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## Figalli, Alessio

Compute Distance To:
 Author ID: figalli.alessio Published as: Figalli, Alessio; Figalli, A. Further Spellings: Фигалли А Homepage: https://people.math.ethz.ch/~afigalli/ External Links: MGP · ORCID · Wikidata · Google Scholar · Math-Net.Ru · dblp · GND · IdRef · theses.fr Awards: EMS Prize (2012) · Fields Medal (2018)
 Documents Indexed: 160 Publications since 2007, including 4 Books 5 Contributions as Editor · 4 Further Contributions Biographic References: 6 Publications Co-Authors: 104 Co-Authors with 143 Joint Publications 2,597 Co-Co-Authors
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### Co-Authors

 23 single-authored 17 Ambrosio, Luigi 13 De Philippis, Guido 13 Rifford, Ludovic 11 Maggi, Francesco 7 Colombo, Maria 7 Villani, Cédric 6 Bonforte, Matteo 6 Ros-Oton, Xavier 5 Jerison, David S. 5 Kim, Young-Heon 5 Pratelli, Aldo 5 Valdinoci, Enrico 4 Caffarelli, Luis Ángel 4 Carrillo de la Plata, José Antonio 4 McCann, Robert J. 4 Serra Montolí, Joaquim 4 Shahgholian, Henrik 3 Barrios Barrera, Begoña 3 Chen, Shibing 3 Ciraolo, Giulio 3 Mooney, Connor 3 Paul, Thierry 3 Vazquez, Juan Luis 3 Wickramasekera, Neshan 2 Abbondandolo, Alberto 2 Bernot, Marc 2 Bjorland, Clayton 2 Bourgain, Jean 2 Brézis, Haïm 2 Cabré, Xavier 2 Carlen, Eric Anders 2 Crippa, Gianluca 2 Dipierro, Serena 2 Fathi, Albert 2 Friesecke, Gero 2 Giannoulis, Johannes 2 Gigli, Nicola 2 Glaudo, Federico 2 Guionnet, Alice 2 Jhaveri, Yash 2 Kang, Moon-Jin 2 Kohn, Robert Vita 2 Laurent, Thomas 2 Pelayo, Alvaro 2 Santambrogio, Filippo 2 Slepčev, Dejan 2 Spinolo, Laura V. 2 Toro, Tatiana 1 Baer, Eric 1 Bekerman, Florent 1 Bench, William 1 Braga, J. Ederson M. 1 Brendle, Simon 1 Cagnetti, Filippo 1 Chang, Sun-Yung Alice 1 Contreras, Gonzalo 1 Cordero-Erausquin, Dario 1 Cozzi, Matteo 1 da Silva, A. Belotto 1 de Franchis, Carlo 1 Del Pino, Manuel A. 1 Deproit, Laurent 1 Desvillettes, Laurent 1 Difrancesco, M. 1 Farina, Alberto 1 Fernández-Real, Xavier 1 Fusco, Nicola 1 Gallouët, Thomas O. 1 Gangbo, Wilfrid 1 Ge, Yuxin 1 Gilles, Sébastien 1 Gomes, Diogo Luís Aguiar 1 Gouëzel, Sébastien 1 Grillo, Gabriele 1 Indrei, Emanuel 1 Jerrard, Robert Leon 1 Juillet, Nicolas 1 Kim, Sunghan 1 Klein, Christian 1 Krummel, Brian 1 Ligabò, Marilena 1 Loeper, Grégoire 1 Mandorino, Vito 1 Marcon, Diego 1 Markowich, Peter Alexander 1 Millot, Vincent 1 Mingione, Giuseppe 1 Morales, Javier 1 Moreira, Diego R. 1 Morini, Massimiliano 1 Neumayer, Robin 1 Novaga, Matteo 1 Oh, Byunghyun 1 Palatucci, Giampiero 1 Palmer, Joseph 1 Parusiński, Adam 1 Patacchini, Francesco Saverio 1 Peral Alonso, Ireneo 1 Persson, Ulf A. 1 Philipowski, Robert ...and 12 more Co-Authors
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### Serials

 13 Communications on Pure and Applied Mathematics 11 Archive for Rational Mechanics and Analysis 7 Discrete and Continuous Dynamical Systems 6 Inventiones Mathematicae 6 Journal of Functional Analysis 6 Calculus of Variations and Partial Differential Equations 4 Advances in Mathematics 4 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 4 Journal de Mathématiques Pures et Appliquées. Neuvième Série 4 SIAM Journal on Mathematical Analysis 4 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 3 Communications in Mathematical Physics 3 Duke Mathematical Journal 3 Journal of Differential Equations 3 Journal für die Reine und Angewandte Mathematik 3 Communications in Partial Differential Equations 3 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 3 Journal of the European Mathematical Society (JEMS) 3 Oberwolfach Reports 3 Analysis & PDE 2 Acta Mathematica 2 American Journal of Mathematics 2 Gazette des Mathématiciens 2 Publications Mathématiques 2 Mathematische Annalen 2 Proceedings of the American Mathematical Society 2 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 2 Geometric and Functional Analysis. GAFA 2 Bulletin of the American Mathematical Society. New Series 2 Methods and Applications of Analysis 2 Comptes Rendus. Mathématique. Académie des Sciences, Paris 2 Lecture Notes in Mathematics 1 Israel Journal of Mathematics 1 Annali di Matematica Pura ed Applicata. Serie Quarta 1 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 1 Indiana University Mathematics Journal 1 Journal of Differential Geometry 1 Journal of Economic Theory 1 Mathematische Zeitschrift 1 Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 Rendiconti del Seminario Matematico della Università di Padova 1 Tohoku Mathematical Journal. Second Series 1 Transactions of the American Mathematical Society 1 Chinese Annals of Mathematics. Series B 1 Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni 1 Differential and Integral Equations 1 Differential Geometry and its Applications 1 IMRN. International Mathematics Research Notices 1 The Journal of Geometric Analysis 1 Notices of the American Mathematical Society 1 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 1 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 1 Journal of Mathematical Sciences (New York) 1 European Mathematical Society Newsletter 1 Advances in Geometry 1 Tesi. Scuola Normale Superiore di Pisa (Nuova Serie) 1 ALEA. Latin American Journal of Probability and Mathematical Statistics 1 Journal of Fixed Point Theory and Applications 1 Discrete and Continuous Dynamical Systems. Series S 1 Advances in Calculus of Variations 1 Cambridge Journal of Mathematics 1 Philosophical Transactions A. Royal Society of London 1 Bollettino dell’Unione Matematica Italiana 1 EMS Textbooks in Mathematics 1 Zurich Lectures in Advanced Mathematics 1 Mathematics in Engineering
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### Fields

 83 Partial differential equations (35-XX) 72 Calculus of variations and optimal control; optimization (49-XX) 26 Differential geometry (53-XX) 14 Probability theory and stochastic processes (60-XX) 13 Dynamical systems and ergodic theory (37-XX) 13 Global analysis, analysis on manifolds (58-XX) 10 Fluid mechanics (76-XX) 8 Real functions (26-XX) 8 Measure and integration (28-XX) 8 Convex and discrete geometry (52-XX) 7 General and overarching topics; collections (00-XX) 7 Functional analysis (46-XX) 5 Operations research, mathematical programming (90-XX) 4 Quantum theory (81-XX) 4 Geophysics (86-XX) 3 History and biography (01-XX) 3 Ordinary differential equations (34-XX) 3 Statistics (62-XX) 3 Mechanics of particles and systems (70-XX) 3 Mechanics of deformable solids (74-XX) 3 Biology and other natural sciences (92-XX) 2 Number theory (11-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Potential theory (31-XX) 2 Difference and functional equations (39-XX) 2 Integral equations (45-XX) 2 Operator theory (47-XX) 2 Statistical mechanics, structure of matter (82-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Systems theory; control (93-XX) 1 Combinatorics (05-XX) 1 Algebraic geometry (14-XX) 1 Topological groups, Lie groups (22-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Manifolds and cell complexes (57-XX)

