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https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_12A_Problems/Problem_6&diff=prev&oldid=74250
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# Difference between revisions of "2014 AMC 12A Problems/Problem 6"
## Problem
The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?
$\textbf{(A) }44\qquad \textbf{(B) }55\qquad \textbf{(C) }77\qquad \textbf{(D) }99\qquad \textbf{(E) }110$
## Solution 1
Let the two digits be $a$ and $b$. Then, $5a + 5b = 10a + b - 10b - a = 9a - 9b$, or $2a = 7b$. This yields $a = 7$ and $b = 2$ because $a, b < 10$. Then, $72 + 27 = \boxed{\textbf{(D) }99}.$
## Solution 2 (Meta)
We start like above. Let the two digits be $a$ and $b$. Therefore, $5(a+b) = 10a+b-10b-a=9(a-b)$. Since we are looking for $10a+b+10b+a=11(a+b)$ and we know that $a+b$ must be a multiple of $9$, the only answer choice that works is $\boxed{\textbf{(D) }99}.$
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2021-10-28 02:22:32
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https://kscript.org/more/compare
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# Language Comparison
## kscript vs. Python
Python and kscript are probably the closest languages in this page. They share a similar object-oriented and duck-typed philosophy. Scoping rules are also similar, as there are 2 kinds of scope: global and function. They also share a lot of the same keywords (as a lot of languages do).
A somewhat significant difference is that Python has syntactically significant whitespace, whereas kscript only requires whitespace between identifiers and some tokens. This is a holy war in and of itself, and many good arguments on both sides. However, kscript ultimately chose {} blocks and non-significant whitespace for (mainly) two reasons (which are related and similar in many ways):
• Copying and pasting code between different indentation levels with significant whitespace causes errors, or worse, changes the semantic meanings (think about cutting something in a 1-indentation deep block into a 4-indentation deep block – it would still be at 1-indentation deep, and thus cause the 4-indentation deep block to cease)
• Autoformatting/autoindenting code is impossible, as changing the indentation would change the semantic meaning, and changing the semantic meaning would change the formatting. Having non-significant whitespace solves this, as an IDE is free to indent/dedent and add newlines as it needs to properly format it
The internals of kscript and Python (specifically, CPython) are similar - both use a bytecode interpreter VM, along with a GIL to manage resources among threads.
## kscript vs. C
C and kscript are very different, even though kscript is written in C. Interfacing between C and kscript is easy (ffi for calling C from kscript, and libks for calling kscript from C), but fundamentally kscript is more dynamic, object-oriented, and cross platform, and C programs typically are more efficient, although many kscript modules end up running compiled C code anyway, so number crunching and other expensive operations end up being similar performance.
kscript code is easier to read, write, and distribute. And, you can write your application/library once and then run on many platforms without modification – that’s rare in C.
## kscript vs. kscript
To disambiguate throughout this section, I will refer to the our kscript as the “good kscript”, and the other kscript (see here: https://github.com/holgerbrandl/kscript) as the “bad kscript”.
Good kscript:
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2021-08-02 12:13:42
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https://www.zbmath.org/authors/?q=ai%3Aanderson.daniel-m
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## Anderson, Daniel M.
Compute Distance To:
Author ID: anderson.daniel-m Published as: Anderson, Daniel M.; Anderson, D. M.; Anderson, Daniel
Documents Indexed: 49 Publications since 1973 Co-Authors: 22 Co-Authors with 19 Joint Publications 426 Co-Co-Authors
all top 5
### Co-Authors
4 single-authored 7 McFadden, Geoffrey Bey 4 Davis, Stephen Howard 4 Wheeler, Adam A. 3 Guba, Peter 3 McLaughlin, Richard M. 3 Miller, Cass T. 2 Hamnen, H. 2 Karoubi, Max 2 Seshaiyer, Padmanabhan 2 Wagoner, John B. 2 Wilhelmsson, Hans 2 Worster, M. Grae 1 Benson, James D. 1 Bondarev, Andrei 1 Bondeson, Anders 1 Braun, Richard J. 1 Cattani, F. 1 Cermelli, Paolo 1 Coriell, Sam R. 1 Corsaro, Maria 1 Droniou, Jérôme 1 Forest, Mark Gregory 1 Fried, Eliot 1 Ghil, Michael 1 Gurtin, Morton Edward 1 Helczynski-Wolf, L. 1 Horton, Jonathan 1 Kearsley, Anthony José 1 Le Bodic, Pierre 1 Lundgren, Lukas 1 Morgan, Kerri 1 Murray, Bruce T. 1 Nishikawa, Kyoji 1 Nong, Kumnit 1 Österberg, Ulf 1 Pekkari, L.-O. 1 Reid, Tim 1 Semenov, V. E. 1 Siddique, Javed I. 1 Superfine, Richard 1 Talagrand, Olivier 1 Talbott, Kevin 1 Tange, Toshio 1 Winter, Katlyn N. 1 Xu, Amber
all top 5
### Serials
11 Journal of Fluid Mechanics 7 Physics of Fluids 5 Physica D 5 Physics of Fluids 3 Physica Scripta 3 Mathematical Medicine and Biology 2 SIAM Journal on Applied Mathematics 1 Journal of the Institute of Mathematics and its Applications 1 Journal of Mathematical Physics 1 Mathematical Biosciences 1 Physics Letters. A 1 Transactions of the American Mathematical Society 1 Mathematical Programming. Series A. Series B 1 SIAM Journal on Scientific Computing 1 American Journal of Physics 1 Journal of Physics A: Mathematical and Theoretical
all top 5
### Fields
32 Fluid mechanics (76-XX) 13 Classical thermodynamics, heat transfer (80-XX) 8 Partial differential equations (35-XX) 5 Biology and other natural sciences (92-XX) 3 Numerical analysis (65-XX) 3 Optics, electromagnetic theory (78-XX) 3 Statistical mechanics, structure of matter (82-XX) 2 Category theory; homological algebra (18-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 Mechanics of deformable solids (74-XX) 1 Integral equations (45-XX) 1 Algebraic topology (55-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Computer science (68-XX) 1 Mechanics of particles and systems (70-XX) 1 Quantum theory (81-XX) 1 Operations research, mathematical programming (90-XX)
### Citations contained in zbMATH Open
34 Publications have been cited 663 times in 521 Documents Cited by Year
Diffuse-interface methods in fluid mechanics. Zbl 1398.76051
Anderson, D. M.; McFadden, G. B.; Wheeler, A. A.
1998
A phase-field model of solidification with convection. Zbl 0951.35112
Anderson, D. M.; McFadden, G. B.; Wheeler, A. A.
2000
The spreading of volatile liquid droplets on heated surfaces. Zbl 0843.76016
Anderson, D. M.; Davis, S. H.
1995
Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys. Zbl 0868.76031
Anderson, D. M.; Worster, M. Grae
1995
A diffuse-interface description of internal waves in a near-critical fluid. Zbl 1185.76467
Anderson, D. M.; McFadden, G. B.
1997
A new oscillatory instability in a mushy layer during the solidification of binary alloys. Zbl 0859.76024
Anderson, D. M.; Worster, M. Grae
1996
A model for wetting and evaporation of a post-blink precorneal tear film. Zbl 1196.92005
Winter, Katlyn N.; Anderson, Daniel M.; Braun, Richard J.
2010
General dynamical sharp-interface conditions for phase transformations in viscous heat-conducting fluids. Zbl 1119.76066
Anderson, Daniel M.; Cermelli, Paolo; Fried, Eliot; Gurtin, Morton E.; McFadden, Geoffrey B.
2007
Thin interface asymptotics for an energy/entropy approach to phase-field models with unequal conductivities. Zbl 0962.35175
McFadden, G. B.; Wheeler, A. A.; Anderson, D. M.
2000
A phase-field model with convection: sharp-inerface asymptotics. Zbl 1076.80504
Anderson, D. M.; McFadden, G. B.; Wheeler, A. A.
2001
Two-fluid viscous flow in a corner. Zbl 0789.76094
Anderson, D. M.; Davis, S. H.
1993
The averaging of gravity currents in porous media. Zbl 1186.76026
Anderson, Daniel M.; Mclaughlin, Richard M.; Miller, Cass T.
2003
A model for diffusion-controlled solidification of ternary alloys in mushy layers. Zbl 1055.76054
Anderson, D. M.
2003
Soliton perturbations: a variational principle for the soliton parameters. Zbl 1063.35528
Bondeson, A.; Lisak, M.; Anderson, D.
1979
Local fluid and heat flow near contact lines. Zbl 0800.76486
Anderson, D. M.; Davis, S. H.
1994
An arbitrary-order scheme on generic meshes for miscible displacements in porous media. Zbl 1395.65068
Anderson, Daniel; Droniou, Jérôme
2018
Linear and nonlinear convection in solidifying ternary alloys. Zbl 1085.76563
Anderson, D. M.; Schulze, T. P.
2005
Thin film evolution over a thin porous layer: modeling a tear film on a contact Lens. Zbl 1429.76034
Nong, Kumnit; Anderson, Daniel M.
2010
Equivalent Lagrangians in generalized mechanics. Zbl 0262.70019
Anderson, D.
1973
Pattern selection in ternary mushy layers. Zbl 1374.76071
Guba, Peter; Anderson, Daniel M.
2017
Capillary rise of a liquid into a deformable porous material. Zbl 1183.76481
Siddique, J. I.; Anderson, D. M.; Bondarev, Andrei
2009
A sharp-interface interpretation of a continuous density model for homogenization of gravity-driven flow in porous media. Zbl 1198.76110
Anderson, Daniel M.; McLaughlin, Richard M.; Miller, Cass T.
2010
Relations between higher algebraic $$K$$-theories. Zbl 0346.18015
Anderson, D.; Karoubi, M.; Wagoner, J.
1973
Higher algebraic K-theories. Zbl 0354.18016
Anderson, D.; Karoubi, M.; Wagoner, J.
1977
Imbibition of a liquid droplet on a deformable porous substrate. Zbl 1187.76023
Anderson, Daniel M.
2005
Convective instabilities during the solidification of an ideal ternary alloy in a mushy layer. Zbl 1189.76195
Anderson, Daniel M.; McFadden, Geoffrey B.; Coriell, Sam R.; Murray, Bruce T.
2010
Diffusive and phase change instabilities in a ternary mushy layer. Zbl 1331.76113
Guba, Peter; Anderson, Daniel M.
2014
Effects of particle trapping on ion-cyclotron resonance heating in a toroidal plasma. Zbl 0594.76121
Anderson, D.; Lisak, M.; Pekkari, L.-O.
1985
A model for a spreading and melting droplet on a heated substrate. Zbl 0995.76094
Anderson, D. M.; Forest, M. G.; Superfine, R.
2001
Fermat’s principle and the variational analysis of an optical model for light propagation exhibiting a critical radius. Zbl 1219.78120
Marklund, M.; Anderson, D.; Cattani, F.; Lisak, M.; Lundgren, L.
2002
Mathematical basis for analysis of partially coherent wave propagation in nonlinear, non-instantaneous, Kerr media. Zbl 1189.78053
Semenov, V.; Lisak, M.; Anderson, D.; Hansson, T.; Helczynski-Wolf, L.; Österberg, U.
2008
Convective phenomena in mushy layers. Zbl 1439.76149
Anderson, Daniel M.; Guba, Peter
2020
Numerical solution of inward solidification of a dilute ternary solution towards a semi-permeable spherical cell. Zbl 1425.92061
Anderson, Daniel M.; Benson, James D.; Kearsley, Anthony J.
2019
Further results on an abstract model for branching and its application to mixed integer programming. Zbl 1478.90061
Anderson, Daniel; Le Bodic, Pierre; Morgan, Kerri
2021
Further results on an abstract model for branching and its application to mixed integer programming. Zbl 1478.90061
Anderson, Daniel; Le Bodic, Pierre; Morgan, Kerri
2021
Convective phenomena in mushy layers. Zbl 1439.76149
Anderson, Daniel M.; Guba, Peter
2020
Numerical solution of inward solidification of a dilute ternary solution towards a semi-permeable spherical cell. Zbl 1425.92061
Anderson, Daniel M.; Benson, James D.; Kearsley, Anthony J.
2019
An arbitrary-order scheme on generic meshes for miscible displacements in porous media. Zbl 1395.65068
Anderson, Daniel; Droniou, Jérôme
2018
Pattern selection in ternary mushy layers. Zbl 1374.76071
Guba, Peter; Anderson, Daniel M.
2017
Diffusive and phase change instabilities in a ternary mushy layer. Zbl 1331.76113
Guba, Peter; Anderson, Daniel M.
2014
A model for wetting and evaporation of a post-blink precorneal tear film. Zbl 1196.92005
Winter, Katlyn N.; Anderson, Daniel M.; Braun, Richard J.
2010
Thin film evolution over a thin porous layer: modeling a tear film on a contact Lens. Zbl 1429.76034
Nong, Kumnit; Anderson, Daniel M.
2010
A sharp-interface interpretation of a continuous density model for homogenization of gravity-driven flow in porous media. Zbl 1198.76110
Anderson, Daniel M.; McLaughlin, Richard M.; Miller, Cass T.
2010
Convective instabilities during the solidification of an ideal ternary alloy in a mushy layer. Zbl 1189.76195
Anderson, Daniel M.; McFadden, Geoffrey B.; Coriell, Sam R.; Murray, Bruce T.
2010
Capillary rise of a liquid into a deformable porous material. Zbl 1183.76481
Siddique, J. I.; Anderson, D. M.; Bondarev, Andrei
2009
Mathematical basis for analysis of partially coherent wave propagation in nonlinear, non-instantaneous, Kerr media. Zbl 1189.78053
Semenov, V.; Lisak, M.; Anderson, D.; Hansson, T.; Helczynski-Wolf, L.; Österberg, U.
2008
General dynamical sharp-interface conditions for phase transformations in viscous heat-conducting fluids. Zbl 1119.76066
Anderson, Daniel M.; Cermelli, Paolo; Fried, Eliot; Gurtin, Morton E.; McFadden, Geoffrey B.
2007
Linear and nonlinear convection in solidifying ternary alloys. Zbl 1085.76563
Anderson, D. M.; Schulze, T. P.
2005
Imbibition of a liquid droplet on a deformable porous substrate. Zbl 1187.76023
Anderson, Daniel M.
2005
The averaging of gravity currents in porous media. Zbl 1186.76026
Anderson, Daniel M.; Mclaughlin, Richard M.; Miller, Cass T.
2003
A model for diffusion-controlled solidification of ternary alloys in mushy layers. Zbl 1055.76054
Anderson, D. M.
2003
Fermat’s principle and the variational analysis of an optical model for light propagation exhibiting a critical radius. Zbl 1219.78120
Marklund, M.; Anderson, D.; Cattani, F.; Lisak, M.; Lundgren, L.
2002
A phase-field model with convection: sharp-inerface asymptotics. Zbl 1076.80504
Anderson, D. M.; McFadden, G. B.; Wheeler, A. A.
2001
A model for a spreading and melting droplet on a heated substrate. Zbl 0995.76094
Anderson, D. M.; Forest, M. G.; Superfine, R.
2001
A phase-field model of solidification with convection. Zbl 0951.35112
Anderson, D. M.; McFadden, G. B.; Wheeler, A. A.
2000
Thin interface asymptotics for an energy/entropy approach to phase-field models with unequal conductivities. Zbl 0962.35175
McFadden, G. B.; Wheeler, A. A.; Anderson, D. M.
2000
Diffuse-interface methods in fluid mechanics. Zbl 1398.76051
Anderson, D. M.; McFadden, G. B.; Wheeler, A. A.
1998
A diffuse-interface description of internal waves in a near-critical fluid. Zbl 1185.76467
Anderson, D. M.; McFadden, G. B.
1997
A new oscillatory instability in a mushy layer during the solidification of binary alloys. Zbl 0859.76024
Anderson, D. M.; Worster, M. Grae
1996
The spreading of volatile liquid droplets on heated surfaces. Zbl 0843.76016
Anderson, D. M.; Davis, S. H.
1995
Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys. Zbl 0868.76031
Anderson, D. M.; Worster, M. Grae
1995
Local fluid and heat flow near contact lines. Zbl 0800.76486
Anderson, D. M.; Davis, S. H.
1994
Two-fluid viscous flow in a corner. Zbl 0789.76094
Anderson, D. M.; Davis, S. H.
1993
Effects of particle trapping on ion-cyclotron resonance heating in a toroidal plasma. Zbl 0594.76121
Anderson, D.; Lisak, M.; Pekkari, L.-O.
1985
Soliton perturbations: a variational principle for the soliton parameters. Zbl 1063.35528
Bondeson, A.; Lisak, M.; Anderson, D.
1979
Higher algebraic K-theories. Zbl 0354.18016
Anderson, D.; Karoubi, M.; Wagoner, J.
1977
Equivalent Lagrangians in generalized mechanics. Zbl 0262.70019
Anderson, D.
1973
Relations between higher algebraic $$K$$-theories. Zbl 0346.18015
Anderson, D.; Karoubi, M.; Wagoner, J.
1973
all top 5
### Cited by 895 Authors
14 Yang, Xiaofeng 12 Lowengrub, John Samuel 11 Shen, Jie 9 Grasselli, Maurizio 8 Mauri, Roberto 7 Braun, Richard J. 7 Liu, Chun 7 Xu, Chuanju 6 Anderson, Daniel M. 6 Frigeri, Sergio 6 Giorgini, Andrea 6 Hughes, Thomas J. R. 6 Kim, Junseok 6 Rohde, Christian 6 Yariv, Ehud 6 Zhao, Jia 5 Abels, Helmut 5 Ardekani, Arezoo M. 5 Balashov, Vladislav 5 Driscoll, Tobin A. 5 Du, Qiang 5 Han, Daozhi 5 Huang, Ziyang 5 Lin, Guang 5 Lin, Ping 5 Prüß, Jan Wilhelm 5 Udaykumar, Holavanahalli S. 4 Anderson, Patrick D. 4 Balashov, V. A. 4 Beckermann, C. 4 Chen, Lizhen 4 Cimmelli, Vito Antonio 4 Desjardins, Benoît 4 Deugoue, Gabriel 4 Dong, Suchuan 4 Feireisl, Eduard 4 Feng, James J. 4 Fried, Eliot 4 Gal, Ciprian Gheorghe Sorin 4 Garcke, Harald 4 Hou, Dianming 4 Lamorgese, Andrea G. 4 Liang, Zhilei 4 McFadden, Geoffrey Bey 4 Neufeld, Jerome A. 4 Ren, Weiqing 4 Shu, Chang 4 Soldati, Alfredo 4 Tachim Medjo, Theodore 4 Thiele, Uwe 4 Wang, Qi 4 Wang, Xiaoming 4 Wise, Steven M. 4 Yang, Zhiguo 4 You, Bo 4 Yu, Haijun 4 Zlotnik, Alexander A. 3 Begley, Carolyn G. 3 Calo, Victor Manuel 3 Cartalade, Alain 3 Di Pietro, Daniele Antonio 3 Ding, Hang 3 Fakhari, Abbas 3 Feng, Xiaobing 3 Fernández-Cara, Enrique 3 Golden, John Murrough 3 Haspot, Boris 3 Huppert, Herbert E. 3 Inamuro, Takaji 3 King-Smith, P. Ewen 3 Levitas, Valery I. 3 Liu, Ju 3 Nestler, Britta 3 Niu, Xiaodong 3 Nogueira, Xesús 3 Oliveri, Francesco 3 Pace, Angelo Raffaele 3 Petcu, Madalina 3 Rahimian, Mohammad Hassan 3 Rivière, Beatrice M. 3 Rocca, Elisabetta 3 Roccon, Alessio 3 Savenkov, E. B. 3 Schimperna, Giulio 3 Schnitzer, Ory 3 Shimizu, Senjo 3 Simonett, Gieri 3 Soligo, Giovanni 3 Sprekels, Jürgen 3 Takada, Naoki 3 van der Zee, Kristoffer George 3 Vasil, Geoffrey M. 3 Wheeler, Adam A. 3 Wilke, Mathias 3 Wu, Hao 3 Yang, Jiang 3 Yue, Pengtao 3 Zhang, Zhen 3 Zhou, Zhi 2 Afkhami, Shahriar ...and 795 more Authors
all top 5
### Cited in 122 Serials
91 Journal of Computational Physics 43 Journal of Fluid Mechanics 34 Physics of Fluids 23 Computer Methods in Applied Mechanics and Engineering 22 Computers and Fluids 17 Journal of Scientific Computing 13 Journal of Mathematical Analysis and Applications 13 Physica D 13 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 12 Computers & Mathematics with Applications 9 International Journal for Numerical Methods in Fluids 8 Journal of Computational and Applied Mathematics 8 Communications in Computational Physics 7 Archive for Rational Mechanics and Analysis 7 Journal of Mathematical Fluid Mechanics 6 Applied Mathematics and Computation 6 Applied Mathematics and Optimization 6 SIAM Journal on Scientific Computing 5 Journal of Engineering Mathematics 5 Bulletin of Mathematical Biology 5 Journal of Differential Equations 5 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 5 SIAM Journal on Mathematical Analysis 5 Continuum Mechanics and Thermodynamics 4 International Journal of Engineering Science 4 Mathematics of Computation 4 Applied Mathematical Modelling 4 SIAM Journal on Applied Mathematics 4 Computational Geosciences 3 Mathematical Methods in the Applied Sciences 3 International Journal for Numerical Methods in Engineering 3 Mathematics and Computers in Simulation 3 Numerical Methods for Partial Differential Equations 3 Applied Mathematics Letters 3 European Journal of Mechanics. B. Fluids 3 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 3 Philosophical Transactions of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 2 Acta Mechanica 2 International Journal of Heat and Mass Transfer 2 Journal of the Mechanics and Physics of Solids 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 Chinese Annals of Mathematics. Series B 2 European Journal of Applied Mathematics 2 Communications in Partial Differential Equations 2 NoDEA. Nonlinear Differential Equations and Applications 2 Russian Journal of Numerical Analysis and Mathematical Modelling 2 Journal of Applied Mechanics and Technical Physics 2 Nonlinear Analysis. Real World Applications 2 Archives of Computational Methods in Engineering 2 International Journal of Numerical Analysis and Modeling 2 Journal of Statistical Mechanics: Theory and Experiment 2 Advances in Applied Mathematics and Mechanics 2 East Asian Journal on Applied Mathematics 1 International Journal of Modern Physics B 1 Applicable Analysis 1 Communications in Mathematical Physics 1 Computer Physics Communications 1 Journal of Mathematical Biology 1 Journal of Statistical Physics 1 Nonlinearity 1 Physica A 1 Physics Reports 1 Theoretical and Computational Fluid Dynamics 1 Meccanica 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Numerische Mathematik 1 SIAM Journal on Control and Optimization 1 SIAM Journal on Numerical Analysis 1 Studies in Applied Mathematics 1 Zeitschrift für Analysis und ihre Anwendungen 1 Stochastic Analysis and Applications 1 Applied Numerical Mathematics 1 Numerical Algorithms 1 Journal of Elasticity 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 SIAM Review 1 Journal of Dynamics and Differential Equations 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Journal of Nonlinear Science 1 Calculus of Variations and Partial Differential Equations 1 Communications in Numerical Methods in Engineering 1 Annales Mathématiques Blaise Pascal 1 International Journal of Numerical Methods for Heat & Fluid Flow 1 International Journal of Computational Fluid Dynamics 1 European Series in Applied and Industrial Mathematics (ESAIM): Proceedings 1 Optimization Methods & Software 1 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 1 Wuhan University Journal of Natural Sciences (WUJNS) 1 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Interfaces and Free Boundaries 1 Flow, Turbulence and Combustion 1 M2AN. Mathematical Modelling and Numerical Analysis. ESAIM, European Series in Applied and Industrial Mathematics 1 Acta Mathematica Sinica. English Series 1 Communications in Nonlinear Science and Numerical Simulation 1 Combustion Theory and Modelling 1 International Journal of Modern Physics C 1 Differential Equations 1 Mathematical Modelling and Analysis 1 Journal of Evolution Equations 1 Computational Methods in Applied Mathematics ...and 22 more Serials
all top 5
### Cited in 24 Fields
420 Fluid mechanics (76-XX) 169 Partial differential equations (35-XX) 150 Numerical analysis (65-XX) 54 Classical thermodynamics, heat transfer (80-XX) 37 Statistical mechanics, structure of matter (82-XX) 34 Mechanics of deformable solids (74-XX) 24 Biology and other natural sciences (92-XX) 10 Calculus of variations and optimal control; optimization (49-XX) 10 Geophysics (86-XX) 8 Dynamical systems and ergodic theory (37-XX) 8 Probability theory and stochastic processes (60-XX) 5 Integral equations (45-XX) 5 Optics, electromagnetic theory (78-XX) 3 Real functions (26-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Systems theory; control (93-XX) 1 General and overarching topics; collections (00-XX) 1 Ordinary differential equations (34-XX) 1 Approximations and expansions (41-XX) 1 Functional analysis (46-XX) 1 Operator theory (47-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Astronomy and astrophysics (85-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)
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2022-05-16 05:24:11
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http://forums.torque3d.org/viewtopic.php?f=18&t=228&sid=844346e2cc5cfca46bc3e68d9b5d5ca3
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Theora Video seems zoomed in
Level design, models, animations, physics, etc.
• 1
• 2
Theora Video seems zoomed in
PaulWeston
Posts: 143
Joined: Thu Apr 23, 2015 7:16 pm
Hi all,
Been using TheoraTextureObject to put videos on objects, works great...
However, and this is true for theora video GUI controls in my menus as well, it seems as if it kind of zooms into the video somewhat. Like the video is larger than the object it is supposed to be mapping to. This results in clipping of a lot of the frame.
The videos were originally AVI, converted to OGV using Miro Video Converter. There were not many options to choose in terms of the end result, but they play fine as OGV files in VLC Player.
I have not seen anything in Torque about how one could scale the video or the resolution, are there perhaps specific sizes/resolutions/aspect ratios that this control needs the videos to be converted to in order to display properly?
Thanks!
Re: Theora Video seems zoomed in
PaulWeston
Posts: 143
Joined: Thu Apr 23, 2015 7:16 pm
Did some more testing with this...
If I set the matchvideosize to 1, then it correctly sets the extent of the video control to be the dimensions of the OGV file, but the video itself seems to be bigger.
It's as if the texture size on the object is set to a value smaller than the video, so you only see the center part of the video on the object and not the whole thing.
For example, if I make the video 320 by 240 the window will size to that correctly but will only show roughly 250 by 200. If I make the video 1024 by 768 the window will take up the whole screen which is fine, except the video again only seems to display about 800 by 600 of the video. I always lose the sides and top.
Rather than having to make videos with a hard coded black border of a couple hundred pixels, is there perhaps some way to play with the texture sizing so the whole video will show up?
Thanks
P
Re: Theora Video seems zoomed in
PaulWeston
Posts: 143
Joined: Thu Apr 23, 2015 7:16 pm
Anyone else have success mapping Theora videos onto objects?
As noted above, everything works mechanically, however the video is not scaling properly to the object.
For example, when using a movie screen (rectangular shape), and trying to stick a video on it, the object only shows a cropped portion of the video. Even if I make the source video wide screen format, it still seems to want to texture on as a square, which doesn't fit the movie screen shape, and we lose the top and bottom of the video as well as a bit on the sides.
Is there some secret I don't know about, that lets me force the video to scale exactly to the size of the object we stick it to?
Thanks
Re: Theora Video seems zoomed in
marauder2k9
Posts: 126
Joined: Wed Feb 18, 2015 7:36 am
can u post a screenshot
Re: Theora Video seems zoomed in
PaulWeston
Posts: 143
Joined: Thu Apr 23, 2015 7:16 pm
Sure...
Taking this video clip:
And putting it onto a Theora video object, will result in this:
Re: Theora Video seems zoomed in
Nils
Posts: 160
Joined: Thu Feb 05, 2015 3:32 am
Hey Paul, perhaps you should post the script (GUI)
try this if it's different then yours:
matchVideoSize = "0";position = "0 0";horizSizing = "width";vertSizing = "height";
Re: Theora Video seems zoomed in
Nils
Posts: 160
Joined: Thu Feb 05, 2015 3:32 am
You could also try ffmpeg2theora, getting good results with this converter (lots of parameters)
Re: Theora Video seems zoomed in
PaulWeston
Posts: 143
Joined: Thu Apr 23, 2015 7:16 pm
I'm just using the straight theora video on texture, using materials:
singleton TheoraTextureObject(WebStarTrek1_Movie)
{
texTargetName = "MyWebStarTrek1TextureName";
theoraFile = "scripts/web/video/startrek1.ogv";
};
singleton Material(WebStarTrek1_WebStarTrek1)
{
mapTo = "WebStarTrek1";
diffuseMap[0] = "#MyWebStarTrek1TextureName";
emissive[0] = "1";
};
I have a .dae object that has the material WebStarTrek1, the TheoraTextureObject calls the video file.
The settings you gave:
matchVideoSize = "0";
position = "0 0";
horizSizing = "width";
vertSizing = "height";
... would be for the Theora Gui controls, no? Can I use them in the TheoraTextureObject as well?
Thanks
Re: Theora Video seems zoomed in
Nils
Posts: 160
Joined: Thu Feb 05, 2015 3:32 am
... would be for the Theora Gui controls, no?
Sorry, I didn't paid enough attention to notice you place it on objects as material.
Yes, it's for GUI's
UV wrap is in place?
Perhaps a guiDynamicTexture would be a solution, but don't know if that's working or not
Re: Theora Video seems zoomed in
PaulWeston
Posts: 143
Joined: Thu Apr 23, 2015 7:16 pm
So, nobody out there is using the Theora on Object stuff?
It's a neat feature, used to be a resource way back when until it was added to stock T3D. So, I figured it should just work, the tutorial video I watched shows the video properly proportioned on the object.
What do you mean by UV wrap in place? Something specific that needs to be done to the DAE model? I thought all that was needed was any basic shape. I just used Blender to make a simple cube with one named material. Didn't know of any specific steps to making an object that will properly display Theora video, there is nothing about that in the tutorials.
I have tried many methods of converting video as well - original video is MP4, convert to MPEG2, then to AVI, then to OGV. Along the way I processed it into standard 320 X 240 size. So, it's not like it's a 16 X 9 aspect ratio widescreen video or anything, it's basic 320 X 240, so why does it no just fill the shape properly?
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2017-04-23 11:50:19
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https://en.wikipedia.org/wiki/Number_needed_to_treat
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# Number needed to treat
Group exposed to a treatment (left) has reduced risk of an adverse outcome (grey) compared to the unexposed group (right). 4 individuals need to be treated to prevent 1 adverse outcome (NNT = 4).
The number needed to treat (NNT) is an epidemiological measure used in communicating the effectiveness of a health-care intervention, typically a treatment with medication. The NNT is the average number of patients who need to be treated to prevent one additional bad outcome (e.g. the number of patients that need to be treated for one of them to benefit compared with a control in a clinical trial). It is defined as the inverse of the absolute risk reduction, and computed as ${\displaystyle 1/(I_{u}-I_{e})}$, where ${\displaystyle I_{e}}$ is the incidence in the treated (exposed) group, and ${\displaystyle I_{u}}$ is the incidence in the control (unexposed) group.[1] [2] This calculation implicitly assumes monotonicity, that is, no individual can be harmed by treatment. The modern approach, based on counterfactual conditionals, relaxes this assumption and yields bounds on NNT.
A type of effect size, the NNT was described in 1988 by McMaster University's Laupacis, Sackett and Roberts.[3] The ideal NNT is 1, where everyone improves with treatment and no one improves with control. A higher NNT indicates that treatment is less effective.[4]
NNT is similar to number needed to harm (NNH), where NNT usually refers to a therapeutic intervention and NNH to a detrimental effect or risk factor.
## Relevance
The NNT is an important measure in pharmacoeconomics. If a clinical endpoint is devastating enough (e.g. death, heart attack), drugs with a high NNT may still be indicated in particular situations. If the endpoint is minor, health insurers may decline to reimburse drugs with a high NNT. NNT is significant to consider when comparing possible side effects of a medication against its benefits. For medications with a high NNT, even a small incidence of adverse effects may outweigh the benefits. Even though NNT is an important measure in a clinical trial, it is infrequently included in medical journal articles reporting the results of clinical trials.[5] There are several important problems with the NNT, involving bias and lack of reliable confidence intervals, as well as difficulties in excluding the possibility of no difference between two treatments or groups.[6]
NNT may vary substantially over time,[7] and hence convey different information as a function of the specific time-point of its calculation. Snapinn and Jiang[8] showed examples where the information conveyed by the NNT may be incomplete or even contradictory compared to the traditional statistics of interest in survival analysis. A comprehensive research on adjustment of the NNT for explanatory variables and accommodation to time-dependent outcomes was conducted by Bender and Blettner,[9] Austin,[10] and Vancak et al.[11]
## Explanation of NNT in practice
There are a number of factors that can affect the meaning of the NNT depending on the situation. The treatment may be a drug in the form of a pill or injection, a surgical procedure, or many other possibilities. The following examples demonstrate how NNT is determined and what it means. In this example, it is important to understand that every participant has the condition being treated, so there are only "diseased" patients who received the treatment or did not. This is typically a type of study that would occur only if both the control and the tested treatment carried significant risks of serious harm, or if the treatment was unethical for a healthy participant (for example, chemotherapy drugs or a new method of appendectomy - surgical removal of the appendix). Most drug trials test both the control and the treatment on both healthy and "diseased" participants. Or, if the treatment's purpose is to prevent a condition that is fairly common (an anticoagulant to prevent heart attack for example), a prospective study may be used. A study which starts with all healthy participants is termed a prospective study, and is in contrast to a retrospective study, in which some participants already have the condition in question. Prospective studies produce much higher quality evidence, but are much more difficult and time-consuming to perform.
In the table below:
• ${\displaystyle I_{e}}$ is the probability of seeing no improvement after receiving the treatment (this is 1 minus the probability of seeing improvement with the treatment). This measure applies only to the treated group.
• ${\displaystyle I_{u}}$ is the probability of seeing no improvement after receiving the control (this is 1 minus the probability of seeing improvement with only the control). This measure applies only to the control (unexposed) group. The control group may receive a placebo treatment, or in cases where the goal is to find evidence that a new treatment is more effective than an existing treatment, the control group will receive the existing treatment. The meaning of the NNT is dependent on whether the control group received a placebo treatment or an existing treatment, and, in cases where a placebo treatment is given, the NNT is also affected to the quality of the placebo (i.e. for participants, is the placebo completely indistinguishable from the tested treatment.
Description ${\displaystyle I_{e}}$ ${\displaystyle I_{u}}$ NNT Interpretation
Perfect treatment, previously untreatable condition with no placebo effect involved 0.0 1.0 1 Half of participants receive the treatment, and half receive a control (which may be simply a placebo, or may be an existing treatment with a known effectiveness). Every person that receives the treatment shows improvement, which may be a reduction or halt in worsening of the condition, an improvement in the condition, or an outright cure of the condition. Every person in the control group shows no improvement, therefore the condition never improves on its own and the control is never effective. NNT is 1/(1.0-0.0), which is 1.
Very effective treatment with large improvement over control 0.1 0.9 1.25 For simplicity, a low number of participants will be used, thought scientific studies almost always require many more. Ten people receive the treatment, and ten receive a control. Of the ten in the treated group, nine show improvement, and one shows no improvement. In the control group, one person shows improvement and nine show none. Since one of those who received the control showed improvement without the treatment, it is said that one of the nine from the treated group would have improved without receiving the treatment. Therefore, one person’s outcome does not represent evidence that the treatment is better than the control. NNT is 1/(0.9-0.1), which is 1.25. The absolute risk reduction is 0.9-0.1, equal to 0.8.
Effective treatment with moderate improvement over control 0.3 0.7 2.5 Ten receive the treatment, and ten receive a control. In the treatment group, seven show improvement and three show none. In the control group, three show improvement and seven show none. Therefore, the treatment was more helpful than the control in four of ten cases (7 treated improved minus 3 controls improved), and was not any more helpful in six of ten cases (3 not improved despite treatment, 3 that would have improved anyway as seen in the control group). NNT is 1/(0.7 – 0.3), which is 2.5.
Effective treatment, but little improvement over control 0.4 0.5 10 Ten receive the treatment, and ten receive a control. In the treatment group six improve with the treatment, and four do not. In the control group, five improve and five do not. Therefore, the treatment was more helpful than the control in only one of ten cases (6 treated improved minus 5 controls improved), and was not helpful is nine of ten (4 not improved despite treatment, 5 that would have improved anyway as seen in the control group). NNT is 1/(0.5 – 0.4), which is 10.
Not very effective treatment with little improvement over control 0.8 0.9 10 Ten receive the treatment, and ten receive a control. Two improve with the treatment and eight do not. In the control group, one improves and nine do not. Therefore, the treatment was more helpful than the control in only one of ten cases, and was not helpful is nine of ten. NNT is 1/(0.9 – 0.8), which is 10.
Apparently very effective treatment, but with little real improvement over control 0.1 0.2 10 Ten receive the treatment, and ten receive a control. Nine improve with the treatment and one does not. In the control group, eight improve and two do not. Therefore, the treatment was more helpful than the control in only one of ten cases, and was not helpful is nine of ten. NNT is 1/(0.2 – 0.1), which is 10.
Treatment is very effective but worse than control 0.2 0.1 −10 Ten receive the treatment, and ten receive a control. Eight improve with the treatment and two do not. In the control group, nine improve and one does not. Therefore, the treatment was less helpful than the control in one of ten cases. NNT is 1/(0.1 – 0.2), which is -10. Notice that, even though the treatment was effective in eight of ten cases (only one less than the previous example) the NNT has shifted from 10 to -10. This is because NNT measures how many patients must be given the treatment instead of the control in order to see improvement in one person. Since giving the treatment to ten people would cause one of those people to be worse than if they had received the control instead, the NNT is -10. If control is placebo, giving placebo appear better than to give treatment.
## Real-life example
ASCOT-LLA manufacturer-sponsored study addressed the benefit of atorvastatin 10 mg (a cholesterol-lowering drug) in patients with hypertension (high blood pressure) but no previous cardiovascular disease (primary prevention). The trial ran for 3.3 years, and during this period the relative risk of a "primary event" (heart attack) was reduced by 36% (relative risk reduction, RRR). The absolute risk reduction (ARR), however, was much smaller, because the study group did not have a very high rate of cardiovascular events over the study period: 2.67% in the control group, compared to 1.65% in the treatment group.[12] Taking atorvastatin for 3.3 years, therefore, would lead to an ARR of only 1.02% (2.67% minus 1.65%). The number needed to treat to prevent one cardiovascular event would then be 98.04 for 3.3 years.[13]
## Numerical example
Example of risk reduction
Quantity Experimental group (E) Control group (C) Total
Events (E) EE = 15 CE = 100 115
Non-events (N) EN = 135 CN = 150 285
Total subjects (S) ES = EE + EN = 150 CS = CE + CN = 250 400
Event rate (ER) EER = EE / ES = 0.1, or 10% CER = CE / CS = 0.4, or 40%
Variable Abbr. Formula Value
Absolute risk reduction ARR CEREER 0.3, or 30%
Number needed to treat NNT 1 / (CEREER) 3.33
Relative risk (risk ratio) RR EER / CER 0.25
Relative risk reduction RRR (CEREER) / CER, or 1 − RR 0.75, or 75%
Preventable fraction among the unexposed PFu (CEREER) / CER 0.75
Odds ratio OR (EE / EN) / (CE / CN) 0.167
## Modern Approach to NNT
The above calculations for NNT are valid under monotonicity, where treatment can't have a negative effect on any individual. However, in the case where the treatment may benefit some individuals and harm others, the NNT as defined above cannot be estimated from a Randomized Controlled Trial (RCT) alone. The inverse of the absolute risk reduction only provides an upper bound, i.e., ${\displaystyle {\text{NNT}}\leqslant 1/(I_{u}-I_{e})}$.
The modern approach defines NNT literally, as the number of patients one needs to treat (on the average) before saving one. However, since "saving" is a counterfactual notion (a patient must recover if treated and not recover if not treated) the logic of counterfactuals must be invoked to estimate this quantity from experimental or observational studies. The probability of "saving" is captured by the Probability of Necessity and Sufficiency (PNS), where ${\displaystyle {\text{PNS}}=P({\text{Recovery if and only if treated}})}$.[14] Once PNS is estimated, NNT is give as ${\displaystyle {\text{NNT}}={\text{PNS}}^{-1}}$. However, due to the counterfactual nature of PNS, only bounds can be computed from an RCT, rather than a precise estimate. Tian and Pearl have derived tight bounds on PNS, based on multiple data sources, and Pearl showed that a combination of observational and experimental data may sometimes make the bounds collapse to a point estimate.[15][16] Mueller and Pearl provide a conceptual interpretation for this phenomenon and illustrate its impact on both individual and policy-makers decisions.[17]
## References
1. ^ Porta, Miquel, ed. (2016-07-21). Dictionary of Epidemiology - Oxford Reference. Oxford University Press. doi:10.1093/acref/9780199976720.001.0001. ISBN 9780199976720. Retrieved 2018-05-09.
2. ^ Vancak, V., Goldberg, Y., Levine, S. Z. (2020). "Systematic analysis of the number needed to treat". Statistical Methods in Medical Research. 29 (9): 2393-2410. doi:10.1177/0962280219890635.{{cite journal}}: CS1 maint: multiple names: authors list (link)
3. ^ Laupacis A, Sackett DL, Roberts RS (1988). "An assessment of clinically useful measures of the consequences of treatment". N. Engl. J. Med. 318 (26): 1728–33. doi:10.1056/NEJM198806303182605. PMID 3374545.
4. ^ "Number Needed to Treat". Bandolier. Retrieved 2017-04-21.
5. ^ Nuovo, J.; Melnikow J.; Chang D. (2002-06-05). "Reporting number needed to treat and absolute risk reduction in randomized controlled trials". JAMA. 287 (21): 2813–4. doi:10.1001/jama.287.21.2813. PMID 12038920.
6. ^ Hutton JL (2010). "Misleading Statistics: The Problems Surrounding Number Needed to Treat and Number Needed to Harm". Pharm Med. 24 (3): 145–9. doi:10.1007/BF03256810. ISSN 1178-2595.
7. ^ Palle Mark Christensen; Kristiansen, IS (2006). "Number-Needed-to-Treat (NNT) – Needs Treatment with Care". Basic & Clinical Pharmacology & Toxicology. 99 (1): 12–16. doi:10.1111/j.1742-7843.2006.pto_412.x. PMID 16867164. Archived from the original on 2013-01-05.
8. ^ Snapinn S, Jiang Q (2011). "On the clinical meaningfulness of a treatment's effect on a time-to-event variable". Stat Med. 30 (19): 2341–2348.
9. ^ Bender R, Blettner M (2002). "Calculating the "number needed to be exposed" with adjustment for confounding variables in epidemiological studies". J Clin Epidemiol. 55 (5): 525–530.
10. ^ Austin PC (2010). "Absolute risk reductions, relative risks, relative risk reductions, and numbers needed to treat can be obtained from a logistic regression model". J Clin Epidemiol. 63 (1): 2–6.
11. ^ Vancak V, Goldberg Y, Levine SZ (2022). "The number needed to treat adjusted for explanatory variables in regression and survival analysis: Theory and application". Stat Med: 1–22. doi:10.1002/sim.9418. PMID 35472818.{{cite journal}}: CS1 maint: multiple names: authors list (link)
12. ^ Sever PS, Dahlöf B, Poulter NR, et al. (2003). "Prevention of coronary and stroke events with atorvastatin in hypertensive patients who have average or lower-than-average cholesterol concentrations, in the Anglo-Scandinavian Cardiac Outcomes Trial—Lipid Lowering Arm (ASCOT-LLA): a multicentre randomised controlled trial". Lancet. 361 (9364): 1149–58. doi:10.1016/S0140-6736(03)12948-0. PMID 12686036.
13. ^ John Carey. "Do Cholesterol Drugs Do Any Good?". Business Week. Archived from the original on December 28, 2014. Retrieved 2008-03-31.
14. ^ Pearl, Judea (1999). "Probabilities of Causation: Three Counterfactual Interpretations and their identification". Synthese. 121: 93–149. doi:10.1023/A:1005233831499.
15. ^ Tian, Jin; Pearl, Judea (2000). "Probabilities of causation: Bounds and identification". Annals of Mathematics and Artificial Intelligence. 28: 287–313. doi:10.1023/A:1018912507879.
16. ^ Pearl, Judea (2009). Causality: Models, Reasoning and Inference. Cambridge University Press. doi:10.1017/CBO9780511803161. ISBN 9780511803161.
17. ^ Mueller, Scott; Judea Pearl (2022). Personalized Decision Making -- A Conceptual Introduction (PDF) (Technical report). UCLA.
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2022-09-27 04:45:54
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https://physics.stackexchange.com/questions/258094/rotating-the-bucket-in-circular-motion-without-spilling-water
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# Rotating the bucket in circular motion without spilling water
In the bucket experiment when the bucket reaches the top of the circle why will it have a normal force acting on the water downwards? Doesn't normal force oppose any other force? There is no force acting upwards... (As we are an observer in an inertial frame so we wont consider the centrifugal force because it is a pseudo force)
• In fact there is a reaction to the normal force and it is given by the contact force the liquid applies on the bucket. This happens even when the bucket is at the top of the trajectory, unless the water falls off. – Diracology May 26 '16 at 12:53
• And what would be the magnitude of this contact force? – oshhh May 26 '16 at 13:01
• Why are you removing the homework tag? – user36790 May 27 '16 at 11:10
• because it isn't my homework and the discussion is more about a concept than the solution of this question in particular – oshhh May 27 '16 at 11:14
• Hi Osheen Sachdev. Welcome to Phys.SE. Echoing @MAFIA36790's above comments, if you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. – Qmechanic May 27 '16 at 13:19
I want to focus on one thing, here. The nature of the normal force.
You write
Doesn't normal force oppose any other force?
which is a easy impression to get when you are introduced to the normal force in the context of things sitting on other things in a gravitational field, but that's not the best way to think about it.
The normal force keeps things from occupying the same space.
So consider a book sitting on the lab bench. It's subject to gravity, and in the absence of other forces would fall. But to fall it would have to occupy the same space as the solid top of the bench. The normal force is the interaction between to two objects that resists their inter-penetrating. In this case it has to provide a force that is equal and opposite to that of gravity to make that happen.
In the case of the water in the bucket, it's inertia would take it in a straight line, gravity modifies that into a parabola, but the sides and bottom of the bucket are moving in a circle for both those things to be true (the bucket goes in a circle and the water goes in a parabola) the water would have to move through the bottom of the bucket. The normal force serves to prevent the water from penetrating the solid material of the bucket and must supply whatever forces (beyond that of gravity) are necessary to cause the motion to be in a circle.
• But shouldn't normal force point sideways then...because water wants to go tangentially but the sides of the bucket prevent that? – oshhh May 26 '16 at 16:37
• The normal from the sides of the bucket points toward the inside of the bucket all the way around. The normal from the bottom of the bucket points (roughly) toward the center of the circle (my shoulder joint when I do this demo in class). Remember that for circular motion $\frac{\mathrm{d}\vec{v}}{\mathrm{d}t}$ points to the middle of the circle and that is the direction of the force that must be imposed on the water to keep it moving in a circle. – dmckee --- ex-moderator kitten May 26 '16 at 16:43
• As there is normal force pushing water from the bottom of the bucket, there must be a force pushing the water towards the bucket. I don't understand when the tension is inwards and water wants to go tangentially due to inertia then what force is pushing it outwards? considering the observer to be in an inertial frame there is no centrifugal force too... – oshhh May 27 '16 at 10:58
• Remember that the water is not in equilibrium (because it is moving in a circle, which means that it is accelerating). The forces not only don't need to cancel out they need to not cancel out. – dmckee --- ex-moderator kitten May 27 '16 at 15:06
Imagine a scenario where the bucket is rotated at just the right speed so that the centripetal acceleration required to keep the water on a circular path is exactly 9.81 $ms^{-1}$. Then at the top of the rotation, all the centripetal acceleration is supplied by gravity.
However the bucket might be rotating faster in any given scenario but it still rotates on a fixed radius. Since the force required to maintain circular motion is given by
$F=\frac{mv^2}{r}$
the force needs to be greater at higher velocities if the radius is constant. This means that gravity is no longer able to supply all the acceleration required to keep the water inside the bucket on its circular path.
The water wants to move to a larger radius but it cannot because the bucket is in the way. Hence the water feels a reaction force from the bucket. This contact force is simply the force required to hold the water at a given radius minus the force of gravity:
$F_N=\frac{mv^2}{r}-F_g$
Another way to think about it is if you desperately tried to move through a wall the wall would exert a force on you and the force would increase the harder you tried.
• I don't understand when the tension is inwards and water wants to go tangentially due to inertia then what force is pushing it outwards? considering the observer to be in an inertial frame there is no centrifugal force too... – oshhh May 27 '16 at 10:55
• You have to remember that the bucket is also following the circular path that the water takes – Jaywalker May 27 '16 at 10:56
• There isn't actually a force acting outwards, it is only perceived that way. As explained in dmckee's answer the normal force is just what is keeping the water from going through the bucket. – Jaywalker May 27 '16 at 10:58
• Yes but think about it. When the water moves tangentially, the bucket has also moved a bit so it is actually hitting the bottom of the bucket. – Jaywalker May 27 '16 at 11:25
• Well that is the explanation. I'll try one more time with a different analogy. Its like you keep catching the water with the bucket by moving it such that it can never go off on its tangent. The whole thing is a constant game of "catch" between bucket and water. – Jaywalker May 27 '16 at 11:49
In the bucket experiment when the bucket reaches the top of the circle why will it have a normal force acting on the water downwards?
The normal contact force is exerted by the bottom of the bucket as explicitly mentioned by the author. So, it is acting downward when the bucket is inverted.
Doesn't normal force oppose any other force? There is no force acting upwards
hmm... as said by Dirocology the bucket is forced by the contact force from the liquid; the normal force is reactive to that force.
Normal force plays the role of centripetal force when the velocity of the liquid is greater than $\sqrt{rg}\;.$
It is sufficient to demonstrate that the vertical acceleration of the bucket is $\ge g$; for this you can consider the curvature and resulting change in velocity vector. Which looks a lot like the derivation of centripetal force, of course...
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2021-06-18 05:18:51
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http://ompl.kavrakilab.org/structompl_1_1control_1_1KPIECE1_1_1TreeData.html
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ompl::control::KPIECE1::TreeData Struct Reference
The data defining a tree of motions for this algorithm. More...
#include <ompl/control/planners/kpiece/KPIECE1.h>
## Public Attributes
Grid grid {0}
A grid containing motions, imposed on a projection of the state space.
unsigned int size {0}
The total number of motions (there can be multiple per cell) in the grid.
unsigned int iteration {1}
The number of iterations performed on this tree.
## Detailed Description
The data defining a tree of motions for this algorithm.
Definition at line 330 of file KPIECE1.h.
The documentation for this struct was generated from the following file:
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2020-11-27 08:13:43
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https://itprospt.com/num/12837877/calculate-the-volume-in-liters-of-0-025um-copper-ii-sulfate
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5
# Calculate the volume in liters of 0.025UM copper(II) sulfate solution that contains 400. mmol of copper(II) sulfate (Cuso4): Round your answer to 2 significant digi...
## Question
###### Calculate the volume in liters of 0.025UM copper(II) sulfate solution that contains 400. mmol of copper(II) sulfate (Cuso4): Round your answer to 2 significant digitsOxo
Calculate the volume in liters of 0.025UM copper(II) sulfate solution that contains 400. mmol of copper(II) sulfate (Cuso4): Round your answer to 2 significant digits Oxo
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2022-09-26 13:44:55
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http://www.nag.com/numeric/FL/nagdoc_fl24/html/F07/f07wsf.html
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F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual
# NAG Library Routine DocumentF07WSF (ZPFTRS)
Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
## 1 Purpose
F07WSF (ZPFTRS) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
$AX=B ,$
using the Cholesky factorization computed by F07WRF (ZPFTRF) stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.
## 2 Specification
SUBROUTINE F07WSF ( TRANSR, UPLO, N, NRHS, A, B, LDB, INFO)
INTEGER N, NRHS, LDB, INFO COMPLEX (KIND=nag_wp) A(N*(N+1)/2), B(LDB,*) CHARACTER(1) TRANSR, UPLO
The routine may be called by its LAPACK name zpftrs.
## 3 Description
F07WSF (ZPFTRS) is used to solve a complex Hermitian positive definite system of linear equations $AX=B$, the routine must be preceded by a call to F07WRF (ZPFTRF) which computes the Cholesky factorization of $A$, stored in RFP format. The solution $X$ is computed by forward and backward substitution.
If ${\mathbf{UPLO}}=\text{'U'}$, $A={U}^{\mathrm{H}}U$, where $U$ is upper triangular; the solution $X$ is computed by solving ${U}^{\mathrm{H}}Y=B$ and then $UX=Y$.
If ${\mathbf{UPLO}}=\text{'L'}$, $A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular; the solution $X$ is computed by solving $LY=B$ and then ${L}^{\mathrm{H}}X=Y$.
## 4 References
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2
## 5 Parameters
1: TRANSR – CHARACTER(1)Input
On entry: specifies whether the normal RFP representation of $A$ or its conjugate transpose is stored.
${\mathbf{TRANSR}}=\text{'N'}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{TRANSR}}=\text{'C'}$
The conjugate transpose of the RFP representation of the matrix $A$ is stored.
Constraint: ${\mathbf{TRANSR}}=\text{'N'}$ or $\text{'C'}$.
2: UPLO – CHARACTER(1)Input
On entry: specifies how $A$ has been factorized.
${\mathbf{UPLO}}=\text{'U'}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{UPLO}}=\text{'L'}$
$A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
3: N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
4: NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{NRHS}}\ge 0$.
5: A(${\mathbf{N}}×\left({\mathbf{N}}+1\right)/2$) – COMPLEX (KIND=nag_wp) arrayInput
On entry: the Cholesky factorization of $A$ stored in RFP format, as returned by F07WRF (ZPFTRF).
6: B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
7: LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07WSF (ZPFTRS) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
8: INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).
## 6 Error Indicators and Warnings
Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
## 7 Accuracy
For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
• if ${\mathbf{UPLO}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|{U}^{\mathrm{H}}\right|\left|U\right|$;
• if ${\mathbf{UPLO}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|L\right|\left|{L}^{\mathrm{H}}\right|$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
$x-x^∞ x∞ ≤cncondA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$ and ${\kappa }_{\infty }\left(A\right)$ is the condition number when using the $\infty$-norm.
Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$.
## 8 Further Comments
The total number of real floating point operations is approximately $8{n}^{2}r$.
The real analogue of this routine is F07WEF (DPFTRS).
## 9 Example
This example solves the system of equations $AX=B$, where
$A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i$
and
$B= 3.93-06.14i 1.48+06.58i 6.17+09.42i 4.65-04.75i -7.17-21.83i -4.91+02.29i 1.99-14.38i 7.64-10.79i .$
Here $A$ is Hermitian positive definite, stored in RFP format, and must first be factorized by F07WRF (ZPFTRF).
### 9.1 Program Text
Program Text (f07wsfe.f90)
### 9.2 Program Data
Program Data (f07wsfe.d)
### 9.3 Program Results
Program Results (f07wsfe.r)
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2016-07-30 16:05:10
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http://ncatlab.org/nlab/show/bireflective+subcategory
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# nLab bireflective subcategory
category theory
## Applications
#### Notions of subcategory
A bireflective subcategory $B\subset C$ is a subcategory which is both reflective and coreflective; i.e. it is a fully faithful functor possessing both left and right adjoints.
Some people use the term in different, weaker meaning, that the unit of the adjunction is a bimorphism.
The adjoint pair induced form the adjoint triple given by reflection, inclusion and coreflection of a bireflective subcategory is an adjoint modality. See there for more.
Revised on March 28, 2015 15:53:40 by David Corfield (86.187.82.215)
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2015-11-28 09:30:14
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https://www.brightstorm.com/math/calculus/limits-and-continuity/continuity-problem-1/
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# Continuity - Problem 1 6,405 views
To check if a function is continuous at a point, first check if the function exists at that point (if for that x, there exists an f(x)). Then, check if the limit as x approaches that point exists. Recall that this can be done by checking if the left- and right-hand limits are the same. Finally, for a function to be continuous, the value of the function at a point must be the same as the limit of the function at that point.
Let's take a look at an example. We have this function graphed here y equals f(x). I want to know why this function is not continuous at x equals -2 right here.
Well, remember that there are three conditions for continuity. The function has to be defined at the number in question. The limit as x approaches that number f(x) has to exist. Finally, that limit has to equal the value of the function at that number.
Now the number here is -2. Our problem is with the first condition f(a) doesn't exist. F(-2) does not exist. It is not defined. So the function is not continuous at -2.
