# Analysis

# Members

Vorname | Nachname | Titel | Zusatz |
---|---|---|---|

Lars | Bugiera | MSc | Doktorand NF |

Gennaro | Ciampa | MSc | Doktorand |

Gianluca | Crippa | Prof. Dr. | |

Azahara | De la Torre Pedraza | Dr. | Wiss. Mitarbeiterin NF |

Francesco | Ghiraldin | Dr. | Wiss. Mitarbeiter |

Dominik | Himmelsbach | Dipl.-Math. | Doktorand |

Yang | Lan | Dr. | Wiss. Mitarbeiter |

Enno | Lenzmann | Prof. Dr. | |

Silvia | Ligabue | MSc | Doktorandin NF |

Renato | Lucà | Dr. | Wiss. Mitarbeiter ERC |

Gabriele | Mancini | Dr. | Wiss. Mitarbeiter NF |

Luca | Martinazzi | Prof. Dr. | Ass. Professor NF |

Cheikh Birahim | Ndiaye | Dr. | Wiss. Mitarbeiter NF |

Camilla | Nobili | Dr. | Wiss. Mitarbeiterin |

Christian | Schulze | MSc | Doktorand |

# Analysis Seminar

Our weekly Seminar in Analysis takes place on

**Wednesdays from 3:15 to 4:15 pm in room 00.003, Spiegelgasse 1, with a tea in room 06.001, Spiegelgasse 1, afterwards.**

For questions please contact Silvia Ligabue. For more information, please visit the webpage of the seminar.

# Publications

2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017

## Monographs

Giaquinta, Mariano; Martinazzi, Luca: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Pisa 2012. edoc |

Crippa, Gianluca: The flow associated to weakly differentiable vector fields, 12, Pisa 2009 (=12). edoc |

Giaquinta, Mariano; Martinazzi, Luca: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Pisa 2005. edoc |

## Journal Articles

Iula, Stefano; Maalaoui, Ali; Martinazzi, Luca: A fractional Moser-Trudinger type inequalitiy in one dimension and its critical points, in: Differential and Integral Equations 29, 2016, H. 5/6, S. 455-492. (Abriss / ) edoc, URL |

Maalaoui, Ali; Schikorra, Armin: Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension, in: Communications Partial Differential Equations 41, 2016, H. 10, S. 1593-1618. |

Boulenger, Thomas; Himmelsbach, Dominik; Lenzmann, Enno: Blowup for fractional NLS, in: J. Funct. Anal. 271, 2016, H. 9, S. 2569-2603. URL |

Crippa, Gianluca; Lopes Filho, Milton C.; Miot, Evelyne; Nussenzveig Lopes, Helena J.: Flows of vector fields with point singularities and the vortex-wave system, in: Discrete and continuous dynamical systems 36, 2016, H. 5, S. 2405-2417. (Abriss) edoc |

Bohun, Anna; Bouchut, Francois; Crippa, Gianluca: Lagrangian flows for vector fields with anisotropic regularity, in: Annales de l'Institut Henri Poincaré (C) Analyse non linéaire 33, 2016, H. 6, S. 1409-1429. (Abriss) edoc |

Bohun, Anna; Bouchut, Francois; Crippa, Gianluca: Lagrangian solutions to the 2D Euler system with L1 vorticity and infinite energy, in: Nonlinear analysis 132, 2016, S. 160-172. (Abriss) edoc |

Bohun, Anna; Bouchut, François; Crippa, Gianluca: Lagrangian solutions to the Vlasov-Poisson system with L1 density, in: Journal of differential equations 260, 2016, H. 4, S. 3576-3597. (Abriss) edoc |

Colombo, Maria; Crippa, Gianluca; Stefano, Spirito: Logarithmic estimates for continuity equations, in: Networks and Heterogeneous Media 11, 2016, H. 2, S. 301-311. (Abriss) edoc (Open Access) |

Mancini, Gabriele: Onofri-type inequalities for singular Liouville equations, in: J. Geom. Anal. 26, 2016, H. 2, S. 1202-1230. |