### Citations contained in zbMATH Open

146 Publications have been cited 2,365 times in 1,526 Documents Cited by Year
A mass transportation approach to quantitative isoperimetric inequalities. Zbl 1196.49033
Figalli, A.; Maggi, F.; Pratelli, A.
2010
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Zbl 1215.35045
Carrillo, J. A.; Difrancesco, M.; Figalli, A.; Laurent, T.; Slepčev, D.
2011
Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. Zbl 1169.60010
Figalli, Alessio
2008
Isoperimetry and stability properties of balls with respect to nonlocal energies. Zbl 1312.49051
Figalli, A.; Fusco, N.; Maggi, F.; Millot, V.; Morini, M.
2015
The Monge-Ampère equation and its applications. Zbl 1435.35003
Figalli, Alessio
2017
Non-local gradient dependent operators. Zbl 1252.35099
Bjorland, C.; Caffarelli, L.; Figalli, A.
2012
Nonlocal tug-of-war and the infinity fractional Laplacian. Zbl 1235.35278
Bjorland, Clayton; Caffarelli, Luis; Figalli, Alessio
2012
Regularity of solutions to the parabolic fractional obstacle problem. Zbl 1277.35088
Caffarelli, Luis; Figalli, Alessio
2013
Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces. Zbl 1316.35061
Barrios, Begoña; Figalli, Alessio; Valdinoci, Enrico
2014
The Monge-Ampère equation and its link to optimal transportation. Zbl 06377770
De Philippis, Guido; Figalli, Alessio
2014
On flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna-Lions. Zbl 1156.60036
Ambrosio, Luigi; Figalli, Alessio
2009
Optimal transportation on non-compact manifolds. Zbl 1198.49044
Fathi, Albert; Figalli, Alessio
2010
Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation. Zbl 1307.26027
Carlen, Eric A.; Figalli, Alessio
2013
A refined Brunn-Minkowski inequality for convex sets. Zbl 1192.52015
Figalli, A.; Maggi, F.; Pratelli, A.
2009
$$W^{2,1}$$ regularity for solutions of the Monge-Ampère equation. Zbl 1286.35107
De Philippis, Guido; Figalli, Alessio
2013
Asymptotics of the $$s$$-perimeter as $$s \searrow 0$$. Zbl 1275.49083
Dipierro, Serena; Figalli, Alessio; Palatucci, Giampiero; Valdinoci, Enrico
2013
The optimal partial transport problem. Zbl 1245.49059
Figalli, Alessio
2010
Transport maps for $$\beta$$-matrix models and universality. Zbl 1330.49046
Bekerman, F.; Figalli, A.; Guionnet, A.
2015
Mass transportation on sub-Riemannian manifolds. Zbl 1201.49048
Figalli, Alessio; Rifford, Ludovic
2010
A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment. Zbl 1405.35206
Figalli, Alessio; Kang, Moon-Jin
2019
$$C^{1}$$ regularity of solutions of the Monge-Ampère equation for optimal transport in dimension two. Zbl 1170.35400
Figalli, Alessio; Loeper, Grégoire
2009
Continuity of optimal transport maps and convexity of injectivity domains on small deformations of $$\mathbb S^2$$. Zbl 1175.49040
Figalli, Alessio; Rifford, Ludovic
2009
Strongly nonlocal dislocation dynamics in crystals. Zbl 1304.35731
Dipierro, Serena; Figalli, Alessio; Valdinoci, Enrico
2014
On the shape of liquid drops and crystals in the small mass regime. Zbl 1279.76005
Figalli, A.; Maggi, F.
2011
A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions. Zbl 1203.35126
Figalli, Alessio; Gigli, Nicola
2010
Nearly round spheres look convex. Zbl 1241.53031
Figalli, Alessio; Rifford, Ludovic; Villani, Cedric
2012
Regularity and Bernstein-type results for nonlocal minimal surfaces. Zbl 1380.49060
Figalli, Alessio; Valdinoci, Enrico
2017
Some new well-posedness results for continuity and transport equations, and applications to the chromatography system. Zbl 1222.35060
Ambrosio, Luigi; Crippa, Gianluca; Figalli, Alessio; Spinolo, Laura V.
2009
A sharp stability result for the relative isoperimetric inequality inside convex cones. Zbl 1262.49045
Figalli, A.; Indrei, E.
2013
Semiclassical limit of quantum dynamics with rough potentials and well-posedness of transport equations with measure initial data. Zbl 1231.35186
Ambrosio, Luigi; Figalli, Alessio; Friesecke, Gero; Giannoulis, Johannes; Paul, Thierry
2011
Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds. Zbl 1262.58013
Figalli, Alessio; Rifford, Ludovic; Villani, Cédric
2011
Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation. Zbl 1286.46035
Figalli, Alessio; Maggi, Francesco; Pratelli, Aldo
2013
When is multidimensional screening a convex program? Zbl 1282.90085
Figalli, Alessio; Kim, Young-Heon; McCann, Robert J.
2011
On the Ma-Trudinger-Wang curvature on surfaces. Zbl 1203.53034
Figalli, Alessio; Rifford, Ludovic; Villani, Cédric
2010
A note on interior $$W^{2,1+ \varepsilon }$$ estimates for the Monge-Ampère equation. Zbl 1280.35153
De Philippis, G.; Figalli, A.; Savin, O.
2013
Hölder continuity and injectivity of optimal maps. Zbl 1281.49037
Figalli, Alessio; Kim, Young-Heon; McCann, Robert J.
2013
Global regularity for the free boundary in the obstacle problem for the fractional Laplacian. Zbl 1409.35044
Barrios, Begoña; Figalli, Alessio; Ros-Oton, Xavier
2018
Existence of Eulerian solutions to the semigeostrophic equations in physical space: The 2-dimensional periodic case. Zbl 1258.35164
Ambrosio, Luigi; Colombo, Maria; De Philippis, Guido; Figalli, Alessio
2012
Geodesics in the space of measure-preserving maps and plans. Zbl 1253.76007
Ambrosio, Luigi; Figalli, Alessio
2009
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains. Zbl 1443.35067
Bonforte, Matteo; Figalli, Alessio; Vázquez, Juan Luis
2018
A general class of free boundary problems for fully nonlinear elliptic equations. Zbl 1326.35137
Figalli, Alessio; Shahgholian, Henrik
2014
Quantitative stability for the Brunn-Minkowski inequality. Zbl 1380.52010
Figalli, Alessio; Jerison, David
2017
Confinement in nonlocal interaction equations. Zbl 1233.35032
Carrillo, J. A.; Di Francesco, M.; Figalli, A.; Laurent, T.; Slepčev, D.
2012
On the Hausdorff dimension of the Mather quotient. Zbl 1172.37018
Fathi, Albert; Rifford, Ludovic; Figalli, Alessio
2009
On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations. Zbl 1138.49031
Ambrosio, Luigi; Figalli, Alessio
2008
Partial regularity of Brenier solutions of the Monge-Ampère equation. Zbl 1193.35087
Figalli, Alessio; Kim, Young-Heon
2010
Regularity of optimal transport maps on multiple products of spheres. Zbl 1268.49053
Figalli, Alessio; Kim, Young-Heon; McCann, Robert J.
2013
Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains. Zbl 1377.35259
Bonforte, Matteo; Figalli, Alessio; Ros-Oton, Xavier
2017
Partial regularity for optimal transport maps. Zbl 1325.49051
De Philippis, Guido; Figalli, Alessio
2015
Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature. Zbl 1411.53030
Ciraolo, Giulio; Figalli, Alessio; Maggi, Francesco; Novaga, Matteo
2018
Total variation flow and sign fast diffusion in one dimension. Zbl 1242.35049
Bonforte, Matteo; Figalli, Alessio
2012
BMO-type norms related to the perimeter of sets. Zbl 1352.46026
Ambrosio, Luigi; Bourgain, Jean; Brezis, Haim; Figalli, Alessio
2016
Existence and uniqueness of maximal regular flows for non-smooth vector fields. Zbl 1348.34039
Ambrosio, Luigi; Colombo, Maria; Figalli, Alessio
2015
On the fine structure of the free boundary for the classical obstacle problem. Zbl 1408.35228
Figalli, Alessio; Serra, Joaquim
2019
Regularity properties of optimal maps between nonconvex domains in the plane. Zbl 1193.35086
Figalli, Alessio
2010
On the isoperimetric problem for radial log-convex densities. Zbl 1307.49046
Figalli, A.; Maggi, F.
2013
A global existence result for the semigeostrophic equations in three dimensional convex domains. Zbl 1287.35064
Ambrosio, Luigi; Colombo, Maria; De Philippis, Guido; Figalli, Alessio
2014
Geometry of minimizers for the interaction energy with mildly repulsive potentials. Zbl 1408.49035
Carrillo, J. A.; Figalli, A.; Patacchini, F. S.
2017
A simple proof of the Morse-Sard theorem in Sobolev spaces. Zbl 1157.58002
Figalli, Alessio
2008
Free boundary regularity in the parabolic fractional obstacle problem. Zbl 1420.35428
Barrios, Begoña; Figalli, Alessio; Ros-Oton, Xavier
2018
On stable solutions for boundary reactions: a De Giorgi-type result in dimension $$4 + 1$$. Zbl 1441.35254
Figalli, Alessio; Serra, Joaquim
2020
Existence, uniqueness, and regularity of optimal transport maps. Zbl 1132.28322
Figalli, Alessio
2007
A note on Cheeger sets. Zbl 1168.39008
Figalli, Alessio; Maggi, Francesco; Pratelli, Aldo
2009
High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces. Zbl 1114.37034
Abbondandolo, Alberto; Figalli, Alessio
2007
Regularity theory for local and nonlocal minimal surfaces: an overview. Zbl 1385.49028
Cozzi, Matteo; Figalli, Alessio
2017
Stable solutions to semilinear elliptic equations are smooth up to dimension $$9$$. Zbl 1467.35172
Cabré, Xavier; Figalli, Alessio; Ros-Oton, Xavier; Serra, Joaquim
2020
Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces. Zbl 1228.60063
Ambrosio, Luigi; Figalli, Alessio
2011
Absolute continuity of Wasserstein geodesics in the Heisenberg group. Zbl 1246.53049
Figalli, A.; Juillet, N.
2008
Universality in several-matrix models via approximate transport maps. Zbl 1402.60009
Figalli, Alessio; Guionnet, Alice
2016
Second order stability for the Monge-Ampère equation and strong Sobolev convergence of optimal transport maps. Zbl 1278.35090
De Philippis, Guido; Figalli, Alessio
2013
On the Lagrangian structure of transport equations: the Vlasov-Poisson system. Zbl 1403.35085
Ambrosio, Luigi; Colombo, Maria; Figalli, Alessio
2017
Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flow. Zbl 1384.35130
Figalli, Alessio; Kang, Moon-Jin; Morales, Javier
2018
Quantitative stability for sumsets in $$\mathbb R^n$$. Zbl 1325.49052
Figalli, Alessio; Jerison, David
2015
Local semiconvexity of Kantorovich potentials on non-compact manifolds. Zbl 1228.49047
Figalli, Alessio; Gigli, Nicola
2011
On flows of $$H ^{3/2}$$-vector fields on the circle. Zbl 1195.46032
Figalli, Alessio
2010
Regularity results for very degenerate elliptic equations. Zbl 1282.35175
Colombo, Maria; Figalli, Alessio
2014
Regularity of optimal transport maps (after Ma-Trudinger-Wang and Loeper). Zbl 1211.49054
Figalli, Alessio
2010
Optimal regularity of the convex envelope. Zbl 1316.49019
De Philippis, Guido; Figalli, Alessio
2015
Symmetry results for critical anisotropic $$p$$-Laplacian equations in convex cones. Zbl 1453.35104
Ciraolo, Giulio; Figalli, Alessio; Roncoroni, Alberto
2020
Generic regularity of free boundaries for the obstacle problem. Zbl 1456.35234
Figalli, Alessio; Ros-Oton, Xavier; Serra, Joaquim
2020
Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations. Zbl 1392.35144
Bonforte, Matteo; Figalli, Alessio; Vázquez, Juan Luis
2018
Generalized solutions for the Euler equations in one and two dimensions. Zbl 1170.35078
Bernot, Marc; Figalli, Alessio; Santambrogio, Filippo
2009
Convergence to the viscous porous medium equation and propagation of chaos. Zbl 1180.35592
Figalli, Alessio; Philipowski, Robert
2008
Rigidity and stability of Caffarelli’s log-concave perturbation theorem. Zbl 1359.60050
De Philippis, Guido; Figalli, Alessio
2017
Gradient stability for the Sobolev inequality: the case $$p\geq 2$$. Zbl 1417.46023
Figalli, Alessio; Neumayer, Robin
2019
Semiclassical limit for mixed states with singular and rough potentials. Zbl 1264.81177
Figalli, Alessio; Ligabò, Marilena; Paul, Thierry
2012
An approximation lemma about the cut locus, with applications in optimal transport theory. Zbl 1172.53022
Figalli, Alessio; Villani, Cédric
2008
Regularity of monotone transport maps between unbounded domains. Zbl 1448.35188
Cordero-Erausquin, Dario; Figalli, Alessio
2019
On the regularity of the free boundary in the $$p$$-Laplacian obstacle problem. Zbl 1387.35631
Figalli, Alessio; Krummel, Brian; Ros-Oton, Xavier
2017
On the continuity of center-outward distribution and quantile functions. Zbl 1433.62132
Figalli, Alessio
2018
A note on the regularity of the free boundaries in the optimal partial transport problem. Zbl 1183.35277
Figalli, Alessio
2009
Almost everywhere well-posedness of continuity equations with measure initial data. (Existence presque partout des équations de continuité avec données initiales mesures.) Zbl 1195.35202
Ambrosio, Luigi; Figalli, Alessio
2010
Boundary $$\varepsilon$$-regularity in optimal transportation. Zbl 1307.49045
Chen, Shibing; Figalli, Alessio
2015
Synchronized traffic plans and stability of optima. Zbl 1148.49039
Bernot, Marc; Figalli, Alessio
2008
An overview of unconstrained free boundary problems. Zbl 1353.35322
Figalli, Alessio; Shahgholian, Henrik
2015
Closing Aubry sets. II. Zbl 1377.37090
Figalli, A.; Rifford, L.
2015
A general class of free boundary problems for fully nonlinear parabolic equations. Zbl 1338.35508
Figalli, Alessio; Shahgholian, Henrik
2015
A variational method for a class of parabolic PDEs. Zbl 1239.35074
Figalli, Alessio; Gangbo, Wilfrid; Yolcu, Turkay
2011
On sets of finite perimeter in Wiener spaces: reduced boundary and convergence to halfspaces. Zbl 1282.28002
Ambrosio, Luigi; Figalli, Alessio; Runa, Eris
2013
Sobolev regularity for Monge-Ampère type equations. Zbl 1271.49031
De Philippis, Guido; Figalli, Alessio
2013
