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## Bhattacharya, Kaushik

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 Author ID: bhattacharya.kaushik Published as: Bhattacharya, Kaushik; Bhattacharya, K. Homepage: https://eas.caltech.edu/people/bhatta External Links: MGP · ORCID · Google Scholar
 Documents Indexed: 63 Publications since 1991, including 1 Book
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#### Co-Authors

 7 single-authored 9 Ortiz, Michael 4 Dondl, Patrick Werner 3 Kohn, Robert Vita 3 Ravichandran, Guruswami 3 Suryanarayana, Phanish 2 Blesgen, Thomas 2 Chenchiah, Isaac Vikram 2 deBotton, Gal 2 Dolzmann, Georg 2 Gavini, Vikram 2 Gavinia, Vikram 2 Givli, Sefi 2 James, Richard D. 2 Knap, Jaroslaw 2 Liu, Liping 2 Plucinsky, Paul 2 Purohit, Prashant K. 2 Schlömerkemper, Anja 2 Shilo, Doron 2 Simha, N. K. 2 Warner, Mark R. E. 1 Abeyaratne, Rohan C. 1 Biggins, J. S. 1 Bourdin, Blaise 1 Brach, Stella 1 Braides, Andrea 1 Burcsu, E. 1 Cesana, Pierluigi 1 Courte, Luca 1 Cracium, Bogdan 1 Craciun, Bogdan 1 Dayal, Kaushik 1 Firoozye, Nikan B. 1 Fonseca, Irene 1 Francfort, Gilles A. 1 Giang, Ha Huong 1 Hosseini, Bamdad 1 Hulikal, Srivatsan 1 Ioffe, Dmitry 1 Jabbour, Michel E. 1 Knowles, James K. 1 Kovachki, Nikola B. 1 Kozlov, Sergeĭ Mikhaĭlovich 1 Kukta, R. V. 1 Lapusta, Nadia 1 Lemm, Marius 1 Lewicka, Marta 1 Li, Bo 1 Luckhaus, Stephan 1 Luskin, Mitchell 1 Modes, C. D. 1 Otto, Felix 1 Rudykh, Stephan 1 Schäffner, Mathias 1 Shen, Ching-Ping 1 Stuart, Andrew M. 1 Suquet, Pierre-Marie 1 Tan, Likun 1 Tanné, Erwan 1 Tevet-Deree, L. 1 Tian, Li-Xiu 1 Wang, Xin-Cindy 1 Xiao, Yu 1 Yavari, Arash 1 Zhang, Rongjing
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#### Serials

 16 Journal of the Mechanics and Physics of Solids 12 Archive for Rational Mechanics and Analysis 7 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 3 International Journal of Solids and Structures 3 Journal of Elasticity 3 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2 Interfaces and Free Boundaries 1 Computer Methods in Applied Mechanics and Engineering 1 Journal of Applied Mechanics 1 Journal of Computational Physics 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 International Journal for Numerical Methods in Engineering 1 Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 1 SIAM Journal on Applied Mathematics 1 Continuum Mechanics and Thermodynamics 1 Oberwolfach Reports 1 Acta Mechanica Sinica 1 PAMM. Proceedings in Applied Mathematics and Mechanics 1 Philosophical Transactions of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 SMAI Journal of Computational Mathematics
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#### Fields

 55 Mechanics of deformable solids (74-XX) 11 Partial differential equations (35-XX) 11 Statistical mechanics, structure of matter (82-XX) 6 Calculus of variations and optimal control; optimization (49-XX) 3 Numerical analysis (65-XX) 3 Quantum theory (81-XX) 2 Real functions (26-XX) 2 Probability theory and stochastic processes (60-XX) 2 Fluid mechanics (76-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Biology and other natural sciences (92-XX) 1 General and overarching topics; collections (00-XX) 1 Statistics (62-XX) 1 Computer science (68-XX)