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2014-04-25 04:00:23
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https://themortgagestudent.com/tag/550a36-bbr3-hybridization-of-central-atom
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bbr3 hybridization of central atom
It is connected to 5 other atoms, one more than the 4 for sp3 hybridization. Thus, the configuration will be sp^3. Lv 5. Before completing the octets, ... BF3 Hybridization . BeBr2 is the only molecule that is linear, so I asssume it is sp hybridized. 6. Adding up the exponents, you get 4. BF3 is SP2 hybridization. Still have questions? 9 years ago. Determine the hybridization. Build a model and draw a 3D perspective drawing using the wedge and dash format. sp Hybridization. 1 Answer. The central atom of PCl5 is P. In this case, phosphorous has an sp3d hybridization. The exponents on the subshells should add up to the number of bonds and lone pairs. Which one of the following molecules has sp hybridization at the central atom BeBr2, SeF6, BF3, PF5, or CF4? Since it is a "tetrahedral" arrangement, the central atom "S" will have a total of four electron groups around it (four things that's holding on to it: two electron groups from Flourine holding on to Sulfur, and two electron groups from each lone pair holding on to Sulfur). They are accommodating to explain molecular geometry and nuclear bonding properties. Doc. Indicate whether the species is polar or nonpolar. 5. This hybridization gives you the trigonal planar geometry around the central atom with the p-orbital sticking in the up and down vertical direction. O2, BBr3, H30+, NOF, IF2 , AsO43-, HCO3-, HBrO3, CH2F2, CH2O2, C2Cl2, C2H3F3, C2H2F2 3. Relevance. Ask Question + 100. Answer Save. 1 0. Favorite Answer. For … Hybridization stands for mixing atomic orbitals into new hybrid orbitals. Join Yahoo Answers and get 100 points today. 4. We will take a pragmatic approach to hybridization theory: if there are two regions of electron density around a central atom, that atom is said to be sp hybridized; if there are three regions of electron density around a central atom, that atom is said to be sp 2 hybridized; if there are four regions of electron density around a central atom, that atom is said to be sp 3 hybridized. Provide the electronic geometry and molecular geometry about the central atom(s). Fluorine has 1 bond and 3 lone pairs giving a total of 4, making the hybridization: sp3. Formula=1/2(valence electron in central atom+atom linked to it by single bond+negative charge-positive charge) BF3 Formula=1/2(3+3+0–0)=3=s+2p=sp2 BF4 Formula=1/2(3+4)=3.5~4=sp3 In the case of the sp hybridization, only one s- and one p-orbital are mixed together to make hybrids. There are several types of hybridization like SP3, SP2, SP. The “unused” p-orbital can make a π-bond or to participate in a complex resonance conjugation. For each of the following molecules, indicate the hybridization requested and whether or not the electrons will be delocalized: (a) ozone $\left(\mathrm{O}_{3}\right)$ central O hybridization (b) carbon dioxide (CO_ ) central Chybridization (c) nitrogen dioxide (NO_) central N hybridization BF3 has a total of 24 valence electrons, which we have to set around the central atom. Since iodine has a total of 5 bonds and 1 lone pair, the hybridization is sp3d2. Provide the hybridization of all central atoms. Get your answers by asking now.
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2021-04-14 07:43:58
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https://terrytao.wordpress.com/2020/10/10/climbing-the-cosmic-distance-ladder-book-announcement/
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Several years ago, I developed a public lecture on the cosmic distance ladder in astronomy from a historical perspective (and emphasising the role of mathematics in building the ladder). I previously blogged about the lecture here; the most recent version of the slides can be found here. Recently, I have begun working with Tanya Klowden (a long time friend with a background in popular writing on a variety of topics, including astronomy) to expand the lecture into a popular science book, with the tentative format being non-technical chapters interspersed with some more mathematical sections to give some technical details. We are still in the middle of the writing process, but we have produced a sample chapter (which deals with what we call the “fourth rung” of the distance ladder – the distances and orbits of the planets – and how the work of Copernicus, Brahe, Kepler and others led to accurate measurements of these orbits, as well as Kepler’s famous laws of planetary motion). As always, any feedback on the chapter is welcome. (Due to various pandemic-related uncertainties, we do not have a definite target deadline for when the book will be completed, but presumably this will occur sometime in the next year.)
The book is currently under contract with Yale University Press. My coauthor Tanya Klowden can be reached at tklowden@gmail.com.
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2021-04-16 01:08:07
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https://zbmath.org/?q=an:0773.68059
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## Rank-$$r$$ decision trees are a subclass of $$r$$-decision lists.(English)Zbl 0773.68059
Summary: We prove that the concept class of rank-$$r$$ decision trees is contained within the class of $$r$$-decision lists. Each class if known to be learnable in polynomial time in the PAC model for constant $$r$$. One result of this note, however, is that the algorithm of R. L. Rivest [Learning decision lists, Machine Learning 2, 229-246 (1987)] can be used for both.
### MSC:
68T05 Learning and adaptive systems in artificial intelligence 68Q25 Analysis of algorithms and problem complexity
### Keywords:
machine learning; decision lists; decision trees
Full Text:
### References:
[1] Ehrenfeucht, A.; Haussler, D., Learning decision trees from random examples, Inform. and comput., 82, 231-246, (1989) · Zbl 0679.68157 [2] Helmbold, D.; Sloan, R.; Warmuth, M.K., Learning nested differences of intersection-closed concept classes, Proc. second ann. workshop on computational learning theory, 41-56, (1989) · Zbl 0746.68072 [3] Kearns, M.; Li, M.; Pitt, L.; Valiant, L., On the learnibility of Boolean formulae, Proc. nineteenth ann. ACM symp. on theory of computing, 285-295, (1987) [4] Littlestone, N., Personal communication (a mistake-bound version of Rivest’s decision-List algorithm), (1989) [5] Rivest, R.L., Learning decision lists, Machine learning, 2, 229-246, (1987) [6] Sakakibara, Y., Algorithmic learning of formal languages and decision trees, () [7] Simon, H.U., On the number of examples and stages needed for learning decision trees, (), 303-313 [8] Valiant, L.G., A theory of the learnable, Comm. ACM, 27, 1134-1142, (1984) · Zbl 0587.68077 [9] Wenocur, R.S.; Dudley, R.M., Some special vapnik-chervonenkis classes, Discrete math., 33, 313-318, (1981) · Zbl 0459.60008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2022-08-08 22:49:27
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https://www.cemc.uwaterloo.ca/pandocs/potw/2022-23/English/POTWC-22-A-N-08-P.html
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# Problem of the Week Problem C Gone Shopping
While grocery shopping, Terry has a way to approximate the total cost of his purchases. He simply approximates that each item will cost $$\3.00$$.
One day, Terry purchased $$20$$ items. He purchased items that each had an actual cost of either $$\1.00$$, $$\3.00$$, or $$\7.50$$. Exactly seven of the purchased items had an actual cost of $$\3.00$$. If the total actual cost of the $$20$$ items was the same as the total approximated cost, how many items had an actual cost of $$\7.50$$?
Themes: Algebra, Number Sense
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2023-03-27 00:08:38
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https://www.andlearning.org/sodium-iodide-formula/
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Connect with us
## What is Sodium Iodide?
Sodium Iodine is a common inorganic salt and taken as the important source of iodine as well. The chemical formula for the compound is NaI and its molecular weight is 149.89 g/mol approximately. This is a simple ionic compound where sodium cation is connected with iodine anion and its chemical structure is also given below.
### Sodium Iodide Structure
The chemical compound adopts a similar octahedral crystal geometry similar to the sodium chloride.
### Sodium Iodide Formula
For the industrial grade, the product is prepared by mixing sodium hydroxide and hydroiodic acid together. It can also be prepared by mixing sodium carbonate and hydroiodic acid together as shown below.
NaOH + HI → NaI + H2O
### Sodium Iodide Molecular
Moving ahead, let us look at the physical properties of the compound. This is a white odorless crystal powder with a density of 3.67 g/mL, the melting point of 651 °C and the boiling point of 1,304 °C that absorbs moisture from the air and turns into solution further.
#### About Sodium Iodide
The product is quite soluble in water and organic solvents too. This is highly sensitive to sir, light, moisture etc. The chemical compound turns to brown color when exposed to light or air due to the formation of iodine flumes. The product reacts violently as a strong oxidizing agent and a strong acid, producing iodine in the end.
The major uses of sodium iodine can be seen in dietary supplements and table salts. It prevents iodine deficiency too in your body. Another major application of the product is nuclear medicine as radioactive sodium iodine can be used for treating thyroid cancer. The product is used as a reagent in organic synthesis process to prepare various alkyl iodides.
Continue Reading
Advertisement
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2023-03-23 21:04:02
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https://repository.uantwerpen.be/link/irua/119910
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Title Measurement of the ratio $B(t\rightarrow Wb)/B(t\rightarrow Wq)$ in pp collisions at $\sqrt{s}$=8 TeV Author Khachatryan, V. Sirunyan, A. M. Tumasyan, A. Alderweireldt, S. Bansal, M. Bansal, S. Cornelis, T. de Wolf, E.A. Janssen, X. Knutsson, A. Luyckx, S. Roland, B. Rougny, R. van de Klundert, M. van Haevermaet, H. Van Mechelen, P. Van Remortel, N. Van Spilbeeck, A. et al. Faculty/Department Faculty of Sciences. Physics Publication type article Publication 2014 Amsterdam , 2014 Subject Physics Source (journal) Physics letters: B. - Amsterdam, 1967, currens Volume/pages 736(2014) , p. 33-57 ISSN 0370-2693 ISI 000341487800007 Carrier E Target language English (eng) Full text (Publishers DOI) Affiliation University of Antwerp Abstract The ratio of the top-quark branching fractions R = B(t --> Wb)/B(t --> Wq), where the denominator includes the sum over all down-type quarks (q = b, s, d), is measured in the t (t) over bar dilepton final state with proton-proton collision data at root s = 8 TeV from an integrated luminosity of 19.7 fb(-1), collected with the CMS detector. In order to quantify the purity of the signal sample, the cross section is measured by fitting the observed jet multiplicity, thereby constraining the signal and background contributions. By counting the number of b jets per event, an unconstrained value of R = 1.014 +/- 0.003 (stat.) +/- 0.032 (syst.) is measured, in a good agreement with current precision measurements in electroweak and flavour sectors. A lower limit R > 0.955 at the 95% confidence level is obtained after requiring R <= 1, and a lower limit on the Cabibbo-Kobayashi-Maskawa matrix element |V-tb| > 0.975 is set at 95% confidence level. The result is combined with a previous CMS measurement of the t-channel single-top-quark cross section to determine the top-quark total decay width, Gamma(t) = 1.36 +/- 0.02 (stat.)(-0.11)(+0.14) (syst.) GeV. (C) 2014 The Authors. Published by Elsevier B.V. Full text (open access) https://repository.uantwerpen.be/docman/irua/3b8a6b/8523.pdf E-info http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000341487800007&DestLinkType=RelatedRecords&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000341487800007&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000341487800007&DestLinkType=CitingArticles&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 Handle
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2017-02-24 12:45:23
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https://neurips.cc/Conferences/2012/ScheduleMultitrack?event=3589
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Timezone: »
Poster
One Permutation Hashing
Ping Li · Art B Owen · Cun-Hui Zhang
Mon Dec 03 07:00 PM -- 12:00 AM (PST) @ Harrah’s Special Events Center 2nd Floor
While minwise hashing is promising for large-scale learning in massive binary data, the preprocessing cost is prohibitive as it requires applying (e.g.,) $k=500$ permutations on the data. The testing time is also expensive if a new data point (e.g., a new document or a new image) has not been processed. In this paper, we develop a simple \textbf{one permutation hashing} scheme to address this important issue. While it is true that the preprocessing step can be parallelized, it comes at the cost of additional hardware and implementation. Also, reducing $k$ permutations to just one would be much more \textbf{energy-efficient}, which might be an important perspective as minwise hashing is commonly deployed in the search industry. While the theoretical probability analysis is interesting, our experiments on similarity estimation and SVM \& logistic regression also confirm the theoretical results.
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2023-02-08 06:44:50
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https://johannesbader.ch/blog/yet-another-bazarloader-dga/
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# Yet Another Bazar Loader DGA
#### Disclaimer
These are just unpolished notes. The content likely lacks clarity and structure; and the results might not be adequately verified and/or incomplete.
#### Aliases
The malware in this blog post is known as BazarBackdoor, Team9Backdoor, BazDor, BazarLoader and BazaLoader
#### Cover Image
Cover Photo by Naim Benjelloun on Pexels
#### Related Posts
This is one of four blog posts on "Bazar Loader". Check out the other parts here:
Bazar Loader decided to change its perfectly fine domain generation algorithm (DGA) once again. The change in the algorithm is very minor, but it yields more domain names.
### Sample
I looked at this sample:
MD5
SHA1
a97893ab95f794cabc261483423f942f552926d0
SHA256
8e244f1a5b4653d6dbb4cc3978c7dd773b227a443361fbc30265b79f102f7eed
Size
288 KB (295616 Bytes)
Compile Timestamp
2021-01-20 19:37:37 UTC
MalwareBazaar, Malpedia, Dropping_sha256, Cape, VirusTotal
Filenames
Preview_report20-01.exe (VirusTotal)
Detections
MalwareBazaar: BazaLoader, Virustotal: 33/76 as of 2021-01-23 07:31:37 - Trojan.Win32.Zenpak.4!c (AegisLab), Backdoor:Win32/KZip.90c5e0b2 (Alibaba), BackDoor.Bazar.55 (DrWeb), Trojan.Win32.Zenpak.bfcu (Kaspersky), Trojan:Win64/Bazarldr.BMB!MSR (Microsoft), Trojan.Win32.Zenpak.bfcu (ZoneAlarm)
it unpacks to this
MD5
63784053ac2f608d94c18b17c46ab5d4
SHA1
e01c814d6a4993c74a2bfb87b1b661fe78c41291
SHA256
c0a087a520fdfb5f1e235618b3a5101969c1de85b498bc4670372c02756efd55
Size
98 KB (100864 Bytes)
Compile Timestamp
2021-01-20 19:10:11 UTC
MalwareBazaar, Malpedia, Dropping_sha256, Dropping_sha256, Cape, VirusTotal
Filenames
none
Detections
MalwareBazaar: BazaLoader, Virustotal: 21/75 as of 2021-01-23 13:37:55 - Gen:Variant.Bulz.163525 (ALYac), Gen:Variant.Bulz.163525 (Ad-Aware), Trojan.Bulz.D27EC5 (Arcabit), Gen:Variant.Bulz.163525 (BitDefender), Gen:Variant.Bulz.163525 (B) (Emsisoft), Gen:Variant.Bulz.163525 (GData), Gen:Variant.Bulz.163525 (MicroWorld-eScan), Trojan:Win32/TrickBot.VSF!MTB (Microsoft), Trojan.TrickBot!8.E313 (TFE:5:6iToUtBEDBC) (Rising)
which finally drops
MD5
7e8eddaef14aa8de2369d1ca6347b06d
SHA1
4543e6da0515bb7d93e930c9f30e40912d495373
SHA256
f29253139dab900b763ef436931213387dc92e860b9d3abb7dcd46040ac28a0e
Size
89 KB (91136 Bytes)
Compile Timestamp
2021-01-18 14:29:29 UTC
MalwareBazaar, Malpedia, Dropped_by_sha256, Cape, VirusTotal
Filenames
none
Detections
MalwareBazaar: None, Virustotal: 19/76 as of 2021-01-23 15:04:35 - Gen:Variant.Bulz.163525 (ALYac), Gen:Variant.Bulz.163525 (Ad-Aware), Trojan.Win32.Bulz.4!c (AegisLab), Trojan.Bulz.D27EC5 (Arcabit), Gen:Variant.Bulz.163525 (BitDefender), Gen:Variant.Bulz.163525 (B) (Emsisoft), Gen:Variant.Bulz.163525 (FireEye), Gen:Variant.Bulz.163525 (GData), Gen:Variant.Bulz.163525 (MicroWorld-eScan)
### Difference from the Last Version
The current version is just a slight modification to the version from December. Like the previous version of the algorithm, this version calculates all ordered pairs of 19 consonants and 6 vowels (including y ). These pairs are then permuted based on a fixed value. This value is the same, so the resulting list of 228 pairs is also the same.
The calculation of the first four letters is the same – that is, the selection of the first two pairs of letters: The permuted list of letters is divided into groups of 19 pairs. Then the two digits of the current month determine which group is selected. From these, one pair at a time is randomly – and unpredictably – selected.
The last four letters (two pairs) are still determined by the two year digits. However, the division of letter pairs into groups is different. Based on the current decade, two letters are chosen from a group of 22 pairs. The groups of 22 pairs partly overlap, so that theoretically after every 10 years identical domains could be generated again. This in contrast to the version from December, where the decade still determined a non-overlapping group of 6 pairs only. The last two letters are picked from groups of 4 — instead of 6 — letter pairs.
The DGA still generates 10'000 domains. But because there are 88 potential monthly combinations for the last for letters instead of just 36 previously, the excepted number of unique domains is larger:
$$\mathbb{E} = 31768 \left( 1 - \left(\frac{31768 -1}{31768 }\right)^{10000} \right) \approx 8579$$
Since domain names partially repeat after each decade, domains can no longer be uniquely assigned to a seed. But since I strongly doubt that the domain generation algorithm will still have any relevance in a few months, let alone 10 years, the domain to seed tool assumes domains are from the 20s.
### Reimplementation in Python
This is the new version reimplemented in Python
from itertools import product
from datetime import datetime
import argparse
from collections import namedtuple
Param = namedtuple('Param', 'mul mod idx')
pool = (
"qeewcaacywemomedekwyuhidontoibeludsocuexvuuftyliaqydhuizuctuiqow"
"agypetehfubitiaziceblaogolryykosuptaymodisahfiybyxcoleafkudarapu"
"qoawyluxqagenanyoxcygyqugiutlyvegahepovyigqyqibaeqynyfkiobpeepby"
"hoevmeburedeviihiravygkemywaerdonoyryqloammoseweesuvfopiriboikuz"
"orruzemuulimyhceukoqiwfexuefgoycwiokitnuneroxepyanbekyixxiuqsias"
"xoapaxmaohezwoildifaluzihipanizoecxyopguakdudyovhaumunuwsusyenko"
"atugabiv"
)
def dga(date):
seed = date.strftime("%m%Y")
params = [
Param(19, 19, 0),
Param(19, 19, 1),
Param(4, 22, 4),
Param(4, 4, 5)
]
ranges = []
for p in params:
s = int(seed[p.idx])
lower = p.mul*s
upper = lower + p.mod
ranges.append(list(range(lower, upper)))
for indices in product(*ranges):
domain = ""
for index in indices:
domain += pool[index*2:index*2 + 2]
domain += ".bazar"
yield domain
if __name__ == "__main__":
parser = argparse.ArgumentParser()
"-d", "--date", help="date used for seeding, e.g., 2020-06-28",
default=datetime.now().strftime('%Y-%m-%d'))
args = parser.parse_args()
d = datetime.strptime(args.date, "%Y-%m-%d")
for domain in dga(d):
print(domain)
### Characteristics
Except for the number of domains per month, the characteristics are the same as for the previous verion:
propertyvalue
typeTDD (time-dependent-deterministic)
generation schemearithmetic
seedcurrent date
domain change frequencyevery month
unique domains per month19·19·22·4 = 31'768
sequencerandom selection, might pick domains multiple times
wait time between domains10 seconds
top level domain.bazar
second level charactersa-z, without j
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2021-03-01 22:03:36
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https://www.nature.com/articles/s41467-019-10377-9
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# Reversal of transmission and reflection based on acoustic metagratings with integer parity design
## Abstract
Phase gradient metagratings (PGMs) have provided unprecedented opportunities for wavefront manipulation. However, this approach suffers from fundamental limits on conversion efficiency; in some cases, higher order diffraction caused by the periodicity can be observed distinctly, while the working mechanism still is not fully understood, especially in refractive-type metagratings. Here we show, analytically and experimentally, a refractive-type metagrating which can enable anomalous reflection and refraction with almost unity efficiency over a wide incident range. A simple physical picture is presented to reveal the underlying diffraction mechanism. Interestingly, it is found that the anomalous transmission and reflection through higher order diffraction can be completely reversed by changing the integer parity of the PGM design, and such phenomenon is very robust. Two refractive acoustic metagratings are designed and fabricated based on this principle and the experimental results verify the theory.
## Introduction
The ability to control at will the propagation of waves, such as electromagnetic waves and acoustic waves, has captured the fascination of scientists. In the past few years, as the 2D version of bulk metamaterials, metasurfaces have provided new paradigms to build devices that direct the flow of waves in a way not possible before1,2,3,4, and have enabled new physics5,6,7,8,9 that are distinctly different from those observed in their 3D counterparts (i.e., metamaterials). Typical examples, include planar lenses5, holograms6, ultrathin cloaking7 in electromagnetics, and other devices in acoustics10,11. By engineering phase shift ϕ(x) along metasurfaces, the scattered wavefronts can be manipulated to achieve anomalous reflection or refraction12,13,14,15,16, which is summarized as the generalized Snell’s law (GSL)12,
$$k_x^{in} = k_x^{r(t)} - \xi ,$$
(1)
where $$k_x^{in}$$ and $$k_x^{r(t)}$$ are tangential wave vectors of incident and reflected (transmitted) wave. For the 2D case, ξ = ∂ϕ(x)/∂x describes the phase gradient along the metasurface. In acoustics, similar wavefront manipulation has been demonstrated using structured phase arrays17,18,19,20,21,22. However, recently some studies23,24 have shown that this kind of phase gradient metasurface is inherently limited in conversion efficiency for wavefront manipulation, due to impedance mismatch at boundaries. Even for an ideal phase gradient metasurface with infinite resolutions (i.e., m → ∞, where m is the number of unit cells in a superlattice; see below), such a limitation is still present.
A few solutions23,24,25 were proposed to successively overcome this inherent limitation to achieve the scattering-free manipulation of anomalous reflected and refracted waves, but the designed metasurfaces require active elements or strong nonlocality, posing challenges for practical implementations26,27. To realize extremely anomalous transmission/reflection with perfect efficiency in a passive and lossless structure, bianisotropic metasurfaces28,29,30 were proposed and experimentally demonstrated in both electromagnetic and acoustic waves. Alternatively, metagratings31, periodic structures with a supercell comprising of several subscatters, were suggested to deliver the output wavefront into the desired direction with unity efficiency. However, this method solely works for a specific incidence angle, as the design of the metastructure is well defined for a specific angle. By electrostatically biasing graphene sheets, reconfigurable metagratings32 can extend the incidence to several discrete angles, but the structures are complex and the working angle is still limited. Therefore, how to realize high-efficient anomalous reflection or/and anomalous refraction, that can cover a wide incidence in a passive structure, is still an open question.
Essentially, phase gradient metasurfaces are periodic structures with a supercell spatially repeated along the interface, because of folded phase profile33. In this way, the GSL is insufficient to determine completely the directions of anomalous reflected or/and refracted waves, in particular for incident angle beyond the so-called critical angle predicted by the GSL. Instead, it is replaced by another formula involving superlattices16,19
$$k_x^{in} = k_x^{r(t)} - nG,$$
(2)
where G = 2π/p is the reciprocal lattice vector, and p is period. Both ξ and G commonly share the identical magnitude, yet with different physical origin; the former is introduced by the phase gradient, whereas the latter is caused by the periodicity of grating. Eq. (2) can not only steer a wavefront as expected from the GSL, but can also exhibit other unique features. In fact, in a large number of aforementioned phase gradient metasurfaces, particularly in acoustic metasurfaces17,18,19,20,21,22,34,35,36,37,38, anomalous reflection or refraction with high-efficiency were obtained through higher order diffraction. For convenience, in this work we call all periodic structures with phase gradient as phase gradient metagratings (PGMs). Normally, there are several diffraction channels simultaneously open for a particular incidence and these propagation channels are available for incident wave to depart from PGM. The diffraction mechanism therein is complex and ambiguous, especially in more complicated refractive-type PGMs, since the refractive and reflective diffraction channels are concurrently included. Eq. (2) fails to predict the primary diffraction order of the scattering waves. For instance, for incident angle beyond the critical angle (the n = 1 order in Eq. (2)), multiple diffraction channels coexist, and only negative refraction stemming from the n = −3 order in Eq. (2) was observed in experiments19. The underlying mechanism is still a puzzle.
In this article, we will investigate theoretically and experimentally a passive and lossless refractive-type PGM, and we will show that the designed PGM can enable anomalous reflection and refraction with near unity conversion efficiencies over a wide angle of incidence. Recently, based on loss-induced suppression of higher order diffraction, acoustic asymmetric transmission39 was demonstrated in the lossy PGMs. Transient simulations revealed that multiple reflections (MRs) are responsible for the energy-loss of higher order diffraction39, which offered a new insight to explore the uncharted diffraction rule. Starting from the MR effect39,40, we will reveal the diffraction mechanism of PGMs. It is found that the diffraction order is relevant to the propagation number of MRs (i.e., the number of times the wave travels inside the PGM) and the number of unit cells m of the PGMs. In particular, the transmission and reflection amplitudes of a particular diffraction order are determined by the integer parity of the propagation number. Consequently, the control of transmission and reflection of the diffraction order can be realized by controlling the integer parity, i.e., oddness or evenness (and hereafter referred to simply as parity), of the number of unit cells in the PGMs. Further explorations show that such parity-dependent phenomena are very robust for any m, implying that the diffraction law in Eq. (2) should be carefully refined according to the integer parity of m. Based on the demonstrated diffraction mechanism, we derive here a new set of formulas that can well explain the complicated diffraction phenomena of our studied PGMs, and can fully predict the parity-dependent perfect anomalous reflection and refraction. The puzzling diffraction phenomena in previous work can also be well understood from our diffraction rule. The experimentally measured results of acoustic waves verify our findings.
## Results
### Models and theory
To demonstrate our idea, let us start from the metagrating structure shown in Fig. 1a, where the PGM is composed of periodically repeated supercells with lengths of p. It should be noted that although this study focuses on acoustic waves, the achieved results are also applicable to the electromagnetic analogs16. The whole system is immersed in a background medium of air with density of ρ0 = 1.21 kg m−3 and speed of sound c0 = 343 ms−1. Figure 1b shows the details of the supercell, which includes m unit cells with widths of a (=p/m), and each unit cell is made of sound-hard material (gray area) perforated by a slit (blue area) with a width of w. The thickness of the metagrating is h. To steer the outgoing wave, the transmitted phase across a supercell covers a complete range of 2π, with a phase gradient of ξ. To begin with, we consider effective medium filled in the subwavelength slits. The effective medium is characterized by different effective refractive indices, and the index profile in the jth unit cell is given as nj = 1 + (j − 1)λ0/mh. For obtaining a specific phase gradient, the period length is set to be constant, and the width of unit cell is determined by the number of unit cells in a supercell. We consider incident wave from air with $$k_x^{in} = k_0\sin \theta _{in}$$, where k0 = 2π/λ0 is wave vector in air and θin is the incident angle. The reflected and transmitted waves obey the diffraction law of Eq. (2), with the maximum diffraction order (N, a negative integer) given as, N = roundup [−2k0/G] + 1. Regardless of the direction of the incident wave, i.e., $$k_x^{in} \in \left[ { - k_0,\;k_0} \right]$$, the existing diffraction orders of the reflected and transmitted waves belong to n [N, 1].
### Diffraction mechanism of PGM
We first provide an intuitive physical picture to reveal the diffraction mechanism. Owing to the sound-hard materials of a PGM (gray area in Fig. 1b), these unit cells could be regarded as acoustic waveguides. The sound-hard material is thick enough to avoid wave coupling between adjacent unit cells. When the width of unit cell is much smaller than the working wavelength (i.e., aλ0), only fundamental mode can be supported inside these unit cells. The forward and backward waves propagating in the unit cells interfere to form standing waves stemming from the MR effect of incident rays in the PGM (see the yellow arrows in Fig. 1c). For simplicity, we define the number of times the waves pass through the medium as L. When incident rays pass directly through the PGM, i.e., L = 1 (see the solid yellow arrows), the phase shift in the jth unit cell is ϕj = k0njh and the phase difference of adjacent unit cells per period is Δϕ = ϕj+1 − ϕj = 2π/m. By analyzing Eq. (2), the phase gradient of the nth diffraction order could be equivalent to ξ = nG, accordingly, the phase difference of two adjacent unit cells is expressed as Δφn = = 2πn/m. For one-pass propagation, the lowest order n = 1 is satisfied for Δϕ = Δφ1, therefore Eq. (2) can be expressed as
$$k_x^{in} = k_x^t - G = k_x^t - \xi ,$$
(3)
which is well-known as GSL. In such a case, the incident wave with $$k_x^{in} \in \left[ { - k_0,\;k_0 - \xi } \right]$$ will follow GSL (the n = 1 order), with kx = k0 − ξ being the critical momentum. When the incident angle is beyond the critical angle ($$k_x^{in} \in \left[ {k_0 - \xi ,\;k_0} \right]$$), the channel of the n = 1 order will close, and normally the incident wave cannot pass through the PGM via direct transmission. Thereupon, waves will undergo another propagation process (L = 2) via internal reflection at the transmitted interface (see the dashed yellow arrows), leading to a phase shift of 2ϕj and a phase difference of Δϕ = 2 × (2π/m) at the reflected interface. As the remaining diffraction orders are n [N, 0], so Δφn = 2πn/m ≤ 0. Therefore, it seems that waves cannot couple to these diffraction orders by means of Δϕ = Δφn. However, when a phase wrap of 2π is applied to Δϕ, i.e., Δϕ − 2π (2π phase wrap is enough for wave to couple to higher-order (N) in a PGM with unit cells supporting fundamental waveguide modes), the phase difference becomes equivalent. Therefore, when Δϕ − 2π = Δφn, the reflected wave with the n-diffraction order will occur (see the red arrows in Fig. 1c); if not, the third time propagation process (L = 3) will emerge in unit cells via internal reflection at the reflected interface (see Fig. 1d). When rays reach the transmitted interface, the phase difference is Δϕ = 3 × (2π/m). Similarly, if it can meet Δϕ − 2π = Δφn, there will be a transmitted wave of the n-diffraction order, otherwise the fourth time propagation (L = 4) will happen and so forth (See Fig. 1d).
Generally, if we consider the oscillating wave inside a PGM coupling to the n-diffraction order via L-time propagation process in unit cells, the corresponding relation can be expressed as 2πL/m − 2π = 2πn/m, i.e.
$$L = m + n.$$
(4)
As L > 0 and the maximum diffraction order is N, the number of unit cells is required to meet m ≥ 1 − N. When L is odd, the incident wave will couple to the corresponding transmitted wave of the higher orders; whereas when L is even, it will couple to the corresponding reflected wave of the higher orders. Consequently, by combining Eqs. (2) and (4), the diffraction law in a PGM is summarized as
$$\left\{ {\begin{array}{*{20}{l}} {k_x = k_x^t - nG,} \hfill & {\left( {L\,{\mathrm{is}\,{\mathrm{odd}}}} \right)} \hfill \\ {k_x = k_x^r - nG,} \hfill & {\left( {L\,{\mathrm{is}\,{\mathrm{even}}}} \right)} \hfill \end{array}} \right..$$
(5)
Using Eqs. (3)–(5), the diffraction phenomena in a PGM can be predicted. For the incident wave below the critical angle, the propagation number is L = 1, the incident wave will couple to the transmitted wave of the lowest order n = 1, which is independent of m. For the incident wave beyond the critical angle, MRs happen inside the PGM in turn (i.e., L = 1 → 2 → 3...) and resonance transmission or reflection can be induced when the path length due to MRs reaches the Fabry–Perot condition. If the wave travels through the slab an odd (even) propagation number, strong transmission (reflection) can be generated, with the diffraction order determined by Eq. (5) (see Fig. 1d). Although several diffraction orders are open for the incident wave, the maximum diffraction order is preferential owing to minimum propagation number that corresponds to minimum geometric path length. Furthermore, if one designs a PGM with odd and even unit cells, caused by the parity transition of the propagation number, the transmission and reflection of the diffraction order can be reversed.
### Analytical and numerical demonstration
Although the above revealed diffraction mechanism and associated diffraction rule are very simple, they are indeed powerful for making complex diffraction phenomena clear. Without loss of generality, we take PGMs with ξ = k0 to verify this point, in which the maximum diffraction order is N = −1 and the critical angle of is θ1 = 0°. When θin < θ1, the propagation number is L = 1, it is mainly the transmitted wave following GSL (the n = 1 order). While for θin > θ1, there are two diffraction orders, i.e., the n = 0 order and the n = −1 order. As we have discussed in Fig. 1d, the higher diffraction order is preferential owing to the minimum propagation number. Hence, for θin > θ1, the effective diffraction order is the n = −1 order and the corresponding propagation number of PGM with m unit cells is L = m − 1. Based on Eqs. (4) and (5), when m is odd, e.g., m = 3, the propagation number L is even, which leads to the reflection of the n = −1 order (see the equifrequency contour in Fig. 2a). On the other hand, when m is even, e.g., m = 4, the propagation number L is odd, which results in the transmission of the n = −1 order (see the equi-frequency contour in Fig. 2b). To demonstrate above theoretical prediction, numerical simulations are performed using COMSOL MULTIPHYSICS. The simulated field patterns of the PGMs with three and four unit cells are respectively displayed in Fig. 2c, d, where these two metagratings share identical phase gradient, since the period length is constant, i.e., p = λ0. When θin = −30°, which is below the critical angle, the incident waves in both cases pass through the PGMs following the n = 1 order (see the lower plots of Fig. 2c, d). However, when θin = 30°, beyond the critical angle, the incident wave is reflected back for the PGM with m = 3 (see the upper plot in Fig. 2c), and passes through the PGM with m = 4 (see the upper plot in Fig. 2d). In both cases, the scattered waves follow the n = −1 diffraction order with nearly perfect conversion efficiency (see the arrows in Fig. 2). Therefore, the theoretical prediction based on Eqs. (4) and (5) is well demonstrated from numerically simulated acoustic field patterns. In addition, by observing the equicontour in Fig. 2a, when the incident angle is within the critical angle, the anomalous transmission can occur by following $$k_x^t = k_x{\mathrm{ + }}\xi$$. While for the incident angle beyond the critical angle, the PGM will generate an equivalent tangential momentum of −ξ, the anomalous reflection will take place by obeying $$k_x^r = k_x - \xi$$. Hence, bounded by the critical angle, the anomalous reflection and transmission can simultaneously exist in a single PGM, which enables potential design for multifunctional acoustic planar devices.
In fact, the reflection (transmission) of the n = −1 order for the PGM with m = 3 (m = 4) not only happens at θin = 30°, but occurs in a wider incident range, which can be observed from the equi-contours in Fig. 2. To quantify angular performance of the PGMs, we analytically and numerically show the relationship between the transmission/reflection of the diffraction orders and the incident angle in Fig. 3, where the analytical results are described by the curves and the numerical results are indicated by the symbols. The analytical results are obtained based on the coupled mode theory18,39 (details shown in Supplementary Note 1), which agree well with numerical results except for very steep incident angles. For the case of the PGM with m = 3 (see Fig. 3a, b), more than 90% transmission of the n = 1 order is observed for θin [−60°, 0°] and the reflection of the n = −1 order is higher than 90% for θin [0°, 60°]. In addition, for the normal incidence, that is, at the critical angle, there is an odd propagation number of L = 3 for the n = 0 order, bringing about higher transmission of the n = 0 order (see the black data in Fig. 3a). For the case of the PGM with m = 4 (see Fig. 3c, d), similar behavior is observed for θin [−60°, 0°]. For angles above the critical angle (θin [0°, 60°]), however, the reflection mode is reversed to transmission mode due to integer parity of the cell number. Furthermore, it is an even propagation number of L = 4 for the n = 0 order, therefore there is higher reflection of the n = 0 order at the critical angle (see the black data in Fig. 3d), which is opposite with that in Fig. 3a. For the incident angle near ±90°, owing to intrinsic limitation of PGMs29,30, the coupling efficiency between incident wave and the n = 1/n = −1 order is extremely low and stronger specular reflection appears (see black data in Fig. 3b, d). In addition, to further demonstrate Eqs. (4) and (5), as a more complicated case, PGMs with ξ = 0.6k0 are used to reveal similar reversal phenomena of the diffraction orders, shown in Supplementary Figs. 1 and 2, and Supplementary Note 2.
### Design of PGMs and experimental verifications
To further confirm the diffraction behavior of the PGMs (ξ = k0) with odd/even number of unit cells, we utilize zigzag microstructures to design two groups of PGMs at 4.0 kHz: one is a PGM with three unit cells and the other has four unit cells. In each case, the transmissions of these designed unit cells are nearly unity and the phase differences between two adjacent cells are Δϕ = ϕj+1 − ϕj = 2π/m (m = 3 and 4). The Supplementary Figs. 3 and 5, and Supplementary Notes 3 and 4 show the physical dimensions of the final designs and design details. The fabricated samples of the two PGMs are shown in Fig. 4a, where one period of the PGM with m = 3 (m = 4) is highlighted by the red (blue) box (see the inset). The experimental setup is shown in Fig. 4b. For the designed PGM with m = 3, we numerically show the corresponding relationship between the transmission/reflection of the main diffraction orders (T1, T0 and R−1) and the incident angle in Fig. 4c, where the transmission/reflection agrees well with that in the ideal case of Fig. 3. In the experiments, to measure the angular performance of the designed PGMs, the Gaussian beam from the speaker array is incident from θin = −60° to θin = 60° with a step of 15°, and the measured results denoted by the stars are also displayed in Fig. 4c. While the measured result has some deviation in amplitude from the numerical result, the variation tendency of the curves agrees well with each other. Indeed, anomalous reflection and transmission can simultaneously exist in such a single PGM. To clearly show the reflection performance of the PGM with m = 3, the simulated scattered field, including reflected field and transmitted field of incident beam with θin = 30° (for other angles, see Supplementary Fig. 4) is shown in Fig. 4d, where a strong reflected wave towards the opposite direction with the incident wave is seen and the transmitted wave is much weaker. The experimentally measured scattered field shows the identical result (see Fig. 4e).
For the designed PGM with m = 4, the corresponding relationship between the transmission/reflection of the main diffraction orders (T1, T−1, and R0) and the incident angle is shown in Fig. 4f, with transmission/reflection agreeing well with prediction (Fig. 3). The corresponding measured result is displayed in Fig. 4f, where there are some small discrepancies between the numerical and measured results, but the overall trend in both cases is consistent. In addition, the field patterns of simulated and measured scattered waves for θin = 30° (for other angles, see Supplementary Fig. 6) are respectively shown in Fig. 4g, h, where both results reveal that strong transmitted waves appear and reflected waves are considerably weaker. Therefore, the parity design of the PGMs can effectively manipulate the switching of reflection and transmission of the higher order diffraction, enabling more flexibility in the design of acoustic planar devices.
### Robust feature of parity-dependent transmission and reflection
We would also like to point out that the phenomenon of anomalous reflection and refraction in the PGMs with parity design is very robust, depending only on the parity of the number m of unit cells. The reversal phenomenon could be observed in a PGM with parity design, even with large m, as long as the wave coupling between adjacent unit cells is negligible. To demonstrate this robust feature, we consider a specific case of a PGM with ξ > k0 as an example, and analyze the behavior at normal incidence. In this way, only the transmission and reflection of the n = 0 order41 need to be taken into consideration. After some mathematical derivations (see Supplementary Note 5), the corresponding transmission and reflection coefficients for the n = 0 order are respectively given as
$$\begin{array}{*{20}{l}} {r_0 = \frac{{\left| {\zeta _1} \right|^2 - \left| {\zeta _2} \right|^2}}{{\zeta _1^2 - \zeta _2^2}},} \hfill & {t_0 = \frac{{\zeta _1\zeta _2^ \ast - \zeta _1^ \ast \zeta _2}}{{\zeta _1^2 - \zeta _2^2}}} \hfill \end{array}{,}$$
(6)
where $$\zeta _1 = \left( {\tilde g - 1} \right)\mathop {\sum}\nolimits_{j = 1}^m {1/\left( {u_j^2 - 1} \right)}$$, $$\zeta _2 = \left( {1 - \tilde g} \right)\mathop {\sum}\nolimits_{j = 1}^m {u_j/\left( {u_j^2 - 1} \right)}$$, $$\tilde g_1 = 2g_1^2/\left( {g_1^2 - \gamma _1} \right)$$,$$g_1 = {\mathrm{sin}}c(Gw/2)$$ and $$u_j = \exp (i\phi _j)$$ is phase shift in the jth unit cell in u-complex plane. From Eq. (6), we know that the reflection and transmission are only determined by two factors: (i) the coefficient $$\tilde g_1$$, which is related to the geometry structure of a PGM and is a constant for a fixed configuration. (ii) the sums of $$Y_1 = \mathop {\sum}\nolimits_{j = 1}^m {1/\left( {u_j^2 - 1} \right)}$$ and $$Y_2 = \mathop {\sum}\nolimits_{j = 1}^m {u_j/(u_j^2 - 1)}$$, which are highly dependent on the phase distribution ϕj in u-complex plane. When m is odd, the phase distribution of ϕj is asymmetric (see Fig. 5a), which results in |Y1| = |Y2|. When m is even, the phase distribution of ϕj is symmetric (see Fig. 5b), and Y2 = 0. The detailed mathematical derivation is shown in Supplementary Note 5. With these results, Eq. (6) becomes
$$r_{0} = 0,\quad t_{0} = \exp(- i\varphi_{T}),\, m\,\,\mathrm{is}\,\,\mathrm{odd};$$
(7)
$$\begin{array}{lll} r_{0} = \exp( - i\varphi _R), & t_{0} = 0, & m\,\,\mathrm{is}\,\,\mathrm{even}; \hfill \end{array}$$
(8)
where $$\varphi _T = \arg(\zeta _1) + \arg(\zeta _2)$$ and $$\varphi _R = 2\arg(\zeta _1)$$, giving rise to a perfect transmission for odd m and a perfect reflection for even m. The results are consistent with these from the theoretical prediction summarized in Eqs. (4) and (5). Based on the generalized theoretical formulas in Eqs. (S11)–(S14), Fig. 5c, d displays the results of $$R = \mathop {\sum}\nolimits_n {\left| {r_n(m)} \right|}$$ and $$T = \mathop {\sum}\nolimits_n {\left| {t_n(m)} \right|}$$ with n = 0, respectively, which agree well with the approximate results of Eqs. (7) and (8). Therefore, it is analytically confirmed that even for larger m, the reversal phenomenon of almost perfect transmission and reflection is preserved, implying the parity-dependent feature is quite robust.
## Discussion
In conclusion, through a combination of analytical calculations and numerical simulations, we have revealed the governing diffraction mechanism of PGMs from the perspective of MRs. We find that the integer parity of the PGMs plays a pivotal role in the higher order diffraction for incident waves beyond the critical angle. To be more precise, the parity transition in the designed unit cells of a PGM enables the relevant odd/even transition of propagation number in the unit cells, which induces the reversal of transmission and reflection for a particular diffraction order. To demonstrate our findings, two acoustic PGMs (ξ = k0) with three and four unit cells are designed using zigzag microstructures, and the reversal phenomenon of higher order diffraction is clearly demonstrated in experiments. In particular, the coexistence of anomalous reflection and anomalous transmission, depending on a critical angle, is achieved in a single PGM with an odd number of unit cells. Compared with previous works in both acoustic waves and electromagnetic waves, the diffraction mechanism proposed here can comprehensively explain almost all the known diffraction behaviors in the metagratings with phase gradient. While our system is designed to work at a specific frequency, the parity-dependent behavior can be observed in a certain bandwidth as the phase gradient along the metagrating is preserved, leading to some tolerances in the frequency response (see Supplementary Fig. 7 and Supplementary Note 6). In addition, if a larger m-integer is designed for the lossy metagratings, the higher diffraction orders will undergo more round-trips, along with more absorption40. As a result, the parity-dependent scattering behavior of the higher diffraction orders will gradually disappear as “m” increases. We believe that our proposed diffraction mechanism can become a new paradigm for the design of acoustic/electromagnetic PGMs and open up new wave manipulation capabilities based on the versatile platform that can offer. For instance, due to achieved anomalous refraction and reflection, our work enables more systematic design of functional planar devices, such as asymmetric and wide-angle absorbers39,40, multifunctional metagratings42, omnidirectional reflector43,44,45. Alternatively, inspired by the phenomenon that an incident wave can be totally transmitted or reflected by a disordered slab46,47,48, one can design a metagrating of disorder with a properly designed combinations of integer m, which might enable some new effects associated with disorder-induced transition.
## Methods
### Numerical simulations
The full wave simulations are performed using COMSOL Multiphysics Pressure Acoustics module. In Fig. 2, the plane wave is incident on the PGM consisting of two supercells, the upper and lower walls are set as periodic boundary conditions and perfectly matched layers (PMLs) are used in the left and right sides to reduce the reflection. In Fig. 4, a spatially modulated Gaussian wave is incident on the designed PGM with 20 supercells, and the surrounding regions are PMLs. The normalized transmission and reflection efficiencies of the diffraction orders are numerically obtained from the port analysis of COMSOL RF module, where the acoustic profiles are replaced by their optical analogs.
### Experimental apparatus
The samples were fabricated with fused deposition modeling in three dimensional printing and the printed material is acrylonitrile butadiene styrene plastic with density of 1180 kg m−3 and speed of sound 2700 ms−1. As the characteristic impedance of the plastic is much larger than that of air, the walls can be considered as acoustically rigid. The fabricated PGM consists of ten supercells and is placed in a two-dimensional waveguide for the measurement. A loudspeaker array with 28 speakers emits a Gaussian modulated beam to the PGM and the reflected and transmitted field is scanned using a moving microphone with a step of 2.0 cm. The acoustic field at each spot is then calculated using Fourier Transform. The overall scanned area is 90 cm by 30 cm and the signal at each position is averaged out of four measurements to reduce noise. The transmission/reflection efficient is calculated by performing Fourier Transform along a line right behind/in front of the PGM.
## Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
## Code availability
The code used for the analyses will be made available upon e-mail request to the corresponding author.
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## Acknowledgements
This work was supported by the National Natural Science Foundation of China (grant Nos. 11604229, 11774252, and 11874311), the Natural Science Foundation of Jiangsu Province (grant Nos. BK20161210 and BK20171206), a Multidisciplinary University Research Initiative grant from the Office of Naval Research (N00014-13-1-0631), an Emerging Frontiers in Research and Innovation grant from the National Science Foundation (grant No. 1641084), the Project funded by China Postdoctoral Science Foundation (grant No. 2018T110540), Hong Kong Research Grants Council (AoE/P-02/12), and the Fundamental Research Funds for the Central Universities (grant No. 20720170015). Fu would like to thank the start-up fund support from Nanjing University of Aeronautics and Astronautics, Xu would like to thank the support from the Collaborative Innovation Center of Suzhou Nano Science and Technology at Soochow University, and Gao thanks the support from the Qing Lan project, “333” project (BRA2015353) and PAPD of Jiangsu Higher Education Institutions. We also thank the helpful discussions with Prof. Z.-Q. Zhang from Hong Kong University of Science and Technology.
## Author information
Authors
### Contributions
Y.X. and Y.F. conceived the idea. Y.F., C.S., Y.C., and Y.X. performed the theoretical calculation and numerical simulations. C.S. and S.A.C. fabricated the samples and performed experiments. L.G. and H.C. helped with the theoretical interpretation. Y.X., C.T.C., and S.A.C. supervised the project. All authors discussed the results and prepared the paper.
### Corresponding authors
Correspondence to C. T. Chan, Steven A. Cummer or Yadong Xu.
## Ethics declarations
### Competing Interests
The authors declare no competing interests.
Journal peer review information: Nature Communications thanks Fabrice Lemoult and other anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Fu, Y., Shen, C., Cao, Y. et al. Reversal of transmission and reflection based on acoustic metagratings with integer parity design. Nat Commun 10, 2326 (2019). https://doi.org/10.1038/s41467-019-10377-9
• Accepted:
• Published:
• DOI: https://doi.org/10.1038/s41467-019-10377-9
• ### Unidirectional acoustic metamaterials based on nonadiabatic holonomic quantum transformations
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• YongYuan Jiang
Science China Physics, Mechanics & Astronomy (2022)
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• Yijia Huang
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• Yang-Yang Fu
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Frontiers of Physics (2020)
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https://projecteuclid.org/euclid.afa/1499824814
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## Annals of Functional Analysis
### Scattered locally $C^{\ast}$-algebras
Maria Joiţa
#### Abstract
In this article, we introduce the notion of a scattered locally $C^{\ast}$-algebra and we give the conditions for a locally $C^{\ast}$-algebra to be scattered. Given an action $\alpha$ of a locally compact group $G$ on a scattered locally $C^{\ast}$-algebra $A[\tau_{\Gamma}]$, it is natural to ask under what conditions the crossed product $A[\tau_{\Gamma}]\times_{\alpha}G$ is also scattered. We obtain some results concerning this question.
#### Note
The current version of this article, posted on 9 August 2017, supersedes the original advance publication version posted on 12 July 2017. The author’s original terminology has been restored.
#### Article information
Source
Ann. Funct. Anal. (2018), 11 pages.
Dates
Accepted: 30 January 2017
First available in Project Euclid: 12 July 2017
https://projecteuclid.org/euclid.afa/1499824814
Digital Object Identifier
doi:10.1215/20088752-2017-0021
#### Citation
Joiţa, Maria. Scattered locally $C^{\ast}$ -algebras. Ann. Funct. Anal., advance publication, 12 July 2017. doi:10.1215/20088752-2017-0021. https://projecteuclid.org/euclid.afa/1499824814
#### References
• [1] B. Blackadar,Operator Algebras: Theory of$C^{\ast}$-Algebras and von Neumann Algebras, Encyclopaedia Math. Sci.122, Springer, Berlin, 2006.
• [2] C. H. Chu,Crossed products of scattered$C^{\ast}$-algebras, J. Lond. Math. Soc. (2)26(1982), no. 2, 317–324.
• [3] M. Fragoulopoulou,Topological Algebras with Involution, North-Holland Math. Stud.200, North-Holland, Amsterdam, 2005.
• [4] M. Haralampidou, “The Krull nature of locally $C^{\ast}$-algebras” inFunction Spaces (Edwardsville, Ill., 2002), Contemp. Math.328, Amer. Math. Soc., Providence, 2003, 195–200.
• [5] T. Huruya,A spectral characterization of a class of$C^{\ast}$-algebras, Sci. Rep. Niigata Univ. Ser. A15(1978), 21–24.
• [6] A. Inoue,Locally$C^{\ast}$-algebra, Mem. Fac. Sci. Kyushu Univ. Ser. A,25(1971), 197–235.
• [7] H. E. Jensen,Scattered$C^{\ast}$-algebras, Math. Scand.41(1977), no. 2, 308–314.
• [8] H. E. Jensen,Scattered$C^{\ast}$-algebras, II, Math. Scand.43(1978), no. 2, 308–310.
• [9] M. Joiţa,Crossed Products of Locally$C^{\ast}$-Algebras, Editura Academiei Române, Bucharest, 2007.
• [10] M. Kusuda,A characterization of scattered$C^{\ast}$-algebras and its application to$C^{\ast}$-crossed products, J. Operator Theory63(2010), no. 2, 417–424.
• [11] M. Kusuda, $C^{\ast}$-algebras in which every$C^{\ast}$-subalgebra is AF, Quart. J. Math.63(2012), no. 3, 675–680.
• [12] A. J. Lazar,On scattered$C^{\ast}$-algebras, in preparation.
• [13] A. Pelczynski and Z. Semadeni,Spaces of continuous functions, III: Spaces $C(\Omega)$ for $\Omega$ without perfect subsets, Studia Math.18(1959), 211–222.
• [14] N. C. Phillips,Inverse limits of$C^{\ast}$-algebras, J. Operator Theory19(1988), no. 1, 159–195.
• [15] M. L. Rothwell,Scattered$C^{\ast}$-algebras, in preparation.
• [16] W. Rudin,Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc.8(1957), 39–42.
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https://en.academic.ru/dic.nsf/enwiki/116960/9_%28number%29
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# 9 (number)
9 (number)
9 Cardinal 9 nine Ordinal 9th ninth Numeral system nonary Factorization 32 Divisors 1, 3, 9 Amharic ፱ Roman numeral IX Roman numeral (Unicode) Ⅸ, ⅸ prefixes ennea- (from Greek) nona- (from Latin) Binary 1001 Octal 11 Duodecimal 9 Hexadecimal 9 Arabic-Indic numeral ٩ Armenian numeral Թ Bengali ৯ Chinese/Japanese numeral 九 玖 (formal writing) Devanāgarī ९ (Nao) Greek numeral θ´ Hebrew numeral ט (Tet) Tamil numeral ௯ Khmer ៩ Telugu numeral ౯ Thai numeral ๙
9 (nine /ˈnn/) is the natural number following 8 and preceding 10. The ordinal adjective is ninth.
## Companies
• Nine Lives cat food; its name is derived from the legend that a cat has nine lives
• Nine Network a.k.a. Channel 9, an Australian free-to-air television station
• Nine West, a clothing brand [1]
## Culture and mythology
### Chinese culture
• Nine is strongly associated with the Chinese dragon, a symbol of magic and power. There are nine forms of the dragon, it is described in terms of nine attributes, and it has nine children. It has 117 scales - 81 yang (masculine, heavenly) and 36 yin (feminine, earthly). All three numbers are multiples of 9 (9x13=117, 9x9=81, 9x4=36)[2] as well adding up individually to 9 (1+1+7=9, 8+1=9, 3+6=9).
• The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City.
• The circular altar platform (Earthly Mount) of the Temple of Heaven has one circular marble plate in the center, surrounded by a ring of nine plates, then by a ring of 18 plates, and so on, for a total of nine rings, with the outermost having 81=9×9 plates.
• The nine-rank system was a civil service nomination system used during certain Chinese dynasties.
### Ancient Egypt
• The nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt
### European culture
• The Nine Worthies are nine historical, or semi-legendary figures who, in the Middle Ages, were believed to personify the ideals of chivalry
### Greek Mythology
• The nine muses in Greek mythology are Calliope (epic poetry), Clio (history), Erato (erotic poetry), Euterpe (lyric poetry), Melpomene (tragedy), Polyhymnia (song), Terpsichore (dance), Thalia (comedy), and Urania (astronomy).
### Japanese culture
• The Japanese consider 9 to be unlucky because it sounds similar to the Japanese word for "pain" or "distress" (苦 kunrei ku)[citation needed].
• The character Cirno from the Touhou series is often called "nine," "⑨," "circle-nine," or "nineball," because in the game manual for "Phantasmagoria of Flower View," she was labeled (9) Idiot (⑨ バカ ⑨ baka?).[3]
## Evolution of the glyph
According to Georges Ifrah, the origin of the 9 integers can be attributed to the ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the 0.[4]
In the beginning, various Indians wrote 9 similar to the modern closing question mark without the bottom dot. The Kshtrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the @ character encircles a lowercase a. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller. Soon, all that was left of the 3-look-alike was a squiggle. The Arabs simply connected that squiggle to the downward stroke at the middle and subsequent European change was purely cosmetic.
While the shape of the 9 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .
This numeral resembles an inverted 6 evolved from the letter "8". To disambiguate the two on objects and documents that can be inverted, the 9 is often underlined, as is done for the 6. Another distinction from the 6 is that it is often handwritten with a straight stem.
## Games
• 9: The Last Resort was a 1995 computer game
• In the game of craps, 9 is known as the center field because it is in the middle of the seven numbers on the field bet
• Magic: The Gathering has a set of nine rare cards, widely regarded as overpowered, known as the Power Nine
• Nine Mens Morris (Nine Men’s Morris) is a European board game known since Roman times.
## Idioms and popular phrases
• "A cat-o'-nine-tails suggests perfect punishment and atonement." --Robert Ripley.
• The word "K-9" pronounces the same as canine and is used in many U.S. police departments to denote the police dog unit.
• Someone dressed "to the nines" is dressed up as much as they can be.
• In urban culture, "nine" is a slang word for a 9mm pistol or homicide, the latter from the Illinois Criminal Code for homicide.
## Internet
• The 9 on Yahoo!, hosted by Maria Sansone, was a daily video compilation show, or vlog, on Yahoo! featuring the nine top "web finds" of the day.
## Literature
• There are nine circles of Hell in Dante's Divine Comedy.
• The Nine Bright Shiners, characters in Garth Nix's Old Kingdom trilogy. The Nine Bright Shiners was a 1930s book of poems by Anne Ridler[5] and a 1988 fiction book by Anthea Fraser;[6] the name derives from "a very curious old semi-pagan, semi-Christian" song.[7]
• The Nine Tailors is a 1934 mystery novel by British writer Dorothy L. Sayers, her ninth featuring sleuth Lord Peter Wimsey
• Nine Unknown Men are, in occult legend, the custodians of the sciences of the world since ancient times
• In J.R.R. Tolkien's Middle-earth, there are nine rings of power given to men, and consequently, nine ringwraiths
## Mathematics
Nine is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and hence the third square number. Nine is a Motzkin number. It is the first composite lucky number, along with the first composite odd number.
Nine is the highest single-digit number in the decimal system. It is the second non-unitary square prime of the form (p2) and the first that is odd. All subsequent squares of this form are odd. It has a unique aliquot sum 4 which is itself a square prime. Nine is; and can be, the only square prime with an aliquot sum of the same form. The aliquot sequence of nine has 5 members (9,4,3,1,0) this number being the second composite member of the 3-aliquot tree. It is the aliquot sum of only one number the discrete semiprime 15.