Strong stability for the Wulff inequality with a crystalline norm. Zbl 07492702
Figalli, Alessio; Zhang, Yi Ru-Ya
2022
Sharp extinction rates for fast diffusion equations on generic bounded domains. Zbl 1467.35047
Bonforte, Matteo; Figalli, Alessio
2021
An invitation to optimal transport, Wasserstein distances, and gradient flows. Zbl 1472.49001
Figalli, Alessio; Glaudo, Federico
2021
On stable solutions for boundary reactions: a De Giorgi-type result in dimension $$4 + 1$$. Zbl 1441.35254
Figalli, Alessio; Serra, Joaquim
2020
Stable solutions to semilinear elliptic equations are smooth up to dimension $$9$$. Zbl 1467.35172
Cabré, Xavier; Figalli, Alessio; Ros-Oton, Xavier; Serra, Joaquim
2020
Symmetry results for critical anisotropic $$p$$-Laplacian equations in convex cones. Zbl 1453.35104
Ciraolo, Giulio; Figalli, Alessio; Roncoroni, Alberto
2020
Generic regularity of free boundaries for the obstacle problem. Zbl 1456.35234
Figalli, Alessio; Ros-Oton, Xavier; Serra, Joaquim
2020
On the sharp stability of critical points of the Sobolev inequality. Zbl 1440.35163
Figalli, Alessio; Glaudo, Federico
2020
Optimal regularity and structure of the free boundary for minimizers in cohesive zone models. Zbl 1442.49005
Caffarelli, L.; Cagnetti, F.; Figalli, A.
2020
A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment. Zbl 1405.35206
Figalli, Alessio; Kang, Moon-Jin
2019
On the fine structure of the free boundary for the classical obstacle problem. Zbl 1408.35228
Figalli, Alessio; Serra, Joaquim
2019
Gradient stability for the Sobolev inequality: the case $$p\geq 2$$. Zbl 1417.46023
Figalli, Alessio; Neumayer, Robin
2019
Regularity of monotone transport maps between unbounded domains. Zbl 1448.35188
Cordero-Erausquin, Dario; Figalli, Alessio
2019
Optimal regularity for the convex envelope and semiconvex functions related to supersolutions of fully nonlinear elliptic equations. Zbl 07046610
Braga, J. Ederson M.; Figalli, Alessio; Moreira, Diego
2019
Global regularity for the free boundary in the obstacle problem for the fractional Laplacian. Zbl 1409.35044
Barrios, Begoña; Figalli, Alessio; Ros-Oton, Xavier
2018
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains. Zbl 1443.35067
Bonforte, Matteo; Figalli, Alessio; Vázquez, Juan Luis
2018
Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature. Zbl 1411.53030
Ciraolo, Giulio; Figalli, Alessio; Maggi, Francesco; Novaga, Matteo
2018
Free boundary regularity in the parabolic fractional obstacle problem. Zbl 1420.35428
Barrios, Begoña; Figalli, Alessio; Ros-Oton, Xavier
2018
Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flow. Zbl 1384.35130
Figalli, Alessio; Kang, Moon-Jin; Morales, Javier
2018
Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations. Zbl 1392.35144
Bonforte, Matteo; Figalli, Alessio; Vázquez, Juan Luis
2018
On the continuity of center-outward distribution and quantile functions. Zbl 1433.62132
Figalli, Alessio
2018
A quantitative analysis of metrics on $$\mathbb R^n$$ with almost constant positive scalar curvature, with applications to fast diffusion flows. Zbl 1408.53018
Ciraolo, Giulio; Figalli, Alessio; Maggi, Francesco
2018
The sharp quantitative Euclidean concentration inequality. Zbl 1385.39005
Figalli, Alessio; Maggi, Francesco; Mooney, Connor
2018
Symplectic $$G$$-capacities and integrable systems. Zbl 1395.57034
Figalli, Alessio; Palmer, Joseph; Pelayo, Álvaro
2018
Global existence for the semigeostrophic equations via Sobolev estimates for Monge-Ampère. Zbl 1406.35273
Figalli, Alessio
2018
The Monge-Ampère equation and its applications. Zbl 1435.35003
Figalli, Alessio
2017
Regularity and Bernstein-type results for nonlocal minimal surfaces. Zbl 1380.49060
Figalli, Alessio; Valdinoci, Enrico
2017
Quantitative stability for the Brunn-Minkowski inequality. Zbl 1380.52010
Figalli, Alessio; Jerison, David
2017
Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains. Zbl 1377.35259
Bonforte, Matteo; Figalli, Alessio; Ros-Oton, Xavier
2017
Geometry of minimizers for the interaction energy with mildly repulsive potentials. Zbl 1408.49035
Carrillo, J. A.; Figalli, A.; Patacchini, F. S.
2017
Regularity theory for local and nonlocal minimal surfaces: an overview. Zbl 1385.49028
Cozzi, Matteo; Figalli, Alessio
2017
On the Lagrangian structure of transport equations: the Vlasov-Poisson system. Zbl 1403.35085
Ambrosio, Luigi; Colombo, Maria; Figalli, Alessio
2017
Rigidity and stability of Caffarelli’s log-concave perturbation theorem. Zbl 1359.60050
De Philippis, Guido; Figalli, Alessio
2017
On the regularity of the free boundary in the $$p$$-Laplacian obstacle problem. Zbl 1387.35631
Figalli, Alessio; Krummel, Brian; Ros-Oton, Xavier
2017
Characterization of isoperimetric sets inside almost-convex cones. Zbl 1355.49035
Figalli, Alessio; Baer, Eric
2017
Quantitative stability of the Brunn-Minkowski inequality for sets of equal volume. Zbl 1369.49065
Figalli, Alessio; Jerison, David
2017
Lipschitz changes of variables between perturbations of log-concave measures. Zbl 1383.60006
Colombo, Maria; Figalli, Alessio; Jhaveri, Yash
2017
Partial $$W^{2,p}$$ regularity for optimal transport maps. Zbl 1364.35058
Chen, Shibing; Figalli, Alessio
2017
Regularity of codimension-1 minimizing currents under minimal assumptions on the integrand. Zbl 1386.49060
Figalli, Alessio
2017
BMO-type norms related to the perimeter of sets. Zbl 1352.46026
Ambrosio, Luigi; Bourgain, Jean; Brezis, Haim; Figalli, Alessio
2016
Universality in several-matrix models via approximate transport maps. Zbl 1402.60009
Figalli, Alessio; Guionnet, Alice
2016
On the density function on moduli spaces of toric 4-manifolds. Zbl 1386.53103
Figalli, Alessio; Pelayo, Álvaro
2016
Nonlinear bounds in Hölder spaces for the Monge-Ampère equation. Zbl 1338.35196
Figalli, Alessio; Jhaveri, Yash; Mooney, Connor
2016
Weak KAM theory for a weakly coupled system of Hamilton-Jacobi equations. Zbl 1355.49023
Figalli, Alessio; Gomes, Diogo; Marcon, Diego
2016
Stability results on the smoothness of optimal transport maps with general costs. Zbl 1406.35120
Chen, Shibing; Figalli, Alessio
2016
Isoperimetry and stability properties of balls with respect to nonlocal energies. Zbl 1312.49051
Figalli, A.; Fusco, N.; Maggi, F.; Millot, V.; Morini, M.
2015
Transport maps for $$\beta$$-matrix models and universality. Zbl 1330.49046
Bekerman, F.; Figalli, A.; Guionnet, A.
2015
Partial regularity for optimal transport maps. Zbl 1325.49051
De Philippis, Guido; Figalli, Alessio
2015
Existence and uniqueness of maximal regular flows for non-smooth vector fields. Zbl 1348.34039
Ambrosio, Luigi; Colombo, Maria; Figalli, Alessio
2015
Quantitative stability for sumsets in $$\mathbb R^n$$. Zbl 1325.49052
Figalli, Alessio; Jerison, David
2015
Optimal regularity of the convex envelope. Zbl 1316.49019
De Philippis, Guido; Figalli, Alessio
2015
Boundary $$\varepsilon$$-regularity in optimal transportation. Zbl 1307.49045
Chen, Shibing; Figalli, Alessio
2015
An overview of unconstrained free boundary problems. Zbl 1353.35322
Figalli, Alessio; Shahgholian, Henrik
2015
Closing Aubry sets. II. Zbl 1377.37090
Figalli, A.; Rifford, L.
2015
A general class of free boundary problems for fully nonlinear parabolic equations. Zbl 1338.35508
Figalli, Alessio; Shahgholian, Henrik
2015
Closing Aubry sets. I. Zbl 1321.37061
Figalli, A.; Rifford, L.
2015
A note on the dimension of the singular set in free interface problems. Zbl 1340.49044
De Philippis, Guido; Figalli, Alessio
2015
Generic hyperbolicity of aubry sets on surfaces. Zbl 1352.37169
Contreras, G.; Figalli, A.; Rifford, L.
2015
Stability results for the Brunn-Minkowski inequality. Zbl 1346.49068
Figalli, Alessio
2015
Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces. Zbl 1316.35061
Barrios, Begoña; Figalli, Alessio; Valdinoci, Enrico
2014
The Monge-Ampère equation and its link to optimal transportation. Zbl 06377770
De Philippis, Guido; Figalli, Alessio
2014
Strongly nonlocal dislocation dynamics in crystals. Zbl 1304.35731
Dipierro, Serena; Figalli, Alessio; Valdinoci, Enrico
2014
A general class of free boundary problems for fully nonlinear elliptic equations. Zbl 1326.35137
Figalli, Alessio; Shahgholian, Henrik
2014
A global existence result for the semigeostrophic equations in three dimensional convex domains. Zbl 1287.35064
Ambrosio, Luigi; Colombo, Maria; De Philippis, Guido; Figalli, Alessio
2014
Regularity results for very degenerate elliptic equations. Zbl 1282.35175
Colombo, Maria; Figalli, Alessio
2014
Perimeter of sets and BMO-type norms. (Périmètre d’ensembles et normes de type BMO.) Zbl 1316.46026
Ambrosio, Luigi; Bourgain, Jean; Brezis, Haïm; Figalli, Alessio
2014
How to recognize convexity of a set from its marginals. Zbl 1294.26022
Figalli, Alessio; Jerison, David
2014
Higher integrability for minimizers of the Mumford-Shah functional. Zbl 1316.49044
De Philippis, Guido; Figalli, Alessio
2014
Quantitative stability results for the Brunn-Minkowski inequality. Zbl 1373.49054
Figalli, Alessio
2014
WKB analysis of Bohmian dynamics. Zbl 1341.35139
Figalli, A.; Klein, C.; Markowich, P.; Sparber, C.
2014
An excess-decay result for a class of degenerate elliptic equations. Zbl 1286.35114
Colombo, Maria; Figalli, Alessio
2014
A geometric approach to correlation inequalities in the plane. Zbl 1288.60024
Figalli, A.; Maggi, F.; Pratelli, A.
2014
Partial regularity results in optimal transportation. Zbl 1323.49023
de Philippis, G.; Figalli, A.
2014
Regularity of solutions to the parabolic fractional obstacle problem. Zbl 1277.35088
Caffarelli, Luis; Figalli, Alessio
2013
Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation. Zbl 1307.26027
Carlen, Eric A.; Figalli, Alessio
2013
$$W^{2,1}$$ regularity for solutions of the Monge-Ampère equation. Zbl 1286.35107
De Philippis, Guido; Figalli, Alessio
2013
Asymptotics of the $$s$$-perimeter as $$s \searrow 0$$. Zbl 1275.49083
Dipierro, Serena; Figalli, Alessio; Palatucci, Giampiero; Valdinoci, Enrico
2013
A sharp stability result for the relative isoperimetric inequality inside convex cones. Zbl 1262.49045
Figalli, A.; Indrei, E.
2013
Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation. Zbl 1286.46035
Figalli, Alessio; Maggi, Francesco; Pratelli, Aldo
2013
A note on interior $$W^{2,1+ \varepsilon }$$ estimates for the Monge-Ampère equation. Zbl 1280.35153
De Philippis, G.; Figalli, A.; Savin, O.
2013
Hölder continuity and injectivity of optimal maps. Zbl 1281.49037
Figalli, Alessio; Kim, Young-Heon; McCann, Robert J.
2013
Regularity of optimal transport maps on multiple products of spheres. Zbl 1268.49053
Figalli, Alessio; Kim, Young-Heon; McCann, Robert J.
2013
On the isoperimetric problem for radial log-convex densities. Zbl 1307.49046
Figalli, A.; Maggi, F.
2013
Second order stability for the Monge-Ampère equation and strong Sobolev convergence of optimal transport maps. Zbl 1278.35090
De Philippis, Guido; Figalli, Alessio
2013
On sets of finite perimeter in Wiener spaces: reduced boundary and convergence to halfspaces. Zbl 1282.28002
Ambrosio, Luigi; Figalli, Alessio; Runa, Eris
2013
Sobolev regularity for Monge-Ampère type equations. Zbl 1271.49031
De Philippis, Guido; Figalli, Alessio
2013
On supporting hyperplanes to convex bodies. Zbl 1288.52001
Figalli, Alessio; Kim, Young-Heon; McCann, Robert J.
2013
Aubry sets, Hamilton-Jacobi equations, and the Mañé conjecture. Zbl 1364.37137
Figalli, A.; Rifford, L.
2013
Stability in geometric and functional inequalities. Zbl 1364.49059
Figalli, Alessio
2013
Non-local gradient dependent operators. Zbl 1252.35099
Bjorland, C.; Caffarelli, L.; Figalli, A.
2012
Nonlocal tug-of-war and the infinity fractional Laplacian. Zbl 1235.35278
Bjorland, Clayton; Caffarelli, Luis; Figalli, Alessio
2012
Nearly round spheres look convex. Zbl 1241.53031
Figalli, Alessio; Rifford, Ludovic; Villani, Cedric
2012
Existence of Eulerian solutions to the semigeostrophic equations in physical space: The 2-dimensional periodic case. Zbl 1258.35164
Ambrosio, Luigi; Colombo, Maria; De Philippis, Guido; Figalli, Alessio
2012
Confinement in nonlocal interaction equations. Zbl 1233.35032
Carrillo, J. A.; Di Francesco, M.; Figalli, A.; Laurent, T.; Slepčev, D.
2012
Total variation flow and sign fast diffusion in one dimension. Zbl 1242.35049
Bonforte, Matteo; Figalli, Alessio
2012
Semiclassical limit for mixed states with singular and rough potentials. Zbl 1264.81177
Figalli, Alessio; Ligabò, Marilena; Paul, Thierry
2012
Isoperimetric-type inequalities on constant curvature manifolds. Zbl 1250.58008
Figalli, Alessio; Ge, Yuxin
2012
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Zbl 1215.35045
Carrillo, J. A.; Difrancesco, M.; Figalli, A.; Laurent, T.; Slepčev, D.
2011
On the shape of liquid drops and crystals in the small mass regime. Zbl 1279.76005
Figalli, A.; Maggi, F.
2011
Semiclassical limit of quantum dynamics with rough potentials and well-posedness of transport equations with measure initial data. Zbl 1231.35186
Ambrosio, Luigi; Figalli, Alessio; Friesecke, Gero; Giannoulis, Johannes; Paul, Thierry
2011
...and 46 more Documents
all top 5