#### Citations contained in zbMATH Open

59 Publications have been cited 716 times in 558 Documents Cited by Year
Microstructure of martensite. Why it forms and how give rise to the shape-memory effect. Zbl 1109.74002
Bhattacharya, Kaushik
2003
A theory of thin films of martensitic materials with applications to microactuators. Zbl 0960.74046
Bhattacharya, K.; James, R. D.
1999
Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Zbl 0780.73005
Bhattacharya, K.
1993
Restrictions on microstructure. Zbl 0808.73063
Bhattacharya, Kaushik; Firoozye, Nikan B.; James, Richard D.; Kohn, Robert V.
1994
Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. Zbl 1120.74690
Dayal, Kaushik; Bhattacharya, Kaushik
2006
Self-accommodation in martensite. Zbl 0771.73007
Bhattacharya, Kaushik
1992
Elastic energy minimization and the recoverable strains of polycrystalline shape-memory materials. Zbl 0894.73225
Bhattacharya, Kaushik; Kohn, Robert V.
1997
Non-periodic finite-element formulation of Kohn-Sham density functional theory. Zbl 1193.81006
Suryanarayana, Phanish; Gavini, Vikram; Blesgen, Thomas; Bhattacharya, Kaushik; Ortiz, Michael
2010
Dielectric elastomer composites. Zbl 1244.74014
Tian, Li-Xiu; Tevet-Deree, L.; Debotton, Gal; Bhattacharya, Kaushik
2012
Plates with incompatible prestrain. Zbl 1382.74081
Bhattacharya, Kaushik; Lewicka, Marta; Schäffner, Mathias
2016
Strain-energy functions with multiple local minima: Modeling phase transformations using finite thermoelasticity. Zbl 0993.74047
Abeyaratne, R.; Bhattacharya, K.; Knowles, J. K.
2001
Kinetics of phase boundaries with edges and junctions. Zbl 1017.74051
Simha, N. K.; Bhattacharya, K.
1998
The simply laminated microstructure in martensitic crystals that undergo a cubic-to-orthorhombic phase transformation. Zbl 0942.74056
Bhattacharya, Kaushik; Li, Bo; Luskin, Mitchell
1999
A model problem concerning recoverable strains of shape-memory polycrystals. Zbl 1186.74028
Bhattacharya, K.; Suquet, P. M.
2005
Quasi-continuum orbital-free density-functional theory: a route to multi-million atom non-periodic DFT calculation. Zbl 1162.74314
Gavinia, Vikram; Bhattacharya, Kaushik; Ortiz, Michael
2007
Phase boundary propagation in a heterogeneous body. Zbl 0990.74047
Bhattacharya, Kaushik
1999
Gaussian curvature from flat elastica sheets. Zbl 1219.74026
Modes, C. D.; Bhattacharya, K.; Warner, M.
2011
Relaxation of some multi-well problems. Zbl 0977.74029
Bhattacharya, Kaushik; Dolzmann, Georg
2001
Non-periodic finite-element formulation of orbital-free density functional theory. Zbl 1162.74461
Gavinia, Vikram; Knap, Jaroslaw; Bhattacharya, Kaushik; Ortiz, Michael
2007
Kinetics of phase boundaries with edges and junctions in a three-dimensional multi-phase body. Zbl 1005.74049
Simha, N. K.; Bhattacharya, K.
2000
The relaxation of two-well energies with possibly unequal moduli. Zbl 1134.74042
Chenchiah, Isaac V.; Bhattacharya, Kaushik
2008
Relaxed constitutive relations for phase transforming materials. Zbl 0966.74053
Bhattacharya, Kaushik; Dolzmann, Georg
2000
An asymptotic study of the debonding of thin films. Zbl 0999.74079
Bhattacharya, Kaushik; Fonseca, Irene; Francfort, Gilles
2002
Dynamics of strings made of phase-transforming materials. Zbl 1020.74020
Purohit, Prashant K.; Bhattacharya, Kaushik
2003
Energy-based model of compressive splitting in heterogeneous brittle solids. Zbl 0968.74056
Bhattacharya, K.; Ortiz, M.; Ravichandran, G.
1998
Multiscale instabilities in soft heterogeneous dielectric elastomers. Zbl 1348.74114
Rudykh, S.; Bhattacharya, K.; deBotton, G.
2014
A micromechanical model of surface steps. Zbl 1116.74323
Kukta, R. V.; Bhattacharya, K.
2002
Effective motion of a curvature-sensitive interface through a heterogeneous medium. Zbl 1061.35148
Craciun, Bogdan; Bhattacharya, Kaushik
2004
A theory of anharmonic lattice statics for analysis of defective crystals. Zbl 1104.74018
Yavari, Arash; Ortiz, Michael; Bhattacharya, Kaushik
2007
Effective behavior of nematic elastomer membranes. Zbl 1457.35078
Cesana, Pierluigi; Plucinsky, Paul; Bhattacharya, Kaushik
2015
Homogenization of a Hamilton-Jacobi equation associated with the geometric motion of an interface. Zbl 1043.35028
Cracium, Bogdan; Bhattacharya, Kaushik
2003
A continuum theory of deformable, semiconducting ferroelectrics. Zbl 1140.74012
Xiao, Yu; Bhattacharya, Kaushik
2008
A mesh-free convex approximation scheme for Kohn-sham density functional theory. Zbl 1219.65058
Suryanarayana, Phanish; Bhattacharya, Kaushik; Ortiz, Michael
2011
Stability of multicomponent biological membranes. Zbl 1322.74040
Givli, Sefi; Giang, Ha; Bhattacharya, Kaushik
2012
Examples of nonlinear homogenization involving degenerate energies. I. Plane strain. Zbl 1370.74129
Chenchiah, Isaac V.; Bhattacharya, Kaushik
2005
Effective behavior of an interface propagating through a periodic elastic medium. Zbl 1342.35027
Dondl, Patrick W.; Bhattacharya, Kaushik
2016
Thin films with many small cracks. Zbl 1011.74042
Bhattacharya, Kaushik; Braides, Andrea
2002
On beams made of a phase-transforming material. Zbl 1019.74022
Purohit, Prashant K.; Bhattacharya, Kaushik
2002
The microstructure of martensite and its implications for the shape- memory effect. Zbl 0813.73056
Bhattacharya, Kaushik
1993
A continuum theory of multispecies thin solid film growth by chemical vapor deposition. Zbl 1079.74044
Jabbour, Michel E.; Bhattacharya, Kaushik
2003
A model for large electrostrictive actuation in ferroelectric single crystals. Zbl 1108.74018
Shilo, D.; Burcsu, E.; Ravichandran, G.; Bhattacharya, K.
2007
Wave propagation in a sandwich structure. Zbl 1167.74457
Liu, Liping; Bhattacharya, Kaushik
2009
A coarse-grained model of the myofibril: Overall dynamics and the evolution of sarcomere non-uniformities. Zbl 1173.74026
Givli, Sefi; Bhattacharya, Kaushik
2009
Stress-induced phase transformations in shape-memory polycrystals. Zbl 1304.74049
Bhattacharya, Kaushik; Schlömerkemper, Anja
2010
Elasticity of polydomain liquid crystal elastomers. Zbl 1244.74011
Biggins, J. S.; Warner, Mark; Bhattacharya, Kaushik
2012
Coarse-graining Kohn-Sham density functional theory. Zbl 1258.81090
Suryanarayana, Phanish; Bhattacharya, Kaushik; Ortiz, Michael
2013
Actuation of thin nematic elastomer sheets with controlled heterogeneity. Zbl 1384.35126
Plucinsky, Paul; Lemm, Marius; Bhattacharya, Kaushik
2018
Some examples of nonlinear homogenization involving nearly degenerate energies. Zbl 0926.74099
Bhattacharya, Kaushik; Kohn, Robert V.; Kozlov, S.
1999
Korn’s inequality for sequences. Zbl 0735.73010
Bhattacharya, Kaushik
1991
A sharp interface model for the propagation of martensitic phase boundaries. Zbl 1261.74025
Dondl, Patrick W.; Bhattacharya, Kaushik
2010
Computational analysis of martensitic thin films using subdivision surfaces. Zbl 1194.74183
Dondl, Patrick W.; Shen, Ching-Ping; Bhattacharya, Kaushik
2007
Existence of surface waves and band gaps in periodic heterogeneous half-spaces. Zbl 1328.74053
Hu, L. X.; Liu, L. P.; Bhattacharya, K.
2012
Phase-field study of crack nucleation and propagation in elastic-perfectly plastic bodies. Zbl 1441.74201
Brach, Stella; Tanné, Erwan; Bourdin, Blaise; Bhattacharya, Kaushik
2019
Bounds on precipitate hardening of line and surface defects in solids. Zbl 1440.35323
Courte, Luca; Bhattacharya, Kaushik; Dondl, Patrick
2020
Mechanical characterization of released thin films by contact loading. Zbl 1111.74740
Zhang, Rongjing; Shilo, Doron; Ravichandran, Guruswami; Bhattacharya, Kaushik
2006
Corrigendum to “Non-periodic finite-element formulation of orbital-free density functional theory” [Journal of the Mechanics and Physics of Solids 55, 669-696 (2007; Zbl 1162.74461)]. Zbl 1225.74088
Gavini, Vikram; Knap, Jaroslaw; Bhattacharya, Kaushik; Ortiz, Michael
2010
Length scales and pinning of interfaces. Zbl 1353.82057
Tan, Likun; Bhattacharya, Kaushik
2016
On the Sachs bound in stress-induced phase transformations in polycrystalline scalar shape-memory alloys. Zbl 1393.74023
Schlömerkemper, Anja; Bhattacharya, Kaushik
2008
Model reduction and neural networks for parametric PDEs. Zbl 07388500
Bhattacharya, Kaushik; Hosseini, Bamdad; Kovachki, Nikola B.; Stuart, Andrew M.
2021
Model reduction and neural networks for parametric PDEs. Zbl 07388500
Bhattacharya, Kaushik; Hosseini, Bamdad; Kovachki, Nikola B.; Stuart, Andrew M.
2021
Bounds on precipitate hardening of line and surface defects in solids. Zbl 1440.35323
Courte, Luca; Bhattacharya, Kaushik; Dondl, Patrick
2020
Phase-field study of crack nucleation and propagation in elastic-perfectly plastic bodies. Zbl 1441.74201
Brach, Stella; Tanné, Erwan; Bourdin, Blaise; Bhattacharya, Kaushik
2019
Actuation of thin nematic elastomer sheets with controlled heterogeneity. Zbl 1384.35126
Plucinsky, Paul; Lemm, Marius; Bhattacharya, Kaushik
2018
Plates with incompatible prestrain. Zbl 1382.74081
Bhattacharya, Kaushik; Lewicka, Marta; Schäffner, Mathias
2016
Effective behavior of an interface propagating through a periodic elastic medium. Zbl 1342.35027
Dondl, Patrick W.; Bhattacharya, Kaushik
2016
Length scales and pinning of interfaces. Zbl 1353.82057
Tan, Likun; Bhattacharya, Kaushik
2016
Effective behavior of nematic elastomer membranes. Zbl 1457.35078
Cesana, Pierluigi; Plucinsky, Paul; Bhattacharya, Kaushik
2015
Multiscale instabilities in soft heterogeneous dielectric elastomers. Zbl 1348.74114
Rudykh, S.; Bhattacharya, K.; deBotton, G.
2014
Coarse-graining Kohn-Sham density functional theory. Zbl 1258.81090
Suryanarayana, Phanish; Bhattacharya, Kaushik; Ortiz, Michael
2013
Dielectric elastomer composites. Zbl 1244.74014
Tian, Li-Xiu; Tevet-Deree, L.; Debotton, Gal; Bhattacharya, Kaushik
2012
Stability of multicomponent biological membranes. Zbl 1322.74040
Givli, Sefi; Giang, Ha; Bhattacharya, Kaushik
2012
Elasticity of polydomain liquid crystal elastomers. Zbl 1244.74011
Biggins, J. S.; Warner, Mark; Bhattacharya, Kaushik
2012
Existence of surface waves and band gaps in periodic heterogeneous half-spaces. Zbl 1328.74053
Hu, L. X.; Liu, L. P.; Bhattacharya, K.
2012
Gaussian curvature from flat elastica sheets. Zbl 1219.74026
Modes, C. D.; Bhattacharya, K.; Warner, M.
2011
A mesh-free convex approximation scheme for Kohn-sham density functional theory. Zbl 1219.65058
Suryanarayana, Phanish; Bhattacharya, Kaushik; Ortiz, Michael
2011
Non-periodic finite-element formulation of Kohn-Sham density functional theory. Zbl 1193.81006
Suryanarayana, Phanish; Gavini, Vikram; Blesgen, Thomas; Bhattacharya, Kaushik; Ortiz, Michael
2010
Stress-induced phase transformations in shape-memory polycrystals. Zbl 1304.74049
Bhattacharya, Kaushik; Schlömerkemper, Anja
2010
A sharp interface model for the propagation of martensitic phase boundaries. Zbl 1261.74025
Dondl, Patrick W.; Bhattacharya, Kaushik
2010
Corrigendum to “Non-periodic finite-element formulation of orbital-free density functional theory” [Journal of the Mechanics and Physics of Solids 55, 669-696 (2007; Zbl 1162.74461)]. Zbl 1225.74088
Gavini, Vikram; Knap, Jaroslaw; Bhattacharya, Kaushik; Ortiz, Michael
2010
Wave propagation in a sandwich structure. Zbl 1167.74457
Liu, Liping; Bhattacharya, Kaushik
2009
A coarse-grained model of the myofibril: Overall dynamics and the evolution of sarcomere non-uniformities. Zbl 1173.74026
Givli, Sefi; Bhattacharya, Kaushik
2009
The relaxation of two-well energies with possibly unequal moduli. Zbl 1134.74042
Chenchiah, Isaac V.; Bhattacharya, Kaushik
2008
A continuum theory of deformable, semiconducting ferroelectrics. Zbl 1140.74012
Xiao, Yu; Bhattacharya, Kaushik
2008
On the Sachs bound in stress-induced phase transformations in polycrystalline scalar shape-memory alloys. Zbl 1393.74023
Schlömerkemper, Anja; Bhattacharya, Kaushik
2008
Quasi-continuum orbital-free density-functional theory: a route to multi-million atom non-periodic DFT calculation. Zbl 1162.74314
Gavinia, Vikram; Bhattacharya, Kaushik; Ortiz, Michael
2007
Non-periodic finite-element formulation of orbital-free density functional theory. Zbl 1162.74461
Gavinia, Vikram; Knap, Jaroslaw; Bhattacharya, Kaushik; Ortiz, Michael
2007
A theory of anharmonic lattice statics for analysis of defective crystals. Zbl 1104.74018
Yavari, Arash; Ortiz, Michael; Bhattacharya, Kaushik
2007
A model for large electrostrictive actuation in ferroelectric single crystals. Zbl 1108.74018
Shilo, D.; Burcsu, E.; Ravichandran, G.; Bhattacharya, K.
2007
Computational analysis of martensitic thin films using subdivision surfaces. Zbl 1194.74183
Dondl, Patrick W.; Shen, Ching-Ping; Bhattacharya, Kaushik
2007
Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. Zbl 1120.74690
Dayal, Kaushik; Bhattacharya, Kaushik
2006
Mechanical characterization of released thin films by contact loading. Zbl 1111.74740
Zhang, Rongjing; Shilo, Doron; Ravichandran, Guruswami; Bhattacharya, Kaushik
2006
A model problem concerning recoverable strains of shape-memory polycrystals. Zbl 1186.74028
Bhattacharya, K.; Suquet, P. M.
2005
Examples of nonlinear homogenization involving degenerate energies. I. Plane strain. Zbl 1370.74129
Chenchiah, Isaac V.; Bhattacharya, Kaushik
2005
Effective motion of a curvature-sensitive interface through a heterogeneous medium. Zbl 1061.35148
Craciun, Bogdan; Bhattacharya, Kaushik
2004
Microstructure of martensite. Why it forms and how give rise to the shape-memory effect. Zbl 1109.74002
Bhattacharya, Kaushik
2003
Dynamics of strings made of phase-transforming materials. Zbl 1020.74020
Purohit, Prashant K.; Bhattacharya, Kaushik
2003
Homogenization of a Hamilton-Jacobi equation associated with the geometric motion of an interface. Zbl 1043.35028
Cracium, Bogdan; Bhattacharya, Kaushik
2003
A continuum theory of multispecies thin solid film growth by chemical vapor deposition. Zbl 1079.74044
Jabbour, Michel E.; Bhattacharya, Kaushik
2003
An asymptotic study of the debonding of thin films. Zbl 0999.74079
Bhattacharya, Kaushik; Fonseca, Irene; Francfort, Gilles
2002
A micromechanical model of surface steps. Zbl 1116.74323
Kukta, R. V.; Bhattacharya, K.
2002
Thin films with many small cracks. Zbl 1011.74042
Bhattacharya, Kaushik; Braides, Andrea
2002
On beams made of a phase-transforming material. Zbl 1019.74022
Purohit, Prashant K.; Bhattacharya, Kaushik
2002
Strain-energy functions with multiple local minima: Modeling phase transformations using finite thermoelasticity. Zbl 0993.74047
Abeyaratne, R.; Bhattacharya, K.; Knowles, J. K.
2001
Relaxation of some multi-well problems. Zbl 0977.74029
Bhattacharya, Kaushik; Dolzmann, Georg
2001
Kinetics of phase boundaries with edges and junctions in a three-dimensional multi-phase body. Zbl 1005.74049
Simha, N. K.; Bhattacharya, K.
2000
Relaxed constitutive relations for phase transforming materials. Zbl 0966.74053
Bhattacharya, Kaushik; Dolzmann, Georg
2000
A theory of thin films of martensitic materials with applications to microactuators. Zbl 0960.74046
Bhattacharya, K.; James, R. D.
1999
The simply laminated microstructure in martensitic crystals that undergo a cubic-to-orthorhombic phase transformation. Zbl 0942.74056
Bhattacharya, Kaushik; Li, Bo; Luskin, Mitchell
1999
Phase boundary propagation in a heterogeneous body. Zbl 0990.74047
Bhattacharya, Kaushik
1999
Some examples of nonlinear homogenization involving nearly degenerate energies. Zbl 0926.74099
Bhattacharya, Kaushik; Kohn, Robert V.; Kozlov, S.
1999
Kinetics of phase boundaries with edges and junctions. Zbl 1017.74051
Simha, N. K.; Bhattacharya, K.
1998
Energy-based model of compressive splitting in heterogeneous brittle solids. Zbl 0968.74056
Bhattacharya, K.; Ortiz, M.; Ravichandran, G.
1998
Elastic energy minimization and the recoverable strains of polycrystalline shape-memory materials. Zbl 0894.73225
Bhattacharya, Kaushik; Kohn, Robert V.
1997
Restrictions on microstructure. Zbl 0808.73063
Bhattacharya, Kaushik; Firoozye, Nikan B.; James, Richard D.; Kohn, Robert V.
1994
Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Zbl 0780.73005
Bhattacharya, K.
1993
The microstructure of martensite and its implications for the shape- memory effect. Zbl 0813.73056
Bhattacharya, Kaushik
1993
Self-accommodation in martensite. Zbl 0771.73007
Bhattacharya, Kaushik
1992
Korn’s inequality for sequences. Zbl 0735.73010
Bhattacharya, Kaushik
1991
all top 5