There are nine Heegner numbers.[8]
Since 9 = 321, 9 is an exponential factorial.
8 and 9 form a Ruth-Aaron pair under the second definition that counts repeated prime factors as often as they occur.
In bases 12, 18 and 24, nine is a 1-automorphic number and in base 6 a 2-automorphic number (displayed as '13').
A polygon with nine sides is called a nonagon or enneagon.[9] A group of nine of anything is called an ennead.
In base 10 a number is evenly divisible by nine if and only if its digital root is 9.[10] That is, if you multiply nine by any natural number, and repeatedly add the digits of the answer until it is just one digit, you will end up with nine:
• 2 × 9 = 18 (1 + 8 = 9)
• 3 × 9 = 27 (2 + 7 = 9)
• 9 × 9 = 81 (8 + 1 = 9)
• 121 × 9 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9)
• 234 × 9 = 2106 (2 + 1 + 0 + 6 = 9)
• 578329 × 9 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = 27; 2 + 7 = 9)
• 482729235601 × 9 = 4344563120409 (4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45; 4 + 5 = 9)
There are other interesting patterns involving multiples of nine:
• 12345679 x 9 = 111111111
• 12345679 x 18 = 222222222
• 12345679 x 81 = 999999999
This works for all the multiples of 9. n = 3 is the only other n > 1 such that a number is divisible by n if and only if its digital root is n. In base N, the divisors of N − 1 have this property. Another consequence of 9 being 10 − 1, is that it is also a Kaprekar number.
The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Examples:
• The sum of the digits of 41 is 5, and 41-5 = 36. The digital root of 36 is 3+6 = 9, which, as explained above, demonstrates that it is evenly divisible by nine.
• The sum of the digits of 35967930 is 3+5+9+6+7+9+3+0 = 42, and 35967930-42 = 35967888. The digital root of 35967888 is 3+5+9+6+7+8+8+8 = 54, 5+4 = 9.
Subtracting two base-10 positive integers that are transpositions of each other yields a number that is a whole multiple of nine. Examples:
• 41 - 14 = 27 (2 + 7 = 9)
• 36957930 - 35967930 = 990000, a multiple of nine.
This works regardless of the number of digits that are transposed. For example, the largest transposition of 35967930 is 99765330 (all digits in descending order) and its smallest transposition is 03356799 (all digits in ascending order); subtracting pairs of these numbers produces:
• 99765330 - 35967930 = 63797400; 6+3+7+9+7+4+0+0 = 36; 3+6 = 9.
• 99765330 - 03356799 = 96408531; 9+6+4+0+8+5+3+1 = 36; 3+6 = 9.
• 35967930 - 03356799 = 32611131; 3+2+6+1+1+1+3+1 = 18; 1+8 = 9.
Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers, known as long ago as the 12th Century.[11]
Every prime in a Cunningham chain of the first kind with a length of 4 or greater is congruent to 9 mod 10 (the only exception being the chain 2, 5, 11, 23, 47).
Six recurring nines appear in the decimal places 762 through 767 of pi. This is known as the Feynman point.
If an odd perfect number is of the form 36k + 9, it has at least nine distinct prime factors.[12]
Nine is the binary complement of number six:
9 = 1001
6 = 0110
### List of basic calculations
Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
$9 \times x$ 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225 450 900 9000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
$9 \div x$ 9 4.5 3 2.25 1.6 1.5 $1.\overline{285714}$ 1.125 1 0.9 $0.\overline{81}$ 0.75 $0.\overline{692307}$ $0.6\overline{428571}$ 0.6
$x \div 9$ $0.\overline{1}$ $0.\overline{2}$ $0.\overline{3}$ $0.\overline{4}$ $0.\overline{5}$ $0.\overline{6}$ $0.\overline{7}$ $0.\overline{8}$ 1 $1.\overline{1}$ $1.\overline{2}$ $1.\overline{3}$ $1.\overline{4}$ $1.\overline{5}$ $1.\overline{6}$
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
$9 ^ x\,$ 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 31381059609 282429536481 2541865828329
$x ^ 9\,$ 1 512 19683 262144 1953125 10077696 40353607 134217728 387420489 1000000000 2357947691 5159780352 10604499373
Radix 1 5 10 15 20 25 30 40 50 60 70 80 90 100
110 120 130 140 150 200 250 500 1000 10000 100000 1000000
$x_{9} \$ 1 5 $11_{9} \$ $16_{9} \$ $22_{9} \$ $27_{9} \$ $33_{9} \$ $44_{9} \$ $55_{9} \$ $66_{9} \$ $77_{9} \$ $88_{9} \$ $110_{9} \$ $121_{9} \$
$132_{9} \$ $143_{9} \$ $154_{9} \$ $165_{9} \$ $176_{9} \$ $242_{9} \$ $307_{9} \$ $615_{9} \$ $1331_{9} \$ $14641_{9} \$ $162151_{9} \$ $1783661_{9} \$
### Numeral systems
Base Numeral system
2 binary 1001
3 ternary 100
4 quaternary 21
5 quinary 14
6 senary 13
7 septenary 12
8 octal 11
9 novenary 10
### Probability
In probability, the nine is a logarithmic measure of probability of an event, defined as the negative of the base-10 logarithm of the probability of the event's complement. For example, an event that is 99% likely to occur has an unlikelihood of 1% or 0.01, which amounts to −log10 0.01 = 2 nines of probability. Zero probability gives zero nines (−log10 1 = 0). A 100% probability is considered to be impossible in most circumstances: that results in infinite improbability. The effectivity of processes and the availability of systems can be expressed (as a rule of thumb, not explictly) as a series of "nines". For example, "five nines" (99.999%) availability implies a total downtime of no more than five minutes per year - typically a very high degree of reliability; but never 100%.
## Organizations
• Divine Nine—The National Pan-Hellenic Council (NPHC) is a collaborative organization of nine historically African American, international Greek lettered fraternities and sororities.
## Religion and philosophy
• Nine, as the highest single-digit number (in base ten), symbolizes completeness in the Bahá'í Faith. In addition, the word Bahá' in the Abjad notation has a value of 9, and a 9-pointed star is used to symbolize the religion.
• The number 9 is revered in Hinduism and considered a complete, perfected and divine number because it represents the end of a cycle in the decimal system, which originated from the Indian subcontinent as early as 3000 BC.
• Important Buddhist rituals usually involve nine monks.
• The first nine days of the Hebrew month of Av are collectively known as "The Nine Days" (Tisha HaYamim), and are a period of semi-mourning leading up to Tisha B'Av, the ninth day of Av on which both Temples in Jerusalem were destroyed.
• Nine is a significant number in Norse Mythology. Odin hung himself on an ash tree for nine days to learn the runes.
• The Fourth Way Enneagram is one system of knowledge which shows the correspondence between the 9 integers and the circle.
• In the Christian angelic hierarchy there are 9 choirs of angels.
• Ramadan, the month of fasting and prayer, is the ninth month of the Islamic calendar.
## Science
### Physiology
A human pregnancy normally lasts nine months, the basis of the Naegele's rule.
## Sports
A Nine-ball rack with the 9 ball at the center
### Auto racing
• A car in the Sprint Cup Series currently owned by Richard Petty Motorsports. The number was most notably borne by the car that Bill Elliott drove to the Cup Series title in 1988 with Melling Racing. Evernham Motorsports, the predecessor team to Richard Petty Motorsports, acquired the number in 2001 when Elliott joined that team after a brief stint as a driver-owner. Elliott used this number again through the 2003 season. Kasey Kahne has driven the 9 car since 2004.
### Billiards
• Nine-ball is the standard professional pocket billiards variant played in the United States.
### Rugby
• In rugby league, the number generally worn by the hooker.
• In rugby union, the number generally worn by the scrum-half.
### Soccer
• In association football (soccer) the centre-forward/striker traditionally (since at least the fifties) wears the number 9 shirt.
### All sports
The jersey number 9 has been retired by several North American sports teams in honor of past playing greats (or in one case, an owner):
## Technology
• ISO 9 is the ISO's standard for the transliteration of Cyrillic characters into Latin characters
• In the Rich Text Format specification, 9 is the language code for the English language. All codes for regional variants of English are congruent to 9 mod 256.
• The seven-segment display allows the number 9 to be constructed two ways, either with a hook at the end of its stem or without one. Most LCD calculators use the former, but some VFD models use the latter.
• The9 Limited (owner of the9.com) is a company in the video-game game industry, including former ties to the extremely popular MMORPG World of Warcraft
## Other fields
International maritime signal flag for 9
Playing cards showing the 9 of all four suits
• Nine justices sit on the United States Supreme Court.
• Stanines, a method of scaling test scores, range from 1 to 9.
## References
2. ^ Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN 1417964294.
3. ^ Cirno - TouhouWiki http://en.touhouwiki.net/wiki/Cirno#Other_Information
4. ^ Georges Ifrah (1985). From One to Zero: A Universal History of Numbers. Viking. ISBN 0-670-37395-8.
5. ^ Jane Dowson (1996). Women's Poetry of the 1930s: A Critical Anthology. Routledge. ISBN 0415130956.
6. ^ Anthea Fraser (1988). The Nine Bright Shiners. Doubleday. ISBN 0385243235.
7. ^ Charles Herbert Malden (1905). Recollections of an Eton Colleger, 1898-1902. Spottiswoode.
8. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
9. ^ Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
10. ^ Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155
11. ^ Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
12. ^ Eyob Delele Yirdaw, "Proving Touchard's Theorem from Euler's Form" ArXiv preprint.
Wikimedia Foundation. 2010.
### Look at other dictionaries:
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2020-08-14 18:02:29
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https://bt.gateoverflow.in/788/gate-bt-2022-question-42
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Emerging viruses such as $\text{SARS-Co V2}$ cause epidemics. Which of the following process(es) contribute to the rise of such viruses?
1. Mutation of existing virus
2. Jumping of existing virus from current to new hosts
3. Spread of virus in the new host population
4. Replication of virus outside a host
1
2
3
4
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2022-06-25 14:30:02
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http://openstudy.com/updates/507ff59de4b0b8b0cacd6c1f
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## lgbasallote 2 years ago Achilles and a tortoise were racing on a 1 km track. Achilles runs on a constant 8 m/s speed and the tortoise's speed is 0.8 m/s. Halfway through the track, Achilles decided to take a nap since the tortoise was so far behind. However, when he woke up, the tortoise was already near the finish line. Achilles started to run again with constant speed of 8 m/s, but he lost to the tortoise by 3 seconds. How long did Achilles sleep? Note: Let the tortoise and Achilles' speed be constant
1. Yahoo!
T(torto) = 625 + 625 = 1250s to Cover 1000m T(achilles) = 62.5 + 62.5 = 125s to cover 1000m Since the Achilles reached 3 Seconds Later .it lost by 3 seconds means 125 + x = 1253 x = 1128s
2. lgbasallote
i don't think that's right.....
3. lgbasallote
what's that 625 on the first line by the way?
4. Yahoo!
$S=\frac{ d }{ t }$
5. lgbasallote
i doubt this is that simple
6. sauravshakya
8(1253-x)=1000 x=1128 seconds Looks correct to me.
7. lgbasallote
1253 - x?
8. lgbasallote
you do know i asked how long Achilles slept and not how long Achilles was running...right?
9. sauravshakya
x is the time for which Achilles sleeps...... So, 1253-x gives how long he ran.
10. lgbasallote
hmm then that's not right
11. sauravshakya
WHY???
12. lgbasallote
like i said...it's not simple
13. lgbasallote
the question is...what's the right solution...
14. Yahoo!
@lgbasallote which part Do u think...i made a Mistake...
15. lgbasallote
16. lgbasallote
because you think you're right?
17. Yahoo!
Toto took 1250s To cover 1000m.... right.?
18. lgbasallote
no need to explain to me your solution..i get it
19. lgbasallote
..it's the book that says it's wrong..not me
20. myininaya
Did he shout? You guys play nice please. People can disagree with answers. I don't find it rude to do so. I'm not saying anyone's answer is right here. I'm just saying people have a right to disagree but please be respectful about it.
21. myininaya
I will be deleting anything I find nasty here. Thanks.
22. myininaya
Keep in mind. Discussion is an awesome tool in learning. :)
23. shubhamsrg
* :P
24. shubhamsrg
though no one asked,,but i guess yahoo is right..
25. lgbasallote
i suppose one needs to get the speeds first...
26. ganeshie8
me too i dont see a mistake in the solution
27. shubhamsrg
whats the ans in the book ?
28. shubhamsrg
and speeds are already given right ?
29. lgbasallote
ahh i think i get what the question is asking
30. alexyn
I got 1128. using a much more complex method.
31. lgbasallote
hmm maybe I can get an idea from that complex method
32. lgbasallote
33. alexyn
Alright. Using your head. visualize the problem. A 1000 meter track. Tortoise and Achilles begin running. Achilles pulls ahead and half way takes a nap. Thats after 62.5 seconds. He's already covered 500 m. At that same time Achilles sleeps, Tortoise has reached 50 m. Now using that knowledge, we know it will take the Tortoise 1187.5 s to finish. It'll take Achilles 62.5 s. So subtract Achilles time from Tortoise's. Which gives you. 1,187.5 s - 62.5 = 1,125 s. Add three seconds. = 1,128 s
34. lgbasallote
35. alexyn
Yep. It makes complete sense. He napped for 1,128 seconds. in order to finish 3 seconds after tortoise.
36. lgbasallote
what if the 3 seconds doesn't mean that Achilles finished 3 seconds later?
37. alexyn
Oh it does. Because at 1,125 s. Achilles finished right with tortoise. But adding 3 seconds means he finished 3 seconds later.
38. lgbasallote
i meant 3 seconds can mean that Achilles' running time is 3 more than the tortoise's
39. alexyn
No. It doesn't mean that here.
40. myininaya
@lgbasallote I agree with Yahoo's answer also. Like this is the way I thought of it: |dw:1350569105212:dw| Since like everyone is saying A finished 3 seconds after T. So that means A finished in 1253 seconds.
41. myininaya
125/2+nap+125/2 =1253
42. myininaya
125+nap=1253
43. lgbasallote
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2015-07-06 20:26:20
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https://physics.stackexchange.com/questions/83587/why-are-orthogonal-functions-and-eigenvalues-functions-so-important-in-quantum-m
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# Why are orthogonal functions and eigenvalues/functions so important in quantum mechanics?
The mathematics and physics we have studied so far at university are heavily focused around the idea of orthogonal functions, orthogonality, sets of solutions, eigenvalues and eigenfunctions.
Why are we so interested in these properties? What are the conceptual aspects of them, mainly in quantum mechanics?
• This is a great question but I think you should confine it to one area -- ie. QM. It's far too broad to include their use in other areas of physics because almost every area has some use for them. – tpg2114 Nov 6 '13 at 1:26
• The short answer (at least as far as I know) is that if you have an orthogonal basis it's very easy to find the components of any vector: just take inner product with the basis elements. This is pretty much what everything Fourier is about. – Javier Nov 6 '13 at 1:39
• The two words that cover @JavierBadia's nice comment completeness and uniqueness: for any given basis there is one and only one decompositions every member of the solution space. – dmckee --- ex-moderator kitten Nov 6 '13 at 2:03
• To expand on the comments here, it is important that we have complete orthonormal basis sets from a practical standpoint because we can often use them in solving the differential equations that show up in mathematical physics. Other than the fact that the functions themselves are beautiful from a functional analysis standpoint, they are very useful in constructing solutions in different orthogonal curvilinear coordinate systems. – codeAndStuff Nov 6 '13 at 13:53
• The eigenfunction of a Hermitian operator is orthogonal to each other. – user26143 Nov 6 '13 at 14:28
2. More generally, it is the class of normal operators (and an important special case self adjoint operators) which the spectral theorem most readily works and is most complete for. The eigenvectors of such operators are always orthogonal. The "Diagonalising" an operator in any linear system theory is an important step for understanding - it means we can decouple the operator's action into the sum of its action on altogether uncoupled eigenvectors. It's an important step in "untangling" a highly coupled problem. In the context of when the Hilbert space concerned is a function space, the relevant Sturm-Liouville theory, e.g. for the quantum harmonic oscillator shows that the linear space of all "practical", normalisable quantum states is spanned by discrete eigenfunctions. In other words, the Hilbert space's dimension is countably infinite, even though we are dealing with spaces of continuous functions and you might intuitively think the dimension cardinality might be $\aleph_1$, and that's just too scary to deal with!
3. We deal often with two important conservation laws: conservation of energy and conservation of probability. These conservation laws are most readily expressed if the basis for the relevant state space is orthogonal - it means that energy, power or probability as appropriate is simply the $\mathbf{L}^2$ length of any vector. We don't have to manage cross coupling terms in our inner product space. Whether it be functions or Cartesian bases for three dimensional Eucliean space, projections and resolution into basis superpositions are always heaps easier and clearer if the basis is orthogonal. You'd be a sucker for punishment if you did an everyday geometric problem in $\mathbb{R}^3$ with a general, linearly independent but nonorthogonal basis, even though this can certainly be done. Exactly the same intellectual work minimalisation principles apply to functions spaces as much as they do to $\mathbb{R}^3$. Energy- or probability-conservative system transformations are then unitary and so on and so forth.
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2021-07-30 18:12:00
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https://tex.stackexchange.com/questions/208740/how-can-i-fix-a-sty-not-found-error-in-lyx-when-it-works-in-pdflatex
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# How can I fix a “*.sty not found” error in Lyx, when it works in pdflatex?
I've just installed a gantt chart package, which I'm trying to use it in LyX.
I attempted to use the first minimal example, but this failed in LyX with LaTeX Error: File 'forloop.sty' not found. However, compiling the pdf from raw LaTeX works fine. I attempted both creating the example file from scratch, and exporting the LaTeX (plain) from LyX, and attempting a manual pdflatex example.tex. This worked in both cases.
Back in LyX, I tried following the instructions on the wiki. Specifically,
$kpsewhich forloop.sty ./forloop.sty$ sudo texhash
texhash: Updating /etc/texmf/ls-R...
texhash: Updating /usr/share/texmf/ls-R...
texhash: Updating /usr/share/texmf-dist/ls-R...
texhash: Updating /var/lib/texmf/ls-R...
texhash: Done.
Then in LyX, Tools→Reconfigure and restart. It still failed with the same error. How can I fix this problem in LyX?
I'm using LyX 2.1.1 and on Arch Linux. pdflatex details are below.
\$ pdflatex --version
pdfTeX 3.14159265-2.6-1.40.15 (TeX Live 2014/Arch Linux)
kpathsea version 6.2.0
...
Compiled with libpng 1.6.13; using libpng 1.6.13
Compiled with zlib 1.2.8; using zlib 1.2.8
Compiled with poppler version 0.26.5
## Details of the LyX file
I created a new LyX file, and put the following into the header.
\usepackage{tikz}
\usepackage{gantt}
I then created ERT and copy pasted from the first minimal example here, from \begin{gantt}{10}{12} to \end{gantt}.
My best guess is that you did not install the forloop.sty file correctly, but just save it to the folder, where you have your project. If this is the case, it will work with pdflatex, as pdflatex will install search you current directory for .sty files, but LyX will not. To install a package you have to choices. You can use the texlive install utitlity, if the package is aviliable in the texlive mirror.
tlmgr install <package1> <package2>
If not you need to create a new texmf folder (typically in ~/texmf/) and then make a subfolder in ~/texmf/tex/latex/forloop/ and save your forloop.sty file to there. Then run
texhash ~/texmf
For TeX to recognize your package
• Sorry, I was halfway typing a response to your comment. locate forloop.sty finds nothing at all on my system. Oddly enough, pdf creation still works with from the command line though. – Sparhawk Oct 24 '14 at 12:27
• Locate only works if you manually updated the system file database (nothing to do with latex, update using sudo updatedb). For latex files use kpsewhich file.sty – daleif Oct 24 '14 at 14:56
• @daleif Yes, sorry, I did sudo updatedb first. However, I've just realised that half my problem was I was running the cli pdflatex from /tmp, which did contain forloop.sty. It's all working now (although I'm still clueless as to why LyX didn't work even after I'd applied the fixes as per my question). – Sparhawk Oct 25 '14 at 2:04
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2019-11-18 13:41:48
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https://www.physicsforums.com/threads/why-am-i-getting-this-relativity-velocity-addition-problem-wrong.660982/
|
# Why am I getting this relativity velocity addition problem wrong?
1. Dec 26, 2012
### DunWorry
1. The problem statement, all variables and given/known data
A spacecraft S2 is capable of firing a missile which can travel 0.98c. S2 is escaping from S1 at a speed of 0.95c when it fires a missile towards S1.
part A) According to the pilot of S2, what speed does the missile approach S1?
Part B) according to pilot of S1, what speed does the missile approach it?
2. Relevant equations
Call the S1 frame x and S2 frame y and speed of missile U
Velocity addition V$_{x}$ = $\frac{v_{y} + U}{1 + \frac{v_{y} U}{C^{2}}}$
3. The attempt at a solution
My problem lies with part A. The answer is just a simple 0.98c - 0.95c = 0.03c. However I cant get this result with the velocity addition formula, why is it in this case the velocity addition formula does not work/ does not apply?
I tried imagining S2 moving to right (positive) and firing the missile backwards towards S1 (left direction which is negative). Taking the frame of reference of S2, the spaceship S2 is stationary and S1 is moving to the left at a velocity of -0.95c, the missile is also moving to left with speed -0.98c
if I try use the velocity addition formula Velocity addition V$_{x}$ = $\frac{-0.98 - 0.95}{1 + \frac{0.98 x 0.95}{C^{2}}}$ I get -0.9994C, which is wrong. The answer is just 0.98c - 0.95C but I cannot see what I am doing wrong with the velocity addition formula or why it is not needed in this case.
I solved part B) using the formula V$_{x}$ = $\frac{0.98 - 0.95}{1 - \frac{0.98 x 0.95}{C^{2}}}$. The signs are as they are as in the frame of S1, the ship S2 is moving in + direction with speed 0.98C and the missile is moving with -0.95C. It seems to work for part B but not for part A and I cannot see why.
Thanks
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. Dec 26, 2012
### TSny
For part (A) you don't need to use the velocity addition formula. You already know how fast the missile travels relative to S2 and also how fast S1 moves relative to S2. You just want to know how fast the missile is "closing" on S1 as measured by S2.
It's the same as asking if S2 rolls a ball at 5 m/s along her x-axis and then rolls a ball at 7 m/s along her x-axis, how fast is the second ball closing on the first ball according to S2? No relativity needed since all measurements are in one inertial frame. You are not switching frames of reference.
3. Dec 26, 2012
### Janus
Staff Emeritus
You use the relativistic formula when you are working between two frames.
For instance, in part B) you add adding two velocities from S2's frame but want an answer for S1 frame.
When you are working just in one frame, it is is not used. For example in part A) you are adding two velocities according to S2 and want an answer for that frame.
Think of it is this way. in part (A, you have a missile traveling away from S2 at .98c
After 1 sec, the missile will be 0.98 light secs further away.
S1 is traveling away a 0.95c, so after 1 sec, it will be 0.98 light sec away.
This means that, according to S2, after 1 sec the missile will be 0.03 light sec closer to each other. which works out to a difference of 0.03 c between S1 and the missile according to S2.
4. Dec 26, 2012
### DunWorry
Ahhh I see thats much clearer now. So its basically because for part A the measurements are given from the frame of S2 and you are working in the same frame of reference because you want an answer for S2 so it is not needed. However in part B you are trying to take the position from S1 and so you are using measurements which were given from the frame in S2 and therefore need to use the velocity addition because you are switching frames of reference
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2018-03-23 02:10:47
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https://sharepoint.stackexchange.com/questions/178489/error-while-adding-user-to-sharepoint-group-using-csom
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# Error while adding user to SharePoint group using CSOM
I'm facing error while trying to add users into the SharePoint groups using CSOM. When I try to add user by passing user name as "domain\V*****" (all users start with "v" character), it through error as "\v" is hexadecimal value which can't pass.
Also I tried again by adding escape character like @domain\v******, but no luck..
But I tried the code with some test users like "domain\user1" which is working fine.
Any help would be appreciated..
• Are you trying to add all users under the domain? You can't pass wild card characters – Amal Hashim May 2 '16 at 14:55
• @AmalHashim No, I'm trying to add single user only.. All username starts with V***** like ID.. – Arun May 2 '16 at 15:34
• Can you post code you tried? – Amal Hashim May 2 '16 at 15:35
• should be @"domain\victor" or "domain\\victor" with two slashes – Mike May 2 '16 at 20:37
I use the code bellow, but like Mike said maybe the key is to provide the string like "domain\user":
UserCreationInformation uci = new UserCreationInformation();
uci.Title = "John Lennon";
uci.Email = "jlennon@thebeatles.com";
Web web = ctx.Web;
GroupCollection groupcoll = web.SiteGroups;
string grouptitle = "My Group";
ctx.Load(groupcoll, grps => grps.Include(g => g.Title).Where(g => g.Title == grouptitle));
ctx.ExecuteQuery();
Group group = null;
User member = web.EnsureUser(uci.Email);
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2019-10-16 16:41:33
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https://math.stackexchange.com/questions/3343004/represent-log-3528-by-log-147-and-log-145
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# represent $\log_{35}(28)$ by $\log_{14}(7)$ and $\log_{14}(5)$
I'm trying to figure out how to express $$\log_{35}(28)$$ with $$a:=\log_{14}(7)$$ and $$b:=\log_{14}(5)$$ (the hint convert the base to 14 was given).
So, $$\log_{35}(28) = \dfrac{\log_{14}(28)}{\log_{14}(35)}$$.
I already figured out the denominator is $$a+b = \log_{14}(7)+\log_{14}(5) = \log_{14}(5\cdot 7) = \log_{14}(35) \Longrightarrow \log_{35}(28) = \dfrac{\log_{14}(28)}{a+b}$$.
But I can't figure out the numerator. My guess is that $$7\cdot 5 -\textbf{7}=28$$ but there's no rule by which I can perform a subtraction in the argument of the log.
My other guess would be to find something like $$x\log_{14}(7)+y\log_{14}(5) =\log_{14}(7^x\cdot 5^y)$$ or $$x\log_{14}(7)-y\log_{14}(5) =\log_{14}\left(\dfrac{7^x}{5^y}\right)$$ so that $$7^x\cdot 5^y=28$$ or $$\dfrac{7^x}{5^y}=28$$.
However, I believe that there must be an easier way.
• $\log_{14}28=2-\log_{14}7$ – J. W. Tanner Sep 3 at 12:53
You are correct that $$\log_{14}28$$ cannot be simplified
by expanding $$\log_{14}(7\cdot5-7)\ne\log_{14}(7\cdot5)-\log_{14}5$$.
Instead, I would suggest using $$\log_{14}28=2-\log_{14}7$$.
To see that, note that $$\log_{14}(28\cdot7)=\log_{14}(14\cdot14)=2$$.
$$a(\log5+\log7)=2\log2+\log7=2\log2+a\log5+(1-a)\log7=0$$
$$b(\log2+\log7)=\log5\iff b\log2-\log5+b\log7=0$$
Let $$c=\log_{14}5,$$ $$\implies c\log2-\log5+c\log7=0$$
$$\implies\det\begin{pmatrix} 2 & a & 1-a \\ b & -1 & b \\ c & -1 &c\end{pmatrix}=0$$
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2019-12-06 21:40:11
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http://annals.math.princeton.edu/articles/PMSC2000/14j28
|
# Articles with primary mathematical subject classification: 14J28
## Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic
Pages 269-313 by Igor Dolgachev, JongHae Keum | From volume 169-1
## Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for $K3$ surfaces, and the Tate conjecture
Pages 487-526 by François Charles | From volume 184-2
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2019-09-16 02:04:03
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https://www.researchgate.net/journal/Physical-Review-Applied-2331-7019
|
Physical Review Applied
Online ISSN: 2331-7019
Publications
Article
We report the existence of confined electronic states at the (110) and (111) surfaces of SrTiO3. Using angle-resolved photoemission spectroscopy, we find that the corresponding Fermi surfaces, subband masses, and orbital ordering are different from the ones at the (001) surface of SrTiO3. This occurs because the crystallographic symmetries of the surface and sub-surface planes, and the electron effective masses along the confinement direction, influence the symmetry of the electronic structure and the orbital ordering of the t2g manifold. Remarkably, our analysis of the data also reveals that the carrier concentration and thickness are similar for all three surface orientations, despite their different polarities. The orientational tuning of the microscopic properties of two-dimensional electron states at the surface of SrTiO3 echoes the tailoring of macroscopic (e.g. transport) properties reported recently in LaAlO3/SrTiO3 (110) and (111) interfaces, and is promising for searching new types of 2D electronic states in correlated-electron oxides.
Article
Modern electronic devices are unthinkable without the well-controlled formation of interfaces at heterostructures. These often involve at least one amorphous material. Modeling such interfaces poses a significant challenge, since a meaningful result can only be expected by using huge models or by drawing from many statistically independent samples. Here we report on the results of high throughput calculations for interfaces between crystalline silicon (c-Si) and amorphous silicon nitride (a-Si$_3$N$_{3.5}$:H), which are omnipresent in commercially available solar cells. The findings reconcile only partly understood key features. At the interface, threefold coordinated Si atoms are present. These are caused by the structural mismatch between the amorphous and crystalline part. The local Fermi level of undoped c-Si lies well below that of a-SiN:H. To align the Fermi levels in the device, charge is transferred from the a-SiN:H part to the c-Si part resulting in an abundance of positively charged, threefold coordinated Si atoms at the interface. This explains the existence of a positive, fixed charge at the interface that repels holes.
Article
We find that the motion of the valley electrons -- electronic states close to the ${\rm K}$ and ${\rm K'}$ points of the Brillouin zone -- is confined into two dimension when the layers of MoS$_{2}$ follow the 3R stacking, while in the 2H polytype the bands have dispersion in all the three dimensions. According to our first-principles band structure calculations, the valley states have no interlayer hopping in 3R-MoS$_{2}$, which is proved to be the consequence of the rotational symmetry of the Bloch functions. By measuring the reflectivity spectra and analyzing an anisotropic hydrogen atomic model, we confirm that the valley excitons in 3R-MoS$_{2}$ have two-dimensional hydrogen-like spectral series, and the spreads of the wave function are smaller than the interlayer distance. In contrast, the valley excitons in 2H-MoS$_{2}$ are well described by the three-dimensional model and thus not confined in a single layer. Our results indicate that the dimensionality of the valley degree of freedom can be controlled simply by the stacking geometry, which can be utilized in future valleytronics.
Article
We fabricated YBa$_2$Cu$_3$O$_7$ (YBCO) direct current (dc) nano superconducting quantum interference devices (nanoSQUIDs) based on grain boundary Josephson junctions by focused ion beam patterning. Characterization of electric transport and noise properties at 4.2$\,$K in magnetically shielded environment yields a very small inductance $L$ of a few pH for an optimized device geometry. This in turn results in very low values of flux noise $<50\,{\rm n}\Phi_0/{\rm Hz}^{1/2}$ in the thermal white noise limit, which yields spin sensitivities of a few $\mu_{\rm B}/{\rm Hz}^{1/2}$ ($\Phi_0$ is the magnetic flux quantum and $\mu_{\rm B}$ is the Bohr magneton). We observe frequency-dependent excess noise up to 7$\,$MHz, which can only partially be eliminated by bias reversal readout. This indicates the presence of fluctuators of unknown origin, possibly related to defect-induced spins in the SrTiO$_3$ substrate. We demonstrate the potential of using YBCO nanoSQUIDs for the investigation of small spin systems, by placing a 39$\,$nm diameter Fe nanowire, encapsulated in a carbon nanotube, on top of a non-optimized YBCO nanoSQUID and by measuring the magnetization reversal of the Fe nanowire via the change of magnetic flux coupled to the nanoSQUID. The measured flux signals upon magnetization reversal of the Fe nanowire are in very good agreement with estimated values, and the determined switching fields indicate magnetization reversal of the nanowire via curling mode.
Article
Despite the lack of reproducible experimental confirmation, group V elements have been considered as possible sources of \textit{p}-type doping in ZnO in the form of simple and complex defects. Using \textit{ab initio} calculations, based on state-of-the-art hybrid exchange-correlation functional, we studied a wide range of defects and defects complexes related with N, P, As and Sb impurities. We show that none of the candidates for \textit{p}-type doping can be considered a good source of holes in the valence band due to deep acceptor levels and low formation energies of compensating donor defects. In addition, we discuss the stability of complexes in different regimes.
Article
High fidelity coherent control of quantum systems is critical to building quantum devices and quantum computers. We provide a general optimal control framework for designing control sequences that account for hardware control distortions while maintaining robustness to environmental noise. We demonstrate the utility of our algorithm by presenting examples of robust quantum gates optimized in the presence of nonlinear distortions. We show that nonlinear classical controllers do not necessarily incur additional computational cost to pulse optimization, enabling more powerful quantum devices.
Article
We describe the coherent manipulation of harmonic oscillator and qubit modes using resonant trains of single flux quantum pulses in place of microwaves. We show that coherent rotations are obtained for pulse-to-pulse spacing equal to the period of the oscillator. We consider a protocol for preparing bright and dark harmonic oscillator pointer states. Next we analyze rotations of a two-state qubit system. We calculate gate errors due to timing jitter of the single flux quantum pulses and due to weak anharmonicity of the qubit. We show that gate fidelities in excess of 99.9% are achievable for sequence lengths of order 20 ns.
Article
Ultrasound-driven oscillating micro-bubbles have been used as active actuators in microfluidic devices to perform manifold tasks such as mixing, sorting and manipulation of microparticles. A common configuration consists on side-bubbles, created by trapping air pockets in blind channels perpendicular to the main channel direction. This configuration consists of acoustically excited bubbles with a semi-cylindrical shape that generate significant streaming flow. Due to the geometry of the channels, such flows have been generally considered as quasi two-dimensional. Similar assumptions are often made in many other microfluidic systems based on \emph{flat} micro-channels. However, in this paper we show that microparticle trajectories actually present a much richer behavior, with particularly strong out-of-plane dynamics in regions close to the microbubble interface. Using Astigmatism Particle Tracking Velocimetry, we reveal that the apparent planar streamlines are actually projections of a \emph{streamsurface} with a pseudo-toroidal shape. We therefore show that acoustic streaming cannot generally be assumed as a two-dimensional phenomenon in confined systems. The results have crucial consequences for most of the applications involving acoustic streaming as particle trapping, sorting and mixing.
Article
We study the thermoelectric effects in arrays of disordered nanowires in parallel, at temperatures where charge transport between localized states is thermally assisted by phonons. We obtain large power factors and electrical figures of merit, when the chemical potential probes the band edges of the nanowires, the large thermopowers self-averaging while the small electrical conductances add. The role of the parasitic phonon heat transport is estimated. We also show that phonon absorption and emission occur at opposite ends of the array in band-edge transport, a phenomenon which could be exploited for cooling hot spots in electronic circuits.
Article
We studied polycrystalline B2-type Co2FeAl (CFA) full-Heusler alloy based magnetic tunnel junctions (MTJs) fabricated on a Si/SiO2 amorphous substrate. Polycrystalline CFA films with a (001) orientation, a high B2 ordering, and a flat surface were achieved using a MgO buffer layer. A tunnel magnetoresistance (TMR) ratio up to 175% was obtained for an MTJ with a CFA/MgO/CoFe structure on a 7.5-nm-thick MgO buffer. Spin-transfer torque induced magnetization switching was achieved in the MTJs with a 2-nm-thick polycrystalline CFA film as a switching layer. Using a thermal activation model, the intrinsic critical current density (Jc0) was determined to be 8.2 x 10^6 A/cm^2, which is lower than 2.9 x 10^7 A/cm^2, the value for epitaxial CFA-MTJs [Appl. Phys. Lett. 100, 182403 (2012)]. We found that the Gilbert damping constant evaluated using ferromagnetic resonance measurements for the polycrystalline CFA film was ~0.015 and was almost independent of the CFA thickness (2~18 nm). The low Jc0 for the polycrystalline MTJ was mainly attributed to the low damping of the CFA layer compared with the value in the epitaxial one (~0.04).
Article
At strong pump powers, a semiconductor optical cavity passes through a Hopf bifurcation and undergoes self-oscillation. We simulate this device using semiclassical Langevin equations and assess the effect of quantum fluctuations on the dynamics. Below threshold, the cavity acts as a phase-insensitive linear amplifier, with noise $\sim 5\times$ larger than the Caves bound. Above threshold, the limit cycle acts as an analog memory, and the phase diffusion is $\sim 10\times$ larger than the bound set by the standard quantum limit. We also simulate entrainment of this oscillator and propose an optical Ising machine and classical CNOT gate based on the effect.
Article
We demonstrate fast readout of a double quantum dot (DQD) that is coupled to a superconducting resonator. Utilizing parametric amplification in a nonlinear operational mode, we improve the signal-to-noise ratio (SNR) by a factor of 2000 compared to the situation with the parametric amplifier turned off. With an integration time of 400 ns we achieve a SNR of 76. By studying SNR as a function of the integration time we extract an equivalent charge sensitivity of 8 x 10^{-5} e/root(Hz). The high SNR allows us to acquire a DQD charge stability diagram in just 20 ms. At such a high data rate, it is possible to acquire charge stability diagrams in a live "video-mode," enabling real time tuning of the DQD confinement potential.
Article
We study microfluidic self digitization in Hele-Shaw cells using pancake droplets anchored to surface tension traps. We show that above a critical flow rate, large anchored droplets break up to form two daughter droplets, one of which remains in the anchor. Below the critical flow velocity for breakup the shape of the anchored drop is given by an elastica equation that depends on the capillary number of the outer fluid. As the velocity crosses the critical value, the equation stops admitting a solution that satisfies the boundary conditions; the drop breaks up in spite of the neck still having finite width. A similar breaking event also takes place between the holes of an array of anchors, which we use to produce a 2D array of stationary drops in situ.
Article
We show that uniaxial color centers in silicon carbide with hexagonal lattice structure can be used to measure not only the strength but also the polar angle of the external magnetic field with respect to the defect axis with high precision. The method is based on the optical detection of multiple spin resonances in the silicon vacancy defect with quadruplet ground state. We achieve a perfect agreement between the experimental and calculated spin resonance spectra without any fitting parameters, providing angle resolution of a few degrees in the magnetic field range up to several millitesla. Our approach is suitable for ensembles as well as for single spin-3/2 color centers, allowing for vector magnetometry on the nanoscale at ambient conditions.
Article
Spin-orbit coupling in ferromagnets gives rise to the anomalous Hall effect and the anisotropic magnetoresistance, both of which can be used to create spin-transfer torques in a similar manner as the spin Hall effect. In this paper we show how these effects can be used to reliably switch perpendicularly magnetized layers and to move domain walls. A drift-diffusion treatment of the anomalous Hall effect and the anisotropic magnetoresistance describes the spin currents that flow in directions perpendicular to the electric field. In systems with two ferromagnetic layers separated by a spacer layer, an in-plane electric field cause spin currents to be injected from one layer into the other, creating spin transfer torques. Unlike the related spin Hall effect in non-magnetic materials, the anomalous Hall effect and the anisotropic magnetoresistance allow control of the orientation of the injected spins, and hence torques, by changing the direction of the magnetization in the injecting layer. The torques on one layer show a rich angular dependence as a function of the orientation of the magnetization in the other layer. The control of the torques afforded by changing the orientation of the magnetization in a fixed layer makes it possible to reliably switch a perpendicularly magnetized free layer. Our calculated critical current densities for a representative CoFe/Cu/FePt structure show that the switching can be efficient for appropriate material choices. Similarly, control of the magnetization direction can drive domain wall motion, as shown for NiFe/Cu/NiFe structures.
Article
Using first-principles calculations, we propose a microscopic model to explain the reversible lithiation/delithiation of tin-oxide anodes in lithium-ion batteries. When the irreversible regime ends, the anode grains consist of layers of Li-oxide separated by Sn bilayers. During the following reversible lithiation, the Li-oxide undergoes two phase transformations that give rise to a Li-enrichment of the oxide and the formation of a SnLi composite. The anode grain structure stays layered and ordered with an effective theoretical reversible capacity of 4.5 Li per Sn atom. The predicted anode volume expansion and voltage profile agree well with experiments, contrary to existing models.
Article
Crystalline organic semiconductors, bonded by weak van der Waals forces, exhibit macroscopic properties that are very similar to those of inorganic semiconductors. While there are many open questions concerning the microscopic nature of charge transport, minimizing the density of trap states (trap DOS) is crucial to elucidate the intrinsic transport mechanism. We explore the limits of state-of-the-art organic crystals by measuring single crystalline rubrene field-effect transistors that show textbook like transfer characteristics, indicating a very low trap DOS. Particularly, the high purity of the crystals and the very clean interface to the gate dielectric are reflected in an unprecedentedly low subthreshold swing of $65$ ${\rm mV / decade}$, remarkably close to the fundamental limit of $58.5\,{\rm mV / decade}$. From the measured subthreshold behavior we have consistently quantified the trap DOS by two different methods, yielding an exceedingly low trap density of $D_{bulk} = 1 \times 10^{13}~{\rm cm^{-3}eV^{-1}}$ at an energy of $\sim0.62~{\rm eV}$. These numbers correspond to one trap per eV in $10^8$ rubrene molecules. The equivalent density of traps located at the interface is $D_{it} = 3 \times 10^{9}~{\rm cm^{-2}eV^{-1}}$ which puts them on par with the best crystalline ${\rm SiO_2/Si}$ field-effect transistors.
Article
Solid-state qubits have recently advanced to the level that enables them, in-principle, to be scaled-up into fault-tolerant quantum computers. As these physical qubits continue to advance, meeting the challenge of realising a quantum machine will also require the engineering of new classical hardware and control architectures with complexity far beyond the systems used in today's few-qubit experiments. Here, we report a micro-architecture for controlling and reading out qubits during the execution of a quantum algorithm such as an error correcting code. We demonstrate the basic principles of this architecture in a configuration that distributes components of the control system across different temperature stages of a dilution refrigerator, as determined by the available cooling power. The combined setup includes a cryogenic field-programmable gate array (FPGA) controlling a switching matrix at 20 millikelvin which, in turn, manipulates a semiconductor qubit.
Article
In this work, an effective quantum model based on the non-equilibrium Green's function formalism is used to investigate a selectively contacted high density quantum dot array in an wide band gap host matrix for operation as a quantum dot-enhanced single junction solar cell. By establishing a direct relation between nanostructure configuration and optoelectronic properties, the investigation reveals the influence of inter-dot and dot-contact coupling strength on the radiative rates and consequently on the ultimate performance of photovoltaic devices with finite quantum dot arrays as the active medium. The dominant effects originate in the dependence of the Joint Density of States on the inter-dot coupling in terms of band width and effective band gap.
Article
We report on experiments with a microfabricated surface trap designed for trapping a chain of ions in a ring. Uniform ion separation over most of the ring is achieved with a rotationally symmetric design and by measuring and suppressing undesired electric fields. After minimizing these fields the ions are confined primarily by an rf trapping pseudo-potential and their mutual Coulomb repulsion. The ring-shaped crystal consists of approximately 400 Ca$^+$ ions with an estimated average separation of 9 $\mu m$.
Article
Switching of the direction of the magnetic moment in a nanomagnet is studied within a modified Slonczewski's model that permits torsional oscillations of the magnet. We show that the latter may inhibit or assist the magnetization switching, depending on parameters. Three regimes have been studied: the switching by torsional oscillations alone, the switching by the spin-polarized current with torsional oscillations permitted, and the magnetization switching by the current combined with the mechanical twist. We show that switching of the magnetic moment is possible in all three cases and that allowing torsional oscillations of the magnet may have certain advantages for applications. Phase diagrams are computed that show the range of parameters required for the switching.
Article
Mass spectrometry is used in a wide range of scientific disciplines including proteomics, pharmaceutics, forensics, and fundamental physics and chemistry. Given this ubiquity, there is a worldwide effort to improve the efficiency and resolution of mass spectrometers. However, the performance of all techniques is ultimately limited by the initial phase-space distribution of the molecules being analyzed. Here, we dramatically reduce the width of this initial phase-space distribution by sympathetically cooling the input molecules with laser-cooled, co-trapped atomic ions, improving both the mass resolution and detection efficiency of a time-of-flight mass spectrometer by over an order of magnitude. Detailed molecular dynamics simulations verify the technique and aid with evaluating its effectiveness. Our technique appears to be applicable to other types of mass spectrometers.
Article
We demonstrate a dual-axis accelerometer and gyroscope atom interferometer, which forms the building blocks of a six-axis inertial measurement unit. By recapturing the atoms after the interferometer sequence, we maintain a large atom number at high data-rates of 50 to 100 measurements per second. Two cold ensembles are formed in trap zones located a few centimeters apart, and are launched toward one-another. During their ballistic trajectory, they are interrogated with a stimulated Raman sequence, detected, and recaptured in the opposing trap zone. We achieve sensitivities at $\mathrm{\mu \mathit{g} / \sqrt{Hz}}$ and $\mathrm{\mu rad / s / \sqrt{Hz}}$ levels, making this a compelling prospect for expanding the use of atom interferometer inertial sensors beyond benign laboratory environments.
Article
We report on transport in the 2$^{\text{nd}}$ Landau level in in-situ back-gated two-dimensional electron gases in GaAs/Al$_x$Ga$_{1-x}$As quantum wells. Minimization of gate leakage is the primary heterostructure design consideration. Leakage currents resulting in dissipation as small as a few pW can cause noticeable heating of the electrons at 10 mK, limiting the formation of novel correlated states. We show that when the heterostructure design is properly optimized, gate voltages as large as 4V can be applied with negligible gate leakage, allowing the density to be tuned over a large range from depletion to over 4 $\times$ 10$^{11}$ cm$^{-2}$. As a result, the strength of the $\nu = 5/2$ state can be continuously tuned from onset at n $\sim 1.2 \times 10^{11}$ cm$^{-2}$ to a maximum $\Delta_{5/2} = 625$ mK at n = $3.35 \times 10^{11}$ cm$^{-2}$. An unusual evolution of the reentrant integer quantum Hall states as a function of density is also reported. These devices can be expected to be useful in experiments aimed at proving the existence of non-Abelian phases useful for topological quantum computation.
Article
The interaction of shear bands with crystalline nanoprecipitates in Cu-Zr-based metallic glasses is investigated by a combination of high-resolution TEM imaging and molecular dynamics computer simulations. Our results reveal different interaction mechanisms: Shear bands can dissolve precipitates, can wrap around crystalline obstacles or can be blocked depending on size and density of the precipitates. If the crystalline phase has a low yield strength, we also observe slip transfer through the precipitate. Based on the computational results and experimental findings a qualitative mechanism map is proposed that categorizes the various processes as a function of the critical stress for dislocation nucleation, precipitate size and distance.
Article
Employing first-principles calculations, we investigate efficiency of spin injection from a ferromagnetic (FM) electrode (Ni) into graphene and possible enhancement by using a barrier between the electrode and graphene. Three types of barriers, h-BN, Cu(111), and graphite, of various thickness (0-3 layers) are considered and the electrically biased conductance of the Ni/Barrier/Graphene junction are calculated. It is found that the minority spin transport channel of graphene can be strongly suppressed by the insulating h-BN barrier, resulting in a high spin injection efficiency. On the other hand, the calculated spin injection efficiencies of Ni/Cu/Graphene and Ni/Graphite/Graphene junctions are low, due to the spin conductance mismatch. Further examination on the electronic structure of the system reveals that the high spin injection efficiency in the presence of a tunnel barrier is due to its asymmetric effects on the two spin states of graphene.
Article
Thermoelectric performance is of interest for numerous applications such as waste heat recovery and solid state energy conversion, and will be seen to be closely connected to topological insulator behavior. In this context we here report first principles transport and defect calculations for Bi$_{2}$Te$_{2}$Se in relation to Bi$_{2}$Te$_{3}$. The two compounds are found to contain remarkably different electronic structures in spite of being isostructural and isoelectronic. We discuss these results in terms of the topological insulator characteristics of these compounds.
Article
We theoretically investigate tunneling magnetoresistance (TMR) devices, which are probing the spin-momentum coupled nature of surface states of the three-dimensional topological insulator Bi2Se3. Theoretical calculations are performed based on a realistic tight-binding model for Bi2Se3. We study both three dimensional devices, which exploit the surface states of Bi2Se3, as well as two-dimensional devices, which exploit the edge states of thin Bi2Se3 strips. We demonstrate that the material properties of Bi2Se3 allow a TMR ratio at room temperature of the order of 1000%. Analytical formulas are derived that allow a quick estimate of the achievable TMR ratio in these devices. The devices can be used to measure the spin polarization of the topological surface states as an alternative to spin-ARPES. Unlike TMR devices based on magnetic tunnel junctions the present devices avoid the use of a second ferromagnetic electrode whose magnetization needs to be pinned.
Article
For two electrically small nonreciprocal scatterers an analytical electromagnetic model of polarizabilities is developed. Both particles are bianisotropic: the so-called Tellegen-omega particle and moving-chiral particle. Analytical results are compared to the full-wave numerical simulations. Both models satisfy to main physical restrictions and leave no doubts in the possibility to realize these particles experimentally. This paper is a necessary step towards applications of nonreciprocal bianisotropic particles such as perfect electromagnetic isolators, twist polarizers, thin-sheet phase shifters, and other devices.
Article
Currently, one of the major nanotechnological challenges is to design thermoelectric devices that have a high figure of merit. To that end, we propose to use bilayer excitons. Bilayer exciton systems are shown to have an improved thermopower and an enhanced electric counterflow and thermal conductivity, with respect to regular semiconductor-based thermoelectrics. Here we present a roadmap towards experimental realization of a bilayer exciton thermocouple. A bilayer exciton heterostructures of $p$- and $n$-doped Bi$_2$Te$_3$ can have a figure of merit $zT \sim 60$. Another material suggestion is to make a bilayer out of electron-doped SrTiO$_3$ and hole-doped Ca$_3$Co$_4$O$_9$.
Article
We report a comprehensive micro-Raman study of a pressurized suspended graphene membrane that hermetically seals a circular pit, etched in a Si/SiO$_2$ substrate. Placing the sample under a uniform pressure load results in bulging of the graphene membrane and subsequent softening of the main Raman features, due to tensile strain. In such a microcavity, the intensity of the Raman features depends very sensitively on the distance between the graphene membrane and the Si substrate, which acts as the bottom mirror of the cavity. Thus, a spatially resolved analysis of the intensity of the G and 2D mode features as a function of the pressure load permits a direct reconstruction of the blister profile. An average strain is then deduced at each pressure. This allows a determination of the Gr\"{u}neisen parameters of $1.8\pm0.2$ and $2.4\pm0.2$ for the Raman G and 2D modes, respectively. The measured blister height is proportional to the cubic root of the pressure load, as predicted theoretically. The validation of this scaling provides a direct and accurate determination the Young's modulus of graphene with a purely optical, hence minimally invasive and contactless approach. We find a Young's modulus of $\left(1.05\pm 0.10\right) \rm TPa$ for monolayer graphene, in perfect match with previous nano-indentation measurements. This all optical approach opens avenues for pressure sensing using graphene and could readily be adapted to other emerging two dimensional membranes.
Article
We address the tunneling current in a graphene-hBN-graphene heterostructure as function of the twisting between the crystals. The twisting induces a modulation of the hopping amplitude between the graphene layers, that provides the extra momentum necessary to satisfy momentum and energy conservation and to activate coherent tunneling between the graphene electrodes. Conservation rules limit the tunneling to states with wavevectors lying at the conic curves defined by the intersection of two Dirac cones shifted in momentum and energy. There is a critical voltage where the intersection is a straight line, and the joint density of states presents a maximum. This reflects in a peak in the tunneling current and in a negative differential conductivity.
Article
We propose a class of linear elastic three-dimensional metamaterials for which the e?ective parameters bulk modulus and mass density can be adjusted independently over a large range|which is not possible for ordinary materials. First, we systematically evaluate the static mechanical properties and the phonon dispersion relations. We show that the two are quantitatively consistent in the long-wavelength limit. To demonstrate the feasibility, corresponding fabricated polymer microstructures are presented. Finally, we discuss calculations for laminates composed of alternating layers of two di?erent metamaterials with equal bulk modulus yet di?erent mass density. This leads to metamaterials with e?ectively anisotropic uniaxial dynamic mass density tensors.
Article
We demonstrate radiofrequency thermometry on a micrometer-sized metallic island below 100 mK. Our device is based on a normal metal-insulator-superconductor tunnel junction coupled to a resonator with transmission readout. In the first generation of the device, we achieve 100 {\mu}K/Hz^1/2 noise-equivalent temperature, limited by the first amplifier, with 10 MHz bandwidth. We measure the thermal relaxation time of the electron gas in the island, which we find to be of the order of 100 {\mu}s. Such a calorimetric detector, upon optimization, can be seamlessly integrated into superconducting circuits, with immediate applications in quantum-thermodynamics experiments down to single quanta of energy.
Article
Layered LiMnO2 and Li2MnO3 are of great interest for lithium-ion battery cathodes because of their high theoretical capacities. The practical application of these materials is, however, limited due to poor electrochemical performance. We herein report a comprehensive first-principles study of defect physics in LiMnO2 and Li2MnO3 using hybrid-density functional calculations. We find that manganese antisites have low formation energies in LiMnO2 and may act as nucleation sites for the formation of impurity phases. The antisites can also occur with high concentrations in Li2MnO3; however, unlike in LiMnO2, they can be eliminated by tuning the experimental conditions during preparation. Other intrinsic point defects may also occur and have an impact on the materials' properties and functioning. An analysis of the formation of lithium vacancies indicates that lithium extraction from LiMnO2 is associated with oxidation at the manganese site, resulting in the formation of manganese small hole polarons; whereas in Li2MnO3 the intrinsic delithiation mechanism involves oxidation at the oxygen site, leading to the formation of bound oxygen hole polarons η+O. The layered oxides are found to have no or negligible bandlike carriers and they cannot be doped n- or p-type. The electronic conduction proceeds through hopping of hole and/or electron polarons; the ionic conduction occurs through lithium monovacancy and/or divacancy migration mechanisms. Since η+O is not stable in the absence of negatively charged lithium vacancies in bulk Li2MnO3, the electronic conduction near the start of delithiation is likely to be poor. We suggest that the electronic conduction associated with η+O and, hence, the electrochemical performance of Li2MnO3 can be improved through nanostructuring and/or ion substitution.
Article
We propose and analyze a hybrid device by integrating a microscale diamond beam with a single built-in nitrogen-vacancy (NV) center spin to a superconducting coplanar waveguide (CPW) cavity. We find that under an ac electric field the quantized motion of the diamond beam can strongly couple to the single cavity photons via polarization interaction. Together with the strong spin-motion interaction via a large magnetic field gradient, it provides a hybrid quantum device where the diamond resonator can strongly couple both to the single microwave cavity photons and to the single NV center spin. This enables coherent information transfer and effective coupling between the NV spin and the CPW cavity via mechanically dark polaritons. This hybrid spin-electromechanical device, with tunable couplings by external fields, offers a realistic platform for implementing quantum information with single NV spins, diamond mechanical resonators, and single microwave photons.
Article
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Article
We investigate the use of guided modes bound to defects in photonic crystals for achieving double resonances. Photoluminescence enhancement by more than three orders of magnitude has been observed when the excitation and emission wavelengths are simultaneously in resonance with the localized guided mode and cavity mode, respectively. We find that the localized guided modes are relatively insensitive to the size of the defect for one of the polarizations, allowing for flexible control over the wavelength combinations. This double resonance technique is expected to enable enhancement of photoluminescence and nonlinear wavelength conversion efficiencies in a wide variety of systems.
Article
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Article
We visualize the formation of fingered flow in dry model sandy soils under different raining conditions using a quasi-2d experimental set-up, and systematically determine the impact of soil grain diameter and surface wetting property on water channelization phenomenon. The model sandy soils we use are random closely-packed glass beads with varied diameters and surface treatments. For hydrophilic sandy soils, our experiments show that rain water infiltrates into a shallow top layer of soil and creates a horizontal water wetting front that grows downward homogeneously until instabilities occur to form fingered flows. For hydrophobic sandy soils, in contrast, we observe that rain water ponds on the top of soil surface until the hydraulic pressure is strong enough to overcome the capillary repellency of soil and create narrow water channels that penetrate the soil packing. Varying the raindrop impinging speed has little influence on water channel formation. However, varying the rain rate causes significant changes in water infiltration depth, water channel width, and water channel separation. At a fixed raining condition, we combine the effects of grain diameter and surface hydrophobicity into a single parameter and determine its influence on water infiltration depth, water channel width, and water channel separation. We also demonstrate the efficiency of several soil water improvement methods that relate to rain water channelization phenomenon, including pre-wetting sandy soils at different level before rainfall, modifying soil surface flatness, and applying superabsorbent hydrogel particles as soil modifiers.
Article
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Article
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Article
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Article
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We present the first demonstration of all-optical squeezing in an on-chip monolithically integrated CMOS-compatible platform. Our device consists of a low loss silicon nitride microring optical parametric oscillator (OPO) with a gigahertz cavity linewidth. We measure 1.7 dB (5 dB corrected for losses) of sub-shot noise quantum correlations between bright twin beams generated in the microring four-wave-mixing OPO pumped above threshold. This experiment demonstrates a compact, robust, and scalable platform for quantum optics and quantum information experiments on-chip.