### Cited by 1,644 Authors

 82 Figalli, Alessio 46 Valdinoci, Enrico 39 Carrillo de la Plata, José Antonio 27 Dipierro, Serena 23 Novaga, Matteo 23 Röckner, Michael 20 Ambrosio, Luigi 19 Santambrogio, Filippo 17 Ros-Oton, Xavier 16 Fusco, Nicola 16 Rifford, Ludovic 15 Savin, Ovidiu V. 15 Shaposhnikov, Stanislav V. 14 Bogachev, Vladimir Igorevich 14 Maggi, Francesco 13 Vazquez, Juan Luis 12 Choi, Young-Pil 12 Ciraolo, Giulio 12 Luo, Dejun 12 Vauchelet, Nicolas 11 Brasco, Lorenzo 11 Caffarelli, Luis Ángel 11 Carlier, Guillaume 11 Indrei, Emanuel 11 McCann, Robert J. 11 Pratelli, Aldo 10 Gigli, Nicola 10 Goldman, Michael 10 Maldonado, Diego 10 Serra Montolí, Joaquim 10 Wang, Xu-Jia 9 Cabré, Xavier 9 Colombo, Maria 9 De Philippis, Guido 9 Fetecau, Razvan C. 9 Le, Nam Q. 9 Neumayer, Robin 8 Barrios Barrera, Begoña 8 Chen, Shibing 8 Cinti, Eleonora 8 Flandoli, Franco 8 Kitagawa, Jun 8 Rossi, Julio Daniel 8 Sire, Yannick 8 Topaloglu, Ihsan 8 Trevisan, Dario 7 Böröczky, Karoly jun. 7 Cesaroni, Annalisa 7 Cozzi, Matteo 7 Da Prato, Giuseppe 7 del Barrio, Eustasio 7 Fall, Mouhamed Moustapha 7 Fathi, Max 7 Jabin, Pierre-Emmanuel 7 Kim, Young-Heon 7 Lee, Paul W. Y. 7 Ohta, Shin-ichi 7 Patacchini, Francesco Saverio 7 Paul, Thierry 7 Ruffini, Berardo 7 Salani, Paolo 7 Shahgholian, Henrik 7 Velichkov, Bozhidar 7 Villani, Cédric 7 Warma, Mahamadi 7 Zhang, Xicheng 6 Abdellaoui, Boumediene 6 Bonforte, Matteo 6 Crippa, Gianluca 6 Guillen, Nestor 6 James, François 6 Lee, Ki-Ahm 6 Liu, Jiakun 6 Medina, Maria 6 Palatucci, Giampiero 6 Park, Jinwan 6 Pass, Brendan W. 6 Patrizi, Stefania 6 Peral Alonso, Ireneo 6 Slepčev, Dejan 6 Spolaor, Luca 5 Balogh, Zoltán M. 5 Bertozzi, Andrea Louise 5 Brézis, Haïm 5 Bucur, Claudia 5 Carlen, Eric Anders 5 Choksi, Rustum 5 Cicalese, Marco 5 Craig, Katy 5 da Silva, João Vítor 5 Del Pino, Manuel A. 5 del Teso, Félix 5 Erdős, László 5 Fang, Shizan 5 Fernández-Real, Xavier 5 Frank, Rupert L. 5 Gallouët, Thomas O. 5 Garofalo, Nicola 5 Guionnet, Alice 5 Julin, Vesa ...and 1,544 more Authors
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### Cited in 264 Serials

 113 Calculus of Variations and Partial Differential Equations 65 Archive for Rational Mechanics and Analysis 63 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 57 Journal of Functional Analysis 54 Journal of Differential Equations 44 Advances in Mathematics 37 Discrete and Continuous Dynamical Systems 35 SIAM Journal on Mathematical Analysis 33 Journal of Mathematical Analysis and Applications 30 Journal de Mathématiques Pures et Appliquées. Neuvième Série 28 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 27 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 26 Communications in Mathematical Physics 20 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 19 Communications in Partial Differential Equations 18 Transactions of the American Mathematical Society 17 Communications on Pure and Applied Mathematics 17 The Journal of Geometric Analysis 16 Communications on Pure and Applied Analysis 15 Mathematische Annalen 15 Probability Theory and Related Fields 15 Journal of the European Mathematical Society (JEMS) 14 Journal of Evolution Equations 13 Annali di Matematica Pura ed Applicata. Serie Quarta 13 Potential Analysis 13 NoDEA. Nonlinear Differential Equations and Applications 12 The Annals of Probability 12 Inventiones Mathematicae 12 Manuscripta Mathematica 12 Revista Matemática Iberoamericana 11 Proceedings of the American Mathematical Society 11 Bulletin des Sciences Mathématiques 11 Comptes Rendus. Mathématique. Académie des Sciences, Paris 10 Nonlinearity 10 Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni 10 Communications in Contemporary Mathematics 10 Advances in Calculus of Variations 9 Geometric and Functional Analysis. GAFA 9 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 9 Science China. Mathematics 8 Journal für die Reine und Angewandte Mathematik 8 Interfaces and Free Boundaries 8 Bulletin of Mathematical Sciences 7 ZAMP. Zeitschrift für angewandte Mathematik und Physik 7 Numerische Mathematik 7 SIAM Journal on Control and Optimization 7 SIAM Journal on Numerical Analysis 7 Acta Applicandae Mathematicae 7 Applied Mathematics Letters 7 Journal of Nonlinear Science 7 Discrete and Continuous Dynamical Systems. Series S 7 Séminaire Laurent Schwartz. EDP et Applications 6 Journal d’Analyse Mathématique 6 Journal of Mathematical Physics 6 Journal of Statistical Physics 6 Mathematics of Computation 6 Applied Mathematics and Optimization 6 Duke Mathematical Journal 6 Mathematische Zeitschrift 6 Journal of Mathematical Sciences (New York) 6 Electronic Journal of Probability 6 Revista Matemática Complutense 6 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 6 Kinetic and Related Models 6 Analysis & PDE 6 Stochastic and Partial Differential Equations. Analysis and Computations 6 Annals of PDE 5 Israel Journal of Mathematics 5 Chinese Annals of Mathematics. Series B 5 Physica D 5 Advanced Nonlinear Studies 5 SIAM Journal on Applied Dynamical Systems 4 Archiv der Mathematik 4 Indiana University Mathematics Journal 4 Publications Mathématiques 4 Memoirs of the American Mathematical Society 4 Quarterly of Applied Mathematics 4 Ricerche di Matematica 4 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 4 Bulletin of the American Mathematical Society. New Series 4 Journal of Dynamics and Differential Equations 4 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 4 SIAM Journal on Scientific Computing 4 Doklady Mathematics 4 Acta Mathematica Sinica. English Series 4 Discrete and Continuous Dynamical Systems. Series B 4 Bollettino dell’Unione Matematica Italiana 3 Journal of Computational Physics 3 Journal of Geometry and Physics 3 The Annals of Statistics 3 Geometriae Dedicata 3 Journal of the Korean Mathematical Society 3 Journal of Optimization Theory and Applications 3 Journal of Theoretical Probability 3 Publicacions Matemàtiques 3 European Journal of Applied Mathematics 3 The Annals of Applied Probability 3 Journal of Mathematical Imaging and Vision 3 St. Petersburg Mathematical Journal 3 Journal of Convex Analysis ...and 164 more Serials
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### Cited in 49 Fields

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text:
# Know Your Overclocking Benchmark and Stress Testing Methods and Tools

## Introduction

Geekbench, 3DMark, Cinebench, Unigine, CPU-Z/GPU-Z, AIDA64, SiSoftware Sandra, Speecy, PassMark, FurMark, MemTest64, PCMark, on and on, finally last but not least Prime95, SuperPi, and y-cruncher. Do you actually use them knowing what these tools actually do? Also for stress testing, are you just going to stay in the same place without hardening your overclocked build that will potentially break down in realistically possible workloads and just keep saying “it will be okay in games”? There are a lot of benchmarks and stress testing applications, but even some professional reviewers either use them without proper utilization or compare different builds without distinguishing the factors correctly. This post explains how benchmarks and stress testing applications should work and also recommend some programs not many have tried to use.