#### Cited by 744 Authors

 22 Bhattacharya, Kaushik 19 Conti, Sergio 13 Steinmann, Paul 12 James, Richard D. 12 Javili, Ali 10 Kružík, Martin 10 Luskin, Mitchell 10 Ortiz, Michael 10 Zhang, Kewei 9 Dolzmann, Georg 8 Müller, Stefan 8 Ponte Castañeda, Pedro 7 Bělík, Pavel 7 Dondl, Patrick Werner 7 Gavini, Vikram 7 Stupkiewicz, Stanisław 6 Della Porta, Francesco 6 Grabovsky, Yury 6 Hackl, Klaus 6 Lewicka, Marta 6 Rüland, Angkana 6 Zhou, Aihui 6 Zwicknagl, Barbara Maria 5 Babadjian, Jean-François 5 Chenchiah, Isaac Vikram 5 Friesecke, Gero 5 Heinen, Rainer 5 Lipton, Robert P. 5 Peigney, Michaël 5 Petryk, Henryk 5 Roubíček, Tomáš 5 Schlömerkemper, Anja 5 Truskinovsky, Lev 4 Acharya, Amit 4 Ball, John M. 4 Bartel, Thorsten 4 Blesgen, Thomas 4 Cesana, Pierluigi 4 Chen, Huajie 4 Dayal, Kaushik 4 deBotton, Gal 4 DeSimone, Antonio 4 Dhote, R. P. 4 Fonseca, Irene 4 Hu, Guanghui 4 Kohn, Robert Vita 4 Melnik, R. N. V. 4 Menzel, Andreas M. 4 Miehe, Christian 4 Neff, Patrizio 4 Ortner, Christoph 4 Pedregal, Pablo 4 Sfyris, Dimitris 4 Simha, N. K. 4 Zillinger, Christian 3 Arroyo, Marino 3 Bao, Gang 3 Benešová, Barbora 3 Berezovski, Arkadi 3 Braides, Andrea 3 Cardaliaguet, Pierre 3 Cicalese, Marco 3 Dal Maso, Gianni 3 Eremeyev, Victor A. 3 Fabrizio, Mauro 3 Fang, Jun 3 Fischer, Franz Dieter 3 Gao, Xingyu 3 Goriely, Alain 3 Govindjee, Sanjay 3 Grandi, Diego 3 Hong, Jung-Wuk 3 Idiart, Martín I. 3 Jabbour, Michel E. 3 Kalpakides, Vassilios K. 3 Kirchheim, Bernd 3 Le Dret, Hervé 3 Leoni, Giovanni 3 Li, Xiantao 3 Lions, Pierre-Louis 3 Liu, Di 3 Liu, Liping 3 Liu, Wenyang 3 Lopez-Pamies, Oscar 3 Maugin, Gérard A. 3 Meunier, Nicolas 3 Michaille, Gérard 3 Mielke, Alexander 3 Milton, Graeme Walter 3 Monchiet, Vincent 3 Motamarri, Phani 3 Raoult, Annie 3 Rudykh, Stephan 3 Rumpf, Martin 3 Schneider, Matti 3 Siboni, Morteza Hakimi 3 Silling, Stewart A. 3 Souganidis, Panagiotis E. 3 Vainchtein, Anna 3 Vallicotti, Daniel ...and 644 more Authors
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#### Cited in 108 Serials

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# Why ordinal target in classification problems need special attention?

I have been working on an ML problem in which I want to predict an interval of money say, a, b, c, d that might be lent to a customer given its credit files, those amounts are represented on ordered bins the i.e a < b < c < d.

First I faced this problem as a multiclass classification problem and even though I did not obtain "good" performance I never thought It could be because of the inherent order on my target.

After googling it I found a paper in which a method to perform classification on ordinal target was developed, but I'm not still sure what are the implications on this scenario and even why it needs special attention.

In that paper is stated:

Standard classification algorithms for nominal classes can be applied to ordinal prediction problems by discarding the ordering information in the class attribute. However, some information is lost when this is done, information that can potentially improve the predictive performance of a classifier.

But it does not clarify the point to me.

Could you please help me to understand the implications of an ordinal target? Might this order, be responsible for poor performance on a multiclass classification task when this order is not considered by applying Logistic Regression, an Ensemble method or any other classification model?

• Let’s say you want to predict a rating: awful, bad, neutral, good, or excellent. Your prediction is neutral when the true label is good, so you got that wrong. However, you didn’t get it as wrong as if you predicted an awful rating. – Dave Oct 23 '20 at 1:09
• So, this is not a matter of the metric you use to evaluate the model rather than the predictions itself? – Julio Jesus Luna Oct 23 '20 at 1:25
• I mean, say you define your classes as 0, 1, 2, 3. If you construct a metric such as average((y_hat - y_true)**2) this would penalize more if the true class is for example 0 and you predict 1 vs if you predict 2 or 3. Am I missing something? – Julio Jesus Luna Oct 23 '20 at 1:31
• As the metric tries to optimize, it doesn’t know that it missed the correct classification by just a little. (There’s more to it than that, since we’d be getting probability predictions, but that’s the gist.) – Dave Oct 23 '20 at 1:32
• Those categories could just as easily be coded as $0$, $1$, $2$, $3$, $400$, so treating them as numbers and using square or absolute loss does not quite do what you want to do. – Dave Oct 23 '20 at 10:00

Dave's comments are on the right track. I'll try to expand on them.

Ordinal regression is half-way between classification and real-valued regression. When you perform multiclass classification of your ordinal data, you are assigning the same penalty whenever your classifier predicts a wrong class, no matter which one.

For example, assume that in your problem for some input vector $$x$$ the right prediction is $$a$$. Assume you are training two classifiers, $$C_1$$ and $$C_2$$. The first one predicts $$b$$, while the other predicts $$d$$. In the multivariate classifier's sense, $$C_1$$ and $$C_2$$ are equally far off, they have missed the correct class. But from the ordinal regression perspective, $$C_1$$ is obviously better than $$C_2$$, since it has missed the correct "class" only by one bin, not by three.

To drive this point into extreme, imagine performing a very-many-classes-classification instead of regression. I.e. you have predictors $$x$$ and a real-valued response variable $$y$$. You can treat values of $$y$$ as classes: $$y = 3.14159$$ would be one class, $$y = 1.4142$$ another, and so on. If you had $$N$$ observations, you're likely to have $$N$$ different classes (assuming all $$y$$'s differ). You could try to train a multiclass classifier, but you'd be likely to fail, as there would be only one observation per class. And even if you succeeded (because you were lucky to have same $$y$$'s repeat multiple times), you'd be essentially having many independent models, where each would only predict its own class and wouldn't care much about the others.

Such an ensemble of models would also be quite complex. If each model has, say, $$M$$ parameters, and if you had $$K$$ classes to predict $$(K < N)$$, your ensemble would have $$M \cdot K$$ parameters. In contrast, the complexity of the regression model is likely to be independent of the number of distinct $$y$$ values. You'd settle in advance for a linear, quadratic, or whatever function to fit through your data and the form of the function would determine the number of parameters.

In ordinal regression, e.g. proportional odds logistic regression, it is common to have one set of parameters (a vector) common to all "classes" (i.e. ordinal values), and a set of scalars to distinguish between the individual ordinal values. The same holds also for support vector ordinal regression (see e.g. http://www.gatsby.ucl.ac.uk/~chuwei/paper/svor.pdf), where you have the same model, consisting of the same $$\alpha$$'s (Lagrange coefficients) for all "classes", and distinguish between the classes only by the corresponding $$b$$'s (one per "class").

I'd argue that there are two potential complications with discarding the ordering information and just running multiclass regression:

1. Model complexity: the parameters of the predictions for each category won't be tied together in any way. So the model is trying to learn how to predict category a, b, c, and d as four separate problems, without realizing that there is some structure (e.g. examples of class a will look more similar to those of class b than those of class c or d). This could lead to poor performance on relatively small datasets.

2. As mentioned in the comments above, the multiclass loss function doesn't take the ordering into account. Presumably in your application mispredicting class a as class b is less bad than mispredicting it as class c.

I can think of a few possible solutions to these issues. One option is to define a similarity matrix across conditions in some way. This could come from some measure of uncertainty between the categories (e.g. a confusion matrix from human labels), or be related to the consequences for misprediction (e.g. if class d is a very elite class that gets a special credit score, that could be designated as less similar to the other classes). This kind of similarity matrix could be used to solve both problems listed above, if it is used for regularization (encouraging similar classes to have similar parameters) and for loss (penalizing mispredictions less harshly for similar classes).

Another possible answer is to just abandon ordinal prediction entirely, and try to predict the amount of money as a continuous value (which you could then discretize into bins if you wanted to). You still might need to think carefully about the loss function (e.g. if these values span multiple orders of magnitude you may want to penalize squared loss of the log of the values, rather than the values themselves).

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# Domain Driven Design : My April Read

At the beginning of this month I challenged myself to learn more about Domain Driven Design. I have come across this topic multiple times but had never focused much on it. Instead of reading blog posts and watching courses on Pluralsight (which I eventually did), I decided to first read Eric Evans’ book in which he introduces the concept. In this post I am going to talk about what I have learnt about Domain Driven Design and how it can help solve complex software problems. I am going to talk about core concepts in Domain Driven Design as I understand them.

## What Is A Domain?

Before we talk about Domain Driven Design we need to understand what a domain is. According to this definition a domain is a specified sphere of activity or knowledge. In software terms a domain is a subject area to which a software program is applied, for instance, e-commerce, insurance, entertainment, etcetera. Every software program seeks to solve a problem in a specific domain.

## Domain Driven Design

Now that we have an understanding of what a domain is let us now talk about what Domain Driven Design is. From an English standpoint the phrase domain-driven design simply means a design that is driven by and revolves around a particular domain. But what is it exactly? Domain Driven Design is an approach to software development that places primary focus on the core domain of the software program. The first point of call for developers should be to understand the problem domain of the software they want to build and then create an abstract model of the domain. If one doesn’t fully understand the problem domain that the software seeks to solve it is virtually impossible to solve it properly.

As software developers we tend to think about every problem we face in technical terms – the data structures, algorithms etc. This however, is not supposed to be the case. Eric Evans points out that the most significant complexity of many software applications is not technical but rather in the domain itself – the activity or business of the user. Without a proper design of the core domain one’s technical prowess is of no value at all.

Domain Driven Design is not a once-off event. It is an iterative process in which the model is refactored and refined as more knowledge and insight of the domain is gained.

## Domain Driven Design Concepts

We have already spoken about the domain which, in my opinion, is the core around which everything revolves. I am going to talk about a few terms I picked up in Eric Evans’ book that I think are important in Domain Driven Design.

