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# zbMATH — the first resource for mathematics

## <NAME>

Compute Distance To:
 Author ID: ashwin.peter Published as: <NAME>.; Ashwin, Peter External Links: ORCID
 Documents Indexed: 125 Publications since 1990
all top 5

#### Co-Authors

 11 single-authored 9 Fu, Xinchu 7 Mei, Zhen 7 Nicol, Matthew 7 Sturman, Rob 6 Melbourne, Ian 5 Podvigina, <NAME> 5 Rucklidge, <NAME>. 5 Stewart, <NAME> 5 Swift, <NAME>. 4 Böhmer, Klaus 4 Field, <NAME>. 4 Goetz, Arek 4 Karabacak, Özkan 4 King, <NAME>. 3 Chossat, Pascal 3 Dangelmayr, Gerhard 3 Hawker, David 3 Orosz, Gábor 3 Postlethwaite, <NAME>. 3 Rodrigues, Ana 3 Terry, <NAME>. 3 Tsaneva-Atanasova, <NAME>. 3 Wieczorek, <NAME>. 2 Aston, <NAME>. 2 Bick, Christian 2 Borresen, Jon 2 Buescu, Jorge 2 Burylko, Oleksandr 2 Cox, <NAME>. 2 Creaser, <NAME>. 2 <NAME>. 2 Trovati, Marcello 1 Aguiar, <NAME>. 1 Alkhayuon, <NAME>. 1 Allgower, <NAME>. 1 Aronson, <NAME>. 1 Azad, <NAME> Al 1 Bartuccelli, <NAME>. 1 Bassom, <NAME>. 1 Benjamin, Oscar 1 Blyuss, <NAME>. 1 Breakspear, Michael 1 Bridges, <NAME>. 1 Broomhead, <NAME>. 1 Byott, <NAME>. 1 Callard, <NAME>. 1 Castro, <NAME>. 1 Chambers, <NAME>. 1 Chawanya, Tsuyoshi 1 Chowdhury, Fahmida N. 1 Coombes, Stephen 1 Covas, Eurico O. 1 Deane, <NAME>. 1 Dias, <NAME>. 1 Duan, Jinqiao 1 Fitzgerald, <NAME>. 1 Glendinning, <NAME> 1 Gourley, <NAME>. 1 Guaschi, John 1 Huguet, Gemma 1 Ikegami, Giko 1 Ismail, Asma 1 Krupa, Martin 1 Lavric, Aureliu 1 Lin, Congping 1 Lohse, Alexander 1 Lu, Weiping 1 Maistrenko, <NAME>. 1 Moehlis, Jeff 1 Montaldi, <NAME>. 1 Moroz, <NAME>. 1 Nicks, Rachel 1 Nijhof, <NAME>. 1 Nishikawa, Takashi 1 Nowotny, Thomas 1 Ochs, Gunter 1 Peres, <NAME>. 1 Pérez-Cervera, Alberto 1 Perryman, Clare 1 Petkov, <NAME>. 1 Phelps, <NAME>. 1 Rankin, James 1 Richardson, <NAME>. 1 Roberts, Richard Mark 1 Roslan, Ummu Atiqah Mohd 1 M-Seara, Tere 1 Smug, Damian 1 Sornette, Didier 1 Stork, <NAME>. 1 Tavakol, Reza 1 Tchizawa, Koyoyuki 1 Thoubaan, Mary 1 Timme, Marc 1 Tomes, Jörg 1 Townley, <NAME>. 1 von der Heydt, <NAME>. 1 Wegelin, Michael 1 Weinberger, Oskar 1 Wordsworth, John 1 <NAME> ...and 1 more Co-Authors
all top 5

#### Serials

 17 Physica D 15 Nonlinearity 9 Journal of Nonlinear Science 9 Chaos 7 Dynamical Systems 6 Physics Letters. A 6 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 6 SIAM Journal on Applied Dynamical Systems 5 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 4 The Journal of Mathematical Neuroscience 3 Mathematical Proceedings of the Cambridge Philosophical Society 3 Chaos, Solitons and Fractals 2 Archive for Rational Mechanics and Analysis 2 Journal of Fluid Mechanics 2 Ergodic Theory and Dynamical Systems 2 Dynamics and Stability of Systems 2 Discrete and Continuous Dynamical Systems 1 Journal of Statistical Physics 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Mathematics of Computation 1 Journal of Computational and Applied Mathematics 1 SIAM Journal on Numerical Analysis 1 Transactions of the American Mathematical Society 1 Journal of Physics A: Mathematical and General 1 Journal of Dynamics and Differential Equations 1 Topological Methods in Nonlinear Analysis 1 Discrete and Continuous Dynamical Systems. Series B 1 Progress of Theoretical Physics
all top 5

#### Fields

 97 Dynamical systems and ergodic theory (37-XX) 47 Ordinary differential equations (34-XX) 13 Partial differential equations (35-XX) 12 Biology and other natural sciences (92-XX) 9 Numerical analysis (65-XX) 8 Global analysis, analysis on manifolds (58-XX) 8 Fluid mechanics (76-XX) 6 Information and communication theory, circuits (94-XX) 4 Statistical mechanics, structure of matter (82-XX) 3 Probability theory and stochastic processes (60-XX) 3 Geophysics (86-XX) 2 Computer science (68-XX) 2 Mechanics of particles and systems (70-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Systems theory; control (93-XX) 1 History and biography (01-XX) 1 Combinatorics (05-XX) 1 Number theory (11-XX) 1 Group theory and generalizations (20-XX) 1 Measure and integration (28-XX) 1 Integral equations (45-XX) 1 Operator theory (47-XX) 1 Convex and discrete geometry (52-XX) 1 Manifolds and cell complexes (57-XX) 1 Astronomy and astrophysics (85-XX)