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A semiclassical simulation approach is presented for studying quantum noise in large-scale photonic circuits incorporating an ideal Kerr nonlinearity. A netlist-based circuit solver is used to generate matrices defining a set of stochastic differential equations, in which the resonator field variables represent random samplings of the Wigner quasi-probability distributions. Although the semiclassical approach involves making a large-photon-number approximation, tests on one- and two-resonator circuits indicate satisfactory agreement between the semiclassical and full-quantum simulation results in the parameter regime of interest. The semiclassical model is used to simulate random errors in a large-scale circuit that contains 88 resonators and hundreds of components in total, and functions as a 4-bit ripple counter. The error rate as a function of on-state photon number is examined, and it is observed that the quantum fluctuation amplitudes do not increase as signals propagate through the circuit, an important property for scalability.
Article
We demonstrate a general non--Derjaguin-Landau-Verwey-Overbeek method to stabilize colloids in liquids. By this method, colloidal particles that initially form unstable suspension and sediment from the liquid are stabilized by the addition of salt to the suspending liquid. Yet, the salt is not expected to adsorb or directly interact with the surface of the colloids. For the method to work, the liquid should be a mixture, and the salt needs to be antagonistic such that each ion is preferentially solvated by a different component of the mixture. The stabilization may depend on the salt content, mixture composition, or distance from the mixture's coexistence line.
Article
Since the introduction of the decoy-state technique, phase-randomised weak coherent light pulses have been the key to increase the practicality of quantum-based communications. Their ultra-fast generation was accomplished via compact gain-switched (GS) lasers, leading to high key rates in quantum key distribution (QKD). Recently, the question arose of whether the same laser could be employed to achieve high-speed measurement-device-independent-QKD, a scheme that promises long-haul quantum communications immune to all detector attacks. For that, a challenging highvisibility interference between independent picosecond optical pulses is required. Here, we answer the above question in the affirmative by demonstrating high-visibility interference from two independent GS lasers triggered at 1GHz. The result is obtained through a careful characterization of the laser frequency chirp and time jitter. By relating these quantities to the interference visibility, we obtain a parameter-free verification of the experimental data and a numerical simulation of the achievable key rates. These findings are beneficial to other applications making use of GS lasers, including random number generation and standard QKD.
Top-cited authors
• Forschungszentrum Jülich
• Tongji University
• University of Oxford
• University of Oxford
• Stanford University
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2022-08-10 15:56:43
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http://math.stackexchange.com/questions/309204/how-to-solve-this-nonstandard-trigonometric-equation
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# How to solve this nonstandard trigonometric equation?
I want to solve this equation $$\sin(\sin(\sin(\sin x)))=\cos(\cos(\cos(\cos x))),$$ but I don't know how to solve.
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"Numerically" is probably the only way forward here. – Henning Makholm Feb 20 '13 at 16:19
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2014-09-22 14:43:41
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https://wisc.pb.unizin.org/minimisgenchem/chapter/bond-length-and-strength-m8q4/
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# 47 Bond Length and Strength (M8Q4)
## Introduction
This section further explores bond length and bond strength, which were introduced in the previous section. A bond’s strength describes how strongly each atom is joined to another atom, and therefore how much energy is required to break the bond between the two atoms. The strength of a given bond is directly related to the length of this bond. The section below provides a more detailed description of these topics, worked examples, practice problems and a glossary of important terms.
Learning Objectives for Bond Length and Bond Strength
## Covalent Bonds
Stable molecules exist because covalent bonds hold the atoms together. We measure the strength of a covalent bond by the energy required to break it, that is, the energy necessary to separate the bonded atoms. Separating any pair of bonded atoms requires energy. The stronger a bond, the greater the energy required to break it.
The energy required to break a specific covalent bond in one mole of gaseous molecules is called the bond energy or the bond dissociation energy. The bond energy for a diatomic molecule is defined as the standard enthalpy change for the endothermic reaction:
XY(g) → X(g) + Y(g) Δ
For example, the bond energy of the pure covalent H–H bond is 436 kJ per mole of H–H bonds broken:
H2(g) → 2 H(g) Δ = 436 kJ
Molecules with three or more atoms have two or more bonds. The sum of all bond energies in such a molecule is equal to the standard enthalpy change for the endothermic reaction that breaks all the bonds in the molecule. For example, the sum of the four C–H bond energies in CH4, 1664 kJ, is equal to the standard enthalpy change of the reaction:
The average C–H bond energy is 1664/4 = 416 kJ/mol because there are four moles of C–H bonds broken per mole of the reaction. Although the four C–H bonds are equivalent in the original molecule, they do not each require the same energy to break; once the first bond is broken (which requires 439 kJ/mol), the remaining bonds are easier to break. The average C-H bond energy is not only averaged over each C-H bond broken within a single molecule, but the average of C-H bond energies of many molecules. For example, ethane, C2H6, has an actual C-H bond energy closer to 423 kJ/mol, lower than the 439 kJ/mol of methane. The 416 kJ/mol value is the average, not the exact value required to break any one bond in any one molecule.
The strength of a bond between two atoms increases as the number of electron pairs in the bond increases. Generally, as the bond strength increases, the bond length decreases. Thus, we find that triple bonds are stronger and shorter than double bonds between the same two atoms; likewise, double bonds are stronger and shorter than single bonds between the same two atoms. Average bond energies for some common bonds appear in Table 1 and Appendix G, and a comparison of bond lengths for some common bonds appears in Table 2 and Appendix G. When one atom bonds to various atoms in a group, the bond strength typically decreases (and bond length increases) as we move down the group. For example, C–F is 485 kJ/mol and 141 pm, C–Cl is 327 kJ/mol and 176 pm, and C–Br is 285 kJ/mol and 191 pm.
## Average Bond Enthalpy
In kJ/mol
Data from Cotton, F.A., Wilkinson, G. and Gaus, P.L., Basic Inorganic Chemistry, 3rd ed., New York: Wiley, 1995. Corrected values for C-C and C-O from Cottrell, T.L., The Strengths of Chemical Bonds, 2ed., London:Butterworths, 1958. I Br Cl S P Si F O N C H H 299 366 431 347 322 323 566 467 391 416 436 C 213 285 327 272 264 301 485 358 285 346 N - - 193 - ~200 335 272 201 160 O 201 - 205 - ~340 368 190 146 F - 238 255 326 490 582 158 Si 234 310 391 226 - 226 P 184 264 319 - 209 S - 213 255 226 Multiple Bonds Multiple Bonds Cl 209 217 242 N=N 418 C=C 598 Br 180 193 N≡N 946 C≡C 813 I 151 C=N 616 C=O in CO2 803 C≡N 866 C=O carbonyl 695 N=O 607 C≡O 1073 O=O in O2 498 N≡O 632
Table 1: Bond Enthalpies in kJ/mol.
## Average Bond Length
In picometers, (pm)
I Br Cl S P Si F O N C H H 161 142 127 132 138 145 92 94 98 110 74 C 210 191 176 181 187 194 141 143 147 154 N 203 184 169 174 180 187 134 136 140 O 199 180 165 170 176 183 130 148 F 197 178 163 168 174 181 128 Si 250 231 216 221 227 234 P 243 224 209 214 220 S 237 218 203 208 Multiple Bonds Multiple Bonds Cl 232 213 200 N=N 120 C=C 134 Br 247 228 N≡N 110 C≡C 121 I 266 C=N 127 C=O in CO2 116 C≡N 115 C=O carbonyl 121 N=O 115 C≡O 113 O=O in O2 121 N≡O 115
Table 2: Average bond lengths in picometers (pm).
We can use bond energies to calculate approximate enthalpy changes for reactions where enthalpies of formation are not available. Calculations of this type will also tell us whether a reaction is exothermic or endothermic. An exothermic reaction (-ΔH, heat released) results when the bonds in the products are stronger than the bonds in the reactants. An endothermic reaction (+ΔH, heat absorbed) results when the bonds in the products are weaker than those in the reactants.
## Calculating Enthalpy Change
The enthalpy change, ΔH, for a chemical reaction is approximately equal to the sum of the energy required to break all bonds in the reactants (energy “in”, positive sign) plus the energy released when all bonds are formed in the products (energy “out,” negative sign). This can be expressed mathematically in the following way:
ΔHreaction = ∑ (energy of bonds broken) – ∑ (energy of bonds formed)
In this expression, the symbol Ʃ means “the sum of” and we sum the bond energy in kilojoules per mole, which is always a positive number. The bond energy is obtained from a table (like Table 1/Appendix G) and will depend on whether the particular bond is a single, double, or triple bond. Thus, in calculating enthalpies in this manner, it is important that we consider the bonding in all reactants and products. Because bond energy values are typically averages for one type of bond in many different molecules, this calculation provides a rough estimate, not an exact value, for the enthalpy of reaction.
Consider the following reaction:
H2(g) + Cl2(g) → 2 HCl(g)
or
H-H(g) + Cl-Cl(g) → 2 H-Cl(g)
To form two moles of HCl, one mole of H–H bonds and one mole of Cl–Cl bonds must be broken. The energy required to break these bonds is the sum of the bond energy of the H–H bond (436 kJ/mol) and the Cl–Cl bond (242 kJ/mol). During the reaction, two moles of H–Cl bonds are formed (bond energy = 431 kJ/mol), releasing 2 × 431 kJ; or 862 kJ. Because the bonds in the products are stronger than those in the reactants, the reaction releases more energy than it consumes:
ΔHreaction = ∑ (energy of bonds broken) – ∑ (energy of bonds formed)
= [(H-H) + (Cl-Cl)] – 2(H-Cl)
= [436 + 242] – 2(431) = -184 kJ
This excess energy is released as heat, so the reaction is exothermic. Appendix F gives a value for the standard molar enthalpy of formation of HCl(g), ΔHf°, of -92.307 kJ/mol. Twice that value is –184.6 kJ, which agrees well with the answer obtained earlier for the formation of two moles of HCl.
### Example 1
Using Bond Energies to Calculate Approximate Enthalpy Changes
Methanol, CH3OH, may be an excellent alternative fuel. The high-temperature reaction of steam and carbon produces a mixture of the gases carbon monoxide, CO, and hydrogen, H2, from which methanol can be produced. Using the bond energies in Table 1, calculate the approximate enthalpy change, ΔH, for the reaction here:
CO(g) + 2 H2(g) → CH3OH(g)
Solution
First, we need to write the Lewis structures of the reactants and the products:
From this, we see that ΔH for this reaction involves the energy required to break a C–O triple bond and two H–H single bonds, as well as the energy produced by the formation of three C–H single bonds, a C–O single bond, and an O–H single bond. We can express this as follows:
ΔH = ∑ (energy of bonds broken) – ∑ (energy of bonds formed)
= [(C≡O) + 2(H-H)] – [3(C-H) + (C-O) + (O-H)]
Using the bond energy values in Table 1, we obtain:
ΔH = [1073 + 2(436)] – [3(416) + 358 + 467]
= -128 kJ
We can compare this value to the value calculated based on ΔHf° data from Appendix F:
ΔH = [ΔHf°(CH3OH(g))] – [ΔHf°(CO(g)) + 2 × ΔHf°(H2)]
= [-200.66] – [-110.525 + 2 × 0]
= -90.135 kJ
Note that there is a fairly significant gap between the values calculated using the two different methods. This occurs because bond energy values are the average of different bond strengths; therefore, they often give only rough agreement with other data.
Ethyl alcohol, CH3CH2OH, was one of the first organic chemicals deliberately synthesized by humans. It has many uses in industry, and it is the alcohol contained in alcoholic beverages. It can be obtained by the fermentation of sugar or synthesized by the hydration of ethylene in the following reaction:
Using the bond energies in Table 1, calculate an approximate enthalpy change, ΔH, for this reaction.
–55 kJ
## Ionic Bond Strength and Lattice Energy
An ionic compound is stable because of the electrostatic attraction between its positive and negative ions. The lattice energy of a compound is a measure of the strength of this attraction. The lattice energy (ΔHlattice) of an ionic compound is defined as the energy required to separate one mole of the solid into its component gaseous ions. For the ionic solid, MX, the lattice energy is the enthalpy change of the process:
MX(s) → Mn+(g) + Xn-(g) ΔHlattice
Note that we are using the convention where the ionic solid is separated into ions, so our lattice energies will be endothermic (positive values). Some texts use the equivalent but opposite convention, defining lattice energy as the energy released when separate ions combine to form a lattice and giving negative (exothermic) values. Thus, if you are looking up lattice energies in another reference, be certain to check which definition is being used. In both cases, a larger magnitude for lattice energy indicates a more stable ionic compound. For sodium chloride, ΔHlattice = 769 kJ. Thus, it requires 769 kJ to separate one mole of solid NaCl into gaseous Na+ and Cl ions. When one mole each of gaseous Na+ and Cl ions form solid NaCl, 769 kJ of heat is released.
The lattice energy ΔHlattice of an ionic crystal can be expressed by the following equation (derived from Coulomb’s law, governing the forces between electric charges):
ΔHlattice = $\frac{\text{C}(\text{Z}^+)(\text{Z}^-)}{\text{R}_0}$
in which C is a constant that depends on the type of crystal structure; Z+ and Z are the charges on the ions; and Ro is the interionic distance (the sum of the radii of the positive and negative ions). Thus, the lattice energy of an ionic crystal increases rapidly as the charges of the ions increase and the sizes of the ions decrease. When all other parameters are kept constant, doubling the charge of both the cation and anion quadruples the lattice energy. For example, the lattice energy of LiF (Z+ and Z = 1) is 1023 kJ/mol, whereas that of MgO (Z+ and Z = 2) is 3900 kJ/mol (Ro is nearly the same—about 200 pm for both compounds).
Different interatomic distances produce different lattice energies. For example, we can compare the lattice energy of MgF2 (2957 kJ/mol) to that of MgI2 (2327 kJ/mol) to observe the effect on lattice energy of the smaller ionic size of F as compared to I.
### Example 2
Lattice Energy Comparisons
The precious gem ruby is aluminum oxide, Al2O3, containing traces of Cr3+. The compound Al2Se3 is used in the fabrication of some semiconductor devices. Which has the larger lattice energy, Al2O3 or Al2Se3?
Solution
In these two ionic compounds, the charges Z+ and Z are the same, so the difference in lattice energy will depend upon Ro. The O2– ion is smaller than the Se2– ion. Thus, Al2O3 would have a shorter interionic distance than Al2Se3, and Al2O3 would have the larger lattice energy.
Zinc oxide, ZnO, is a very effective sunscreen. How would the lattice energy of ZnO compare to that of NaCl?
ZnO would have the larger lattice energy because the Z values of both the cation and the anion in ZnO are greater, and the interionic distance of ZnO is smaller than that of NaCl.
Lattice energies calculated for ionic compounds are typically much higher than bond dissociation energies measured for covalent bonds. Whereas lattice energies typically fall in the range of 600–4000 kJ/mol (some even higher), covalent bond dissociation energies are typically between 150–400 kJ/mol for single bonds. Keep in mind, however, that these are not directly comparable values. For ionic compounds, lattice energies are associated with many interactions, as cations and anions pack together in an extended lattice. For covalent bonds, the bond dissociation energy is associated with the interaction of just two atoms.
## The Born-Haber Cycle
It is not possible to measure lattice energies directly. However, the lattice energy can be calculated using the equation given above or by using a thermochemical cycle. The Born-Haber cycle is an application of Hess’s law that breaks down the formation of an ionic solid into a series of individual steps:
• ΔHf°, the standard enthalpy of formation of the compound
• IE, the ionization energy of the metal
• EA, the electron affinity of the nonmetal
• ΔH°sublimation, the enthalpy of sublimation of the metal
• D, the bond dissociation energy of the nonmetal
• ΔH°lattice, the lattice energy of the compound
Figure 1 diagrams the Born-Haber cycle for the formation of solid cesium fluoride.
We begin with the elements in their most common states, Cs(s) and F2(g). The ΔH°sublimation represents the conversion of solid cesium into a gas, and then the ionization energy converts the gaseous cesium atoms into cations. In the next step, we account for the energy required to break the F–F bond to produce fluorine atoms. Converting one mole of fluorine atoms into fluoride ions is an exothermic process, so this step gives off energy (the electron affinity) and is shown as decreasing along the y-axis. We now have one mole of Cs cations and one mole of F anions. These ions combine to produce solid cesium fluoride. The enthalpy change in this step is the negative of the lattice energy, so it is also an exothermic quantity. The total energy involved in this conversion is equal to the experimentally determined enthalpy of formation, ΔHf°, of the compound from its elements. In this case, the overall change is exothermic.
Hess’s law can also be used to show the relationship between the enthalpies of the individual steps and the enthalpy of formation. Table 3 shows this for sodium chloride, NaCl.
Enthalpy of sublimation of Na(s) Na(s) → Na(g) ΔH = ΔHºsublimation = 109 kJ One-half of the bond energy of Cl2 $\frac{1}{2}$Cl2(g) → Cl(g) ΔH = $\frac{1}{2}$D = 121 kJ Ionization energy of Na(g) Na(g) → Na+(g) + e– ΔH = IE = 496 kJ Negative of the electron affinity of Cl Cl(g) + e– → Cl–(g) ΔH = -EA = -368 kJ Negative of the lattice energy of NaCl(s) Na+(g) + Cl–(g) → NaCl(s) ΔH = -ΔHlattice = ? Enthalpy of formation of NaCl(s), add steps 1–5 and find ΔHf º in Appendix F Na(s) + $\frac{1}{2}$Cl2(g) → NaCl(s) ΔHf º = -411 kJ Table 3.
We can write:
ΔH = ΔHf° ΔH°sublimation + $\frac{1}{2}$D + IE + (-EA) + (-ΔHlattice)
Thus, the lattice energy can be calculated from other values. For sodium chloride, using this data, the lattice energy is:
ΔHlattice = (411 + 109 + 121 + 496 + 368) kJ = 769 kJ
The Born-Haber cycle may also be used to calculate any one of the other quantities in the equation for lattice energy, provided that the remainder is known. For example, if the relevant enthalpy of sublimation ΔH°sublimation, ionization energy (IE), bond dissociation enthalpy, lattice energy ΔHlattice, and standard enthalpy of formation ΔHf° are known, the Born-Haber cycle can be used to determine the electron affinity of an atom.
## Key Concepts and Summary
The strength of a covalent bond is measured by its bond dissociation energy, that is, the amount of energy required to break that particular bond in a mole of molecules. Multiple bonds are stronger than single bonds between the same atoms. The enthalpy of a reaction can be estimated based on the energy input required to break bonds and the energy released when new bonds are formed. For ionic bonds, the lattice energy is the energy required to separate one mole of a compound into its gas phase ions. Lattice energy increases for ions with higher charges and shorter distances between ions. Lattice energies are often calculated using the Born-Haber cycle, a thermochemical cycle including all of the energetic steps involved in converting elements into an ionic compound.
## Key Equations
• Bond energy for a diatomic molecule: XY(g) → X(g) + Y(g) Δ
• Enthalpy change:
ΔHreaction = ∑ (energy of bonds broken) – ∑ (energy of bonds formed)
• Lattice energy for a solid MX: MX(s) → Mn+(g) + Xn-(g) ΔHlattice
• Lattice energy for an ionic crystal:
ΔHlattice = $\frac{\text{C}(\text{Z}^+)(\text{Z}^-)}{\text{R}_0}$
## Glossary
bond energy
(also, bond dissociation energy) energy required to break a covalent bond in a gaseous substance
Born-Haber cycle
thermochemical cycle relating the various energetic steps involved in the formation of an ionic solid from the relevant elements
lattice energy (ΔHlattice)
energy required to separate one mole of an ionic solid into its component gaseous ions
### Chemistry End of Section Exercises
1. Which bond in each of the following pairs of bonds is the strongest?
1. C–C or C=C
2. C–N or C≡N
3. C≡O or C=O
4. H–F or H–Cl
5. C–H or O–H
6. C–N or C–O
2. Using the bond energies in Table 1, determine the approximate enthalpy change for each of the following reactions:
1. H2(g) + Br2(g) ⟶ 2HBr(g)
2. CH4(g) + I2(g) ⟶ CH3I(g) + HI(g)
3. C2H4(g) + 3O2(g) ⟶ 2CO2(g) + 2H2O(g)
3. When a chemical formula can generate two different molecular structures (isomers), the structure with the stronger bonds is usually the more stable form. Use bond energies to predict the more stable molecule with the formula H3NO, hydroxylamine:
4. How does the bond energy of HCl(g) differ from the standard enthalpy of formation of HCl(g)?
5. Using the standard enthalpy of formation data in Appendix F, show how the standard enthalpy of formation of HCl(g) can be used to determine the bond energy.
6. Using the standard enthalpy of formation data in Appendix F, determine which bond is stronger: the S–F bond in SF4(g) or in SF6(g)?
7. Complete the following Lewis structure by adding bonds (not atoms), and then indicate the longest bond:
8. Use the bond energy to calculate an approximate value of ΔH for the following reaction. Which is the more stable form of FNO2?
9. Use principles of atomic structure to answer each of the following:[1]
1. The radius of the Ca atom is 197 pm; the radius of the Ca2+ ion is 99 pm. Account for the difference.
2. The lattice energy of CaO(s) is –3460 kJ/mol; the lattice energy of K2O is –2240 kJ/mol. Account for the difference.
3. Given these ionization values, explain the difference between Ca and K with regard to their first and second ionization energies.
Element First Ionization Energy (kJ/mol) Second Ionization Energy (kJ/mol)
K 419 3050
Ca 590 1140
Table 4.
4. The first ionization energy of Mg is 738 kJ/mol and that of Al is 578 kJ/mol. Account for this difference.
10. The lattice energy of LiF is 1023 kJ/mol, and the Li–F distance is 200.8 pm. NaF crystallizes in the same structure as LiF but with a Na–F distance of 231 pm. Which of the following values most closely approximates the lattice energy of NaF: 510, 890, 1023, 1175, or 4090 kJ/mol? Explain your choice.
11. For which of the following substances is the least energy required to convert one mole of the solid into separate ions?
1. MgO
2. SrO
3. KF
4. CsF
5. MgF2
12. The reaction of a metal, M, with a halogen, X2, proceeds by an exothermic reaction as indicated by this equation: M(s) + X2(g) → MX2(s). For each of the following, indicate which option will make the reaction more exothermic. Explain your answers.
2. a high ionization energy vs. a low ionization energy for M
3. an higher bond energy for the halogen vs. a lower bond energy for the halogen
4. a lower electron affinity for the halogen vs. a higher electron affinity for the halogen
5. a larger size of the anion formed by the halogen vs. a smaller size of the anion formed by the halogen
### Answers to Chemistry End of Section Exercises
1. (a) C=C; (b) C≡N; (c) C≡O; (d) H-F; (e) O-H; (f) C-O
2. (a) −103 kJ; (b) 55 kJ; (c) −1324 kJ
3. The greater bond energy is in the figure on the left. It is the more stable form.
4. The enthalpy of formation is -431.6 kJ, while the bond energy of H-Cl is -431 kJ. They are practically the same.
5. $\begin{array}{l l} \text{HCl}(g) \longrightarrow \frac{1}{2} \text{H}_2(g) + \frac{1}{2} \text{Cl}_2(g) & \Delta H^{\circ}_1 = - \Delta H^{\circ}_{\text{f}[\text{HCl}(g)]} \\[1em] \frac{1}{2} \text{H}_2(g) \longrightarrow \text{H}(g) & \Delta H^{\circ}_2 = \Delta H^{\circ}_{\text{f}[\text{H}(g)]} \\[1em] \rule[-1.3ex]{23em}{0.1ex}\hspace{-23em}\frac{1}{2} \text{Cl}_2(g) \longrightarrow \text{Cl}(g) & \Delta H^{\circ}_3 = \Delta H^{\circ}_{\text{f}[\text{Cl}(g)]} \\[1em] \text{HCl}(g) \longrightarrow \text{H}(g) + \text{Cl}(g) & \Delta H^{\circ}_{298} = \Delta H^{\circ}_{1} + \Delta H^{\circ}_{2} + \Delta H^{\circ}_{3} \end{array}$
$\begin{array}{l @{{}={}} l} D_{\text{HCl}} = \Delta H^{\circ}_{298} &= -\Delta H^{\circ}_{\text{f}[\text{HCl}(g)]} + \Delta H^{\circ}_{\text{f}[\text{H}(g)]} + \Delta H^{\circ}_{\text{f}[\text{Cl}(g)]} \\[1em] &= -(-92.307 \;\text{kJ}) + 217.97 \;\text{kJ} + 121.3 \;\text{kJ} \\[1em] &= 431.6 \;\text{kJ} \end{array}$
6. The S–F bond in SF4 is stronger.
7. , the C–C single bonds are longest.
8. ΔH = 82 kJ, the left one is more stable.
9. (a) When two electrons are removed from the valence shell, the Ca radius loses the outermost energy level and reverts to the lower n = 3 level, which is much smaller in radius.
(b) The +2 charge on calcium pulls the oxygen much closer compared with K, thereby increasing the lattice energy relative to a less charged ion.
(c) Removal of the 4s electron in Ca requires more energy than removal of the 4s electron in K because of the stronger attraction of the nucleus and the extra energy required to break the pairing of the electrons. The second ionization energy for K requires that an electron be removed from a lower energy level, where the attraction is much stronger from the nucleus for the electron. In addition, energy is required to unpair two electrons in a full orbital. For Ca, the second ionization potential requires removing only a lone electron in the exposed outer energy level.
(d) In Al, the removed electron is relatively unprotected and unpaired in a p orbital. The higher energy for Mg mainly reflects the unpairing of the 2s electron.
10. 890 kJ/mol. The lattice energy of NaF is less than that that of LiF because of the longer Na-F distance. Using ΔHlattice = $\frac{\text{C}(\text{Z}^+)(\text{Z}^-)}{\text{R}_0}$, the lattice energy of NaF is calculated to be 889 kJ/mol.
11. (d)
12. ΔH = ΔHf° = ΔH°sublimation + ½D + IE + (-EA) + (-ΔHlattice)
a) small M+2 radius will make the reaction more exothermic; lattice energies increases with shorter distances between the ions;
b) low M ionization energy will make the reaction more exothermic (+IE);
c) lower X2 bond energy will make the reaction more exothermic (+½D);
d) higher halogen electron affinity will make the reaction more exothermic (-EA);
e) a smaller size of the anion will make the reaction more exothermic; lattice energies increases with shorter distances between the ions (-ΔHlattice)
1. This question is taken from the Chemistry Advanced Placement Examination and is used with the permission of the Educational Testing Service.
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https://garotadeluxo.com/do-homework-39
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## DOMAIN AND RANGE OF EXPONENTIAL FUNCTIONS
The range of an exponential function depends on the values of a and b: For b = 1, the range of f (x) = abx is simply {a}. For b other than 1 and a > 0, the range of f (x) = abx is (0, infinity). For b other than 1 and a < 0, the range of f
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## Introduction to Exponential Functions
In general, the graph of the basic exponential function y = a x drops from ∞ to 0 when 0 1 . The exponential function y = a x , can be shifted k units vertically and h units
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https://ftp.aimsciences.org/article/doi/10.3934/dcdsb.2013.18.2211
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# American Institute of Mathematical Sciences
November 2013, 18(9): 2211-2238. doi: 10.3934/dcdsb.2013.18.2211
## A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver
1 Statistical and Applied Mathematical Sciences Institute (SAMSI), Research Triangle Park, NC 27709, United States 2 Department of Mathematics, University of Tennessee, Knoxville, TN 37996, United States 3 Department of Mathematics, The University of Tennessee, Knoxville, TN 37996-0612, United States
Received June 2013 Revised August 2013 Published September 2013
In this paper we devise and analyze a mixed discontinuous Galerkin finite element method for a modified Cahn-Hilliard equation that models phase separation in diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system, unconditionally uniquely solvable, and convergent in the natural energy norm with optimal rates. We describe an efficient nonlinear multigrid solver for advancing our semi-implicit scheme in time and conclude the paper with numerical tests confirming the predicted convergence rates and suggesting the near-optimal complexity of the solver.
Citation: Andreas C. Aristotelous, Ohannes Karakashian, Steven M. Wise. A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2211-2238. doi: 10.3934/dcdsb.2013.18.2211
##### References:
[1] A. Aristotelous, "Adaptive Discontinuous Galerkin Finite Element Methods for a Diffuse Interface Model of Biological Growth," PhD thesis, University of Tennessee, 2011. [2] D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM Journal on Numerical Analysis, 19 (1982), 742-760. doi: 10.1137/0719052. [3] D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, 39 (2002), 1749-1779. doi: 10.1137/S0036142901384162. [4] I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty, SIAM Journal on Numerical Analysis, 10 (1973), 863-875. doi: 10.1137/0710071. [5] G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp., 31:45-59, 1977. doi: 10.1090/S0025-5718-1977-0431742-5. [6] A. Baskaran, Z. Hu, J. S. Lowengrub, C. Wang, S. M. Wise and P. Zhou, Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation, Journal of Computational Physics, 250 (2013), 270-292. doi: 10.1016/j.jcp.2013.04.024. [7] J. H. Bramble, "Multigrid Methods," Research Notes in Mathematics Series. Chapman and Hall/CRC, London, 1993. [8] J. W. Cahn, On spinodal decomposition, Acta Metallurgica, 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1. [9] R. Choksi, M. Maras and J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM Journal on Applied Dynamical Systems, 10 (2011), 1344-1362. doi: 10.1137/100784497. [10] R. Choksi and X. Ren, On a derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176. doi: 10.1023/A:1025722804873. [11] P. G. Ciarlet, "Introduction to Numerical Linear Algebra and Optimisation," Cambridge University Press, Cambridge, UK, 1989. [12] C. Collins, J. Shen and S. M. Wise, Unconditionally stable finite difference multigrid schemes for the Cahn-Hilliard-Brinkman equation, Commun. Comput. Phys., 13 (2013), 929-957. doi: 10.4208/cicp.171211.130412a. [13] J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, In "Computing Methods in Applied Sciences," pages 207-216. Springer, Berlin, 1976. [14] C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Archive for Rational Mechanics and Analysis, 96 (1986) ,339-357. doi: 10.1007/BF00251803. [15] C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, In J.F. Rodrigues, editor, Mathematical Models for Phase Change Problems: Proceedings of the European Workshop held at Óbidos, Portugal, October 1-3, 1988, International Series of Numerical Mathematics, 35-73, Berlin, 1989. Birkhäuser Verlag. [16] D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, In J. W. Bullard, R. Kalia, M. Stoneham, and L.Q. Chen, editors, Computational and Mathematical Models of Microstructural Evolution, volume 53, pages 1686-1712, Warrendale, PA, USA, 1998. Materials Research Society. [17] X. Feng and O. A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition, Math. Comput., 76 (2007), 1093-1117. doi: 10.1090/S0025-5718-07-01985-0. [18] X. Feng and S. M. Wise, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM Journal on Numerical Analysis, 50 (2012), 1320-1343. doi: 10.1137/110827119. [19] J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numerische Mathematik, 95 (2003), 527-550. doi: 10.1007/s002110200392. [20] W. Hackbusch, "Multi-Grid Methods and Applications," Springer Series in Computational Mathematics. Springer, Berlin, 2010. [21] M. R. Hanisch, Multigrid preconditioning for the biharmonic Dirichlet problem, SIAM Journal on Numerical Analysis, 30 (1993), 184-214. doi: 10.1137/0730009. [22] Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, Journal of Computational Physics, 228 (2009), 5323-5339. doi: 10.1016/j.jcp.2009.04.020. [23] O. A. Karakashian and W. N. Jureidini, A nonconforming finite element method for the stationary Navier-Stokes equations, SIAM Journal on Numerical Analysis, 35 (1998), 93-120. doi: 10.1137/S0036142996297199. [24] D. Kay, V. Styles and E. Süli, Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection, SIAM Journal on Numerical Analysis, 47 (2009), 2660-2685. doi: 10.1137/080726768. [25] T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. doi: 10.1021/ma00164a028. [26] P. Percell and M. F. Wheeler, A local residual finite element procedure for elliptic equations, SIAM Journal on Numerical Analysis, 15 (1978), 705-714. doi: 10.1137/0715047. [27] U. Trottenberg, C. W. Oosterlee and A. Schüller, "Multigrid," Academic Press, New York, 2005. [28] C. Wang, X. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 28 (2010), 405-423. doi: 10.3934/dcds.2010.28.405. [29] C. Wang and S. M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM Journal on Numerical Analysis, 49 (2011), 945-969. doi: 10.1137/090752675. [30] G. N. Wells, E. Kuhl and K. Garikipati, A discontinuous Galerkin method for the Cahn-Hilliard equation, Journal of Computational Physics, 218 (2006), 860-877. doi: 10.1016/j.jcp.2006.03.010. [31] M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15 (1978), 152-161. doi: 10.1137/0715010. [32] S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, Journal of Scientific Computing, 44 (2010), 38-68. doi: 10.1007/s10915-010-9363-4. [33] S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM Journal on Numerical Analysis, 47 (2009), 2269-2288. doi: 10.1137/080738143. [34] Y. Xia, Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for the Cahn-Hilliard type equations, Journal of Computational Physics, 227 (2007), 472-491. doi: 10.1016/j.jcp.2007.08.001.
show all references
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Zhou, Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation, Journal of Computational Physics, 250 (2013), 270-292. doi: 10.1016/j.jcp.2013.04.024. [7] J. H. Bramble, "Multigrid Methods," Research Notes in Mathematics Series. Chapman and Hall/CRC, London, 1993. [8] J. W. Cahn, On spinodal decomposition, Acta Metallurgica, 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1. [9] R. Choksi, M. Maras and J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM Journal on Applied Dynamical Systems, 10 (2011), 1344-1362. doi: 10.1137/100784497. [10] R. Choksi and X. Ren, On a derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176. doi: 10.1023/A:1025722804873. [11] P. G. Ciarlet, "Introduction to Numerical Linear Algebra and Optimisation," Cambridge University Press, Cambridge, UK, 1989. [12] C. Collins, J. Shen and S. M. Wise, Unconditionally stable finite difference multigrid schemes for the Cahn-Hilliard-Brinkman equation, Commun. Comput. Phys., 13 (2013), 929-957. doi: 10.4208/cicp.171211.130412a. [13] J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, In "Computing Methods in Applied Sciences," pages 207-216. Springer, Berlin, 1976. [14] C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Archive for Rational Mechanics and Analysis, 96 (1986) ,339-357. doi: 10.1007/BF00251803. [15] C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, In J.F. Rodrigues, editor, Mathematical Models for Phase Change Problems: Proceedings of the European Workshop held at Óbidos, Portugal, October 1-3, 1988, International Series of Numerical Mathematics, 35-73, Berlin, 1989. Birkhäuser Verlag. [16] D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, In J. W. Bullard, R. Kalia, M. Stoneham, and L.Q. Chen, editors, Computational and Mathematical Models of Microstructural Evolution, volume 53, pages 1686-1712, Warrendale, PA, USA, 1998. Materials Research Society. [17] X. Feng and O. A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition, Math. Comput., 76 (2007), 1093-1117. doi: 10.1090/S0025-5718-07-01985-0. [18] X. Feng and S. M. Wise, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM Journal on Numerical Analysis, 50 (2012), 1320-1343. doi: 10.1137/110827119. [19] J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numerische Mathematik, 95 (2003), 527-550. doi: 10.1007/s002110200392. [20] W. Hackbusch, "Multi-Grid Methods and Applications," Springer Series in Computational Mathematics. Springer, Berlin, 2010. [21] M. R. Hanisch, Multigrid preconditioning for the biharmonic Dirichlet problem, SIAM Journal on Numerical Analysis, 30 (1993), 184-214. doi: 10.1137/0730009. [22] Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, Journal of Computational Physics, 228 (2009), 5323-5339. doi: 10.1016/j.jcp.2009.04.020. [23] O. A. Karakashian and W. N. Jureidini, A nonconforming finite element method for the stationary Navier-Stokes equations, SIAM Journal on Numerical Analysis, 35 (1998), 93-120. doi: 10.1137/S0036142996297199. [24] D. Kay, V. Styles and E. Süli, Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection, SIAM Journal on Numerical Analysis, 47 (2009), 2660-2685. doi: 10.1137/080726768. [25] T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. doi: 10.1021/ma00164a028. [26] P. Percell and M. F. Wheeler, A local residual finite element procedure for elliptic equations, SIAM Journal on Numerical Analysis, 15 (1978), 705-714. doi: 10.1137/0715047. [27] U. Trottenberg, C. W. Oosterlee and A. Schüller, "Multigrid," Academic Press, New York, 2005. [28] C. Wang, X. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 28 (2010), 405-423. doi: 10.3934/dcds.2010.28.405. [29] C. Wang and S. M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM Journal on Numerical Analysis, 49 (2011), 945-969. doi: 10.1137/090752675. [30] G. N. Wells, E. Kuhl and K. Garikipati, A discontinuous Galerkin method for the Cahn-Hilliard equation, Journal of Computational Physics, 218 (2006), 860-877. doi: 10.1016/j.jcp.2006.03.010. [31] M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15 (1978), 152-161. doi: 10.1137/0715010. [32] S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, Journal of Scientific Computing, 44 (2010), 38-68. doi: 10.1007/s10915-010-9363-4. [33] S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM Journal on Numerical Analysis, 47 (2009), 2269-2288. doi: 10.1137/080738143. [34] Y. Xia, Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for the Cahn-Hilliard type equations, Journal of Computational Physics, 227 (2007), 472-491. doi: 10.1016/j.jcp.2007.08.001.
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## Swipe to navigate through the articles of this issue
07-11-2017 | Issue 3/2018 Open Access
# A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes
Journal:
Journal of Scientific Computing > Issue 3/2018
Authors:
José A. Carrillo, Bertram Düring, Daniel Matthes, David S. McCormick
Important notes
JAC acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) under Grant No. EP/P031587/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award and by the National Science Foundation (NSF) under Grant No. RNMS11-07444 (KI-Net). DM was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”. BD and DSMcC were supported by the Leverhulme Trust research project grant “Novel discretisations for higher-order nonlinear PDE” (RPG-2015-69).
## 1 Introduction
### 1.1 Nonlinear Fokker–Planck Equations
We study a variational Lagrangian discretization of the following type of initial value problem:
\begin{aligned} \partial _{t} \rho&= \Delta P(\rho ) + \nabla \cdot (\rho \,\nabla V)&\text {on } {\mathbb {R}}_{>0}\times {\mathbb {R}}^d, \end{aligned}
(1.1a)
\begin{aligned} \rho (\cdot ,0)&= \rho ^0&\text {on }{\mathbb {R}}^d. \end{aligned}
(1.1b)
This problem is posed for the time-dependent probability density function $$\rho :{\mathbb {R}}_{\ge 0}\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}_{\ge 0}$$, with a given initial density $$\rho ^0$$. We assume that the pressure $$P :{\mathbb {R}}_{\ge 0}\rightarrow {\mathbb {R}}_{\ge 0}$$ can be written in the form
\begin{aligned} P(r) = rh'(r)-h(r) \quad \text {for all }r \ge 0, \end{aligned}
(1.2)
for some non-negative and convex $$h \in C^1({\mathbb {R}}_{\ge 0})\cap C^\infty ({\mathbb {R}}_{>0})$$, and that $$V\in C^2({\mathbb {R}}^d)$$ is a non-negative potential without loss of generality. Problem (1.1) encompasses a large class of diffusion equations, such as—for power-type nonlinearities $$P(r)=r^m$$ and vanishing potential $$V\equiv 0$$—the heat equation ($$m=1$$), porous medium equations ($$m>1$$) and fast diffusion equations ($$m<1$$). By a slight abuse of notation, we refer to (1.1) with more general P and non-vanishing V as nonlinear Fokker–Planck equations. In this paper, we assume a degenerate diffusion, that is $$h(0)=h'(0)=0$$, and a confining potential, that is V is convex, not necessarily strict. For technical reasons, we further need to assume that
\begin{aligned} \lim _{s\rightarrow \infty }sh''(s)=+\infty . \end{aligned}
(1.3)
Since our particular spatio-temporal discretization of the initial value problem (1.1) is based on the Lagrangian representation of its dynamics, and on its variational formulation, we briefly recall both of them now.
### 1.2 Lagrangian Formulation
Equation (1.1) can be written as a transport equation,
\begin{aligned} \partial _t\rho + \nabla \cdot \big (\rho \,\mathbf {v}[\rho ]\big ) = 0, \end{aligned}
(1.4a)
with a velocity field $$\mathbf {v}$$ that depends on the solution $$\rho$$ itself,
\begin{aligned} \mathbf {v}[\rho ] = -\nabla \big (h' (\rho )+V\big ). \end{aligned}
(1.4b)
Various further evolution equations can be written in the form (1.4a), such as non-local aggregation equations (see, e.g., Ambrosio et al. [1]); Keller–Segel type models (see, e.g., Blanchet et al. [5]); and also fourth order thin film equations (see, e.g., Otto [34]) or quantum equations (see, e.g., Gianazza et al. [21]). To simplify the presentation, we stick to equations of nonlinear Fokker–Planck type (1.1a).
The system (1.4) naturally induces a Lagrangian representation of the dynamics, which can be summarized as follows. Below, the reference density $${\overline{\rho }}$$ is a probability density supported on some compact set $$K\subset {\mathbb {R}}^d$$, and we use the notation $$G_\#{\overline{\rho }}$$ for the push-forward of $${\overline{\rho }}$$ under a map $$G :K\rightarrow {\mathbb {R}}^d$$; the definition is recalled in (2.1).
Lemma 1.1
Assume that $$\rho :[0,T]\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}_{\ge 0}$$ is a smooth positive solution of (1.1). Let $$G^0 :K\rightarrow {\mathbb {R}}^d$$ be a given map such that $$G^0_\#{\overline{\rho }}=\rho ^0$$. Further, let $$G :[0,T]\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$$ be the flow map associated to (1.4b), satisfying
\begin{aligned} \partial _tG_t = \mathbf {v}[\rho _t]\circ G_t, \quad G(0,\cdot )=G^0, \end{aligned}
(1.5)
where $$\rho _t:=\rho (t,\cdot )$$ and $$G_t:=G(t,\cdot ) :{\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$$. Then, at any $$t\in [0,T]$$,
\begin{aligned} \rho _t = (G_t)_{\#} {\overline{\rho }}. \end{aligned}
(1.6)
In short, the solution G to (1.5) is a Lagrangian map for the solution $$\rho$$ to (1.1). This fact is an immediate consequence of (1.4a); for convenience of the reader, we recall the proof in “Appendix A”. Subsequently, (1.6) can be substituted for $$\rho$$ in the expression (1.4b) for the velocity, which makes (1.5) an autonomous evolution equation for G:
\begin{aligned} \partial _tG_t = -\nabla \left[ h'\left( \frac{{\overline{\rho }}}{\det \mathrm {D}G_t}\right) \right] \circ G_t-\nabla V\circ G_t. \end{aligned}
(1.7)
A more explicit form of (1.7) is derived in (5.2).
### 1.3 Variational Structure
It is well-known (see Otto [35] or Ambrosio et al. [1]) that (1.1) is a gradient flow for the relative Renyi entropy functional
\begin{aligned} {\mathcal {E}}(\rho ) = \int _{{\mathbb {R}}^d}\big [ h(\rho (x))+V(x)\rho (x)\big ]\,\mathrm {d}x, \end{aligned}
(1.8)
with respect to the $$L^2$$-Wasserstein metric on the space $${\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$ of probability densities on $${{\mathbb {R}}^d}$$ with finite second moment. It appears to be less well known (see Evans et al. [20], Carrillo and Moll [13], or Carrillo and Lisini [12]) that also (1.7) is a gradient flow, namely for the functional
\begin{aligned} {\mathbf {E}}(G|{\overline{\rho }}) := {\mathcal {E}}(G_\#{\overline{\rho }}) = \int _K \left[ \widetilde{h}\left( \frac{\det \mathrm {D}G}{{\overline{\rho }}}\right) + V\circ G\right] {\overline{\rho }}\,\mathrm {d}\omega , \quad \widetilde{h}(s):=s\,h(s^{-1}), \end{aligned}
(1.9)
on the Hilbert space $$L^2(K\rightarrow {{\mathbb {R}}^d};{\overline{\rho }})$$ of square integrable maps from K to $${{\mathbb {R}}^d}$$. We shall discuss these gradient flow structures in more detail in Sect. 2 below.
### 1.4 Discretization and Approximation Results
Our discretization in space is based on the Lagrangian formulation. Instead of numerically integrating (1.1a) to obtain the density $$\rho$$ directly, we approximate the associated Lagrangian maps G that satisfy (1.7): specifically, we assume that a simplicial decomposition $${\mathscr {T}}$$ of K is given, and we restrict G to the finite dimensional subspace $${\mathcal {A}}_{\mathscr {T}}$$ of continuous maps from K to $${\mathbb {R}}^d$$ that are piecewise linear with respect to $${\mathscr {T}}$$. A posteriori, we recover an approximation of $$\rho$$ via (1.6). That ansatz for the Lagrangian maps corresponds to a simple geometric picture: the induced densities are piecewise constant on triangles whose vertices move in time.
For the discretization in time, we exploit the aforementioned variational structure of (1.7): namely, we adopt the celebrated minimizing movement scheme that is known to provide a robust approximation of gradient flows. In the context at hand, this scheme reads as follows: let a time step $$\tau >0$$ and an initial condition $$G_\boxplus ^0\in {\mathcal {A}}_{\mathscr {T}}$$ be given. (Here and below, $$\boxplus$$ symbolizes the space-time mesh generated by $${\mathscr {T}}$$ on K and $$\tau$$ on $${\mathbb {R}}_{>0}$$.) Then the nth time iterate $$G_\boxplus ^n\in {\mathcal {A}}_{\mathscr {T}}$$—that serves as our approximation of $$G(n\tau ;\cdot )$$—is chosen inductively for $$n=1,2,\ldots$$ as the minimizer in the respective problem
\begin{aligned} \frac{1}{2\tau }\Vert G-G_\boxplus ^n\Vert _{L^2(K\rightarrow {\mathbb {R}}^d;{\overline{\rho }})}^2+{\mathbf {E}}(G|{\overline{\rho }}) \quad \longrightarrow \quad \min , \end{aligned}
(1.10)
where the minimization is carried out over the finite dimensional space $${\mathcal {A}}_{\mathscr {T}}$$. With the sequence $$(G_\boxplus ^n)_{n=0,1,\ldots }$$ of approximating Lagrangian maps at hand, we define piecewise-constant-in-time interpolations for the derived density $${\widetilde{\rho }}_\boxplus$$ and velocity $${\widetilde{\mathbf {v}}}_\boxplus$$ as usual via
\begin{aligned} {\widetilde{\rho }}_\boxplus (t) = (G_\boxplus ^n)_{\#}{\overline{\rho }}, \quad {\widetilde{\mathbf {v}}}_\boxplus (t) = \frac{G_\boxplus ^n-G_\boxplus ^{n-1}}{\tau }\quad \text {with }n \text { such that } t\in ((n-1)\tau ,n\tau ]. \end{aligned}
Our analytical results on the scheme can be summarized as follows.
• The sequence of fully discrete minimization problems (1.10) is well-posed: see Lemma 3.1. We thus obtain a sequence $$(G_\boxplus ^n)_{n=0,1,\ldots }$$ for each sufficiently fine discretization $$\boxplus$$.
• The $$G_\boxplus ^n$$ are entropy-diminishing and are $$\boxplus$$-uniformly Hölder continuous: see Lemma 4.1.
• Consequently, the induced densities $${\widetilde{\rho }}_\boxplus$$ converge weakly to an absolutely continuous limit trajectory $$\rho$$, and the fluxes $${\widetilde{\rho }}_\boxplus {\widetilde{\mathbf {v}}}_\boxplus$$ converge weakly to a limit of the form $$\rho \mathbf {v}$$: see Theorem 4.2. The identification of the limit velocity $$\mathbf {v}$$, however, is only possible under strong additional hypotheses: see Corollary 4.5.
• In $$d=2$$ dimensions, we prove numerical consistency in the sense that, if G is a smooth solution to (1.7), then its restriction to the mesh $$\boxplus$$ satisfies the fully discrete Euler–Lagrange equations associated to (1.10), with a quantifiable error that vanishes in a suitable continuous limit: see Theorem 5.2.
• Our previously mentioned consistency results requires that the triangulation $${\mathscr {T}}$$ of K is almost ideally hexagonal: see Eq. (5.7). We discuss why consistency cannot be expected if that condition is violated: see Remark 5.4.
### 1.5 Comparison with Results in the Literature
The approach presented in this paper is an alternative to the one developed by Carrillo et al. [13, 15], where G is obtained by directly solving the PDE (1.7) numerically with finite differences or Galerkin approximation via finite element methods. In other words, while Carrillo et al. [13, 15] follows the strategy minimize first then discretize, our present approach is to discretize first then minimize. In the former approach, the minimization (1.10) is performed on the spatially continuous level, yielding Euler–Lagrange equations that are then discretized in space; in the present approach, the space of Lagrangian maps is approximated by the finite dimensional subspace $${\mathcal {A}}_{\mathscr {T}}$$, and the minimization problem (1.10) on $${\mathcal {A}}_{\mathscr {T}}$$ yields a nonlinear system of Euler–Lagrange equations that are directly solvable numerically.
Let us mention that other numerical methods have been developed to conserve particular properties of solutions of the gradient flow (1.1). Finite volume methods preserving the decay of energy at the semi-discrete level, along with other important properties like non-negativity and mass conservation, were proposed in the papers [4, 8, 10]. Particle methods based on suitable regularizations of the flux of the continuity Eq. (1.1) have been proposed in the papers [18, 27, 28, 37]. A particle method based on the steepest descent of a regularized internal part of the energy $${\mathcal {E}}$$ in (1.8) by substituting particles by non-overlapping blobs was proposed and analysed in Carrillo et al. [11, 14]. Deterministic particle methods for diffusions have been recently explored, see [9] and the references therein. High-order relaxation schemes for nonlinear diffusion problems have been proposed in Cavalli et al. [16], while high-resolution schemes for nonlinear convection-diffusion problems are introduced in Kurganov et al. [26]. Moreover, the numerical approximation of the JKO variational scheme has already been tackled by different methods using pseudo-inverse distributions in one dimension (see [5, 7, 23, 40]) or solving for the optimal map in a JKO step (see [3, 25]). Finally, note that gradient-flow-based Lagrangian methods in one dimension for higher-order, drift diffusion and Fokker–Planck equations have recently been proposed in the papers [19, 3133].
There are two main arguments in favour of our taking this indirect approach of solving (1.7) instead of solving (1.1). The first is our interest in structure-preserving discretizations: the scheme that we present builds on the non-obvious “secondary” gradient flow representation of (1.1) in terms of Lagrangian maps. The benefits include monotonicity of the transformed entropy functional $${\mathbf {E}}$$ and $$L^2$$ control on the metric velocity for our fully discrete solutions, that eventually lead to weak compactness of the trajectories in the continuous limit. We remark that our long-term goal is to design a numerical scheme that makes full use of the much richer “primary” variational structure of (1.1) in the Wasserstein distance, which is reviewed in Sect. 2 below. However, despite significant effort in the recent past—see, e.g., the references [3, 5, 14, 15, 19, 22, 25, 29, 36, 40]—it has not been possible so far to preserve features like metric contractivity of the flow under the discretization, except in the rather special situation of one space dimension (see Matthes and Osberger [29]). This is mainly due to the non-existence of finite-dimensional submanifolds of $${\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$ that are complete with respect to generalized geodesics.
The second motivation is that Lagrangian schemes are a natural choice for numerical front tracking, see, e.g., Budd [6] for first results on the numerical approximation of self-similar solutions to the porous medium equation. We recall that, due to the assumed degeneracy $$P'(0)=0$$ of the diffusion in (1.1), solutions that are compactly supported initially remain compactly supported for all times. A numerically accurate calculation of the moving edge of support is challenging, since the solution can have a very complex behavior near that edge, like the waiting time phenomenon (see Vazquez [38]). Our simulation results for $$\partial _t\rho =\Delta (\rho ^3)$$ — which possesses an analytically known, compactly supported, self-similar Barenblatt solution — indicate that our discretization is indeed able to track the edge of support quite accurately.
The expected convergence of our scheme, with implicit Euler stepping in time and piecewise linear approximation of the Lagrangian maps, is of first order in both space and time. This is confirmed in our experiments. For an improved approximation, particularly of the moving fronts, numerical schemes with a higher order of consistency would be desirable. In principle, such schemes could be constructed along the same lines, for example, by replacing the implicit Euler method by a Runge–Kutta method in time, and the piecewise constant ansatz space $${\mathcal {A}}_{\mathscr {T}}$$ by finite elements with functions of higher global regularity in space. However, it is unclear if a similar degree of structure preservation can be achieved for these schemes, and their analysis would be very different from the one presented here.
### 1.6 Structure of the Paper
This work is organized as follows. In Sect. 2, we present an overview of previous results in gradient flows pertaining our work. Section 3 is devoted to the introduction of the linear set of Lagragian maps and the derivation of the numerical scheme. Section 4 shows the compactness of the approximated sequences of discretizations and we give conditions leading to the eventual convergence of the scheme towards (1.1). Section 5 deals with the consistency of the scheme in two dimensions, while Sect. 6 gives several numerical tests showing the performance of this scheme.
### 2.1 Notations from Probability Theory
$${\mathcal {P}}(X)$$ is the space of probability measures on a given base set X. We say that a sequence $$(\mu _n)$$ of measures in $${\mathcal {P}}(X)$$ converges narrowly to a limit $$\mu$$ in that space if
\begin{aligned} \int _Xf(x)\,\mathrm {d}\mu _n(x)\rightarrow \int _Xf(x)\,\mathrm {d}\mu (x) \end{aligned}
for all bounded and continuous functions $$f\in C^0_b(X)$$. The push-forward $$T_\#\mu$$ of a measure $$\mu \in {\mathcal {P}}(X)$$ under a measurable map $$T :X\rightarrow Y$$ is the uniquely determined measure $$\nu \in {\mathcal {P}}(Y)$$ such that, for all $$g\in C^0_b(Y)$$,
\begin{aligned} \int _Xg\circ T(x)\,\mathrm {d}\mu (x) = \int _Yg(y)\,\mathrm {d}\nu (y). \end{aligned}
With a slight abuse of notation — identifying absolutely continuous measures with their densities—we denote the space of probability densities on $${{\mathbb {R}}^d}$$ of finite second moment by
\begin{aligned} {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)= \left\{ \rho \in L^1({{\mathbb {R}}^d})\,;\,\rho \ge 0,\,\int _{{\mathbb {R}}^d}\rho (x)\,\mathrm {d}x=1,\,\int _{{\mathbb {R}}^d}\Vert x\Vert ^2\rho (x)\,\mathrm {d}x<\infty \right\} . \end{aligned}
Clearly, the reference density $${\overline{\rho }}$$, which is supported on the compact set $$K\subset {\mathbb {R}}^d$$, belongs to $${\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$. If $$G :K\rightarrow {\mathbb {R}}^d$$ is a diffeomorphism onto its image (which is again compact), then the push-forward of $${\overline{\rho }}$$’s measure produces again a density $$G_\#{\overline{\rho }}\in {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$, given by
\begin{aligned} G_\#{\overline{\rho }}= \frac{{\overline{\rho }}}{\det \mathrm {D}G}\circ G^{-1}. \end{aligned}
(2.1)
### 2.2 Gradient Flow in the Wasserstein Metric
Below, some basic facts about the Wasserstein metric and the formulation of (1.1) as gradient flow in that metric are briefly reviewed. For more detailed information, we refer the reader to the monographs of Ambrosio et al. [1] and Villani [39].
One of the many equivalent ways to define the $$L^2$$-Wasserstein distance between $$\rho _0,\rho _1\in {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$ is as follows:
\begin{aligned} \mathrm {W}_2(\rho _0,\rho _1) := \inf \left\{ \int _{{\mathbb {R}}^d} \Vert T(x)-x\Vert ^2\rho _0(x)\,\mathrm {d}x\,;\,T:{{\mathbb {R}}^d}\rightarrow {{\mathbb {R}}^d}\ \text {measurable},\,T_\#\rho _0=\rho _1\right\} ^{\frac{1}{2}}. \end{aligned}
(2.2)
The infimum above is in fact a minimum, and the — essentially unique — optimal map $$T^*$$ is characterized by Brenier’s criterion; see, e.g., Villani [39, Section 2.1]. A trivial but essential observation is that if $${\overline{\rho }}\in {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$ is a reference density with support $$K\subset {{\mathbb {R}}^d}$$, and $$\rho _0=(G_0)_\#{\overline{\rho }}$$ with a measurable $$G_0 :K\rightarrow {{\mathbb {R}}^d}$$, then (2.2) can be re-written as follows:
\begin{aligned} \mathrm {W}_2(\rho _0,\rho _1) = \inf \left\{ \int _K \Vert G(\omega )-G_0(\omega )\Vert ^2{\overline{\rho }}(\omega )\,\mathrm {d}\omega \,;\,G :K\rightarrow {{\mathbb {R}}^d}\ \text {measurable},\,G_\#{\overline{\rho }}=\rho _1\right\} ^{\frac{1}{2}}, \end{aligned}
(2.3)
and the essentially unique minimizer $$G^*$$ in (2.3) is related to the optimal map $$T^*$$ in (2.2) via $$G^*=T^*\circ G_0$$.
$$\mathrm {W}_2$$ is a metric on $${\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$; convergence in $$\mathrm {W}_2$$ is equivalent to weak-$$\star$$ convergence in $$L^1({{\mathbb {R}}^d})$$ and convergence of the second moment. Since P and hence also h are of super-linear growth at infinity, each sublevel set $${\mathcal {E}}$$ is weak-$$\star$$ closed and thus complete with respect to $$\mathrm {W}_2$$.