Caravenna, Laura; Crippa, Gianluca: Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation, in: Comptes rendus mathematique 354, 2016, H. 12, S. 1168-1173. (Abriss) edoc (Open Access) |

Frank, Rupert L.; Lenzmann, Enno; Silvestre, Luis: Uniqueness of radial solutions for the fractional Laplacian, in: Comm. Pure Appl. Math. 69, 2016, H. 9, S. 1671-1726. URL |

Battaglia, Luca; Mancini, Gabriele: A note on compactness properties of the singular Toda system, in: Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 26, 2015, H. 3, S. 299-307. |

Schikorra, Armin; Spector, Daniel; Van Schaftingen, Jean: An L1-type estimate for Riesz potentials, in: Revista Matemática Iberoamericana , 2015, S. 4 pages. |

Da Lio, Francesca; Martinazzi, Luca; Riviere, Tristan: Blow-up analysis of a nonlocal Liouville-type equation, in: Analysis & PDE 8, 2015, H. 7, S. 1757-1805. edoc |

Martinazzi, Luca; Hyder, Ali: Conformal metrics on R^{2m} with constant Q-curvature, prescribed volume and asymptotic behavior, in: Discrete and continuous dynamical systems. Series A 35, 2015, H. 1, S. 283-299. edoc |

Jin, Tianling; Maalaoui, Ali; Martinazzi, Luca; Xiong, Jingang: Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three, in: Calculus of variations and partial differential equations 52, 2015, H. 3-4, S. 469-488. (Abriss) edoc |

Crippa, Gianluca; Gusev, Nikolay; Spirito, Stefano: Failure of the chain rule for the divergence of bounded vector fields, in: Annali della Scuola normale superiore di Pisa, Classe di scienze , 2015, S. 16. (Abriss) |

Martinazzi, Luca: Fractional Adams-Moser-Trudinger type inequalities, in: Nonlinear Analysis 127, 2015, S. 263-278. (Abriss) edoc |

Schikorra, Armin; Shieh, Tien-Tsan; Spector, Daniel: Lp-theory for fractional gradient PDE with VMO coefficients, in: Atti della Accademia nazionale dei Lincei. Classe di scienze fisiche, matematiche e naturali. Rendiconti lincei. Matematica e applicazioni 26, 2015, H. 4, S. 433-443. (Abriss) edoc |

Crippa, Gianluca; Gusev, Nikolay; Spirito, Stefano; Wiedemann, Emil: Non-uniqueness and prescribed energy for the continuity equation, in: Communications in mathematical sciences 13, 2015, H. 7, S. 1937-1947. (Abriss) edoc (Open Access) |

Schikorra, Armin: Nonlinear commutators for the fractional p-Laplacian and applications, in: Mathematische Annalen , 2015, S. 27. (Abriss) |

Crippa, Gianluca; Spirito, Stefano: Renormalized solutions of the 2D Euler equations, in: Communications in mathematical physics 339, 2015, H. 1, S. 191-198. (Abriss) edoc (Open Access) |

Colombo, Maria; Crippa, Gianluca; Spirito, Stefano: Renormalized solutions to the continuity equation with an integrable damping term, in: Calculus of variations and partial differential equations 54, 2015, H. 2, S. 1831-1845. (Abriss) edoc (Open Access) |

Crippa, Gianluca; Semenova, Elizaveta; Spirito, Stefano: Strong continuity for the 2D Euler equations, in: Kinetic and related models 8, 2015, H. 4, S. 685-689. (Abriss) edoc |

Alberti, Giovanni; Bianchini, Stefano; Crippa, Gianluca: A uniqueness result for the continuity equation in two dimensions, in: Journal of the European Mathematical Society 16, 2014, H. 2, S. 201-234. (Abriss) edoc (Open Access) |

Ambrosio, Luigi; Crippa, Gianluca: Continuity equations and ODE flows with non-smooth velocity, in: Proceedings of the Royal Society of Edinburgh. Section A, Mathematics 144, 2014, H. 6, S. 1191-1244. (Abriss) edoc |