#### Citations contained in zbMATH Open

115 Publications have been cited 1,175 times in 690 Documents Cited by Year
From attractor to chaotic saddle: A tale of transverse instability. Zbl 0887.58034
<NAME>; <NAME>; <NAME>
1996
The dynamics of $$n$$ weakly coupled identical oscillators. Zbl 0872.58049
<NAME>.; <NAME>.
1992
Travelling fronts for the KPP equation with spatio-temporal delay. Zbl 1005.92024
<NAME>.; <NAME>.; <NAME>.; <NAME>.
2002
Bubbling of attractors and synchronisation of chaotic oscillators. Zbl 0959.37508
<NAME>; <NAME>; <NAME>
1994
Mathematical frameworks for oscillatory network dynamics in neuroscience. Zbl 1356.92015
<NAME>; <NAME>; <NAME>
2016
Heteroclinic networks in coupled cell systems. Zbl 0947.37012
Ashwin, Peter; Field, Michael
1999
Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation. Zbl 1254.37053
<NAME>.; <NAME>.; <NAME>.; <NAME>.
2011
Weak chimeras in minimal networks of coupled phase oscillators. Zbl 1345.34052
Ashwin, Peter; Burylko, Oleksandr
2015
Three identical oscillators with symmetric coupling. Zbl 0708.58020
Ashwin, Peter; <NAME>.; <NAME>.
1990
Dynamics on networks of cluster states for globally coupled phase oscillators. Zbl 1167.34326
<NAME>; <NAME>; <NAME>; <NAME>
2007
On designing heteroclinic networks from graphs. Zbl 1417.37098
<NAME>; Postlethwaite, Claire
2013
On local attraction properties and a stability index for heteroclinic connections. Zbl 1219.37021
<NAME>; <NAME>
2011
Elliptic behaviour in the sawtooth standard map. Zbl 1053.37520
<NAME>
1997
Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators. Zbl 1178.34041
Ashwin, Peter; Burylko, Oleksandr; <NAME>
2008
Noncompact drift for relative equilibria and relative periodic orbits. Zbl 0912.35079
Ashwin, Peter; Melbourne, Ian
1997
Lossless digital filter overflow oscillations; approximation of invariant fractals. Zbl 0902.65100
Ashwin, Peter; <NAME>.; <NAME>.
1997
On the geometry of orientation-preserving planar piecewise isometries. Zbl 1100.37027
Ashwin, P.; <NAME>.
2002
Unstable attractors: existence and robustness in networks of oscillators with delayed pulse coupling. Zbl 1084.37061
Ashwin, Peter; <NAME>
2005
Transverse instability for non-normal parameters. Zbl 0984.37053
<NAME>; <NAME>; <NAME>
1999
Tangencies in invariant disc packings for certain planar piecewise isometries are rare. Zbl 1032.37026
<NAME>; <NAME>
2001
Attractors for robust heteroclinic cycles with continua of connections. Zbl 0901.58045
<NAME>.; <NAME>.
1998
Chaotic weak chimeras and their persistence in coupled populations of phase oscillators. Zbl 1355.37041
<NAME>; <NAME>
2016
Excitability in ramped systems: the compost-bomb instability. Zbl 1219.86006
<NAME>.; <NAME>.; <NAME>.; <NAME>.
2011
Parameter shifts for nonautonomous systems in low dimension: bifurcation- and rate-induced tipping. Zbl 1435.37104
<NAME>; <NAME>; <NAME>
2017
Polygonal invariant curves for a planar piecewise isometry. Zbl 1080.37009
Ashwin, Peter; <NAME>
2006
Discrete computation using a perturbed heteroclinic network. Zbl 1195.34053
Ashwin, Peter; <NAME>
2005
Invariant curves and explosion of periodic islands in systems of piecewise rotations. Zbl 1090.37029
Ashwin, Peter; <NAME>
2005
Symbolic analysis for some planar piecewise linear maps. Zbl 1042.37009
<NAME>; <NAME>
2003
Riddling and invariance for discontinuous maps preserving Lebesgue measure. Zbl 1073.37009
Ashwin, Peter; <NAME>; <NAME>.
2002
Euclidean extensions of dynamical systems. Zbl 1183.37043
<NAME>; <NAME>; <NAME>
2001
On riddling and weak attractors. Zbl 0954.37015
Ashwin, Peter; <NAME>.
2000
Hopf normal form with $$S_N$$ symmetry and reduction to systems of nonlinearly coupled phase oscillators. Zbl 1364.34041
Ashwin, Peter; <NAME>
2016
Criteria for robustness of heteroclinic cycles in neural microcircuits. Zbl 1259.37054
Ashwin, Peter; <NAME>; <NAME>
2011
Designing the dynamics of globally coupled oscillators. Zbl 1196.34046
Orosz, Gábor; <NAME>; <NAME>
2009
Hopf bifurcation with cubic symmetry and instability of ABC flow. Zbl 1116.37310
Ashwin, Peter; <NAME>
2003
Symmetric chaos in systems of three and four forced oscillators. Zbl 0711.94031
Ashwin, Peter
1990
Chaos in generically coupled phase oscillator networks with nonpairwise interactions. Zbl 1382.34038
Bick, Christian; Ashwin, Peter; <NAME>
2016
Properties of the invariant disk packing in a model bandpass sigma-delta modulator. Zbl 1056.37059
Ashwin, Peter; <NAME>; <NAME>
2003
On the unfolding of a blowout bifurcation. Zbl 0930.37013
<NAME>; <NAME>.; <NAME>
1998
Symbolic representations of iterated maps. Zbl 1021.37007
<NAME>; <NAME>; <NAME>; <NAME>
2001
A Hopf bifurcation with Robin boundary conditions. Zbl 0807.35013
Ashwin, Peter; <NAME>
1994
Symmetry groups of attractors. Zbl 0817.58020
Ashwin, Peter; <NAME>
1994
A study of particle paths in non-axisymmetric Taylor-Couette flows. Zbl 0889.76089
<NAME>.; <NAME>.
1997
A phenomenological model of seizure initiation suggests network structure may explain seizure frequency in idiopathic generalised epilepsy. Zbl 1291.92022
<NAME>; Fitzgerald, <NAME>.; <NAME>; Tsaneva-Atanasova, Krasimira; <NAME>; Richardson, <NAME>.; <NAME>.
2012
Heteroclinic ratchets in networks of coupled oscillators. Zbl 1208.34057
Karabacak, Özkan; Ashwin, Peter
2010
Cycling chaos: Its creation, persistence and loss of stability in a model of nonlinear magnetoconvection. Zbl 0955.76035
Ashwin, Peter; Rucklidge, <NAME>.
1998
Rate-induced tipping from periodic attractors: partial tipping and connecting orbits. Zbl 1392.37026
Alkhayuon, <NAME>.; <NAME>
2018
Designing heteroclinic and excitable networks in phase space using two populations of coupled cells. Zbl 1362.37058
Ashwin, Peter; Postlethwaite, Claire
2016
Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory. Zbl 1051.37041
Ashwin, Peter; Rucklidge, <NAME>.; <NAME>
2004
Phase resetting effects for robust cycles between chaotic sets. Zbl 1080.37518
Ashwin, Peter; Field, Michael; Rucklidge, <NAME>.; <NAME>
2003
On the dynamics of some nonhyperbolic area-preserving piecewise linear maps. Zbl 1013.37039
Ashwin, Peter; Fu, Xin-Chu
2002
Attractors of a randomly forced electronic oscillator. Zbl 1065.94567
Ashwin, Peter
1999
Torus doubling in four weakly coupled oscillators. Zbl 0885.58056
Ashwin, Peter; <NAME>.
1995
Normal form for Hopf bifurcation of partial differential equations on the square. Zbl 0839.35010
Ashwin, Peter; <NAME>.
1995
A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square. Zbl 0829.65075
Ashwin, Peter; <NAME>; <NAME>
1995
Within-burst synchrony changes for coupled elliptic bursters. Zbl 1195.37032
<NAME>; <NAME>
2010
The $$1:\sqrt 2$$ mode interaction and heteroclinic networks in Boussinesq convection. Zbl 1141.76453
<NAME>; <NAME>
2007
Convergence to local random attractors. Zbl 1136.37345
<NAME>; <NAME>
2003
Drift bifurcations of relative equilibria and transitions of spiral waves. Zbl 0958.37017
<NAME>; <NAME>; <NAME>
1999
Boundary of two frequency behaviour in a system of three weakly coupled electronic oscillators. Zbl 0956.37013
<NAME>
1998
Streamline topology in eccentric Taylor vortex flow. Zbl 0882.76097
<NAME>.; <NAME>.
1995
High corank steady-state mode interactions on a rectangle. Zbl 0769.35006
Ashwin, Peter
1992
From coupled networks of systems to networks of states in phase space. Zbl 1395.37061
<NAME>; Ashwin, Peter
2018
Quantifying noisy attractors: from heteroclinic to excitable networks. Zbl 1360.37141
Ashwin, Peter; Postlethwaite, Claire
2016
A low-dimensional model of binocular rivalry using winnerless competition. Zbl 1194.91167
Ashwin, Peter; <NAME>
2010
Tangency properties of a pentagonal tiling generated by a piecewise isometry. Zbl 1163.37379
<NAME>; <NAME>
2007
Hypermeander of spirals: local bifurcations and statistical properties. Zbl 1049.92001
<NAME>.; <NAME>.; <NAME>.
2001
Invariant sets for discontinuous parabolic area-preserving torus maps. Zbl 0952.37007
<NAME>; <NAME>; <NAME>; <NAME>
2000
Minimal attractors and bifurcations of random dynamical systems. Zbl 0934.37049
<NAME>
1999
Detection of symmetry of attractors from observations. I: Theory. Zbl 0894.58037
Ashwin, Peter; <NAME>
1997
Semirotors in the Josephson junction equations. Zbl 0843.34041
<NAME>.; <NAME>.; <NAME>.
1996
Permissible symmetries of coupled cell networks. Zbl 0816.58035
<NAME>; <NAME>
1994
A numerical Liapunov-Schmidt method for finitely determined problems. Zbl 0787.58044
<NAME>.; <NAME>.; <NAME>.
1993
Rotation sets and phase-locking in an electronic three oscillator system. Zbl 0784.34027
Ashwin, Peter; <NAME>; <NAME>.
1993
Local and global stability indices for a riddled basin attractor of a piecewise linear map. Zbl 1370.37057
Roslan, Ummu Atiqah Mohd; <NAME>
2016
Robust heteroclinic behaviour, synchronization, and ratcheting of coupled oscillators. Zbl 1266.34059
<NAME>; <NAME>
2011
On statistical attractors and the convergence of time averages. Zbl 1214.37024
<NAME>; <NAME>
2011
Noise-induced switching near a depth two heteroclinic network and an application to Boussinesq convection. Zbl 1311.37013
<NAME>; <NAME>
2010
On planar piecewise and two-torus parabolic maps. Zbl 1176.37017
Ashwin, Peter; <NAME>; <NAME>
2009
Modelling instability $$ABC$$ flow using a mode interaction between steady and Hopf bifurcations with rotational symmetries of the cube. Zbl 1091.35520
<NAME>; <NAME>; <NAME>
2006
Product dynamics for homoclinic attractors. Zbl 1145.37307
Ashwin, Peter; <NAME>
2005
Reduced dynamics and symmetric solutions for globally coupled weakly dissipative oscillators. Zbl 1086.37041
<NAME>.; <NAME>.
2005
Isochronicity-induced bifurcations in systems of weakly dissipative coupled oscillators. Zbl 0976.34033
Ashwin, Peter; Dangelmayr, Gerhard
2000
Cycles homoclinic to chaotic sets; robustness and resonance. Zbl 1012.37020
Ashwin, Peter
1997
Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry. Zbl 0858.65123
<NAME>; <NAME>; <NAME>.
1996
Bifurcations of stationary, standing and travelling waves in triply diffusive convection. Zbl 0899.76165
<NAME>; <NAME>; <NAME>
1995
Unfolding the torus: Oscillator geometry from time delays. Zbl 0804.34036
<NAME>.; <NAME>.
1993
Existence and stability of Chimera states in a minimal system of phase oscillators. Zbl 1400.34059
Thoubaan, Mary; <NAME>
2018
Sequential noise-induced escapes for oscillatory network dynamics. Zbl 1387.60088
Creaser, Jennifer; Tsaneva-Atanasova, Krasimira; <NAME>
2018
Heteroclinic switching in coupled oscillator networks: dynamics on odd graphs. Zbl 1221.37044
<NAME>; <NAME>; <NAME>
2010
Cone exchange transformations and boundedness of orbits. Zbl 1213.37063
Ashwin, Peter; <NAME>
2010
Packings induced by piecewise isometries cannot contain the Arbelos. Zbl 1156.52013
Trovati, Marcello; Ashwin, Peter; <NAME>
2008
Robust bursting to the origin: heteroclinic cycles with maximal symmetry equilibria. Zbl 1093.37509
Hawker, David; Ashwin, Peter
2005
Cycling chaotic attractors in two models for dynamics with invariant subspaces. Zbl 1080.37026
Ashwin, Peter; Rucklidge, <NAME>.; <NAME>
2004
Group theoretic conditions for existence of robust relative homoclinic trajectories. Zbl 1039.37015
<NAME>; <NAME>
2002
Chaotic intermittency of patterns in symmetric systems. Zbl 0999.37033
<NAME>
1999
Blowout bifurcations of codimension two. Zbl 1044.37515
<NAME>; Aston, <NAME>.
1998
Oscillator networks with the symmetry of the unit quaternion group. Zbl 0817.58036
Ashwin, Peter; Dangelmayr, Gerhard; <NAME>; <NAME>
1994
Liapunov-Schmidt reduction for a bifurcation problem with periodic boundary conditions on a square domain. Zbl 0787.58043
<NAME>.; <NAME>.; <NAME>.; <NAME>.
1993
Extreme sensitivity and climate tipping points. Zbl 1446.37075
Ashwin, Peter; <NAME>, <NAME>.
2020
Extreme sensitivity and climate tipping points. Zbl 1446.37075
Ashwin, Peter; <NAME>, <NAME>.