### Model

In Domain Driven Design a model is a set of abstractions that describe the aspects and processes of the domain. It is a simplification of the ‘bigger picture’ of the problem domain and describes the relationships among the entities within the domain.

### Ubiquitous Language

This is the common language structured around the domain model and is used by everyone involved with the project. In Domain Driven Design everyone working on the project should speak the same language that is free of ambiguities. Both the domain experts and the software developers should understand each other when talking about terms in the domain.

Domain experts have limited understanding of technical terms used in software development and, on the other hand, developers may struggle to understand what the domain experts are saying. Without a common language some meaning will be lost in ‘translation’ which might lead to problems with the design of the solution.

### Bounded Context

This is a description of a boundary within which a particular model is defined and applicable. In a large system you often find the same name of a model being used in different subsystems. For instance, a Customer can be a model used in both the sales subsystem as well as the support subsytem. These two models, though they have the same name, may mean completely diffent things and thus behave differently. If there are no clear boundaries between these subsytems developers may fall into the temptation of using the same Customer model in both subsystems which might lead to problems.

There should be a clear seperation between models in different contexts. Even though they may share the same name they usually carry different meanings and thus should be treated as completely different things to allow them to evolve indepent of each other. Points of contact between contexts should then be defined, outlining explicit translation for any communication and highlighting any sharing.

## The Role Of Domain Experts

Domain experts play a significant role in Domain Driven Design. They are the ones who fully understand the problem domain and, in most cases, they are the ones who will use the software. There should be constant communication between developers and the domain experts to iron out all ambiguities and develop a ubiquitous language.

## When To Use Domain Driven Design

Domain Driven Design, in my opinion, can be used in any software application. However, it shines the brightest when you are faced with a problem that has complex domain logic and business rules. Using Domain Driven Design on simple CRUD applications may be an overkill. I’m certainly no expert in DDD so don’t take my word for it. This is just the observation I made when I was reading Eric’s book.

## Conclusion

In this post I spoke about what I understand about Domain Driven Design after finishing Eric Evans’ book on the subject. There are a lot of things I left out such as the use of entities, value objects, services etc. I may talk about them in a later post. The aim of this post was to talk about what DDD is and what it entails. Once again thank you so much for taking your time to read. If you’re reading this during the COVID-19 pandemic of 2020 stay safe and observe social distancing.


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# Example of a pattern that generates numbers that slowly round themselves?

So here is my question:

Is there an algorithm that generates a pattern, in which when comparing each subsequent number, every number is a rounded version of the previous number, rounded down 1 place-value.

For example, the pattern generated from this algorithm could be:

1848  185  19  2

Every number after the first number in the pattern is a rounded form of the previous number.

My research online has held no results to such an algorithm that can generate a series of number like this. Do any of you possibly know?

EDIT

In response to a mistake on my question, I am seeking an algorithm in which the numbers generated don't have to be floored. Or modified after being generated.

Optimally the algorithm I am seeking should generate the numbers already in integer form.

Along the same lines as Yujie's answer is (but without explicitly using floor function): $x_{n+1} = (x_n + 5 - (x_n+5 \mod 10))/10$.

Alternatively, since the OP asked for "an algorithm", I'd infer that means more latitude than a recurrence relation. For example an algorithm could be:

$x_{n+1} = \lfloor \frac{x_n + 5}{10} \rfloor$, here the division is integral division (i.e. floor of the divided result).


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Opuscula Math. 26, no. 3 (2006), 499-506

Opuscula Mathematica

# On the asymptotics of the difference equation with a proportional delay

Petr Kundrát

Abstract. This paper deals with asymptotic properties of a vector difference equation with delayed argument $\Delta x_k=Ax_k+Bx_{\lfloor\lambda k\rfloor},\qquad 0\lt\lambda\lt 1,\quad k=0,1,2,\dots,$ where $$A$$, $$B$$ are constant matrices and the term $$\lfloor\lambda k\rfloor$$ is the integer part of $$\lambda k$$. Our aim is to emphasize some resemblances between the asymptotic behaviour of this delay difference equation and its continuous counterpart.

Keywords: asymptotics of difference equations, approximation methods for dynamical systems.

Mathematics Subject Classification: 39A11, 37M99.

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Petr Kundrát, On the asymptotics of the difference equation with a proportional delay, Opuscula Math. 26, no. 3 (2006), 499-506

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# Electric Field Strength due to a Uniformly Charged Rod at a General Point

##### Electric Field Strength due to a Uniformly Charged Rod at a General Point

To find electric field strength at any general point P, let the point P be at a perpendicular distance R from the rod and let Q be the total charge (For specificity, Q is taken as positive, but the results apply to either sign of charge on the rod) on the rod. Let the linear charge density of the rod is λ (=Q/L).

The right end of the rod is located at angle θR measured counter-clockwise from the y axis and the left end of the rod is located at angle θL measured clockwise from the y axis. Using sign convention θR will be taken as positive and θL will be taken as negative.

To find electric field strength at P, we consider an element on rod of length dx at a distance x from point O as shown in figure. Now if dE is the electric field at P due to the element, then

$\displaystyle dE=\frac{kdq}{{{r}^{2}}}$, here $\displaystyle dq=\frac{Q}{L}dx$ = λdx

The x component of the electric field is seen from this diagram will be :-

$\displaystyle {{E}_{x}}=-\int{\frac{kdQ}{{{r}^{2}}}}\sin \theta =-\int{\frac{k\lambda dx}{{{r}^{2}}}}\sin \theta$     …..(1)

As dQ = λdx

[ Negative sign is because dEsinθ is in negative x-direction ]

and y component :-

$\displaystyle {{E}_{y}}=\int{\frac{kdQ}{{{r}^{2}}}}\cos \theta =\int{\frac{k\lambda dx}{{{r}^{2}}}}\cos \theta$    …..(2)

Now from above figure x = R tanθ ⇒ dx = R sec2θ dθ    …..(3)

and r2 = R2 + x= R2 + (R tanθ)2 = R2 (1 + tan2θ) = R2sec2θ    …..(4)

Using the values of equations (3) and (4), in equations (1) and (2), we get

$\displaystyle {{E}_{x}}=-\int{\frac{k\lambda (R{{\sec }^{2}}\theta d\theta )}{{{R}^{2}}{{\sec }^{2}}\theta }}\sin \theta =-\frac{k\lambda }{R}\int\limits_{{{\theta }_{L}}}^{{{\theta }_{R}}}{\sin \theta d\theta }$

$\displaystyle {{E}_{x}}=-\frac{k\lambda }{R}(-\cos \theta )_{{{\theta }_{L}}}^{{{\theta }_{R}}}=\frac{k\lambda }{R}(\cos {{\theta }_{R}}-\cos {{\theta }_{L}})$    …..(5)

and similarly…

$\displaystyle {{E}_{y}}=\int{\frac{k\lambda (R{{\sec }^{2}}\theta d\theta )}{{{R}^{2}}{{\sec }^{2}}\theta }}\cos \theta =\frac{k\lambda }{R}\int\limits_{{{\theta }_{L}}}^{{{\theta }_{R}}}{\cos \theta d\theta }$

$\displaystyle {{E}_{y}}=\frac{k\lambda }{R}(sin\theta )_{{{\theta }_{L}}}^{{{\theta }_{R}}}=\frac{k\lambda }{R}(sin{{\theta }_{R}}-\sin {{\theta }_{L}})$    …..(6)

Note :- Equation (5) and (6) are the general results. The sign convention for θ is not applied till now. We will use these results in various interesting cases with sign convention for θ. Here are the different cases…..

Case I :- To find Electric Field Strength when the point P lies on the perpendicular bisector of the rod

On the perpendicular bisector of a the finite rod, θR = +θ and θL = -θ.

Then From Eq. (5), Ex = 0,  and Eq. (6) becomes

$\displaystyle {{E}_{y}}=\frac{k\lambda }{R}[sin(+\theta )-\sin (-\theta )]=\frac{k\lambda }{R}(2\sin \theta )$

$\displaystyle {{E}_{y}}=\frac{2k\lambda }{R}\frac{L/2}{\sqrt{{{R}^{2}}+{{(L/2)}^{2}}}}$

Or $\displaystyle {{E}_{y}}=\frac{kQ}{R\sqrt{{{R}^{2}}+{{L}^{2}}/4}}$    …..(7)

In this case also if R >>> L, then the result becomes $\displaystyle {{E}_{y}}=\frac{kQ}{{{R}^{2}}}$, this is nothing but Electric Filed Intensity due to a point charge.

Case II :- To find Electric Field Strength for an infinitely long rod

When the rod is of infinite length, then θR = +90° and θL = -90°.

From Eq. (5), Ex = 0,  and Eq. (6) gives

$\displaystyle {{E}_{y}}=\frac{k\lambda }{R}[sin(+90)-\sin (-90)]=\frac{k\lambda }{R}(1+1)$

$\displaystyle {{E}_{y}}=\frac{2k\lambda }{R}$    …..(8)

Case III :- To find Electric Field Strength when the point P lies exactly above the right end of the finite rod

In this case θR = +0° and θL = -θ°(Let).

Then equation (5) gives…

$\displaystyle {{E}_{x}}=\frac{k\lambda }{R}[(\cos 0-\cos (-\theta )]=\frac{k\lambda }{R}(1-\cos \theta )$

$\displaystyle {{E}_{x}}=\frac{k\lambda }{R}\left[ 1-\frac{R}{\sqrt{{{R}^{2}}+{{L}^{2}}}} \right]$    …..(9)

And equation (6) gives…

$\displaystyle {{E}_{y}}=\frac{k\lambda }{R}[sin0-\sin (-\theta )]=\frac{k\lambda }{R}\sin \theta$

$\displaystyle {{E}_{y}}=\frac{k\lambda }{R}\frac{L}{\sqrt{{{R}^{2}}+{{L}^{2}}}}$    …..(10)

Case IV :- To find Electric Field Strength when the point P lies exactly above the right end of a semi- infinite rod

In that case θR = +0° and θL = -90°.

Equation (5) gives…

$\displaystyle {{E}_{x}}=\frac{k\lambda }{R}[(\cos 0-\cos (-90)]=\frac{k\lambda }{R}(1-0)$

$\displaystyle {{E}_{x}}=\frac{k\lambda }{R}$    …..(11)

And from equation (6)…

$\displaystyle {{E}_{y}}=\frac{k\lambda }{R}[sin0-\sin (-90)]=\frac{k\lambda }{R}(0+1)$

$\displaystyle {{E}_{y}}=\frac{k\lambda }{R}$    …..(12)

Note that equations (11) and (12) gives Ex = Ey, which means that…

1. “electric field due to a charged rod of semi-infinite length at a point vertically above its right end (also true if point P lies vertically above the left end of semi-infinite rod), always points 45° away from the axis of the rod, regardless of how near or far the point is from the rod.” and
2. The value of equations (11) and (12) is half of the equation (8), which is obvious because if we join together the ends of two equally-charged semi-infinite rods, we get an infinite rod. The x components of the two rods are equal and opposite, so they cancel out and we get equation (8).