Alberti, Giovanni; Crippa, Gianluca; Mazzucato, Anna L.: Exponential self-similar mixing and loss of regularity for continuity equations, in: Comptes rendus mathematique 352, 2014, H. 11, S. 901-906. (Abriss) edoc (Open Access) |

Crippa, Gianluca; Donadello, Carlotta; Spinolo, Laura V.: Initial-boundary value problems for continuity equations with BV coefficients, in: Journal de mathématiques pures et appliquées 102, 2014, H. 1, S. 79-98. (Abriss) edoc (Open Access) |

Alberti, Giovanni; Bianchini, Stefano; Crippa, Gianluca: On the Lp-differentiability of certain classes of functions, in: Revista matemática Iberoamericana 30, 2014, H. 1, S. 349-367. (Abriss) edoc (Open Access) |

Herr, Sebastian; Lenzmann, Enno: The Boson star equation with initial data of low regularity, in: Nonlinear analysis 97, 2014, S. 125-137. (Abriss) edoc |

Martinazzi, Luca: Conformal metrics on R^2m with constant Q-curvature and large volume, in: Ann. Inst. Henri Poincaré (C) , 2013, H. 30, S. 969-982. |

Lenzmann, Enno; Lewin, Mathieu: Dynamical ionization bounds for atoms, in: Analysis of PDEs 6, 2013, H. 5, S. 1183-1211. (Abriss) edoc |

Crippa, Gianluca; Lecureux-Mercier, Magali: Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, in: NoDEA: nonlinear differential equations and applications 20, 2013, H. 3, S. 523-537. (Abriss) edoc (Open Access) |

Bouchut, Francois; Crippa, Gianluca: Lagrangian flows for vector fields with gradient given by a singular integral, in: Journal of hyperbolic differential equations 10, 2013, H. 2, S. 235-282. (Abriss) edoc |

Krieger, Joachim; Lenzmann, Enno; Raphael, Pierre: Nondispersive solutions to the L²-critical half-wave equation, in: Archive for rational mechanics and analysis 209, 2013, H. 1, S. 61-129. (Abriss) edoc |

Kirkpatrick, Kay; Lenzmann, Enno; Staffilani, Gigliola: On the continuum limit for discrete NLS with long-range lattice interactions, in: Communications in mathematical physics 317, 2013, H. 3, S. 563-591. (Abriss) edoc |

Battaglia, Luca; Mancini, Gabriele: Remarks on the Moser-Trudinger inequality, in: Advances in Nonlinear Analysis 2, 2013, H. 4, S. 389-425. |

Alberti, Giovanni; Bianchini, Stefano; Crippa, Gianluca: Structure of level sets and Sard-type properties of Lipschitz maps, in: Annali della Scuola Normale di Pisa - Classe di Scienze 12, 2013, H. 4, S. 863-902. (Abriss) edoc (Open Access) |

Frank, Rupert L.; Lenzmann, Enno: Uniqueness of non-linear ground states for fractional Laplacians in $BbbR, in: Acta mathematica 210, 2013, H. 2, S. 261-318. (Abriss) edoc |

Mantegazza, Carlo; Martinazzi, Luca: A note on quasilinear parabolic equations on manifolds, in: Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (5) , 2012, H. XI, S. 1-18. |

Acerbi, Emilio; Crippa, Gianluca; Mucci, Domenico: A variational problem for multifunctions with interaction between leaves, in: ESAIM: Control, Optimisation and Calculus of Variations 18, 2012, H. 4, S. 1178-1206. (Abriss) edoc (Open Access) |

Malchiodi, Andrea; Martinazzi, Luca: Critical points of the Moser-Trudinger functional on a disk, in: J. European Mathematical Society , 2012, S. (to appear). |

Martinazzi, Luca; Struwe, Michael: Quantization for an elliptic equation of order 2m with critical exponential non-linearity, in: Mathematische Zeitschrift , 2012, H. 270, S. 453-487. |

Martinazzi, Luca: A note on n-axially symmetric harmonic maps from B^3 into S^2 minimizing the relaxed energy, in: Journal of functional analysis , 2011, H. 261, S. 3099-3117. |