2020
Almost complete and equable heteroclinic networks. Zbl 1431.34058
Ashwin, Peter; Castro, <NAME>.; <NAME>
2020
The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability. Zbl 1439.92051
Pérez-Cervera, Alberto; Ashwin, Peter; <NAME>; <NAME>.; <NAME>
2019
Rate-induced tipping from periodic attractors: partial tipping and connecting orbits. Zbl 1392.37026
Alkhayuon, <NAME>.; <NAME>
2018
From coupled networks of systems to networks of states in phase space. Zbl 1395.37061
Weinberger, Oskar; <NAME>
2018
Existence and stability of Chimera states in a minimal system of phase oscillators. Zbl 1400.34059
Thoubaan, Mary; <NAME>
2018
Sequential noise-induced escapes for oscillatory network dynamics. Zbl 1387.60088
Creaser, Jennifer; Tsaneva-Atanasova, Krasimira; <NAME>
2018
A generalized 2D-dynamical mean-field Ising model with a rich set of bifurcations (inspired and applied to financial crises). Zbl 1392.37098
<NAME>; <NAME>; <NAME>
2018
Parameter shifts for nonautonomous systems in low dimension: bifurcation- and rate-induced tipping. Zbl 1435.37104
Ashwin, Peter; Perryman, Clare; Wieczorek, Sebastian
2017
Mathematical frameworks for oscillatory network dynamics in neuroscience. Zbl 1356.92015
Ashwin, Peter; Coombes, Stephen; <NAME>
2016
Chaotic weak chimeras and their persistence in coupled populations of phase oscillators. Zbl 1355.37041
Bick, Christian; Ashwin, Peter
2016
Hopf normal form with $$S_N$$ symmetry and reduction to systems of nonlinearly coupled phase oscillators. Zbl 1364.34041
Ashwin, Peter; <NAME>
2016
Chaos in generically coupled phase oscillator networks with nonpairwise interactions. Zbl 1382.34038
Bick, Christian; <NAME>; <NAME>
2016
Designing heteroclinic and excitable networks in phase space using two populations of coupled cells. Zbl 1362.37058
Ashwin, Peter; Postlethwaite, Claire
2016
Quantifying noisy attractors: from heteroclinic to excitable networks. Zbl 1360.37141
Ashwin, Peter; Postlethwaite, Claire
2016
Local and global stability indices for a riddled basin attractor of a piecewise linear map. Zbl 1370.37057
Roslan, Ummu Atiqah Mohd; <NAME>
2016
Weak chimeras in minimal networks of coupled phase oscillators. Zbl 1345.34052
Ashwin, Peter; Burylko, Oleksandr
2015
Multi-cluster dynamics in coupled phase oscillator networks. Zbl 1351.37206
Ismail, Asma; <NAME>
2015
On designing heteroclinic networks from graphs. Zbl 1417.37098
Ashwin, Peter; Postlethwaite, Claire
2013
A phenomenological model of seizure initiation suggests network structure may explain seizure frequency in idiopathic generalised epilepsy. Zbl 1291.92022
<NAME>; Fitzgerald, <NAME>.; <NAME>; Tsaneva-Atanasova, Krasimira; <NAME>; <NAME>.; <NAME>.
2012
Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation. Zbl 1254.37053
<NAME>.; <NAME>.; <NAME>.; <NAME>.
2011
On local attraction properties and a stability index for heteroclinic connections. Zbl 1219.37021
<NAME>; <NAME>
2011
Excitability in ramped systems: the compost-bomb instability. Zbl 1219.86006
<NAME>.; <NAME>.; <NAME>.; <NAME>.
2011
Criteria for robustness of heteroclinic cycles in neural microcircuits. Zbl 1259.37054
<NAME>; <NAME>; <NAME>
2011
Robust heteroclinic behaviour, synchronization, and ratcheting of coupled oscillators. Zbl 1266.34059
<NAME>; <NAME>
2011
On statistical attractors and the convergence of time averages. Zbl 1214.37024
Karabacak, Özkan; <NAME>
2011
Correction to: “Excitability in ramped systems: the compost-bomb instability”. Zbl 1221.86005
<NAME>.; <NAME>.; <NAME>.; <NAME>.
2011
Heteroclinic ratchets in networks of coupled oscillators. Zbl 1208.34057
Karabacak, Özkan; Ashwin, Peter
2010
Within-burst synchrony changes for coupled elliptic bursters. Zbl 1195.37032
<NAME>; <NAME>
2010
A low-dimensional model of binocular rivalry using winnerless competition. Zbl 1194.91167
<NAME>; <NAME>
2010
Noise-induced switching near a depth two heteroclinic network and an application to Boussinesq convection. Zbl 1311.37013
<NAME>; <NAME>
2010
Heteroclinic switching in coupled oscillator networks: dynamics on odd graphs. Zbl 1221.37044
Ashwin, Peter; <NAME>; <NAME>
2010
Cone exchange transformations and boundedness of orbits. Zbl 1213.37063
Ashwin, Peter; <NAME>
2010
A minimal system with a depth-two heteroclinic network. Zbl 1204.37027
Chawanya, Tsuyoshi; <NAME>
2010
Designing the dynamics of globally coupled oscillators. Zbl 1196.34046
<NAME>; <NAME>; <NAME>
2009
On planar piecewise and two-torus parabolic maps. Zbl 1176.37017
Ashwin, Peter; <NAME>; <NAME>
2009
Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators. Zbl 1178.34041
Ashwin, Peter; Burylko, Oleksandr; Maistrenko, Yuri
2008
Packings induced by piecewise isometries cannot contain the Arbelos. Zbl 1156.52013
Trovati, Marcello; Ashwin, Peter; <NAME>
2008
Dynamics on networks of cluster states for globally coupled phase oscillators. Zbl 1167.34326
Ashwin, Peter; <NAME>; <NAME>; <NAME>
2007
The $$1:\sqrt 2$$ mode interaction and heteroclinic networks in Boussinesq convection. Zbl 1141.76453
<NAME>; <NAME>
2007
Tangency properties of a pentagonal tiling generated by a piecewise isometry. Zbl 1163.37379
Trovati, Marcello; Ashwin, Peter
2007
Polygonal invariant curves for a planar piecewise isometry. Zbl 1080.37009
Ashwin, Peter; <NAME>
2006
Modelling instability $$ABC$$ flow using a mode interaction between steady and Hopf bifurcations with rotational symmetries of the cube. Zbl 1091.35520
<NAME>; <NAME>; <NAME>
2006
Unstable attractors: existence and robustness in networks of oscillators with delayed pulse coupling. Zbl 1084.37061
Ashwin, Peter; <NAME>
2005
Discrete computation using a perturbed heteroclinic network. Zbl 1195.34053
Ashwin, Peter; <NAME>
2005
Invariant curves and explosion of periodic islands in systems of piecewise rotations. Zbl 1090.37029
Ashwin, Peter; <NAME>
2005
Product dynamics for homoclinic attractors. Zbl 1145.37307
Ashwin, Peter; Field, Michael
2005
Reduced dynamics and symmetric solutions for globally coupled weakly dissipative oscillators. Zbl 1086.37041
<NAME>.; <NAME>.
2005
Robust bursting to the origin: heteroclinic cycles with maximal symmetry equilibria. Zbl 1093.37509
<NAME>; <NAME>
2005
Riddled basins and coupled dynamical systems. Zbl 1103.37019
<NAME>.
2005
Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory. Zbl 1051.37041
<NAME>; Rucklidge, <NAME>.; <NAME>
2004
Cycling chaotic attractors in two models for dynamics with invariant subspaces. Zbl 1080.37026
Ashwin, Peter; Rucklidge, <NAME>.; <NAME>
2004
Internal dynamics of intermittency. Zbl 1080.37036
Sturman, Rob; <NAME>
2004
Symbolic analysis for some planar piecewise linear maps. Zbl 1042.37009
<NAME>; <NAME>
2003
Hopf bifurcation with cubic symmetry and instability of ABC flow. Zbl 1116.37310
Ashwin, Peter; <NAME>
2003
Properties of the invariant disk packing in a model bandpass sigma-delta modulator. Zbl 1056.37059
Ashwin, Peter; <NAME>; <NAME>
2003
Phase resetting effects for robust cycles between chaotic sets. Zbl 1080.37518
Ashwin, Peter; Field, Michael; Rucklidge, <NAME>.; <NAME>
2003
Convergence to local random attractors. Zbl 1136.37345
<NAME>; <NAME>
2003
Cycling attractors of coupled cell systems and dynamics with symmetry. Zbl 1050.37047
<NAME>; Rucklidge, <NAME>.; <NAME>
2003
Patchwork patterns: Dynamics on unbounded domains. Zbl 1032.37013
<NAME>
2003
Travelling fronts for the KPP equation with spatio-temporal delay. Zbl 1005.92024
<NAME>.; <NAME>.; <NAME>.; <NAME>.
2002
On the geometry of orientation-preserving planar piecewise isometries. Zbl 1100.37027
<NAME>.; <NAME>.
2002
Riddling and invariance for discontinuous maps preserving Lebesgue measure. Zbl 1073.37009
<NAME>; <NAME>; <NAME>.
2002
On the dynamics of some nonhyperbolic area-preserving piecewise linear maps. Zbl 1013.37039
Ashwin, Peter; <NAME>
2002
Group theoretic conditions for existence of robust relative homoclinic trajectories. Zbl 1039.37015
Ashwin, Peter; Montaldi, James
2002
Tangencies in invariant disc packings for certain planar piecewise isometries are rare. Zbl 1032.37026
Ashwin, Peter; Fu, Xin-Chu
2001
Euclidean extensions of dynamical systems. Zbl 1183.37043
Nicol, Matthew; Melbourne, Ian; <NAME>
2001
Symbolic representations of iterated maps. Zbl 1021.37007
<NAME>; <NAME>; <NAME>; <NAME>
2001
Hypermeander of spirals: local bifurcations and statistical properties. Zbl 1049.92001
<NAME>.; <NAME>.; <NAME>.
2001
On riddling and weak attractors. Zbl 0954.37015
Ashwin, Peter; <NAME>.
2000
Invariant sets for discontinuous parabolic area-preserving torus maps. Zbl 0952.37007
Ashwin, Peter; <NAME>; <NAME>; <NAME>
2000
Isochronicity-induced bifurcations in systems of weakly dissipative coupled oscillators. Zbl 0976.34033
Ashwin, Peter; Dangelmayr, Gerhard
2000
Drift for Euclidean extensions of dynamical systems. Zbl 0976.37011
Ashwin, Peter; Melbourne, Ian; <NAME>
2000
Heteroclinic networks in coupled cell systems. Zbl 0947.37012
Ashwin, Peter; <NAME>
1999
Transverse instability for non-normal parameters. Zbl 0984.37053
Ashwin, Peter; <NAME>; <NAME>
1999
Attractors of a randomly forced electronic oscillator. Zbl 1065.94567
Ashwin, Peter
1999
Drift bifurcations of relative equilibria and transitions of spiral waves. Zbl 0958.37017
<NAME>; <NAME>; <NAME>
1999
Minimal attractors and bifurcations of random dynamical systems. Zbl 0934.37049
<NAME>
1999
Chaotic intermittency of patterns in symmetric systems. Zbl 0999.37033
<NAME>
1999
Attractors for robust heteroclinic cycles with continua of connections. Zbl 0901.58045
<NAME>.; <NAME>.
1998
On the unfolding of a blowout bifurcation. Zbl 0930.37013
Ashwin, Peter; <NAME>.; <NAME>
1998
Cycling chaos: Its creation, persistence and loss of stability in a model of nonlinear magnetoconvection. Zbl 0955.76035
Ashwin, Peter; <NAME>.
1998
Boundary of two frequency behaviour in a system of three weakly coupled electronic oscillators. Zbl 0956.37013
<NAME>
1998
Blowout bifurcations of codimension two. Zbl 1044.37515
Ashwin, Peter; <NAME>.
1998
A numerical bifurcation function for homoclinic orbits. Zbl 0911.34046
Ashwin, Peter; <NAME>
1998
Elliptic behaviour in the sawtooth standard map. Zbl 1053.37520
Ashwin, Peter
1997
Noncompact drift for relative equilibria and relative periodic orbits. Zbl 0912.35079
Ashwin, Peter; Melbourne, Ian
1997
Lossless digital filter overflow oscillations; approximation of invariant fractals. Zbl 0902.65100
<NAME>; <NAME>.; <NAME>.
1997
A study of particle paths in non-axisymmetric Taylor-Couette flows. Zbl 0889.76089
<NAME>.; <NAME>.
1997
Detection of symmetry of attractors from observations. I: Theory. Zbl 0894.58037
<NAME>; <NAME>
1997
Cycles homoclinic to chaotic sets; robustness and resonance. Zbl 1012.37020
<NAME>
1997
Detection of symmetry of attractors from observations. II: An experiment with S$$_4$$ symmetry. Zbl 0894.58038
Ashwin, Peter; <NAME>
1997
From attractor to chaotic saddle: A tale of transverse instability. Zbl 0887.58034
<NAME>; <NAME>; <NAME>
1996
Semirotors in the Josephson junction equations. Zbl 0843.34041
<NAME>.; <NAME>.; <NAME>.
1996
Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry. Zbl 0858.65123
Ashwin, Peter; Böhmer, Klaus; <NAME>.
1996
Torus doubling in four weakly coupled oscillators. Zbl 0885.58056
Ashwin, Peter; <NAME>.
1995
Normal form for Hopf bifurcation of partial differential equations on the square. Zbl 0839.35010
Ashwin, Peter; <NAME>.
1995
A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square. Zbl 0829.65075
<NAME>; <NAME>; <NAME>
1995
Streamline topology in eccentric Taylor vortex flow. Zbl 0882.76097
<NAME>.; <NAME>.
1995
Bifurcations of stationary, standing and travelling waves in triply diffusive convection. Zbl 0899.76165
Ashwin, Peter; <NAME>; <NAME>
1995
...and 15 more Documents
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#### Cited by 1,068 Authors