As already mentioned above, solutions $$\rho$$ to (1.1) constitute a gradient flow for the functional $${\mathcal {E}}$$ from (1.8) in the metric space $$({\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d);\mathrm {W}_2)$$. In fact, assuming that the potential V is $$\lambda$$-convex (i.e., $$\nabla ^2V\ge \lambda \mathbb {1}$$), the flow is even $$\lambda$$-contractive as a semi-group, thanks to the $$\lambda$$-uniform displacement convexity of $${\mathcal {E}}$$ (see McCann [30], or Daneri and Savaré [17]), which is a strengthened form of $$\lambda$$-uniform convexity along geodesics. The $$\lambda$$-contractivity of the flow implies various properties (see Ambrosio et al. [1, Section 11.2]) like global existence, uniqueness and regularity of the flow, monotonicity of $${\mathcal {E}}$$ and its sub-differential, uniform exponential estimates on the convergence (if $$\lambda >0$$) or divergence (if $$\lambda \le 0$$) of trajectories, quantified exponential rates for the approach to equilibrium (if $$\lambda >0$$) and the like.
An important further consequence is that the unique flow can be obtained as the limit for $$\tau \searrow 0$$ of the time-discrete minimizing movement scheme (see Ambrosio et al. [1] and Jordan, Kinderlehrer and Otto [24]):
\begin{aligned} \rho _\tau ^{n} := \mathop {{{\mathrm{argmin}}}}\limits _{\rho \in {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)}{\mathcal {E}}_\tau (\rho ;\rho _\tau ^{n-1}), \quad {\mathcal {E}}_\tau (\rho ,\hat{\rho }):=\frac{1}{2\tau }\mathrm {W}_2(\rho ,\hat{\rho })^{2} + {\mathcal {E}}(\rho ). \end{aligned}
(2.4)
This time discretization is well-adapted to approximate $$\lambda$$-contractive gradient flows. All of the properties of mentioned above are already reflected on the level of these time-discrete solutions.
### 2.3 Gradient Flow in $$L^2$$
Equation (1.7) is the gradient flow of $${\mathbf {E}}$$ on the space $$L^2(K\rightarrow {{\mathbb {R}}^d};{\overline{\rho }})$$ of square integrable (with respect to $${\overline{\rho }}$$) maps $$G :K\rightarrow {{\mathbb {R}}^d}$$ (see Evans et al. [20] or Jordan et al. [25]). However, the variational structure behind this gradient flow is much weaker than above: most notably, $${\mathbf {E}}$$ is only poly-convex, but not $$\lambda$$-uniformly convex. Therefore, the abstract machinery for $$\lambda$$-contractive gradient flows in Ambrosio et al. [1] does not apply here. Clearly, by equivalence of (1.1) and (1.7) at least for sufficiently smooth solutions, certain properties of the primary gradient flow are necessarily inherited by this secondary flow, but for instance $$\lambda$$-contractivity of the flow in the $$L^2$$-norm seems to fail.
Nevertheless, it can be proven (see Ambrosio, Lisini and Savaré [2]) that the gradient flow is globally well-defined, and it can again be approximated by the minimizing movement scheme:
\begin{aligned} G_\tau ^n:=\mathop {{{\mathrm{argmin}}}}\limits _{G\in L^2(K\rightarrow {{\mathbb {R}}^d};{\overline{\rho }})}{\mathbf {E}}_\tau \big (G;G_\tau ^{n-1}\big ), \quad {\mathbf {E}}_\tau (G;\hat{G})= \frac{1}{2\tau }\int _K\Vert G-\hat{G}\Vert ^2\,\,\mathrm {d}{\overline{\rho }}+ {\mathbf {E}}(G|{\overline{\rho }}). \end{aligned}
(2.5)
In fact, there is an equivalence between (2.5) and (2.4): simply substitute $$(G_\tau ^{n-1})_\#{\overline{\rho }}$$ for $$\rho _\tau ^{n-1}$$ and $$G_\#{\overline{\rho }}$$ for $$\rho$$ in (2.4); notice that any $$\rho \in {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$ can be written as $$G_\#{\overline{\rho }}$$ with a suitable (highly non-unique) choice of $$G\in L^2(K\rightarrow {{\mathbb {R}}^d};{\overline{\rho }})$$. This equivalence was already exploited in Carrillo et al. [13, 15]. Thanks to the equality (2.3), the minimization with respect to $$\rho =G_\#{\overline{\rho }}$$ can be relaxed to a minimization with respect to G. Consequently, if $$(G_\tau ^0)_\#{\overline{\rho }}=\rho _\tau ^0$$, then $$(G_\tau ^n)_\#{\overline{\rho }}=\rho _\tau ^n$$ at all discrete times $$n=1,2,\ldots$$. However, while the functional $${\mathcal {E}}_\tau (\cdot ;\rho _\tau ^{n-1})$$ in (2.4) is $$(\lambda +\tau ^{-1})$$-uniformly convex in $$\rho$$ along geodesics in $$\mathrm {W}_2$$, the functional $${\mathbf {E}}_\tau (\cdot ;G_\tau ^{n-1})$$ in (2.5) has apparently no useful convexity properties in G on $$L^2(K\rightarrow {{\mathbb {R}}^d};{\overline{\rho }})$$.
## 3 Definition of the Numerical Scheme
Recall the Lagrangian formulation of (1.1) that has been given in Lemma 1.1. For definiteness, fix a reference density $${\overline{\rho }}\in {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$, whose support $$K\subset {{\mathbb {R}}^d}$$ is a compact, convex polytope.
### 3.1 Discretization in Space
Our spatial discretization is performed using a finite subspace of linear maps for the Lagrangian maps G. More specifically: let $${\mathscr {T}}$$ be some (finite) simplicial decomposition of K with nodes $$\omega _{1}$$ to $$\omega _{L}$$ and n-simplices $$\Delta _{1}$$ to $$\Delta _{M}$$. In the case $$d=2$$, which is of primary interest here, $${\mathscr {T}}$$ is a triangulation, with triangles $$\Delta _m$$. The reference density $${\overline{\rho }}$$ is approximated by a density $${\overline{\rho }}_{\mathscr {T}}\in {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$ that is piecewise constant on the simplices of $${\mathscr {T}}$$, with respective values
\begin{aligned} {\overline{\rho }}_{\mathscr {T}}^m := \frac{\mu _{\mathscr {T}}^m}{|\Delta _m|} \quad \text {for the simplex masses}\quad \mu _{\mathscr {T}}^m:=\int _{\Delta _m}{\overline{\rho }}(\omega )\,\mathrm {d}\omega . \end{aligned}
(3.1)
The finite dimensional ansatz space $${\mathcal {A}}_{\mathscr {T}}$$ is now defined as the set of maps $$G:K\rightarrow {{\mathbb {R}}^d}$$ that are globally continous, affine on each of the simplices $$\Delta _m\in {\mathscr {T}}$$, and orientation preserving. That is, on each $$\Delta _m\subset {\mathscr {T}}$$, the map $$G\in {\mathcal {A}}_{\mathscr {T}}$$ can be written as follows:
\begin{aligned} G(\omega ) = A_{m} \omega + b_{m} \qquad \text {for all } \omega \in \Delta _{m}, \end{aligned}
(3.2)
with a suitable matrix $$A_{m} \in {\mathbb {R}}^{d\times d}$$ of positive determinant and a vector $$b_{m} \in {\mathbb {R}}^d$$.
For the calculations that follow, we shall use a more geometric way to describe the maps $$G\in {\mathcal {A}}_{\mathscr {T}}$$, namely by the positions $$G_\ell =G(\omega _\ell )$$ of the images of each node $$\omega _\ell$$. Denote by $$({\mathbb {R}}^d)_{\mathscr {T}}^L\subset {\mathbb {R}}^{L\cdot d}$$ the space of L-tuples $$\mathbf {G}=(G_\ell )_{\ell =1}^L$$ of points $$G_\ell \in {{\mathbb {R}}^d}$$ with the same simplicial combinatorics (including orientation) as the $$\omega _\ell$$ in $${\mathscr {T}}$$. Clearly, any $$G\in {\mathcal {A}}_{\mathscr {T}}$$ is uniquely characterized by the L-tuple $$\mathbf {G}$$ of its values, and moreover, any $$\mathbf {G}\in ({\mathbb {R}}^d)_{\mathscr {T}}^L$$ defines a $$G\in {\mathcal {A}}_{\mathscr {T}}$$.
More explicitly, fix a $$\Delta _m\in {\mathscr {T}}$$, with nodes labelled $$\omega _{m,0}$$ to $$\omega _{m,d}$$ in some orientation preserving order, and respective image points $$G_{m,0}$$ to $$G_{m,d}$$. With the standard d-simplex given by
\begin{aligned} \mathord {\bigtriangleup }^d:= \left\{ \xi =(\xi _1,\ldots ,\xi _d)\in {\mathbb {R}}_{\ge 0}^d\,;\,\sum _{j=1}^d\xi _j\le 1\right\} , \end{aligned}
introduce the linear interpolation maps $$r_m :\mathord {\bigtriangleup }^d\rightarrow K$$ and $$q_m :\mathord {\bigtriangleup }^d\rightarrow {{\mathbb {R}}^d}$$ by
\begin{aligned} r_m(\xi )&= \omega _{m,0} + \sum _{j=1}^d(\omega _{m,j}-\omega _{m,0})\xi _j, \\ q_m(\xi )&= G_{m,0} + \sum _{j=1}^d(G_{m,j}-G_{m,0})\xi _j. \end{aligned}
Then the affine map (3.2) equals to $$q_m\circ r_m^{-1}$$; this is shown schematically in Fig. 1 for the case $$d=2$$. In particular, we obtain that
\begin{aligned} \det A_m = \frac{\det \mathrm {D}q_m}{\det \mathrm {D}r_m} = \frac{\det Q_{\mathscr {T}}^m[G]}{2|\Delta _m|} \quad \text {where}\quad Q_{\mathscr {T}}^m[G]:=\big (G_{m,1}-G_{m,0}\big |\cdots \big |G_{m,d}-G_{m,0}\big ). \end{aligned}
(3.3)
For later reference, we give a more explicit representation for the transformed entropy $${\mathbf {E}}$$ for $$G\in {\mathcal {A}}_{\mathscr {T}}$$, and for the $$L^2$$-distance between two maps $$G,\hat{G}\in {\mathcal {A}}_{\mathscr {T}}$$. Substitution of the special form (3.2) into (1.9) produces
\begin{aligned} {\mathbf {E}}(G|{\overline{\rho }}_{\mathscr {T}}) =\sum _{\Delta _m\in {\mathscr {T}}}\mu _{\mathscr {T}}^m \big [{\mathbb {H}}_{\mathscr {T}}^m(G)+{\mathbb {V}}_{\mathscr {T}}^m(G)\big ] \end{aligned}
(3.4)
with the internal energy [recall the definition of $$\widetilde{h}$$ from (1.9)]
\begin{aligned} {\mathbb {H}}_{\mathscr {T}}^m(G) :=\widetilde{h} \left( \frac{\det A_m}{{\overline{\rho }}_{\mathscr {T}}^m} \right) = \widetilde{h}\left( \frac{\det Q_{\mathscr {T}}^m[G]}{2\mu _{\mathscr {T}}^m}\right) \end{aligned}
and the potential energy
For the $$L^2$$-difference of G and $$G^*$$, we have
\begin{aligned} \Vert G-G^*\Vert _{L^2(K;{\overline{\rho }}_{\mathscr {T}})}^2 =\int _K\Vert G-G^*\Vert ^2{\overline{\rho }}_{\mathscr {T}}\,\mathrm {d}\omega = \sum _{\Delta _m\in {\mathscr {T}}}\mu _{\mathscr {T}}^m {\mathbb {L}}_{\mathscr {T}}^m(G,G^*). \end{aligned}
(3.5)
Using Lemma B.1, we obtain on each simplex $$\Delta _m$$:
(3.6)
### 3.2 Discretization in Time
Let a time step $$\tau >0$$ be given; in the following, we symbolize the spatio-temporal discretization by $$\boxplus$$, and we write $$\boxplus \rightarrow 0$$ for the joint limit of $$\tau \rightarrow 0$$ and vanishing mesh size in $${\mathscr {T}}$$.
The discretization in time is performed in accordance with (2.5): we modify $${\mathbf {E}}_\tau$$ from (2.5) by restriction to the ansatz space $${\mathcal {A}}_{\mathscr {T}}$$. This leads to the minimization problem
\begin{aligned} G_\boxplus ^n:&=\mathop {{{\mathrm{argmin}}}}\limits _{G\in {\mathcal {A}}_{\mathscr {T}}}{\mathbf {E}}_\boxplus \big (G;G_\boxplus ^{n-1}\big ) \quad \text {where}\quad {\mathbf {E}}_\boxplus (G;G^*)\nonumber \\&= \frac{1}{2\tau }\Vert G-G^*\Vert _{L^2(K;{\overline{\rho }}_{\mathscr {T}})}^2 + {\mathbf {E}}(G|{\overline{\rho }}_{\mathscr {T}}). \end{aligned}
(3.7)
For a fixed discretization $$\boxplus$$, the fully discrete scheme is well-posed in the sense that for a given initial map $$G_\boxplus ^0\in {\mathcal {A}}_{\mathscr {T}}$$, an associated sequence $$(G_\boxplus ^n)_{n\ge 0}$$ can be determined by successive solution of the minimization problems (3.7). One only needs to verify:
Lemma 3.1
For each given $$G^*\in {\mathcal {A}}_{\mathscr {T}}$$, there exists at least one global minimizer $$G\in {\mathcal {A}}_{\mathscr {T}}$$ of $${\mathbf {E}}_\boxplus (\cdot ;G^*)$$.
Remark 3.2
We do not claim uniqueness of the minimizers. Unfortunately, the minimization problem (3.7) inherits the lack of convexity from (2.5), whereas the correspondence between (2.5) and the convex problem (2.4) is lost under spatial discretization. A detailed discussion of $${\mathbf {E}}_\boxplus$$’s (non-)convexity is provided in “Appendix C”.
Proof of Lemma 3.1
We only sketch the main arguments. For definiteness, let us choose (just for this proof) one of the infinitely many equivalent norm-induced metrics on the dL-dimensional vector space $$V_{\mathscr {T}}$$ of all continuous maps $$G :K\rightarrow {\mathbb {R}}^d$$ that are piecewise affine with respect to the fixed simplicial decomposition $${\mathscr {T}}$$: given $$G,G'\in V_{\mathscr {T}}$$ with their respective point locations $$\mathbf {G},\mathbf {G}'\in {\mathbb {R}}^{dL}$$, i.e., $$\mathbf {G}=(G_\ell )_{\ell =1}^L$$ for $$G_\ell =G(\omega _\ell )$$, define the distance between these maps as the maximal $${\mathbb {R}}^d$$-distance $$\Vert G_\ell -G'_\ell \Vert$$ of corresponding points $$G_\ell \in \mathbf {G}$$, $$G'_\ell \in \mathbf {G}'$$. Clearly, this metric makes $$V_{\mathscr {T}}$$ a complete space.
It is easily seen that the subset $${\mathcal {A}}_{\mathscr {T}}$$ — which is singled out by requiring orientation preservation of the G’s—is an open subset of $$V_{\mathscr {T}}$$. It is further obvious that the map $$G\mapsto {\mathbf {E}}_\boxplus (G;G^*)$$ is continuous with respect to the metric. The claim of the lemma thus follows if we can show that the sub-level
\begin{aligned} S_c:=\left\{ G\in {\mathcal {A}}_{\mathscr {T}}\,;\,{\mathbf {E}}_\boxplus (G;G^*)\le c\right\} \quad \text {with}\quad c:={\mathbf {E}}(G^*|{\overline{\rho }}_{\mathscr {T}}) \end{aligned}
is a non-empty compact subset of $$V_{\mathscr {T}}$$. Clearly, $$G^*\in S_c$$, so it suffices to verify compactness.
$$S_c$$ is bounded. We are going to show that there is a radius $$R>0$$ such that no $$G\in S_c$$ has a distance larger than R to $$G^*$$. From non-negativity of $${\mathbf {E}}$$, and from the representations (3.5) and (3.6), it follows that
\begin{aligned} c\ge \frac{1}{2\tau }\Vert G-G^*\Vert _{L^2(K;{\overline{\rho }}_{\mathscr {T}})}^2&\ge \frac{\underline{\mu }_{\mathscr {T}}}{2\tau }\sum _{\Delta _m\in {\mathscr {T}}}{\mathbb {L}}_{\mathscr {T}}^m(G,G^*)\\&= \frac{\underline{\mu }_{\mathscr {T}}}{(d+1)(d+2) \tau }\sum _{0\le i\le j\le d}(G_{m,i}-G^*_{m,i})\cdot (G_{m,j}-G^*_{m,j})\\&\ge \frac{\underline{\mu }_{\mathscr {T}}}{2(d+1)(d+2) \tau }\sum _{\ell =1}^L\Vert G_\ell -G_\ell ^*\Vert ^2, \end{aligned}
where $$\underline{\mu }_{\mathscr {T}}=\min _{\Delta _m}\mu _{\mathscr {T}}^m$$. It is now easy to compute a suitable value for the radius R.
$$S_c$$ is a closed subset of $$V_{\mathscr {T}}$$. It suffices to show that the limit $$\overline{G}\in V_{\mathscr {T}}$$ of any sequence $$(G^{(k)})_{k=1}^\infty$$ of maps $$G^{(k)}\in S_c$$ belongs to $${\mathcal {A}}_{\mathscr {T}}$$. By definition of our metric on $$V_{\mathscr {T}}$$, global continuity and piecewise linearity of the $$G^{(k)}$$ trivially pass to the limit $$\overline{G}$$. We still need to verify that $$\overline{G}$$ is orientation-preserving. Fix a simplex $$\Delta _m$$ and consider the corresponding matrices $$A_m^{(k)}$$ and $$\overline{A}_m$$ from (3.2). Since the $$G^{(k)}$$ converge to $$\overline{G}$$ in the metric, also $$A_m^{(k)}\rightarrow \overline{A}_m$$ entry-wise. Now, by non-negativity of $$\widetilde{h}$$, we have for all k that
\begin{aligned} c\ge {\mathbf {E}}(G^{(k)}|{\overline{\rho }}_{\mathscr {T}}) \ge \mu _{\mathscr {T}}^m\widetilde{h}\left( \frac{\det A^{(k)}_m}{{\overline{\rho }}_{\mathscr {T}}^m}\right) , \end{aligned}
and since $$\widetilde{h}(s)\rightarrow +\infty$$ as $$s\downarrow 0$$, it follows that $$\det A^{(k)}_m>0$$ is bounded away from zero, uniformly in k. But then also $$\det \overline{A}_m>0$$, i.e., the mth linear map piece of the limit $$\overline{G}$$ preserves orientation. $$\square$$
### 3.3 Fully Discrete Equations
We shall now derive the Euler–Lagrange equations associated to the minimization problem (3.7), i.e., for each given $$G^*:=G_\boxplus ^{n-1}\in {\mathcal {A}}_{\mathscr {T}}$$, we calculate the variations of $${\mathbf {E}}_\boxplus (G;G^*)$$ with respect to the degrees of freedom of $$G\in {\mathcal {A}}_{\mathscr {T}}$$. Since that function is a weighted sum over the triangles $$\Delta _m\in {\mathscr {T}}$$, it suffices to perform the calculations for one fixed triangle $$\Delta _m$$, with respective nodes $$\omega _{m,0}$$ to $$\omega _{m,d}$$, in positive orientation. The associated image points are $$G_{m,0}$$ to $$G_{m,d}$$. Since we may choose any vertex to be labelled $$\omega _{m,0}$$, it will suffice to perform the calculations at one fixed image point $$G_{m,0}$$.
• mass term:
\begin{aligned} \frac{\partial }{\partial G_{m,0}}{\mathbb {L}}_{\mathscr {T}}^m(G,G^*)&=\frac{2}{(d+1)(d+2)}\frac{\partial }{\partial G_{m,0}}\sum _{0\le i\le j\le d}(G_{m,i}-G^*_{m,i})\cdot (G_{m,j}-G^*_{m,j}) \\&=\frac{2}{(d+1)(d+2)}\left( 2(G_{m,0}-G^*_{m,0})+\sum _{j=1}^d(G_{m,j}-G^*_{m,j})\right) \end{aligned}
• internal energy: observing that—recall (1.2) —
\begin{aligned} \widetilde{h}'(s) = \frac{\mathrm {d}}{\,\mathrm {d}s}\big [sh(s^{-1})\big ] = h(s^{-1})-s^{-1}h'(s^{-1}) = -P(s^{-1}), \end{aligned}
(3.8)
we obtain
\begin{aligned} \frac{\partial }{\partial G_{m,0}}{\mathbb {H}}_{\mathscr {T}}^m(G) = \frac{\partial }{\partial G_{m,0}} \widetilde{h}\left( \frac{\det Q_{\mathscr {T}}^m[G]}{2\mu _{\mathscr {T}}^m}\right) = \frac{1}{2\mu _{\mathscr {T}}^m}P\left( \frac{2\mu _{\mathscr {T}}^m}{\det Q_{\mathscr {T}}^m[G]}\right) \nu _{\mathscr {T}}^m[G], \end{aligned}
where
\begin{aligned} \nu _{\mathscr {T}}^m[G] := - \frac{\partial }{\partial G_{m,0}} \det Q_{\mathscr {T}}^m[G] = (\det Q_{\mathscr {T}}^m[G])\, (Q_{\mathscr {T}}^m[G])^{-T}\sum _{j=1}^d\mathrm {e}_j \end{aligned}
(3.9)
is the uniquely determined vector in $${\mathbb {R}}^d$$ that is orthogonal to the $$(d-1)$$-simplex with corners $$G_{m,1}$$ to $$G_{m,d}$$ (pointing away from $$G_{m,0}$$) and whose length equals the $$(d-1)$$-volume of that simplex.
• potential energy:
Now let $$\omega _\ell$$ be a fixed vertex of $${\mathscr {T}}$$. Summing over all simplices $$\Delta _m$$ that have $$\omega _\ell$$ as a vertex, and choosing vertex labels in accordance with above, i.e., such that $$\omega _{m,0}=\omega _\ell$$ in $$\Delta _m$$, produces the following Euler–Lagrange equation:
(3.10)
### 3.4 Approximation of the Initial Condition
For the approximation $$\rho ^0_\boxplus =(G^0_\boxplus )_\#{\overline{\rho }}_{\mathscr {T}}$$ of the initial datum $$\rho ^0=G^0_\#{\overline{\rho }}$$, we require:
• $$\rho ^0_\boxplus$$ converges to $$\rho ^0$$ narrowly;
• $${\mathcal {E}}(\rho ^0_\boxplus )$$ is $$\boxplus$$-uniformly bounded, i.e.,
\begin{aligned} \overline{{\mathcal {E}}}:=\sup {\mathcal {E}}(\rho _\boxplus ^0) < \infty . \end{aligned}
(3.11)
In our numerical experiments, we always choose $${\overline{\rho }}:=\rho ^0$$, in which case $$G^0 :K\rightarrow {\mathbb {R}}^d$$ can be taken as the identity on K, and we choose accordingly $$G^0_\boxplus$$ as the identity as well. Hence $$\rho ^0_\boxplus ={\overline{\rho }}_{\mathscr {T}}$$, which converges to $$\rho ^0={\overline{\rho }}$$ even strongly in $$L^1(K)$$. Moreover, since h is convex, it easily follows from Jensen’s inequality that
\begin{aligned} \int _{\Delta _m} h\big ({\overline{\rho }}(x)\big )\,\mathrm {d}x \ge |\Delta _m|h({\overline{\rho }}_{\mathscr {T}}^m), \end{aligned}
and therefore,
\begin{aligned} {\mathcal {E}}(\rho _\boxplus ^0) \le {\mathcal {E}}(\rho ^0). \end{aligned}
In more general situations, in which $$G^0$$ is not the identity, a sequence of approximations $$G^0_\boxplus$$ of $$G^0$$ is needed. Pointwise convergence $$G^0_\boxplus \rightarrow G^0$$ is more than sufficient to guarantee narrow convergence of $$\rho _\boxplus ^0$$ to $$\rho ^0$$, but the uniform bound (3.11) might require a well-adapted approximation, especially for non-smooth $$G^0$$’s.
## 4 Limit Trajectory
In this section, we assume that a sequence of vanishing discretizations $$\boxplus \rightarrow 0$$ is given, and we study the respective limit of the fully discrete solutions $$(G_\boxplus ^n)_{n\ge 0}$$ that are produced by the inductive minimization procedure (3.7). For the analysis of that limit trajectory, it is more natural to work with the induced densities and velocities,
\begin{aligned} \rho _\boxplus ^n:=(G_\boxplus ^n)_\#{\overline{\rho }}, \quad \mathbf {v}_\boxplus ^n:=\frac{{\mathrm {id}}-G_\boxplus ^{n-1}\circ (G_\boxplus ^n)^{-1}}{\tau }, \end{aligned}
instead of the Lagrangian maps $$G_\boxplus ^n$$ themselves. Note that $$\mathbf {v}_\boxplus ^n$$ is only well-defined on the support of $$\rho _\boxplus ^n$$—that is, on the image of $$G_\boxplus ^n$$—and can be assigned arbitrary values outside. Let us introduce the piecewise constant in time interpolations $${\widetilde{\rho }}_\boxplus :[0,T]\times {{\mathbb {R}}^d}\rightarrow {\mathbb {R}}_{\ge 0}$$, and $${\widetilde{\mathbf {v}}}_\boxplus :[0,T]\times {{\mathbb {R}}^d}\rightarrow {\mathbb {R}}^d$$ as usual,
\begin{aligned} {\widetilde{\rho }}_\boxplus (t) = \rho _\boxplus ^n, \quad {\widetilde{\mathbf {v}}}_\boxplus (t) = \mathbf {v}_\boxplus ^n \quad \text {with }n \text { such that }t\in ((n-1)\tau ,n\tau ]. \end{aligned}
Note that $${\widetilde{\rho }}(t,\cdot )\in {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$ and $${\widetilde{\mathbf {v}}}_\boxplus (t,\cdot )\in L^2({{\mathbb {R}}^d}\rightarrow {\mathbb {R}}^d;{\widetilde{\rho }}_\boxplus (t,\cdot ))$$ at each $$t\ge 0$$.
### 4.1 Energy Estimates
We start by proving the classical energy estimates on minimizing movements for our fully discrete scheme.
Lemma 4.1
For each discretization $$\boxplus$$ and for any indices $$\overline{n}>\underline{n}\ge 0$$, one has the a priori estimate
\begin{aligned} {\mathcal {E}}(\rho _\boxplus ^{\overline{n}}) +\frac{\tau }{2}\sum _{n=\underline{n}+1}^{\overline{n}}\left( \frac{\mathrm {W}_2(\rho _\boxplus ^n,\rho _\boxplus ^{n-1})}{\tau }\right) ^2 \le {\mathcal {E}}(\rho ^{\underline{n}}). \end{aligned}
(4.1)
Consequently:
(1)
$${\mathbf {E}}$$ is monotonically decreasing, i.e., $${\mathcal {E}}({\widetilde{\rho }}_\boxplus (t))\le {\mathcal {E}}({\widetilde{\rho }}_\boxplus (s))$$ for all $$t\ge s\ge 0$$;
(2)
$${\widetilde{\rho }}_\boxplus$$ is Hölder-$$\frac{1}{2}$$-continuous in $$\mathrm {W}_2$$, up to an error $$\tau$$,
\begin{aligned} \mathrm {W}_2\big ({\widetilde{\rho }}_\boxplus (t),{\widetilde{\rho }}_\boxplus (s)\big ) \le \sqrt{2{\mathcal {E}}(\rho _\boxplus ^0)}\big (|t-s|^{\frac{1}{2}}+\tau ^{\frac{1}{2}}\big ) \quad \text { for all }t\ge s\ge 0. \end{aligned}
(4.2)
(3)
$${\widetilde{\mathbf {v}}}_\boxplus$$ is square integrable with respect to $${\widetilde{\rho }}_\boxplus$$,
\begin{aligned} \int _0^T\int _{{\mathbb {R}}^d}\Vert {\widetilde{\mathbf {v}}}_\boxplus \Vert ^2{\widetilde{\rho }}_\boxplus \,\mathrm {d}x\,\mathrm {d}t \le 2{\mathcal {E}}(\rho _\boxplus ^0). \end{aligned}
(4.3)
Proof
By the definition of $$G_\boxplus ^n$$ as a minimizer, we know that $${\mathbf {E}}_\boxplus (G_\boxplus ^n;G_\boxplus ^{n-1})\le {\mathbf {E}}_\boxplus (G;G_\boxplus ^{n-1})$$ for any $$G\in {\mathcal {A}}_{\mathscr {T}}$$, and in particular for the choice $$G:=G_\boxplus ^{n-1}$$, which yields:
\begin{aligned} \frac{1}{2\tau }\int _K\Vert G_\boxplus ^n-G_\boxplus ^{n-1}\Vert ^2{\overline{\rho }}_{\mathscr {T}}\,\mathrm {d}\omega +{\mathbf {E}}(G_\boxplus ^n|{\overline{\rho }}_{\mathscr {T}}) \le {\mathbf {E}}(G_\boxplus ^{n-1}|{\overline{\rho }}_{\mathscr {T}}). \end{aligned}
(4.4)
Summing these inequalies for $$n=\underline{n}+1,\ldots ,\overline{n}$$, recalling that $${\mathcal {E}}(\rho _\boxplus ^n)={\mathbf {E}}(G_\boxplus ^n|{\overline{\rho }}_{\mathscr {T}})$$ by (1.9) and that $$\mathrm {W}_2(\rho _\boxplus ^n,\rho _\boxplus ^{n-1})^2\le \int _K|G_\boxplus ^n-G_\boxplus ^{n-1}|^2{\overline{\rho }}\,\mathrm {d}\omega$$ by (2.3), produces (4.1).
Monotonicity of $${\mathcal {E}}$$ in time is obvious.
To prove (4.2), choose $$\underline{n}\le \overline{n}$$ such that $$s\in ((\underline{n}-1)\tau ,\underline{n}\tau ]$$ and $$t\in ((\overline{n}-1)\tau ,\overline{n}\tau ]$$. Notice that $$\tau (\overline{n}-\underline{n})\le t-s+\tau$$. If $$\underline{n}=\overline{n}$$, the claim (4.2) is obviously true; let $$\underline{n}<\overline{n}$$ in the following. Combining the triangle inequality for the metric $$\mathrm {W}_2$$, estimate (4.1) above and Hölder’s inequality for sums, we arrive at
\begin{aligned} \mathrm {W}_2\big ({\widetilde{\rho }}_\boxplus (t),{\widetilde{\rho }}_\boxplus (s)\big )&= \mathrm {W}_2(\rho _\boxplus ^{\overline{n}},\rho _\boxplus ^{\underline{n}}) \le \sum _{n=\underline{n}+1}^{\overline{n}}\mathrm {W}_2(\rho _\boxplus ^n,\rho _\boxplus ^{n-1}) \\&\le \left[ \sum _{n=\underline{n}+1}^{\overline{n}}\tau \right] ^{\frac{1}{2}} \left[ \sum _{n=\underline{n}+1}^{\overline{n}}\frac{\mathrm {W}_2(\rho _\boxplus ^n,\rho _\boxplus ^{n-1})^2}{\tau }\right] ^{\frac{1}{2}}\\&= \big [\tau (\overline{n}-\underline{n}) \big ]^{\frac{1}{2}} \left[ \tau \sum _{n=\underline{n}+1}^{\overline{n}}\left( \frac{\mathrm {W}_2(\rho _\boxplus ^n,\rho _\boxplus ^{n-1})}{\tau }\right) ^2\right] ^{\frac{1}{2}}\\&\le [t-s+\tau ]^{\frac{1}{2}} \left[ 2\big ({\mathcal {E}}(\rho _\boxplus ^{\underline{n}}) - {\mathcal {E}}(\rho _\boxplus ^{\overline{n}})\big )\right] ^{\frac{1}{2}} \\&\le \big [|t-s|^{\frac{1}{2}}+\tau ^{\frac{1}{2}}\big ]{\mathcal {E}}(\rho _\boxplus ^0)^{\frac{1}{2}}. \end{aligned}
Finally, changing variables using $$x=G_\boxplus ^n(\omega )$$ in (4.4) yields
\begin{aligned} \frac{\tau }{2}\int _{{\mathbb {R}}^d}\Vert \mathbf {v}_\boxplus ^n\Vert ^2\rho _\boxplus ^n\,\mathrm {d}x + {\mathbf {E}}(G_\boxplus ^n) \le {\mathbf {E}}(G_\boxplus ^{n-1}), \end{aligned}
and summing these inequalities from $$n=1$$ to $$n=N_\tau$$ yields (4.3). $$\square$$
### 4.2 Compactness of the Trajectories and Weak Formulation
Our main result on the weak limit of $${\widetilde{\rho }}_\boxplus$$ is the following.
Theorem 4.2
Along a suitable sequence $$\boxplus \rightarrow 0$$, the curves $${\widetilde{\rho }}_\boxplus :{\mathbb {R}}_{\ge 0}\rightarrow {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$ convergence pointwise in time, i.e., $${\widetilde{\rho }}_\boxplus (t)\rightarrow \rho _*(t)$$ narrowly for each $$t>0$$, towards a Hölder-$$\frac{1}{2}$$-continuous limit trajectory $$\rho _* :{\mathbb {R}}_{\ge 0}\rightarrow {\mathcal {P}}_2^\text {ac}({\mathbb {R}}^d)$$.
Moreover, the discrete velocities $${\widetilde{\mathbf {v}}}_\boxplus$$ possess a limit $$\mathbf {v}_*\in L^2({\mathbb {R}}_{\ge 0}\times {{\mathbb {R}}^d};\rho _*)$$ such that $${\widetilde{\mathbf {v}}}_\boxplus {\widetilde{\rho }}_\boxplus \overset{*}{\rightharpoonup }\mathbf {v}_*\rho _*$$ in $$L^1({\mathbb {R}}_{\ge 0}\times {{\mathbb {R}}^d})$$, and the continuity equation
\begin{aligned} \partial _t\rho _* + \nabla \cdot (\rho _*\mathbf {v}_*) = 0 \end{aligned}
(4.5)
holds in the sense of distributions.
Remark 4.3
The Hölder continuity of $$\rho _*$$ implies that $$\rho _*$$ satisfies the initial condition (1.1b) in the sense that $$\rho _*(t)\rightarrow \rho ^0$$ narrowly as $$t\downarrow 0$$.
Proof of Theorem 4.2
We closely follow an argument that is part of the general convergence proof for the minimizing movement scheme as given in Ambrosio et al. [1, Section 11.1.3]. Below, convergence is shown for some arbitrary but fixed time horizon $$T>0$$; a standard diagonal argument implies convergence at arbitrary times.
First observe that by estimate (4.2)—applied with $$0=s\le t\le T$$—it follows that $$\mathrm {W}_2({\widetilde{\rho }}_\boxplus (t),\rho _\boxplus ^0)$$ is bounded, uniformly in $$t\in [0,T]$$ and in $$\boxplus$$. Since further $$\rho ^0_\boxplus$$ converges narrowly to $$\rho ^0$$ by our hypotheses on the initial approximation, we conclude that all densities $${\widetilde{\rho }}_\boxplus (t)$$ belong to a sequentially compact subset for the narrow convergence. The second observation is that the term on the right hand side of (4.2) simplifies to $$(2\overline{{\mathcal {E}}})^\frac{1}{2}|t-s|^\frac{1}{2}$$ in the limit $$\boxplus \rightarrow 0$$. A straightforward application of the “refined version” of the Ascoli-Arzelà theorem (Proposition 3.3.1 in Ambrosio et al. [1]) yields the first part of the claim, namely the pointwise narrow convergence of $${\widetilde{\rho }}_\boxplus$$ towards a Hölder continuous limit curve $$\rho _*$$.
It remains to pass to the limit with the velocity $${\widetilde{\mathbf {v}}}_\boxplus$$. Towards that end, we define a probability measure $$\widetilde{\gamma }_\boxplus \in {\mathcal {P}}(Z_T)$$ on the set $$Z_T:=[0,T]\times {\mathbb {R}}^d\times {\mathbb {R}}^d$$ as follows:
\begin{aligned} \int _{Z_T}\varphi (t,x,v)\,\mathrm {d}\widetilde{\gamma }_\boxplus (t,x,v) = \int _0^T\int _{{\mathbb {R}}^d} \varphi \big (t,x,{\widetilde{\mathbf {v}}}_\boxplus (t,x)\big )\,{\widetilde{\rho }}_\boxplus (t,x)\,\mathrm {d}x\frac{\mathrm {d}t}{T}, \end{aligned}
for every bounded and continuous function $$\varphi \in C^0_b(Z_T)$$. For brevity, let $$\widetilde{M}_\boxplus \in {\mathcal {P}}([0,T]\times {\mathbb {R}}^d)$$ be the (tx)-marginals of $$\widetilde{\gamma }_\boxplus$$, that have respective Lebesgue densities $$\frac{\rho _\boxplus (t,x)}{T}$$ on $$[0,T]\times {\mathbb {R}}^d$$. Thanks to the result from the first part of the proof, $$\widetilde{M}_\boxplus$$ converges narrowly to a limit $$M_*$$, which has density $$\frac{\rho _*(t,x)}{T}$$. On the other hand, the estimate (4.3) implies that
\begin{aligned} \int _{Z_T} |v|^2\,\mathrm {d}\widetilde{\gamma }_\boxplus (t,x,v) = \int _{[0,T]\times {\mathbb {R}}^d}|{\widetilde{\mathbf {v}}}_\boxplus (t,x)|^2\,\mathrm {d}\widetilde{M}_\boxplus (t,x) \le 2\overline{{\mathcal {E}}}. \end{aligned}
We are thus in the position to apply Theorem 5.4.4 in Ambrosio et al. [1], which yields the narrow convergence of $$\widetilde{\gamma }_\boxplus$$ towards a limit $$\gamma _*$$. Clearly, the (tx)-marginal of $$\gamma _*$$ is $$M_*$$. Accordingly, we introduce the disintegration $$\gamma _{(t,x)}$$ of $$\gamma _*$$ with respect to $$M_*$$, which is well-defined $$M_*$$-a.e.. Below, it will turn out that $$\gamma _*$$’s v-barycenter,
\begin{aligned} \mathbf {v}_*(t,x) := \int _{{\mathbb {R}}^d} v \,\mathrm {d}\gamma _{(t,x)}(v), \end{aligned}
(4.6)
is the sought-for weak limit of $${\widetilde{\mathbf {v}}}_\boxplus$$. The convergence $${\widetilde{\mathbf {v}}}_\boxplus {\widetilde{\rho }}_\boxplus \overset{*}{\rightharpoonup }\mathbf {v}_*\rho _*$$ and the inheritance of the uniform $$L^2$$-bound (4.3) to the limit $$\mathbf {v}_*$$ are further direct consequences of Theorem 5.4.4 in Ambrosio et al. [1].
The key step to establish the continuity equation for the just-defined $$\mathbf {v}_*$$ is to evaluate the limit as $$\boxplus \rightarrow 0$$ of
\begin{aligned} J_\boxplus [\phi ] := \frac{1}{\tau }\left[ \int _0^T\int _{{\mathbb {R}}^d}\phi (t,x){\widetilde{\rho }}_\boxplus (t,x)\,\mathrm {d}x\,\mathrm {d}t - \int _0^T\int _{{\mathbb {R}}^d}\phi (t,x){\widetilde{\rho }}_\boxplus (t-\tau ,x)\,\mathrm {d}x\,\mathrm {d}t \right] \end{aligned}
for any given test function $$\phi \in C^\infty _c((0,T)\times {\mathbb {R}}^d)$$ in two different ways. First, we change variables $$t\mapsto t+\tau$$ in the second integral, which gives
\begin{aligned} J_\boxplus [\phi ]&= \int _0^T\int _{{\mathbb {R}}^d} \frac{\phi (t,x)-\phi (t+\tau ,x)}{\tau }{\widetilde{\rho }}_\boxplus (t,x)\,\mathrm {d}x\,\mathrm {d}t {\mathop {\longrightarrow }\limits ^{\boxplus \rightarrow 0}}\\&\quad -\int _0^T\int _{{\mathbb {R}}^d} \partial _t\phi (t,x)\,\rho _*(t,x)\,\mathrm {d}x\,\mathrm {d}t. \end{aligned}
For the second evaluation, we write
\begin{aligned} \rho _\boxplus ^{n-1} = \big (G_\boxplus ^{n-1}\circ (G_\boxplus ^n)^{-1}\big )_\#\rho _\boxplus ^n = \big ({\mathrm {id}}-\tau \mathbf {v}_\boxplus ^n\big )_\#\rho _\boxplus ^n, \end{aligned}
and substitute accordingly $$x\mapsto x-\tau {\widetilde{\mathbf {v}}}_\boxplus (t,x)$$ in the second integral, leading to
\begin{aligned} J_\boxplus [\phi ]&= \int _0^T\int _{{\mathbb {R}}^d} \frac{\phi (t,x)-\phi \big (t,x-\tau {\widetilde{\mathbf {v}}}_\boxplus (t,x)\big )}{\tau }{\widetilde{\rho }}_\boxplus (t,x)\,\mathrm {d}x\,\mathrm {d}t \\&= \int _0^T\int _{{\mathbb {R}}^d} \nabla \phi (t,x)\cdot {\widetilde{\mathbf {v}}}_\boxplus (t,x){\widetilde{\rho }}_\boxplus (t,x)\,\mathrm {d}x\,\mathrm {d}t + \mathfrak {e}_\boxplus [\phi ]\\&= \int _{Z_T} \nabla \phi (t,x)\cdot v\,\mathrm {d}\widetilde{\gamma }_\boxplus (t,x,v) + \mathfrak {e}_\boxplus [\phi ] \\&\quad {\mathop {\longrightarrow }\limits ^{\boxplus \rightarrow 0}} \int _{Z_T} \nabla \phi (t,x)\cdot v\,\mathrm {d}\gamma _*(t,x,v) \\&= \int _{[0,T]\times {\mathbb {R}}^d}\nabla \phi (t,x)\cdot \left[ \int _{{\mathbb {R}}^d}v\,\mathrm {d}\gamma _{(t,x)}(v)\right] \,\mathrm {d}M_*(t,x) \\&= \int _0^T\int _{{\mathbb {R}}^d}\nabla \phi (t,x)\cdot \mathbf {v}_*(t,x)\rho _*(t,x)\,\mathrm {d}x\,\mathrm {d}t. \end{aligned}
The error term $$\mathfrak {e}_\boxplus [\phi ]$$ above is controlled via Taylor expansion of $$\phi$$ and by using (4.3),
\begin{aligned} \big |\mathfrak {e}_\boxplus [\phi ]\big | \le \int _0^T\int _{{\mathbb {R}}^d}\frac{\tau }{2}\Vert \phi \Vert _{C^2}\big \Vert {\widetilde{\mathbf {v}}}_\boxplus (t,x)\big \Vert ^2{\widetilde{\rho }}_\boxplus (t,x)\,\mathrm {d}x\,\mathrm {d}t \le \overline{{\mathcal {E}}}\Vert \phi \Vert _{C^2}T\;\tau . \end{aligned}
Equality of the limits for both evaluations of $$J_\boxplus [\phi ]$$ for arbitrary test functions $$\phi$$ shows the continuity Eq. (4.5). $$\square$$
Unfortunately, the convergence provided by Theorem 4.2 is generally not sufficient to conclude that $$\rho _*$$ is a weak solution to (1.1), since we are not able to identify $$\mathbf {v}_*$$ as $$\mathbf {v}[\rho _*]$$ from (1.4b). The problem is two-fold: first, weak-$$\star$$ convergence of $${\widetilde{\rho }}_\boxplus$$ is insufficient to pass to the limit inside the nonlinear function P. Second, even if we would know that, for instance, $$P({\widetilde{\rho }}_\boxplus )\overset{*}{\rightharpoonup }P(\rho _*)$$, we would still need a $$\boxplus$$-independent a priori control on the regularity (e.g., maximal diameter of triangles) of the meshes generated by the $$G_\boxplus ^n$$ to justify the passage to limit in the weak formulation below.
The main difficulty in the weak formulation that we derive now is that we can only use “test functions” that are piecewise affine with respect to the changing meshes generated by the $$G_\boxplus ^n$$. For definiteness, we introduce the space
\begin{aligned} \mathcal {D}({\mathscr {T}}):=\left\{ \Gamma :K\rightarrow {\mathbb {R}}^d\,;\,\Gamma \text { is globally continuous, and is piecewise affine w.r.t. }\Delta _m\right\} . \end{aligned}
Lemma 4.4
Assume $$S :{{\mathbb {R}}^d}\rightarrow {\mathbb {R}}^d$$ is such that $$S\circ G_\boxplus ^n\in \mathcal {D}({\mathscr {T}})$$. Then:
\begin{aligned} \int _{{{\mathbb {R}}^d}} P(\rho _\boxplus ^{n}) \, \nabla \cdot S \,\mathrm {d}x - \int _{{{\mathbb {R}}^d}} \nabla V \cdot S\, \rho _\boxplus ^{n} \,\mathrm {d}x = \int _{{{\mathbb {R}}^d}} S\cdot \mathbf {v}_\boxplus ^n \rho _\boxplus ^{n} \,\mathrm {d}x. \end{aligned}
(4.7)
Proof
For all sufficiently small $$\varepsilon >0$$, let $$G_\varepsilon = ({\mathrm {id}}+S)\circ G_\boxplus ^n$$. By definition of $$G_\boxplus ^n$$ as a minimizer, we have that $${\mathbf {E}}_\boxplus (G_\varepsilon ;G_\boxplus ^{n-1})\ge {\mathbf {E}}_\boxplus (G_\boxplus ^n;G_\boxplus ^{n-1})$$. This implies that
\begin{aligned} 0&\le \frac{1}{\varepsilon }\int _K\bigg (\frac{1}{2\tau }\big [\Vert G_\varepsilon -G_\boxplus ^{n-1}\Vert ^2-\Vert G_\boxplus ^n-G_\boxplus ^{n-1}\Vert ^2\big ] \nonumber \\&\qquad + \left[ \widetilde{h}\left( \frac{\det \mathrm {D}G_\varepsilon }{{\overline{\rho }}_{\mathscr {T}}}\right) -\widetilde{h}\left( \frac{\det \mathrm {D}G_\boxplus ^n}{{\overline{\rho }}_{\mathscr {T}}}\right) \right] + \big [V\circ G_\varepsilon -V\big ]\bigg ) {\overline{\rho }}_{\mathscr {T}}\,\mathrm {d}\omega . \end{aligned}
(4.8)
We discuss limits of the three terms under the integral for $$\varepsilon \searrow 0$$. For the metric term:
\begin{aligned} \frac{1}{2\tau \varepsilon }\left[ \Vert G_\varepsilon -G_\boxplus ^{n-1}\Vert ^2-\Vert G_\boxplus ^n-G_\boxplus ^{n-1}\Vert ^2\right]&= \frac{G_\boxplus ^n-G_\boxplus ^{n-1}}{\tau }\cdot \frac{G_\varepsilon -G_\boxplus ^n}{\varepsilon }+ \frac{1}{2\tau \varepsilon }\Vert G_\varepsilon -G_\boxplus ^n\Vert ^2 \\&= \left[ \left( \frac{{\mathrm {id}}-T_\boxplus ^n}{\tau }\right) \cdot S\right] \circ G_\boxplus ^n + \frac{\varepsilon }{2\tau }\Vert S\Vert ^2\circ G_\boxplus ^n, \end{aligned}
and since S is bounded, the last term vanishes uniformly on K for $$\varepsilon \searrow 0$$. For the internal energy, since $$\mathrm {D}G_\varepsilon =\mathrm {D}({\mathrm {id}}+\varepsilon S)\circ G_\boxplus ^n\cdot \mathrm {D}G_\boxplus ^n$$, and recalling (3.8),
\begin{aligned}&\frac{1}{\varepsilon }\left[ \widetilde{h}\left( \frac{\det \mathrm {D}G_\varepsilon }{{\overline{\rho }}_{\mathscr {T}}}\right) -\widetilde{h}\left( \frac{\det \mathrm {D}G_\boxplus ^n}{{\overline{\rho }}_{\mathscr {T}}}\right) \right] \\&\quad = \frac{1}{\varepsilon }\left[ \widetilde{h}\left( \frac{\det \mathrm {D}G_\boxplus ^n}{{\overline{\rho }}_{\mathscr {T}}}\det ({\mathbb {1}}+\varepsilon \mathrm {D}S)\circ G_\boxplus ^n\right) -\widetilde{h}\left( \frac{\det \mathrm {D}G_\boxplus ^n}{{\overline{\rho }}_{\mathscr {T}}}\right) \right] \\&\quad {\mathop {\longrightarrow }\limits ^{\varepsilon \searrow 0}} \frac{\det \mathrm {D}G_\boxplus ^n}{{\overline{\rho }}_{\mathscr {T}}}\widetilde{h}'\left( \frac{\det \mathrm {D}G_\boxplus ^n}{{\overline{\rho }}_{\mathscr {T}}}\right) \left( \lim _{\varepsilon \searrow 0}\frac{\det ({\mathbb {1}}+\varepsilon \mathrm {D}S)}{\varepsilon }\right) \circ G_\boxplus ^n \\&\quad = -\frac{\det \mathrm {D}G_\boxplus ^n}{{\overline{\rho }}_{\mathscr {T}}} P\left( \frac{{\overline{\rho }}_{\mathscr {T}}}{\det \mathrm {D}G_\boxplus ^n}\right) {{\mathrm{tr}}}[\mathrm {D}S]\circ G_\boxplus ^n \\&\quad = - \frac{\det \mathrm {D}G_\boxplus ^n}{{\overline{\rho }}_{\mathscr {T}}}\big [P(\rho ^n)\,\nabla \cdot S\big ]\circ G_\boxplus ^n. \end{aligned}
Since the piecewise constant function $$\det \mathrm {D}G_\boxplus ^n$$ has a positive lower bound, the convergence as $$\varepsilon \searrow 0$$ is uniform on K. Finally, for the potential energy,
\begin{aligned} \frac{1}{\varepsilon }\big [V\circ ({\mathrm {id}}+\varepsilon S)\circ G_\boxplus ^n-V\circ G_\boxplus ^n\big ] {\mathop {\longrightarrow }\limits ^{\varepsilon \searrow 0}} \big [\nabla V\cdot S\big ]\circ G_\boxplus ^n. \end{aligned}
Again, the convergence is uniform on K. Passing to the limit in the integral (4.8) yields
\begin{aligned} 0&\le \int _K \left[ \left( \frac{{\mathrm {id}}-T_\boxplus ^n}{\tau }\right) \cdot S\right] \circ G_\boxplus ^n{\overline{\rho }}_{\mathscr {T}}\,\mathrm {d}\omega \\&\qquad - \int _K \big [P(\rho ^n)\,\nabla \cdot S\big ]\circ G_\boxplus ^n \det \mathrm {D}G_\boxplus ^n\,\mathrm {d}\omega + \int _K \big [\nabla V\cdot S\big ]\circ G_\boxplus ^n{\overline{\rho }}_{\mathscr {T}}\,\mathrm {d}\omega . \end{aligned}
The same inequality is true with $$-S$$ in place of S, hence this inequality is actually an equality. Since $$\rho _\boxplus ^n=(G_\boxplus ^n)_\#{\overline{\rho }}_{\mathscr {T}}$$, a change of variables $$x=S_\boxplus ^n(\omega )$$ produces (4.7). $$\square$$
Corollary 4.5
In addition to the hypotheses of Theorem 4.2, assume that
(1)
$$P({\widetilde{\rho }}_\boxplus )\overset{*}{\rightharpoonup }p_*$$ in $$L^1([0,T]\times \Omega )$$;
(2)
each $$G_\boxplus ^n$$ is injective;
(3)
as $$\boxplus \rightarrow 0$$, all simplices in the images of $${\mathscr {T}}$$ under $$G_\boxplus ^n$$ have non-degenerate interior angles and tend to zero in diameter, uniformly w.r.t. n.
Then $$\rho _*$$ satisfies the PDE
\begin{aligned} \partial _t\rho _* = \Delta p_* + \nabla \cdot (\rho _*\nabla V) \end{aligned}
(4.9)
in the sense of distributions.
Proof
Let a smooth test function $$\zeta \in C^\infty _c({\mathbb {R}}^d\rightarrow {\mathbb {R}}^d)$$ be given. For each $$\boxplus$$ and each n, a $$\zeta _\boxplus ^n :{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$$ with $$\zeta _\boxplus ^n\circ G_\boxplus ^n\in \mathcal {D}({\mathscr {T}})$$ can be constructed in such a way that
\begin{aligned} \zeta _\boxplus ^n\rightarrow \zeta , \quad \nabla \cdot \zeta _\boxplus ^n\rightarrow \nabla \cdot \zeta \end{aligned}
(4.10)
uniformly on $${\mathbb {R}}^d$$, and uniformly in n as $$\boxplus \rightarrow 0$$. This follows from our hypotheses on the $$\boxplus$$-uniform regularity of the Lagrangian meshes: inside the image of $$G_\boxplus ^n$$, one can simply choose $$\zeta _\boxplus ^n$$ as the affine interpolation of the values of $$\zeta$$ at the points $$G_\boxplus ^n(\omega _\ell )$$. Outside, one can take an arbitrary approximation of $$\zeta$$ that is compatible with the piecewise-affine approximation on the boundary of $$G_\boxplus ^n$$’s image; one may even choose $$\zeta _\boxplus ^n\equiv \zeta$$ at sufficient distance to that boundary. The uniform convergences (4.10) then follow by standard finite element analysis.
Further, let $$\eta \in C^\infty _c(0,T)$$ be given. For each $$t\in ((n-1)\tau ,n\tau ]$$, substitute $$S(t,x):=\eta (t)\zeta _\boxplus ^n(x)$$ into (4.7). Integration of these equalities with respect to $$t\in (0,T)$$ yields
\begin{aligned} \int _0^T\int _{{\mathbb {R}}^d} P({\widetilde{\rho }}_\boxplus )\nabla \cdot S\,\mathrm {d}x\,\mathrm {d}t - \int _0^T\int _{{\mathbb {R}}^d} \nabla V\cdot S\,\mathrm {d}x\,\mathrm {d}t = \int _0^T\int _{{\mathbb {R}}^d} S\cdot {\widetilde{\mathbf {v}}}_\boxplus {\widetilde{\rho }}_\boxplus \,\mathrm {d}x\,\mathrm {d}t. \end{aligned}
We pass to the limit $$\boxplus \rightarrow 0$$ in these integrals. For the first, we use that $$P({\widetilde{\rho }})\overset{*}{\rightharpoonup }p_*$$ by hypothesis, for the last, we use Theorem 4.2 above. Since any test function $$S\in C^\infty _c((0,T)\times \Omega )$$ can be approximated in $$C^1$$ by linear combinations of products $$\eta (t)\zeta (x)$$ as above, we thus obtain the weak formulation of
\begin{aligned} \rho _*\mathbf {v}_* = \nabla p_* + \rho _*\nabla V. \end{aligned}
In combination with the continuity Eq. (4.5), we arrive at (4.9). $$\square$$
Remark 4.6
In principle, our discretization can also be applied to the linear Fokker–Planck equation with $$P(r)=r$$ and $$h(r)=r\log r$$. In that case, one automatically has $$P({\widetilde{\rho }})\overset{*}{\rightharpoonup }p_*\equiv P(\rho _*)$$ thanks to Theorem 4.2. Corollary 4.5 above then provides an a posteriori criterion for convergence: if the Lagrangian mesh does not deform too wildly under the dynamics as the discretization is refined, then the discrete solutions converge to the genuine solution.
## 5 Consistency in 2D
In this section, we prove consistency of our discretization in the following sense. Under certain conditions on the spatial discretization $${\mathscr {T}}$$, any smooth and positive solution $$\rho$$ to the initial value problem (1.1) projects to a discrete solution that satisfies the Euler–Lagrange equations up to a controlled error. We restrict ourselves to $$d=2$$ dimensions.
### 5.1 Smooth Lagrangian Evolution
First, we derive an alternative form of the velocity field $$\mathbf {v}$$ from (1.4b) in terms of G.
