Optimization hardness as transient chaos in an analog approach to constraint satisfaction

Boolean satisfiability [1] (k-SAT) is one of the most studied optimization problems, as an efficient (that is, polynomial-time) solution to k-SAT (for $k\geq 3$) implies efficient solutions to a large number of hard optimization problems [2,3]. Here we propose a mapping of k-SAT into a deterministic continuous-time dynamical system with a unique correspondence between its attractors and the k-SAT solution clusters. We show that beyond a constraint density threshold, the analog trajectories become transiently chaotic [4-7], and the boundaries between the basins of attraction [8] of the solution clusters become fractal [7-9], signaling the appearance of optimization hardness [10]. Analytical arguments and simulations indicate that the system always finds solutions for satisfiable formulae even in the frozen regimes of random 3-SAT [11] and of locked occupation problems [12] (considered among the hardest algorithmic benchmarks); a property partly due to the system's hyperbolic [4,13] character. The system finds solutions in polynomial continuous-time, however, at the expense of exponential fluctuations in its energy function.Comment: 27 pages, 14 figure

GEANT4 : a simulation toolkit

Abstract Geant4 is a toolkit for simulating the passage of particles through matter. It includes a complete range of functionality including tracking, geometry, physics models and hits. The physics processes offered cover a comprehensive range, including electromagnetic, hadronic and optical processes, a large set of long-lived particles, materials and elements, over a wide energy range starting, in some cases, from 250 eV and extending in others to the TeV energy range. It has been designed and constructed to expose the physics models utilised, to handle complex geometries, and to enable its easy adaptation for optimal use in different sets of applications. The toolkit is the result of a worldwide collaboration of physicists and software engineers. It has been created exploiting software engineering and object-oriented technology and implemented in the C++ programming language. It has been used in applications in particle physics, nuclear physics, accelerator design, space engineering and medical physics. PACS: 07.05.Tp; 13; 2

GEANT4--a simulation toolkikt

Geant4 is a toolkit for simulating the passage of particles through matter. It includes a complete range of functionality including tracking, geometry, physics models and hits. The physics processes offered cover a comprehensive range, including electromagnetic, hadronic and optical processes, a large set of long-lived particles, materials and elements, over a wide energy range starting, in some cases, from 250 eV and extending in others to the TeV energy range. It has been designed and constructed to expose the physics models utilised, to handle complex geometries, and to enable its easy adaptation for optimal use in different sets of applications. The toolkit is the result of a worldwide collaboration of physicists and software engineers. It has been created exploiting software engineering and object-oriented technology and implemented in the C++ programming language. It has been used in applications in particle physics, nuclear physics, accelerator design, space engineering and medical physics

First Monte Carlo simulation study of Galeras volcano structure using muon tomography

Geant4 is a toolkit for simulating the passage of particles through matter. It includes a complete range of functionality including tracking, geometry, physics models and hits. The physics processes offered cover a comprehensive range, including electromagnetic, hadronic and optical processes, a large set of long-lived particles, materials and elements, over a wide energy range starting, in some cases, from View the MathML source and extending in others to the TeV energy range. It has been designed and constructed to expose the physics models utilised, to handle complex geometries, and to enable its easy adaptation for optimal use in different sets of applications. The toolkit is the result of a worldwide collaboration of physicists and software engineers. It has been created exploiting software engineering and object-oriented technology and implemented in the C++ programming language. It has been used in applications in particle physics, nuclear physics, accelerator design, space engineering and medical physics