## Benchmarks

First, let’s look at benchmarks. Although people confuse the concept of benchmarks and stress testing tools and lots of them are designed to do both stuffs together, the methodology must not be same. The reason and objective of doing two different things with diferent purposes aren’t same just because the tools are same. Formal benchmarks that are meant to compare one new component of a computer compared to other new ones must be designed to have only one quantitatively different factor (which is the component itself) in the computer (this is middle school science… ). Other variables must be same including all the other components in a computer and the OS/Environment/Software used to benchmark. This is the very fundamental principle in scientific research or investigation, which is the relation of independent variables, dependent variables, and control variables. It is similar to a function. Let’s assume a linear function $y=ax+b$. Independent variables ($x$) are what can be changed. There must only be normally one independent variable in a series of experiments (unless another computer is simultaneously managing multiple variables in that investigation but even so the computer is doing one investigation with one independent variable at a time just pretty fast so people think it is simultaneous), because more than one means that you don’t know which independent variable change how much of the results you see, and thus not being able to deduce a meaningful conclusion. Dependent variables ($y$) are variables that changes inside the system during an investigation as one alters the independent variables. It is dependent to the independent variable, so if the value of the independent variable and control variable are same, it should get the same dependent variable value, so it is called a dependent variable. Control variables ($a$ and $b$) are everything else in that environment that can change the dependent variable value in a way different from the independent variable, thus making the whole experiment invalid if changed anyhow. There are so many amateur-level computer reviewers (I have seen even those that officially receive products from official manufacturers to review, this means they have recognition to manufacturers as being somewhat trustworthy) that ignore this and do not take care of the control variables (such as using different RAM, motherboard, secondary disk, even cooling, etc.), or those that try to take care of these but miss critical aspects that may change results drastically.

So benchmarks, especially those designed to compare one component to others, have to be experiments in an equal environment except that component itself, and changes that result by changing that components represent the subjective performance compared to others. If environments can’t be exactly equal, one should do the same experiment multiple times and make a trustable average value that cancels out abnormalities. But, not everyone is a PC reviewer. Most of you guys just want to check and increase the performance of your system in your room, so compared to optimal experimentation environments people don’t really have to change your components to define independent and dependent variables (again unless you are a reviewer). The CPU performance score in your machine is close to the real maximum performance you will get when you do everyday workloads. In this case, your system on whole is the independent variable if you want to compare with other complete systems. You still have to do it multiple times and average all of them to deduce a conclusion that you know you can trust. But if you want to identify which component is bottlenecking inside your system, you still have to do what the reviewers should do, albeit in a more casual qualitative fashion.

Benchmarks do not necessarily have to squeeze your system to the maximum as most people normally do more casual workloads. Some people want the best performance in word processing. Some want the highest numerical performance in supercomputers. A comparably cheaper (but still quite expensive to most people) gaming computer is better at gaming (usually measured in rendered frames) than a server costing ten times more that is only meant to do numerical computation because the most intensive games matter in latency (of which a server is slower at) of the system than distributed multiprocessing performance.

### Types of Benchmarks

The types of benchmarks: 1. Just using the real-world program you know you will use the most and the metrics you are most sensitive at inside that program. Word, Excel, CAD, Photoshop, Video Editors and Renderers, Electronic Stock Traders, Games, the code you wrote on your own that you execute frequently, etc. These don’t have to pull the system to the extreme, and just compare on the things that they each do best at. But the thing I want to emphasize about graphics and games is that you may be changing your control variables while you don’t know it. The benchmarks should be automated in the exact same conditions and if this is really impossible you should try your best to keep the environment (= how the game is rendered so you shouldn’t make stuff different) as equal as possible and also repeat the same test multiple times to make a trustable average value of the benchmark. 2. Synthetic benchmarks: if you want to build a system that does well in many different types of workloads, synthetic benchmarks generate computational load that resembles, but does not fully represent real-life workloads. Since this is not attached to one type of workload, it is good for getting an overall outline of how this system will work like in a variety of real-world workloads. The focus of most benchmarks aren’t evaluating one component to its maximum, instead getting a peek on how some types of programs will perform. 3. Kernel benchmarks: These test how the OS and its underlying layers and kernels themselves perform, which is the maximum performance of the system plus the OS itself instead of real-world workloads. This tests the maximum possible underlying performance, and thus is more intensive to the system itself. It does not necesarily mean that it will stress test all components in the system, since only the bottlenecked components will run in maximum and others may be more idle (such as if the RAM was not 100% allocated and motherboards do not supply their maximum power designed). This still is a more accurate benchmark if focusing more on the system itself, because it cares less of individual software optimization. Examples are LINPACK, which is used in benchmarking TOP500 Supercomputers and LAPACK, which were developed decades ago but the same algorithm of solving systems of large linear equations and matrices (as in the college linear algebra courses if you took those) tests the very raw performance of a system since computers have first been developed to do best in solving linear problems and it is still the most fundamental underlying role of the computer. 4. I/O benchmarks focus on one or more of the communication throughput and latency of the L1/L2/L3 cache, RAM, the secondary storage systems of the system such as SSDs or HDDs, or even the interconnects between other nodes in a cluster or supercomputer. 5. Database benchmarks test the performance of database servers with Database Management Systems (DBMS) such as SQL DBMSs including MySQL and PostgreSQL, Oracle, Cassandra, Ignite, MongoDB, ArrangoDB, Microsoft Access, and etc. 6. Browser benchmarks test browser rendering and loading in each system such as WebGL rendering of 3D objects in the web or loading speed of websites and web components in a browser. It represents the performance of browsing the modern web which has various static and dynamic contents client-side. 7. Component Benchmarks or Microbenchmarks: these are as simple as the output you get in cat /proc/cpuinfo in Linux or when you open Task Manager in Windows. They are very short benchmarks used for identifying the basic features of a component such as frequency or memory bandwidth.

### Best Practices

Since benchmarking programs are volatile and since the definition of “best” is subjective as many different programs can be used for many different desired workloads, I won’t point them out one by one. I do have to quote to avoid benchmarking tools that have weird scoring metrics that score systems in a questionable way to presumably make some CPU models reach higher scores than others, for example setting a weight of over 80% on single core scores and the remaining 20% on multicore scores. Since even most games nowadays are using multiprocessing in their engines, this scoring weight is really peculiar. The list of tools in the Introduction is a non-exhaustive list of common tools used. Any program is okay as long as you quantitatively compare the correct metrics. Always keep independent variables, dependent variables, and control variables in mind. This is a very easy method to get a trustable result. It is also highly recommended to consult the methodologies that the most trustworthy reviewers use. They can give you a new idea or method of performing benchmarks than the one you currently use as well as the programs for each workloads and you can learn the rigor of experienced computer experimentalists.

Dr. Ian Cutress (look at his Twitter and TechTechPotato Youtube too) is a professional high-performance computing specialist that I can assert is currently the most skillful PC reviewer both in server and desktop levels. I have worked with him before with the world record of Euler-Mascheroni Constant. Because he has years of experience computational chemistry research and simulation during his Ph.D. times in Oxford, his reviews are very reasonable and accurate, tolerating no errors at all. Phoronix is a very good website for Linux benchmarks and methodologies (they even have their own Linux benchmark suite). While the reviews itself are very slightly less rigorous than Dr. Ian Cutress, it still has the great professionalism in every review and is even better on reviews related to development environments and lower-level implementations inside the Linux kernel. Other reviewers aren’t really worse than both of them, but you must filter out unprofessional reviewers that don’t disclose details of the environment (= control variables) used in the benchmarks clearly and also those that just pop out conclusions out of nowhere without releasing the quantitative data that supports that conclusion. Benchmarks should only be based on facts and not be based on opinions.

## Stress Tests

Therefore, this is not an evaluation of “how does this system work in my workload of choice?” and instead being “is the system stablized enough for any kind of stressed scenarios it can meet coincidentally that happens even if users didn’t mean to run such workloads?” evaluation. Therefore the lighter benchmark programs in the section before won’t work. As in the term “light” common graphics and CPU benchmark programs such as 3DMark, Cinebench, PCMark, and Passmark as well as RAM benchmarking/stress testing programs such as Memtest86+, MemTest64 or AIDA64 may not expose instabilities that actually exist but is invisible in most uses. These invisible instabilities are decreasing the lifetime of your components and they are the main reasons your computer gets an error, from as small as a failure of rendering one pixel to as serious as showing a blue screen every hour. But people tend to think the common benchmarking programs are already intensive enough to test stability in the environments they normally use because their CPUs and GPUs already gets hotter than ever (which they deem uncomparably intensive to games they normally do) and coolers spin like mad. There are lots of cases by enthusiast overclockers in the internet that showed what they thought as a “stable overclock” is not stable at all when introduced to more sophisticated and fragile workloads.

So what I want to say is that errors are not a definitive thing. As a system or a component becomes unstable, the probability of generating errors increase and there are undetected or silent errors. I will now introduce overclockers to which stress testing tool they are supposed to use to ensure their overclocked builds will be stable on use-cases up to very fragile numerical computations.

I personally don’t recommend Prime95 (although this is the most used program) because it gives people a false sense of security but I’ll go through this program to start. This program mainly runs an algorithmic test called the Lucas-Lehmer Primality Test or the Fermat Probable Prime Test to find new world record Mersenne primes that can provide insight on encryption methods such as RSA. This requires a fair bit of precision and also stresses the systems a fair bit (this is one of a very few programs that have implemented the new AVX-512 vector instructions). If you run this program and you see errors or a blue screen, it means your overclock is not stable. But the downside is that the stress test using this program cannot ensure 100% that the CPU and RAM has numerical precision in all situations. There has been cases where runs of Prime95 for hours were “stable” but with the benchmarks I am introducing now they corrupted very fast.

