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### Serials

 20 Journal of Differential Equations 16 Archive for Rational Mechanics and Analysis 16 Communications in Partial Differential Equations 15 Mathematical Methods in the Applied Sciences 10 Discrete and Continuous Dynamical Systems 9 Journal of Mathematical Fluid Mechanics 8 Nonlinearity 8 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 8 Discrete and Continuous Dynamical Systems. Series S 8 Nonlinear Analysis. Theory, Methods & Applications 7 Communications in Mathematical Physics 7 Aplikace Matematiky 7 Applications of Mathematics 7 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 7 SIAM Journal on Mathematical Analysis 6 Czechoslovak Mathematical Journal 5 Indiana University Mathematics Journal 5 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 5 Asymptotic Analysis 5 Journal of Dynamics and Differential Equations 5 Journal of Evolution Equations 4 Annali di Matematica Pura ed Applicata. Serie Quarta 4 Comptes Rendus de l’Académie des Sciences. Série I 4 NoDEA. Nonlinear Differential Equations and Applications 4 Nonlinear Analysis. Real World Applications 3 Applicable Analysis 3 IMA Journal of Numerical Analysis 3 Applied Mathematics and Optimization 3 Mathematische Annalen 3 Journal of Integral Equations and Applications 3 Differential and Integral Equations 3 Journal de Mathématiques Pures et Appliquées. Neuvième Série 3 Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 3 Advances in Mathematical Fluid Mechanics 3 Handbook of Differential Equations 2 Journal of Statistical Physics 2 Russian Mathematical Surveys 2 Commentationes Mathematicae Universitatis Carolinae 2 Journal of the London Mathematical Society. Second Series 2 Manuscripta Mathematica 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Numerische Mathematik 2 Quarterly of Applied Mathematics 2 Physica D 2 Bollettino della Unione Matemàtica Italiana. Serie VII. A 2 Journal of Nonlinear Science 2 Calculus of Variations and Partial Differential Equations 2 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 2 Foundations of Computational Mathematics 2 Discrete and Continuous Dynamical Systems. Series B 2 Communications in Mathematical Sciences 2 Journal of Hyperbolic Differential Equations 2 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 2 Jindřich Nečas Center for Mathematical Modeling Lecture Notes 2 Nečas Center Series 1 Computers & Mathematics with Applications 1 IMA Journal of Applied Mathematics 1 Journal of Mathematical Analysis and Applications 1 Journal of Mathematical Physics 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1 Annali dell’Università di Ferrara. Nuova Serie. Sezione VII 1 Funkcialaj Ekvacioj. Serio Internacia 1 Hiroshima Mathematical Journal 1 Journal of the Mathematical Society of Japan 1 Journal für die Reine und Angewandte Mathematik 1 Meccanica 1 Monatshefte für Mathematik 1 Portugaliae Mathematica 1 SIAM Journal on Numerical Analysis 1 Transactions of the American Mathematical Society 1 Zeitschrift für Analysis und ihre Anwendungen 1 Acta Applicandae Mathematicae 1 Probability Theory and Related Fields 1 Numerical Methods for Partial Differential Equations 1 Applied Mathematics Letters 1 Journal of Scientific Computing 1 European Journal of Applied Mathematics 1 Mathematica Bohemica 1 The Annals of Applied Probability 1 Siberian Advances in Mathematics 1 Dynamic Systems and Applications 1 Matemática Aplicada e Computacional 1 Tatra Mountains Mathematical Publications 1 Methods and Applications of Analysis 1 Advances in Differential Equations 1 Acta Mathematica et Informatica Universitatis Ostraviensis 1 Revista Matemática Complutense 1 Analysis (München) 1 Communications on Pure and Applied Analysis 1 Milan Journal of Mathematics 1 Multiscale Modeling & Simulation 1 Differential Equations & Nonlinear Mechanics 1 Lecture Notes in Mathematics 1 Oxford Lecture Series in Mathematics and its Applications 1 RIMS Kôkyûroku Bessatsu 1 Analysis & PDE 1 MS&A. Modeling, Simulation and Applications 1 Science China. Mathematics 1 Annals of PDE ...and 3 more Serials
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### Fields

 324 Partial differential equations (35-XX) 222 Fluid mechanics (76-XX) 30 Classical thermodynamics, heat transfer (80-XX) 25 Mechanics of deformable solids (74-XX) 18 Dynamical systems and ergodic theory (37-XX) 18 Numerical analysis (65-XX) 11 Integral equations (45-XX) 11 Probability theory and stochastic processes (60-XX) 10 General and overarching topics; collections (00-XX) 9 Operator theory (47-XX) 7 Astronomy and astrophysics (85-XX) 5 Ordinary differential equations (34-XX) 4 Mechanics of particles and systems (70-XX) 4 Statistical mechanics, structure of matter (82-XX) 4 Biology and other natural sciences (92-XX) 4 Systems theory; control (93-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Optics, electromagnetic theory (78-XX) 2 Geophysics (86-XX) 1 Measure and integration (28-XX) 1 Functional analysis (46-XX) 1 Differential geometry (53-XX)