Lemma 5.1
For $$\rho =G_\#{\overline{\rho }}$$ with a smooth diffemorphism $$G :K\rightarrow {{\mathbb {R}}^d}$$, we have
\begin{aligned} \mathbf {v}[\rho ]\circ G=\mathbf {V}[G] := P'\left( \frac{{\overline{\rho }}}{\det \mathrm {D}G}\right) \, (\mathrm {D}G)^{-T}\left( {{\mathrm{tr}}}_{12}\big [(\mathrm {D}G)^{-1}\mathrm {D}^2G\big ]^T-\frac{\nabla {\overline{\rho }}}{{\overline{\rho }}}\right) - \nabla V \circ G. \end{aligned}
(5.1)
Consequently, the Lagrangian map G—relative to the reference density $${\overline{\rho }}$$ — for a smooth solution $$\rho$$ to (1.1) satisfies
\begin{aligned} \partial _t G = \mathbf {V}[G]. \end{aligned}
(5.2)
Proof
On the one hand,
\begin{aligned} \mathrm {D}\big [h'(\rho )\circ G\big ] = \big [\mathrm {D}h'(\rho )\big ]\circ G\,\mathrm {D}G, \end{aligned}
and on the other hand, by definition of the push forward,
\begin{aligned} \mathrm {D}\big [h'(\rho )\circ G\big ]&= \mathrm {D}h'\left( \frac{{\overline{\rho }}}{\det \mathrm {D}G}\right) \\&= h''\left( \frac{{\overline{\rho }}}{\det \mathrm {D}G}\right) \,\left( \frac{{\overline{\rho }}}{\det \mathrm {D}G}\right) \,\left( \frac{\mathrm {D}{\overline{\rho }}}{{\overline{\rho }}}-{{\mathrm{tr}}}_{12}\big [(\mathrm {D}G)^{-1}\mathrm {D}^2G \big ]\right) \\&= \big [\rho h''(\rho )\big ]\circ G \,\left( \frac{\mathrm {D}{\overline{\rho }}}{{\overline{\rho }}}-{{\mathrm{tr}}}_{12}\big [(\mathrm {D}G)^{-1}\mathrm {D}^2G \big ]\right) . \end{aligned}
Hence
\begin{aligned} \nabla h'(\rho ) \circ G = \big [\rho h''(\rho )\big ]\circ G \,(\mathrm {D}G)^{-T}\left( \frac{\nabla {\overline{\rho }}}{{\overline{\rho }}}-{{\mathrm{tr}}}_{12}\big [(\mathrm {D}G)^{-1}\mathrm {D}^2G \big ]^T\right) . \end{aligned}
Observing that (1.2) implies that $$rh''(r)=P'(r)$$, we conclude (5.2) directly from (1.4b). $$\square$$
### 5.2 Discrete Euler–Lagrange Equations in Dimension $$d=2$$
In the planar case $$d=2$$, the Euler–Lagrange equation (3.10) above can be rewritten in a more convenient way.
In the following, fix some vertex $$\omega _\times$$ of the triangulation, which is incident to precisely six triangles. For convenience, we assume that these are labelled $$\Delta _0$$ to $$\Delta _5$$ in counter-clockwise order. Similarly, the six neighboring vertices are labeled $$\omega _0$$ to $$\omega _5$$ in counter-clockwise order, so that $$\Delta _k$$ has vertices $$\omega _{k}$$ and $$\omega _{k+1}$$, where we set $$\omega _6:=\omega _0$$.
Using these conventions and recalling Lemma B.2, the expression for the vector $$\nu$$ in (3.9) simplifies to
\begin{aligned} \nu _{\mathscr {T}}^k = - {\mathbb {J}}(G_{k+1}-G_{k}), \quad \text {where}\quad {\mathbb {J}}= \begin{pmatrix} 0 &{} -1 \\ 1 &{} 0 \end{pmatrix}. \end{aligned}
Summing the Euler–Lagrange equation (3.10) over $$\Delta _0$$ to $$\Delta _5$$, we obtain
\begin{aligned} {\mathbf {p}}_\times = {\mathbf {J}}_\times , \end{aligned}
(5.3)
where the momentum term $${\mathbf {p}}_\times$$ and the impulse $${\mathbf {J}}_\times$$, respectively, are given by
\begin{aligned} {\mathbf {p}}_\times&= \frac{1}{12}\sum _{k=0}^5\mu _{\mathscr {T}}^k \left[ 2\left( \frac{G_\times -G^*_\times }{\tau }\right) +\left( \frac{G_k-G^*_k}{\tau }\right) +\left( \frac{G_{k+1}-G^*_{k+1}}{\tau }\right) \right] \end{aligned}
(5.4)
\begin{aligned} {\mathbf {J}}_\times&= \sum _{k=0}^5 \mu _{\mathscr {T}}^{k}\bigg [ \frac{1}{2 \mu _{\mathscr {T}}^{k}}P\left( \frac{2\mu _{\mathscr {T}}^k}{\det (G_k-G_\times |G_{k+1}-G_\times )}\right) {\mathbb {J}}(G_{k+1}-G_k) \end{aligned}
(5.5)
(5.6)
We shall now prove our main result on consistency. The setup is the following: a sequence of triangulations $${\mathscr {T}}_\varepsilon$$ on K, parametrized by $$\varepsilon >0$$, and a sequence of time steps $$\tau _\varepsilon ={\mathcal {O}}(\varepsilon )$$ are given. We assume that there is an $$\varepsilon$$-independent region $$K'\subset K$$ on which the $${\mathscr {T}}_\varepsilon$$ are almost hexagonal in the following sense: each node $$\omega _\times \in K'$$ of $${\mathscr {T}}_\varepsilon$$ has precisely six neighbors—labelled $$\omega _0$$ to $$\omega _5$$ in counter-clockwise order—and there exists a rotation $$R\in \mathrm {SO}(2)$$ such that
\begin{aligned} R(\omega _k-\omega _\times ) = \varepsilon \sigma _k + {\mathcal {O}}(\varepsilon ^2) \quad \text {with}\quad \sigma _k = \begin{pmatrix} \cos \frac{\pi }{3}k \\ \sin \frac{\pi }{3}k \end{pmatrix} \end{aligned}
(5.7)
for $$k=0,1,\ldots ,5$$.
Now, let $$G :[0,T]\times K\rightarrow {{\mathbb {R}}^d}$$ be a given smooth solution to the Lagrangian evolution Eq. (5.2), and fix a time $$t\in (0,T)$$. For all sufficiently small $$\varepsilon >0$$, we define maps $$G_\varepsilon ,G_\varepsilon ^*\in {\mathcal {A}}_{{\mathscr {T}}_\varepsilon }$$ by linear interpolation of the values of $$G(t;\cdot )$$ and $$G(t-\tau ;\cdot )$$, respectively, on $${\mathscr {T}}_\varepsilon$$. That is, $$G_\varepsilon (\omega _\ell )=G(t;\omega _\ell )$$ and $$G^*_\varepsilon (\omega _\ell )=G(t-\tau ;\omega _\ell )$$, at all nodes $$\omega _\ell$$ in $${\mathscr {T}}_\varepsilon$$. Theorem 5.2 below states that the pair $$G_\varepsilon ,G_\varepsilon ^*$$ is an approximate solution to the discrete Euler–Lagrange equations (5.3) at all nodes $$\omega _\times$$ of the respective triangulation $${\mathscr {T}}_\varepsilon$$ that lie in $$K'$$.
The hexagonality hypothesis on the $${\mathscr {T}}_\varepsilon$$ is strong, but some very strong restriction of $${\mathcal {A}}_{{\mathscr {T}}_\varepsilon }$$’s geometry is apparently necessary. See Remark 5.4 following the proof for further discussion.
Theorem 5.2
Under the hypotheses and with the notations introduced above, the Euler–Lagrange equation (5.3) admits the following asymptotic expansion:
\begin{aligned} {\mathbf {p}}_\times&= \frac{\sqrt{3}}{2}\varepsilon ^2\,{\overline{\rho }}(\omega _\times )\partial _tG(t;\omega _\times )+{\mathcal {O}}(\varepsilon ^3), \end{aligned}
(5.8a)
\begin{aligned} {\mathbf {J}}_\times&= \frac{\sqrt{3}}{2}\varepsilon ^2\,{\overline{\rho }}(\omega _\times )\mathbf {V}[G](t;\omega _\times )+{\mathcal {O}}(\varepsilon ^3), \end{aligned}
(5.8b)
as $$\varepsilon \rightarrow 0$$, uniformly at the nodes $$\omega _\times \in K'$$ of the respective $${\mathscr {T}}_\varepsilon$$.
Remark 5.3
Up to an error $${\mathcal {O}}(\varepsilon ^3)$$, the geometric pre-factor $$\frac{\sqrt{3}}{2}\varepsilon ^2$$ equals to one third of the total area of the hexagon with vertices $$\omega _0$$ to $$\omega _5$$, and is thus equal to the integral of the piecewise affine hat function with peak at $$\omega _\times$$.
Proof of Theorem 5.2
Throughout the proof, let $$\varepsilon >0$$ be fixed; we shall omit the $$\varepsilon$$-index for $${\mathscr {T}}_\varepsilon$$ and $$\tau _\varepsilon$$. First, we fix a node $$\omega _\times$$ of $${\mathscr {T}}\cap K'$$. Thanks to the equivariance of both (5.2) and (5.3) under rigid motions of the domain, we may assume that R in (5.7) is the identity, and that $$\omega _\times =0$$.
We collect some relations that are helpful for the calculations that follow. Trivially,
\begin{aligned} \sum _{k=0}^5\sigma _k=0, \quad \sum _{k=0}^5\omega _k={\mathcal {O}}(\varepsilon ^2). \end{aligned}
(5.9)
Moreover, we have that
\begin{aligned} |\Delta _k| = \det (\omega _k|\omega _{k+1}) = \varepsilon ^2\det (\sigma _k|\sigma _{k+1}) + {\mathcal {O}}(\varepsilon ^3) = \frac{\sqrt{3}}{4}\varepsilon ^2+{\mathcal {O}}(\varepsilon ^3). \end{aligned}
(5.10)
On the other hand, by definition of $$\mu _{\mathscr {T}}^k$$ in (3.1), it follows that
(5.11)
Combining (5.10) and (5.11) yields
\begin{aligned} \mu _{\mathscr {T}}^k = \varepsilon ^2\left( \frac{\sqrt{3}}{4}{\overline{\rho }}_\times +{\mathcal {O}}(\varepsilon )\right) . \end{aligned}
(5.12)
In accordance with the definition of $$G_\varepsilon$$ and $$G_\varepsilon ^*$$ from G detailed above, let $$G_\times :=G(t,\omega _\times )$$ and $$G^*_\times =G(t-\tau ,\omega _\times )$$, and define $$G_k$$, $$G_k^*$$ for $$k=0,\ldots ,5$$ in the analogous way. Further, we introduce $$\mathrm {D}G_\times =\mathrm {D}G(t,\omega _\times )$$, $$\mathrm {D}^2G_\times =\mathrm {D}^2G(t,\omega _\times )$$, $$\partial _tG_\times =\partial _tG(t,\omega _\times )$$.
To perform an expansion in the momentum term, first observe that
\begin{aligned} G(t-\tau ;\omega _k) = G(t;\omega _k) - \tau \partial _t G(t;\omega _k) + {\mathcal {O}}(\tau ^2), \end{aligned}
for each $$k=0,1,\ldots ,5$$, and so, using that $$\tau ={\mathcal {O}}(\varepsilon )$$ by hypothesis,
\begin{aligned} \frac{G_k-G_k^*}{\tau }= \partial _tG(t;\omega _k) + {\mathcal {O}}(\tau ) = \partial _tG_\times + {\mathcal {O}}(\varepsilon ) + {\mathcal {O}}(\tau ) = \partial _tG_\times + {\mathcal {O}}(\varepsilon ). \end{aligned}
Using (5.12) and then (5.9) yields
\begin{aligned} {\mathbf {p}}_\times&=\frac{1}{12\tau }\sum _{k=0}^5 \varepsilon ^2\left( \frac{\sqrt{3}}{4}{\overline{\rho }}_\times +{\mathcal {O}}(\varepsilon )\right) \big [4\partial _tG_\times +{\mathcal {O}}(\varepsilon )\big ] \\&=\frac{\sqrt{3}}{2}\varepsilon ^2\,{\overline{\rho }}_\times \partial _tG_\times + {\mathcal {O}}(\varepsilon ^3). \end{aligned}
This is (5.8a).
For the impulse term, we start with a Taylor expansion to second order in space:
\begin{aligned} G_k = G_\times + \mathrm {D}G_\times \omega _k + \frac{1}{2} \mathrm {D}^2G_\times :[\omega _k]^2 + {\mathcal {O}}(\varepsilon ^3). \end{aligned}
We combine this with the observation that $$(\omega _k|\omega _{k+1})^{-1}={\mathcal {O}}(\varepsilon ^{-1})$$ to obtain:
\begin{aligned}&\frac{\mu _{\mathscr {T}}^k}{\det (G_k-G_\times |G_{k+1}-G_\times )} \\&\quad = \frac{\det (\omega _k|\omega _{k+1})}{\det \mathrm {D}G_\times } \frac{{\overline{\rho }}_\times +\varepsilon \nabla {\overline{\rho }}_\times \cdot \frac{\sigma _k+\sigma _{k+1}}{3}+{\mathcal {O}}(\varepsilon ^2)}{\det \big [(\omega _k|\omega _{k+1})+\frac{1}{2}(\mathrm {D}G_\times )^{-1}\big (\mathrm {D}^2G_\times :[\omega _k]^2\big |\mathrm {D}^2G_\times :[\omega _{k+1}]^2\big )+{\mathcal {O}}(\varepsilon ^3)\big ]} \\&\quad = \frac{{\overline{\rho }}_\times }{\det \mathrm {D}G_\times } \frac{1+\displaystyle {\varepsilon \frac{\nabla {\overline{\rho }}_\times }{{\overline{\rho }}_\times }\cdot \frac{\sigma _k+\sigma _{k+1}}{3}}+{\mathcal {O}}(\varepsilon ^2)}{\det \big [{\mathbb {1}}+\frac{1}{2}(\mathrm {D}G_\times )^{-1}\big (\mathrm {D}^2G_\times :[\omega _k]^2\big |\mathrm {D}^2G_\times :[\omega _{k-1}]^2\big ) \,(\omega _k|\omega _{k+1})^{-1}+{\mathcal {O}}(\varepsilon ^2) \big ]} \\&\quad =\frac{{\overline{\rho }}_\times }{\det \mathrm {D}G_\times }\left( 1+\varepsilon \left\{ \chi _k-\frac{1}{2}\vartheta _k\right\} +{\mathcal {O}}(\varepsilon ^2)\right) , \end{aligned}
where
\begin{aligned} \chi _k&= \frac{\nabla {\overline{\rho }}_\times }{{\overline{\rho }}_\times }\cdot \frac{\sigma _k+\sigma _{k+1}}{3}, \\ \vartheta _k&= {{\mathrm{tr}}}\left[ \big ((\mathrm {D}G_\times )^{-1}\mathrm {D}^2G_\times :[\sigma _k]^2\big |(\mathrm {D}G_\times )^{-1}\mathrm {D}^2G_\times :[\sigma _{k+1}]^2\big )\,(\sigma _k|\sigma _{k+1})^{-1}\right] . \end{aligned}
\begin{aligned}&\sum _{k=0}^5\left\{ \frac{1}{2} P\left( \frac{{\overline{\rho }}_\times }{\det \mathrm {D}G_\times }\right) + \frac{\varepsilon }{2}P'\left( \frac{{\overline{\rho }}_\times }{\det \mathrm {D}G_\times }\right) \left\{ \chi _k-\frac{1}{2}\vartheta _k\right\} + {\mathcal {O}}(\varepsilon ^2) \right\} {\mathbb {J}}\mathrm {D}G_\times (\omega _{k+1}-\omega _{k}) \\&=\frac{1}{2} P\left( \frac{{\overline{\rho }}_0}{\det \mathrm {D}G_\times }\right) {\mathbb {J}}\mathrm {D}G_\times \left( \sum _{k=0}^5 (\omega _{k+1}-\omega _{k})\right) \\&\quad + \frac{\varepsilon ^2}{4}P'\left( \frac{{\overline{\rho }}_\times }{\det \mathrm {D}G_\times }\right) {\mathbb {J}}\mathrm {D}G_\times {\mathbb {J}}^T\left( \sum _{k=0}^5 \left\{ 2\chi _k-\vartheta _k\right\} {\mathbb {J}}(\sigma _{k+1}-\sigma _{k})\right) + {\mathcal {O}}(\varepsilon ^3) \\&= 0 + \frac{\sqrt{3}}{2}\varepsilon ^2P'\left( \frac{{\overline{\rho }}_\times }{\det \mathrm {D}G_\times }\right) \, (\mathrm {D}G_\times )^{-T}\left\{ {{\mathrm{tr}}}_{12}\big [(\mathrm {D}G_\times )^{-1}\mathrm {D}^2G_\times \big ]^T-\frac{\nabla \rho _\times }{\rho _\times }\right\} + {\mathcal {O}}(\varepsilon ^3), \end{aligned}
where we have use the auxiliary algebraic results from Lemmas B.2, B.3 and B.4.
For the remaining part of the impulse term, a very rough approximation is sufficient:
\begin{aligned} \nabla V(g) = \nabla V(G_\times ) + {\mathcal {O}}(\varepsilon ) \end{aligned}
holds for any g that is a convex combination of $$G_\times ,G_0,\ldots ,G_5$$, where the implicit constant is controlled in terms of the supremum of $$\mathrm {D}^2V$$ and $$\mathrm {D}G$$ on $$K'$$. With that, we simply have, using again (5.12):
Together, this yields (5.8b). $$\square$$
Remark 5.4
The hypotheses of Theorem (5.2) require that the $${\mathscr {T}}_\varepsilon$$ are almost hexagonal on $$K'$$. This seems like a technical hypothesis that simplifies calculations, but apparently, some strong symmetry property of the $${\mathscr {T}}_\varepsilon$$ is necessary for the validity of the result.
To illustrate the failure of consistency—at least in the specific form considered here—assume that $$V\equiv 0$$ and $${\overline{\rho }}\equiv 1$$, and consider a sequence of triangulations $${\mathscr {T}}_\varepsilon$$ for which there is a node $$\omega _\times$$ such that (5.7) holds with the $$\sigma _k$$ being replaced by a different six-tuple of vectors $$\sigma '_k$$. Repeating the steps of the proof above, it is easily seen that $${\mathbf {p}}_\times =a\varepsilon ^2\,\partial _tG(t;\omega _\times )+{\mathcal {O}}(\varepsilon ^3)$$, with an $$\varepsilon$$-independent constant $$a>0$$ in place of $$\sqrt{3}/2$$, and that
\begin{aligned} {\mathbf {J}}_\times =-\frac{\varepsilon ^2}{4} P'\left( \frac{1}{\det \mathrm {D}G_\times }\right) \,(\mathrm {D}G_\times )^{-T} \sum _{k=0}^5\vartheta '_k{\mathbb {J}}(\sigma _{k+1}'-\sigma _k')+{\mathcal {O}}(\varepsilon ^3), \end{aligned}
with
\begin{aligned} \vartheta _k' = {{\mathrm{tr}}}\left[ \big ((\mathrm {D}G_\times )^{-1}\mathrm {D}^2G_\times :[\sigma _k']^2\big |(\mathrm {D}G_\times )^{-1}\mathrm {D}^2G_\times :[\sigma _{k+1}']^2\big )\,(\sigma _k'|\sigma _{k+1}')^{-1}\right] . \end{aligned}
If a result of the form (5.8b)—with $$\sqrt{3}/2$$ replaced by a—was true, then this implies in particular that
\begin{aligned} \sum _{k=0}^5\vartheta '_k{\mathbb {J}}(\sigma _{k+1}'-\sigma _k') = a'{{\mathrm{tr}}}_{12}\big [(\mathrm {D}G_\times )^{-1}\mathrm {D}^2G_\times \big ] \end{aligned}
(5.13)
holds with some constant $$a'>0$$ for arbitrary matrices $$\mathrm {D}G_\times \in {\mathbb {R}}^{2\times 2}$$ of positive determinant and tensors $$\mathrm {D}^2 G_\times \in {\mathbb {R}}^{2\times 2\times 2}$$ that are symmetric in the second and third component. A specific example for which (5.13) is not true is given by
\begin{aligned} \sigma _0' = {1\atopwithdelims ()0} = -\sigma _3',\quad \sigma _1' = {\frac{1}{2}\atopwithdelims ()\frac{1}{2}} = -\sigma _4',\quad \sigma _2' = {0\atopwithdelims ()1} = -\sigma _5', \end{aligned}
(5.14)
in combination with $$\mathrm {D}G_\times ={\mathbb {1}}$$, and a $$\mathrm {D}^2G_\times$$ that is zero except for two ones, at the positions (1, 2, 2) and (2, 1, 1). In Lemma B.5, we show that the left-hand side in (5.13) equals to $$1\atopwithdelims ()1$$; on the other hand, the right-hand side is clearly zero.
Note that this counter-example is significant, insofar as the skew (in fact, degenerate) hexagon described by the $$\sigma _k'$$ in (5.14) corresponds to a popular method for triangulation of the plane.
## 6 Numerical Simulations in $$d=2$$
### 6.1 Implementation
The Euler–Lagrange equations for the $$d=2$$-dimensional case have been derived in (5.3). We perfom a small modification in the potential term in order to simplify calculations with presumably minimal loss in accuracy:
\begin{aligned} \mathbf {Z}_\times [G;G^*]&= \sum _{k=0}^5\frac{\mu _{\mathscr {T}}^k}{12} \left[ 2\left( \frac{G_\times -G^*_\times }{\tau }\right) +\left( \frac{G_k-G^*_k}{\tau }\right) +\left( \frac{G_{k+1}-G^*_{k+1}}{\tau }\right) \right] \\&\quad + \sum _{k=0}^5 \bigg [ \frac{1}{2}\widetilde{h}'\left( \frac{\det (G_k-G_\times |G_{k+1}-G_\times )}{2\mu _{\mathscr {T}}^k}\right) {\mathbb {J}}(G_{k+1}-G_k)\\&\quad + \frac{\mu _{\mathscr {T}}^{k}}{6}\nabla V(G_{k+\frac{1}{2}})\bigg ], \end{aligned}
with the short-hand notation
\begin{aligned} G_{k+\frac{1}{2}} = \frac{1}{3}(G_\times +G_k+G_{k+1}). \end{aligned}
On the main diagonal, the Hessian amounts to
\begin{aligned} \mathbf {H}_{\times \times }[G]&= \left( \sum _{k=0}^5\frac{\mu _{\mathscr {T}}^k}{6\tau }\right) \mathbb {1}_2 \\&\quad + \sum _{k=0}^5\frac{1}{4\mu _{\mathscr {T}}^k}\widetilde{h}''\left( \frac{\det (G_k-G_\times |G_{k+1}-G_\times )}{2\mu _{\mathscr {T}}^k}\right) \big [{\mathbb {J}}(G_{k+1}-G_k)\big ]\big [{\mathbb {J}}(G_{k+1}-G_k)\big ]^\top \!\\&\quad + \sum _{k=0}^5 \frac{\mu _{\mathscr {T}}^{k}}{18}\nabla ^2 V(G_{k+\frac{1}{2}}) \end{aligned}
Off the main diagonal, the entries of the Hessian are given by
\begin{aligned} \mathbf {H}_{\times k}[G]&= \frac{\mu _{\mathscr {T}}^k+\mu _{\mathscr {T}}^{k-1}}{12\tau }\mathbb {1}_2 \\&\quad + \frac{1}{4\mu _{\mathscr {T}}^k}\widetilde{h}''\left( \frac{\det (G_k-G_\times |G_{k+1}-G_\times )}{2\mu _{\mathscr {T}}^k}\right) \big [{\mathbb {J}}(G_{k+1}-G_k)\big ]\big [{\mathbb {J}}(G_{k+1}-G_\times )\big ]^\top \!\\&\quad - \frac{1}{4\mu _{\mathscr {T}}^{k-1}}\widetilde{h}''\left( \frac{\det (G_{k-1}-G_\times |G_{k}-G_\times )}{2\mu _{\mathscr {T}}^{k-1}}\right) \big [{\mathbb {J}}(G_{k}-G_{k-1})\big ]\big [{\mathbb {J}}(G_{k-1}-G_\times )\big ]^\top \!\\&\quad + \frac{\mu _{\mathscr {T}}^{k}}{18}\nabla ^2 V(G_{k+\frac{1}{2}}) + \frac{\mu _{\mathscr {T}}^{k-1}}{18}\nabla ^2 V(G_{k-\frac{1}{2}}). \end{aligned}
The scheme consists of an inner (Newton) and an outer (time stepping) iteration. We start from a given initial density $$\rho _0$$ and define the solution at the next time step inductively by applying Newton’s method in the inner iteration. To this end we initialise $$G^{(0)}:=G^n$$ with $$G^n$$, the solution at the nth time step, and define inductively
\begin{aligned} G^{(s+1)} := G^{(s)} + \delta G^{(s+1)} , \end{aligned}
where the update $$\delta G^{(s+1)}$$ is the solution to the linear system
\begin{aligned} \mathbf {H}[G^{(s)}] \delta G^{(s+1)} = -\mathbf {Z}[G^{(s)};G^n] . \end{aligned}
The effort of each inner iteration step is essentially determined by the effort to invert the sparse matrix $$\mathbf {H}[G^{(s)}]$$. As soon as the norm of $$\delta G^{(s+1)}$$ drops below a given stopping threshold, define $$G^{n+1}:=G^{(s+1)}$$ as approximate solution in the $$n+1$$st time step.
In all experiments the stopping criterion in the Newton iteration is set to $$10^{-9}$$.
### 6.2 Numerical Experiments
In this section we present results of our numerical experiments for (1.1) with a cubic porous-medium nonlinearity $$P(r)=r^3$$ and different choices for the external potential V,
\begin{aligned} \partial _t\rho = \Delta (u^3) + \nabla \cdot (u\nabla V). \end{aligned}
(6.1)
#### 6.2.1 Numerical experiment 1: unconfined evolution of Barenblatt profile
As a first example, we consider the “free” cubic porous medium equation, that is (6.1) with $$V\equiv 0$$. It is well-known (see, e.g., Vazquez [38]) that in the long-time limit $$t\rightarrow \infty$$, arbitrary solutions approach a self-similar one,
\begin{aligned} \rho ^*(t,x) = t^{-d\alpha }{\mathcal {B}}_3\big (t^{-\alpha }x\big ) \quad \text {with}\quad \alpha =\frac{1}{6}, \end{aligned}
(6.2)
where $${\mathcal {B}}_3$$ is the associated Barenblatt profile
\begin{aligned} {\mathcal {B}}_3(z) = \left( C_3-\frac{1}{3}\Vert z\Vert ^2\right) _+^{\frac{1}{2}}, \end{aligned}
(6.3)
where $$C_3=(2\pi )^{-\frac{2}{3}}\approx 0.29$$ is chosen to normalize $${\mathcal {B}}_3$$’s mass to unity.
In this experiment, we are only interested in the quality of the numerical approximation for the self-similar solution (6.2). To reduce numerical effort, we impose a four-fold symmetry of the approximation: we use the quarter circle as computational domain K, and interprete the discrete function thereon as one of four symmetric pieces of the full discrete solution. To preserve reflection symmetry over time, homogeneous Neumann conditions are imposed on the artificial boundaries. This is implemented by reducing the degrees of freedom of the nodes along the x- and y-axes to tangential motion. We initialize our simulation with a piecewise constant approximation of the profile of $$\rho ^*$$ from (6.3) at time $$t=0.01$$. We choose a time step $$\tau =0.001$$ and the final time $$T=2$$. In Fig. 2, we have collected snapshots of the approximated density at different instances of time. The Barenblatt profile of the solution is very well pertained over time.
Remark 6.1
It takes less than 2 min to complete this simulation on standard laptop (Matlab code on a mid-2013 MacBook Air 11” with 1.7 GHz Intel Core i7 processor).
Figure 3 shows surface plots of the discrete solution at different times in comparison with the Barenblatt profile at the respective time. By construction of the scheme, the initial mass is exactly conserved in time as the discrete solution propagates. The left plot in Fig. 4 shows the decay in the energy and gives quantitative information about the difference of the discrete solution to the analytical Barenblatt solution. The numerical solution shows good agreement with the analytical energy decay rate $$c=2/3$$.
We also compute the $$l_1$$-error of the discrete solution to the exact Barenblatt profile and observe that it remains within the order of the fineness of the triangulation. The mass of the discrete solution is perfectly conserved, as guaranteed by the construction of our method.
To estimate the convergence order of our method, we run several experiments with the above initial data on different meshes. We fix the ratio $$\tau /h_\mathrm{max}^2=0.4$$ and compute the $$l_1$$-error at time $$T=0.2$$ on triangulations with $$h_\mathrm{max}=0.2,\,0.1,\,0.05,\,0.025.$$ We expect the error to decay as a power of $$h_\mathrm{max}$$. The double logarithmic plot should reveal a line with its slope indicating the numerical convergence order. The right plot in Fig. 4 shows the result, the estimated numerical convergence order which is obtained from a least-squares fitted line through the points is equal to 1.18. This indicates first order convergence of the scheme with respect to the spatial discretisation parameter $$h_\mathrm{max}$$.
#### 6.2.2 Numerical experiment 2: Asymptotic self-similarity
In our second example, we are still concerned with the free cubic porous medium Eq. (6.1) with $$V\equiv 0$$. This time, we wish to give an indication that the discrete approximation of the self-similar solution from (6.2) from the previous experiment might inherit the global attractivity of its continuous counterpart. More specifically, we track the discrete evolution for the initial datum
\begin{aligned} \rho _0(x,y)= 3000(x^2+y^2)\exp [-5(|x|+|y|)]+0.1 \end{aligned}
(6.4)
until time $$T=0.1$$ and observe that it appears to approach the self-similar solution from above. Snapshots of the simulation are collected in Fig. 5.
#### 6.2.3 Numerical experiment 3: two peaks merging into one under the influence of a confining potential
In this example we consider as initial condition two peaks, connected by a thin layer of mass, given by
\begin{aligned} \rho _0(x,y)= & {} \exp [-\,20((x\,-\,0.35)^2\,+\,(y\,-\,0.35)^2)]\,+\,\exp [-\,20((x\,+\,0.35)^2\nonumber \\&+\,(y\,+\,0.35)^2 )]\,+\,0.001. \end{aligned}
(6.5)
We choose a triangulation of the square $$[-1.5,1.5]^2$$ and initialise the discrete solution piecewise constant in each triangle, with a value corresponding to (6.5), evaluated in the centre of mass of each triangle. We solve the porous medium equation with a confining potential, i.e. (1.1) with $$P(r)=r^m$$ and $$V(x,y)=5(x^2+y^2)/2$$. The time step is $$\tau =0.001$$ and the final time is $$T=0.2.$$
Figure 6 shows the evolution from the initial density. As time increases the peaks smoothly merge into each other. As the thin layer around the peaks is also subject to the potential the triangulated domain shrinks in time. Even if we do not know how to prevent theoretically the intersection of the images of the discrete Lagrangian maps, this seems not to be a problem in practice. As time evolves, the discrete solution approaches the steady state Barenblatt profile given by
\begin{aligned} {\mathcal {B}}(z) = \left( C-\frac{5}{3} ||z||^2 \right) _+^{\frac{1}{2}}, \end{aligned}
(6.6)
where C is chosen as the mass of the density. The plot in Fig. 7 shows the exponential decay of the $$l_1$$-distance of the discrete solution to the steady state Barenblatt profile (6.6). We observe that the decay agrees very well with the analytically predicted decay $$\exp (-5t)$$ until $$t=0.08$$. For larger times, one would monitor triangle quality numerically, and re-mesh, locally coarsening the triangulation where necessary.
#### 6.2.4 Numerical experiment 4: one peak splitting under the influence of a quartic potential
We consider as the initial condition
\begin{aligned} \rho _0(x,y)=1-(x^2+y^2). \end{aligned}
(6.7)
We choose a triangulation of the unit circle and initialise the discrete solution piecewise constant in each triangle, with a value corresponding to (6.7), evaluated in the centre of mass of each triangle. We solve the porous medium equation with a quartic potential, i.e. (1.1) with $$P(r)=r^m$$ and $$V(x)=5(x^2+(1-y^2)^2)/2$$. The time step is $$\tau =0.005$$ and the final time is $$T=0.02.$$
Figure 8 shows the evolution of the initial density. As time increases the initial density is progressively split, until two new maxima emerge which are connected by a thin layer. For larger times, when certain triangles become excessively distorted, one would monitor triangle quality numerically, and re-mesh, locally refining the triangulation where necessary.
## Appendix A: Proof of the Lagrangian representation
Proof of Lemma 1.1
We verify that the density function given by $$(G_t^{-1})_{\#}\rho _t$$ on $$K\subset {{\mathbb {R}}^d}$$ is constant with respect to time t; the identity (1.6) then follows since
\begin{aligned} \rho _t = (G_t\circ G_t^{-1})_\#\rho _t = (G_t)_\#\big [(G_t^{-1})_{\#}\rho _t\big ] = (G_t)_\#\big [(G_0^{-1})_\#\rho ^0\big ] = (G_t)_\#{\overline{\rho }}. \end{aligned}
Firstly, from the definition of the inverse,
\begin{aligned} G_t^{-1}\circ G_t = {\mathrm {id}}\end{aligned}
for all t, differentiating with respect to time yields
\begin{aligned} \mathrm {D}(G_t^{-1}) \circ G_t\, \partial _{t} G_t + \partial _{t} (G_t^{-1}) \circ G_t = 0, \end{aligned}
and so, using (1.5) and (1.4b),
\begin{aligned} \partial _{t} (G_t^{-1}) = - \mathrm {D}(G_t^{-1}) (\partial _{t} G_t \circ G_t^{-1}) = - \mathrm {D}(G_t^{-1})\mathbf {v}[\rho _t]. \end{aligned}
(A.1)
Now, let $$\varphi$$ be a smooth test function, and consider
\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t} \int \varphi \,(G_t^{-1})_{\#} \rho _t = \frac{\mathrm {d}}{\mathrm {d}t} \int (\varphi \circ G_t^{-1}) \rho _t \\&\qquad = \int (\varphi \circ G_t^{-1}) \partial _{t} \rho _t + \int \mathrm {D}\varphi \circ G_t^{-1} \partial _{t} (G_t^{-1}) \rho _t \\&\qquad = -\int (\varphi \circ G_t^{-1}) [\nabla \cdot (\rho _t v(\rho _t))] \\&\qquad \quad - \int (\mathrm {D}\varphi \circ G_t^{-1})\,\mathrm {D}(G_t^{-1})\,v(\rho _t) \rho _t&\text {by }(1.1) \text { and }(1.5) \\&\qquad = \int (\mathrm {D}\varphi \circ G_t^{-1}) \mathrm {D}(G_t^{-1}) \,[v(\rho _t) - v(\rho _t)] \rho _t&\text {integrating by parts} \\&\qquad = 0. \end{aligned}
As $$\varphi$$ was arbitrary, $$(G_t^{-1})_{\#}\rho _t$$ is constant with respect to time. $$\square$$
## Appendix B: Technical lemmas
Lemma B.1
Given $$g_0,g_1,\ldots ,g_d\in {\mathbb {R}}^d$$, then
(B.1)
Proof
Thanks to the symmetry of the integral with respect to the exchange of the components $$\omega _j$$, the left-hand side of (B.1) equals to
(B.2)
We calculate the integrals, using Fubini’s theorem. First integral:
Second integral:
Third integral:
Substitute this into (B.2):
\begin{aligned}&\left( 1-\frac{2}{d+1}+\frac{d^2+d}{(d+1)(d+2)}\right) \Vert g_0\Vert ^2 +\left( \frac{2}{d+1}-\frac{2d+2}{(d+1)(d+2)}\right) \sum _{1\le j\le d}g_0\cdot g_j \\&\qquad +\frac{2}{(d+1)(d+2)}\sum _{1\le j\le d}\Vert g_j\Vert ^2 +\frac{2}{(d+1)(d+2)}\sum _{1\le i<j\le d}g_i\cdot g_j\\&\quad =\frac{2}{(d+1)(d+2)}\left( \Vert g_0\Vert ^2 + \sum _{1\le j\le d}g_0\cdot g_j + \sum _{1\le j\le d}\Vert g_j\Vert ^2 + \sum _{1\le i<j\le d}g_i\cdot g_j \right) . \end{aligned}
Collecting terms yields the right-hand side of (B.1). $$\square$$
Lemma B.2
For each $$A\in {\mathbb {R}}^{2\times 2}$$, we have $${\mathbb {J}}A{\mathbb {J}}^T=(\det A)\,A^{-T}$$.
Proof
This is verified by direct calculation:
\begin{aligned} {\mathbb {J}}A{\mathbb {J}}^T = \begin{pmatrix} 0 &{} -1 \\ 1 &{} 0 \end{pmatrix} \begin{pmatrix} a_{11} &{} a_{12} \\ a_{21} &{} a_{22} \end{pmatrix} \begin{pmatrix} 0 &{} 1 \\ -1 &{} 0 \end{pmatrix} = \begin{pmatrix} a_{22} &{} -a_{21} \\ -a_{12} &{} a_{11} \end{pmatrix} = (\det A)\, A^{-T}. \end{aligned}
$$\square$$
Lemma B.3
With $$\sigma _k\in {\mathbb {R}}^2$$ defined as in (5.7), we have that
\begin{aligned} \sum _{k=0}^5{\mathbb {J}}(\sigma _{k}-\sigma _{k+1}) \left( \frac{\sigma _k+\sigma _{k+1}}{3}\right) ^T = \sqrt{3}\ {\mathbb {1}}. \end{aligned}
Proof
With the abbreviations $$\phi _x=\frac{\pi }{3}x$$ and $$\psi =\frac{\pi }{3}$$:
\begin{aligned} \sum _{k=0}^5{\mathbb {J}}(\sigma _{k}-\sigma _{k+1}) \left( \frac{\sigma _k+\sigma _{k+1}}{3}\right) ^T&= \frac{1}{3}\sum _{k=0}^5 {\sin \phi _{k+1}-\sin \phi _k\atopwithdelims ()\cos \phi _k-\cos \phi _{k+1}}{\cos \phi _k+\cos \phi _{k+1}\atopwithdelims ()\sin \phi _k+\sin \phi _{k+1}}^T \\&= \frac{1}{3}\sum _{k=0}^5\left( 2\sin \frac{\psi }{2}\right) {\cos \phi _{k+\frac{1}{2}}\atopwithdelims ()\sin \phi _{k+\frac{1}{2}}} \,\left( 2\cos \frac{\psi }{2}\right) {\cos \phi _{k+\frac{1}{2}}\atopwithdelims ()\sin \phi _{k+\frac{1}{2}}}^T \\&= \frac{\sin \psi }{3}\sum _{k=0}^5 \begin{pmatrix} 2\cos ^2\phi _{k+\frac{1}{2}} &{} 2\cos \phi _{k+\frac{1}{2}}\sin \phi _{k+\frac{1}{2}} \\ 2\cos \phi _{k+\frac{1}{2}}\sin \phi _{k+\frac{1}{2}} &{} 2\sin ^2\phi _{k+\frac{1}{2}} \end{pmatrix}\\&= \frac{\sqrt{3}}{6} \sum _{k=0}^5\left[ {\mathbb {1}}+ \begin{pmatrix} \cos \phi _{2k+1} &{} \sin \phi _{2k+1} \\ \sin \phi _{2k+1} &{} -\cos \phi _{2k+1} \end{pmatrix} \right] = \sqrt{3}\ {\mathbb {1}}. \end{aligned}
$$\square$$
Lemma B.4
Let the scheme $$B:=(b_{pqr})_{p,q,r\in \{1,2\}}\in {\mathbb {R}}^{2\times 2\times 2}$$ of eight numbers $$b_{pqr}\in {\mathbb {R}}$$ be symmetric in the last two indices, $$b_{pqr}=b_{prq}$$. With $$\sigma _k\in {\mathbb {R}}^2$$ defined as in (5.7), we have that
\begin{aligned} \sum _{k=0}^5 {{\mathrm{tr}}}\big [\big (\sigma _k\big |\sigma _{k+1}\big )^{-1}\big (B:[\sigma _k]^2\big |B:[\sigma _{k+1}]^2\big )\big ]\,{\mathbb {J}}(\sigma _{k}-\sigma _{k+1}) =2\sqrt{3}{{\mathrm{tr}}}_{12}[B]^T. \end{aligned}
(B.3)
Proof
In principle, this lemma can be verified by a direct calculation, by writing out the six terms in the sum explicitly and using trigonometric identities. Below, we give a slightly more conceptual proof, in which we use symmetry arguments to reduce the number of expressions significantly.
For the matrix involving B, we obtain
\begin{aligned}&\big (B:[\sigma _k]^2\big |B:[\sigma _{k+1}]^2\big ) \\&= \begin{pmatrix} b_{111}\sigma _{k,1}^2+b_{122}\sigma _{k,2}^2+2b_{112}\sigma _{k,1}\sigma _{k,2} &{} b_{111}\sigma _{k+1,1}^2+b_{122}\sigma _{k+1,2}^2+2b_{112}\sigma _{k+1,1}\sigma _{k+1,2} \\ b_{211}\sigma _{k,1}^2+b_{222}\sigma _{k,2}^2+2b_{212}\sigma _{k,1}\sigma _{k,2} &{} b_{211}\sigma _{k+1,1}^2+b_{222}\sigma _{k+1,2}^2+2b_{212}\sigma _{k+1,1}\sigma _{k+1,2} \end{pmatrix}, \end{aligned}
while clearly
\begin{aligned} \big (\sigma _k\big |\sigma _{k+1}\big )^{-1} = \frac{2}{\sqrt{3}} \begin{pmatrix} \sigma _{k+1,2} &{} -\sigma _{k+1,1} \\ -\sigma _{k,2} &{} \sigma _{k,1} \end{pmatrix}. \end{aligned}
The sum of the diagonal entries of the matrix product are easily calculated,
\begin{aligned} T_k:={{\mathrm{tr}}}\big [\big (\sigma _k\big |\sigma _{k+1}\big )^{-1}\big (B:[\sigma _k]^2\big |B:[\sigma _{k+1}]^2\big )\big ] = \frac{2}{\sqrt{3}}\sum _{p,q,r=1}^2 b_{pqr}\gamma _{pqr,k}, \end{aligned}
with the trigonometric expressions
\begin{aligned}&\gamma _{111,k}=\sigma _{k,1}^2\sigma _{k+1,2}-\sigma _{k+1,1}^2\sigma _{k,2},\quad \gamma _{122,k}=\sigma _{k,2}^2\sigma _{k+1,2}-\sigma _{k+1,2}^2\sigma _{k,2}, \\&\gamma _{112,k}=\gamma _{121,k}=\sigma _{k,1}\sigma _{k,2}\sigma _{k+1,2}-\sigma _{k+1,1}\sigma _{k+1,2}\sigma _{k,2}, \\&\gamma _{211,k}=\sigma _{k+1,1}^2\sigma _{k,1}-\sigma _{k,1}^2\sigma _{k+1,1},\quad \gamma _{222,k}=\sigma _{k+1,2}^2\sigma _{k,1}-\sigma _{k,2}^2\sigma _{k+1,1},\\&\gamma _{212,k}=\gamma _{221,k}=\sigma _{k+1,1}\sigma _{k+1,2}\sigma _{k,1}-\sigma _{k,1}\sigma _{k,2}\sigma _{k+1,1}. \end{aligned}
To key step is to calculate the sum over $$k=0,1,\ldots ,5$$ of the products of $$T_k$$ with the respective vector
\begin{aligned} \eta _k ={\mathbb {J}}(\sigma _{k}-\sigma _{k+1})= \begin{pmatrix} \sigma _{k+1,2}-\sigma _{k,2}\\ \sigma _{k,1}-\sigma _{k+1,1} \end{pmatrix}. \end{aligned}
Several simplifications of this sum can be performed, thanks to the particular form of the $$\gamma _{pqr,k}$$ and elementary trigonometric identities. First, observe that $$\sigma _{k+3}=-\sigma _k$$, and hence that $$\gamma _{pqr,k+3}=-\gamma _{pqr,k}$$. Since further $$\eta _{k+3}=-\eta _k$$, it follows that
\begin{aligned} \gamma _{pqr,k+3}\eta _{k+3}=\gamma _{pqr,k}\eta _k. \end{aligned}
(B.4)
Second, $$\eta$$ can be evaluated explicitly for $$k=1,2,3$$:
\begin{aligned} \eta _0 =\frac{1}{2} \begin{pmatrix} \sqrt{3} \\ 1 \end{pmatrix}, \quad \eta _1 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \quad \eta _2 =\frac{1}{2} \begin{pmatrix} -\sqrt{3} \\ 1 \end{pmatrix}. \end{aligned}
(B.5)
Third, since $$\sigma _{0,1}=-\sigma _{3,1}$$ and $$\sigma _{1,1}=-\sigma _{2,1}$$, as well as $$\sigma _{0,2}=\sigma _{3,2}$$ and $$\sigma _{1,2}=\sigma _{2,2}$$, we obtain that
\begin{aligned} \gamma _{pqr,1} = 0 \quad \text {if }p+q+r\text { is odd}, \quad \text {and}\quad \gamma _{pqr,2} = (-1)^{p+q+r}\gamma _{pqr,0}. \end{aligned}
(B.6)
By putting this together, we arrive at
\begin{aligned} \sum _{k=0}^5\gamma _{pqr,k}\eta _k&{\mathop {=}\limits ^{\mathrm{B.4}}}2\sum _{k=0}^2\gamma _{pqr,k}\eta _k \\&{\mathop {=}\limits ^{\mathrm{B.5}}} \begin{pmatrix} \sqrt{3}\big (\gamma _{pqr,0}-\gamma _{pqr,2}\big ) \\ \gamma _{pqr,0}+2\gamma _{pqr,1}+\gamma _{pqr,2} \end{pmatrix} \\&{\mathop {=}\limits ^{\mathrm{B.6}}} \begin{pmatrix} \sqrt{3}\big (1-(-1)^{p+q+r}\big )\gamma _{pqr,0}\\ \big (1+(-1)^{p+q+r}\big )\big (\gamma _{pqr,0}+\gamma _{pqr,1}\big ) \end{pmatrix} = \begin{pmatrix} 2\sqrt{3}\,\gamma _{pqr,0}\,(1-\mathfrak {e}_{pqr})\\ 2\big (\gamma _{pqr,0}+\gamma _{pqr,1}\big )\,\mathfrak {e}_{pqr} \end{pmatrix}, \end{aligned}
where $$\mathfrak {e}_{pqr}=1$$ if $$p+q+r$$ is even, and $$\mathfrak {e}_{pqr}=0$$ if $$p+q+r$$ is odd. By elementary computations,
\begin{aligned} \begin{array}{llll} p+q+r\text { odd, }k=0: &{}\gamma _{111,0}=\frac{\sqrt{3}}{2}, &{}\gamma _{122,0}=0, &{}\gamma _{212,0}=\gamma _{221,0}=\frac{\sqrt{3}}{4};\\ p+q+r\text { even, }k=0: &{}\gamma _{211,0}=-\frac{1}{4}, &{}\gamma _{222,0}=\frac{3}{4}, &{}\gamma _{112,0}=\gamma _{121,0}=0;\\ p+q+r\text { even, }k=1: &{}\gamma _{211,1}=\frac{1}{4}, &{}\gamma _{222,1}=\frac{3}{4}, &{}\gamma _{112,1}=\gamma _{121,1}=\frac{3}{4}, \end{array} \end{aligned}
and so the final result is:
\begin{aligned}&\sum _{k=0}^5 {{\mathrm{tr}}}\big [\big (\sigma _k\big |\sigma _{k+1}\big )^{-1}\big (B:[\sigma _k]^2\big |B:[\sigma _{k+1}]^2\big )\big ]\,{\mathbb {J}}(\sigma _{k}-\sigma _{k+1}) \\&\quad =\sum _{k=0}^5T_k\eta _k =\frac{2}{\sqrt{3}}\sum _{p,q,r=1}^2\left( b_{pqr}\sum _{k=0}^5\gamma _{pqr,k}\eta _k\right) =2\sqrt{3} \begin{pmatrix} b_{111}+b_{212} \\ b_{222}+b_{112} \end{pmatrix}, \end{aligned}
which is (B.3). $$\square$$
Lemma B.5
With $$\sigma _k'\in {\mathbb {R}}^2$$ defined as in (5.14), and with $$B=(b_{pqr})_{p,q,r\in \{1,2\}}\in {\mathbb {R}}^{2\times 2\times 2}$$ such that $$b_{pqr}=0$$ except for $$b_{122}=b_{211}=1$$, we have that
\begin{aligned} \sum _{k=0}^5 {{\mathrm{tr}}}\big [\big (\sigma _k'\big |\sigma _{k+1}'\big )^{-1}\big (B:[\sigma _k']^2\big |B:[\sigma _{k+1}']^2\big )\big ]\,{\mathbb {J}}(\sigma _{k}-\sigma _{k+1}) =-{1\atopwithdelims ()1}. \end{aligned}
(B.7)
Proof
This is a slightly tedious, but straightforward calculation. First, by the choice of B,
\begin{aligned} \beta _k:=\big (B:[\sigma _k']^2\big |B:[\sigma _{k+1}']^2\big ) = \begin{pmatrix} (\sigma _{k,2}')^2 &{} (\sigma _{k+1,2}')^2 \\ (\sigma _{k,1}')^2 &{} (\sigma _{k+1,1}')^2 \end{pmatrix}, \end{aligned}
and so, by definition of the $$\sigma _k'$$ in (5.14),
\begin{aligned} \beta _0 =\beta _3 = \begin{pmatrix} 0 &{} \frac{1}{4} \\ 1 &{} \frac{1}{4} \end{pmatrix}, \quad \beta _0=\beta _3 = \begin{pmatrix} \frac{1}{4} &{} 1 \\ \frac{1}{4} &{} 0 \end{pmatrix}, \quad \beta _0=\beta _3 = \begin{pmatrix} 1 &{} 0 \\ 0 &{} 1 \end{pmatrix}. \end{aligned}
For the inverse matrices $$S_k:=\big (\sigma _k'\big |\sigma _{k+1}'\big )^{-1}$$, we obtain
\begin{aligned} S_0 = \begin{pmatrix} 1&{}-1 \\ 0 &{} 2 \end{pmatrix} = -S_3, \quad S_1 = \begin{pmatrix} 2&{}0 \\ -1&{}1 \end{pmatrix} = -S_4, S_2 = \begin{pmatrix} 0&{}1 \\ -1&{}0 \end{pmatrix} = -S_5. \end{aligned}
For the traces $$T_k:={{\mathrm{tr}}}\big [S_k\beta _k\big ]$$, we thus obtain the values:
\begin{aligned} T_0=T_1=-\frac{1}{2},\quad T_3=T_4=\frac{1}{2}, \quad T_2=T_5=0. \end{aligned}
In conclusion,
\begin{aligned} \sum _{k=0}^5 T_k\,{\mathbb {J}}(\sigma _{k}-\sigma _{k+1}) = {\mathbb {J}}\left[ -\frac{1}{2} (\sigma _0-\sigma _2) +\frac{1}{2} (\sigma _3-\sigma _5) \right] = {\mathbb {J}}{-1\atopwithdelims ()1} = -{1\atopwithdelims ()1}, \end{aligned}
which is (B.7). $$\square$$
## Appendix C: Lack of convexity
Below, we discuss why the minimization problem (3.7) is not convex. More precisely, we show that $$G\mapsto {\mathbf {E}}_\boxplus (G;\hat{G})$$ is not convex as a function of G on the affine ansatz space $${\mathcal {A}}_{\mathscr {T}}$$. Since $${\mathbf {E}}_\boxplus (G;\hat{G})$$ is a convex combination of the expressions $${\mathbb {H}}_m\big ((A_m|b_m);(\hat{A}_m|\hat{b}_m)\big )$$, it clearly suffices to discuss the convexity of the latter.
We consider a curve $$s\mapsto (A_m+s\alpha _m|b_m+s\beta _m)$$ and evaluate the second derivatives of the components of the functional at $$s=0$$. First,
Second,
If we assume that $$\nabla ^{2} V \ge \lambda {\mathbb {1}}$$, then we obtain for the sum of these two contributions that
For the remaining term, however, we obtain — using the abbreviations $$\widetilde{g}(s)=s\widetilde{h}'(s)$$ and $$\widetilde{f}(s)=s\widetilde{g}'(s)$$ — that
\begin{aligned} \frac{\mathrm {d}^2}{\mathrm {d}^2 s}\bigg |_{s=0}&\widetilde{h}\left( \frac{\det (A_{m} + s \alpha _{m})}{{\overline{\rho }}_m}\right) \\&=\frac{\mathrm {d}}{\mathrm {d}s}\bigg |_{s=0} \left\{ \widetilde{g}\left( \frac{\det (A_{m} + s \alpha _{m})}{{\overline{\rho }}_m}\right) \,{{\mathrm{tr}}}\big [(A_{m} + s \alpha _{m})^{-1} \alpha _{m}\big ] \right\} \\&=\widetilde{f}\left( \frac{\det A_{m}}{{\overline{\rho }}_m}\right) \,\big ({{\mathrm{tr}}}\big [A_m^{-1}\alpha _m\big ]\big )^2 - \widetilde{g}\left( \frac{\det A_{m}}{{\overline{\rho }}_m}\right) \,{{\mathrm{tr}}}\big [\big ( A_m^{-1}\alpha _m \big )^2\big ] . \end{aligned}
Now observe that $$\widetilde{f}(s)=P'(1/s)-sP(1/s)$$ is a non-negative, and $$\widetilde{g}(s) = -sP(1/s)$$ is a non-positive function. Thus, from the two terms in the final sum, the first one is generally non-negative whereas the second one is of indefinite sign. Choosing
\begin{aligned} \alpha _m:=A_m \begin{pmatrix} 0 &{} 1 \\ 1 &{} 0 \end{pmatrix}, \quad \text {such that}\quad \big ({{\mathrm{tr}}}\big [A_m^{-1}\alpha _m\big ]\big )^2=0, \, {{\mathrm{tr}}}\big [\big ( A_m^{-1}\alpha _m \big )^2\big ]=2, \end{aligned}
the sum is obviously negative.
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2019-09-17 10:54:05
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https://ictp.acad.ro/note-inexact-secant-methods/
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# A note on inexact secant methods
## Abstract
The inexact secant method
$$[x_{k-1},x_{k};F]s_{k}=-F(x_k) +r_k$$,
$$x_{k+1}=x_k+s_k$$, $$k=1,2,\ldots$$, $$x_0,x_1 \in {\mathbb R}^n$$ is considered for solving the nonlinear system $$F(x)=0$$, where $$F:{\mathbb R}^n \rightarrow {\mathbb R}^n$$ is a nonlinear mapping.
We study the similar setting of the inexact Newton method, i.e., when the linear system (involving the divided differences) at each step is not solved exactly, and an error term $$r_k$$ (called residual) is considered.
Under certain standard assumptions, we characterize the superlinear convergence and the r-convergence order of the secant method in terms of the residuals. We also give a sufficient result for linear convergence.
## Authors
E. Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)
## Keywords
nonlinear system of equations; secant method; r-convergence order; inexact method; residual; error term; linear convergence.
## Cite this paper as:
E. Cătinaş, A note on inexact secant methods, Rev. Anal. Numér. Théor. Approx., 25 (1996) nos. 1-2, pp. 33-41.
2457-6794
2501-059X
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2021-09-17 00:18:00
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https://socratic.org/questions/how-do-you-calculate-the-arcsin-sqrt-3-2
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# How do you calculate the arcsin (sqrt(3)/2)?
Calculate $\arcsin \left(\frac{\sqrt{3}}{2}\right)$
Ans: $\frac{\pi}{6} \mathmr{and} \frac{5 \pi}{6}$
$\sin x = \frac{\sqrt{3}}{2}$ --> arc $x = \frac{\pi}{6}$.
Trig unit circle gives another arc $x = \frac{5 \pi}{6}$ that has the same sin value $\left(\frac{\sqrt{3}}{2}\right) .$
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2020-02-22 20:10:51
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https://socratic.org/questions/what-is-the-trigonometric-form-of-7-e-5-pi-12-i
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# What is the trigonometric form of 7 e^( ( 5 pi)/12 i ) ?
Aug 2, 2018
$7 {e}^{\left(\frac{5 \pi}{12}\right) i} = 7 \cdot \left(\cos \left(\frac{5 \pi}{12}\right) + i \sin \left(\frac{5 \pi}{12}\right)\right)$
#### Explanation:
Trigonometric form of ${e}^{i x}$, using Euler's Equation, is given by
${e}^{i x} = \cos x + i \sin x$
$z = | z | {e}^{i x} = | z | \cdot \left(\cos \theta + i \sin \theta\right)$
$7 {e}^{\left(\frac{5 \pi}{12}\right) i} = 7 \cdot \left(\cos \left(\frac{5 \pi}{12}\right) + i \sin \left(\frac{5 \pi}{12}\right)\right)$
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2021-06-21 06:11:35
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http://mathcenter.oxford.emory.edu/site/home/futurePages/rProjectAnthrax/
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## R Project: Testing for Anthrax
You are designing an automated test to see if a 10 mm circular collection plate has accumulated a lethal dose of anthrax (Bacillus Anthracis). A lethal dose of anthrax is 4130 spores, each of which is 0.0001 mm wide and roughly spherical.