Case V :- To find Electric Field Strength when the point P lies vertically above a point S, which is at a distance d from the right end, on the axis of the finite charged rod

From the above figure…

$\displaystyle \sin {{\theta }_{R}}=\frac{-d}{\sqrt{{{R}^{2}}+{{d}^{2}}}}$

$\displaystyle \sin {{\theta }_{L}}=\frac{-(d+L)}{\sqrt{{{R}^{2}}+{{(d+L)}^{2}}}}$

$\displaystyle \cos {{\theta }_{R}}=\frac{R}{\sqrt{{{R}^{2}}+{{d}^{2}}}}$       and

$\displaystyle \cos {{\theta }_{L}}=\frac{R}{\sqrt{{{R}^{2}}+{{(d+L)}^{2}}}}$

Using the value of cosθL and cosθR, in equation (5), we get…

$\displaystyle {{E}_{x}}=\frac{k\lambda }{R}\left[ \frac{R}{\sqrt{{{R}^{2}}+{{d}^{2}}}}-\frac{R}{\sqrt{{{R}^{2}}+{{(d+L)}^{2}}}} \right]$

$\displaystyle {{E}_{x}}=k\lambda \left[ \frac{1}{\sqrt{{{R}^{2}}+{{d}^{2}}}}-\frac{1}{\sqrt{{{R}^{2}}+{{(d+L)}^{2}}}} \right]$    …..(13)

Similarly, using the value of sinθL and sinθR, in equation (6), we get…

$\displaystyle {{E}_{y}}=\frac{k\lambda }{R}\left[ \frac{-d}{\sqrt{{{R}^{2}}+{{d}^{2}}}}-\frac{-(d+L)}{\sqrt{{{R}^{2}}+{{(d+L)}^{2}}}} \right]$

$\displaystyle {{E}_{y}}=\frac{k\lambda }{R}\left[ \frac{(d+L)}{\sqrt{{{R}^{2}}+{{(d+L)}^{2}}}}-\frac{d}{\sqrt{{{R}^{2}}+{{d}^{2}}}} \right]$    …..(14)

Case VI :- To find Electric Field Strength when the point P lies on the axis of the finite charged rod at a distance d from the right end

From above figure as the point P lies on the axis of the rod so, R→0

Taking limit as R→0, sinθR and sinθR becomes….

$\displaystyle \underset{R\to 0}{\mathop{\text{Lim}}}\,\left[ \sin {{\theta }_{R}}=\frac{-d}{\sqrt{{{R}^{2}}+{{d}^{2}}}} \right]\Rightarrow \sin {{\theta }_{R}}\to -1\text{ }$

$\displaystyle \underset{R\to 0}{\mathop{\text{Lim}}}\,\left[ \sin {{\theta }_{L}}=\frac{-(d+L)}{\sqrt{{{R}^{2}}+{{(d+L)}^{2}}}} \right]\Rightarrow \sin {{\theta }_{L}}\to -1$

From equation (6), we get Ey = 0.

Similarly applying limit, R→0, in equation(5) we get…

$\displaystyle \underset{R\to 0}{\mathop{Lim}}\,\left[ {{E}_{x}} \right]=\underset{R\to 0}{\mathop{Lim}}\,\left[ \frac{k\lambda }{R}\left[ \frac{R}{\sqrt{{{R}^{2}}+{{d}^{2}}}}-\frac{R}{\sqrt{{{R}^{2}}+{{(d+L)}^{2}}}} \right] \right]=\underset{R\to 0}{\mathop{Lim}}\,\left[ k\lambda \left[ \frac{1}{\sqrt{{{R}^{2}}+{{d}^{2}}}}-\frac{1}{\sqrt{{{R}^{2}}+{{(d+L)}^{2}}}} \right] \right]$

$\displaystyle \Rightarrow {{E}_{x}}=k\lambda \left[ \frac{1}{d}-\frac{1}{d+L} \right]=\frac{k\lambda L}{d\left( d+L \right)}$

$\displaystyle \Rightarrow {{E}_{x}}=\frac{kQ}{d\left( d+L \right)}$    …..(15)

[ Because λL = Q, total charge on the rod]

Note :- If d >>>>L, then, equation (15) becomes, $\displaystyle {{E}_{x}}=\frac{kQ}{{{d}^{2}}}$, which is nothing but Field Intensity due to a point charge Q at a distance d. Because if d >>>>L, then the rod will behave as a point charge.


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# Question : In the video Exercise 29 -  Equations In...

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Week 3 Quiz

1 .   Question : In the video Exercise 29 - Equations Involving Radicals , the presenter first eliminates the radicals by __________.
Student Answer : Squaring the equation
Multiplying both sides of the equation by 3
Cubing the equation
Multiplying both sides of the equation by the square root
Question 2 .   Question : In the slideshow Product Rule For Radicals , under media in Chapter 9 , in order to simplify the radical , the presenter works to find perfect square factors of 45 .


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## Thursday, October 29, 2009

### Zero Crossing Rate (ZCR) in Audio

ZCR is a very useful audio feature, it is defined as the number of times that the audio waveform crosses the zero axis.

fs is the sampling rate. ZCR is used to discern unvoiced speech. Usually unvoiced speech has a low short-term energy but a high zero crossing rate. Combining ZCR and STE to prevent low energy unvoiced speech frames from being classified as silent. MATLAB code:

% assume the window size is 2 seconds% there are overlaps in windowing% assume the step of shif is 1 second[wav fs] = wavread('DEMO.wav');wav = wav / max(max(wav));window_length = 2 * fs;step = 1 * fs; frame_num = floor((length(wav)-window_length)/step) + 1;zcr_ = zeros(frame_num, 1);wav_window2 = zeros(window_length, 1);pos = 1;for i=1:frame_num   wav_window = wav(pos:pos + window_length-1);   wav_window2(2:end) = wav_window(1:end-1);   zcr_(i) = 1/2 * sum(abs(sign(wav_window) - ...       sign(wav_window2))) * fs / window_length;   pos = pos + step;end

1. Anonymous7:01 AM

Hi Weis,

A very good work, but, seems to me that it doesn't work. For you it's ok?

2. is the result wrong or the program cannot run?

3. Anonymous10:21 AM

the program cannot run

4. Anonymous10:23 AM

Hi Weis,

olease i need this program but the program cannot run

5. I think you should use:

wav = wav / max(abs(wav));


text:
# Number of firms in Long Run

In a competitive market, is it possible to know the change (increase/decrease) in number of firms in the Long run with a positive shift in demand for increasing costs case?

## 1 Answer

Yes, it is possible. In the long run, firms enter until they break even. Suppose firms are symmetric. Then for each firm the break even condition is that the average costs equal the price. This is because the price is equal to the average revenue. The average revenue is given by $px/x=p$ where $p$ is price and $x$ is quantity. If revenue equals costs on average, then there are no profits left to be made and this is the equilibrium number of firms.

Further, in a competitive market, price equals marginal costs. Hence, the break even condition there is also price equals marginal costs. Alternatively, in this case the condition is Average Costs equal Marginal Costs. From this condition you can find the break even quantity. From there find the break even price, by setting price=marginal costs. Then plug in the break even price and quantity into the demand function to determine how many firms must enter until the market equilibrium price and quantity are equal to the break even price and quantity. The result is the long run equilibrium number of firms.

Note that the question is only reasonable in the presence of fixed costs. If there are no fixed costs, the long run number of firms is infinite and we have a trivial answer. However, the treatment of this topic in textbooks as well as academic papers alwa assumes the presence of fixed costs.

• Is this independent of the returns to scale of firms' technology? I had the idea that CRS does not allow you to pin down number of firms, as scale is irrelevant. Mar 30, 2017 at 18:56
• It is possible to pin down if there are fixed costs, which are usually assumed in such models. Otherwise, with CRS there is an infinite (or undefined) number of firms. Mar 30, 2017 at 21:02
• "Yes, it is possible" - Shouldn't this be "Yes, it is possible in some circumstances"? The circumstances being symmetric firms with some fixed costs. Apr 1, 2017 at 9:08
• The firms need not necessarily be symmetric. This is just the easiest case to illustrate and the only case used in undergraduate studies, which is the impression I got from OP. Questions about the number of firms always include fixed costs. Papers on the subject always assume it. Technically, it is even easier to find the long run number of firms without fixed costs. It is infinite. I can add my comment on this issue to the answer. Apr 1, 2017 at 9:19
• Actually, the solution of this question lies in this other question posed by me: economics.stackexchange.com/questions/16028/… Apr 2, 2017 at 18:43


text:
### Scientific Thinking

We live in an exciting time!

Science is part of our culture more today than ever before. Black holes, rocketry, and quantum computing are all fields of advanced science that have made their way onto YouTube, the news, and pop culture. But if you want to learn more about science than you can get from a news clipping, the last thousand years of scientific ideas stand between you and mastery.

The puzzles in this course will start to bridge that gap. You'll develop physical insight while you play. From gears to electric circuits to laser tag, you'll find scientific principles where you least expect them.

# Nature is a Puzzle

We've all had one of those satisfying "aha!" moments while poring over a crossword, finding the right arrangement of Sudoku numbers, or considering the logical plights of knights and knaves.

Puzzles present a challenge for your brain — they all have a simple structure, and a clear set of rules for solving them. But in this course, we won't be solving puzzles involving general knowledge, simple arithmetic, or logical contradictions.

We will be tackling puzzles in the natural world.

# Nature is a Puzzle

You can't enjoy a puzzle unless you know its rules. Similarly you can't really appreciate the world around you until you understand the rules of nature. The physical sciences are the study of these rules, whether it is in the form of chemistry, biology, or physics.

In particular, physics is more than just a part of the physical sciences. Unlike the others, it explains the basic building blocks of the natural world: matter in motion, forces, energy, heat, light, and time.
Everything follows these same rules — the world is beautifully connected by the laws of physics that you'll find in this course.

# Nature is a Puzzle

Physics and engineering are often said to be about precision and mathematics, but here we will dispense with both these tenets in search of something else: physical insight.

While mathematical analysis can explain the exact range of a $\SI{500}{\kilo \pascal}$ firehose at a $30 ^\circ$ angle, the precision gained by specifying all the variables pays a steep price in generality.

Answering simple questions can become hard when you become reliant on mathematics: What happens to the range of the firehose when it doubles in length? What about when it is raised to the top of a ladder?

# Nature is a Puzzle

On a journey to physical insight, we'll need to start from scratch.

When you approach an unfamiliar physics problem you want to first learn the main ideas and the important rules at play: what is pressure, and does it change with the length and height of the fire hose? Only by answering these questions can you make new predictions about the physics at play.

# Nature is a Puzzle

You might be surprised to learn that a fire hose can be made as long as you'd like, without any drop in pressure.$^*$ But for every meter you raise the fire hose, the pressure of the water and its range will dwindle.

These ideas and rules structure your understanding of the problem. Precision isn't important, and neither is mathematics. It's easier to refine this intuitive understanding than it is to create the fully-formed mathematical analysis. This course will show you how to build insight and understanding first, so you don't end up getting lost in number-crunching.

$^*$As long as the water flows frictionlessly. In extremely long hoses, friction may have a noticeable effect on the pressure.

# Nature is a Puzzle

The thinking involved in solving puzzles is a unique blend of following rules with imaginative association. Only by combining both can you see a pattern or a twist that the puzzle conceals. If all goes well, you're able to use that hidden insight to form new information and ideas.

The process will be exactly the same with physics: we'll bring the rules, you bring the imagination.