### Citations contained in zbMATH Open

274 Publications have been cited 4,517 times in 2,263 Documents Cited by Year
Dynamics of viscous compressible fluids. Zbl 1080.76001
Feireisl, Eduard
2004
On the existence of globally defined weak solutions to the Navier-Stokes equations. Zbl 0997.35043
Feireisl, Eduard; Novotný, Antonín; Petzeltová, Hana
2001
Singular limits in thermodynamics of viscous fluids. Zbl 1176.35126
Feireisl, Eduard; Novotný, Antonín
2009
The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars. Zbl 1113.76098
Ducomet, Bernard; Feireisl, Eduard
2006
Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. Zbl 1256.35054
Feireisl, Eduard; Jin, Bum Ja; Novotný, Antonín
2012
On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable. Zbl 1115.35096
Feireisl, Eduard
2001
Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Zbl 1285.76034
Feireisl, Eduard; Novotný, Antonín
2012
On the motion of a viscous, compressible, and heat conducting fluid. Zbl 1087.35078
Feireisl, Eduard
2004
Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids. Zbl 1248.35143
Feireisl, Eduard; Novotný, Antonín; Sun, Yongzhong
2011
On a diffuse interface model for a two-phase flow of compressible viscous fluids. Zbl 1144.35041
Abels, Helmut; Feireisl, Eduard
2008
Analysis of a phase-field model for two-phase compressible fluids. Zbl 1200.76155
Feireisl, Eduard; Petzeltová, Hana; Rocca, Elisabetta; Schimperna, Giulio
2010
Convergence for semilinear degenerate parabolic equations in several space dimensions. Zbl 0977.35069
Feireisl, Eduard; Simondon, Frédérique
2000
On the dynamics of gaseous stars. Zbl 1085.76061
Ducomet, Bernard; Feireisl, Eduard
2004
Global attractors for semilinear damped wave equations with supercritical exponent. Zbl 0819.35097
Feireisl, Eduard
1995
Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow. Zbl 0946.35058
Feireisl, Eduard; Petzeltová, Hana
1999
A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. Zbl 1167.76316
Bulíček, Miroslav; Feireisl, Eduard; Málek, Josef
2009
Compressible Navier-Stokes equations with a non-monotone pressure law. Zbl 1012.76079
Feireisl, Eduard
2002
The low Mach number limit for the full Navier-Stokes-Fourier system. Zbl 1147.76049
Feireisl, Eduard; Novotný, Antonín
2007
On convergence to equilibria for the Keller-Segel chemotaxis model. Zbl 1139.35018
Feireisl, Eduard; Laurençot, Philippe; Petzeltová, Hana
2007
On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries. Zbl 1143.35080
Bucur, Dorin; Feireisl, Eduard; Nečasová, Šárka; Wolf, Joerg
2008
On the motion of rigid bodies in a viscous compressible fluid. Zbl 1090.76061
Feireisl, Eduard
2003
Global in time weak solutions for compressible barotropic self-gravitating fluids. Zbl 1080.35068
Ducomet, Bernard; Feireisl, Eduard; Petzeltová, Hana; Straškraba, Ivan
2004
A new approach to non-isothermal models for nematic liquid crystals. Zbl 1282.76059
Feireisl, Eduard; Frémond, Michel; Rocca, Elisabetta; Schimperna, Giulio
2012
On the Navier-Stokes equations with temperature-dependent transport coefficients. Zbl 1133.35419
Feireisl, Eduard; Málek, Josef
2006
Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent. Zbl 0815.35069
Feireisl, Eduard; Zuazua, Enrique
1993
Long-time convergence of solutions to a phase-field system. Zbl 0984.35026
Aizicovici, Sergiu; Feireisl, Eduard; Issard-Roch, Françoise
2001
Dissipative measure-valued solutions to the compressible Navier-Stokes system. Zbl 1360.35143
Feireisl, Eduard; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka; Wiedemann, Emil
2016
On the motion of rigid bodies in a viscous incompressible fluid. Zbl 1039.76071
Feireisl, Eduard
2003
On integrability up to the boundary of the weak solutions of the Navier-Stokes equations of compressible flow. Zbl 0966.35096
Feireisl, Eduard; Petzeltová, Hana
2000
Well/ill posedness for the Euler-Korteweg-Poisson system and related problems. Zbl 1326.35253
Donatelli, Donatella; Feireisl, Eduard; Marcati, Pierangelo
2015
Long-time stabilization of solutions to a phase-field model with memory. Zbl 0973.35037
Aizicovici, Sergiu; Feireisl, Eduard
2001
On a model in radiation hydrodynamics. Zbl 1328.76074
Ducomet, Bernard; Feireisl, Eduard; Nečasová, Šárka
2011
Regularity and energy conservation for the compressible Euler equations. Zbl 1365.35113
Feireisl, Eduard; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka; Wiedemann, Emil
2017
On the domain dependence of solutions to the compressible Navier-Stokes equations of a barotropic fluid. Zbl 0996.35051
Feireisl, Eduard; Novotný, Antonín; Petzeltová, Hana
2002
Analysis of a diffuse interface model of multispecies tumor growth. Zbl 1367.35185
Dai, Mimi; Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Schonbek, Maria E.
2017
Boundary behavior of viscous fluids: Influence of wall roughness and friction-driven boundary conditions. Zbl 1273.76073
Bucur, Dorin; Feireisl, Eduard; Nečasová, Šárka
2010
Multicomponent reactive flows: global-in-time existence for large data. Zbl 1323.76091
Feireisl, Eduard; Petzeltová, Hana; Trivisa, Konstantina
2008
On a non-isothermal model for nematic liquid crystals. Zbl 1372.76011
Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio
2011
A singular limit for compressible rotating fluids. Zbl 1263.76077
Feireisl, Eduard; Gallagher, Isabelle; Novotný, Antonín
2012
Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities. Zbl 0784.35117
Feireisl, Eduard; Norbury, John
1991
On the weak solutions to the equations of a compressible heat conducting gas. Zbl 1315.35160
Chiodaroli, Elisabetta; Feireisl, Eduard; Kreml, Ondřej
2015
Attractors for semilinear damped wave equations on $$\mathbb{R}^ 3$$. Zbl 0819.35096
Feireisl, Eduard
1994
Inviscid incompressible limits of the full Navier-Stokes-Fourier system. Zbl 1291.35178
Feireisl, Eduard; Novotný, Antonín
2013
Multi-scale analysis of compressible viscous and rotating fluids. Zbl 1251.35069
Feireisl, Eduard; Gallagher, Isabelle; Gerard-Varet, David; Novotný, Antonín
2012
Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential. Zbl 1330.76012
Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Zarnescu, Arghir
2014
Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent. Zbl 0838.35078
Feireisl, Eduard
1995
Global attractors for degenerate parabolic equations on unbounded domains. Zbl 0862.35058
Feireisl, E.; Laurençot, Ph.; Simondon, F.
1996
Stochastically forced compressible fluid flows. Zbl 1387.76001
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2018
A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems. Zbl 1062.35152
Feireisl, Eduard; Issard-Roch, Françoise; Petzeltová, Hana
2004
Mathematical theory of compressible, viscous, and heat conducting fluids. Zbl 1122.76075
Feireisl, Eduard
2007
Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy. Zbl 1446.76067
Feireisl, Eduard; Schimperna, Giulio; Rocca, Elisabetta; Zarnescu, Arghir
2015
Uniqueness of rarefaction waves in multidimensional compressible Euler system. Zbl 1327.35230
Feireisl, Eduard; Kreml, Ondřej
2015
Singular limits in thermodynamics of viscous fluids. 2nd edition. Zbl 1432.76002
Feireisl, Eduard; Novotný, Antonín
2017
Bounded, locally compact global attractors for semilinear damped wave equations on $$\mathbb{R}^ N$$. Zbl 0858.35084
Feireisl, Eduard
1996
Existence of solutions to a phase transition model with microscopic movements. Zbl 1194.35210
Feireisl, Eduard; Petzeltová, Hana; Rocca, Elisabetta
2009
On the motion of a viscous compressible fluid driven by a time-periodic external force. Zbl 0937.35131
Feireisl, Eduard; Matušů-Nečasová, Šárka; Petzeltová, Hana; Straškraba, Ivan
1999
Local strong solutions to the stochastic compressible Navier-Stokes system. Zbl 1390.60226
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2018
On the motion of several rigid bodies in an incompressible non-Newtonian fluid. Zbl 1166.35362
Feireisl, Eduard; Hillairet, Matthieu; Nečasová, Šárka
2008
The Oberbeck-Boussinesq approximation as a singular limit of the full Navier-Stokes-Fourier system. Zbl 1214.76008
Feireisl, Eduard; Novotný, Antonin
2009
Time-periodic solutions to the full Navier-Stokes-Fourier system. Zbl 1287.76187
Feireisl, Eduard; Mucha, Piotr B.; Novotný, Antonín; Pokorný, Milan
2012
Dimension reduction for compressible viscous fluids. Zbl 1306.35093
Bella, Peter; Feireisl, Eduard; Novotný, Antonín
2014
Maximal dissipation and well-posedness for the compressible Euler system. Zbl 1308.35190
Feireisl, Eduard
2014
Structure of periodic solutions and asymptotic behavior for time-periodic reaction-diffusion equations on $$\mathbb{R}$$. Zbl 0987.35079
Feireisl, Eduard; Poláčik, Peter
2000
Convergence of a Brinkman-type penalization for compressible fluid flows. Zbl 1206.35194
Feireisl, Eduard; Neustupa, Jiří; Stebel, Jan
2011
Measure-valued solutions to the complete Euler system. Zbl 1408.35134
Březina, Jan; Feireisl, Eduard
2018
On a simple model of reacting compressible flows arising in astrophysics. Zbl 1130.35108
Feireisl, Eduard; Novotný, Antonín
2005
On the motion of rigid bodies in a viscous fluid. Zbl 1090.35137
Feireisl, Eduard
2002
On the zero-velocity-limit solutions to the Navier-Stokes equations of compressible flow. Zbl 0910.35097
Feireisl, Eduard; Petzeltová, Hana
1998
Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations. Zbl 0879.35023
Feireisl, E.; Petzeltová, H.
1997
Scale analysis of a hydrodynamic model of plasma. Zbl 1308.76339
Donatelli, Donatella; Feireisl, Eduard; Novotný, Antonín
2015
Compact attractors for reaction-diffusion equations in $$\mathbb{R}^ N$$. Zbl 0806.35075
Feireisl, Eduard; Laurençot, Philippe; Simondon, Frédérique; Touré, Hamidou
1994
On the asymptotic limit of flows past a ribbed boundary. Zbl 1189.35219
Bucur, Dorin; Feireisl, Eduard; Nečasová, Šárka
2008
Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains. Zbl 1256.35055
Feireisl, Eduard; Kreml, Ondřej; Nečasová, Šárka; Neustupa, Jiří; Stebel, Jan
2013
The long-time behaviour of solutions to parabolic problems on unbounded intervals: The influence of boundary conditions. Zbl 0933.35101
Fašangová, Eva; Feireisl, Eduard
1999
On the incompressible limit for the Navier-Stokes-Fourier system in domains with wavy bottoms. Zbl 1158.35072
Feireisl, Eduard; Novotný, Antonín; Petzeltová, Hana
2008
Weak solutions to problems involving inviscid fluids. Zbl 1371.35204
Feireisl, Eduard
2016
Stability of the isentropic Riemann solutions of the full multidimensional Euler system. Zbl 1325.35148
Feireisl, E.; Kreml, O.; Vasseur, A.
2015
Scale interactions in compressible rotating fluids. Zbl 1302.76200
Feireisl, Eduard; Novotný, Antonín
2014
On singular limits arising in the scale analysis of stratified fluid flows. Zbl 1339.35210
Feireisl, Eduard; Klein, Rupert; Novotný, Antonín; Zatorska, Ewelina
2016
Compressible fluids driven by stochastic forcing: the relative energy inequality and applications. Zbl 1362.35228
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2017
Global existence for compressible barotropic self-gravitating fluids. (Existence globale pour un fluide barotrope autogravitant.) Zbl 0986.35086
Ducomet, Bernard; Feireisl, Eduard; Petzeltová, Hana; Straškraba, Ivan
2001
Long-time stabilization of solutions to the Ginzburg-Landau equations of superconductivity. Zbl 0993.35045
Feireisl, Eduard; Takáč, Peter
2001
Time periodic solutions to a semilinear beam equation. Zbl 0657.35089
Feireisl, Eduard
1988
Convergence of a mixed finite element-finite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions. Zbl 1402.65096
Feireisl, Eduard; Lukáčová-Medvid’ová, Mária
2018
On a hyperbolic system arising in liquid crystals modeling. Zbl 1388.76023
Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Zarnescu, Arghir
2018
On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Zbl 1194.35304
Donatelli, Donatella; Feireisl, Eduard; Novotný, Antonín
2010
Compressible fluid flows driven by stochastic forcing. Zbl 1336.35374
Feireisl, Eduard; Maslowski, Bohdan; Novotný, Antonín
2013
Incompressible limit for compressible fluids with stochastic forcing. Zbl 1360.35137
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2016
Attractors for wave equations with nonlinear dissipation and critical exponent. Zbl 0792.35123
Feireisl, Eduard
1992
On oscillatory solutions to the complete Euler system. Zbl 1437.35557
Feireisl, Eduard; Klingenberg, Christian; Kreml, Ondřej; Markfelder, Simon
2020
Dissipative solutions and semiflow selection for the complete Euler system. Zbl 1441.35188
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2020
The incompressible limit of the full Navier-Stokes-Fourier system on domains with rough boundaries. Zbl 1162.76048
Bucur, Dorin; Feireisl, Eduard
2009
Relative entropies in thermodynamics of complete fluid systems. Zbl 1245.80002
Feireisl, Eduard
2012
Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains. Zbl 1284.35310
Feireisl, Eduard; Novotny, Antonin; Sun, Yongzhong
2014
New perspectives in fluid dynamics: mathematical analysis of a model proposed by Howard Brenner. Zbl 1221.35276
Feireisl, Eduard; Vasseur, Alexis
2010
Multiple scales and singular limits for compressible rotating fluids with general initial data. Zbl 1307.35220
Feireisl, Eduard; Novotný, Antonín
2014
A note on uniqueness for parabolic problems with discontinuous nonlinearities. Zbl 0736.35060
Feireisl, Eduard
1991
Influence of wall roughness on the slip behaviour of viscous fluids. Zbl 1151.76004
Bucur, Dorin; Feireisl, Eduard; Nečasová, Šárka
2008
Mathematical theory of compressible viscous fluids. Analysis and numerics. Zbl 1356.76001
Feireisl, Eduard; Karper, Trygve G.; Pokorný, Milan
2016
On the existence of periodic solutions of a semilinear wave equation with a superlinear forcing term. Zbl 0665.35050
Feireisl, Eduard
1988
Generalized solutions to models of compressible viscous fluids. Zbl 1460.35274
Abbatiello, Anna; Feireisl, Eduard; Novotný, Antoní
2021
Numerical analysis of compressible fluid flows (to appear). Zbl 07349496
Feireisl, Eduard; Lukáčová-Medviďová, Mária; Mizerová, Hana; She, Bangwei
2021
On the long-time behavior of dissipative solutions to models of non-Newtonian compressible fluids. Zbl 1472.35294
Feireisl, Eduard; Kwon, Young-Sam; Novotný, Antonín
2021
Weak-strong uniqueness property for models of compressible viscous fluids near vacuum. Zbl 1479.35609
Feireisl, Eduard; Novotný, Antonín