The first program is y-cruncher. This is the program that computed world records of various important mathematical constants such as Pi (Google has been a recent world record holder and did a great amount of promotion with this record). This is a program that focuses on the highest precision and does not tolerate any kind of errors because one has to calculate hundreds of billions to tens of trillions of digits correctly. One error in the computation means that all digits generated later than the error are corrupted with it and outputting incorrect digits, disqualifying them from being world records. Because of this, the same constant has to be computed in two different independent algorithms to be recognized as the correct accurate world record. This is the same for discovered Mersenne primes in Prime95, but the required precision and accuracy for y-cruncher is far higher than Prime95. The AVX-512 vector instructions are also a core part of y-cruncher and are used in the stress tests if the CPU supports each instruction.

The stress testing mode tests your system to withstand certain Fast Fourier Transform operations and other very heavy linear algebra computations comparable to the Linpack Xtreme tool coming up next. If you also run the Benchmark Pi RAM-only mode and set the number of digits to a size that will saturate your RAM almost totally, you can verify that the output is the same as the correct results saved in the y-cruncher program while the CPU and RAM are utilized in the most intensive way a real-world computation can go. Consult This Post for more details on how y-cruncher should be operated and also for an in-depth explanation of optimizing the configurations of y-cruncher to compute world records as well, and my posts in This Category on examples of how intensive in terms of precision y-cruncher gets. I strongly recommend this program along with Linpack Xtreme as I have used this multiple times to set world records. It is really not easy to get more intensive than this.

Linpack Xtreme for Windows and Linux is another good program that detects corruptions faster than many other programs. As I said in Types of Benchmarks, LINPACK focuses on the purpose of the computer when it was first manufactured; solving complex linear algebra problems. This is a very good stress testing program provided that this is an algorithm used in huge supercomputers and it really utilizes both the CPU and RAM to a scale not possible by most viable workloads. This is equally as good as or even better than y-cruncher can also verify precision computing workloads if you run the Benchmark Pi in RAM option and saturate the RAM. On the contrary, Linpack Xtreme is more intensive and also serves a purpose as a standardized benchmarking program so it is up to your choice on what you prefer. Why not both?

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## Cryptology ePrint Archive: Report 2006/029

Authenticated Hybrid Encryption for Multiple Recipients

Stéphanie Alt

Abstract: Authenticated encryption schemes used in order to send one message to one recipient have received considerable attention in the last years. We investigate the case of schemes, we call authenticated $\mathtt{1{\to}n}$ schemes, that allow one to encrypt efficiently in a public-key setting a message for several, say $n$, recipients in an authenticated manner. We propose formal security definitions for such schemes that work also for $n=1$ and which are stronger and/or more general than those currently proposed. We then present a flexible mode of operation that transforms any $\mathtt{1{\to}1}$ authenticated encryption scheme working on small messages into a $\mathtt{1{\to}n}$ authenticated encryption scheme working on longer messages. We show that it allows the construction of efficient $\mathtt{1{\to}n}$ schemes that are proved secure for the strongest security notion.

Category / Keywords: public-key cryptography / Public key encryption, Authentication,


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### Translations

Code Language Translator Run
es es es

### Software Requirements

SoftwareRequirements

 Android iOS Windows MacOS with best with Chrome Chrome Chrome Chrome support full-screen? Yes. Chrome/Opera No. Firefox/ Samsung Internet Not yet Yes Yes cannot work on some mobile browser that don't understand JavaScript such as..... cannot work on Internet Explorer 9 and below

### Credits

Wolfgang Christian - Davidson College; lookang

### Three State Decay Theory

A three-state radioactive sample. Radionuclides change from red to green to blue as they decay. If the number of nuclides is large, the upper state N1, intermediate state N2, and final state N3 populations obey the following coupled ordinary differential equations (ODEs)

$$\frac{dN_{1}}{dt} = - \lambda_{1}N_{1}$$

$$\frac{dN_{2}}{dt} = + \lambda_{1}N_{1} - \lambda_{2}N_{2}$$

$$\frac{dN_{3}}{dt} = - \lambda_{2}N_{2}$$

where λ1 is the upper state decay rate and λ2 is the intermediate state decay rate. The nature of the decay is governed by these decay constants. The Three State Nuclear Decay Model displays both the continuous ODE solution for these nuclide populations as well as a stochastic (probabilistic) solution that assumes integer nuclide populations and uses probability to determine if a radionuclide survives for a time Δt without decaying.

### Discrete decay model

The differential equation model produces continuous decay curves but hides the random and discrete nature of the underlying processes. A stochastic model that "rolls the dice" to determine if a radionuclide decays shows that the continuous model is only an approximation. The simulation computes the probably of decay between states 1->2, 2->3, and 1->3 during the finite time interval Δt. When the simulation is run, these probabilities are applied to each radioactive nucleus to determine if it decays. Note that there is a finite probability that a nuclide decays directly from state 1 to the stable state 3 within the time interval . Nature does not, however, skip state 2. We merely waited too long and the nuclide finished its decay before it was observed in state 2.

### Introduction

The Three State Nuclear Decay JavaScript Model extends the Two State Nuclear Decay JavaScript Model to simulate the radioactive decay of atomic nuclei in which the parent nucleus first decays into an intermediate state before decaying into a stable state. Although the decay of both the parent and intermediate nucleus (radionuclides) is spontaneous and unpredictable, the probability of decay of each radionuclide is constant and is usually known. The model displays a color-coded sample with N1 parent nuclides, N2 intermediate state nuclides, and N3 stable state nuclides. Before the simulation starts, users can set the initial numbers N1 and N2, the decay constants k1 and k2, and the time interval between measurements Δt. The simulation counts the number of decay events ΔN1 and ΔN2 within Δt and stops when all nuclides are in the stable state.

The plot shows the time evolution of the nuclides as well as the number of decay events in each Δt time interval to allow users to compare the data generated by the random decay model with a differential equation-based model as described on the Theory page. The Radioactivity check box switches the view to show the number of decay events ΔN within the Δt interval as described on the Theory page. The Particles check box enlarges the nuclide view.

### Credits:

The Three State Nuclear Decay model was developed by Wolfgang Christian using version 5.2 of the Easy Java/JavaScript Simulations (EjsS) authoring and modeling tool. Although EjsS is a Java program, it creates ready-to-run JavaScript simulations and ePubs. Information about EJS is available at: < http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection < http://www.compadre.org/OSP/>.

### Version:

1. http://weelookang.blogspot.sg/2011/01/ejs-open-source-decay-parent-daughter-g.html Blogpost remixing from Java to JavaScript by Loo Kang Wee
2. http://www.compadre.org/osp/items/detail.cfm?ID=13991 by JavaScript Version by Wolfgang Christian
3. http://www.compadre.org/osp/items/detail.cfm?ID=10577 by Java Version by Wolfgang Christian

[text]


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# ct.category theory: are quotients by equivalence relations "better" than assumptions?

The nLab page you are looking for is called factoring systems. Here is my favorite, which I think answers your question. In any category with limits and finite colimitos, all morphism. $$f: X to Y$$ has a canonical factorization

$$X to text {coim} (f) to text {im} (f) to Y$$

where $$text {im} (f)$$, the regular image, is the equalizer of the cokernel pair of $$f$$ (this is the "nonabelian" version of "kernel of the cokernel") and $$text {coim} (f)$$, the regular coimage, is the coequalizer of the kernel pair of $$f$$ (again, the "nonabelian" version of "kernel cokernel"). These two constructions are categorically dual and, therefore, among other things, the image-coimage factorization of $$f$$ in the opposite category is the same sequence of maps but in the opposite order.

In $$text {Set}$$, the coimage and the image are the image of a function in the usual sense, but they are calculated in different ways, which I think coincide with the distinction that is obtained. $$text {coim} (f)$$ is calculated, more or less, by building the equivalence relation in $$X$$ defined by $$x_1 sim x_2 Leftrightarrow f (x_1) = f (x_2)$$, then quoting $$X$$ For this. $$text {im} (f)$$ it is calculated categorically double, although at the beginning it seems a little strange: when building the pushout first $$Y sqcup_X Y$$, then isolating the subset of $$Y$$ of elements that are sent to the same element through both canonical maps $$Y a Y sqcup_X Y$$.

In particular, the factorization you want for an injection is the normal image factorization, and the factorization you want for an injection is the normal image factorization, so in fact they are categorically dual. The full coimage image factorization combines these.

It is a nontrivial theorem that the map $$text {coim} (f) to text {im} (f)$$ it's an isomorphism in $$text {Set}$$. It is also an isomorphism in any Abelian category and in $$text {Grp}$$ (This is an abstract form of the first isomorphism theorem), but in general it is just a monomorphism and an epimorphism. A very instructive example is $$text {Up}$$, where $$text {coim} (f)$$ is the image of topologized set theory as a quotient of $$X$$Y $$text {im} (f)$$ is the image of topologized set theory as a subspace of $$Y$$. (Note that these match for compact Hausdorff spaces!)


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## Tsukuba Journal of Mathematics

### Analytic properties of generalized Mordell-Tornheim type of multiple zeta-functions and $L$-functions

Takashi Miyagawa

#### Abstract

Analytic properties of three types of multiple zeta functions, that is, the Euler-Zagier type, the Mordell-Tornheim type and the Apostol-Vu type have been studied by a lot of authors. In particular, in the study of multiple zeta functions of the Apostol-Vu type, a generalized multiple zeta function, including both the Euler-Zagier type and the Apostol-Vu type, was introduced. In this paper, similarly we consider generalized multiple zeta-functions and $L$-functions, which include both the Euler-Zagier type and the Mordell-Tornheim type as special cases. We prove the meromorphic continuation to the multi-dimensional complex space, and give the results on possible singularities.

#### Article information

Source
Tsukuba J. Math., Volume 40, Number 1 (2016), 81-100.

Dates
Revised: 14 January 2016
First available in Project Euclid: 24 September 2016

https://projecteuclid.org/euclid.tkbjm/1474747488

Digital Object Identifier
doi:10.21099/tkbjm/1474747488

Mathematical Reviews number (MathSciNet)
MR3550933

Zentralblatt MATH identifier
06642042

#### Citation

Miyagawa, Takashi. Analytic properties of generalized Mordell-Tornheim type of multiple zeta-functions and $L$-functions. Tsukuba J. Math. 40 (2016), no. 1, 81--100. doi:10.21099/tkbjm/1474747488. https://projecteuclid.org/euclid.tkbjm/1474747488


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# Deriving symmetries of a Gauge theory

Hello, I don't know if this is a good place for exposing my problem but I'll try...