×


text:
### From Theory to Practice

The summation in Eq.(4.13) cannot be implemented in practice because the ``ideal lowpass filter'' impulse response actually extends from minus infinity to infinity. It is necessary in practice to window the ideal impulse response so as to make it finite. This is the basis of the window method for digital filter design [115,362]. While many other filter design techniques exist, the window method is simple and robust, especially for very long impulse responses. In the case of the algorithm presented below, the filter impulse response is very long because it is heavily oversampled. Another approach is to design optimal decimated ``sub-phases'' of the filter impulse response, which are then interpolated to provide the ``continuous'' impulse response needed for the algorithm [358].

Figure 4.23 shows the frequency response of the ideal lowpass filter. This is just the Fourier transform of .

If we truncate at the fifth zero-crossing to the left and the right of the origin, we obtain the frequency response shown in Fig.4.24. Note that the stopband exhibits only slightly more than 20 dB rejection.

If we instead use the Kaiser window [221,438] to taper to zero by the fifth zero-crossing to the left and the right of the origin, we obtain the frequency response shown in Fig.4.25. Note that now the stopband starts out close to dB. The Kaiser window has a single parameter which can be used to modify the stop-band attenuation, trading it against the transition width from pass-band to stop-band.

Next Section:
Implementation
Previous Section:
Theory of Ideal Bandlimited Interpolation


text:
# Mathematical Recreations: Tweetable Game Of Life10

Posted by Loren Shure,

Today I’d like to introduce a guest blogger, Matt, who is consultant over in The MathWorks UK. Some of you may have come across his customer projects and series of demos on using MATLAB in Physics. He’s going to talk about an interesting one line MATLAB program.

### Introduction

One Saturday afternoon I saw a link on Hacker News to a YouTube video showing Conway's Game of Life in one line of APL. This made me wonder whether MATLAB® could perform as well.

The answer: maybe not quite a single line, but definitely a tweet.

The code is reproduced below:

s=[1 1 1]';
n=@(a)conv2(s,s',1*a,'same')-a;
lf=@(a)n(a)==2&a|n(a)==3;
a=rand(128)>0.8;
for ii=1:500,spy(a);drawnow;a=lf(a);end

In the code below I've commented out the population initialisation and visualisation code and replaced it with various alternatives for this published version, together with an animated gif of the output.

A = accumarray([3 3; 3 4; 4 2; 4 4; 5 4],1,[7 7]);
imagesc(~A); colormap(gray);
for ii=1:9,imagesc(~A);drawnow;A=lf(A);end

In this post I'll show the evolution of the code from initial conception to finished version, and how thinking in matrices can lead to new algorithms.

### Conway's Game Of Life

Conway's Game of Life is a two-dimensional cellular automation that demonstrates complex behaviour.

It consists of a grid of cells, each of which may be in one of two possible states, 0 ('dead') or 1 ('alive'). The state of each cell in the next iteration of the game is dependent on the state of the surrounding eight cells as follows:

• a live cell with 2 neighbours remains alive
• any cell with 3 neighbours becomes or remains alive
• any other cell dies of loneliness or overcrowding

The core of the algorithm consists of two steps:

• calculate the number of live neighbours of each cell
• using this, determine the new state of each cell based on its current state

### Calculating The Number Of Neighbours

To calculate the number of neighbours for each cell one approach would be to loop over each cell in the array, add up the number of live cells in the surrounding 3x3 square, and finally subtract the state of the cell itself.

The YouTube video used the different approach of creating 8 copies of the original array, offseting each by a step horizontally and/or vertically and then summing the arrays.

The key part of development of my algorithm was to realise that this looping-with-sum-over-neighbourhood is just a 2D convolution. In MATLAB we can do a 2D convolution using conv2.

S = [1 1 1; 1 0 1; 1 1 1]
neighbours=@(a) conv2(double(a),S,'same')
S =
1     1     1
1     0     1
1     1     1
neighbours =
@(a)conv2(double(a),S,'same')

The 'same' argument is to restrict the output of conv2 to be the same size as the original input matrix.

For example:

b = rand(5)>0.5
neighbours(b)
b =
0     0     1     1     1
1     1     0     1     0
1     0     1     0     1
0     1     1     0     0
0     0     1     0     1
ans =
2     3     3     3     2
2     4     5     5     4
3     6     4     4     1
2     4     3     5     2
1     3     2     3     0

### Calculating The Next State

After calculating the number of neighbours, the next step is to use it to determine the next state using the rules of Conway's Game of Life.

life = @(a) (neighbours(a)==2 & a)...  % Rule for 2 neighbours
| (neighbours(a)==3)               % Rule for 3 neighbours
b
life(b)
life =
@(a)(neighbours(a)==2&a)|(neighbours(a)==3)
b =
0     0     1     1     1
1     1     0     1     0
1     0     1     0     1
0     1     1     0     0
0     0     1     0     1
ans =
0     1     1     1     1
1     0     0     0     0
1     0     0     0     0
0     0     1     0     0
0     1     1     1     0

Let's use this as a reference implementation to test that modifications to the function do the same thing:

t = @(fOrig,f,a) assert(all(all(fOrig(a) == f(a))));

### Reducing The Size

Twitter allows users to post 'tweets' of up to 140 characters in length. Fitting in the Game of Life evolution function and visualisation in this limit required a few tricks:

Use the syntax of conv2 that uses 1D vectors for the filter. This happens to also be faster but requires an extra element-wise matrix subtraction:

s = [1 1 1]';
n2 = @(a) conv2(s,s',double(a),'same')-a;
t(neighbours, n2, b)

Drop some more characters by using an implicit cast to double by multiplying a by 1. Also remove spaces:

n=@(a)conv2(s,s',1*a,'same')-a;
t(neighbours, n2, b)

Now shorten the Life rule function by removing spaces and parentheses, instead relying on the operator precedence rules:

lf=@(a)n(a)==2&a|n(a)==3;
t(life,lf,b)

Finally, initialise a random population and visualise using spy in a loop:

a=rand(128)>0.8;for ii=1:25,spy(a);drawnow;a=lf(a);end

### Modifications

It is possible to lose a few more characters by changing the 'for' loop to a 'while 1' loop (to go forever), changing the definition of s to be a row vector, changing the 0.8 to .8, etc. I stopped at the point where I did since it allowed me to fit the code into a tweet, with room for a #MATLAB tag.

A modification that would be worth making is to introduce periodic boundary conditions:

makeperiodic = @(a) [a(end,end), a(end,:),a(end,1);...
a(:,end),a,a(:,1);...
a(1,end),a(1,:),a(1,1)];
unperiodic = @(a) a(2:end-1,2:end-1);
np = @(a) unperiodic(n(makeperiodic(a)));
lfp = @(a) np(a)==2&a|np(a)==3;

(an alternative is to use the Image Processing Toolbox imfilter command which has an option to use periodic boundaries)

... and making the visualisation nicer:

A = accumarray([3 3; 3 4; 4 2; 4 4; 5 4],1,[7 7]);
imagesc(~A); colormap(gray);
for ii=1:25,imagesc(~A);drawnow;A=lfp(A);end

Other things to try are left as an exercise for the reader:

### Conclusion

I've shown here how thinking in terms of operations on matrices rather than looping constructs can result in compact code. In this case the motivation was just for fun, but in general it can make use of the built-in MATLAB functionality and avoid having to reimplement existing functions.

Do you have any good one line (or one tweet) MATLAB programs? I'd love to hear from you here.

Get the MATLAB code

Published with MATLAB® 7.9

Hello

I am a big fan of your blog, and usually I spent some time trying to figure out the ways you reduce in one line what I would spend like 4.

Keep up the good work, and keep good programs coming.

Leon

Gabe replied on : 2 of 10

Is Lauren on Twitter btw? Failed to find her.

Loren replied on : 3 of 10

Gabe-

Spell my name as my parents did and you will find me on twitter. I am not very active there. Matt was the real author of this particular post.

–loren

Fun :-). Just tried a few tricks to shorten it further… This is what i got:

a=rand(128)>0.8;
for ii=1:500,spy(a);drawnow;a=abs(filter2(ones(3),a)*2-6-a)<2;end

And a high-life version:

a=rand(256)>0.8;
for ii=1:500,spy(a);drawnow;f=filter2(ones(3),a);a=(f-6)+2*a==0|f==3;end

Test with this initial “a”:

a=zeros(200);
a(101:105,101:105)=[0 0 1 1 1;0 1 0 0 1; 1 0 0 0 1;1 0 0 1 0;1 1 1 0 0];

Aslak-
Very nice! I like your abs(filter2(…)…) approach, although I think the conv2 with 1D vectors will still win in terms of speed for large arrays.

I’m working on an approach using accumarray and the integral image of ‘a’ (google Viola-Jones for reference), I’ll add it to the thread when I get it working.

Now, if we can just get HashLife into a tweet…

-Matt

Here’s another fun piece of code inspired by cellular automata. This one generates what looks like a labyrinth… I made it as a music visualizer for winamp many years ago:
http://www.winamp.com/visualization/phunck-2nd-pack/142505

And now i re-implemented it in matlab.

c=-ones(5); c(2:4,2:4)=2;
c(3,3)=13; c=c/8;
A=rand(300)>.99;
close all
colormap gray
for ii=1:100; A=max(min(filter2(c,A),1),0); imagesc(A==0); drawnow; end;

Hello,
here is simple code to calculate for an optional vector a[i], i=1,..,n the sum by i of products, containing multipliers (for all j!=i) such as a[j]/(a[j]-a[i]). It can be proved that for all vectors this expression equals to 1. The code does not contain any loops:

A=x'*ones(size(x));
B=A./(A-A'+diag(x));
res=sum(prod(B))

These postings are the author's and don't necessarily represent the opinions of MathWorks.


text:
How do you solve 0.5(2x+8)=x-4?

Jan 8, 2017

See explanation.

$0.5 \left(x + 8\right) = x - 4$

First step is to multiply the expression in brackets on the left side by $0.5$:

$x + 4 = x - 4$

Now we can move the unknown $x$ to left side and the numbers to right side:

$0 = - 8$

The last equality is false, so the original equation has no solutions.


text:
New interface

# Polymer Schwarzschild black hole: An effective metric

Abstract : We consider the modified Einstein equations obtained in the framework of effective spherically symmetric polymer models inspired by Loop Quantum Gravity. When one takes into account the anomaly free point-wise holonomy quantum corrections, the modification of Einstein equations is parametrized by a function $f(x)$ of one phase space variable. We solve explicitly these equations for a static interior black hole geometry and find the effective metric describing the trapped region, inside the black hole, for any $f(x)$. This general resolution allows to take into account a standard ambiguity inherent to the polymer regularization: namely the choice of the spin $j$ labelling the $\SU(2)$-representation of the holonomy corrections. When $j=1/2$, the function $f(x)$ is the usual sine function used in the polymer litterature. For this simple case, the effective exterior metric remains the classical Schwarzschild's one but acquires modifications inside the hole. The interior metric describes a regular trapped region and presents strong similarities with the Reissner-Nordstr\"om metric, with a new inner horizon generated by quantum effects. We discuss the gluing of our interior solution to the exterior Schwarzschild metric and the challenge to extend the solution outside the trapped region due to covariance requirement. By starting from the anomaly free polymer regularization for inhomogeneous spherically symmetric geometry, and then reducing to the homogeneous interior problem, we provide an alternative treatment to existing polymer interior black hole models which focus directly on the interior geometry, ignoring covariance issue when introducing the polymer regularization.
Keywords :
Document type :
Journal articles
Complete list of metadata

https://hal.archives-ouvertes.fr/hal-01757976
Contributor : INSPIRE HEP Connect in order to contact the contributor
Submitted on : Wednesday, April 4, 2018 - 10:52:37 AM
Last modification on : Friday, November 25, 2022 - 1:06:24 AM