 58 Ashwin, Peter 19 Fu, Xinchu 12 Aguiar, <NAME>. 12 Dias, <NAME>. 11 Podvigina, <NAME> 11 Stewart, <NAME> 9 Bick, Christian 9 Castro, <NAME>. 9 Laing, <NAME>. 9 Li, Wan-Tong 9 Lohse, Alexander 9 Melbourne, Ian 9 Nicol, Matthew 9 Trofimchuk, <NAME>. 9 Wang, <NAME>eng 8 Aston, <NAME>. 8 Chen, Tianping 8 Field, <NAME>. 7 Dangelmayr, Gerhard 7 Labouriau, <NAME>. 7 Lamb, <NAME>. 7 Yu, Rongzhong 6 Bischi, Gian-Italo 6 Coombes, Stephen 6 Lu, Wenlian 6 Lueptow, <NAME>. 6 Maistrenko, <NAME>. 6 Moehlis, Jeff 6 Ruan, Shigui 6 Rucklidge, <NAME>. 6 Timme, Marc 5 Bressloff, <NAME>. 5 Burylko, Oleksandr 5 Chossat, Pascal 5 Du, Zengji 5 Goetz, Arek 5 Golubitsky, <NAME>. 5 Grebogi, Celso 5 Ha, Seung-Yeal 5 Knobloch, Edgar 5 Omel’chenko, <NAME>. 5 Ott, Edward 5 Pecora, <NAME>. 5 Pikovsky, <NAME>. 5 Postlethwaite, <NAME>. 5 Rodrigues, <NAME>. 5 Swift, <NAME>. 4 Aihara, Kazuyuki 4 Blyuss, <NAME>. 4 Buono, Pietro-Luciano 4 Carroll, <NAME>. 4 Collins, <NAME>. 4 Gardini, Laura 4 Gottwald, <NAME>. 4 Hasík, Karel 4 Huguet, Gemma 4 Inaba, Naohiko 4 Johnson, <NAME>. 4 Kamiyama, Kyohei 4 Kapitaniak, Tomasz 4 MacLaurin, <NAME>. 4 Mei, Zhen 4 Rodrigues, Ana 4 Sturman, Rob 4 Viana, <NAME> 4 Wechselberger, Martin 4 Wulff, Claudia 4 Yanchuk, Sergiĭ Volodymyrovych 4 Yuan, Liguo 3 Afraimovich, <NAME>. 3 Aminzare, Zahra 3 Belhaq, Mohamed 3 Belykh, <NAME>. 3 Belykh, <NAME>. 3 Bloch, <NAME>. 3 Böhmer, Klaus 3 Borisyuk, <NAME>. 3 Bruin, Henk 3 Cánovas, <NAME>. 3 Chawanya, Tsuyoshi 3 Chen, Zhanhe 3 Chien, Cheng-Sheng 3 Crauel, Hans 3 Deane, <NAME>. 3 Dellnitz, Michael 3 Duan, Jinqiao 3 Dudkowski, Dawid 3 Endo, Tetsuro 3 Fahsi, Abdelhak 3 Feng, Zhaosheng 3 Homburg, <NAME> 3 Kahng, Byungik 3 Kim, Yongduck 3 Kirk, Vivien 3 Krawiecki, A. 3 Krupa, Martin 3 Kurths, Jürgen 3 Lai, Yingcheng 3 Li, Kun 3 Li, Zhuchun ...and 968 more Authors
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#### Cited in 128 Serials