Assume you have available to you a computer program that can take pictures of randomly selected regions of the plate using an Olympus Compound microscope set to 1000x magnification (where the field of view is a circle 0.184 mm across), and count the number of anthrax spores present in each picture.
Even working under the assumption that the spores will be uniformly distributed on the plate, given the small field of view you are concerned that a single picture (or perhaps even several pictures) might not reveal the presence of any spores, when they are indeed present on the plate (just not in the places you looked).
1. What is the probability that a single spore on the plate is seen in a randomly selected picture? Write a function in R, named prob.single.spore.seen(p,f,a) that will calculate this value as a function of the plate diameter, $p$, the diameter of the field of view for this microscope, $f$, and the diameter of an anthrax spore, $a$. Assume a spore will be "seen" if any part of it is visible in the field of view.
2. Let X be the number of spores seen in a single randomly selected picture if there are $n$ spores present on the collection plate. What type of distribution does X follow? One of your colleagues insists that for what you will be using it for, the distribution is approximately Poisson in nature. However, given the life-and-death decisions that your work may be called upon to make, you are worried about the potential errors that could be introduced by using an approximating distribution. Consequently, you decide to NOT use a Poisson distribution to model this situation.
Under this restraint, write a function in R, called simulated.data(x,n,p), that simulates the numbers of spores seen in $x$ different randomly selected pictures of a collection plate containing exactly $n$ spores, where the probability of seeing any particular spore is $p$. The result should be a vector of length $x$.
3. You wish to visualize the distribution associated with your simulated data to get a better feel for it. Knowing that the chances of seeing many spores in any one picture will be remote, you wish to plot the left portion of the corresponding frequency histogram where there is a separate bar for each outcome from 0 to 10, but no others.
In the interests of making a nice graphic, you also wish to have a custom title, labels for your x and y axes, and the rectangles of your histogram to be filled with some color other than white or black.
To get a feel for how much variability there is in the frequencies related to any given number of spores seen, you would also like to add points -- one for each rectangle of your histogram -- horizontally centered with respect to their corresponding rectangles and with heights indicating the expected frequencies of seeing the associated number of spores in a given single picture, in accordance with the underlying distribution.
Lastly, as a concession to your colleague, you would like to do the same for the expected frequencies of seeing the associated number of spores in any one picture using the approximating Poisson distribution (to see just how good a job it does in approximating things). So that these Poisson-based frequencies can be distinguished from the true expected frequencies, you decide to plot the former as "plus signs", and the latter as small circles, and to include a legend to indicate which is which.
Write an R function named num.spores.hist(data,n,p,max) that will produce the plot described above, where:
1. data is a vector like that produced by simulated.data(p)
2. n is the number of spores on the collection plate
3. p is the probability of seeing any particular spore in a randomly selected picture, and
4. max is the maximum number of spores you want to account for in your histogram (any simulated numbers of spores seen in your data that are greater than max should be discarded). So for our purposes, we will be interested in running this function when max = 10
Then create the corresponding histogram for 1000 random pictures when there is a lethal dose of anthrax present on the plate (i.e. 4130 spores) by running the following:
p = prob.single.spore.seen(10,0.184,0.0001)
data = simulated.data(1000,4130,p)
num.spores.hist(data,4130,p,10)
4. What would happen if a different magnification level was used -- one that resulted in the field of view being 400 times larger by area, and the number of spores present on the plate was only 10 (i.e., almost 400 times smaller)? Use the function you created previously to produce a plot corresponding to this new situation.
5. Examples of the histograms generated in questions #3 and #4 above are given below. Noticing that the Poisson distribution does a better job at approximating the true distribution in one of these two instances, how might your colleague (correctly) explain why this is the case?
6. You wish to know how many different pictures must be examined so that one has at least a 99% chance of seeing at least one spore when a lethal dose (4130 spores) is present. Write an R function, num.pics.needed(n,p) that will determine this value, assuming the answer is not more than 100 pictures (an incredibly conservative estimate). The argument $n$ should be the number of spores on the plate, and $p$ should be the probability of seeing any single particular spore in a randomly selected picture.
p = prob.single.spore.seen(10,0.184,0.0001)
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2021-07-30 07:50:33
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http://phdthes.is/html-edition/thesis-emeijse19.html
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Up Next Tail
4.1 Estimating the Importance of Feedback Documents
In Section 2.3.2 we have introduced core relevance feedback models in the language modeling approach to information retrieval (IR). In Eq. 2.14 we have indicated a means by which to emphasize the importance of each individual feedback document, $P\left(D|R\right)$. In this section, we turn to different ways of estimating this relative importance. When we know (or assume) that a given set of documents, $R=\left\{{D}_{1},\dots ,{D}_{|R|}\right\}$, is relevant to a query, we posit that documents therein that are more similar to $R$ are more topically relevant and should thus receive a higher probability of being picked. We thus propose two models that base the estimate of $P\left(D|R\right)$ on the divergence between $D$ and $R$. They are introduced in this section.
4.1.1 MLgen: A Generative Model
The first model rewards documents that contain terms that are frequent in the set of feedback documents. Using this model, we determine $P\left(D|R\right)$ by determining the generative probability of $D$ given $R$, i.e., the probability that the set of relevant documents generated the terms in the current document, similar to the query likelihood approach (cf. Eq. 2.3). More formally: $\begin{array}{rcll}P\left(D|R\right)& \propto & \prod _{t\in D}P{\left(t|{\stackrel{̃}{\theta }}_{R}\right)}^{n\left(t,D\right)}.& \text{(4.1)}\text{}\text{}\end{array}$
Here, $P\left(t|{\stackrel{̃}{\theta }}_{R}\right)$ is determined using Eq. 2.13; below, we refer to this model as MLgen.
4.1.2 Normalized Log-likelihood Ratio
The second method measures the divergence between $R$ and each $D$ by determining the log-likelihood ratio, normalized by the collection $C$. Interpreted loosely, this measure indicates the average surprise of observing document $D$ when we have $R$ in mind, normalized using a background collection, $C$. That is, terms that are “well-explained” by the collection (i.e., that have a high frequency in the collection) do not contribute as much to the comparison as terms that are not. It quantifies how much better one language model is than another in modeling an observed text in comparison with modeling by a collection model. More formally: $\begin{array}{rcll}P\left(D|R\right)& \propto & H\left({\theta }_{D},{\theta }_{C}\right)-H\left({\theta }_{D},{\theta }_{R}\right)& \text{}\\ & =& Z\sum _{t\in \mathsc{V}}P\left(t|{\theta }_{D}\right)log\frac{P\left(t|{\theta }_{R}\right)}{P\left(t|{\theta }_{C}\right)}.& \text{(4.2)}\text{}\text{}\end{array}$
The measure has the attractive property that it is high for documents for which $H\left({\theta }_{D},{\theta }_{C}\right)$ is high and $H\left({\theta }_{D},{\theta }_{R}\right)$ is low. So, in order to receive a high score, documents should contain specific terminology, i.e., they should be dissimilar from the collection model but similar to the topical model of relevance. Since we do not know the actual parameters of ${\theta }_{R}$ by which we could calculate this, we use $R$ as a surrogate and linearly interpolate it with the collection model (cf. Eq. 2.13). This is similar to the intuitions behind MBF (cf. Eq. 2.16): $\begin{array}{rcll}P\left(t|{\stackrel{̂}{\theta }}_{R}\right)=\left(1-{\lambda }_{R}\right)P\left(t|{\stackrel{̃}{\theta }}_{R}\right)+{\lambda }_{R}P\left(t|{\theta }_{C}\right).& & & \text{(4.3)}\text{}\text{}\end{array}$
This interpolation also ensures that zero-frequency issues are avoided and that the sum in Eq. 4.2 is over the same event space for all language models involved. Then, in order to use this discriminative measure as a probability, we define a normalization factor $Z=1∕{\sum }_{D\in R}P\left(D|R\right)$.
Finally, by putting Eq. 2.15 and Eq. 4.2 together, we obtain an estimate of our expanded query model: $\begin{array}{rcll}P\left({t}_{1},\dots ,{t}_{|\mathsc{V}|}|\theta Q\right)=\prod _{i=1}^{|\mathsc{V}|}\sum _{D\in R}\left\{Z\sum _{{t}^{\prime }\in \mathsc{V}}P\left({t}^{\prime }|{\theta }_{D}\right)log\frac{P\left({t}^{\prime }|{\stackrel{̂}{\theta }}_{R}\right)}{P\left({t}^{\prime }|{\theta }_{C}\right)}\right\}P\left({t}_{i}|{\theta }_{D}\right).& & & \text{(4.4)}\text{}\text{}\end{array}$
This model, to which we refer as NLLR, effectively determines the query model based on information from each individual relevant document and the most representative sample we have of $\theta Q$, namely $R$.
4.1.3 Models Related to MLgen and NLLR
As an aside, other ways of estimating $P\left(D|R\right)$ have been proposed. Examples include simply assuming a uniform distribution, the retrieval score of a document (or the inverse thereof), or information from clustered documents [24170]. One could also apply machine learning to select documents to use for relevance feedback, and use the machine learner’s confidence level as a substitute for $P\left(D|R\right)$ [124].
The surface form of NLLR seems reminiscent of a model introduced in [60]. Carpineto et al. [60] propose to use the KL-divergence between $R$ and the collection to select and weight expansion terms for Rocchio feedback [267]. Their model is also highly similar to the query clarity score that uses this measure to predict the difficulty of a query [84]. Besides the fact that we do not use a VSMCarpineto et al. also ignore the individual document models by assuming independence between relevant documents, similar to MLE.
Ponte’s [247] log ratio method is also related to NLLR. He uses the log of the ratio between a term’s probability given each relevant document and its probability given the collection, summed over all the relevant documents. However, Ponte [247] views the query as a set—as opposed to a generative model—and, moreover, he uses the log ratio only for thresholding the terms to be added to the initial query.
MBF is related to NLLR in that it also uses information from both the set of relevant documents and the collection in its estimations, although the estimation method is different. Moreover, NLLR leverages information from each individual relevant document. When we apply this intuition underlying NLLR to MBF, we should let go of the full document independence assumption in MBF and change the M-step (cf. Eq. 2.18) to: $\begin{array}{rcll}P\left(t|{\stackrel{̂}{\theta }}_{R}\right)& =& \frac{1}{|R|}\sum _{D\in R}\frac{{e}_{t}}{\sum _{{t}^{\prime }}{e}_{{t}^{\prime }}}.& \text{(4.5)}\text{}\text{}\end{array}$
Under the assumption that we exclude the collection estimate, we set ${\lambda }_{R}=0$ (cf. Eq. 2.16) and obtain: $\begin{array}{rcll}P\left(t|{\stackrel{̂}{\theta }}_{R}\right)& =& \frac{1}{|R|}\sum _{D\in R}\frac{n\left(t,D\right)}{\sum _{{t}^{\prime }}n\left({t}^{\prime },D\right)}& \text{(4.6)}\text{}\text{}\\ & =& \frac{1}{|R|}\sum _{D\in R}P\left(t|{\stackrel{̃}{\theta }}_{D}\right),& \text{}\end{array}$
which is a simplified version of NLLR, using a uniform probability of selecting a document. Moreover, this is in fact the same as the relevance model in situation 1 (when the full set of relevant documents is known, cf. Section 2.3.2): RM-0.
The relevance modeling approach to relevance feedback can be viewed as a simplification of MLgen and NLLR, since it assumes that each document has an equal probability of being selected (RM-0) or that this probability is dependent on the query (RM-1 and RM-2). The latter models explicitly consider the initial query by first gathering evidence from each document for a query term and, next, combining the evidence for all query terms (RM-2) or vice versa (RM-1), as detailed in Section 2.3.2. Using the probability that a document generated the query (as is the case with RM-1 and RM-2) is a much simpler implementation of leveraging the notion that documents should be weighted according to their “relative” level of relevance, essentially replacing $R$ in the MLgen and NLLR models with only the query ${\stackrel{̃}{\theta }}_{Q}$. And, since the query is quite sparse compared to $R$, our models avoid overfitting to obtain an improved estimate.
Up Next Front
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2023-03-22 15:26:20
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https://brilliant.org/problems/not-your-favourite-circles-xd/
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# A Weird Cut Of Concentric Circles
Geometry Level 3
In the figure above, the circles are concentric with centre $$O$$. $$B$$ is a point on the larger circle. $$D$$ is a point on the smaller circle. $$BD$$ is joined, provided that $$BD$$ touches the circle at only one point. Now $$OB$$ is joined and extended to meet the larger circle at $$A$$. Find the distance between points $$A$$ and $$D$$ if the radii of the circles are $$4\sqrt{13}$$ and $$8$$ units respectively.
Clarification: Concentric circles are circles with a common centre.
×
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2017-01-18 16:32:44
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https://math.stackexchange.com/questions/2131904/maximum-point-of-continuous-function-in-metric-space
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# Maximum point of continuous function in metric space.
Let $(X,d)$ be a metric space and $f: X \rightarrow [0, \infty )$ be a continuous function. Assume that for any $\epsilon > 0$, there exists a compact set $K_\epsilon \subseteq X$ such that $f(x) < \epsilon$ whenever $x \notin K_\epsilon.$ Show that $f$ has a maximum point.
Well, for any $\epsilon > 0$, we have a compact set $K_\epsilon$ such that if $f(x) \geq \epsilon$ then $x \in K_\epsilon$. Since each $K_\epsilon$ is compact and since $f$ is continuous, I know that $f$ obtains a maximum on each non-empty $K_\epsilon$. One of these must be the biggest, so $f$ has a maximum point? Is there a better way?
• I'm afraid you are wrong. It is not always true that $f(x)\ge\varepsilon$ whenever $x\in K_{\varepsilon}$. Consider zero-function. You have made a classical logical mistake with contraposition. – szw1710 Feb 6 '17 at 15:26
• We know however that if $f(x) \geq \epsilon$, then $x \in K_\epsilon$ – Ben Grossmann Feb 6 '17 at 15:26
• @Omnomnomnom, this is, of course, correct. :) – szw1710 Feb 6 '17 at 15:27
• Whoops. Math before coffee does not work so well. :) – user389056 Feb 6 '17 at 15:28
• How do you know that $f(x)\geq\epsilon$ for some $x\in K_\epsilon$ – user178826 Feb 6 '17 at 15:32
Here's a proof: suppose that $f$ is non-zero. Let $x_0 \in X$ be such that $f(x_0) > 0$. Take $\epsilon = f(x_0)$. We know that $x_0 \in K_\epsilon$, so $K_\epsilon$ is non-empty. Moreover, $f(x) < f(x_0)$ for all $x \notin K_\epsilon$. We conclude that if the restriction $f|_{K_\epsilon}$ attains a maximum, then $f$ attains this same maximum.
So, consider the restriction $f|_{K_\epsilon}$. This is a continuous map on a compact set, so it attains a maximum.
Suppose for contradiction that $f$ attains no maximum. Then, we may construct a sequence $\{x_i\}_{i \in \Bbb N}$ such that $\{f(x_i)\}$ is strictly increasing towards a supremum. However, $x_i$ must have a subsequence with a limit in $K_{f(x_1)}$. The value of $f$ at this limit must be an overall maximum, which is a contradiction.
• It seems that no matter how you approach the problem, there is no advantage from several $K_\epsilon$s, which I find surprising. – Ben Grossmann Feb 6 '17 at 15:43
• I was considering using the finite intersection property, but in that case one still must argue that $f$ attains the desired maximum on the intersection of the $K_\epsilon$s in question. – Ben Grossmann Feb 6 '17 at 15:45
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2021-04-18 12:04:57
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https://rpg.meta.stackexchange.com/questions/1537/2012-moderator-election-town-hall-chat-digest/1546
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# 2012 Moderator Election - Town Hall Chat Digest
The following is a "digest" version of the 2012 Moderator Election Town Hall Chat. The format, as described on Meta Stack Overflow, is one answer to this question for every question asked in the Town Hall, containing all the candidate's answers to that question.
If you see any corrections which need to be made to this digest, or if you were a candidate who was unable to attend the town hall and would like your answers included, please @GraceNote or @TimStone in the chat room and let us know!
## Rob asked:My question: What do you see as the biggest problem at the moment for RPG Stack Exchange and how will you tackle/deal with it
wax eagle answered: how we deal with campaign research and system recommendation questions. I think we need to renew some meta discussion on campaign research specifically. (I've got a post floating in my head, but haven't put it down to screen yet)
wax eagle continued: Also the fact that we didn't get a Lawful Evil moderator candidate.
Brian Ballsun‑Stanton answered: The huge numbers of edge-case games. Part of the way I'm solving the problem is making sure we're actually running games (and a high proportion of indie games too) to build expertise and attract non-mainstream gamers. Functionally, we need to make sure to not be "all D&D all the time." and the best way to do that is to have activities that appeal to those players to attract experts to generate and answer questions with real standing.
AceCalhoon answered: Finding ways to get people to explain the context of their problems, rather than having them try to "purify" the questions into blandness.
DForck42 answered: going through the review queue, one major thing that i've noticed is a tendacy towards keeping shorter answers, even awarding them as the accepted answer, even if there's very little context to the answer itself. My solution would be to encourage the community to encourge more detailed answers, such as many that @brian-ballsunstanton usually provides
• Brian Ballsun‑Stanton agreed: Short answers, especially ones that aren't grounded in the literature, aren't particularly useful. Comments (and the occasional downvote) are a great way to motivate people to cite.
C. Ross answered: Bringing in new users in a friendly and productive way. We're starting to be a little bit known, but there's a big difference between being a positive contributor here and on your generic RPG forum. We need to welcome new users actively, and gently point them to the FAQ and good examples. We need to close/delete bad examples when encouraging the user to change doesn't help. We can't turn away new users, and we can't allow new users to change what makes the site work.
Jadasc answered: Right now, the biggest problems for RPG.StackExchange are the perceptions that it's not open to new members or new players and that it's actually the D&D Stack in disguise. The first part is covered by a later question; the second, I think, can be handled with active curation of other games by interested posters.
mxyzplk answered: Attracting new users and not scaring them off. Our stats aren't all that good and aren't growing consistently, which means we are not attracting and keeping people effectively. I think reaching out to game stores, advertising on gaming sites and gaming con bulletins will help get eyes and then balancing enough moderation to not have total junk with being friendly enough that a new guy doesn't ask an honest question and get mod-closed in 30 seconds and driven off will keep them.
## Brian Ballsun‑Stanton asked:One of the big problems was the "physics question" category. How would you have dealt with those slew of questions?
wax eagle answered: I was for them initially, but at this point my stance on them is similar to my stance on other setting questions. If you can relate them actual mechanics then they are ok (land speed record builds etc). But if they are just speculative, or purely joke questions they've gotta go.
AceCalhoon answered: "Real world stuff" seems to be where things get trickiest. They're best when they are tied to a concrete scenario.
C. Ross answered: If the question is essentially "solve a physics problem in RPG trappings" it's off topic, and should be closed as such, deleted if necessary. If the question is "How do I model X aspect of reality in Y game", that's ok, and probably just needs a little comment nudge to keep it focused and on topic.
DForck42 answered: I agree with Brian's answer, if they're practical and actually serve a purpose, keep them. The rest is junk and too localized.
Jadasc answered: I'm in favor of allowing them to remain open, so long as the question has applicability to more than one game and it doesn't seem like they're simply thought-experiments or "whittling."
mxyzplk answered: Generally agree with @CRoss' answer - physics for the sake of physics or as a joke or an intellectual exercise is off topic. I was strongly for closing the goblin-bag of holding question for example, despite later protestations of "well it could happen...". Relevant to real gameplay is fine.
## F. Randall Farmer http://www.gravatar.com/avatar/6c38e00d92cd9bd3ada3392b15015553?s=16&d=identicon&r=PG F. Randall Farmer asked:In order for any online community to thrive, it must grow, which means retaining new users. As a moderator, you play a huge role in converting a first-time contributor into a recurring one. What actions will you take to nurture our new users?
Brian Ballsun‑Stanton http://www.gravatar.com/avatar/a45d963e487add0f6096d1d723d9dcc7?s=16&d=identicon&r=PG Brian Ballsun‑Stanton answered: Immediate positive and negative feedback. Beyond voting, comments are a great way to shape behaviour. An "attaboy" comment immediately after a good first post links the validation of the comment and the rep with the quality of the post. Just like in education, reward the actions you like, punish the actions you dislike. Comments are one of the best ways to do that. Beyond that, expanding the reach of the site through recommendations in appropriate venues and game stores is a job for all users.
C. Ross http://www.gravatar.com/avatar/8b464e72261a39bd540f7c7c9b037adf?s=16&d=identicon&r=PG C. Ross answered: Get a welcome in as early as possible, and try to point them to the FAQ and good examples. It's important with new users to give positive re-inforcement for things that are even basically good (upvotes, positive comments). You have to make clear the rules early, or set the site up for a more painful breakup later.
C. Ross http://www.gravatar.com/avatar/8b464e72261a39bd540f7c7c9b037adf?s=16&d=identicon&r=PG C. Ross continued: Also in the past, we've had some success inviting people who make it to 20 rep, but are confused to chat to talk about it. It tends to give people a better view, and a quicker intro into the culture. We should probably bring that back.
wax eagle http://www.gravatar.com/avatar/4320ac0987d82025b454bcee57d708d1?s=16&d=identicon&r=PG wax eagle answered: I think an initial welcome comment is key, even if that comment isn't exactly positive towards their question and is accompanied by a down or close vote, a friendly welcoming "hey this is why this is good" or "hi, this is why this is bad" is key.
AceCalhoon http://www.gravatar.com/avatar/013fbfb6411dbf971dc8623af1096ddb?s=16&d=identicon&r=PG AceCalhoon answered: This is one I've not entirely figured out. My main response at the moment: 1. Communicate as much as possible (whys, specifically, especially if a close or clarification is needed); 2. Go the extra mile to make their posts shine; 3. Weight upvoting a bit higher for new users.
DForck42 http://www.gravatar.com/avatar/026f3abfbd6ac89b6dbabbd4cef2e83d?s=16&d=identicon&r=PG DForck42 answered: one of the big issues with retaining new users is that, their first question or two usually sucks, or iust off-topic for the site. just closing their question saying "blargh this doesn't belong here!" will immediately upset them. what i liek to do is to take a personal approach, being nice and saying that i regretfully have to close their question (and i honestly do hate doing it), but i also give them guidelines on what to do and where the resources are.
DForck42 http://www.gravatar.com/avatar/026f3abfbd6ac89b6dbabbd4cef2e83d?s=16&d=identicon&r=PG DForck42 continued: i also encourge them to visit our meta if they don't understand why their question was closed
mxyzplk http://www.gravatar.com/avatar/9640c5561e9b03dfc501bda1aec101a0?s=16&d=identicon&r=PG mxyzplk answered: As mentioned in my previous answer I think this is the #1 issue for our SE. I make it a goal to spend as much time constructively improving questions as I do closing them/arguing about whether they should be closed. I also wait for community close votes before mod-closing (except in egregious cases). Being welcoming - not just by saying "hi" but by shepherding questions and answers short of closing and deleting - is the way to do this IMO.
## F. Randall Farmer asked:Jeff Atwood, co-creator of Stack Exchange, suggested that this platform isn't a perfect fit for all communities. Personally, I see some clear differences between this community and the IT-related communities, such as Stack Overflow. What differences do you see and how would you work to adapt our policies (and possibly suggest technology improvements?) to improve the "fit"?
Brian Ballsun‑Stanton answered: As a tech improvement, getting a gaming venue more integrated will help build an active chat community. (though this is something that's solvable with various other apps out there, it means that people aren't on the site gaming.) One of the best ways to generate questions is to have active-games with a mindset towards question asking. As a policy improvement? We've actually had great success with the good-subjective/bad-subjective policies from parenting.
Brian Ballsun‑Stanton continued: It helps us to define what gives a question standing.
wax eagle answered: Every community is different, and the stack model isn't right for everyone. But at the same time we've adapted it to our community fairly well. It works exceptionally well for rules question. Less well for sys-rec/DM advice questions. But with the Good Subjective/Bad Subjective criteria that have come out thanks to sites like programmers, it's much easier to run a more subjective site like this
C. Ross answered: It definitely isn't a perfect fit for every community, but I think one of the main advantages of RPG.SE is the blending of the RPG community and the SE community, it provides a community with the wit of the RPG community, but is results and format oriented like the SE community. As far as tech improvements, the biggest thing I'd like to see is more flare like gaming has. RP'ers are big on our achievements, and I can see some of the Gaming contests going over well here.
DForck42 answered: the first major difference that i've seen is that a lot of RPG users are very chatty. that's to be expected since you ahve to talk to explain all of your actions in your game. Honestly to help cut down the chatter I thinkwe need to push more users to chat.
mxyzplk answered: I believe that the format is a perfect fit. Only non-programmers or non-gamers believe that programming is more objective than RPGing. I think that with "Good Subjective, Bad Subjective" SE has given us everything we need. If it works for parenting it'll work for us!
## Tim Stone http://www.gravatar.com/avatar/3981cd271c302f5cba628c6b6d2b32ee?s=16&d=identicon&r=PG Tim Stone asked:Do you feel like a representative percentage of the community participates in your site's meta? Based on that, how strongly do you think feedback presented on meta should factor into your decision making as a moderator?
AceCalhoon http://www.gravatar.com/avatar/013fbfb6411dbf971dc8623af1096ddb?s=16&d=identicon&r=PG AceCalhoon answered: Meta is visited dramatically less than the main site. What's really important about meta is that it gives people a place to articulate their sites, and make a case for or against something.
AceCalhoon http://www.gravatar.com/avatar/013fbfb6411dbf971dc8623af1096ddb?s=16&d=identicon&r=PG AceCalhoon continued: When I participate in Meta I pay much less attention to the votes (because a "huge difference" is, like, ten people) and much more to the arguments presented.
Brian Ballsun‑Stanton http://www.gravatar.com/avatar/a45d963e487add0f6096d1d723d9dcc7?s=16&d=identicon&r=PG Brian Ballsun‑Stanton answered: No, but it's the closest thing that we have to a policy consensus. If there is an answer with ten or more upvotes with few competing answers, I like to take it as "policy until future discussion."
wax eagle http://www.gravatar.com/avatar/4320ac0987d82025b454bcee57d708d1?s=16&d=identicon&r=PG wax eagle answered: Meta participation (like on many SE 2.0 sites) is not as good as we'd like it. However it's the only place we have for deciding site policy. If people don't like it they need to come to meta and participate. One of my goals will be driving more meta participation.
C. Ross http://www.gravatar.com/avatar/8b464e72261a39bd540f7c7c9b037adf?s=16&d=identicon&r=PG C. Ross answered: No, but close enough. Still, this is not a pure democracy. I think the history of Gaming.SE shows that you can't govern based on whatever happens on meta. I also think we have seen RPG.SE's culture change from the early days, and not every two year old post on meta is an indicator of current state.
mxyzplk http://www.gravatar.com/avatar/9640c5561e9b03dfc501bda1aec101a0?s=16&d=identicon&r=PG mxyzplk answered: Not enough people participate in meta. The new "Community Bulletin" box is helping with that. I consider meta q&a more binding if there's more than a couple answerers. Are campaign research questions on topic?, I disagree with the consensus but the Q has 19 votes and the dissenting answer 12, so I consider myself bound by it. Something with 2 votes... I take it into account but will act differently if my mod experience tells me so.
## casperOne asked:One of the things that moderators on smaller SE 2.0 sites play a key role in that moderators on larger sites don't is promotion. With RPG being classified as one of the "smaller" sites, how do you envision your role in growing the site, and what are your current specific strategies, if any?
Brian Ballsun‑Stanton answered: I'm already running weekly games. Getting what amounts to a constantcon for us would be a fantastic win in terms of a question generating resource. I'm also asking game-authors of indie games when questions explicity concern their games. We've had good success with Vincent Baker answering questions with dogs in the vineyard, and the various references to blogs I've posted seem to reflect question-visiting rates.
Brian Ballsun‑Stanton continued: However, that kind of infrastructure (for constantly running games) would take some involvement in other platforms and active solicitation of integration. Still, that solicitation of tools will, itself, lead to more interaction.
Brian Ballsun‑Stanton concluded: In an academic sense, I'm using the site to provide research material for myself (thereby promoting it among academics) and plan to make a book on the philosophy of rpgs from my answers on this site.
DForck42 answered: Asking questions, especially for the lesser covered questions. That's the easiest way to cover our search engine footprint. I've also been promotoing the site amongst friends. I've actualyl gottena couple of them to join the site, even if they aren't very active.
DForck42 continued: also on this point, we're running a topic of the week event on movies to encourge users to ask questions about either current topics, or to help fill in some of the holes the site has. right now it's mostly run by us mods, but it's open for anyone to provide input. it's too early to tell if it's had a positive effect, but i think it has
AceCalhoon answered: I'm not that great of a promoter, unfortunately! :) Mostly, I just try to participate and set a good example.
wax eagle answered: I see the role of moderators in this as both instigators and facilitators. On gardening right now We are working on an anniversary contest, one of our moderators instigated that and is following up on it. On C.SE however one of our users really wanted to get a blog kicked off, I'm currently facilitating that by writing the monthly topic posts and helping with scheduling etc.
C. Ross answered: I'm one of the more active promoters of the site (see my badges). I've had the most success by sharing some of the great link-bait question the site has. I would continue to do this, and attempt to organize this activity as well. I also think it's well past time that we move some of our promotion out into meatspace, but that needs some more details worked out. The important thing is to keep people positive and motivated, and have fun with it.
Jadasc answered: I believe that moderators can play a role in growing the site through curation of tags — expanding the scope and breadth of the knowledge available. The recent blog post on self-answering offers some cues in how this can work.
## F. Randall Farmer asked:"-1 votes" are anonymous and discourage new users. Agree/Disagree? If you agree, what would you counteract/fix this?
Brian Ballsun‑Stanton answered: They certainly discourage me. But a good comment of what's wrong or a positive comment and upvote can help mitigate the problem.
AceCalhoon answered: Yeah, I'd say a -1 discourages anyone. I do my best to explain downvotes (even if they aren't mine) if I can and encourage the user to edit their post.
wax eagle answered: I agree wholeheartedly that we should protect the anonymity of the voting system. However, I almost always leave a comment when I downvote (unless there is already a negative comment I agree with). It can be hard for a new user (which is why usually for a new user I'm more likely to flag and answer/cast a close vote)
C. Ross answered: They are anonymous and can discourage a new user. They can also help tweak the behavior of experienced users. As already discussed, we need to be leaving active feedback on new users posts, explaining what they're doing for good or ill. I do not see any reason to make downvotes not anonymous.
DForck42 answered: drive-by downvotes are discouring to almost everyone. but, a couple of upvotes vastly outweight a single downvote. i like to get people to upvote good questions. if people are voting on good questions, and that question is indeed a good one, then the new user shouldn't be as discouraged. also, comments to help the user make their question better are also good.
mxyzplk answered: They are discouraging, though certainly not as much as a close. People react better IME to "I don't like what you're saying" than "and I want to stop you from saying it."
mxyzplk continued: Fixes have to be indirect, as votes are community action, but we certainly encourage comments with downvotes.
## Tim Stone asked:Your site has relatively low traffic compared to most other graduated sites on the network, though it also has an excellent answered rate. In light of this, do you feel like your site is experiencing any growing pains, and is there any aspect of how the site is currently run that you feel negatively impacts continued growth?
Brian Ballsun‑Stanton answered: Growing pains? Not really. We're niche. Our questions cover products with a very long release cycle. This is something that we've learned to deal with. Dealing with D&D next will prove to be a very interesting time, especially considering their modularity. I see nothing wrong with how the site is currently run.
AceCalhoon answered: I think most of the negative light on the community right now is the echo of past growing pains. I think right now we're in a very good place, with some room for improvement (mostly in terms of communication). We do show steady growth, just not in terms of massive spikes.
wax eagle answered: I think the SE learning curve might be the only growing pain we really have. Our recent promotion with Obsidian Portal brought us a wealth of new users. But most RPG types are very used to the forum model and have to be indoctrinated into the SE way of doing things. This was rather evident with teh new users who came in from OP
C. Ross answered: Growing pains? Not really. how the site is currently run: We've been in an awkward space for a while with many moderators, of varying styles and levels of commitment, some new, some old. I think the election will firm that up, and help us get on the same page with a new staff all dedicated and engaged.
## Tim Stone asked:Two highly respected members of the community get in a comment war on a question. They both flag each other's comments and are cussing and it is clear that this is beyond a heated argument. What do you do, what don't you do?
wax eagle answered: I'd nuke the comments, lock the question, then try to snag them both into individual private chats, failing that probably a very polite mod message with a 24 hour cool down attached.
Brian Ballsun‑Stanton answered: Good question. Lock down comments and bring them both to a (probably private) chat. This is something that needs moderation and cooling down. By changing the situation and being able to have people state grievances, it takes the problem outside the public eye. If that persisted, I would impose cooling off periods (equitable) for both of them, with an attempt to have dialogue in the venues that were still open.
Brian Ballsun‑Stanton continued: Engaging in their comment stream beyond a simple "Let's take this to chat." or "We're getting off topic." only adds to the problem.
C. Ross answered: Do: Delete comments, protect contentious posts. Talk to both of them about it, and suspend if necessary. Don't Take sides, or give the appearance there of. Don't suspend people out of hand.
DForck42 answered: I wouldn't jump into the argument for either side. I would comment that both need to take their argument to chat, then clear the comments. If they both get too heated and start to actually take it out on eachother (downvoting, etc.) then they'll both get suspended (probably for a day).
mxyzplk answered: Delete comments, lock the question temporarily, try to get them into chat. Most folks cool down when the comments start disappearing and further intervention is seldom needed.
## Grace Note asked:How would you deal with a user who produced a steady stream of valuable answers, but tends to generate a large number of arguments/flags from comments?
Brian Ballsun‑Stanton answered: Valuable is a function of acceptance by the community. The best recent edge-case of that was the rash of backticking proper nouns. While the highlighting is somewhat useful, the edit-spam and the... "let's code-indicate everything" eventually caused me as normal user to comment with a "hey, can you only format according to our recent meta discussion?"
C. Ross answered: Delete the comments. Send them a message stating what the problem is, while recognizing their contributions. Make a point to upvote their valuable, non-flamy comments.
wax eagle answered: Figure out what the problem is and ask them to address that part of their posts. Either in the comments or in chat. Failing either of those two a message outlining what I think is the problem
DForck42 answered: Talk with them personally about what's going on by trying to get them into chat.
mxyzplk answered: Start by pruning comments and posting the standard "this isn't a discussion forum, comments are for clarifying answers only, please post your own answer and let voting work if you have a strong opinion" verbiage. If it persists, send them a mod message explaining the problem.
## Kalamane asked DForck42: You were a mod on Literature.Stackexchange which failed. What did you learn from this that you can apply to this site? This applies to other candidates that have had similar situations.
DForck42 answered: the first thing that i leanred is that, there has to be community involvement with the site when making decisions. part of the issue we had was that we couldn't get anyone interested in the meta discussions after the first couple of months. the second thing i learned is that the fun questions (the one's that arent' very deep) are good to attract traffic, but you have to have deeper questions to keep most of yoru traffic
wax eagle added: I'm a mod on Chrisianity stackexchange, and we had a huge turnout initially with some serious quality issues, however the way we handled it knocked our traffic off rather dramatically. One of the things I learned through that was that you have to address major site issues carefully and effectively.
## Tim Stone http://www.gravatar.com/avatar/3981cd271c302f5cba628c6b6d2b32ee?s=16&d=identicon&r=PG Tim Stone asked:When you see a question with major issues (poorly-written, argumentative, etc.), what tool do you reach for first?
C. Ross http://www.gravatar.com/avatar/8b464e72261a39bd540f7c7c9b037adf?s=16&d=identicon&r=PG C. Ross continued: The corollary to this is often when the mods see it, one of the site grognards has already left a great constructive comment, and we don't need to pile on.
Brian Ballsun‑Stanton http://www.gravatar.com/avatar/a45d963e487add0f6096d1d723d9dcc7?s=16&d=identicon&r=PG Brian Ballsun‑Stanton answered: Major issues is defined as "I can't edit this into shape." So therefore the comment function. I tend to request for clairification often, especially when the requirements are unclear. We should have a discussion on meta, however, about preemptive closing to avoid getting bad-answers that then lock the question into a bad form.
AceCalhoon http://www.gravatar.com/avatar/013fbfb6411dbf971dc8623af1096ddb?s=16&d=identicon&r=PG AceCalhoon answered: Edit if reasonably possible (usually for quality), comment, then close.
wax eagle http://www.gravatar.com/avatar/4320ac0987d82025b454bcee57d708d1?s=16&d=identicon&r=PG wax eagle answered: This is a tough one. Major issues are a comment followed by a close. If the issue is easily fixed then I might edit instead of closing, but mostly the user needs to come back and learn from their mistake so closing is the right call
DForck42 http://www.gravatar.com/avatar/026f3abfbd6ac89b6dbabbd4cef2e83d?s=16&d=identicon&r=PG DForck42 answered: if it's poorly written, i'll just edit it to clean it up. if it's argumentative, i'll usualyl close the question (and edit it if necessary) saying why it was closed.
mxyzplk http://www.gravatar.com/avatar/9640c5561e9b03dfc501bda1aec101a0?s=16&d=identicon&r=PG mxyzplk answered: The comment. You can always escalate - but recently when this question was posted - and completely sucked - I commented and it got converted perfectly without having to edit or close: Combat-centric 1st or 2nd level adventure with a Native American theme? I believe in starting with the light touch - you can always edit or close in a couple hours instead of RIGHT NOW.
## Rob asked:How often do you expect to be able to do moderation stuff for RPG Stack Exchange?
Brian Ballsun‑Stanton answered: Functionally every day. I find that this site provides significant validation for myself, and therefore I'm engaged every day.
C. Ross answered: At least half an hour a day. Some days obviously a bit more.
wax eagle answered: nearly every day. I'm in front of a computer 9/10 work days and often on weekends
AceCalhoon answered: It's in the background most of the day during the week. I check in a couple times on the weekends (when it isn't as busy anyway)
DForck42 answered: i usually spend about 2-3 hours on movies a day. if i start covering rpg it'll probably be about 2 hours for movies and about 3-4 hours for rpg (depending on how things are going)
## Aarthi asked wax eagle:You are already moderator of two other, growing sites on the Stack Exchange network: Gardening and Christianity. Why do you believe adding a third, RPG, will not be overburdening yourself?
wax eagle answered: Great question. Does the answer "anything ChrisF can do I can do" work? ;). Seriously though the moderation load on Gardening is fairly light, and while C.SE is heavier it's not overwhelming. I'm already on RPG more than the other two sites I moderate so I don't see a disconnect here
## Kalamane asked:What will you do when you come across a question that has been edited to ask something completely different than the answers are answering?
Brian Ballsun‑Stanton answered: Revert, comment, and ask the poster to post the edit in a new question.
DForck42 answered: revert. if the asker changed the question, prompt them to ask their new question as a seperate question.
wax eagle answered: . I think the question that must be asked is "do the answers hold any value" if they do then attempting to return to the original question is the right thing to do. If they don't then either closing and asking the Op to start over or removing the answers and starting with a clean slate is the right thing to do
C. Ross answered: There's a fine line there. Sometimes the "new question" is the one the poster obviously intended to ask all along, and the answerers are confused. In that case comment on the answers to encourage them to get in line, downvoting if necessary. In teh other case it's a new question b/c they thought of an additional one. I would encourage them to post it separately, reminding them they can get more rep for it ;-).
AceCalhoon answered: I think one of the purposes of closing questions early is to prevent this (although communicating that is a challenge). As C.Ross noted, if the "new question" is a refinement, poke the existing answers for an update. If it's a secondary question, encourage splitting it into a new question.
Jadasc answered: If possible, edit the question to include the original request. Otherwise, revert with commentary.
mxyzplk answered: Discuss in comments; possibly arrange a reversion (we did this at least once) or ask the other question as well and migrate answers. Don't let it stand though.
## Rob asked:As a moderator what do you consider your special attack and special weakness, so to speak; how do you counter the latter?
Brian Ballsun‑Stanton answered: Special Attack: wall of academic text. I can generally cite at whatever depth of recursion necessary to provide necessary argumentation. Special Weakness: People who don't care. If there are people who don't respond well to reasoned-arguments or discussion... I don't really like bringing down punative measures save in extremis.
AceCalhoon answered: Special attack: blather; Special weakness: borderline posts. I try to counteract that weakness by talking through the issues.
C. Ross answered: Special attack: "Calm down guys", helping bring a situation under control. Special weakness: questioner. I'm personally heavier in the questions than I am in the answers, and this sometimes creates a weird dynamic as moderator.
wax eagle answered: special weakness: over aggressive closer. Something I'm working on here as a normal user, and I regularly think twice about when I'm running around with a diamond. Special attack: trolls bane flame strike - Moderating C.se I've become adept at dealing with trolls.
DForck42 answered: i think my special attack is my editing abilities. my weakness is indecisiveness on closing some questions. i counter this by talkign with fellow mods to get opinions (that's why my trouble is devil's advocate)
Jadasc answered: Special Attack: Subtle Cut. Weakness: Soft-Hearted. (I ameliorate that through conferring with colleagues to see whether the harder approach is warranted.)
## wraith808 asked:It seems that there is a focus on reputation not equivalent to it's true function related to the community rather than the individual. How will you emphasize the community aspect as opposed to the individual aspect- or do you see that as a problem?
wax eagle answered: I don't see these as competing interests most of the time. Gaining reputation can only be done through posting content. This is a positive feedback loop. You post good content, the community benefits and rewards you, making you want to post more good content and improve the community.
Brian Ballsun‑Stanton answered: Reputation is a function of community acceptance and trust of your answers. It has a personal validation function (as @waxeagle pointed out) and a community-measurment function. From a game-theoretic behaviour modification point of view, I see absolutely nothing wrong with it.
Jadasc answered: I don't see it as a problem. We are a community of people with diverse interests — which means that, often, we don't have the ability to measure the value of a given person's contribution to the group as a whole. Instead, we trust that highly reputed individuals are qualified and generous in their areas of expertise, and generalize from there.
C. Ross answered: I've not seen that as a particular problem. I see problems with community stemming from the clash of our two community sources, and the usual flaming that goes on in forums (You're playing the game wrong man, you're ruining my life!). The clash of the community sources happens when we have people with RPG forum experience and SE experience disagreeing on usage.
C. Ross continued: Rep is there to help us improve content. We have to work to improve community in the usual ways (working together, sharing, communication, etc).
DForck42 continued: the reputation is a system that allows users to express what content they approve of and disapprove of. it also shows which users are providing valuable content to the site. i honestly think the rep systems works as it should. are there bugs? sure, but for the most part it works.
## Kalamane asked:What would you do if you had real life circumstances prevent you from accessing the site for any extended period of time? (Say, over a week)
Brian Ballsun‑Stanton answered: Notify my fellow mods and try to hop on with my phone when and if I can.
Jadasc answered: As with @Kalamane, although I'd be happy to appoint a deputy to serve in my stead, if that were legal.
wax eagle answered: a quick note in the TL and (do mods have a private room here? if not hten main chat) a mod room here letting mods know of my absence.
DForck42 answered: same as @BrianBallsunStanton, let my fellow mods know what's going on, usually with some time before i will be gone.
• Brian Ballsun‑Stanton added: Just to amplify that. Consistency and communication are critical to both alliteration and good governance. Having protocols in place and back-channel communication methods for the mods, as well as a common understanding of what the issues of the day are makes for individual mod "absence" unnoticeable or less problematic.
DForck42 agreed: If the mods aren't talking... then we've got problems. that's part of what happened on literature, us mods never talked with eachother, but not from lack of me trying.
Brian Ballsun‑Stanton responded: Above all else, mods must have mental prediction models of the others' behavour in their heads, so that we don't get one mod just reversing another mod's decisions in public without a very clear and important reason.
DForck42 added: yup. if i see something a mod did and don't agree, i'll usually hash it out with them first becasue maybe i'm wrong?
AceCalhoon answered: Notify other mods that I'm away, check in when I can.
mxyzplk answered: Let my fellow mods know (though I have this on my phone too so it only really happens when on extended vacation in darkest Siberia or getting hit by a bus). There's enough mods that it shouldn't be a crisis.
## Kalamane asked: The avatars of @CRoss, @BrianBallsunStanton, and @waxeagle all show men with nice beards. Do you have a beard, and if not - if elected will you grow one?
Brian Ballsun‑Stanton answered: Yes, yes I do. And it's now less... gorse brushy. The ability to clean pots with my beard is not necessarily a feature.
Jadasc answered: I do not have a beard, or plans to grow one. However, I believe the length of my head-hair more than makes up for this deficiency.
• heh, I missed this one. I sadly can't grow a real beard.... :-( – DForck42 Jun 14 '12 at 14:00
• Didn't comment, but I have had the beard for 6 years this coming Saturday. I have no plans to shave it off any time soon although it currently does need a trim – wax eagle Jun 14 '12 at 18:17
• Oh! That's a qualification? I could have run! :-) – F. Randall Farmer Jun 14 '12 at 22:43
C. Ross answered: I'm a 20th level Cleric with the Law and Community domains. Happy?
Brian Ballsun‑Stanton answered: I'm a gamist/narrativist DM. I believe the rules exist to provide structure and inspiration. The rules should never be ignored on the spur of the moment, because that weakens the ability of players to function within the world by imagining future outcomes. At the same time, as a pragmatist, use the right system/tool for the job. There is no one "holy" system above all others. From an in-game PoV, I'm a highly pre-constructed character designed to meet specific goals.
Brian Ballsun‑Stanton continued: Best to describe me as a Bonisagus Trianormii with a specialization in Intellego and Mentem.
DForck42 answered: chaotic good. the rules are there to give us reference points, but they're not perfect and cover every aspect of life, so we have to make judgement calls to promote the greater good. also, consistancy is key.
wax eagle answered: I'm a fighter. I lead when I have to, but prefer to be on the front lines dealing with things myself. I value rule of law, but try to be mindful that sometimes you have to throw the rule book out the window and just hit things with a sword.
AceCalhoon answered: Mystic Theurge. All ways of doing things are interesting.
mxyzplk answered: Neutral good. I am concerned with both site rule/precedent and the unique case in promoting questions that solve people's problems. The overweening goal is for questions that solve people's real problems. Site guidelines are helpful but not sovereign in that regard.
## wraith808 asked: The function of .SE sites tends not to foster community in a lot of cases because of the focus on answer the questions, rather than learning about each other through conversation. Someone asked earlier about promotion- but do you have any specific ideas in terms of helping to build the community around RPG.SE?
Brian Ballsun‑Stanton answered: Running games and making sure chat is a good and welcoming place.
wax eagle answered: I think regular gaming either in chat, on vid conf and others is a good way to foster that community. SE doesn't do "community" all that well, but it can through chat. More chat participation can help with that feeling of community
## Grace Note asked for final thoughts from the candidates
C. Ross answered: I'm very pleased with everyone who is running. Best of luck to everyone, and thanks to everyone for the great experience so far!
Brian Ballsun‑Stanton answered: Thanks for hosting this chat, oh brave DM. Also, drop by my game sometime, we need more people :)
Jadasc answered: I'm pleased and privileged to have undergone this ritual. I feel like I've learned a lot today.
wax eagle answered: Very excited to see who wins.
mxyzplk answered: I've enjoyed moderating the site so far and hope that I've struck the right balance between mod action when needed and mod inaction and letting the community judge when not needed.
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2020-01-26 13:58:28
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http://www.numdam.org/item/M2AN_1981__15_3_231_0/
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Simultaneous approximation in negative norms of arbitrary order
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 15 (1981) no. 3, p. 231-235
@article{M2AN_1981__15_3_231_0,
author = {Helfrich, Hans-Peter},
title = {Simultaneous approximation in negative norms of arbitrary order},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {Dunod},
volume = {15},
number = {3},
year = {1981},
pages = {231-235},
zbl = {0495.41010},
mrnumber = {631677},
language = {en},
url = {http://www.numdam.org/item/M2AN_1981__15_3_231_0}
}
Helfrich, Hans-Peter. Simultaneous approximation in negative norms of arbitrary order. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 15 (1981) no. 3, pp. 231-235. http://www.numdam.org/item/M2AN_1981__15_3_231_0/
[1] I. Babuska and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Part I. (Ed. A. K. Aziz) Academic Press, New York, London, 1972. | MR 421106 | Zbl 0268.65052
[2] J. H. Bramble and A. H. Schatz, Least squares methods for 2 m th order elliptic boundary-value problems, Math. Comp., 25 (1971), 1-32. | MR 295591 | Zbl 0216.49202
[3] J. H. Bramble, A. H. Schatz, V. Thomée and L. H. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Analysis, 14 (1977), 218-241. | MR 448926 | Zbl 0364.65084
[4] J. H. Bramble and R. Scott, Simultaneous approximation in scales of Banach spaces, Math. Comp. 32 (1978), 947-954. | MR 501990 | Zbl 0404.41005
[5] S. G. Krein, Linear Differential Equations in Banach space, American Math. Soc., Providence, 1971. | MR 342804 | Zbl 0229.34050
[6] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. I, Springer Verlag, Berlin and New York, 1972. | Zbl 0223.35039
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2020-01-22 15:42:28
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http://math.stackexchange.com/questions/201941/last-digit-of-n5-and-n-is-the-same-digit?answertab=active
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last digit of $n^5$ and $n$ is the same digit [duplicate]
Basically, this is the same question as Why is the last digit of $n^5$ equal to the last digit of $n$?
What I want to prove is
$n^5 ≡ n$ mod 10
Since I'm studying Euler's Phi Function, I know that the proof of this is related to it. So I'm looking to prove this using the Phi function. A comment on the original question suggests $φ(10)=4$ but I don't see how I can use this. Anyone can point me in the right direction? Thanks
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This is answered in the following answer to the linked question: math.stackexchange.com/a/184623/19379 – M Turgeon Sep 25 '12 at 1:24
If it's the same question, why ask again? – lhf Sep 25 '12 at 1:28
add comment
marked as duplicate by lhf, M Turgeon, Douglas S. Stones, Steven Stadnicki, Pedro TamaroffSep 25 '12 at 1:56
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
2 Answers
Euler's theorem states that if $a$ and $n$ are relatively prime, then
$$a^{\varphi(n)} ≡ 1 \mbox{ mod } n$$
Multiply by $a$ on both sides,
$$a^{\varphi(n) + 1} ≡ a \mbox{ mod } n$$
Now set $n = 10$. Does this help?
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Don't $a$ and $n$ have to be coprime ? – Belgi Sep 25 '12 at 1:26
Ahh yes that was very clear. Thank you. – MinaHany Sep 25 '12 at 1:31
add comment
When $n\equiv 0,1\pmod 2\implies n^5\equiv 0,1\pmod 2\implies n^5\equiv n\pmod {2}$
Also, $n^5\equiv n\pmod {5}$ (By Fermat's little theorem)
By Chinese remainder theorem,
These both $\implies n^5-n\equiv 0\pmod {10}\implies n^5\equiv n\pmod {10}$
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Can you please add detaild on your third paragraph ? Why is the condition about $a$ and $n$ being coprime can be remived ? – Belgi Sep 25 '12 at 1:37
That was not true in general; i removed it. – Aang Sep 25 '12 at 1:44
add comment
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2014-03-11 15:23:45
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http://hackage.haskell.org/package/lens-4.19.2/docs/Control-Lens-Review.html
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lens-4.19.2: Lenses, Folds and Traversals
Control.Lens.Review
Contents
Description
A Review is a type-restricted form of a Prism that can only be used for writing back via re, review, reuse.
Synopsis
# Reviewing
type Review t b = forall p f. (Choice p, Bifunctor p, Settable f) => Optic' p f t b Source #
This is a limited form of a Prism that can only be used for re operations.
Like with a Getter, there are no laws to state for a Review.
You can generate a Review by using unto. You can also use any Prism or Iso directly as a Review.
type AReview t b = Optic' Tagged Identity t b Source #
If you see this in a signature for a function, the function is expecting a Review (in practice, this usually means a Prism).
unto :: (Profunctor p, Bifunctor p, Functor f) => (b -> t) -> Optic p f s t a b Source #
An analogue of to for review.
unto :: (b -> t) -> Review' t b
unto = un . to
un :: (Profunctor p, Bifunctor p, Functor f) => Getting a s a -> Optic' p f a s Source #
Turn a Getter around to get a Review
un = unto . view
unto = un . to
>>> un (to length) # [1,2,3]
3
re :: AReview t b -> Getter b t Source #
Turn a Prism or Iso around to build a Getter.
If you have an Iso, from is a more powerful version of this function that will return an Iso instead of a mere Getter.
>>> 5 ^.re _Left
Left 5
>>> 6 ^.re (_Left.unto succ)
Left 7
review ≡ view . re
reviews ≡ views . re
reuse ≡ use . re
reuses ≡ uses . re
re :: Prism s t a b -> Getter b t
re :: Iso s t a b -> Getter b t
review :: MonadReader b m => AReview t b -> m t Source #
This can be used to turn an Iso or Prism around and view a value (or the current environment) through it the other way.
review ≡ view . re
review . unto ≡ id
>>> review _Left "mustard"
Left "mustard"
>>> review (unto succ) 5
6
Usually review is used in the (->) Monad with a Prism or Iso, in which case it may be useful to think of it as having one of these more restricted type signatures:
review :: Iso' s a -> a -> s
review :: Prism' s a -> a -> s
However, when working with a Monad transformer stack, it is sometimes useful to be able to review the current environment, in which case it may be beneficial to think of it as having one of these slightly more liberal type signatures:
review :: MonadReader a m => Iso' s a -> m s
review :: MonadReader a m => Prism' s a -> m s
reviews :: MonadReader b m => AReview t b -> (t -> r) -> m r Source #
This can be used to turn an Iso or Prism around and view a value (or the current environment) through it the other way, applying a function.
reviews ≡ views . re
reviews (unto f) g ≡ g . f
>>> reviews _Left isRight "mustard"
False
>>> reviews (unto succ) (*2) 3
8
Usually this function is used in the (->) Monad with a Prism or Iso, in which case it may be useful to think of it as having one of these more restricted type signatures:
reviews :: Iso' s a -> (s -> r) -> a -> r
reviews :: Prism' s a -> (s -> r) -> a -> r
However, when working with a Monad transformer stack, it is sometimes useful to be able to review the current environment, in which case it may be beneficial to think of it as having one of these slightly more liberal type signatures:
reviews :: MonadReader a m => Iso' s a -> (s -> r) -> m r
reviews :: MonadReader a m => Prism' s a -> (s -> r) -> m r
reuse :: MonadState b m => AReview t b -> m t Source #
This can be used to turn an Iso or Prism around and use a value (or the current environment) through it the other way.
reuse ≡ use . re
reuse . unto ≡ gets
>>> evalState (reuse _Left) 5
Left 5
>>> evalState (reuse (unto succ)) 5
6
reuse :: MonadState a m => Prism' s a -> m s
reuse :: MonadState a m => Iso' s a -> m s
reuses :: MonadState b m => AReview t b -> (t -> r) -> m r Source #
This can be used to turn an Iso or Prism around and use the current state through it the other way, applying a function.
reuses ≡ uses . re
reuses (unto f) g ≡ gets (g . f)
>>> evalState (reuses _Left isLeft) (5 :: Int)
True
reuses :: MonadState a m => Prism' s a -> (s -> r) -> m r
reuses :: MonadState a m => Iso' s a -> (s -> r) -> m r
(#) :: AReview t b -> b -> t infixr 8 Source #
An infix alias for review.
unto f # x ≡ f x
l # x ≡ x ^. re l
This is commonly used when using a Prism as a smart constructor.
>>> _Left # 4
Left 4
But it can be used for any Prism
>>> base 16 # 123
"7b"
(#) :: Iso' s a -> a -> s
(#) :: Prism' s a -> a -> s
(#) :: Review s a -> a -> s
(#) :: Equality' s a -> a -> s
class Bifunctor (p :: Type -> Type -> Type) where #
A bifunctor is a type constructor that takes two type arguments and is a functor in both arguments. That is, unlike with Functor, a type constructor such as Either does not need to be partially applied for a Bifunctor instance, and the methods in this class permit mapping functions over the Left value or the Right value, or both at the same time.
Formally, the class Bifunctor represents a bifunctor from Hask -> Hask.
Intuitively it is a bifunctor where both the first and second arguments are covariant.
You can define a Bifunctor by either defining bimap or by defining both first and second.