### Citation

Jibril Ben Achour, Frédéric Lamy, Hongguang Liu, Karim Noui. Polymer Schwarzschild black hole: An effective metric. EPL, 2018, 123 (2), pp.20006. ⟨10.1209/0295-5075/123/20006⟩. ⟨hal-01757976⟩

Record views


text:
# MathSciDoc: An Archive for Mathematician ∫

#### Lecture Notesmathscidoc:1707.37001

Probability Survey, 12, 55-103, 2015
This survey paper is based on the lecture notes for the mini course in the summer school at Yau Mathematics Science Center, Tsinghua University, 2014. We describe and characterize all random subsets $K$ of simply connected domain which satisfy the conformal restriction" property. There are two different types of random sets: the chordal case and the radial case. In the chordal case, the random set $K$ in the upper half-plane $\HH$ connects two fixed boundary points, say 0 and $\infty$, and given that $K$ stays in a simply connected open subset $H$ of $\HH$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is any conformal map from $H$ onto $\HH$ fixing 0 and $\infty$. In the radial case, the random set $K$ in the upper half-plane $\HH$ connects one fixed boundary points, say 0, and one fixed interior point, say $i$, and given that $K$ stays in a simply connected open subset $H$ of $\HH$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is the conformal map from $H$ onto $\HH$ fixing 0 and $i$. It turns out that the random set with conformal restriction property are closely related to the intersection exponents of Brownian motion. The construction of these random sets relies on Schramm Loewner Evolution with parameter $\kappa=8/3$ and Poisson point processes of Brownian excursions and Brownian loops.
Brownian motion, Schramm Loewner Evolution
@inproceedings{hao2015conformal,
title={Conformal restriction and Brownian motion},
author={Hao Wu},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170717103353538636795},
booktitle={Probability Survey},
volume={12},
pages={55-103},
year={2015},
}

Hao Wu. Conformal restriction and Brownian motion. 2015. Vol. 12. In Probability Survey. pp.55-103. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170717103353538636795.


text:
# Finding the mass of the earth's atmosphere

1. Feb 22, 2004

### deltabourne

A barometer having a cross-sectional area of 1.00 cm^2 at sea level measures a pressure of 76.0 cm of mercury. The pressure exerted by this column of mercury is equal to the pressure exerted by all the air on 1 cm^2 of Earth's surface. Given that the density of mercury is 13.6 g/mL and the average radius of Earth is 6371 km, calculate the total mass of Earth's atmosphere in kilograms (Hint: The surface area of a sphere is 4pir^2)

I have no idea where to start on this one, and how to relate the mercury to find the mass of the atmosphere.. can anyone point me in the right direction?

2. Feb 23, 2004

### Tom Mattson

Staff Emeritus
I don't think you even need the density of mercury. You know the pressure and the area over which the pressure acts. Since P=F/A, you can determine the force that exerts the pressure (which is the weight of the air above the barometer). That gives you the mass of a column of air with a cross section of 1cm2. To get the mass of the entire atmosphere, you need to use the formula for the area of a sphere that was given.


text:
2 messages
19/06/2013 at 13:12

I planted a magnolia last autumn ( sorry don't know the name but grows to about 3m and has pink & white blossom ).                                                          It has heardly grown and has only developed 6 - 7 leaves, yellow in colour and edges of leaf turning brown. There doesn't seem to be any new buds developing eiither.                                                                                I've planted it in open ground, in an exposed position in neutral soil.


text:
<< Wednesday, May 01, 2019 >>

## Wednesday, May 1, 2019

### CANCELED: From planarians to parasites: stem cells and developmental plasticity in flatworms

Seminar | May 1 | 11 a.m.-12 p.m. | 100 Genetics & Plant Biology Building | Canceled

Phil Newmark, Morgridge Institute for Research

### Foreign student emigration to the United States: pathways of entry, demographic antecedents, and origin-country contexts: A Brown Bag Talk

Colloquium | May 1 | 12-1 p.m. | 2232 Piedmont, Seminar Room

Kevin Thomas, Professor, Sociology, Demography, African Studies, Penn State University

In this study, we use information from a uniquely developed database to examine recent trends in international student migration to the United States. Our results highlight the differential contributions to these trends made by various entry pathways.

### Plant and Microbial Biology Seminar: "Predicting Antibiotic Resistance"

Seminar | May 1 | 12-1 p.m. | 101 Barker Hall

Roy Kishony, Israel Institute of Technology

Roy Kishony is the Marilyn and Henry Taub Professor of Life Sciences, the director of the Lorry I. Lokey Interdisciplinary Center for Life Sciences and Engineering, and a Faculty of Biology. Combining novel quantitative experimental techniques and clinical studies with mathematical modeling and advanced data analysis, Kishony's lab is studying microbial evolution with a specific focus on...   More >

### MVZ LUNCH SEMINAR - Noelle Bittner (CANCELLED): Tba

Seminar | May 1 | 12-1 p.m. | Valley Life Sciences Building, 3101 VLSB, Grinnell-Miller Library

Noelle Bittner

Museum of Vertebrate Zoology

MVZ Lunch is a graduate level seminar series (IB264) based on current and recent vertebrate research. Professors, graduate students, staff, and visiting researchers present on current and past research projects. The seminar meets every Wednesday from 12- 1pm in the Grinnell-Miller Library. Enter through the MVZ's Main Office, 3101 Valley Life Sciences Building, and please let the receptionist...   More >

### Moving Towards Open Scholarship: UC, Elsevier and All the Rest

Presentation | May 1 | 12-1:30 p.m. | Barrows Hall, Social Science Matrix, 8th Floor

Jeff MacKie-Mason

The movement to make new scholarship freely available to all readers began at least by 1994 with Stevan Harnad's "Subversive proposal". In 2013 the UC Academic Senate adopted one of the first mandatory OA policies in the US, requiring that a copy of all newly authored research be deposited in an open archive regardless of where it is published. In Winter 2018 the University Libraries published an...   More >

### Nanomaterials Enable Delivery of Functional Genetic Material Without Transgenic DNA Integration in Mature Plants

Seminar | May 1 | 12-1 p.m. | 106 Stanley Hall

Markita Landry, University of California, Berkeley

Bioengineering (BioE)

Genetic engineering of plants is at the core of sustainability efforts, natural product synthesis, and agricultural crop engineering. The plant cell wall is a barrier that limits the ease and throughput with which exogenous biomolecules can be delivered to plants. Current delivery methods either suffer from host range limitations, low transformation efficiencies, tissue regenerability, tissue...   More >

### Long Distance Caregiving (BEUHS121)

Workshop | May 1 | 12:10-1:30 p.m. | Tang Center, University Health Services, Class of '42

Maureen Kelly, ED.D., LCSW, Elder Care Program, Be Well @ Work

Elder Care Program

Being a long distance caregiver to an elder brings its own set of challenges. Assessing needs and navigating resources from afar can be stressful and complicated. This interactive workshop examines when and how to help, how to locate resources in your elder’s area, and how to seek support for yourself.

### Harmonic Analysis Seminar: The multilinear Kakeya inequality

Seminar | May 1 | 1:10-2 p.m. | 736 Evans Hall

Kubrat Danailov, UC Berkeley

Department of Mathematics

A short proof of the multilinear Kakeya inequality of Bennett-Carbery-Tao will be presented. This proof (due to Guth, 2015) is based on induction on scales and the Loomis-Whitney inequality, without the nonlinear heat flow of the original proof. In a future lecture, this result will serve as an ingredient in the proof of the multilinear restriction and decoupling inequalities.

### Topology Seminar (Introductory Talk): Background for main topology talk

Seminar | May 1 | 2:10-3 p.m. | 740 Evans Hall

Claudia Scheimbauer, NTNU

Department of Mathematics

I will start by motivating cobordism categories by recalling the notion of topological field theories. Then I will explain why “higher” categories appear naturally in this context (and what they are).

### Rapidly mixing random walks on matroids and related objectsidly mixing random walks on matroids and related objects

Seminar | May 1 | 3-4 p.m. | 1011 Evans Hall

Nima Anari, Stanford University

Department of Statistics

A central question in randomized algorithm design is what kind of distributions can we sample from efficiently? On the continuous side, uniform distributions over convex sets and more generally log-concave distributions constitute the main tractable class. We will build a parallel theory on the discrete side, that yields tractability for a large class of discrete distributions. We will use this...   More >

### Number Theory Seminar: Algebraicity of values of the Gamma function II

Seminar | May 1 | 3:40-5 p.m. | 748 Evans Hall

Rahul Dalal, UC Berkeley

Department of Mathematics

### CBE Colloquium

Colloquium | May 1 | 4-6 p.m. | 180 Tan Hall

Marc Martin Casas, Ph.D. student in the Mesbah Group; James Lincoff, Ph.D. student in the Head-Gordon Group

Bayesian experiment design and estimation for probabilistic modeling of biological systems/Development and Application of a Novel Enhanced Sampling Method and Bayesian Analysis for Characterizing Intrinsically Disordered Proteins

### Solving Hard Computational Problems using Oscillator Networks

Seminar | May 1 | 4-5:30 p.m. | 560 Evans Hall

Tianshi Wang, EECS, UC Berkeley

Over the last few years, there has been considerable interest in Ising machines, ie, analog hardware for solving difficult (NP hard/complete) computational problems effectively. We present a new way to make Ising machines using networks of coupled self-sustaining nonlinear oscillators. Our scheme is theoretically rooted in a novel result that connects the phase dynamics of coupled oscillator...   More >

### Japan's Imperial Underworlds: Intimate Encounters at the Borders of Empire

Colloquium | May 1 | 4-6 p.m. | 180 Doe Library

David Ambaras, North Carolina State University

In this lecture, David Ambaras reveals how the movement of migrants, smugglers, pirates, and trafficked people between China and Japan -- and their sensationalization in the popular press -- created surprising cross-currents in the politics of Sino-Japanese relations during the years of Japanese imperial expansion.

### Topology Seminar (Main Talk): A complete model for the cobordism category

Seminar | May 1 | 4:10-5 p.m. | 3 Evans Hall

Claudia Scheimbauer, NTNU

Department of Mathematics

Lurie’s approach to the Cobordism Hypothesis builds upon a suitable higher category of cobordisms. The model of $$(\infty,1)$$-categories given by complete Segal spaces (and their higher analogs) are a very natural choice for constructing cobordism categories. A drawback is that the first natural definitions only give Segal spaces, which, for high dimensions, are not complete. This follows...   More >

### Center for Computational Biology Seminar: Sohini Ramachandran, Associate Professor, Brown University

Seminar | May 1 | 4:30-5:30 p.m. | 125 Li Ka Shing Center

Center for Computational Biology

Abstract:
Correlation among genotypes in human population-genetic datasets complicates the localization of both adaptive mutations and disease-causing mutations. I will describe our latest efforts to develop new methods for localizing adaptive and disease-causing mutations, motivated by (1) incorporating...   More >

### Reporting in India: Berkeley Journalism Students Present their Stories

Presentation | May 1 | 5-7 p.m. | Stephens Hall, 10 (ISAS Conf. Room)

Geeta Anand, Acting Professor of Reporting, UC Berkeley School of Journalism

Showcasing UC Berkeley Graduate School of Journalism students reporting on India.