 104 Chaos 85 Physica D 60 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 39 Dynamical Systems 27 Journal of Nonlinear Science 27 SIAM Journal on Applied Dynamical Systems 26 Chaos, Solitons and Fractals 18 Journal of Differential Equations 16 Nonlinearity 15 Communications in Nonlinear Science and Numerical Simulation 12 The Journal of Mathematical Neuroscience 9 Ergodic Theory and Dynamical Systems 8 Physics Letters. A 8 Journal of Dynamics and Differential Equations 8 Nonlinear Dynamics 7 Journal of Mathematical Biology 7 Dynamics and Stability of Systems 6 Journal of Mathematical Analysis and Applications 6 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 5 Biological Cybernetics 5 Computer Methods in Applied Mechanics and Engineering 5 Journal of Statistical Physics 5 Applied Mathematics and Computation 5 Transactions of the American Mathematical Society 5 Physics of Fluids 5 Journal of Difference Equations and Applications 5 Communications on Pure and Applied Analysis 4 ZAMP. Zeitschrift für angewandte Mathematik und Physik 4 Journal of Computational and Applied Mathematics 4 Proceedings of the American Mathematical Society 4 Nonlinear Analysis. Real World Applications 4 Journal of Theoretical Biology 3 Israel Journal of Mathematics 3 Mathematics and Computers in Simulation 3 Theoretical Population Biology 3 European Journal of Applied Mathematics 3 Neural Computation 3 Applied Mathematical Modelling 3 Journal of Mathematical Sciences (New York) 3 Abstract and Applied Analysis 3 Journal of Shanghai University 3 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 3 Regular and Chaotic Dynamics 3 Journal of Dynamical and Control Systems 3 Discrete and Continuous Dynamical Systems. Series B 2 International Journal of Modern Physics B 2 Computers & Mathematics with Applications 2 Journal of Fluid Mechanics 2 Mathematical Methods in the Applied Sciences 2 Mathematical Proceedings of the Cambridge Philosophical Society 2 Journal of Geometry and Physics 2 Kybernetika 2 Mathematical and Computer Modelling 2 Japan Journal of Industrial and Applied Mathematics 2 Proceedings of the National Academy of Sciences of the United States of America 2 SIAM Review 2 Journal of Computational Neuroscience 2 Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 2 Qualitative Theory of Dynamical Systems 2 Stochastics and Dynamics 2 Discrete and Continuous Dynamical Systems. Series S 1 Modern Physics Letters B 1 Archive for Rational Mechanics and Analysis 1 Communications in Mathematical Physics 1 International Journal of Systems Science 1 Journal of Computational Physics 1 Journal of Mathematical Physics 1 Mathematical Biosciences 1 Physica A 1 Rocky Mountain Journal of Mathematics 1 Mathematics of Computation 1 Bulletin of Mathematical Biology 1 Theoretical and Computational Fluid Dynamics 1 Advances in Mathematics 1 Automatica 1 Computing 1 International Journal of Circuit Theory and Applications 1 International Journal of Game Theory 1 Inventiones Mathematicae 1 Journal of Approximation Theory 1 Journal of Functional Analysis 1 Journal of the London Mathematical Society. Second Series 1 Meccanica 1 Chinese Annals of Mathematics. Series B 1 Bulletin of the Iranian Mathematical Society 1 Boletim da Sociedade Portuguesa de Matemática 1 Journal of Symbolic Computation 1 Applied Mathematics Letters 1 Science in China. Series A 1 Neural Networks 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 International Journal of Computer Mathematics 1 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1 SIAM Journal on Applied Mathematics 1 SIAM Journal on Mathematical Analysis 1 Stochastic Processes and their Applications 1 Bulletin of the American Mathematical Society. New Series 1 Indagationes Mathematicae. New Series 1 Cybernetics and Systems Analysis 1 Experimental Mathematics ...and 28 more Serials
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text:
# Using C with the Kubos SDK¶

## Creating a Project¶

Create a folder for your project with the mkdir command followed by the name of your project.

Inside of this folder you will create three new folders:

• project-name - This is where any header (*.h) files will go.
• source - This is where any source (*.c) files will go.
• test - This is where any test files will go.

This is the skeleton folder structure of your C project.

You also need to create a text file called CMakeLists.txt. This is where the build instructions for your project will go.

Here is a quick rundown of the files needed to start your project:

Your final directory structure should look like this:

project-name
|_ CMakeLists.txt
|_ project-name
|_ source
|_ test

Kubos uses the CMake build system, along with our own folder conventions, for C projects. You can read more about CMake here.

## Working with CMakeLists.txt¶

The CMakeLists.txt file provides instructions to cmake on how to build the C project.

All CMakeLists.txt files need to start with these two lines:

cmake_minimum_required(VERSION 3.5)
project(project-name VERSION 0.1.0)

Next, cmake must be told what type of project it is building and which source files to compile.

If the project is a library, then the add_library command will be used:

add_library(project-name
source/lib.c
)

If the project is an executable, then the add_executable command will be used:

add_executable(project-name
source/main.c
)

Lastly, the header folder needs to be added using the target_include_directories command:

target_include_directories(project-name
PUBLIC "${project-name_SOURCE_DIR}/project-name" )  ## Building a Project¶ To build a Kubos C project, the CMake build mechanism needs to be invoked. First, a folder will be created to store the build artifacts: $ mkdir build
$cd build  Next, CMake will be called and told where to find the build instructions: $ cmake ..

Lastly, make will be called to execute the build setup by Cmake:

$make  All build artifacts will be generated in the build folder. If the project is an executable, then the binary will be named project-name. If the project is a library, then the library file will be named libproject-name.a. To build a project from scratch, run make clean to remove all prior build artifacts followed by make. To pick up on any changes to the projects CMake files, run cmake .. again, followed by make. To build a project without any of the prior make or cmake artifacts, remove the build directory with all of its contents and start the build process over. ## Cross Compiling¶ By default, CMake will compile with the local gcc toolchain found on the PATH. If you are working inside of the SDK VM, then the native target is x86 Linux. The SDK also provides cross-compiling toolchains for several different targets: Vendor Toolchain Description ISIS /usr/bin/iobc_toolchain/usr/bin/arm-linux-gcc ISIS-OBC Pumpkin /usr/bin/bbb_toolchain/usr/bin/arm-linux-gcc Pumpkin Motherboard Module 2 Beaglebone /usr/bin/bbb_toolchain/usr/bin/arm-linux-gcc Beaglebone Black, Rev. C (Vagrant) /usr/bin/gcc Native x86 Linux If you would like to cross-compile for one of the supported embedded boards, then CMake will need to be informed about which cross-compiling toolchain to use. CMake looks at two environment variables when compiling to determine which toolchain it should use. These variables are CC and CXX. For example: $ mkdir build && cd build
$export CC=/usr/bin/bbb_toolchain/usr/bin/arm-linux-gcc$ export CXX=/usr/bin/bbb_toolchain/usr/bin/arm-linux-g++
\$ cmake .. && make
`

## Transferring¶

Executables generated from C projects can be transferred to the target OBC via a supported file transfer method.

Binaries may be transferred to any location on the target board, however, they should be copied to /home/system/usr/bin if you would like them to be automatically accessible via the system PATH.

format: plain


text:
# FNGR 2 | Lesson 3 | Practice (Building Functions) Solutions

Solutions

1. $$5x^2-6x-8$$
2. $$6x+2$$
3. $$x^2+7x+7$$
4. $$-x^2-5x-11$$
5. $$x^2+11x+13$$
6. $$x^2+12x+11$$
7. $$4x^2-12x-17$$

Back to Practice (Building Functions)


text:
# What sort of questions are appropriate for Bioinformatics SE?

I have spent a lot of time on StackOverflow and am pretty familiar with the types of questions that are and are not allowed on the site. I am a little unsure as to how well the strict question rules transfer over to this site, however. Typically, questions that do not have concrete answers or are opinion-based are discouraged, as are requests for listing of resources. I don't think that this level of stringency is appropriate for this site. An example:

What is the best tool for computing gene-level expression?

This might immediately be flagged for deletion because it is both a resource request and opinion based, but keep in mind:

• It can be difficult to find scientific tools, especially if the analysis or function they perform is not very common or new.
• While people might have opinions on what tool is the best, there are quantifiable ways of demonstrating that some scientific tools are better or worse, which can be important to capture an upvote for future reference.
• There are a lot of unknowns in science, and therefore a lot of opinions, so one could argue a lot of things as opinion-based. Therefore, some leniency should be expected in question asking.

Thoughts?