2021
Ill posedness for the full Euler system driven by multiplicative white noise. Zbl 07403319
Chiodaroli, Elisabetta; Feireisl, Eduard; Flandoli, Franco
2021
Stability of planar rarefaction waves under general viscosity perturbation of the isentropic Euler system. Zbl 07411896
Feireisl, Eduard; Novotný, Antonín
2021
A diffuse interface model of a two-phase flow with thermal fluctuations. Zbl 1473.60097
2021
On oscillatory solutions to the complete Euler system. Zbl 1437.35557
Feireisl, Eduard; Klingenberg, Christian; Kreml, Ondřej; Markfelder, Simon
2020
Dissipative solutions and semiflow selection for the complete Euler system. Zbl 1441.35188
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2020
Solution semiflow to the isentropic Euler system. Zbl 1441.35164
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2020
On a class of generalized solutions to equations describing incompressible viscous fluids. Zbl 1434.35082
Abbatiello, Anna; Feireisl, Eduard
2020
A finite volume scheme for the Euler system inspired by the two velocities approach. Zbl 1435.65137
Feireisl, Eduard; Lukáčová-Medvid’Ová, Mária; Mizerová, Hana
2020
On solvability and ill-posedness of the compressible Euler system subject to stochastic forces. Zbl 1435.35289
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2020
On global-in-time weak solutions to the magnetohydrodynamic system of compressible inviscid fluids. Zbl 1425.35150
Feireisl, Eduard; Li, Yang
2020
$$\mathscr{K}$$-convergence as a new tool in numerical analysis. Zbl 1465.65045
Feireisl, Eduard; Lukáčová-Medviďová, Mária; Mizerová, Hana
2020
On convergence of approximate solutions to the compressible Euler system. Zbl 1448.35375
Feireisl, Eduard; Hofmanová, Martina
2020
On the density of “wild” initial data for the compressible Euler system. Zbl 1448.76126
Feireisl, Eduard; Klingenberg, Christian; Markfelder, Simon
2020
Generalized solutions to models of inviscid fluids. Zbl 1452.35136
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2020
Convergence of finite volume schemes for the Euler equations via dissipative measure-valued solutions. Zbl 1447.65050
Feireisl, Eduard; Lukáčová-Medvid’ová, Mária; Mizerová, Hana
2020
Statistical solutions to the barotropic Navier-Stokes system. Zbl 1455.35172
Fanelli, Francesco; Feireisl, Eduard
2020
Stability of strong solutions to the Navier-Stokes-Fourier system. Zbl 1439.35365
Březina, Jan; Feireisl, Eduard; Novotný, Antonín
2020
Stochastic Navier-Stokes-Fourier equations. Zbl 1452.35265
Breit, Dominic; Feireisl, Eduard
2020
Stationary solutions of the Navier-Stokes-Fourier system in planar domains with impermeable boundary. Zbl 1446.35122
Ciuperca, Ionel Sorin; Feireisl, Eduard; Jai, Mohammed; Petrov, Adrien
2020
Globally bounded trajectories for the barotropic Navier-Stokes system with general boundary conditions. Zbl 1462.35239
Březina, Jan; Feireisl, Eduard; Novotný, Antonín
2020
On the low Mach number limit for the compressible Euler system. Zbl 1420.35209
Feireisl, Eduard; Klingenberg, Christian; Markfelder, Simon
2019
On weak-strong uniqueness for the compressible Navier-Stokes system with non-monotone pressure law. Zbl 1416.35184
Feireisl, Eduard
2019
Stability of strong solutions for a model of incompressible two-phase flow under thermal fluctuations. Zbl 1416.35204
2019
Stationary solutions to the compressible Navier-Stokes system driven by stochastic forces. Zbl 1447.35241
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina; Maslowski, Bohdan
2019
Convergence of a finite volume scheme for the compressible Navier-Stokes system. Zbl 1447.35243
Feireisl, Eduard; Lukáčová-Medvid’ová, Mária; Mizerová, Hana; She, Bangwei
2019
On uniqueness of dissipative solutions to the isentropic Euler system. Zbl 1428.35325
Feireisl, Eduard; Ghoshal, Shyam Sundar; Jana, Animesh
2019
On a singular limit for the stratified compressible Euler system. Zbl 1467.35251
Feireisl, Eduard; Tang, Tong
2019
Relative energy approach to a diffuse interface model of a compressible two-phase flow. Zbl 1420.35185
Feireisl, Eduard; Petcu, Madalina; Pražák, Dalibor
2019
On a class of compressible viscoelastic rate-type fluids with stress-diffusion. Zbl 1423.76032
Bulíček, Miroslav; Feireisl, Eduard; Málek, Josef
2019
Erratum and corrigendum to: “Existence of solutions to some models of phase changes with microscopic movements”. Zbl 1430.35118
Feireisl, Eduard; Petzeltová, Hana; Rocca, Elisabetta
2019
Stochastically forced compressible fluid flows. Zbl 1387.76001
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2018
Local strong solutions to the stochastic compressible Navier-Stokes system. Zbl 1390.60226
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2018
Measure-valued solutions to the complete Euler system. Zbl 1408.35134
Březina, Jan; Feireisl, Eduard
2018
Convergence of a mixed finite element-finite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions. Zbl 1402.65096
Feireisl, Eduard; Lukáčová-Medvid’ová, Mária
2018
On a hyperbolic system arising in liquid crystals modeling. Zbl 1388.76023
Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Zarnescu, Arghir
2018
Stationary solutions to the compressible Navier-Stokes system with general boundary conditions. Zbl 1405.35138
Feireisl, Eduard; Novotný, Antonín
2018
Measure-valued solutions to the complete Euler system revisited. Zbl 1394.35336
Březina, Jan; Feireisl, Eduard
2018
Weak-strong uniqueness for the compressible Navier-Stokes equations with a hard-sphere pressure law. Zbl 1402.35222
Feireisl, Eduard; Lu, Yong; Novotný, Antonín
2018
Asymptotic preserving error estimates for numerical solutions of compressible Navier-Stokes equations in the low Mach number regime. Zbl 1397.35171
Feireisl, Eduard; Lukáčová-Medviďová, Mária; Nečasová, Šárka; Novotný, Antonín; She, Bangwei
2018
Analysis of the adiabatic piston problem via methods of continuum mechanics. Zbl 1393.35168
Feireisl, Eduard; Mácha, Václav; Nečasová, Šárka; Tucsnak, Marius
2018
On a singular limit for stratified compressible fluids. Zbl 1406.35243
Bruell, Gabriele; Feireisl, Eduard
2018
A rigorous derivation of the stationary compressible Reynolds equation via the Navier-Stokes equations. Zbl 1393.35142
Ciuperca, I. S.; Feireisl, E.; Jai, M.; Petrov, A.
2018
Regularity and energy conservation for the compressible Euler equations. Zbl 1365.35113
Feireisl, Eduard; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka; Wiedemann, Emil
2017
Analysis of a diffuse interface model of multispecies tumor growth. Zbl 1367.35185
Dai, Mimi; Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Schonbek, Maria E.
2017
Singular limits in thermodynamics of viscous fluids. 2nd edition. Zbl 1432.76002
Feireisl, Eduard; Novotný, Antonín
2017
Compressible fluids driven by stochastic forcing: the relative energy inequality and applications. Zbl 1362.35228
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2017
Weak solutions for Euler systems with non-local interactions. Zbl 1375.35341
Carrillo, José A.; Feireisl, Eduard; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka
2017
$${\mathcal {A}}$$-free rigidity and applications to the compressible Euler system. Zbl 1382.35201
Chiodaroli, Elisabetta; Feireisl, Eduard; Kreml, Ondřej; Wiedemann, Emil
2017
The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier-Stokes system. Zbl 1375.35026
Diening, Lars; Feireisl, Eduard; Lu, Yong
2017
Error estimates for a numerical method for the compressible Navier-Stokes system on sufficiently smooth domains. Zbl 1360.35144
Feireisl, Eduard; Hošek, Radim; Maltese, David; Novotný, Antonín
2017
Transition to thermohydrodynamic lubrication problem. Zbl 1364.35235
Ciuperca, I. S.; Feireisl, E.; Jai, M.; Petrov, A.
2017
An anelastic approximation arising in astrophysics. Zbl 1379.35320
Donatelli, Donatella; Feireisl, Eduard
2017
Unconditional convergence and error estimates for bounded numerical solutions of the barotropic Navier-Stokes system. Zbl 1439.76067
Feireisl, Eduard; Hošek, Radim; Maltese, David; Novotný, Antonín
2017
On the motion of chemically reacting fluids through porous medium. Zbl 1391.76720
Feireisl, Eduard; Mikyška, Jiří; Petzeltová, Hana; Takáč, Peter
2017
Dissipative measure-valued solutions to the compressible Navier-Stokes system. Zbl 1360.35143
Feireisl, Eduard; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka; Wiedemann, Emil
2016
Weak solutions to problems involving inviscid fluids. Zbl 1371.35204
Feireisl, Eduard
2016
On singular limits arising in the scale analysis of stratified fluid flows. Zbl 1339.35210
Feireisl, Eduard; Klein, Rupert; Novotný, Antonín; Zatorska, Ewelina
2016
Incompressible limit for compressible fluids with stochastic forcing. Zbl 1360.35137
Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina
2016
Mathematical theory of compressible viscous fluids. Analysis and numerics. Zbl 1356.76001
Feireisl, Eduard; Karper, Trygve G.; Pokorný, Milan
2016
Homogenization of the evolutionary Navier-Stokes system. Zbl 1338.35332
Feireisl, Eduard; Namlyeyeva, Yuliya; Nečasová, Šárka
2016
A convergent numerical method for the Navier-Stokes-Fourier system. Zbl 1433.76107
Feireisl, Eduard; Karper, Trygve; Novotný, Antonín
2016
On asymptotic isotropy for a hydrodynamic model of liquid crystals. Zbl 1345.35073
Dai, Mimi; Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Schonbek, Maria E.
2016
Convergence of a numerical method for the compressible Navier-Stokes system on general domains. Zbl 1388.76131
Feireisl, Eduard; Karper, Trygve; Michálek, Martin
2016
On weak solutions to the 2D Savage-Hutter model of the motion of a gravity-driven avalanche flow. Zbl 1346.35196
Feireisl, Eduard; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka
2016
Vanishing dissipation limit for the Navier-Stokes-Fourier system. Zbl 1353.35225
Feireisl, Eduard
2016
Dissipative weak solutions to compressible Navier-Stokes-Fokker-Planck systems with variable viscosity coefficients. Zbl 1433.35270
Feireisl, Eduard; Lu, Yong; Süli, Endre
2016
On weak solutions to a diffuse interface model of a binary mixture of compressible fluids. Zbl 1342.35249
Feireisl, Eduard
2016
A convergent numerical method for the full Navier-Stokes-Fourier system in smooth physical domains. Zbl 1457.65070
Feireisl, Eduard; Hošek, Radim; Michálek, Martin
2016
Conditional regularity of very weak solutions to the Navier-Stokes-Fourier system. Zbl 1349.35269
Feireisl, Eduard; Sun, Yongzhong
2016
On the low Mach number limit of compressible flows in exterior moving domains. Zbl 1369.35058
Feireisl, Eduard; Kreml, Ondřej; Mácha, Václav; Nečasová, Šárka
2016
Mathematical analysis of variable density flows in porous media. Zbl 1376.35090
Feireisl, Eduard; Hilhorst, Danielle; Petzeltová, Hana; Takáč, Peter
2016
On the motion of viscous, compressible, and heat-conducting liquids. Zbl 1346.76178
Feireisl, Eduard; Novotný, Antonín; Sun, Yongzhong
2016
A rigorous justification of the Euler and Navier-Stokes equations with geometric effects. Zbl 1406.35258
Bella, Peter; Feireisl, Eduard; Lewicka, Marta; Novotný, Antonín
2016
Well/ill posedness for the Euler-Korteweg-Poisson system and related problems. Zbl 1326.35253
Donatelli, Donatella; Feireisl, Eduard; Marcati, Pierangelo
2015
On the weak solutions to the equations of a compressible heat conducting gas. Zbl 1315.35160
Chiodaroli, Elisabetta; Feireisl, Eduard; Kreml, Ondřej
2015
Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy. Zbl 1446.76067
Feireisl, Eduard; Schimperna, Giulio; Rocca, Elisabetta; Zarnescu, Arghir
2015
Uniqueness of rarefaction waves in multidimensional compressible Euler system. Zbl 1327.35230
Feireisl, Eduard; Kreml, Ondřej
2015
Scale analysis of a hydrodynamic model of plasma. Zbl 1308.76339
Donatelli, Donatella; Feireisl, Eduard; Novotný, Antonín
2015
Stability of the isentropic Riemann solutions of the full multidimensional Euler system. Zbl 1325.35148
Feireisl, E.; Kreml, O.; Vasseur, A.
2015
Homogenization of stationary Navier-Stokes equations in domains with tiny holes. Zbl 1330.35026
Feireisl, Eduard; Lu, Yong
2015
Global weak solutions to a class of non-Newtonian compressible fluids. Zbl 1335.35180
Feireisl, E.; Liao, X.; Málek, J.
2015
Robustness of strong solutions to the compressible Navier-Stokes system. Zbl 1326.35234
Bella, Peter; Feireisl, Eduard; Jin, Bum Ja; Novotný, Antonín
2015
On the energy inequality for weak solutions to the Navier-Stokes equations of compressible fluids on unbounded domains. Zbl 1328.35148
Dell’Oro, Filippo; Feireisl, Eduard
2015
Asymptotic analysis of compressible, viscous and heat conducting fluids. Zbl 1335.35181
Feireisl, Eduard
2015
Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential. Zbl 1330.76012
Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Zarnescu, Arghir
2014
Dimension reduction for compressible viscous fluids. Zbl 1306.35093
Bella, Peter; Feireisl, Eduard; Novotný, Antonín
2014
Maximal dissipation and well-posedness for the compressible Euler system. Zbl 1308.35190
Feireisl, Eduard
2014
Scale interactions in compressible rotating fluids. Zbl 1302.76200
Feireisl, Eduard; Novotný, Antonín
2014
Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains. Zbl 1284.35310
Feireisl, Eduard; Novotny, Antonin; Sun, Yongzhong
2014
Multiple scales and singular limits for compressible rotating fluids with general initial data. Zbl 1307.35220
Feireisl, Eduard; Novotný, Antonín
2014
Rotating compressible fluids under strong stratification. Zbl 1367.76063
Feireisl, Eduard; Lu, Yong; Novotný, Antonín
2014
Inviscid incompressible limits of strongly stratified fluids. Zbl 1304.35540
Feireisl, Eduard; Jin, Bum Ja; Novotný, Antonín
2014
A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system. Zbl 1293.35239
Feireisl, Eduard; Novotný, Antonín; Sun, Yongzhong
2014
Inviscid incompressible limits on expanding domains. Zbl 1298.76032
Feireisl, Eduard; Nečasová, Šárka; Sun, Yongzhong
2014
Incompressible limits of fluids excited by moving boundaries. Zbl 1300.35072
Feireisl, Eduard; Kreml, Ondřej; Nečasová, Šárka; Neustupa, Jiří; Stebel, Jan
2014
Inviscid incompressible limits under mild stratification: a rigorous derivation of the Euler-Boussinesq system. Zbl 1308.76231
Feireisl, Eduard; Novotný, Antonín
2014
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text:
# Delta functional representation in response field formalism