### Linguistic Anthropology and Literary and Cultural Studies: A Mellon Foundation Sawyer Seminar: Session 7: Ethics and History of Linguistic Anthropology

Conference/Symposium | May 1 – 2, 2019 every day | 5-7 p.m. | 370 Dwinelle Hall

Charles Briggs, UC Berkeley; Mary Bucholtz, UC Santa Barbara; Jillian Cavanaugh, CUNY-Brooklyn College; Laura Graham, University of Iowa; Barbra Meek, University of Michigan; Beth Piatote, UC Berkeley

This is the last of seven two-day meetings of a Mellon Foundation Sawyer Seminar taking place throughout 2018-2019. The seminar aims to explore the potential of a set of concepts, tools, and critical practices developed in the field of linguistic anthropology for work being done in the fields of literary and cultural criticism.


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# it-e-50 What's actually involved in programming

What's actually involved in programming the actual process of writing programs? Here's a
quick overview of the process:
Write a program.
Compile the program.
Run the program.
Debug the program.
Repeat the whole process until the program is finished.
Let's discuss those steps one by one.
I have a small amount of bad news for you: you can't write programs in English. It would be

nice indeed to be able to type "count the number of words in a sentence" into your computer and
have it actually understand, but that's not going to happen for a while (unless someone writes a
program to make a computer do that, of course). Instead, you have to learn a programming
language.
Much of a programming language is indeed in English. Programming languages commonly
use words like "if", "repeat", "end" and such. Also, they use the familiar mathematical operators
like "+" and "=". It's just a matter of learning the "grammar" of the language; how to say things
properly.
So, we said "Write a program". This means: write the steps needed to perform the task,
using the programming language you know. You'll do the typing in a programming environment
(an application program which lets you write programs, which is an interesting thought in itself).
A common programming environment is CodeWarrior, and another common one is InterDev, but
you don't need to worry about those just yet. Some programming environments are free, and
some you have to buy just like any other application program. Commercial (non-free)
programming environments cost anything from $50 to$500+, and you'll almost always get a
huge discount if you're a student or teacher of some kind.
Incidentally, the stuff you type to create a program is usually called source code, or just code.
Programmers also sometimes call programming coding. We think it sounds slightly cooler.
In order to use a program, you usually have to compile it first. When you write a program
(in a programming language, using a programming environment, as we mentioned a moment
ago), it's not yet in a form that the computer can use. This isn't hard to understand, given that
computers actually only understand lots of 1s and 0s in long streams. You can't very well write
programs using only vast amounts of 1s and 0s, so you write it in a more easily-understood form
(a programming language), then you convert it to a form that the computer can actually use. This
conversion process is called compiling, or compilation. Not surprisingly, a program called a
compiler does the compiling.
It's worth mentioning that if your program has problems which the compiler can't deal with,
it won't be able to compile your program.
You'll be pleased to hear that your programming environment will include a suitable compiler
(or maybe more than one compiler: each different programming language your programming
environment allows you to use requires its own compiler). Compilers are just fancy programs, so
they too are written by programmers. Programmers who write compilers are a bit like gods; they
make it possible for everyone else to program.
Now that you've compiled the program into a form that the computer can use, you want to
see if it works: you want to make the computer perform the steps that you specified. This is
called running the program, or sometimes executing it. Just the same as how a car isn't of much

use if you don't drive it, a program isn't of much use if you don't run it. Your programming
environment will allow you to run your program too.
You've probably heard the term "debug" before (it's pronounced just as you might expect:
"dee-bug"). It refers to fixing errors and problems with your program. As I'm sure you know, the
term came about because the earliest computers were huge building-sized contraptions, and
actual real-life insects sometimes flew into the machinery and caused havoc with the circuits and
valves. Hence, those first computer engineers had to physically "debug" the computers they had
to scrape the toasted remains of various kinds of flying insects out of the inner workings of their
machines. The term became used to describe any kind of problem-solving process in relation to
computers, and we use it today to refer purely to fixing errors in our code.
You may also have heard the phrase "it's not a bug, it's a feature". Programmers sometimes
say this when someone points out a problem with their programs; they're saying that it's not a bug,
but rather a deliberate design choice (which is almost always a lie). This is rather like
accidentally spilling coffee all over yourself whilst simultaneously falling down some stairs, then
getting up and saying "I meant to do that".
you'll often find the picture of an insect shown in your programming environment to indicate
debugging). You usually debug your program by stepping through it. This means just what it
sounds like: you go through your program one step at a time, watching how things are going and
what's happening. Sooner or later (usually later), you'll see what's going wrong, and slap yourself
And then you repeat the whole process until you're happy with the program. This is trickier
than it might sound, since programmers are never happy with their programs. You see,
programmers are perfectionists never satisfied until absolutely everything is complete and
elegant and powerful and just gorgeous. Programmers will commonly release a new version of
their program every day for a couple of weeks after the initial release.
As you can imagine, enjoying an intellectual challenge is an important trait to have when
you're going back to correct and enhance your code many times over. You'll actually find that you
can't wait to get back into your program and fix the bugs, make improvements, and refine the
existing code.
And that's the basic process of programming. Note that most programming environments
will make a lot of it much easier for you, by doing such things as:
Taking you to the specific bit of code which is causing the compiler to puke
Letting you quickly look up documentation on the programming language you're using
Letting you just choose to run the program, and compiling it automatically first

Colouring parts of your code to make it easier to read (for example, making numbers a
different colour from other text)
And many other things
So, don't worry too much about the specifics of compiling then running then debugging or
whatever. The purpose of this section was mostly to make you aware of the cyclical nature of
programming: you write code, test it, fix it, write more, test it, fix, and so on.

1, fancy  ['fænsi]
n. 幻想；想像力；爱好
vt. 想象；喜爱；设想；自负
vi. 幻想；想象

2, contraption  [kən'træpʃən]
n. 奇妙的装置
3, havoc  ['hævək]
n. 大破坏,蹂躏
4, valves
n. 阀门；瓣膜（valve的复数）；真空管
v. 装阀于…；以活门调节（valve的三单形式）
5, insects

6, circuits
n. 电路
7, scrape  [skreip]
n. 刮掉,擦掉
v. 刮掉,擦掉
8, toasted
v. 烤（toast的过去分词）；敬酒
9, spilling  ['spiliŋ]

10, whilst  [wailst]
conj. 当...时(可是,虽然,只要)
11, slap  [slæp]
n. 掴,侮辱,拍击声
v. 拍击,侮辱,申斥
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# Derivation of Laplace Operator in Spherical and Cylindrical Coordinates

1. Sep 11, 2012

### terryphi

Hey Guys,

Does anyone know where I can find a derivation of the laplace operator in spherical and cylidrical coordinates?

2. Sep 11, 2012

### mathman

3. Sep 11, 2012

### terryphi

I don't see a derviation there. I've tried several textbooks as well.

4. Sep 12, 2012


text:
# zbMATH — the first resource for mathematics

Validity of the local thermal equilibrium assumption in a natural convection from a vertical plate embedded in a porous medium. (English) Zbl 1159.76380
Summary: The validity of the local thermal equilibrium assumption in natural convection over a vertical flat plate embedded in a porous medium is investigated analytically. The study is based on the two-phase (Schumann) model, using the Darcy model (slip condition) to govern the flow. It is found that there are three dimensionless parameters controlling the local thermal equilibrium assumption: the Biot number (Bi), the modified Rayleigh number (Ra$$_m$$), and the Darcy number (Da). The effects of these parameters are investigated and a correlation equation is developed in order to determine the region where the local thermal equilibrium assumption is valid.

##### MSC:
 76R10 Free convection 76S05 Flows in porous media; filtration; seepage 80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text:


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# Hiatus

I am visiting India, so I will probably not post too much for the month. Assuming I don’t get eaten alive by the mosquitoes (a distinct possibility), I’ll post more in December.

Maybe I’ll post a wedding photo or two (not my wedding).

# My new favorite word

Graunt’s 1662 book on the London Bills of Mortality claims to be written in “succinct paragraphs, without any long series of multiloquious deductions.”

“Multiloquious” is my new favorite word.

I posted earlier about the mean absolute deviation (MAD) of a binomial variable $S_n$ with parameters $(n,p)$. Here’s a little follow-up with plots. This is a plot of $\mathbb{E}|S_n - np|$ versus $p$ for different values of $n$.

The first is for $n = 10$. Looks beautifully scalloped, no? As we’d expect, the MAD is symmetric about $p = 1/2$ and monotonically increasing for the first half of the unit interval. Unfortunately, it’s clearly not concave (although it is piecewise concave), which means I have to do a bit more algebra later on.

When $n = 100$ the scallops turn into a finely serrated dome.

By the time you get to $n = 1000$ the thing might as well be concave for all that your eye can tell. But you would be deceived. Like a shark’s skin, the tiny denticles can abrade your proof, damaging it beyond repair.

Why do I care about this? If you take $n$ samples from a Bernoulli variable with parameter $p$, then the empirical distribution (unnormalized) is $(n - S_n, S_n)$. So $\frac{1}{n} \mathbb{E}|S_n - np|$ is the expected total variational distance between the empirical distribution and its mean. More generally, the expected total variational distance for finite-alphabet distributions is a sum of MAD terms.


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# Stretches of Graphs

Stretch Rule 1 For $y=pf(x)$, $p \gt 0$, the effect of $p$ is to vertically stretch the graph by a factor by $p$. If $p \gt 1$, it moves points of $y=f(x)$ further away from the $x$-axis. If $0 \lt p \lt 1$, it moves points of $y=f(x)$ closer to the $x$-axis. Stretch Rule 2 […]

# Reflections of Graphs

For $y=-f(x)$, we reflect $y=f(x)$ in the $x$-axis. For $y=f(-x)$, we reflect $y=f(x)$ in the $y$-axis. Example 1 Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of $y=f(x)$ and $y=-f(x)$. \begin{align} \displaystyle f(x) &= x^3-4x^2+4x \\ &= x(x^2 – 4x + 4) \\ &= x(x-2)^2 \end{align} Example 2 Consider $f(x)=x^3-4x^2+4x$. On the […]

# Must-Know 10 Basic Translations of Rational Functions Explained

Rational functions are characterised by the presence of both a horizontal asymptote and a vertical asymptote. Any graph of a rational function can be obtained from the reciprocal function $f(x)=\dfrac{1}{x}$ by a combination of transformations including a translation, stretches and compressions. Type 1: Vertical Compression $y=\dfrac{a}{x}, \ 0 \lt a \lt 1$ The […]

# Translations of Graphs

Translation Rule 1 For $y=f(x)+b$, the effect of $b$ is to translate the graph vertically through $b$ units. If $b \gt 0$, it moves upwards. If $b \lt 0$, it moves downwards. Translation Rule 2 For $y=f(x-a)$, the effect of $a$ is to translate the graph horizontally through $a$ units. If $a \gt 0$, it […]

Sketching quadratic graphs are drawn based on $y=x^2$ graph for transforming and translating. Question 1 $f(x) = (x-3)^2$ is drawn and sketch the following graphs by transforming. (a)   $y = f(x)+2$; Transforming upwards by $2$ units (b)   $y=f(x)-3$; Transforming dowanwards by $3$ […]