• Another point could be that some communities might have some general form of agreement on the most appropriate tool for a specific analysis (i.e. blast for simple homology search), despite having many other alternatives. – mgalardini May 16 '17 at 20:15
• While I agree in principle with your bullets, I think that the appropriateness of the example question you posted would depend a lot on the body of the question. If it was just that heading, it's too vague, but if the questioner lays out specific features, there might actually be an objective answer. – kevbonham May 16 '17 at 20:37
• And there's another thing to consider - answers to questions like this will fluctuate wildly over time. We'll need to have pretty strong norms around modifying previous answers or marking old questions as out of date. – kevbonham May 16 '17 at 20:39
• @kevbonham: this is something that occurred to me as well. Just because an answer was the best possible solution at a given time doesn't mean it will even be close to correct 5 years from now. – woemler May 16 '17 at 20:43
• @woemler right. This would be an issue on SO as well, but most of the questions I see there are referring to specific languages/versions. Are there questions like "what's the best framework for designing a website?" on SO? How do they handle that? – kevbonham May 16 '17 at 20:46
• @kevbonham that answer simply wouldn't be accepted on SO, I don't think. On the other hand, if you said "I'm interested in creating a website with these features that runs like this..." then perhaps it would be okay. – <NAME> May 17 '17 at 6:56
• I suggest that bioinformatics 'experts' should ask what they consider to be good questions and provide their own good answers to get this site moving in the right direction! This practice is encouraged on other sites like SO stackoverflow.com/help/self-answer – Chris_Rands May 17 '17 at 8:43

I also don't think that "What is the best foo" can ever be a good question. I would instead suggest something like:

Is there a comprehensive review and benchmark of the various tools for FOO? What tools had the best results?

Something like that can be answered by citing a specific review paper and the measurements there. In any case, the "best" usually depends on what you're doing. For example, while toolA might be the best for computing expression in speciesA or tissueA, toolB might fare better in another species/tissue.

So no, let's not allow "what is the best foo" questions, but instead try and make them as specific as possible while still being general enough to be worth asking.

• Great answer. This would be a very good way to frame this sort of question. – woemler May 20 '17 at 0:35

Such open-ended question should be closed in my opinion. A better question for the site would be

What is better tool for computing gene-level expression A or B?

This question is still open to opinionated personal experience using the tools but can be answered with articles comparing both tools.

This way we can have a tag for and group there questions comparing . The reader after reading the questions and articles might end up with a good knowledge of the tools available and used by the community.

As suggested in the comments:

What are the pros/cons of computing gene level expression with A or B?

This might be a better alternative

• Perhaps even better: What are the pros/cons of computing gene level expression with A or B? This moves it slightly further away from opinion and towards objective fact. And is probably a better way to think about tools anyway. – kevbonham May 17 '17 at 13:15
• But what about cases where there is an objectively better tool, C, that is not specifically part of the question? – woemler May 17 '17 at 14:06
• @woemier If a better tool C exists, you can suggest so in comments either in the question or in any answer you want, and then the next question would be "pros/cons of C vs A?" and "pros/cons of C vs B?". – llrs May 17 '17 at 14:09
• @Llopis agreed. I've expanded my comment into an answer below. – kevbonham May 17 '17 at 14:35

I have a proposal - tool questions/answers should be limited to discussions of pros/cons for a particular tool or tools. In other words, the question you posed (and bouncing off of Llopis' answer):

What are the pros/cons of computing gene level expression with A vs B?

This approach has a number of benefits:

1. Answers to this question can be more objective. People may still quibble about whether a particular approach is better or worse, but the upsides/downsides of any approach shouldn't generate as much disagreement
2. This is probably a better way to think about such questions anyway, and answers to such questions are more generalizable to different experiments/approaches. In other words, even if a questioner asks about a particular experimental setup, good answers about the pros/cons of a particular tool will likely apply to other experimental setups.
3. These answers are less likely to go out of date, since new software will not change the pros/cons of particular approaches, but might change what the best approach is. And even old questions about out of date software will retain value for learners (since they will contain expert thought about how to assess the value of an approach).

I'm not sure if this approach is applicable to all questions of this type, but it could certainly be applied to most. It will take more effort from questioners, since they will have to do some research to actually bring up a tool or tools, but this will probably make for better questions anyway, and there's no reason an answer couldn't bring up some new tool that wasn't asked.


text:
Choosing rings for PLWE

In [ELOS15], the authors give an attack on RLWE, and claim that "the hardness of Ring-LWE is... dependent on special properties of the number field" chosen; whereas, responding to prior attacks on RLWE (including the one cited above), in [P16] the author claims that 'the rings themselves are not the source of insecurity in the vulnerable instantiations.' Which is the case? Are certain rings (uniquely?) vulnerable to certain attacks, or is it merely the parameters chosen which make a construction insecure (e.g. a poorly chosen error distribution)?

Section 4 of [P16] is perhaps the key section to read. I quote it below:

We stress that all these insecure instantiations—excepting [EHL14], for which the following conclusions still apply—are for the “non-dual” version of Ring-LWE with spherical Gaussian errors relative to $$R$$ (in the canonical embedding). By contrast, the definition of Ring-LWE from [LPR10], and the instantiations having worst-case hardness, involve spherical errors relative to the dual ideal $$R^\vee$$ (see Section 2.3.2). When the insecure and hard instantiations are transformed to be directly comparable, the resulting error distributions turn out to have very different widths and shapes. We return to this point in Section 5, where we show that the hard instantiations are immune to the attacks from Section 3

In fact, this can be understood to be the key difference between Ring-LWE and Poly-LWE --- whether LWE samples:

$$(a(x), a(x)s(x) + e(x))\in\begin{cases}R_q\times R_q^\vee & \text{RLWE}\\ R_q\times R_q & \text{PolyLWE}\end{cases}$$

Note that these are almost equivalent. In particular, at the end of section 2 of [P16] there is a heading called Equivalence of dual and non-dual forms where it is shown that they are equivalent up to the resulting choice of error distribution. And for power-of-two cyclotomics, it is known that this change is multiplication by a small constant, i.e. by a transformation that simply scales the covariance matrix of the underlying error distribution. In general the distributions are related by a certain linear transformation, but it need not be this simple.

[P16] points out that this means that the attacks on weak instances of PolyLWE are implicitly attacks on RLWE, and that the attacks are induced by not accounting for the aforementioned change in the underlying error distribution (which does vary depending on the underlying choice of ring). This means on the RLWE side of things, the error was too small, so the worst-case to average-case reduction was not valid (and therefore the PolyLWE attacks do not imply IdealSVP algorithms, as you might have otherwise expected).

• All this is true and useful to address the question, but there is another, simpler, way to see what’s wrong with these weak PLWE parameters without having to get into “dual” RLWE and all the rest. Specifically, the attacked error distributions for the PLWE problems are not “well spread” relative to the polynomial ring $R$. For example, some coefficients of the error polynomials are so small that they lie in (-1/2,1/2) with high probability, and hence round to zero. This is essentially “LWE with no error,” and is trivially insecure. (Feel free use this in your answer if you wish.) May 19 at 20:46


text:
Contents

# Contents

## Idea

For $\mathcal{G}$ a topological group or simplicial group, or generally an ∞-group, the homotopy type of the free loop space of its classifying space/delooping is equivalent to the homotopy quotient of the conjugation ∞-action of $\mathcal{G}$ on iself:

$\mathcal{L} \mathbf{B}\mathcal{G} \;\; \simeq \;\; \mathcal{G} \sslash_{ad} \mathcal{G} \,.$

For discrete groups (in particular for finite groups) this is seen by elementary inspection (Example below) and as such is familiar from many related constructions, such as that of inertia groupoids.

In more generality the statement is folklore (as witnessed by parts of the MO discussion) but rarely argued in detail.

For topological groups of the homotopy type of a CW-complex there is a point-set argument (Gruher 2007) with continuous paths in the universal principal bundle which, with some care, largely mimics the naive example .

More generally, such as for simplicial groups, there is a more abstract computation of the defining homotopy pullback by fibrant resolution, which is spelled out as Prop. below, following an analogous argument for topological groups indicated in Klein, Schochet & Smith 2009.

Finally, since, in the full generality of ∞-groups internal to any (∞,1)-topos, hence for ∞-group ∞-stacks $\mathcal{G}$, the homotopy quotient of an action, regarded in the slice over the delooping/moduli stack $\mathbf{B}\mathcal{G}$, in fact characterizes and defines the action as an ∞-action

$\mathcal{G} \,\in\, Groups(\mathbf{H}) \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \mathcal{G}Actions(\mathbf{H}) \;\; \underoverset {\;\;\;\;\simeq\;\;\;\;} {(-)\sslash \mathcal{G}} {\longrightarrow} \;\; \mathbf{H}_{/\mathbf{B}\mathcal{G}}$

one may turn this around (NSS 12a, Exp. 4.13 (3), SS 20, Exp. 2.82) and define the adjoint action of an ∞-group ∞-stack to be that whose homotopy quotient over $\mathbf{B}\mathcal{G}$ is the left vertical morphism (1) in the defining homotopy pullback-construction of the free loop space object of $\mathbf{B}\mathcal{G}$.

## Proof

The statement is folklore, but complete proofs in the literature are rare.