A general way of obtaining a field-theoretical description of Langevin dynamics is via the Martin-Siggia-Rose (MSR) response fields. This is essentially just representing the identity - up to some Jacobian - in terms of a path integral that takes into account the dynamical equation of interest (as explained is section 5.1 (eq 32) of this article (title: Field theoretic methods, by Tauber)):

\begin{align} 1 &= \int \mathcal{D} [S] \prod_{x,t} \delta \left( \partial_t S(x,t) - F[S] (x,t) - \zeta (x,t) \right) \\ &= \int \mathcal{D}[S] \mathcal{D}[i\tilde{S}] \exp\left[ -\int_{x,t} \tilde{S}(x,t) \left( \partial_t S(x,t) - F (x,t) - \zeta (x,t) \right) \right], \end{align} where $$F$$ and $$\zeta$$ are the deterministic force and noise, respectively. This identity can then be used to compute averages of various quantities w.r.t. to the noise realizations.

I have two questions regarding this:

1. in going from the first line to the second line, I would expect to get something like $$\int \mathcal{D}[\tilde{S}] e^{-i\int_{x,t} \tilde{S} (\partial_t S - F - \zeta)}$$ in similarity with the 1d delta function $$\delta(x) = \int \frac{dk}{2\pi} e^{ikx}$$. But it seems that here we are using a different representation, $$\delta(x) = \int \frac{d \, ik}{2\pi}e^{-kx}$$.
I have seen it stated* that in order to ensure the convergence of the path integral, the integration needs to be performed along the imaginary axis, hence the $$\mathcal{D}[i\tilde{S}]$$ measure in the second line. I don't quite understand this statement; in particular, how is it that if the path integral has convergence issues if we integrate over real fields with complex weights (i.e., $$\int \mathcal{D}[\tilde{S}] e^{i\int \tilde{S}(\ldots)}$$), it suddenly becomes well-defined by integrating along the imaginary axis via a change of variables (i.e., $$\int \mathcal{D}[i\tilde{S}] e^{-\int \tilde{S}(\ldots)}$$) ?

2. this probably has a very simple answer, but does it matter if in the above formula I switch $$\tilde{S}$$ with $$-\tilde{S}$$ (such that we have $$e^{\int \tilde{S}(\ldots)}$$ instead of $$e^{-\int \tilde{S}(\ldots)}$$ for the weights)?