Ambrosio, Luigi; De Philippis, Guido; Martinazzi, Luca: Gamma-convergence of nonlocal perimeter functionals, in: Manuscripta mathematica , 2011, H. 134, S. 377-403. |

Crippa, Gianluca: Lagrangian flows and the one-dimensional Peano phenomenon for ODEs, in: J. Differential Equations 250, 2011, H. 7, S. 3135-3149. |

Lenzmann, Enno; Lewin, Mathieu: On singularity formation for the $L^2$-critical Boson star equation, in: Nonlinearity 24, 2011, H. 12, S. 3515-3540. |

Martinazzi, Luca: Quantization for the prescribed Q-curvature equation on open domains, in: Communications in contemporary mathematics , 2011, H. 13, S. 533-551. |

Martinazzi, Luca; Petrache Mircea: Asymptotics and quantization for a mean-field equation of higher order, in: Comm. Partial Differential Equations , 2010, H. 35, S. 1-22. |

Martinazzi, Luca; Petrache Mircea: Existence of solutions to a higher dimensional mean-field equation on manifolds, in: Manuscripta mathematica , 2010, H. 133, S. 115-130. |

Lenzmann, Enno; Lewin, Mathieu: Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs, in: Duke Math. J. 152, 2010, H. 2, S. 257-315. |

Hainzl, Christian; Lenzmann, Enno; Lewin, Mathieu; Schlein, Benjamin: On blowup for time-dependent generalized Hartree-Fock equations, in: Ann. Henri Poincar`e 11, 2010, H. 6, S. 1023-1052. |

Martinazzi, Luca: A threshold phenomenon for embeddings of H^m_0 into Orlicz spaces, in: Calculus of variations and partial differential equations , 2009, H. 36, S. 493-506. |

Martinazzi, Luca: An application of Q-curvature to an embedding of critical type, in: Oberwolfach Reports , 2009, H. 6, S. 1997-2000. |

Martinazzi, Luca: Classification of solutions to the higher order Liouville's equation on R^{2m}, in: Mathematische Zeitschrift , 2009, H. 263, S. 307-329. |

Martinazzi, Luca: Concentration-compactness phenomena in the higher order Liouville's equation, in: Journal of functional analysis , 2009, H. 256, S. 3743-3771. |

Krieger, Joachim; Lenzmann, Enno; Raphaël, Pierre: On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS, in: Ann. Henri Poincar`e 10, 2009, H. 6, S. 1159-1205. |

Crippa, Gianluca; Jimenez, Chloé; Pratelli, Aldo: Optimum and equilibrium in a transport problem with queue penalization effect, in: Adv. Calc. Var. 2, 2009, H. 3, S. 207-246. |

Ambrosio, Luigi; Crippa, Gianluca; Figalli, Alessio; Spinolo, Laura V.: Some new well-posedness results for continuity and transport equations, and applications to the chromatography system, in: SIAM J. Math. Anal. 41, 2009, H. 5, S. 1890-1920. |

Lenzmann, Enno: Uniqueness of ground states for pseudorelativistic Hartree equations, in: Anal. PDE 2, 2009, H. 1, S. 1-27. |

Martinazzi, Luca: Conformal metrics on R^{2m} with constant Q-curvature, in: Rendiconti Lincei Mat. Appl. , 2008, H. 19, S. 279-292. |

Crippa, Gianluca; De Lellis, Camillo: Estimates and regularity results for the DiPerna-Lions flow, in: J. Reine Angew. Math. 616, 2008, S. 15-46. |

Ambrosio, Luigi; Crippa, Gianluca; LeFloch, Philippe G.: Leaf superposition property for integer rectifiable currents, in: Netw. Heterog. Media 3, 2008, H. 1, S. 85-95. |

Fröhlich, Jürg; Lenzmann, Enno: Blowup for nonlinear wave equations describing boson stars, in: Comm. Pure Appl. Math. 60, 2007, H. 11, S. 1691-1705. |

Fröhlich, Jürg; Jonsson, B. Lars G.; Lenzmann, Enno: Boson stars as solitary waves, in: Comm. Math. Phys. 274, 2007, H. 1, S. 1-30. |