I have a gauge theory with action:

$S=\int\;dt L=\int d^4 x \;\epsilon^{\mu\nu\rho\sigma} B_{\mu\nu\;IJ} F_{\mu\nu}^{\;\;IJ}$

Where $B$ is an antisymmetric tensor of rank two and $F$ is the curvature of a connection $A$ i.e: $F=dA+A\wedge A$, $\mu,\nu...$ are space-time indices and $I,J...$ are Lie Algebra indices (internal indices) I would like to find its symmetries. So I rewrite the Lagrangian by splitting time and space indices $\{\mu,\nu...=0..3\}\equiv \{O; i,j,...=1..3\}$ I find:

$L = \int d^3 x\;(P^i_{\;IJ}\dot{A}_i+B_i^{\,IJ}\Pi^i_{\,IJ}+A_0^{\;IJ}\Pi_{IJ})$

Where $\dot{A}_i = \partial_0 A_i$, $P^i_{\;IJ} = 2\epsilon^{ijk}B_{jk\,IJ}$ is hence the conjugate momentum of $A_i^{\,IJ}$

$B_i^{\,IJ}$ and $A_0^{\;IJ}$ being Lagrange multipliers we obtain respectively two primary and two secondary constraints:

$\Phi_{IJ} = P^0_{\;IJ} \approx0$

$\Phi_{\;\;IJ}^{\mu\nu} = P^{\mu\nu}_{\;\;IJ} \approx0$

$\Pi^i_{\,IJ} = 2\epsilon^{ijk}F_{jk\,IJ} \approx0$

$\Pi_{IJ}=(D_i P^i)_{IJ} \approx0$

Where $P^0_{\;IJ}$ are the conjugate momentums of $A_0^{\,IJ}$ and $P^{\mu\nu}_{\;\;IJ}$ those of $B_{\mu\nu}^{\;\;IJ}$. Making these constraints constant in time produces no further constraints.

Whiche gives us a general constraint:

$\Phi = \int d^3 x \;(\epsilon^{IJ}P^0_{\,IJ}+\epsilon_{\mu\nu}^{IJ}\;P^{\mu\nu}_{\;\;IJ}+\eta^{IJ}\Pi_{IJ}+\eta_i^{IJ}\Pi^i_{\;IJ})$

Each quantity $F$ have thus a Gauge transformation $\delta F = \{F,\Phi\}$ where $\{...\}$ denotes the Poisson bracket.

Knowing that this theory have the following Gauge symmetry:

$\delta A = D\omega$

$\delta B = [B,\omega]$

Where $\omega$ is a 0-form, I would like to retrieve these transformations using the relation below. (where $\Phi$ is considered as the generator of the Gauge symmetry) but my problem is that I don't know how to proceed, I already did this with a Yang-Mills theory and it worked... but for this theory it seems to le intractable! Someone to guide me?

-

Unless for some reason you absolutely must work within the Hamiltonian approach, you can just directly look for the complete set of (infinitesimal Lie point) symmetries of the Euler--Lagrange equations or of the action itself. The procedure is standard and described in many good books. For instance, you can look into those by Olver (more math-y) or Stephani (somewhat closer to physics). Using the theory from these books you can also verify whether the transformation at the end of your question is indeed a symmetry.

-
Although the second book looks very interesting, I hesitate to start reading for the following reason: - Symmetries in physics are related to the indetermination generated by the fact that the Hamiltonian contains constraints times arbitrary parameters, if one tries to write down two possible future states corresponding to two different parameters (using the Poisson bracket with the Hamiltonian), these two states will be physically equivalent whereas different by a term consisting of constraints times arbitrary parameters. Suggesting that constraints are the generators of these Gauge transf. –  Pedro Mar 14 '10 at 11:19
So I don't thing that a pure algebraic method will exhibit these "Gauge transformations"!! On the other hand, I confess I misunderstand the fact that a Lagrangian which did not ask anyone find itself invariant under symmetries exhibited by the passage to the Hamiltonian formalism!! I mean, a priori the form of a Lagrangian has no relation with will be done in the Hamiltonian framework. –  Pedro Mar 14 '10 at 11:28
PS: I don't want to verify that this is a symmetry, it's already done. I want to derive this symmetry, actually I have a little more complicated action which I need to analyse... –  Pedro Mar 14 '10 at 12:26
First of all, the method outlined above is systematic and it should yield, inter alia, the gauge symmetries (and also the obvious symmetries like, say, Poincare or conformal ones). Moreover, while the notion of symmetry can be considered within the Hamiltonian framework, the notion of symmetry (including but not restricted to the gauge symmetry) does make perfect sense without reference to this framework, see e.g. Olver's book I linked above. I hope that perhaps some other people can better understand the menaing of your comments and be of more help. –  mathphysicist Mar 15 '10 at 23:29

Although this question is over a year old, some readers may be interested in the following comments.

1. The symmetries of the Lagrangian and the action are not identical.

2. I don't know why you cannot recover the gauge transformations from the constraints, perhaps it has to do with not including the $B_{0i}$ fields in the second expression for the Lagrangian?

3. The constraints give infinitesimal transformations leading to the usual gauge transformations, but there seem to be symmetries which are non-trivial extensions of these.

4. For example, for a gauge group SU(n), the transformations $$A' = UAU^\dagger + \phi$$

$$B' = UBU^\dagger + UAU^\dagger\wedge\phi + \phi\wedge UAU^\dagger + \phi\wedge\phi$$

where $U(x) \in$ SU(n), and $\phi = -dUU^\dagger$, leaves the action invariant.

-
Thank you for these notes, I just want to say that I finally found how to derive these Gauge transformations (more or less because I arrived to derive them for spatial components only), the demonstration s here docs.google.com/… until page 10 (in french) –  Pedro Apr 30 '11 at 8:38
You are welcome. Getting the transformations for the spatial components should be sufficient for the dynamics, and you can extend them to all the components. –  Amitabha Lahiri May 2 '11 at 7:32


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## Maxout Networks

Researching for my master thesis I tried to understand the paper by Goodfellow et al. on the Maxout Units. I found it very hard understanding the details and thought a clear explanation in combination with a nice figure would be really helpful. So this is my shot at doing so.

Please note, that everything explaned here was not developed by me, but is just an explanation of the paper by Goodfellow et al.

• Maxout is an activation function
• supposed to be combined with dropout
• that minimizes the model averaging approximation error when using dropout
• is a piecewise linear approximation to an arbitrary convex function

### Definition

\begin{aligned} h_{i} \left( x \right) &= \max_{j\in\left[1,k\right]}\left(z_{ij}\right) \\ z_{ij} &= x^{T} W_{\dots ij} + b_{ij} \\ \end{aligned}

$h$Maxout function
$x$Input ($\in \mathbb{R}^{d}$)
$W$4D tensor of learned weights ($\in \mathbb{R}^{d\times m \times k}$)
$d$Number of input units (length of x)
$m$Number of units in each linear feature extractor (complexity)
$k$Number of linear feature extractors
$b$Matrix of learned biases ($\in \mathbb{R}^{m\times k}$)
$i$Runs over the number of Maxout units ($\in \left[1,m \right]$)
$j$Runs over the number of feature extractors ($\in \left[1,k \right]$)

### Illustration

Now, here is how a single layer with five Maxout units and three hidden linear feature extractors looks like. Try hovering units with the mouse to better see the connection scheme.

“But wait, this looks more like at least two layers!”

Yes indeed, this is very important and probably confusing about Maxout. The activation function is implemented using a small sub-network who’s parameters are learned aswell (“Did somebody say ‘Network in Network’?”).

So if we don’t count the input layer, a single layer of Maxout units actually consists of two layers itself (Although I referred to the first layer as input layer, this doesn’t necessarily mean that it is the very first layer in the whole network, but can be the output of a previous layer, too).

Let’s call the first layer the hidden layer. It implements the linear part of the Maxout units. It is a set of fully-connected layers (the columns in the image) with no activation function (which is referred to as affine), thus each unit in this layer just computes the weighted sum of all inputs, which is defined in the second part of the Maxout definition above:

Don’t get confused about the biases. In this definition they form a matrix, however usually biases are implicit by just adding an additional 1 to the inputs, so that the weight matrix is slightly bigger than for the regular inputs only. So actually the bias is an additional weight in the matrix $W$. Maybe think of the bias matrix in this definition as the slice of weights in the weight matrix.

Now the three-dimensional tensor $W$ contains the weights of this first part. The dots in the equation mean that all elements from the first dimension are taken like W[:, i, j] in Python or W(:, i, j) in Matlab. Consequently, $W_{\dots ij}$ is the weight vector of the unit in row $i$ and column $j$.

In the figure above the units in this first part are aranged in a two-dimensional grid. The first dimension of this grid (number of rows) doesn’t have to matches the number of input units, both $j$ and the second dimension $k$ (number of columns) are hyperparameters, which are chosen when the whole architecture is defined. These two parameters control the complexity of the Maxout activation function. The higher the $k$ and $j$ the more accurately any convex function can be approximated. Basically each column of units in this first part performs linear regression.

The second part is much easier. It is just doing max-pooling over each row of the first part, i.e. taking the maximum of the output along each row.

### A simple example

Consider the function $f\left(x\right)=x^{2}$.

We can approximate this function with a single Maxout unit that uses three linear pieces $k=3$. So it uses three hidden units.

This Maxout unit would look like this (biases included this time):

Each hidden unit calculates:

This is a simple linear function. The max-pooling unit takes the maximum of these three linear functions.

Take a look at this picture. It shows the $x^{2}$ function and three linear functions that could be learned by the Maxout unit.

Finally, try imagining how this would look like with 4, 5, 6 or an arbitrary number of linear functions. That’s right, it would be a nice approximation that is linear everywhere, except for the connection points of the linear parts.

### Where is Dropout in all this?

Dropout is a regularization mechanism. It simulates the training of a bag of networks with different architectures. However, this bag of networks contains only those networks that can be created by dropping an arbitrary number of units from the original network. In practice this is implemented by randomly dropping neurons with a certain probability during training. As a result in each training pass a kind of different network is trained, however – and this is important – all networks share the same weights, because acutally only a single network is used and just the weights of this single network are used all the time.

When applying bagging, i.e. using $n$ models for prediction rather than just one, it is necessary to combine the predictions of all models, which can be easily done by calculating the mean of all predictions. In the dropout case this can not be done, because we actually have only one model. (Ok, it could be done but it doesn’t make sense, since the number of models is way too big). Instead, a much better approach is to just scale the weights the whole network proportional to the drop ratio. The issue then is, that this is only accurate for a single linear layer with the Softmax function applied. Thus, for deeper models that use a non-linear activation function it is not accurate anymore.

And that is why Maxout works as we have seen above. Dropout can be applied to the first part of Maxout and model averaging is still accurate, because there are no non-linearities involved. Pretty clever.


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# Find ∂/∂y ∂F/dx (0,0) given F(x,y)

Attached.

## Homework Equations

Just taking derivatives.

## The Attempt at a Solution

Basically, at first I was thinking I can differentiate because the original function is continuous at (0,0) so I do and get: http://www.wolframalpha.com/input/?i=d/dy+(d/dx+(xy(8x^2+%2B+4y^2)/(x^2%2By^2))). Correct me if I am wrong but I was thinking I had to differentiate this twice partially as the Wolfram Alpha link shows and then plug in (0,0) to the answer that Wolfram Alpha gave but that is also not defined. I tried taking the limit with y = 0 and x = 0 respectively and found that the limit as (x,y) => (0,0) of the differentiated part does not exist. I then thought that wait, having F(0,0) = 0 be defined does not mean that it is continuous so I shouldn't be able to differentiate in the first place. I am now completely confused as to what I need to do.