*Page 134 of "Critical Dynamics - A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior"

As written above, yes, the integral needs to be taken along the imaginary axis. The reason for writing it this way is not because it improves convergence with respect to your second representation, but rather because when finding saddle-points, one often finds that (in your original representation) $$\tilde S$$ is real [1]. So it is convenient just to omit the $$i$$ by shoving it into the measure.
OP's questions seem to boil down to the fact that $$\delta(x) ~=~ \delta(\pm x) ~=~\int_{\mathbb{R}} \frac{dk}{2\pi} e^{\pm ikx}~\stackrel{\kappa=ik}{=}~\int_{i\mathbb{R}} \frac{d\kappa}{2\pi i} e^{\pm \kappa x}.$$
• $\uparrow$ 1. No. 2. No. May 31 at 14:46

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text:
# I Maxwell's equations and tensor notation

1. Dec 7, 2016

### Incand

In one of our lectures we wrote Maxwell's equations as (with $c=1$)
$\partial_\mu F^{\mu \nu} = 4\pi J^\nu$
$\partial_\mu F_{\nu \rho} + \partial_\nu F_{\rho \mu} + \partial_\rho F_{\mu \nu} = 0$

where the E.M. tensor is
$F^{\mu \nu} = \begin{pmatrix} 0 & -B_3 & B_2 & E_1\\ B_3 & 0 & -B_1 & E_2\\ -B_2 & B_1 & 0 & E_3\\ -E_1 & -E_2 & -E_3 & 0 \end{pmatrix}$

and $J^\nu = (\mathbf J ,\rho)$,
$\partial_\mu = (\nabla, \frac{\partial}{\partial t} )$
(allowing the notation with the equal sign between the component and the 4-vector.)

Now from this I'm trying to recreate our classic Maxwell's equations.
If I set $\nu = 4$ in the first equation we get the L.H.S. as $4\pi \rho$
The R.H.S. becomes
$\partial_\mu F^{\mu 4} = -\frac{\partial}{\partial x}E_1-\frac{\partial}{\partial y}E_2-\frac{\partial}{\partial z}E_3 = -\nabla \cdot E$.
Now this is a sign error since we should get $\nabla \cdot E = 4\pi \rho$.

So I guess maybe I shouldn't put a minus sign before those but that goes against what I learned by using the Minkowski metric, $g_{\mu \nu} = diag(-1,-1,-1,1)$. Similarly I get the same sign error in the continuity equation. Should I always get + signs?

2. Dec 7, 2016

### vanhees71

You don't need the metric here, because the index of the four-gradient is already a lower index (covariant components of a vector), i.e.
$$\partial_{\mu}=\frac{\partial}{\partial x^{\mu}}.$$
So you get the correct Gauss law (in Gaussian units)
$$\vec{\nabla} \cdot \vec{E}=4\pi \rho.$$
The same is true for the continuity equation:
$$\partial_{\mu}j^{\mu}=\vec{\nabla} \cdot \vec{j}+\partial_t \rho=0.$$

3. Dec 7, 2016

### Incand

Then I end up with another question. Consider 4-momentum vectors
$P^\mu = (P_x,E_1)$
$Q^\mu=(Q_x,E_2)$
Then $(P^\mu+Q^\mu)^2= (E_1+E_2)^2-(Q_x+P_x)^2 = E_1^2+E_2^2-Q_x^2-P_x^2+2E_1E_2-2Q_xP_x$.
However we also have
$(P^\mu+Q^\mu)^2 = P^2+Q^2+2P_\mu Q^\mu = E_1^2-P_x^2+E_2^2-Q_x^2+2(E_1E_2+P_xQ_x)$

So In this case it's obviously a minus sign. Why?
So it should be something like this i guess
$(P+Q)^2 = P^2+Q^2+2g_{\mu \nu}P^\nu Q^\mu$?
But $g_{\mu \nu}P^\nu = P_{\mu}$ by lowering of index.

4. Dec 7, 2016

### Staff: Mentor

Set $U=(P+Q)$ and remember that $(P+Q)^2=U^2=g_{\mu\nu}U^{\mu}U^{\nu}$

Write out the summation explicitly if that doesn't make it clear.

5. Dec 7, 2016

### Incand

Yes that's what I did in the "first calculation".

However a lot of calculations rely on using the second approach looking at $(P+Q)^2 = P^2+Q^2+\dots$. This approach is something that showed up a lot in our class for solving dynamics problem and when deriving things like Compton scattering or threshold energies.

6. Dec 8, 2016

### Incand

Perhaps this can be thought of as it's always the normal scalar product but we have
$U^\mu = (\mathbf p , E)$ and with lowered index
$U_\mu = g_{\mu \nu} U^\nu = (-\mathbf p,E)$.
In that case the normal scalar product of these two always gives the Lorentzian scalar product.

So my confusion is only that every other 4-vector I've seen I'm used to seeing $U^\mu = (\mathbf p , E)$ and $U_\mu = (-\mathbf p,E)$ while for the differentiation operator we defined it the 3-vector part positive for the lowered index.

I guess the problem I had is that in class we always spoke of
$U_\mu U^\mu$ or $A_\mu B^\mu$ as the Lorentz scalar product.

7. Dec 8, 2016

### vanhees71

The Minkowski product is a bilinear form, and it is always good to distinguish vectors and their components although physists always talk about $V^{\mu}$ is a vector, but really right is to say that these are tensor components with respect to a pseudo-orthogonal basis with four basis vectors $\boldsymbol{e}_{\mu}$. The vector is
$$\boldsymbol{V}=V^{\mu} \boldsymbol{e}_{\mu}$$
with
$$\boldsymbol{e}_{\mu} \cdot \boldsymbol{e}_{\nu}=\eta_{\mu \nu}.$$
You have the somewhat unusual convention to have the spatial components as the first three and the temporal component as the fourth component. So in your case, written in a matrix notation you have $(\eta_{\mu \nu})=\mathrm{diag}(-1,-1,-1,1)$ (west-coast convention).

A Lorentz transformation describes the change of one pseudo-orthogonal basis to another, i.e.,
$$\boldsymbol{e}_{\mu} = {\Lambda^{\nu}}_{\mu} \boldsymbol{e}_{\nu}',$$
which implies
$$\boldsymbol{e}_{\mu} \cdot \boldsymbol{e}_{\rho}=\eta_{\mu \rho} ={\Lambda^{\nu}}_{\mu} {\Lambda^{\sigma}}_{\rho} \boldsymbol{e}_{\nu}' \cdot \boldsymbol{e}_{\sigma}'={\Lambda^{\nu}}_{\mu} {\Lambda^{\sigma}}_{\rho} \eta_{\nu \sigma}.$$
Now it is conventient to introduce also the inverse metric components via
$$(\eta^{\mu \nu})=(\eta_{\mu \nu})^{-1}=\mathrm{diag}(-1,-1,-1,1)$$
and covariant vector components and contravariant basis vectors via
$$V_{\mu} = \eta_{\mu \rho} V^{\rho}, \quad \boldsymbol{e}^{\mu}=\eta^{\mu \rho} \boldsymbol{e}_{\rho},$$
and then you have, e.g.,
$$\boldsymbol{V}=\boldsymbol{e}^{\mu} V_{\mu}=\eta^{\mu \nu} \boldsymbol{e}_{\nu} V_{\mu}$$
and
$$\boldsymbol{V} \cdot \boldsymbol{W}=\eta_{\mu \nu} V^{\mu} W^{\nu}=V^{\mu} W_{\mu}=V_{\mu} W^{\mu},$$
etc.

For the Lorentz transformation properties of the components you now get in a straight forward way
$$\boldsymbol{V}=V^{\mu} \boldsymbol{e}_{\mu} = V^{\mu} {\Lambda^{\nu}}_{\mu} \boldsymbol{e}_{\nu}',$$
and thus since the decomposition of a vector in terms of a basis is unique (i.e., the basis vectors are a linearly independent set of vectors)
$$V^{\prime \nu}={\Lambda^{\nu}}_{\mu} V^{\mu}.$$
For the covariant components you find
$$V_{\rho} ' =\eta_{\rho \nu} V^{\prime \nu}=\eta_{\rho \nu} {\Lambda^{\nu}}_{\mu} V^{\mu} = \eta_{\rho \nu} {\Lambda^{\nu}}_{\mu} \eta^{\mu \sigma} V_{\sigma} = {\Lambda_{\rho}}^{\sigma} V_{\sigma}.$$
I hope this clarifies your confusion about how to handle all the signs from the pseudo-scalar-product of Minkowski four-vector space!

8. Dec 8, 2016

### Incand

Cheers! That was very well explained and helped me a lot!


text:
• D K CHOUDHURY

Articles written in Pramana – Journal of Physics

• SU(4), broken Zweig rule and the null plane

Using Glauber’s theory of eikonal approximation, an algebraic representation of the broken Zweig rule in the constituent quark model is obtained. Null plane language is then elaborated. It is pointed out that the exotic current associated with the disconnected quark diagrams, which break the Zweig rule, can have canonical dimension.

• Bare quark masses from the transverse cut offs ine+e interactions

From the transverse cuts off of the electron positron annihilation in the SPEAR II regime, we obtain positivity domains of bare quark masses.

• Scale breaking in neutrino scattering

We consider two types of scale breaking models—one based on asymptotically free gauge theory and the other phenomenological—which have already been fitted to the electron and muon data. Their predictions for the neutrino scattering are worked out and the sensitivity on the model parameters and the input electron data studied. The effects of scale breaking at very lowQ2 and of a finite longitudinal cross-section are incorporated in analogy withep data. The predictions are compared with the recent neutrino scattering data.

• Phenomenological structure functions and Gribov-Lipatov relation

We make an analysis of the Gribov-Lipatov relation using the phenomenological forms of the structure functionF2ep. Our analysis indicate breakdown of the relation at PETRA energies. Plausible reasons of the breakdown of Gribov-Lipatov relation are discussed together with its phenomenological form.

• An analysis of a quantum chromodynamic structure function

We obtain an approximate solution of Altarelli-Parisi equations yielding a sample of quantum chromodynamic structure function. The SLAC-MIT data are analysed with it. Possible effects of intrinsic charm and higher twist are also included. Agreement is found to be good forx⩾0.25.