Fröhlich, Jürg; Lenzmann, Enno: Dynamical collapse of white dwarfs in Hartree- and Hartree-Fock theory, in: Comm. Math. Phys. 274, 2007, H. 3, S. 737-750. |

Fröhlich, J.; Jonsson, B. L. G.; Lenzmann, Enno: Effective dynamics for boson stars, in: Nonlinearity 20, 2007, H. 5, S. 1031-1075. |

Fröhlich, Jürg; Knowles, Antti; Lenzmann, Enno: Semi-classical dynamics in quantum spin systems, in: Lett. Math. Phys. 82, 2007, H. 2-3, S. 275-296. |

Lenzmann, Enno: Well-posedness for semi-relativistic Hartree equations of critical type, in: Math. Phys. Anal. Geom. 10, 2007, H. 1, S. 43-64. |

Colombini, Ferruccio; Crippa, Gianluca; Rauch, Jeffrey: A note on two-dimensional transport with bounded divergence, in: Comm. Partial Differential Equations 31, 2006, H. 7-9, S. 1109-1115. |

Crippa, Gianluca; De Lellis, Camillo: Oscillatory solutions to transport equations, in: Indiana University Mathematics Journal 55, 2006, H. 1, S. 1-13. (Abriss) edoc (Open Access) |

Bouchut, François; Crippa, Gianluca: Uniqueness, renormalization, and smooth approximations for linear transport equations, in: SIAM J. Math. Anal. 38, 2006, H. 4, S. 1316-1328. |

Ambrosio, Luigi; Crippa, Gianluca; Maniglia, Stefania: Traces and fine properties of a BD class of vector fields and applications, in: Ann. Fac. Sci. Toulouse Math. (6) 14, 2005, H. 4, S. 527-561. |

## Essays in Anthologies

Crippa, Gianluca; Donadello, Carlotta; Spinolo, Laura V.: A note on the initial-boundary value problem for continuity equations with rough coefficients, in: Hyperbolic problems: theory, numerics, applications, 8, Padova, Italy 2014 (= 8). (Abriss / ) edoc, URL |

Crippa, Gianluca: Ordinary differential equations and singular integrals, in: Hyperbolic problems: theory, numerics, applications, 8, Padova, Italy 2014 (= 8). (Abriss / ) edoc, URL |

Ambrosio, Luigi; Crippa, Gianluca: Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, in: Ancona, Fabio; Bianchini, Stefano; Colombo, Rinaldo M.; De Lellis, Camillo; Marson, Andrea; Montanari, Annamaria (Hg.): Transport equations and multi-D hyperbolic conservation laws, Berlin 2008, S. 3-57. |

Crippa, Gianluca; Otto, Felix; Westdickenberg, Michael: Regularizing effect of nonlinearity in multidimensional scalar conservation laws, in: Ancona, Fabio; Bianchini, Stefano; Colombo, Rinaldo M.; De Lellis, Camillo; Marson, Andrea; Montanari, Annamaria (Hg.): Transport equations and multi-D hyperbolic conservation laws, Berlin 2008, S. 77-128. |

## Internet Publications

Gérard, Patrick; Lenzmann, Enno; Pocovnicu, Oana; Raphael, Pierre: A Two-Soliton with Transient Turbulent Regime for the Cubic Half-wave Equation on The Real Line, 11.16.2016. URL |

Boulenger, Thomas; Lenzmann, Enno: Blowup for Biharmonic NLS, 05.03.2015. URL |

## Unpublished Talks

Bohun, Anna: Stability Loss Delay in Dynamical Bifurcations, 2011. (Download) |

## Preprints

Crippa, Gianluca; Nobili, Camilla; Seis, Christian; Spirito, Stefano: Eulerian and Lagrangian solutions to the continuity and Euler equations with L1 vorticity, 2017. URL |

Iula, Stefano: A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one, 2016. (Download) |

Hyder, Ali: Conformally Euclidean metrics on R^n with arbitrary total Q-curvature, 2016. URL |