#### Attachments

• 19.8 KB Views: 353

Related Calculus and Beyond Homework Help News on Phys.org
Try
d/dx (x*y*(8*x^2+4*y^2)/(x^2+y^2)) for x=0
and you get 4y
Next try
d/dy (x*y*(8*x^2+4*y^2)/(x^2+y^2)) for y=0
and you get 8x
So the second derivatives will give you different results.


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• On February 17 2016, the IceCube real-time neutrino search identified, for the first time, three muon neutrino candidates arriving within 100 s of one another, consistent with coming from the same point in the sky. Such a triplet is expected once every 13.7 years as a random coincidence of background events. However, considering the lifetime of the follow-up program the probability of detecting at least one triplet from atmospheric background is 32%. Follow-up observatories were notified in order to search for an electromagnetic counterpart. Observations were obtained by Swift's X-ray telescope, by ASAS-SN, LCO and MASTER at optical wavelengths, and by VERITAS in the very-high-energy gamma-ray regime. Moreover, the Swift BAT serendipitously observed the location 100 s after the first neutrino was detected, and data from the Fermi LAT and HAWC were analyzed. We present details of the neutrino triplet and the follow-up observations. No likely electromagnetic counterpart was detected, and we discuss the implications of these constraints on candidate neutrino sources such as gamma-ray bursts, core-collapse supernovae and active galactic nucleus flares. This study illustrates the potential of and challenges for future follow-up campaigns.
• ### The curtain remains open: NGC 2617 continues in a high state(1701.05042)

Optical and near-infrared photometry, optical spectroscopy, and soft X-ray and UV monitoring of the changing look active galactic nucleus NGC 2617 show that it continues to have the appearance of a type-1 Seyfert galaxy. An optical light curve for 2010-2016 indicates that the change of type probably occurred between 2010 October and 2012 February and was not related to the brightening in 2013. In 2016 NGC 2617 brightened again to a level of activity close to that in 2013 April. We find variations in all passbands and in both the intensities and profiles of the broad Balmer lines. A new displaced emission peak has appeared in H$\beta$. X-ray variations are well correlated with UV-optical variability and possibly lead by $\sim$ 2-3 d. The $K$ band lags the $J$ band by about 21.5 $\pm$ 2.5 d. and lags the combined $B+J$ filters by $\sim$ 25 d. $J$ lags $B$ by about 3 d. This could be because $J$-band variability arises from the outer part of the accretion disc, while $K$-band variability comes from thermal re-emission by dust. We propose that spectral-type changes are a result of increasing central luminosity causing sublimation of the innermost dust in the hollow biconical outflow. We briefly discuss various other possible reasons that might explain the dramatic changes in NGC 2617.
• ### Early polarization observations of the optical emission of gamma-ray bursts: GRB150301B and GRB150413A(1511.02641)

Nov. 9, 2015 astro-ph.HE
We report early optical linear polarization observations of two gamma-ray bursts made with the MASTER robotic telescope network. We found the minimum polar- ization for GRB150301B to be 8% at the beginning of the initial stage, whereas we detected no polarization for GRB150413A either at the rising branch or after the burst reached the power-law afterglow stage. This is the earliest measurement of the polarization (in cosmological rest frame) of gamma-ray bursts. The primary intent of the paper is to discover optical emission and publish extremely rare (unique) high- quality light curves of the prompt optical emission of gamma-ray bursts during the non-monotonic stage of their evolution. We report that our team has discovered the optical counterpart of one of the bursts, GRB150413A.
• ### The optical identifcation of events with poorly defined locations: The case of the Fermi GBM GRB140801A(1510.07807)

Oct. 27, 2015 astro-ph.HE
We report the early discovery of the optical afterglow of gamma-ray burst (GRB) 140801A in the 137 deg$^2$ 3-$\sigma$ error-box of the Fermi Gamma-ray Burst Monitor (GBM). MASTER is the only observatory that automatically react to all Fermi alerts. GRB 140801A is one of the few GRBs whose optical counterpart was discovered solely from its GBM localization. The optical afterglow of GRB 140801A was found by MASTER Global Robotic Net 53 sec after receiving the alert, making it the fastest optical detection of a GRB from a GBM error-box. Spectroscopy obtained with the 10.4-m Gran Telescopio Canarias and the 6-m BTA of SAO RAS reveals a redshift of $z=1.32$. We performed optical and near-infrared photometry of GRB 140801A using different telescopes with apertures ranging from 0.4-m to 10.4-m. GRB 140801A is a typical burst in many ways. The rest-frame bolometric isotropic energy release and peak energy of the burst is $E_\mathrm{iso} = 5.54_{-0.24}^{+0.26} \times 10^{52}$ erg and $E_\mathrm{p, rest}\simeq280$ keV, respectively, which is consistent with the Amati relation. The absence of a jet break in the optical light curve provides a lower limit on the half-opening angle of the jet $\theta=6.1$ deg. The observed $E_\mathrm{peak}$ is consistent with the limit derived from the Ghirlanda relation. The joint Fermi GBM and Konus-Wind analysis shows that GRB 140801A could belong to the class of intermediate duration. The rapid detection of the optical counterpart of GRB 140801A is especially important regarding the upcoming experiments with large coordinate error-box areas.
• ### Optical polarization observations with the MASTER robotic net(1401.3221)

April 16, 2014 astro-ph.HE
We present results of optical polarization observations performed with the MASTER robotic net for three types of objects: gamma-ray bursts, supernovae, and blazars. For the Swift gamma-ray bursts GRB100906A, GRB110422A, GRB121011A, polarization observations were obtained during very early stages of optical emission. For GRB100906A it was the first prompt optical polarization observation in the world. Photometry in polarizers is presented for Type Ia Supernova 2012bh during 20 days, starting on March 27, 2012. We find that the linear polarization of SN 2012bh at the early stage of the envelope expansion was less than 3%. Polarization measurements for the blazars OC 457, 3C 454.3, QSO B1215+303, 87GB 165943.2+395846 at single nights are presented. We infer the degree of the linear polarization and polarization angle. The blazars OC 457 and 3C 454.3 were observed during their periods of activity. The results show that MASTER is able to measure substantially polarized light; at the same time it is not suitable for determining weak polarization (less than 5%) of dim objects (fainter than 16$^m$). Polarimetric observations of the optical emission from gamma-ray bursts and supernovae are necessary to investigate the nature of these transient objects.
• ### The MASTER-II Network of Robotic Optical Telescopes. First Results(1305.1620)

May 8, 2013 astro-ph.HE
We present the results of observations obtained using the MASTER robotic telescope in 2005 - 2006, including the earliest observations of the optical emission of the gamma-ray bursts GRB 050824 and GRB 060926. Together with later observations, these data yield the brightness-variation law t^{-0.55+-0.05} for GRB 050824. An optical flare was detected in GRB 060926 - a brightness enhancement that repeated the behavior observed in the X-ray variations. The spectrum of GRB 060926 is found to be F_E ~ E^-\beta, where \beta = 1.0+-0.2. Limits on the optical brightnesses of 26 gamma-ray bursts have been derived, 9 of these for the first time. Data for more than 90% of the accessible sky down to $19^m$ were taken and reduced in real time during the survey. A database has been composed based on these data. Limits have been placed on the rate of optical flares that are not associated with detected gamma-ray bursts, and on the opening angle for the beams of gamma-ray bursts. Three new supernovae have been discovered: SN 2005bv (type Ia) - the first to be discovered on Russian territory, SN 2005ee - one of the most powerful type II supernovae known, and SN 2006ak (type Ia). We have obtained an image of SN 2006X during the growth stage and a light curve that fully describes the brightness maximum and exponential decay. A new method for searching for optical transients of gamma-ray bursts detected using triangulation from various spacecraft is proposed and tested.


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# Maximising a multivariate linear function that includes the rounding in SAGE

Let's say I define the following function in SAGE

$$f(x,y,z,w)=|x+y-\lfloor x+y\rceil|+|y-z-\lfloor y-z \rceil|+|z+w-\lfloor z+w \rceil | + |w-x-\lfloor w-x \rceil | - |x|-|y|-|z|-|w|,$$ and want to find the maximum of $$f$$ for $$x \in I_x, y \in I_y, z \in I_z, w \in I_w$$, some predefined intervals. I am aware of the function

minimize_constrained,

which can be used to minimise multivariate differentiable functions, but since the rounding function is non-differentiable, SAGE returns an error. Is there any way to get SAGE to maximise such a function (or any linear function involving rounding) without having to consider each interval piecewise?

I don't think you should be using optimizers that depend on gradients here. For example, in Python you could so something like that using a global optimizer like differential evolution:

import numpy as np
from scipy.optimize import rosen, differential_evolution
def f(theta):
x = theta[0]
y = theta[1]
z = theta[2]
w = theta[3]
return abs(x+y - round(x+y)) + abs(y-z - round(y-z)) + abs(z+w - round(z+w)) + abs(w-x - round(w-x)) - abs(x) - abs(y) - abs(z)  -abs(w)

bounds = [(-100,100) for i in range(4)]

result = differential_evolution(f, bounds)
print(result)

Which gives the results:

     fun: -400.0
jac: array([-2.99999858,  2.99999858,  2.99999858, -3.00000426])
message: 'Optimization terminated successfully.'
nfev: 3345
nit: 54
success: True
x: array([ 100., -100., -100.,  100.])

So your problem comes probably from the solver you used. I don't know what types of solvers are available in sagemath, though.

• I see - thank you! the package seems to be available for SAGE too, since copying your code also worked in my notebook. Quick question about "differential_evolution" - how do I retrieve the numbers from the array returned in "result"? Jul 26, 2021 at 18:27
• You can use result[x] to return the optimal solution. Other optional parameters are available in the link in my answer. I also suggests looking at the scipy.optimize page for the other solvers available, since maybe there are others more suitable to your problem. Jul 26, 2021 at 18:32


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The P-block Elements

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Hybridisation of carbon atoms ...

Updated On: 27-06-2022

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sp^(3) hybridised sp hybridisation sp^(2) hybridisation sp^(3) d^(2) hybridisation

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students in the given question we need to work on the hybridization of carbon atoms in diamond diamond diamond is risky and a lot of form of carbon so it is basically carbon containing electrons 1234 valence electron adding present so that means it can complete it sorted by sharing this four electrons that covalently bonded with other carbon and technician properties also shown by connecting with the same ka so what happened in the Diamond we're going to have a turtle type of structure it is being connected to this is going to be carbon and carbon atoms are being connected using Technology Centre and similarly for this one was carbon is now connected with other four other three sorry so one then when third similarly for this 123 similarly for the 112 country so this is basically as pending network of solid

CBC with carbon atom it is being linked with 4 1 2 3 4 point that means Kitni total of four orbitals 12345 protection is going to be type because this is going to be a total of four of the time so that it can form a bond with the other carbon atoms respectively and more precisely if you want to see the detective which is of 3D structure through its being represented like this there carbons are the dot Ambika of keeping the turtle positions of the respective portion of this is being extended to other three shown that this and this is what is being expanding with connector with other connected with the other set and this is been connected with this one Subah

City this is connected like this is formed which are linked with each other so this was the solution for the given problem this is the 3D structure with respect to the state address and others on serving 1 2 3 and 4 this is the Trade Centre Tower 1 2 3 and 4 so basically it carbon is connected with one two three four other carbon so that's right habitations comes out to be a what is optionally SP3 hybridization