• Gribov-Lipatov inequality and inclusivee+e processes

We study the inclusivee+e processes at the PETRA energy range within QCD and a fixed point theory using the phenomenological Gribov-Lipatov inequality suggested in an earlier analysis. Theoretical justification is provided within QCD and its possible implication in hadronization is discussed.

• Masses in structure functions

We derive an approximate formula for the quark mass in ξ-formalism. Using the QCD-based form of structure function, we analyse the SLAC-MIT data to estimate the experimentally allowed ranges of quark mass. The possible variation of our results with QCD cut-off parameter, gluon distribution, hard intrinsic charm component of the proton wavefunction and the higher twist effects are also discussed.

• An improved approximate solution of Altarelli-Parisi equations

We have presented here an improved solution to Altarelli-Parisi equations, and it is found to be in good agreement with SLAC-MIT data up tox⩾0.1.

• Bounds on non-leading QCD effects in a quark model for inclusive reactions

We study the possible effect of QCD in the proton wave function in a quark model for inclusive processesA+B→C+X pursued by us. The assumption is the validity of the conjecture of Lepage and Brodsky in QCD on such effects. Our results obey the perturbative expectation, |Rg tan2φ|<1. Symmetric version of the model is, however, found to be at variance with most of the inclusive data as well as with some known phenomenology. If the dynamics of the underlying theory generate Regge-like symmetry breaking approximately, the model is phenomenologically viable, and the non-leading QCD effects become reasonable in size. Phenomenological necessity of the admixtures of (56, 0+)*, (70, 0+) and (56, 2+) in the nucleon wave function is also discussed in the present analysis.

• Comment on a solution of Altarelli Parisi equations

We show that the assumptions of factorizability of the structure functionF(x, t) inx andt and the equality oft evolution of the singlet and gluon distributions are not necessary to obtain the approximate solutions of Altarelli-Parisi equations derived by us recently.

• An analysis of quantum chromodynamic structure function beyond leading order

We obtain approximate solutions of Altarelli-Parisi equations beyond leading logarithmic approximation. Our results suggest quantitative utility of higher order terms in the structure function analysis of deep inelastic scattering.

• Perturbative and non-perturbative evolutions of structure functions at low-x

Approximate solutions of Altarelli-Parisi equations are obtained in low-x limit and have tested them in EMC low-x and low-Q2 data. The results are compared with the phenomenological non-perturbative evolutions. Data conform to perturbative as well as non-perturbative evolutions but do not conform to the predictions of non-linear evolution.

• Comments on structure functions at low-x

We present further analysis of the structure functions at low-x using the approximate solutions of Altarelli-Parisi equations recently reported by us. We also compare our results with non-perturbative and non-linear evolutions.

• Cross-section sum rules and inequalities in a quark model for light and heavy flavour productions

The$$qq\bar q$$ model for inclusive processes is reformulated to consider the production of heavy flavours (c, b andt) and higher order flavour exchange effects. Predictions are made in terms of sum rules and inequalities for various inclusive cross-sections. Plausible parametrization of flavour symmetry breaking is also suggested.

• Low x structure functions—HERA physics

• Perturbative and nonperturbative quark masses in structure functions

Flavour symmetry breaking is introduced in the approximate formula for quark mass reported earlier. We use SLAC-MIT as well as NMC data to estimate the up, down and strange quark masses from structure functions. Using the approximateQ2 evolutions of Altarelli-Parisi equations, we also estimate its perturbative and nonperturbative components.

• Static and dynamic properties of pseudoscalar mesons

We report the results of form factors, charge radii and decay constants of both light and heavy flavoured pseudoscalar mesons in a QCD inspired quark model. We use the quantum mechanical perturbation theory and discuss its limitations in the present problem. Several predictions are also made for bottom and top flavours.

• Static and dynamic properties of heavy light mesons in infinite mass limit

We summarize the consequences of the infinite limit of heavy quark mass in the results of form factors, charge radii and decay constants of heavy light mesons within a QCD inspired quark model recently reported.

• Root mean square radii of heavy flavoured mesons in a quantum chromodynamics potential model

We report the results of root mean square (r.m.s.) radii of heavy flavoured mesons in a QCD model with the potential $V (r) = −(4\alpha_{s}/3r) + br + c$. As the potential is not analytically solvable, we first obtain the results in the absence of confinement and Coulomb terms respectively. Confinement and Coulomb effects are then introduced successively in the approach using the Dalgarno’s method of perturbation. We explicitly consider the following two quantum mechanical aspects in the analysis: (a) The scale factor $c$ in the potential should not effect the wave function of the system even while applying the perturbation theory. (b) Choice of perturbative piece of the Hamiltonian (confinement or linear) should determine the effective radial separation between the quarks and antiquarks. The results are then compared with the available theoretical values of r.m.s. radii.

• Isgur–Wise function in a QCD-inspired potential model with WKB approximation

We use Wentzel–Kramers–Brillouin (WKB) approximation for calculating the slope and curvature of Isgur–Wise function in a QCD-inspired potential model. This work is an extension of the approximation methods to the QCD-inspired potential model. The approach hints at an effective range of distance for calculating the slope and curvature of Isgur–Wise function. Comparison is also made with those of Dalgarno method and variationallyimproved perturbation theory (VIPT) as well as other models to show the advantages of using WKB approximation.

• # Pramana – Journal of Physics

Volume 97, 2023
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
Well a workaround is to install the caffeine gnome extension https://extensions.gnome.org/extension/517/caffeine/ so that is disable dimming with the applications you choose (you can do it in the caffeine settings)
Well a workaround is to install the caffeine gnome extension https://extensions.gnome.org/extension/517/caffeine/ so that is disable it disables dimming with the applications you choose (you can do it in the caffeine settings)


text:
# How to split latex page horizontally, and have a different colour background to the two halves?

I am writing a book and would like each page to be cut in half with red at the top and blue at the bottom (not including front page of initial front book matter). and then add separate text to each of the half of the page (was thinking minipage might be able to do this? - imagine bad advice top half and good advice bottom half of the page. Is this possible?

You could use eso-pic.

The following assumes, that the half of the text field is at the same height as the half of the paper. This is not necessarily true (and is false for default book page layout, roughly half of the last line of the first half of the text field lies in the second half of the page).

We first define an environment (advisepage in the following) for a coloured page. This environment does only colour a single page (top half red, bottom half blue). In this environment we use another environment (advise in the following) that is essentially just a minipage of fixed height .5\textheight and width \textwidth.

Code:

\documentclass[]{book}

\usepackage{xcolor}
\usepackage{environ}
\usepackage{eso-pic}

\makeatletter
{%
\clearpage
{%
\AtPageUpperLeft
{%
\raisebox{-.5\paperheight}{\rule{\paperwidth}{.5\paperheight}}%
}%
\rule{\paperwidth}{.5\paperheight}%
}%
}
{%
\clearpage
}
{%
\noindent
\begin{minipage}[t][.5\textheight]{\textwidth}
}
{%
\end{minipage}%
}
\usepackage{duckuments}
%\expandafter\show\csname set@page@color\endcsname

\begin{document}
Another page
\end{document}

An option using the package background and tikz shadings, code derived from: How to change the background color of a page?.

RESULT:

MWE:

% arara: pdflatex: {synctex: yes, action: nonstopmode}
% arara: pdflatex: {synctex: yes, action: nonstopmode}

\documentclass{article}
\usepackage[bottom=2cm,top=2cm]{geometry}
\usepackage{fontawesome}
\usepackage{graphicx}
\usepackage[
pages=some,
firstpage=false,
opacity=0.8,
scale=1,
placement=center,
angle=0,
hshift=0,
vshift=0
]{background}
\usepackage{lipsum}
\usepackage{tikz}

\begin{document}
\backgroundsetup{contents={}}
\section{First page}
\lipsum[1-4]

%Second page
\newpage
\backgroundsetup{contents={
\begin{tikzpicture}
\fill[bottom color=black!90, top color=blue!20,middle color=blue!70] (current page.west) rectangle (current page.south east);
\fill[bottom color=red!50, top color=black!90,middle color=red] (current page.west) rectangle (current page.north east);
\end{tikzpicture}
}
}
\vbox{
\begin{minipage}[t][0.5\textheight][t]{\textwidth}
\begin{tikzpicture}
\draw[
draw=white,fill=white, line width =1.5pt,
path picture={
\node at (path picture bounding box.center){
\includegraphics[height=6cm]{example-image-A}
};
}
] (0,0) circle (3);
\end{tikzpicture} \par
\vspace{2\baselineskip}
\lipsum[5]
\end{minipage}

\nointerlineskip
\begin{minipage}[b][0.5\textheight][t]{\textwidth}
\vspace{3\baselineskip}
\section{Bottom page}
\lipsum[6-7]

\textcolor{blue!20}{\section{Change text color} \lipsum[8]}

\end{minipage}
}

\newpage

\vbox{
\begin{minipage}[t][0.5\textheight][t]{\textwidth}
\vspace{3\baselineskip}
\lipsum[5-7]
\end{minipage}

\nointerlineskip
\begin{minipage}[b][0.5\textheight][t]{\textwidth}
\vspace{3\baselineskip}
\lipsum[8-9]
\end{minipage}
}
\newpage

\vbox{
\begin{minipage}[t][0.5\textheight][t]{\textwidth}
\vspace{3\baselineskip}
\lipsum[5-7]
\end{minipage}

\nointerlineskip
\begin{minipage}[b][0.5\textheight][t]{\textwidth}
\vspace{3\baselineskip}
\lipsum[8-9]
\end{minipage}
}
\end{document}


text:
Quick Tip: Even Quicker Markup with Sparkup

Read Time: 3 min

Most of us are familiar with the popular Zen Coding utility by now. For those who aren't, it turns the process of creating complex markup into a CSS-like selector. This can save you an extraordinary amount of time; however, it does have a few short-comings. Luckily, another tool, Sparkup, inspired by Zen Coding, remedies these issues, and pushes your productivity even higher!