Geant4 - A simulation toolkit

none127Geant4 is a toolkit for simulating the passage of particles through matter. It includes a complete range of functionality including tracking, geometry, physics models and hits. The physics processes offered cover a comprehensive range, including electromagnetic, hadronic and optical processes, a large set of long-lived particles, materials and elements, over a wide energy range starting, in some cases, from View the MathML source and extending in others to the TeV energy range. It has been designed and constructed to expose the physics models utilised, to handle complex geometries, and to enable its easy adaptation for optimal use in different sets of applications. The toolkit is the result of a worldwide collaboration of physicists and software engineers. It has been created exploiting software engineering and object-oriented technology and implemented in the C++ programming language. It has been used in applications in particle physics, nuclear physics, accelerator design, space engineering and medical physics.noneS. Agostinelli;J. Allison;K. Amako;J. Apostolakis;H. Araujo;P. Arce;M. Asai;D. Axen;S. Banerjee;G. Barrand;F. Behner;L. Bellagamba;J. Boudreau;L. Broglia;A. Brunengo;H. Burkhardt;S. Chauvie;J. Chuma;R. Chytracek;G. Cooperman;G. Cosmo;P. Degtyarenko;A. Dell'Acqua;G. Depaola;D. Dietrich;R. Enami;A. Feliciello;C. Ferguson;H. Fesefeldt;G. Folger;F. Foppiano;A. Forti;S. Garelli;S. Giani;R. Giannitrapani;D. Gibin;J.J. Gómez Cadenas;I. González;G. Gracia Abril;G. Greeniaus;W. Greiner;V. Grichine;A. Grossheim;S. Guatelli;P. Gumplinger;R. Hamatsu;K. Hashimoto;H. Hasui;A. Heikkinen;A. Howard;V. Ivanchenko;A. Johnson;F.W. Jones;J. Kallenbach;N. Kanaya;M. Kawabata;Y. Kawabata;M. Kawaguti;S. Kelner;P. Kent;A. Kimura;T. Kodama;R. Kokoulin;M. Kossov;H. Kurashige;E. Lamanna;T. Lampén;V. Lara;V. Lefebure;F. Lei;M. Liendl;W. Lockman;F. Longo;S. Magni;M. Maire;E. Medernach;K. Minamimoto;P. Mora de Freitas;Y. Morita;K. Murakami;M. Nagamatu;R. Nartallo;P. Nieminen;T. Nishimura;K. Ohtsubo;M. Okamura;S. O'Neale;Y. Oohata;K. Paech;J. Perl;A. Pfeiffer;M.G. Pia;F. Ranjard;A. Rybin;S. Sadilov;E. Di Salvo;G. Santin;T. Sasaki;N. Savvas;Y. Sawada;S. Scherer;S. Sei;V. Sirotenko;D. Smith;N. Starkov;H. Stoecker;J. Sulkimo;M. Takahata;S. Tanaka;E. Tcherniaev;E. Safai Tehrani;M. Tropeano;P. Truscott;H. Uno;L. Urban;P. Urban;M. Verderi;A. Walkden;W. Wander;H. Weber;J.P. Wellisch;T. Wenaus;D.C. Williams;D. Wright;T. Yamada;H. Yoshida;D. ZschiescheS., Agostinelli; J., Allison; K., Amako; J., Apostolakis; H., Araujo; P., Arce; M., Asai; D., Axen; S., Banerjee; G., Barrand; F., Behner; L., Bellagamba; J., Boudreau; L., Broglia; A., Brunengo; H., Burkhardt; S., Chauvie; J., Chuma; R., Chytracek; G., Cooperman; G., Cosmo; P., Degtyarenko; A., Dell'Acqua; G., Depaola; D., Dietrich; R., Enami; A., Feliciello; C., Ferguson; H., Fesefeldt; G., Folger; F., Foppiano; A., Forti; S., Garelli; S., Giani; R., Giannitrapani; Gibin, Daniele; J. J., Gómez Cadenas; I., González; G., Gracia Abril; G., Greeniaus; W., Greiner; V., Grichine; A., Grossheim; S., Guatelli; P., Gumplinger; R., Hamatsu; K., Hashimoto; H., Hasui; A., Heikkinen; A., Howard; V., Ivanchenko; A., Johnson; F. W., Jones; J., Kallenbach; N., Kanaya; M., Kawabata; Y., Kawabata; M., Kawaguti; S., Kelner; P., Kent; A., Kimura; T., Kodama; R., Kokoulin; M., Kossov; H., Kurashige; E., Lamanna; T., Lampén; V., Lara; V., Lefebure; F., Lei; M., Liendl; W., Lockman; F., Longo; S., Magni; M., Maire; E., Medernach; K., Minamimoto; P., Mora de Freitas; Y., Morita; K., Murakami; M., Nagamatu; R., Nartallo; P., Nieminen; T., Nishimura; K., Ohtsubo; M., Okamura; S., O'Neale; Y., Oohata; K., Paech; J., Perl; A., Pfeiffer; M. G., Pia; F., Ranjard; A., Rybin; S., Sadilov; E., Di Salvo; G., Santin; T., Sasaki; N., Savvas; Y., Sawada; S., Scherer; S., Sei; V., Sirotenko; D., Smith; N., Starkov; H., Stoecker; J., Sulkimo; M., Takahata; S., Tanaka; E., Tcherniaev; E., Safai Tehrani; M., Tropeano; P., Truscott; H., Uno; L., Urban; P., Urban; M., Verderi; A., Walkden; W., Wander; H., Weber; J. P., Wellisch; T., Wenaus; D. C., Williams; D., Wright; T., Yamada; H., Yoshida; D., Zschiesch