Alberti, Giovanni; Crippa, Gianluca; Mazzucato, Anna L.: Exponential self-similar mixing by incompressible flows, 2016. (Download) |

Iula, Stefano; Mancini, Gabriele: Extremal functions for singular Moser-Trudinger embeddings, 2016. URL |

Hyder, Ali; Iula, Stefano; Martinazzi, Luca: Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space, 2016. (Download) |

Nobili, Camilla; Otto, Felix: Limitations of the background field method applied to Rayleigh-Bénard convection, 2016. (Download) |

Heck, Horst; Pal Choudhury, Anupam: Stability of the inverse boundary value problem for the biharmonic operator: Logarithmic estimates, 2016. URL |

Mancini, Gabriele; Martinazzi, Luca: The Moser-Trudinger inequality and its extremals on a disk via energy estimates, 2016. URL |

Da Lio, Francesca; Martinazzi, Luca: The nonlocal Liouville-type equation in R and conformal immersions of the disk with boundary singularities, 2016. URL |

Choffrut, Antoine; Nobili, Camilla; Otto, Felix: A maximal regularity estimate for the non-stationary Stokes equation in the strip In: Journal of differential equations, 2015. URL |

Iula, Stefano; Maalaoui, Ali; Martinazzi, Luca: A fractional Moser-Trudinger type inequality in one dimension and its critical points, 2015. URL |

Martinazzi, Luca; Riviere, Tristan; Da Lio, Francesca: Blow-up analysis of a nonlocal Liouville-type equation, 2015. (Download) |

Maalaoui, Ali; Martinazzi, Luca; Schikorra, Armin: Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension , 2015. URL |

Boulenger, Thomas; Lenzmann, Enno: Blowup for Biharmonic NLS , 2015. URL |

Boulenger, Thomas; Himmelsbach, Dominik; Lenzmann, Enno: Blowup for fractional NLS, 2015. URL |

Hyder, Ali: Existence of entire solutions to a fractional Liouville equation in R^n , 2015. (Download) |

Martinazzi, Luca: Fractional Adams-Moser-Trudinger type inequalities, 2015. URL |

Hyder, Ali: Moser functions and fractional Moser-Trudinger type inequalities, 2015. URL |

Schikorra, Armin: Nonlinear Comutators for the fractional p-Laplacian and applications, 2015. URL |

Hyder, Ali: Structure of conformal metrics on R^n with constant Q-curvature, 2015. URL |

Hyder, Ali; Martinazzi, Luca: Conformal Metrics on R2m with Constant Q-Curvature, Prescribed Volume and Asymptotic Behavior, 2014. URL |

Choffrut, Antoine; Nobili, Camilla; Otto, Felix: Upper bounds on Nusselt number at finite Prandtl number, 2014. URL |

Maalaoui, Ali; Martino, Vittorio: Homology computation for a class of contact structures on T3 and equivariant reductions, To appear in Calculus of variations and PDE. 2013. |

Maalaoui, Ali: Prescribing the Q-curvature On the Sphere with Conical singularities. 2013. |

Maalaoui, Ali: Rabinowitz-Floer Homology for Super-quadratic Dirac Equations On spin Manifolds, Journal of Fixed Point Theory and Applications March 2013, Volume 13, Issue 1, pp 175-19, 2013. URL |

Maalaoui, Ali; Martino, Vittorio: The topology of a subspace of the Legendrian curves in a closed contact 3-manifold, to appear in Advanced nonLinear Studies, 2013. |

Frank, Rupert L.; Lenzmann, Enno; Silvestre, Luis: Uniqueness of Radial Solutions for the Fractional Laplacian, 2013. URL |

Lenzmann, Enno; Frank, Rupert L.: Uniqueness of ground states for fractional Laplacians in R. To appear in Acta Math. 45 pages, 2012. URL |

Lenzmann, Enno; Staffilani, Gigliola; Kirkpatrick, Kay: On the continuum limit for discrete NLS with long-range lattice interactions, 25 pages, preprint, 2011. URL |

Lenzmann, Enno; Lewin, Mathieu: Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs. Duke Math. Journal 152, no. 2, 157-315, 2010. URL |