Wait...What's Zen Coding?

Consider the following markup:

This is fairly common markup that you'll create for every new website. Assuming that you don't already have this saved as a snippet, it can easily be recreated within seconds, with Zen Coding.

• Target ids and classes with the # and . selectors.
• Use the > to filter down the tree and create children elements.
• Request multiple elements by using the * symbol.
• Use the + symbol to create siblings.

Room for Improvement

Unfortunately, there are a few issues that I have with Zen Coding.

• To the best of my knowledge, there's no easy way to travel back up the tree. For example, if I create a ul > li*4 selector, I'm not aware of a way to filter back up to the ul, and then create a sibling element.
• There isn't a way to assign innerHTML to an element. Wouldn't it be great if I could type, ul>li {Some Text Here}?
• I don't believe there's a way to tab between stops, after you expand.

Luckily, a utility, inspired by Zen Coding, fixes every one of the issues listed above. It's forward compatible -- meaning, all of your Zen Coding knowledge will transfer over seamlessly.

Availability

At the time of this writing, Sparkup is available for TextMate, Vim, and for general usage, via the command line. It's fairly trivial to install. For more information, refer to the screencast above.

Climb up the Tree

With Sparkup, we can travel back up the tree.

The Syntax

Notice how we're using the < to travel back up the tree. From the anchor tag, one < will take us to the list item, and another will bring us back to the unordered list, at which point we can create a sibling with the + symbol. Very helpful!

Add Text to Elements

By using curly braces, we can assign values, or innerHTML to the elements we create. This feature was badly needed.

Tab Stops

With Sparkup, we can tab over the necessary stops, so that we can manually insert our attributes/values as quickly as possible. With MacVim, as shown in the screencast, you can use Control + N and Control + P to switch between the stops.

This way, rather than having to resort to numerous directional key presses, a single command will take you where you need to be. Refer to the screencast for an example.

So what do you think? Will you be making the switch?


text:
Here's the question you clicked on:

## Waynex On the supplemental problems for pset 8, question 4A-4, it says the force field has a magnitude of k/r. The "expression" for this force field has the scalar multiplier k/r^2. I don't understand the reasoning for r to become r^2 when k stays the same. one year ago one year ago

• This Question is Closed
1. Waynex

<-y,x> has a magnitude of: $y ^{2}+x ^{2}.$ And that magnitude is an equation for a circle. So we can represent that magnitude by $r ^{2}.$ In order to scale that magnitude to the required magnitude of:$\frac{ 1 }{ r },$ we need to multiply the vector <-y,x> by $\frac{ 1 }{ r ^{2} }.$


text:
java.net.URL类的几个细节问题。

# 不支持重定向

HTTP返回码301，即发生了重定向。可在浏览器上这个过程太快以至于我们看不到这个301界面的出现。这里需要说明的是URL.readText()是Kotlin中一个扩展函数，本质还是调用了URL类的openStream方法，部分源码如下：

# 不稳定的equals方法

Compares this URL for equality with another object.
If the given object is not a URL then this method immediately returns false.
Two URL objects are equal if they have the same protocol, reference equivalent hosts, have the same port number on the host, and the same file and fragment of the file.
Two hosts are considered equivalent if both host names can be resolved into the same IP addresses; else if either host name can’t be resolved, the host names must be equal without regard to case; or both host names equal to null.
Since hosts comparison requires name resolution, this operation is a blocking operation.
Note: The defined behavior for equals is known to be inconsistent with virtual hosting in HTTP.

sameFile源码：

# 耗时的equals方法

handler对象的hashCode()方法：


text:
# What DO We Know About High Tc?

Many papers on cuprate superconductors start out by saying that we don’t know much about them. About ten years ago, A.J. Leggett wrote down a laundry list of things we do know. Looking at this list, it seems like things are not as bleak as the introductions to papers on cuprates make them out to be! Here is his list of things we knew back in 2006:

1. Superconductivity in the cuprates is a result of Cooper pairing.
2. The main driver of superconductivity in the cuprates is the copper-oxide plane.
3. To a good approximation, Cooper pairs form independently on each layer (even in a multilayered compound) in the cuprates.
4. The net saving of energy in the transition to the superconducting state is not from ionic kinetic energy as in the classic BCS superconductors. (This surprising result in BCS superconductors was shown to be true in this paper.)
5. The spin state is a singlet.
6. The order parameter is of $d_{x^2-y^2}$ symmetry.
7. The size of Cooper pairs is somewhere between 10-30$\text{\AA}$.
8. The pairs are formed from time-reversed partners as in BCS theory.

There is one more aspect of this paper that I think is significant. Leggett stresses the importance of asking questions that are model-independent, such as, (1) What is the pairing symmetry? (2) Does the macroscopic Ginzburg-Landau theory work? and (3) Where is the energy saved in the transition to the superconducting state? Posing questions like these are the “tortoise” method versus developing microscopic theories, which are the “hare’s” method. With the existence of so many microscopic models, it seems to me that taking the “tortoise” path may yield fruit in the long term.

I have been told by a number of my peers that I apparently like lists, which is probably why this article sticks out in my mind. Between the time this article was written and the present, are there any additional items to add to the list? I’m not an expert in high-Tc, so I’m wondering if there is more we now know. Many experimental signatures spring to mind, but none that I would necessarily say that we know “for sure”.


text:
Project deliverable Open Access

# BioExcel Deliverable 3.4 - Efficacy of Feedback Processes

Carter, Adam; Harrow, Ian; Abraham, Mark; Hospital, Adam; Matser, Vera; Noack, Holger; Apostolov, Rossen

### JSON-LD (schema.org) Export

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This document is the fourth deliverable from WP3 Consultancy and User Groups. The project\u2019s structure requires that there is a good flow of feedback between end-users and the many activities spread amongst the project\u2019s work packages(WPs). It provides an opportunity to take stock and report on the efficacy of feedback in the project.
\n\n
Overall, feedback in the project is good, but it is important to continue to monitor both external and internal feedback. With the former, we need to broaden the areas in which this occurs beyond the current scope, which is focused on the project\u2019s main codes. Internal feedback processes could be more explicit to maintain good feedback as the centre grows.
\n\n
The document considers each work package in turn and depicts graphically the strength of feedback with other parts of the project and with external stakeholders. We look at the current state, a desirable future state, and then consider what could be done to reach the future state.
\n\n
Feedback relating to WP1(software)is broadly considered to be satisfactory, but there are links which could be improved, for example with WP2, the SAB, and particularly WP5. WP1 feedback directly with users through mailing lists and forums is good.
\n\n
Strengths of WP2 (portable environments/workflows) in feedback relate to collaborations with organisations like ELIXIR and Open PHACTS. Possible areas for improvement were identified for all WPs except forWP6. WP2\u2019s work is approaching the stage where end-user feedback will become much more relevant.
\n\n
WP3 (user groups & community) generally has sufficient feedback, although possible improvements relate to WP2, WP5 and other infrastructures. WP3\u2019s strengths have been in supporting feedback to other WPs through provision of forums, webinars, IGs, etc.
\n\n
WP4 (training & dissemination) feedback is broadly sufficient, but areas for improvement relate mainly to the technical WPs. Technical WP members already contribute to training, but coordination between these WPs\u2019 activities could be improved.
\n\n
WP5\u2019s feedback to date has come to a great extent through WP3 and WP4. It is hoped that in future WP5 could connect more directly to other parts of the project, but that WP3 (and WP4) will still provide the main links with end-users.WP6\u2019s feedback is sufficient. The similar future state for WP6 reflects the intention that WP6 should coordinate, but not act as a point through which most project communication and feedback should take place.
\n\n
Areas for improvement are set out in Section 3. Decisions relating to which of these improvements can be actioned will ultimately fall to WP leaders and the EB, but WP3 will continue to track work being done to improve feedback through Task 3.7, Feedback and Improvement.
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text:
# American Institute of Mathematical Sciences

April  2010, 4(2): 393-418. doi: 10.3934/jmd.2010.4.393

## Dynamics of the Teichmüller flow on compact invariant sets

 1 Mathematisches Institut der Rheinischen Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Received  May 2010 Published  August 2010

Let $S$ be an oriented surface of genus $g\geq 0$ with $m\geq 0$ punctures and $3g-3+m\geq 2$. For a compact subset $K$ of the moduli space of area-one holomorphic quadratic differentials for $S$, let $\delta(K)$ be the asymptotic growth rate of the number of periodic orbits for the Teichmüller flow $\Phi^t$ which are contained in $K$. We relate $\delta(K)$ to the topological entropy of the restriction of $\Phi^t$ to $K$. Moreover, we show that sup$_K\delta(K)=6g-6+2m$.
Citation: Ursula Hamenstädt. Dynamics of the Teichmüller flow on compact invariant sets. Journal of Modern Dynamics, 2010, 4 (2) : 393-418. doi: 10.3934/jmd.2010.4.393
 [1] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [2] Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

2019 Impact Factor: 0.465


text:
# $k$-coloring problem which minimizes the number of conflicting vertices

Classical $k$-coloring problem (k-GCP) is to assign a color selected from $k$ colors to each vertex of graph $G$ so that the number of conflicting edges (the edges with same color endpoints) is minimized. I am interested in the algorithms that obtain the largest subset of vertices that are legal colored.

Two major differences of my problem

1. The number of conflicting vertices is minimized instead of the number of conflicting edges.
2. The graph may not be $k$-colorable.

Is there any algorithms that can solve my problem?