Recall that the free loop space object of $\mathbf{B}\mathcal{G}$ (in particular) is defined to be homotopy pullback/homotopy fiber product of the diagonal morphism on $\mathbf{B}\mathcal{G}$ along/with itself:

(1)$\array{ \mathcal{L} \mathbf{B}\mathcal{G} &\longrightarrow& \mathbf{B}\mathcal{G} \\ \big\downarrow &\swArrow_{\mathrlap{pb}}& \big\downarrow {}^{\mathrlap{diag}} \\ \mathbf{B}\mathcal{G} &\xrightarrow{\;\;\;diag\;\;\;}& \mathbf{B}\mathcal{G} \times \mathbf{B} \mathcal{G} }$

### For simplicial sets

We discuss the presentation of the phenomenon in the classical model structure on simplicial sets.

Let

and write

• $\mathcal{G}\Actions(sSet)$ for the category of $\mathcal{G}$-action objects internal to SimplicialSetsl

• $W \mathcal{G} \in \mathcal{G}Actions(sSet)$ for its universal principal simplicial complex;

• $\overline{W}\mathcal{G} \,=\, \frac{W \mathcal{G}}{\mathcal{G}} \in sSet$ for the simplicial classifying space;

• $\mathcal{G}_{ad} \in \mathcal{G}Actions(sSet)$ for the adjoint action of $\mathcal{G}$ on itself:

(2)$\array{ \mathcal{G}_{ad} \times \mathcal{G} &\xrightarrow{\;\;\;}& \mathcal{G}_{ad} \\ (g_k,h_k) &\mapsto& h_k \cdot g_k \cdot h_k^{-1} }$

which we may understand as the restriction along the diagonal morphism $\mathcal{G} \xrightarrow{diag} \mathcal{G} \times \mathcal{G}$ of the following action of the direct product group:

$\array{ \mathcal{G}_{ad} \times (\mathcal{G} \times \mathcal{G}) &\xrightarrow{\;\;\;}& \mathcal{G}_{ad} \\ (g_k, (h'_k, h_k)) &\mapsto& h'_k \cdot g_k \cdot h^{-1}_k \mathrlap{\,.} }$

###### Proposition

The free loop space object of the simplicial classifying space $\overline{W} \mathcal{G}$ is isomorphic in the classical homotopy category to the Borel construction of the adjoint action (2):

$\mathcal{L} \big( \overline{W}\mathcal{G} \big) \;\; \simeq \;\; \mathcal{G}_{ad} \sslash \mathcal{G} \;\;\;\;\;\; \in \;\; Ho\big( sSet_{Qu} \big)$

###### Proof

Consider the following commuting diagram of SimplicialSets:

Here:

One verifies by inspection that the commuting square shown is an ordinary pullback in SimplicialSets.

Finally notice that the simplicial classifying space-construction is indeed a model for the delooping

$\overline{W}\mathcal{G} \;\; \simeq \;\; \mathbf{B}\mathcal{G} \;\;\; \in \; Ho(\infty Groupoids)$

(essentially by its Quillen adjunction to the simplicial loop space-functor (here) and the May recognition theorem, see NSS 12a, Cor. 3.33).

In summary this means (by this Prop.) that this ordinary pullback-square represents the homotopy pullback (1) that defines the free loop space object.

### For topological spaces

A point-set argument for topological spaces/topological groups is spelled out in Gruher 2007, App. A. Discussion of the presentation of the phenomenon in the classical model structure on topological spaces in the abstract style above is indicated in Klein, Schochet & Smith 2009.

### In homotopy type theory

Recall that in homotopy type theory, an ∞-group $G$ is represented by its delooping type, a pointed connected type $\mathbf{B} G$.

(The categorical semantics of this is a groupal A-∞ ∞-stack, in some (∞,1)-topos, so that the following is actually about free loop spaces in the generality of inertia ∞-stacks, not though in the yet further generality of free loop ∞-stacks.)

An ∞-action of $G$ on a term $a$ of type $A$ is given by a group homomorphism $G \to Aut_A(a)$, represented by a morphism of pointed homotopy types $\mathbf{B} G \to_* (A,a)$. (Since $\mathbf{B} G$ is connected, it doesn’t matter whether we restrict the codomain to the connected component of $A$ at $a$.)

We thus see that the type of all ∞-actions of $G$ on objects of $A$ is the function type $\mathbf{B} G \to A$. In particular, the type of $U$-small $G$-types, where $U$ is a type universe, is $\mathbf{B} G \to U$.

By adjointness, the homotopy orbit type of a $G$-type $X \colon \mathbf{B} G \to U$ is given by the dependent sum-type,

$X \!\sslash\! G \;\coloneqq\; \sum_{t : \mathbf{B} G} X(t)$

(The homotopy fixed points are given by the dependent product-type, as discussed at ∞-action.)

Now, the adjoint action of $G$ on itself is given by the morphism

$\array{ G^{ad} \colon & \mathbf{B} G &\to& U \\ & t &\mapsto& (t =_{\mathbf{B} G} t) \,. }$

Indeed, given any path $p : t =_{\mathbf{B} G} u$ in $\mathbf{B} G$ and any element $q : G^{ad}(t)$, the transport of $q$ along $p$ in the family $G^{ad}$ is equal to the conjugate $p^{-1} \cdot q \cdot p : G^{ad}(u)$, as proven by path induction. (Here, we write path composition in diagrammatic order.)

Putting this together, we get that the homotopy orbits of the adjoint action are

\begin{aligned} G^{ad} \!\sslash\! G & \;=\; \sum_{t: \mathbf{B} G} (t =_{\mathbf{B} G} t) \\ & \;\simeq\; (ʃS^1 \to \mathbf{B} G), \end{aligned}

where in the last step we used the universal property of the homotopy type (shape) of the circle type, $ʃS^1 \simeq \mathbf{B}\mathbb{Z}$, defined as a higher inductive type (here) with a point constructor $base : ʃS^1$ and a path constructor $loop : base =_{ʃS^1} base$.

The function type $ʃS^1 \to B G$ is the representation of the free loop type $\mathcal{L}(\mathbf{B} G)$ of $\mathbf{B} G$, completing the argument.

## Examples

### For discrete (finite) groups

###### Example

(free loop space of classifying space of discrete groups)

The archetypical example has $\mathcal{G} = G \in Groups(Sets)$ a discrete group, such as (but not necessarily) a finite group. In this case the classifying space $B \mathcal{G} \simeq K(G,1)$ is an Eilenberg-MacLane space whose homotopy type is represented in the classical homotopy category simply by the delooping groupoid $G \rightrightarrows \ast$ of $G$. With this, the analysis of its free loop space follows from elementary inspection:

The hom-groupoid

$\big[ \mathbf{B}\mathbb{Z} ,\, \mathbf{B}G \big] \;\simeq\; \big( G \times G \underoverset {pr_2} { Ad } {\rightrightarrows} G \big)$

has

• as objects the functors $\mathbf{B}\mathbb{Z} \longrightarrow \mathbf{B}G$, hence equivalently group homomorphisms $\mathbb{Z} \to G$, hence equivalently elements $g \in G$;

• as morphisms the natural transformations between these, whose single component $h \in G$ must make the naturality square commute

$\array{ \mathbf{B}\mathbb{Z} &\xrightarrow{\;\;\;}& \mathbf{B}G \\ \\ \bullet && \bullet &\xrightarrow{\;\;\;h\;\;\;}& \bullet \\ \big\downarrow {}^{\mathrlap{1}} && \big\downarrow {}^{\mathrlap{g}} && \big\downarrow {}^{\mathrlap{g'}} \\ \bullet && \bullet &\xrightarrow{\;\;\;h\;\;\;}& \bullet }$

The condition that this commutes means equivalently that $g'$ is the image of $g$ under the conjugation action by $h$:

$h \cdot g' \;=\; g \cdot h \;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\; g' = h^{-1} \cdot g \cdot h \;=\; Ad_h(g) \,.$

This shows that the hom-groupoid is the action groupoid of the conjugation action

$\mathcal{L} \mathbf{B}G \;\simeq\; \big[ \mathbf{B}\mathbb{Z} ,\, \mathbf{B}G \big] \;\simeq\; G \sslash_{ad} G \,.$

This simple example essentially re-appears in the discussion of inertia groupoids.

### For simplicial abelian groups

As a direct corollary of the general statement we have:

###### Proposition

For $\mathcal{A} \in AbGroup(sSet)$ a simplicial abelian group, the homotopy type of the free loop space of its simplicial classifying space is the homotopy product (Cartesian product) of $\mathcal{A}$ with its delooping:

$\big[ \mathbf{B}\mathbb{Z}, \, \mathbf{B}\mathcal{A} \big] \;\; \simeq \;\; \mathcal{A} \,\times\, \mathbf{B}\mathcal{A} \,.$

###### Proof

This is the composite of the following sequence of equivalences (isomorphisms in the classical homotopy category):

\begin{aligned} \big[ \mathbf{B}\mathbb{Z}, \, \mathbf{B}\mathcal{A} \big] & \;\simeq\; \frac{ \mathcal{A}_{\mathrm{ad}} \times W\mathcal{A} }{ \mathcal{A} } \\ & \;\simeq\; \frac{ \mathcal{A}_{\mathrm{triv}} \times W\mathcal{A} }{ \mathcal{A} } \\ & \;\simeq\; \mathcal{A} \times \frac{ W\mathcal{A} }{ \mathcal{A} } \\ & \;\simeq\; \mathcal{A} \times \overline{W}\mathcal{A} \\ & \;\simeq\; \mathcal{A} \times \mathbf{B}\mathcal{A} \end{aligned}

Here:

As an immediate special cases of Prop. we have:

###### Example

For $n \in \mathbb{N}$ and

$\mathbf{B}^{n+1} \mathbb{Z} \;\simeq\; ʃ \mathbf{B}^n S^1$

the shape of the circle (n+1)-group, we have

$\mathcal{L} \mathbf{B}^{n+1}\mathbb{Z} \;\simeq\; \mathbf{B}^n\mathbb{Z} \,\times\, \mathbf{B}^{n+1} \mathbb{Z} \,.$

More generally, for $A \in AbGroups(Set)$ any discrete abelian group we have

$\mathcal{L} \mathbf{B}^{n+1}A \;\simeq\; \mathbf{B}^n A \,\times\, \mathbf{B}^{n+1}A \,,$

where the $n$-fold delooping

$\mathbf{B}^n A \;\simeq\; K(n,A)$

is equivalently the homotopy type of the Eilenberg-MacLane space for $A$ in degree $n$ (classifying ordinary cohomology in degree $n$ with coefficients in $A$).

###### Remark

Example pertains to the discussion of double dimensional reduction of brane charges in ordinary cohomology by the discussion here at geometry of physics – fundamental super p-branes.

## References

A point-set proof in TopologicalSpaces is given in

• <NAME>, A proof of $L B G \simeq Ad(E G)$, Appendix A in: String Topology of Classifying Spaces, 2007 (pdf, proquest:304826261)

An abstract proof in the above style, for topological groups/topological spaces, is indicated in:


text:
# Capital Lambda Symbol (Λ)

The capital Greek letter (capital lambda).

Symbol Format Data
Λ Unicode
923
TeX
\Lambda
SVG

## Usage

The Greek Alphabet is used throughout math to represent variables, constants, and coeffecients within math expressions, formulas, and equations.

## Related Symbols

The Greek letter lambda (λ) is used throught mathematics, computer science, and physics. The symbol is used to represent the wavelength of a wave. In computer science, the symbol is used in the study of "lambda calculus" and anonymous functions.


text: Steve is thinking of two numbers, $a$ and $b$. If their least common multiple is $100!$ and their greatest common divisor is $24$, how many possible unordered pairs of integers could he be thinking of?


text:
# Accuracy of empirical cumulative distribution function

I have a random variable with an unknown distribution and I want to find its cumulative distribution function. I sample the distribution $N$ times, with $$X_1, \dots, X_N$$ being iid random variables. I then plot the empirical cumulative distribution function.

Am I correct in assuming that if I let $N$ be such that $$N \geq \frac{1}{2\epsilon^2}\ln(\frac{2}{\alpha})$$ Then my empirical distribution function is within $\epsilon$ of the actual cumulative distribution function at all points with a $1-\alpha$ confidence?

Yes, if $$N \geq \frac{1}{2\epsilon^2}\ln(\frac{2}{\alpha})$$, then $$\mathbb{P}\left(\sup_{x \in \mathbb{R}}|F_N(x) - F(x)| > \epsilon \right) \le 2 e^{-2N\epsilon^2} \le \alpha,$$ which is what you want.


text:
# Customizing plots

This vignette exemplifies different ways to plot the specification curve. For most cases, the function plot_specs() takes care of the overall process. However, more specific customization is possible if we use the more specific functions plot_curve() and plot_choices. Furthermore, we can extend the overall plot with the additional function plot_samplesizes(). All of these functions produce objects of the class ggplot that can be customized further using the grammar of graphics provided by the package ggplot2.

## 1. Run the specification curve analysis

In order to have some data to work with, we run the minimal example included in the package.

library(specr)
library(dplyr)
library(ggplot2)
library(cowplot)

# run spec analysis
results <- run_specs(example_data,
y = c("y1", "y2"),
x = c("x1", "x2"),
model = "lm",
controls = c("c1", "c2"),
subset = list(group1 = unique(example_data$group1), group2 = unique(example_data$group2)))

Let’s quickly get some ideas about the specification curve by using summarise_specs()

summarise_specs(results)
## # A tibble: 1 x 7
##   median   mad   min   max   q25   q75   obs
##    <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1   3.59  4.56 -2.05  9.58  1.03  7.63   123
summarise_specs(results, x)
## # A tibble: 2 x 8
##   x     median   mad    min   max   q25   q75   obs
##   <chr>  <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 x1      3.59  4.42 -2.05   9.28 0.510  6.51   123
## 2 x2      4.40  4.84 -0.258  9.58 1.31   7.79   123

We see that it makes quite a difference whether x1 or x2 is used as independent variable.

## 2. A simple way to visualize the results

The simplest way to visualize most of the information contained in the results data frame is by using the plot_specs() function.

plot_specs(results)

We can further customize that function, e.g., by removing unnecessary information (in this case we only specified one model, this analytical choice is hence useless) or by reordering/transforming the analytical choices (and thereby visualize specific contrasts).

plot_specs(results,
choices = c("x", "y", "controls", "subsets"),  # "model is not plotted
rel_heights = c(1, 2))                         # changing relative heights

# Investigating specific contrasts
results %>%
mutate(group1 = ifelse(grepl("group1 = 0", subsets), "0", "1"),
group2 = ifelse(grepl("group2 = A", subsets), "A", "B & C")) %>%
plot_specs(choices = c("x", "y", "controls", "group1", "group2"))

## 3. An alternative way to visualize the results

### 3.1. Plot curve and choices seperately

The overall function plot_specs() is not very flexible. Alternatively, we can plot the specification curve and the choice panel individually and bind them together afterwards. This is useful as it allows us to customize and change both individual plots.

# Plot specification curve
p1 <- plot_curve(results,
ci = FALSE,
ribbon = TRUE) +
geom_hline(yintercept = 0,
linetype = "dashed",
color = "black") +
ylim(-8, 12) +
labs(x = "", y = "unstandarized regression coefficient") +
theme_half_open()

# Plot choices
p2 <- plot_choices(results,
choices = c("x", "y", "controls", "subsets")) +
labs(x = "specifications (ranked)") +
theme_half_open() +
theme(strip.text.x = element_blank())

# Combine plots
plot_specs(plot_a = p1,
plot_b = p2,
labels = c("", ""),      # remove plot labels
rel_height = c(2, 2.5))  # adjust relative heights

### 3.2. Include sample size histogram

By default, we do not know how many participants were included in each specification. If you remove missing values listwise, this may not be a big problem as all models are based on the same subsample. If you have missing values in your dataset and you did not impute them or delete them listwise, we should investigate how many participants were included in each specification. The function plot_samplesizes() provides an additional panel that can then be added to the overall plot using cowplot::plot_grid()

p3 <- plot_samplesizes(results) +
theme_half_open()

# Combine via cowplot
plot_grid(p1, p2, p3,
ncol = 1,
align = "v",
rel_heights = c(1.5, 2, 0.8),
axis = "rbl")


text:
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absolute value of complex numbers worksheet pdf 2021


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Conformal Relativistic Viscous Hydrodynamics: Applications to RHIC results at sqrt(s_NN) = 200 GeV

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A new set of equations for relativistic viscous hydrodynamics that captures both weak-coupling and strong-coupling physics to second order in gradients has been developed recently. We apply this framework to bulk physics at RHIC, both for standard (Glauber-type) as well as for Color-Glass-Condensate initial conditions and show that the results do not depend strongly on the values for the second-order transport coefficients. Results for multiplicity, radial flow and elliptic flow are presented and we quote the ratio of viscosity over entropy density for which our hydrodynamic model is consistent with experimental data. For Color-Glass-Condensate initial conditions, early thermalization does not seem to be required in order for hydrodynamics to describe charged hadron elliptic flow.

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Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics

(2004)
The ratio of shear viscosity to volume density of entropy can be used to characterize how close a given fluid is to being perfect. Using string theory methods, we show that this ratio is equal to a universal value of $$\hbar/4\pi k_B$$ for a large class of strongly interacting quantum field theories whose dual description involves black holes in anti--de Sitter space. We provide evidence that this value may serve as a lower bound for a wide class of systems, thus suggesting that black hole horizons are dual to the most ideal fluids.
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The order of the quantum chromodynamics transition predicted by the standard model of particle physics

(2006)
We determine the nature of the QCD transition using lattice calculations for physical quark masses. Susceptibilities are extrapolated to vanishing lattice spacing for three physical volumes, the smallest and largest of which differ by a factor of five. This ensures that a true transition should result in a dramatic increase of the susceptibilities.No such behaviour is observed: our finite-size scaling analysis shows that the finite-temperature QCD transition in the hot early Universe was not a real phase transition, but an analytic crossover (involving a rapid change, as opposed to a jump, as the temperature varied). As such, it will be difficult to find experimental evidence of this transition from astronomical observations.
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Anisotropy as a signature of transverse collective flow

(1992)
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Author and article information

Journal
2008-04-25
2009-04-21
0804.4015
10.1103/PhysRevC.78.034915