• This question seems a little too general? – muzzlator Jun 9 '14 at 16:12
• If you want an answer which works for ALL graphs, the answer is $|V|-k$. – N. S. Jun 9 '14 at 16:16
• For example, if $G$ is complete and $k=\mid V\mid$, then the number you are looking for is $0$. – Jika Jun 9 '14 at 16:16
• I agree - more constraints on $G$ and $k$ are needed to give a satisfying answer. If, for example $G$ is $k$-colorable, then 0 vertices need to be deleted. But if $G$ is a complete graph on more than $k$ vertices, then you must delete $|V(G)|-k$ vertices. – Perry Elliott-Iverson Jun 9 '14 at 16:18


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# Chapter P - Prerequisites: Fundamental Concepts of Algebra - Concept and Vocabulary Check: 8

$(3x + 4)$

#### Work Step by Step

To find equivalent expressions of fractions, we can only multiply by $1$, and any fraction that solves into $1$. Therefore, to find an equivalent expression for $\frac{3x + 2}{x - 5}$ with a denominator $(3x + 4)(x - 5)$, we multiply by $1$: $$\frac{3x + 2}{x - 5} \times 1 = \frac{3x + 2}{x - 5} \times \frac{3x + 4}{3x + 4}$$ $= \frac{(3x + 2)(3x + 4)}{(x - 5)(3x + 4)}$ which satisfies the conditions of the exercise.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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# What is the sum of the integers from 100 to 200 inclusive

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What is the sum of the integers from 100 to 200 inclusive  [#permalink]

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28 Aug 2018, 02:53
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What is the sum of the integers from 100 to 200 inclusive which are divisible from 6 but not from 12?

1) 1200
2) 1350
3) 1850
4) 2550
5) 2650

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What is the sum of the integers from 100 to 200 inclusive  [#permalink]

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28 Aug 2018, 04:14
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Afc0892 wrote:
What is the sum of the integers from 100 to 200 inclusive which are divisible from 6 but not from 12?

1) 1200
2) 1350
3) 1850
4) 2550
5) 2650

What is the sum of the integers from 100 to 200 inclusive which are divisible by 6 but not by 12?

If you observe the pattern of the numbers that are divisible by 6 but not by 12 in the given range, they lie at a distance of 12 from each other consecutively.

So the series of the terms starts at 102 and ends at 198, they form an AP with common difference of 12.

$$t_{n}=a+(n-1)*d=102+(n-1)*12$$
Or, 198=102+(n-1)*12
Or, n=9

Sum, $$S_{9}=\frac{n}{2}*(a+l)$$=$$\frac{9}{2}*(102+198)=\frac{9}{2}*300=9*150=1350$$

Ans. (B)
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Re: What is the sum of the integers from 100 to 200 inclusive  [#permalink]

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28 Aug 2018, 04:27
Afc0892 wrote:
What is the sum of the integers from 100 to 200 inclusive which are divisible from 6 but not from 12?

1) 1200
2) 1350
3) 1850
4) 2550
5) 2650

We get the following numbers
102, 114,126...

a common difference of 12

a1 =102
tn =200
200 = 102 + (n-1) 12
98 =12n -12
n = 110/12 = 9.6
n = 9

sum = 9/2 ( 2*102 + (8)12) = 150 * 9 = 1350
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Re: What is the sum of the integers from 100 to 200 inclusive  [#permalink]

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30 Aug 2018, 17:09
Afc0892 wrote:
What is the sum of the integers from 100 to 200 inclusive which are divisible from 6 but not from 12?

1) 1200
2) 1350
3) 1850
4) 2550
5) 2650

This means we need to find the sum of the odd multiples of 6 from 100 to 200, inclusive, since even multiples of 6 will be divisible by 12. The first odd multiple of 6 within the range is 6 x 17 = 102, the next one is 6 x 19 = 114, etc. The last one is 6 x 33 = 198. So the odd multiples of 6 within the range are: 102, 114, 126, …, 198.

This is an arithmetic sequence of integers with a common difference of 12, so the number of integers is

(198 - 102)/12 + 1 = 96/12 + 1 = 9

The average of the integers is (198 + 102)/2 = 300/2 = 150. Thus, the sum is

150 x 9 = 1350

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Re: What is the sum of the integers from 100 to 200 inclusive &nbs [#permalink] 30 Aug 2018, 17:09
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text:
For every rational number, does there exist a sequence of irrationals which converges to it?

I can think of of examples where a sequence of irrationals converges to $0$. But if we pick any rational will there always exist a sequence of irrationals which converges to it?

I cannot find a straight answer to this question.

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Let $r$ be our rational. Look at $r+\frac{\sqrt{2}}{n}$. This may be the example you had in mind, "shifted" by $r$. –  André Nicolas Mar 11 '14 at 18:47

Assume your number is $\frac{p}{q}$. Then the sequence $$a_n=\frac{\pi}{n}+\frac{p}{q}$$ converges to the given number and is irrational (any irrational number in the place of $\pi$ would do).

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this is assuming $\lim_{n\to\inf}$, right? –  Cole Johnson Mar 11 '14 at 23:47
@ColeJohnson Yes, right! –  Stef Mar 11 '14 at 23:49

Yes, take a sequence consisting of your sequence of irrationals converging to $0$ plus your desired rational limit.

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Yes: If $r\in\Bbb Q$, then $\forall n\in\Bbb N$: ${rn\over n+\sqrt2}\in{\Bbb Q}^c$ and $$\lim_{n\to\infty}{rn\over n+\sqrt2}=r.$$

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For any rational number $x=\frac{p}{q}$ with $\gcd(p,q)=1$, just consider: $$x_n = \frac{p}{q}\cdot\frac{n}{\sqrt{n^2+1}}.$$ Clearly any $x_n$ belong to $\mathbb{R}\setminus\mathbb{Q}$ and we have $\lim_{n\to +\infty} x_n = x$.

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Yes. Consider

$\frac{p}{q} - \frac{\sqrt{2}}{n}$

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# What are virtual excitations in the context of superconductors?

In a superconductor I have read that the energy gap can be defined as the energy difference between the ground state and the virtual excitations of the system. Is this the energy required to form a cooper pair?

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## 1 Answer

In a superconductor (in BCS theory, at the very least), the ground state exists of a condensate of Cooper pairs (and regular unpaired electrons). The first excited state is the state where one of these Cooper pairs is broken up into two regular unpaired electrons. Hence, rather than the energy to make a pair, it is the energy needed to break up a the first Cooper pair.

In case you're studying superconductivity at a QFT level: This is precisely the content of the Bogoliubov transformation of the BCS Hamiltonian: rather than study the Hamiltonian in terms of the electron creation and annihilation operators $\hat a_k$ and $\hat a_k^\dagger$, one can define new fields $\alpha_k$ and $\alpha_k^\dagger$, where the creation operator $\alpha_k^\dagger$ exactly creates a broken up electron pair.

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This answer is correct enough as far as I understand BCS Theory :-) – user17581 Jan 17 '13 at 23:04


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# How to explain that (a^b)^c is not equal to a^(b^c) [duplicate]

Possible Duplicate:
Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(b*c)}$

Its been a while since I worked much with exponents, and I got confused and thought that $(a^b)^c = a^{(b^c)}$

I wasn't sure, so I tried my usual strategy of decompressing the equation (eg. $x^2 = (x)(x)...(x)$) but that didn't do the trick in this case (maybe I did it wrong). How can I show the difference between the two expressions in a simple and intuitive way?

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## marked as duplicate by MJD, amWhy, Marvis, Henry, EuYuNov 18 '12 at 22:37

Recall that if $y \in \mathbb{Z}^+$, then $$x^y = \underbrace{x \cdot x \cdot x \cdots x}_{y \text{ times}}$$ Hence, \begin{align} \left(a^b \right)^c & = \underbrace{a^b \cdot a^b \cdot a^b \cdots a^b}_{c \text{ times}}\\ & = \underbrace{\underbrace{a \cdot a \cdots a}_{b \text{ times}} \cdot \underbrace{a \cdot a \cdots a}_{b \text{ times}} \cdot \underbrace{a \cdot a \cdots a}_{b \text{ times}} \cdots \underbrace{a \cdot a \cdots a}_{b \text{ times}}}_{c \text{ times}}\\ & = \underbrace{a \cdot a \cdot a \cdots a}_{bc \text{ times}}\\ & = a^{bc} \end{align} Similarly, \begin{align} a^{(b^c)} & = \underbrace{a \cdot a \cdot a \cdots a}_{b^c \text{ times}} \end{align} Hence, $\left(a^b \right)^c$ is $a$ multiplied $bc$ times together, whereas $a^{(b^c)}$ is $a$ multiplied $b^c$ times together.

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For simplicity, assume that $b$ and $c$ are positive integers. Then $$(a^b)^c = \underbrace{a^b \cdot a^b \cdots a^b}_{c \text{ factors}} = \underbrace{\underbrace{a \cdot a \cdots a}_{b \text{ factors}} \cdot \underbrace{a \cdot a \cdots a}_{b \text{ factors}} \cdots \underbrace{a \cdot a \cdots a}_{b \text{ factors}}}_{c \text{ factors}}.$$

How many factors of $a$ do you have in the final expression?

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What is $(2^2)^3$?

What is $2^{(2^3)}$?

Frankly, there's no obvious reason why these two things should be equal. I'm not sure what you mean by "the usual strategy": if you have a specific rationale that makes you think those are the same, share it and we can figure out why it misleads you.

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For a quick check just take a simple base and exponent that you know and check it. For example, let $a=2$, $b=2$, and $c=3$. Now you have $(2^2)^3=64$ and $2^{2^3}=256$. So now you can clearly see that they aren't equal. Just be careful not to use something that does coincidently equal the same. For example I almost set $a,b,c$ all equal to 2 but realized they would all equal the same. So just try out a few different ones if you get them mixed up.

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$(a^b)^c = a^{bc}$. Now, you just have to convince yourself that $bc \neq b^c$ in general, and so $a^{bc} \neq a^{b^c}$ since the two expressions have different, unequal exponents.

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