If you supply bimap, you should ensure that:
bimap id id ≡ id
If you supply first and second, ensure:
first id ≡ id
second id ≡ id
If you supply both, you should also ensure:
bimap f g ≡ first f . second g
These ensure by parametricity:
bimap (f . g) (h . i) ≡ bimap f h . bimap g i
first (f . g) ≡ first f . first g
second (f . g) ≡ second f . second g
Since: base-4.8.0.0
Minimal complete definition
Methods
bimap :: (a -> b) -> (c -> d) -> p a c -> p b d #
Map over both arguments at the same time.
bimap f g ≡ first f . second g
#### Examples
Expand
>>> bimap toUpper (+1) ('j', 3)
('J',4)
>>> bimap toUpper (+1) (Left 'j')
Left 'J'
>>> bimap toUpper (+1) (Right 3)
Right 4
Instances
Since: base-4.8.0.0 Instance detailsDefined in Data.Bifunctor Methodsbimap :: (a -> b) -> (c -> d) -> Either a c -> Either b d #first :: (a -> b) -> Either a c -> Either b c #second :: (b -> c) -> Either a b -> Either a c # Since: base-4.8.0.0 Instance detailsDefined in Data.Bifunctor Methodsbimap :: (a -> b) -> (c -> d) -> (a, c) -> (b, d) #first :: (a -> b) -> (a, c) -> (b, c) #second :: (b -> c) -> (a, b) -> (a, c) # Since: base-4.9.0.0 Instance detailsDefined in Data.Semigroup Methodsbimap :: (a -> b) -> (c -> d) -> Arg a c -> Arg b d #first :: (a -> b) -> Arg a c -> Arg b c #second :: (b -> c) -> Arg a b -> Arg a c # Bifunctor ((,,) x1) Since: base-4.8.0.0 Instance detailsDefined in Data.Bifunctor Methodsbimap :: (a -> b) -> (c -> d) -> (x1, a, c) -> (x1, b, d) #first :: (a -> b) -> (x1, a, c) -> (x1, b, c) #second :: (b -> c) -> (x1, a, b) -> (x1, a, c) # Bifunctor (Const :: Type -> Type -> Type) Since: base-4.8.0.0 Instance detailsDefined in Data.Bifunctor Methodsbimap :: (a -> b) -> (c -> d) -> Const a c -> Const b d #first :: (a -> b) -> Const a c -> Const b c #second :: (b -> c) -> Const a b -> Const a c # Functor f => Bifunctor (FreeF f) Instance detailsDefined in Control.Monad.Trans.Free Methodsbimap :: (a -> b) -> (c -> d) -> FreeF f a c -> FreeF f b d #first :: (a -> b) -> FreeF f a c -> FreeF f b c #second :: (b -> c) -> FreeF f a b -> FreeF f a c # Functor f => Bifunctor (CofreeF f) Instance detailsDefined in Control.Comonad.Trans.Cofree Methodsbimap :: (a -> b) -> (c -> d) -> CofreeF f a c -> CofreeF f b d #first :: (a -> b) -> CofreeF f a c -> CofreeF f b c #second :: (b -> c) -> CofreeF f a b -> CofreeF f a c # Bifunctor (Tagged :: Type -> Type -> Type) Instance detailsDefined in Data.Tagged Methodsbimap :: (a -> b) -> (c -> d) -> Tagged a c -> Tagged b d #first :: (a -> b) -> Tagged a c -> Tagged b c #second :: (b -> c) -> Tagged a b -> Tagged a c # Bifunctor (Constant :: Type -> Type -> Type) Instance detailsDefined in Data.Functor.Constant Methodsbimap :: (a -> b) -> (c -> d) -> Constant a c -> Constant b d #first :: (a -> b) -> Constant a c -> Constant b c #second :: (b -> c) -> Constant a b -> Constant a c # Source # Instance detailsDefined in Control.Lens.Internal.Getter Methodsbimap :: (a -> b) -> (c -> d) -> AlongsideRight f a c -> AlongsideRight f b d #first :: (a -> b) -> AlongsideRight f a c -> AlongsideRight f b c #second :: (b -> c) -> AlongsideRight f a b -> AlongsideRight f a c # Source # Instance detailsDefined in Control.Lens.Internal.Getter Methodsbimap :: (a -> b) -> (c -> d) -> AlongsideLeft f a c -> AlongsideLeft f b d #first :: (a -> b) -> AlongsideLeft f a c -> AlongsideLeft f b c #second :: (b -> c) -> AlongsideLeft f a b -> AlongsideLeft f a c # Bifunctor (K1 i :: Type -> Type -> Type) Since: base-4.9.0.0 Instance detailsDefined in Data.Bifunctor Methodsbimap :: (a -> b) -> (c -> d) -> K1 i a c -> K1 i b d #first :: (a -> b) -> K1 i a c -> K1 i b c #second :: (b -> c) -> K1 i a b -> K1 i a c # Bifunctor ((,,,) x1 x2) Since: base-4.8.0.0 Instance detailsDefined in Data.Bifunctor Methodsbimap :: (a -> b) -> (c -> d) -> (x1, x2, a, c) -> (x1, x2, b, d) #first :: (a -> b) -> (x1, x2, a, c) -> (x1, x2, b, c) #second :: (b -> c) -> (x1, x2, a, b) -> (x1, x2, a, c) # Bifunctor ((,,,,) x1 x2 x3) Since: base-4.8.0.0 Instance detailsDefined in Data.Bifunctor Methodsbimap :: (a -> b) -> (c -> d) -> (x1, x2, x3, a, c) -> (x1, x2, x3, b, d) #first :: (a -> b) -> (x1, x2, x3, a, c) -> (x1, x2, x3, b, c) #second :: (b -> c) -> (x1, x2, x3, a, b) -> (x1, x2, x3, a, c) # Instance detailsDefined in Data.Bifunctor.Wrapped Methodsbimap :: (a -> b) -> (c -> d) -> WrappedBifunctor p a c -> WrappedBifunctor p b d #first :: (a -> b) -> WrappedBifunctor p a c -> WrappedBifunctor p b c #second :: (b -> c) -> WrappedBifunctor p a b -> WrappedBifunctor p a c # Functor g => Bifunctor (Joker g :: Type -> Type -> Type) Instance detailsDefined in Data.Bifunctor.Joker Methodsbimap :: (a -> b) -> (c -> d) -> Joker g a c -> Joker g b d #first :: (a -> b) -> Joker g a c -> Joker g b c #second :: (b -> c) -> Joker g a b -> Joker g a c # Bifunctor p => Bifunctor (Flip p) Instance detailsDefined in Data.Bifunctor.Flip Methodsbimap :: (a -> b) -> (c -> d) -> Flip p a c -> Flip p b d #first :: (a -> b) -> Flip p a c -> Flip p b c #second :: (b -> c) -> Flip p a b -> Flip p a c # Functor f => Bifunctor (Clown f :: Type -> Type -> Type) Instance detailsDefined in Data.Bifunctor.Clown Methodsbimap :: (a -> b) -> (c -> d) -> Clown f a c -> Clown f b d #first :: (a -> b) -> Clown f a c -> Clown f b c #second :: (b -> c) -> Clown f a b -> Clown f a c # Bifunctor ((,,,,,) x1 x2 x3 x4) Since: base-4.8.0.0 Instance detailsDefined in Data.Bifunctor Methodsbimap :: (a -> b) -> (c -> d) -> (x1, x2, x3, x4, a, c) -> (x1, x2, x3, x4, b, d) #first :: (a -> b) -> (x1, x2, x3, x4, a, c) -> (x1, x2, x3, x4, b, c) #second :: (b -> c) -> (x1, x2, x3, x4, a, b) -> (x1, x2, x3, x4, a, c) # (Bifunctor p, Bifunctor q) => Bifunctor (Sum p q) Instance detailsDefined in Data.Bifunctor.Sum Methodsbimap :: (a -> b) -> (c -> d) -> Sum p q a c -> Sum p q b d #first :: (a -> b) -> Sum p q a c -> Sum p q b c #second :: (b -> c) -> Sum p q a b -> Sum p q a c # (Bifunctor f, Bifunctor g) => Bifunctor (Product f g) Instance detailsDefined in Data.Bifunctor.Product Methodsbimap :: (a -> b) -> (c -> d) -> Product f g a c -> Product f g b d #first :: (a -> b) -> Product f g a c -> Product f g b c #second :: (b -> c) -> Product f g a b -> Product f g a c # Bifunctor ((,,,,,,) x1 x2 x3 x4 x5) Since: base-4.8.0.0 Instance detailsDefined in Data.Bifunctor Methodsbimap :: (a -> b) -> (c -> d) -> (x1, x2, x3, x4, x5, a, c) -> (x1, x2, x3, x4, x5, b, d) #first :: (a -> b) -> (x1, x2, x3, x4, x5, a, c) -> (x1, x2, x3, x4, x5, b, c) #second :: (b -> c) -> (x1, x2, x3, x4, x5, a, b) -> (x1, x2, x3, x4, x5, a, c) # (Functor f, Bifunctor p) => Bifunctor (Tannen f p) Instance detailsDefined in Data.Bifunctor.Tannen Methodsbimap :: (a -> b) -> (c -> d) -> Tannen f p a c -> Tannen f p b d #first :: (a -> b) -> Tannen f p a c -> Tannen f p b c #second :: (b -> c) -> Tannen f p a b -> Tannen f p a c # (Bifunctor p, Functor f, Functor g) => Bifunctor (Biff p f g) Instance detailsDefined in Data.Bifunctor.Biff Methodsbimap :: (a -> b) -> (c -> d) -> Biff p f g a c -> Biff p f g b d #first :: (a -> b) -> Biff p f g a c -> Biff p f g b c #second :: (b -> c) -> Biff p f g a b -> Biff p f g a c #
retagged :: (Profunctor p, Bifunctor p) => p a b -> p s b Source #
This is a profunctor used internally to implement Review
It plays a role similar to that of Accessor or Const do for Control.Lens.Getter
class (Profunctor p, Bifunctor p) => Reviewable p Source #
This class is provided mostly for backwards compatibility with lens 3.8, but it can also shorten type signatures.
Instances
(Profunctor p, Bifunctor p) => Reviewable p Source # Instance detailsDefined in Control.Lens.Internal.Review
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2020-10-01 16:25:45
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https://www.physicsforums.com/threads/ode-oscillations.717042/
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# ODE Oscillations
1. Oct 16, 2013
### djh101
1. A spherical buoy of radius r floats half-submerged in water. If it is depressed slightly, a restoring force equal to the weight of the displaced water presses it upward; and if it is then released, it will bob up and down. Find the period of oscillation if the friction of the water is neglected.
T = 2π/ω
y'' + ω2y = 0
F = my'' = .5Vwaterρwaterg
y = cos(ωt) & y'' = ω2cos(ωt)
So ω needs to be found, which can be done by setting t = 0 and amax = ω2 = .5Vρg.
.5V = 2/3πr3 and ρ = 1, so ω2 = 2/3πr3g.
All that is left is to take the square root and divide from 2 to get T. However, the book gives the answer 2π√(2r/3g)s.
Summary: $2\pi\sqrt{\frac{3}{2\pi r^{3}g}} \neq 2\pi\sqrt{\frac{2r}{3g}}$
2. Suppose that a straight tunnel is drilled through the earth between any two points on the surface. If tracks are laid, then a train placed in the tunnel at one end will roll through the earth under its own weight, stop at the other end, and return. Estimate the value of the time required to complete one round trip.
y'' = g = GM/y2
M = 4/3 πR3ρ
y'' = 4/3 πGρy, which gives ω2 = 4/3 πGρ. The book, however, gives T = 2π√(R/g).
Summary: $2\pi\sqrt{\frac{3}{4πGρ}} \neq 2\pi\sqrt{\frac{R}{g}}$
Am I over thinking these? The answers in the book are pretty simple, but I'm having a little bit of a hard time figuring out where they came from.
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2017-10-19 02:48:23
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https://gmatclub.com/forum/if-x-5-999-then-the-value-of-2x-5x-1-is-closest-to-which-of-302332.html
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# If x > 5,999 , then the value of 2x / (5x+1) is closest to which of
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If x > 5,999 , then the value of 2x / (5x+1) is closest to which of [#permalink]
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If x > 5,999 , then the value of $$\frac{2x}{(5x+1)}$$ is closest to which of the following?
(A) $$\frac{2}{7}$$
(B) $$\frac{2}{5}$$
(C) $$\frac{12}{21}$$
(D) $$\frac{12}{6}$$
(E) $$\frac{12}{5}$$
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Re: If x > 5,999 , then the value of 2x / (5x+1) is closest to which of [#permalink]
### Show Tags
10 Aug 2019, 15:13
For higher values of x, 5x+1 ≈ 5x
$$\frac{2x}{5x+1}$$ ≈ $$\frac{2x}{5x}$$= $$\frac{2}{5}$$
MikeScarn wrote:
If x > 5,999 , then the value of $$\frac{2x}{(5x+1)}$$ is closest to which of the following?
(A) $$\frac{2}{7}$$
(B) $$\frac{2}{5}$$
(C) $$\frac{12}{21}$$
(D) $$\frac{12}{6}$$
(E) $$\frac{12}{5}$$
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If x > 5,999 , then the value of 2x / (5x+1) is closest to which of [#permalink]
### Show Tags
10 Aug 2019, 15:59
3
x > 5,999, let's take x=10,000.
2x/(5x+1) = 20,000/50,001 is about 2/5.
+1 kudo if you like my explanation
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Posts: 64248
Re: If x > 5,999 , then the value of 2x / (5x+1) is closest to which of [#permalink]
### Show Tags
11 Aug 2019, 01:47
MikeScarn wrote:
If x > 5,999 , then the value of $$\frac{2x}{(5x+1)}$$ is closest to which of the following?
(A) $$\frac{2}{7}$$
(B) $$\frac{2}{5}$$
(C) $$\frac{12}{21}$$
(D) $$\frac{12}{6}$$
(E) $$\frac{12}{5}$$
Similar question from OG: https://gmatclub.com/forum/if-x-3-000-t ... 66128.html
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Re: If x > 5,999 , then the value of 2x / (5x+1) is closest to which of [#permalink] 11 Aug 2019, 01:47
# If x > 5,999 , then the value of 2x / (5x+1) is closest to which of
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2020-05-31 08:12:23
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http://www.physicsforums.com/showpost.php?p=3793447&postcount=14
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PF Gold
P: 4,781
Quote by Tantalos Michelson made a mistake in his calculations, he assumed the phase shift was only proportional to the time the wave needs to travel the distance L. But in reality phase shift is proportional to the distance from the source (how many wavelengths fit in that distance) It is proportional to the time only in stationary case in which the distance is given by c*t in case of ligth.
Since there was no significant phase shift at any time, how could a mistake in calculation have any effect on the outcome of the experiment?
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2014-09-16 13:33:50
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https://math.stackexchange.com/questions/564617/how-can-i-solve-this-linear-recurrence-relation
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# How can I solve this linear recurrence relation?
My problem is: this given recurrence relation: $$y_{n+1}-\frac{n+2}{2}\cdot y_n = (n+1)(n+2)\cdot 3^n$$ for all: $n\ge 0$ and $y_0 = 0$
I need to find the explicit form and the general solution.
My Approach was: I can see, this is a linear recurrence relation. Due to the right side of the equation, it must be a inhomogeneous equation. So I think the way should be to find a general solution for the homogeneous equation first. And then finding a Special solution for the inhomogenous equation and finally finding my answer in adding both solutions.
And here starts the trouble. I've read my seminar literature twice, but I keep failing in performing the described steps.
In my next try, I concentrated on the given $y_0 = 0$. Through inserting I found out for $n = 0$ $$y_{0+1}-\frac{0+2}{2}\cdot 0 = (0+1)(0+2)\cdot 3^0$$ $$y_{1} = 2$$ And inserting again for $n = 1$ i get: $$y_{1+1}-\frac{1+2}{2}\cdot 2 = (1+1)(1+2)\cdot 3^1$$ $$y_{2}-\frac{3}{2}\cdot 2 = 6\cdot 3$$ $$y_{2} = 21$$
That might be nice, but it seems it doesnt bring me to any solution for the original Task. How can I get the solutions I need?
Update: bounty set... rather than just a solution I could use a explanation how to derive the result.
What would be the way to get the solution for the homogeneous relation and how to get the Special (or it is called specific?) solution for the inhomogeneous relation.
## Direct Method
Putting $y_n=(n+1)!2^{1-n}x_n$ the recurrence relation $$y_{n+1}−\frac{n+2}{2}y_n=(n+1)(n+2)3^n\tag 1$$ with initial condition $y_0=0$ forall $n\ge 0$, becomes $$x_{n+1}=x_n+\frac{6^n}{n!}. \tag 2$$ with initial condition $x_0=\frac{y_0}{2}=0$. This is a non-homogeneous recurrence relation with non-homogeneous term $\xi_n=\frac{6^n}{n!}$ that can be solved directly in a very simple way.
We know that the general solution to a non-homogeneous recurrence relation is the sum of the general solution to the associated homogeneous recurrence and any particular solution to the non-homogeneous recurence. So if $a_n$ is a solution to the associated homogeneous recurrence $x_{n+1}-x_n=0$, and $b_n$ is a particular solution to the non-homogeneous recurrence $x_{n+1}-x_n=\xi_n$, then $a_n+b_n$ is the general solution to the non-homogeneous recurrence.
The general solution $a_n$ of the associated homogeneous recurrence relation $x_{n+1}-x_n=0$ is trivial, i.e. $a_n=0$ for all $n\ge0$. Infact the characteristic equation is $\lambda-1=0$, so that $a_n=\alpha^1$ and from the initial condition $x_0=0$, we have $\alpha=0$, i.e $a_n=0$.
Now we have to find a particular solution $x_n=b_n$ of the relation (2). Observing that the term $\xi_n=\frac{6^n}{n!}$ is the $n$-th term of the exponential series, and that the difference between the $n+1$-th term $b_{n+1}$ and the $n$-th term $b_n$ must be the $n$-th term of the exponential series, we may take a particular solution as a truncated exponential series $$b_n=\sum_{n=0}^{n-1}\frac{6^k}{k!}.$$ It's evident that $b_n$ satisfies the relation(2): \begin{align} b_{n+1}-b_n&=\sum_{k=0}^{n}\frac{6^k}{k!}-\sum_{k=0}^{n-1}\frac{6^k}{k!}\\ &=\left(1+\tfrac{6^1}{1!}+\cdots+\tfrac{6^{n-1}}{(n-1)!}+\frac{6^{n}}{n!}\right)-\left(1+\tfrac{6^1}{1!}+\cdots+\tfrac{6^{n-1}}{(n-1)!}\right)\\ &=\frac{6^{n}}{n!}. \end{align}
So the general solution of (2) is $$x_n=a_n+b_n=\sum_{k=0}^{n-1}\frac{6^k}{k!}.\tag 3$$
Finally the general solution of (1) is $$y_n=(n+1)!2^{1-n}\sum_{k=0}^{n-1}\frac{6^k}{k!}\tag 4$$ for $n\ge 1$ and $y_0=0$.
## Ordinary Generating Function
If you prefer to work with generating function, multiply the recurrence (2) by $z^n$ and sum over $n$ $$\sum_{n=0}^\infty x_{n+1}z^n-\sum_{n=0}^\infty x_{n}z^n=\sum_{n=0}^\infty \frac{6^n}{n!}z^n$$ and put $X(z)=\sum_{n=0}^{\infty} x_nz^n$ for $|z|<1$; we have $$\frac{1}{z}(X(z)-x_0)-X(z)=\operatorname{e}^{6z}$$ that is $$X(z)=\frac{z}{1-z}\cdot \operatorname{e}^{6z}.$$ Observing that $\frac{z}{1-z}=\sum_{n=1}^{\infty}z^n$, then $x_n$ is the discrete convolution of the discrete Heaviside step function $h_n$ and the sequence $\xi_n=\frac{6^n}{n!}$, that is $$x_n=\sum_{k=0}^{n-1}\frac{6^k}{k!}$$
## Other Representations
Recalling that the incomplete gamma function $\Gamma(\alpha,x)$ is given by $$\Gamma(\alpha,x)=\int_x^\infty t^{\alpha-1}\operatorname{e}^{-t}\operatorname{d}t$$ and that for $\alpha$ an integer $n$ $$\Gamma(n,x)=(n-1)!\operatorname{e}^{-x}\sum_{k=0}^{n-1}\frac{x^k}{k!}=(n-1)!\operatorname{e}^{-x}e_{n-1}(x),$$ where $e_n(x)=\displaystyle\sum_{k=0}^{n-1}\frac{x^k}{k!}$ is the exponential sum function, the general solution (4) can be expressed as
$$y_n=(n+1)!2^{1-n}e_{n-1}(6) =\operatorname{e}^{6}2^{1-n}(n+1)n\Gamma(n,6)\tag 5$$
Using the Generalized Exponential Integral $E_n$ function defined as $$E_n(x)= \int_1^\infty\frac{\operatorname{e}^{-xt}}{t^n}\operatorname{d}t= x^{n-1}\Gamma(1-n,x)$$ the solution (5) can be expressed as
$$y_n=\operatorname{e}^{6}2^{1-n}(n+1)n6^nE_{1-n}(6).\tag 6$$
Finally, putting all together the general solution of (1) can be represented as
$$y_n=\operatorname{e}^{6}2^{1-n}(n+1)n\Gamma(n,6)=\operatorname{e}^{6}2^{1-n}(n+1)n6^nE_{1-n}(6).\tag 7$$
Even though this has been answered with a useful hint that basically says it all, the OP seems to ask for more detail, so here are a few additional steps.
The trick is to use exponential generating functions. First divide the recurrence by $(n+2)(n+1),$ to get
$$\frac{y_{n+1}}{(n+2)(n+1)} - \frac{1}{2} \frac{y_n}{(n+1)} = 3^n.$$ The exponential generating function that we will use is $$A(q) = \sum_{n\ge 1} \frac{y_n}{(n+1)!} q^n.$$ We build a functional equation for $A(q)$ by multiplying the recurrence by $q^{n+1}/n!$ and summing over $n\ge 1$ to get $$\sum_{n\ge 1} \frac{y_{n+1}}{(n+2)!} q^{n+1} - \frac{1}{2} q\sum_{n\ge 1} \frac{y_n}{(n+1)!} q^n = \sum_{n\ge 1} \frac{3^n}{n!} q^{n+1}.$$ This is $$A(q) - q - \frac{1}{2} q A(q) = q \sum_{n\ge 1} \frac{3^n}{n!} q^n = q \left(e^{3q} - 1\right).$$ This gives $$\left(1-\frac{1}{2}q\right) A(q) = q \times e^{3q} \quad\text{or}\quad A(q) = q \frac{e^{3q}}{1-\frac{1}{2} q}.$$ To conclude note that $y_n = (n+1)! [q^n] A(q)$ so that $$y_0 = [q^0] q \frac{e^{3q}}{1-\frac{1}{2} q} = 0$$ and for $n\ge 1$ we have $$(n+1)! [q^n] q \frac{e^{3q}}{1-\frac{1}{2} q} = (n+1)! [q^{n-1}] \frac{e^{3q}}{1-\frac{1}{2} q} = (n+1)! \sum_{k=0}^{n-1} \frac{3^k}{k!} \frac{1}{2^{n-1-k}}$$ yielding the final answer $$y_n = \frac{(n+1)!}{2^{n-1}} \sum_{k=0}^{n-1} \frac{6^k}{k!} \sim \frac{(n+1)!}{2^{n-1}} \exp(6)$$ where we have used the Cauchy product of two sequences. Examining this result we see why ordinary generating functions don't work as well here.
Set $x_n=y_n\frac{2^n}{(n+1)!}$. Then $$x_{n+1}=x_n+\frac{2^{n+1}3^n}{n!}$$ etc.
At "how-to-do": I'd try to look at the list of equations for decreasing $n$ and try to make a telescoping sum out of it because that would reduce it to an equation, where on the lhs we have only $y_{n+1}$ and on the rhs an expression free of $y$ which hopefully can be made in a nicer expression depending on the index $n$ such that we can write the general term more easily:
$$\small \begin{array} {} &\; y_{n+1} &-\frac {n+2}2\cdot y_{n } &\;=& 1\cdot &(n+1)(n+2)\cdot 3^n \\ \frac {n+2}2 &( y_{n } &-\frac {n+1}2\cdot y_{n-1} &)= & \frac {n+2}2 \cdot &(n )(n+1)\cdot 3^{n-1} \\ \frac {n+2}2\frac {n+1}2 & ( y_{n-1} &-\frac {n }2\cdot y_{n-2} &)=& \frac {n+2}2 \frac {n+1}2\cdot & (n-1)(n )\cdot 3^{n-2} \\ \\ \vdots \\ \\ \frac {(n+2)!/2!}{2 ^n} & ( y_{1} &-\frac {2 }2\cdot y_{0} &)=& \frac {(n+2)!/2!}{2^n} \cdot & (1)(2 )\cdot 3^{0} \\ \end{array}$$ and this, when column-wise summed reduces to $y_{n+1}$ on the lhs and some sum on the rhs whose terms are easily recognizable consisting mainly of the binomial-coefficents $$y_{n+1} = \sum \small \text{<something depending on n >}$$
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2019-12-06 19:01:00
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https://docs.lammps.org/dihedral_style.html
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# dihedral_style command
## Syntax
dihedral_style style
• style = none or hybrid or charmm or class2 or harmonic or helix or multi/harmonic or opls
## Examples
dihedral_style harmonic
dihedral_style multi/harmonic
dihedral_style hybrid harmonic charmm
## Description
Set the formula(s) LAMMPS uses to compute dihedral interactions between quadruplets of atoms, which remain in force for the duration of the simulation. The list of dihedral quadruplets is read in by a read_data or read_restart command from a data or restart file.
Hybrid models where dihedrals are computed using different dihedral potentials can be setup using the hybrid dihedral style.
The coefficients associated with a dihedral style can be specified in a data or restart file or via the dihedral_coeff command.
All dihedral potentials store their coefficient data in binary restart files which means dihedral_style and dihedral_coeff commands do not need to be re-specified in an input script that restarts a simulation. See the read_restart command for details on how to do this. The one exception is that dihedral_style hybrid only stores the list of sub-styles in the restart file; dihedral coefficients need to be re-specified.
Note
When both a dihedral and pair style is defined, the special_bonds command often needs to be used to turn off (or weight) the pairwise interaction that would otherwise exist between 4 bonded atoms.
In the formulas listed for each dihedral style, phi is the torsional angle defined by the quadruplet of atoms. This angle has a sign convention as shown in this diagram:
where the I,J,K,L ordering of the 4 atoms that define the dihedral is from left to right.
This sign convention effects several of the dihedral styles listed below (e.g. charmm, helix) in the sense that the energy formula depends on the sign of phi, which may be reflected in the value of the coefficients you specify.
Note
When comparing the formulas and coefficients for various LAMMPS dihedral styles with dihedral equations defined by other force fields, note that some force field implementations divide/multiply the energy prefactor K by the multiple number of torsions that contain the J-K bond in an I-J-K-L torsion. LAMMPS does not do this, i.e. the listed dihedral equation applies to each individual dihedral. Thus you need to define K appropriately via the dihedral_coeff command to account for this difference if necessary.
Here is an alphabetic list of dihedral styles defined in LAMMPS. Click on the style to display the formula it computes and coefficients specified by the associated dihedral_coeff command.
Click on the style to display the formula it computes, any additional arguments specified in the dihedral_style command, and coefficients specified by the associated dihedral_coeff command.
There are also additional accelerated pair styles included in the LAMMPS distribution for faster performance on CPUs, GPUs, and KNLs. The individual style names on the Commands dihedral doc page are followed by one or more of (g,i,k,o,t) to indicate which accelerated styles exist.
• none - turn off dihedral interactions
• zero - topology but no interactions
• hybrid - define multiple styles of dihedral interactions
• charmm - CHARMM dihedral
• charmmfsw - CHARMM dihedral with force switching
• class2 - COMPASS (class 2) dihedral
• cosine/shift/exp - dihedral with exponential in spring constant
• fourier - dihedral with multiple cosine terms
• harmonic - harmonic dihedral
• helix - helix dihedral
• multi/harmonic - dihedral with 5 harmonic terms
• nharmonic - same as multi-harmonic with N terms
• opls - OPLS dihedral
• spherical - dihedral which includes angle terms to avoid singularities
• table - tabulated dihedral
• table/cut - tabulated dihedral with analytic cutoff
## Restrictions
Dihedral styles can only be set for atom styles that allow dihedrals to be defined.
Most dihedral styles are part of the MOLECULE package. They are only enabled if LAMMPS was built with that package. See the Build package doc page for more info. The doc pages for individual dihedral potentials tell if it is part of a package.
## Default
dihedral_style none
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2020-11-26 17:55:04
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https://www.physicsforums.com/threads/optimizing-fractions-and-lagrange-multiplier.922775/
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I Optimizing fractions and Lagrange Multiplier
1. Aug 14, 2017
joshmccraney
Hi PF!
When minimizing some fraction $f(x)/g(x)$ can we use Lagrange multipliers and say we are trying to optimize $f$ subject to the constraint $g=1$?
Thanks
2. Aug 14, 2017
Krylov
No, I don't think so, if I understand your question correctly. Why would that be true? Consider $f(x) = e^{-x}$ and $g(x) = 1 + x^2$.
3. Aug 15, 2017
joshmccraney
Yea those are good counter examples. I am curious because I am reading about the Ritz method for approximating eigenvalues. My book says: consider the eigenvalue problem $Lu=\lambda Mu$ where $L$ and $M$ are linear operators, $u$ is a function and $\lambda$ is the eigenvalue. Then the smallest eigenvalue $\lambda_1$ is given by $$\lambda_1=\min\frac{(Lu,u)}{(Mu,u)}$$ or equivalently $$\lambda_1=\min\limits_{(Mu,u)=1}(Lu,u)$$ where $(f,g)=\int_a^bfg\,dx$.
4. Aug 15, 2017
Krylov
Ah ok, yes, but this problem has a special structure. Let's write
$$\lambda_1 := \min_{(Mu,u) \neq 0}\frac{(Lu,u)}{(Mu,u)}$$
and
$$\mu_1 := \min_{(Mu,u)=1}(Lu,u)$$
Your book now claims that $\lambda_1 = \mu_1$. How would you prove this directly (i.e. no Lagrange multipliers or something like that)?
5. Aug 16, 2017
joshmccraney
Take a derivative of what we're trying to minimize w.r.t...hmmm well now I'm not so sure. Any ideas?
6. Aug 16, 2017
Krylov
Yes, but I would like you to puzzle a little bit, too.
You have not given much of the precise context. Let's assume that
• $H$ is a real Hilbert space with inner product $(\cdot,\cdot)$,
• $L$ and $M$ are operators with domains $D(L) = D(M) = H$,
• $M$ is a symmetric and positive operator, i.e. $0 \le (Mu, u)$ for all $u \in H$.
To be safe, I also replace your minima by infima. Let's make the minimization problems a bit more explicit using set notation. Write
$$\Lambda_1 := \left\{\frac{(Lu,u)}{(Mu,u)}\,:\, u \in H,\, (Mu,u) > 0 \right\}, \qquad M_1 := \left\{(Lv,v)\,:\, v \in H,\, (Mv,v) = 1 \right\},$$
so $\lambda_1 = \inf{\Lambda_1}$ and $\mu_1 = \inf{M_1}$. (Note that the positivity of $M$ allowed me to replace the condition $(Mu,u) \neq 0$ by $(Mu,u) > 0$ in $\Lambda_1$.)
Now observe that $M_1 \subseteq \Lambda_1$. (This already gives $\lambda_1 \le \mu_1$.) With a little bit (but not much) more work, you also show the reverse inclusion:
$$M_1 \supseteq \Lambda_1 \qquad (\ast)$$
Once this is done, you have $\Lambda_1 = M_1$, so $\lambda_1 = \mu_1$ follows. If you like, try to deduce $(\ast)$ yourself.
(The essential property you will use, is that both the numerator and the denominator of the original function
$$u \mapsto \frac{(Lu,u)}{(Mu,u)}$$
are homogeneous of degree two, so any scaling of $u$ does not change the function value.)
7. Aug 16, 2017
joshmccraney
Ok, so I'm thinking if the denominator was $(u,u)$ (a less general case) then all we would have to do is let $v = u/||u||$. But with the $M$ operator present, I'm unsure how to proceed...I'll think more on this, but feel free to give the spoiler.
On a separate note, I'm interested in this entire process of finding the lowest eigenvalues because I am trying to solve $Lu=\lambda Mu$. If we instead look at a simpler problem $Lu=\lambda u$ I know the lowest eigenvalue is $$\lambda_1=\inf \frac{(Lu,u)}{(u,u)}.$$ Letting $$u=\sum_{i=1}^N a_i\psi_i$$ where $\psi_i$ are orthonormal basis vectors that satisfy boundary conditions, we deduce
$$(Lu,u) = \sum_{i,k=1}^NF_{ik}a_ia_k:F_{ik}\equiv (L\psi_i,\psi_k)$$
To find the infimum from above, we can reduce the minimizing problem to minimizing $(Lu,u)$ subject to the constraint $(u,u)=1$. This evokes Lagrange multipliers. Letting $\Lambda$ be the Lagrange multiplier, take $$\frac{\partial}{\partial a_j}\left( \sum_{i,k=1}^NF_{ik}a_ia_k + \Lambda \sum_{i,k=1}^Na_ia_k(\psi_i,\psi_k)\right)=0\implies\\ \frac{\partial}{\partial a_j}\left( \sum_{i,k=1}^NF_{ik}a_ia_k + \Lambda \sum_{i}^Na_i^2\right)=0\implies\\ \sum_{k=1}^NF_{ki}a_k + \Lambda a_i=0\implies\\ \det(F_{ki}-\Lambda\delta_{ki})=0.$$
Similarly, when solving the problem $Lu=\lambda M u$, if we let $(Mu,u) = \sum_{i,k=1}^ND_{ik}a_ia_k:D_{ik}\equiv (M\psi_i,\psi_k)$ then the solution should be $$\det(F_{ki}-\Lambda D_{ki})=0$$ where $\lambda_1$ is the lowest root $\Lambda$. Does this look correct to you?
I'm under the impression this technique is called the Rayleigh-Ritz variational approach.
8. Aug 17, 2017
Krylov
The symmetry and - in particular - the positivity of the operator $M$ imply that $(u_1,u_2) \mapsto (Mu_1,u_2)$ defines a bilinear form on $H \times H$ that has all the properties of an inner product, with the exception that $(Mu,u) = 0$ may not imply $u = 0$. In any case, for the problem at hand you can just let
$$v =\frac{1}{\sqrt{(Mu,u)}}u$$
(In other words, you scale by $(Mu,u)^{-\frac{1}{2}}$ instead of $(u,u)^{-\frac{1}{2}}$.)
9. Aug 17, 2017
joshmccraney
Thanks!
10. Aug 17, 2017
WWGD
Id say, if $f,g$ are differentiable, you can use the constraint $f'g-fg'=g^2$ when $g \neq 0 ; g(x) \neq 0$ or not defined , since extremes are reached at critical points.
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2018-07-18 13:32:01
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https://blog.zilin.one/21-355-spring-2012/homework-5/?replytocom=4
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# Homework 5
Exercise: Use the definition of convergence to determine the following limits in $\mathbb{R}$:
1. $\lim_{n\rightarrow\infty}\frac{3n+1}{n+5}$
2. $\lim_{n\rightarrow\infty}(\sqrt{n^2+n}-n)$
Proof.
1. $\lim_{n\rightarrow\infty}\frac{3n+1}{n+5}=3$
2. $\lim_{n\rightarrow\infty}(\sqrt{n^2+n}-n)=\frac{n}{\sqrt{n^2+n}+n}=\frac{1}{2}$
Exercise:
1. Give an example of a convergent sequence $\{s_n\}$ of positive real numbers such that $\lim_{n\rightarrow\infty}\frac{s_{n+1}}{s_n}=1$
2. Give an example of a divergent sequence $\{s_n\}$ of positive real numbers such that $\lim_{n\rightarrow\infty}\frac{s_{n+1}}{s_n}=1$
Proof.
1. $s_n=1$
2. $s_n=n$
Exercise:
1. Give an example of a convergent sequence $\{s_n\}$ of real numbers such that the set $\{s_n:n\in\mathbb{N}\}$ has exactly one limit point.
2. Give an example of a convergent sequence $\{s_n\}$ of real numbers such that the set $\{s_n:n\in\mathbb{N}\}$ has no limit point.
3. Prove: If a sequence $\{s_n\}$ of real numbers converges, then the set $\{s_n:n\in\mathbb{N}\}$ has at most one limit point.
Proof.
1. $s_n=1/n$
2. $s_n=1$
3. Suppose $\lim_{n\rightarrow\infty}s_n=s$. Enough to show that if there is a limit point $t$ for the set $\{s_n:n\in\mathbb{N}\}$, then $s=t$. Deny. Then pick any positive real number $\epsilon<|s-t|/2$. There is an integer $N$ such that $n\geq N$ implies $|s-s_n|<\epsilon$. Observe that now $t$ is still the limit point of the set $\{s_n:n\geq N\}$, because chopping off finitely many points doesn’t change the limit point. However, by the choice of $\epsilon$ and $N$, for all $n\geq N$, $|t-s_n|>|s-t|/2>0$. A contradiction.
Exercise: Suppose $X\neq\emptyset$ is equipped with the discrete metric. Characterize all convergent sequences in $X$.
Proof.
We say a sequence $\{s_n:n\in\mathbb{N}\}$ in $X$ is eventually constant if and only if there are $N\in\mathbb{N}$ and $s\in\ X$ such that for all $n>N$, $s_n=s$.
We claim that all eventually constant sequences are all convergent sequences in $X$. Easy to see they are convergent. Left to see that every convergent sequence is eventually constant. Suppose $\{s_n:n\in\mathbb{N}\}$ convergences to $s$. By definition of convergence, there is an integer $N$ such that $n>N$ implies that $d(s_n,s)<1/2$, which, indeed, implies $s_n=s$ because of the discrete metric.
Exercise:
1. Prove: If a sequence $\{s_n\}\subset\mathbb{R}$ converges in $\mathbb{R}$ then the sequence $\{|s_n|\}$ converges in $\mathbb{R}$.
2. Give an example of a sequence $\{s_n\}\subset\mathbb{R}$ such that $\{|s_n|\}$ converges but $\{s_n\}$ does not converge.
3. For a sequence $\{s_n\}\subset\mathbb{R}$ we define the sequences $\{s_n^+\}$ and $\{s_n^-\}$ where $s_n^+:=\max\{s_n, 0\}$ and $s_n^-:=\min\{s_n,0\}$ for all $n\in\mathbb{N}$. Prove: the sequence $\{s_n\}$ converges if and only if $\{s_n^+\}$ and $\{s_n^-\}$ converge.
Proof.
1. Suppose $\{s_n\}$ converges to $s$. Then for every $\epsilon > 0$, there is an integer $N$ such that $n\geq N$ implies $|s_n-s| < \epsilon$. Notice that $||s_n|-|s||\leq |s_n-s| < \epsilon$. Hence $\{|s_n|\}$ converges to $|s|$.
2. For example, $s_n=(-1)^n$.
3. Suppose $\{s_n\}$ converges. Notice that $s_n^+=(s_n+|s_n|)/2$ and $s_n^-=(s_n-|s_n|)/2$. By (a) and theorem 3.4, we know $\{s_n^+\}$ and $\{s_n^-\}$ also converge. On the other hand, if $\{s_n^+\}$ and $\{s_n^-\}$ converge, $\{s_n\}$ converges since $s_n=s_n^++s_n^-$.
## 2 replies on “Homework 5”
markwongsksays:
I think there’s a typo for 1(a).
My mistake. Thank you, Mark. I’ve corrected it.
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2020-01-19 15:30:43
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http://mathoverflow.net/feeds/question/84591
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subalgebra of a matrix algebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:45:46Z http://mathoverflow.net/feeds/question/84591 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84591/subalgebra-of-a-matrix-algebra subalgebra of a matrix algebra Miguel 2011-12-30T14:33:29Z 2011-12-30T21:18:56Z <p>Let $K$ be an algebraic closed field and $M_n(K)$ the $K$-algebra of all matrices $n\times n$ over $K$. If $L$ and $M$ are two commutative isomorphic subalgebras of $M_n(K)$ it is true that there exista a regular matrix $S\in M_n(K)$ such that $SLS^{-1}=M$. That is the isomorphism is inner? </p> http://mathoverflow.net/questions/84591/subalgebra-of-a-matrix-algebra/84600#84600 Answer by Denis Serre for subalgebra of a matrix algebra Denis Serre 2011-12-30T17:06:23Z 2011-12-30T21:18:56Z <p>Two isomorphic subalgebra of $M_n(K)$ <strong>do not need to be conjugated</strong>. The following example is taken from Exercise 161 of my web site <a href="http://www.umpa.ens-lyon.fr/~serre/DPF/exobis.pdf" rel="nofollow">http://www.umpa.ens-lyon.fr/~serre/DPF/exobis.pdf</a></p> <p>Set $n=p+q$ with $q>p>0$. Then define $\mathcal A$ as the subset of $M_n(k)$ made of the matrices with block form $$\left(\begin{array}{cc} 0_p & 0_{p\times q} \\ A & 0_q \end{array}\right).$$ Likewise, ${\cal B}$ is made of the matrices $$\left(\begin{array}{cc} 0_q & 0_{q\times p} \\ B & 0_p \end{array}\right).$$ Both $\cal A$ and $\cal B$ are subalgebras of $M_n(k)$, with dimension $pq$ and the property that $MN=0_n$ for every two elements (of the same algebra). They are obviously isomorphic, because the algebra structure is trivial. But ${\cal A}$ and $\cal B$ are not conjugated in $M_n(k)$. However $\cal B$ is conjugated to ${\cal A}^T$ in $M_n(k)$.</p>
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2013-06-20 12:45:44
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https://web1.eng.famu.fsu.edu/~dommelen/quantum/solman/herm-g.html
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#### 2.6.7 Solution herm-g
Question:
Show that if is a Hermitian operator, then so is . As a result, under the conditions of the previous question, is a Hermitian operator too. (And so is just , of course, but is the one with the positive eigenvalues, the squares of the eigenvalues of .)
Answer:
To show that is Hermitian, just move the two operators to the other side of the inner product one by one. As far as the eigenvalues are concerned, each application of to one of its eigenfunctions multiplies by the eigenvalue, so two applications of multiplies by the square eigenvalue.
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2023-02-04 05:39:57
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https://solvedlib.com/4-110revie-w-conslants-poncdc-quot-part-aif-the-diameter,6038998
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2023-03-27 17:44:02
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http://stats.stackexchange.com/questions/44475/is-there-a-statistical-test-to-compare-two-samples-of-size-1-and-3
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# Is there a statistical test to compare two samples of size 1 and 3?
For an ecology project, my lab group added vinegar to 4 tanks containing equal volumes of pond water, 1 control with no elodea (an aquatic plant) and 3 treatments with the same amount of elodea in each. The purpose of adding the vinegar was to reduce the pH. The hypothesis was that the tanks with elodea would go back to their normal pH quicker. This was indeed the case. We measured the pH of each tank daily for about two weeks. All the tanks eventually returned to their natural pH, but the length of time that this took was much shorter for the tanks with elodea.
When we told our professor about our experimental design, he said that there exists no statistical test that can be performed on the data to compare the control to the treatment. That because there was no replicate for the control (we only used one control tank) we cannot calculate variance and so we can't compare the sample means of the control and the treatment. So my question is, is this true? I definitely understand what he means. For example, if you took the height of one man and one woman, you can't draw conclusions about their respective populations. But we did 3 treatments, and the variance was small. It seems reasonable to assume that the variance would be similar in the control?
Update:
Thank you for the excellent answer. We got more water and elodea from the wetland and decided we would run the experiment again with smaller tanks but this time with 5 controls and 5 treatments. We were going to combine this with our original data but the starting pH of the tanks was different enough that it doesn't seem valid to consider the new experiment to be sampled from the same population as the original experiment.
We considered adding different amounts of elodea and trying to correlate speed of pH remediation (measured as time elapsed until pH returned to its original value) with amount of elodea, but we decided that wasn't necessary. Our objective is only to show that the elodea makes a positive difference, not to construct some kind of predictive model for exactly how the pH responds to differing amounts of elodea. It would be interesting to determine the optimal amount of elodea, but that's probably just the maximum amount that can survive. Trying to fit a regression curve to the data wouldn't be especially illuminating because of the various complicated changes that occur to the community when adding a large amount. The elodea dies, decomposes, new organisms start to dominate, and so on.
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Did you add the same amount of Elodea to each of the 3 'treatment' tanks? – gung Nov 27 '12 at 1:53
Yes, we added the same amount of Elodea to each treatment. – Simon Hunt Nov 27 '12 at 2:51
Note @gung's question; it matters. I will assume that the treatment was the same for every tank in the treatment group.
If you can argue the variance would be equal for the two groups (which you would typically assume for a two sample t-test anyway), you can do a test. You just can't check that assumption, no matter how badly violated it might be.
The concerns expressed in this answer to a related question are even more relevant to your situation, but there's less you can do about it.
[You ask about it being reasonable to assume the variances are equal. We can't answer that for you, that's something you'd have to convince subject matter experts (i.e. ecologists) was a reasonable assumption. Are there other studies where such levels have been measured under both treatment and control? Others where similar tests (t-tests or anova especially - I bet you can find a better precedent) have been done or similar assumptions made? Some form of general reasoning you can see to apply?]
If $\bar{x}$ is the sample mean of the treatment and $\bar{y}$ is the mean of the control, and both are from normal distributions with variance $\sigma^2$, then $\bar{x}-\bar{y}$ will have mean $\mu_x - \mu_y$ and variance $\sigma^2 (1/n_x + 1/n_y)$ irrespective of whether one of the $n$'s is 1.
So when $n_y$ is 1,
$$\frac{(\bar{x}-\bar{y})}{s_x\sqrt{1/n_x+1}}$$
(where $s_x$ is the standard deviation computed from the treatments) will be $t$-distributed (with $n_x - 1$ degrees of freedom) under the null.
You may notice that with the best available estimate of $\sigma$, $s_x$ used for $s_p$, this is exactly like the ordinary two-sample t-test formula with $n_y$ set to 1.
Edit:
Here's a simulated power curve for this test. The sample size at the null was 10000, at the other points was 1000. As you see, the rejection rate at the null is 0.05, and the power curve, while it requires a large difference in population means to have decent power, has the right shape. That is, this test does what it is supposed to.
(End edit)
With sample sizes so small, this will be somewhat sensitive to distributional assumptions, however.
If you're prepared to make different assumptions, or want to test equality of some other population quantity, some test may still be possible.
So all is not lost... but where possible, it's generally better to have at least some replication in both groups.
-
Note you will need to follow the formulas @Glen_b outlined. Both Excel and Minitab won't compute this. – zbicyclist Nov 27 '12 at 4:01
(+1) An equivalent approach (using the same formula)--and therefore more ammunition for justifying this answer--is that you can compute a prediction interval for one future value from the treatment group. If the control value does not fall within that prediction interval, you have significant evidence of a difference between the two groups. The difference could be some combination of a difference in mean or a difference in variances, but there is (likely) a difference. – whuber Nov 27 '12 at 8:23
@bdemarest An interesting thought, but it doesn't work. It asserts that the value '12' is known exactly (instead of being a random observation from a distribution with s.d. $\sigma$), leading to the formula $\frac{(\bar{x}-12)}{s_x\sqrt{1/n_x}}$ - in this particular case making the value under the square root a quarter of what it should and so making the $t$ value twice as large as it should be. – Glen_b Nov 28 '12 at 1:19
@Glen_b: Not sure if this has changed this last November, but R 3.0 will do a pooled t-test when one of the sample sizes is one, and gives the same answer as an anova. – Aaron Aug 20 '13 at 1:02
For anyone wanting to try it in R: t.test(x=c(4.5,4.8,4.6),y=5.2, var.equal=TRUE) -- it looks like this works in both R2.15.2 and R3.0.0 (the only two versions I have handy). – Glen_b Aug 20 '13 at 2:22
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2014-07-29 20:52:13
|
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https://academia.stackexchange.com/questions/13180/good-responses-to-ivory-tower-accusations/13212
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Good responses to “Ivory Tower” accusations?
Recently I have come across a few articles on Google that are really distressing.
I typed in "professor real world" and it just mentions how professors have lost touch with the real world, how they need to step down from their ivory towers, etc.
Why is all that bad? It doesn't make me reconsider my dreams a little, but it does hurt.
Is it ultimately subjective from the pessimists' point of view? I have come to terms with the fact nothing ultimately matters, we are just grains of (smart) sand in the universe. Whether I did industry or not wouldn't matter, since I love academia.
How can I help others see that academia is a 'real job' too?
• Some people are just jerks. Ignore them. – JeffE Oct 4 '13 at 11:03
• Welcome to AC.SE. It is not really clear to me what you are asking. – StrongBad Oct 4 '13 at 12:49
• I think this is just an invitation to discussion. Academia Chat is the right place for that. I don't see what sort of objective answer could address this question as posed. – 410 gone Oct 4 '13 at 13:04
• Speaking from experience in Computer Science: "Ivory Tower" professors who have "lost touch with the real world" are unfortunately quite common. They're the ones that don't have experience outside of academia, but treat their way as the One True Way, when those of us who do have more experience know that their way not only isn't the best, but might actually be detrimental in the long run. – Izkata Oct 4 '13 at 16:11
• The good news is that people with extreme opinions often say foolish things. For example, "If a medical or dental-school professor has never had to perform a type of surgery he is teaching, how does he know the best way to teach it?" (Sure, this could be a problem in principle, but does anyone actually think it's a widespread problem in practice?) If you want to argue with them, this will give you an advantage. However, I've never actually seen such an argument change anyone's mind. If someone doesn't like academia, you won't convince them to like it by telling them their facts are wrong. – Anonymous Mathematician Oct 5 '13 at 12:28
There is no point talking to people about the job they cannot do themselves or about the parts of reality they cannot even see, not to say enter. Most of them are, probably, just envious (that's why the words "easy life", "good salary", and "tenure" appear in such articles more often than not). The rest are frustrated that our work doesn't provide any immediate personal benefit for them. As to "stepping down from ivory towers", in Soviet Union the students and professors were sent for a month to collective farms to help with harvesting every year. It turned out that we could do the farm work. Sending help in the other direction wasn't considered practical.
• I think there is a big different between an academic planting seeds and an academic promoting a business model when the academic has never experienced the life of a businessperson. If someone claims a theory works in real life, there should be evidence to back up that statement. – earthling Oct 5 '13 at 0:14
• @earthling To be honest, I have never tried to promote any "business model", so I cannot comment on this, but I was involved in a few engineering consulting projects without ever experiencing the life of an engineer and believe it or not, my math. worked the way it was supposed to. – fedja Oct 5 '13 at 2:22
• This answer could be improved with reference to a proof overturning basic sociological arguments about the knowability of social relations. – Samuel Russell Oct 7 '13 at 8:37
• Plenty of "ivory tower" complaints come well reasoned from within academia itself. Responses like this that simply presume the complainant is unqualified and bitter are not constructive to the overall conversation. – Nick Bastin Oct 7 '13 at 9:19
• @Nick Bastin When they come well-reasoned, the answer is different. Unfortunately, when somebody says that "one cannot understand business without being a businessmen himself", it just reminds me of "you cannot tell anything about penguins without spending a few months in Antarctica walking with your hands tight against your hips and diving for fish". Observation and deduction are not less powerful than the first hand experience and the fact that they are used in a wrong way occasionally or even often does not disqualify the general idea. – fedja Oct 7 '13 at 11:32
I can only speak to my personal experience, but it stems from the fact that professors are supposed to be training in a new generation of productive people in the work force. If they have not spent a significant amount of time being productive themselves, they are at a disadvantage when it comes to passing on useful education to future productive people. This is not to say that such education is impossible, or even uncommon, simply that the education is typically based on applied theory as opposed to experience.
As an example, I once was in a lecture about computational complexity. My professor had said that if I could take an algorithm from 4n^2 to 4n that my boss would be happy, but if I could take it from 4n^2 to 2n^2 that he would not be happy. I told her the entire business model of the firm I interned at relied on running our computations as fast as possible, and that if I could cut the time in half, my boss would be thrilled out of his pants. She immediately dismissed this as naive, as such a performance gain would be insignificant: only a reduction in the order of complexity would be noticed - simply reducing the coefficient would not.
I called my boss after the lecture, and he said if I could cut our simulation time in half, he would fly me back to work and double my pay (since it would still be cheaper than the expansion of our beowulf cluster we were planning). We looked up the professors credentials, and despite having a PhD and over 20 years experience in academia, her only real world experience was a 6 month internship that, according to the description, consisted mostly of paperwork.
Now, is this representative of most people in academia? I don't think it is. But it does happen, and it's more common than it would seem from the inside looking out. And because it does happen, it feeds the stereotype of academics who couldn't engineer their way out of a paper bag. There are stereotypes all over all industries. Software developers have a stereotype of being nerds who couldn't possibly get a date, and yet in North America 70% of developers are married, with only a 3% divorce rate (compared to 40% of the population). While I certainly know some nerds in my line of work, and yes they do feed the stereotype, they really aren't represented by the majority of the population.
The conclusions I would make is that the concerns raised by those yelling about "Ivory Towers" I think are valid concerns. They do not apply to all academics, and of course, research should be judged on the actual research, not the researchers. But you should keep in mind that the there are certain individuals who speak with authority based on experience, while others speak with authority based on the assumption that they have the experience. When it comes to published, peer reviewed research, it's easy to separate the wheat from the chaff. In the classroom settings, where such authority is not to be questioned, it can be very dangerous for young, impressionable students.
So how do you respond to an ivory tower accusation? Well, clearly if you have industry experience, put it forth. If you lack industry experience, make it clear that you have no intention of trying to pass off your education as being backed by industry experience. When it comes to your research, encourage skeptics to review your research on its own merits. If they're true scientists, they will.
• Computaional complexity theory and software engineering are NOT the same thing. Complexity theory is a branch of mathematics which studies the limitation and nature of computations. It is studied for pure intellectual curiosity just like number theory or theoretical physics (even though it has practical applications just like physics and number theory). A complexity theorist is not working to increase revenues of software companies. And what she said is absolutely right a factor of 2 is irrelevant for complexity theorist. – user774025 Oct 6 '13 at 4:37
• @user774025: complexity theory is of incredible use to software engineers. It should not, however, be a handcuff to the practical concerns of actual runtime. – Nick Bastin Oct 7 '13 at 9:12
• "I could take an algorithm from 4n^2 to 4n that my boss would be happy, but if I could take it from 4n^2 to 2n^2 that he would not be happy". You misunderstand the idea. The point is that if you can do it with $n=10^6$ now and reduce the running time to $2n^2$, then you'll be able to do it just with $n=1.4\cdot 10^6$. However $4n$ will enable you to jump to $n=10^12$. "Happy boss" means "a real breakthrough", not "bringing your particular application down to acceptable running time". For the latter, sometimes even a 10% reduction may be enough if you are close to the goal already. – fedja Oct 7 '13 at 11:50
• @fedja You seem to misunderstand the idea that if you're a firm that does nothing but thermal modelling and you can double the number of thermal models you run in a given amount of time, you essentially double the revenue of the company. Last time I checked, doubling the revenue of the company is a happy boss. – corsiKa Oct 7 '13 at 14:54
• @corsiKa We have a pure communication problem here. My point is that both the professor in question (and I) understand all this perfectly well. However, it seems that she has long given up trying to explain the ideas of "order of magnitude", "asymptotics", etc. to her students and decided to do the (very sloppy) translation to the language they can comprehend instead. What I tried to do was to translate back and my point was that the construct "happy boss" in her language had nothing to do with the company revenue, the promotion, or anything else in that venue whatsoever. – fedja Oct 11 '13 at 0:56
Firstly, you do matter, we all matter in our own way and we have no idea just how far our influence will extend. Never let yourself otherwise.
Secondly, as JeffE said in the comment, some people are just jerks - for whatever reason, the authors of those articles are venting, and as they can not possibly know every professor - they probably have had an awful experience and are venting, generalising across the board. (or they can just be jerks).
They obviously do not know my professors - the most dedicated educators I have ever had the privilege of working with.
Fundamentally, you do not have to prove yourself to anyone, but yourself. So, be yourself.
• They just mention how it's not a real job and I want some good comebacks to that. – Jossie Oct 4 '13 at 11:35
• I am also a high school teacher, and similar comments are levelled at us - comebacks? you do not need to - you don't answer to their jibes. – user7130 Oct 4 '13 at 11:41
• +1 for "You do not have to prove yourself to anyone." And if you want comebacks, I recommend "I know you are but what am I?" Or, "How appropriate, you fight like a cow". – Nate Eldredge Oct 4 '13 at 15:49
I typed in the same Google query, read the article I think the OP was referring to (this one?), and was filled with a similar hot burst of indignation. (How dare they!) But while the author of that article paints with a broad brush, I think it might not be inappropriate for us, as scholars, to examine ourselves with the same brutal honesty that we take to our intellectual disciplines.
Some of the complaints I've heard about ivory-tower academics simply reflects a misunderstanding of what higher education is about -- for example, as a computer science professor I've heard students complain about not being taught how to use Excel spreadsheets and the like. But I've also seen colleagues who have grown complacent and uncaring, whose courses really do shortchange the students. So yes, the author of that article paints with a very broad brush, but I think that we'd do ourselves a disservice if we just blithely ignored him.
I think it's important for us to be able to articulate why we do what we do. I routinely explain to my reviewers why my research matters -- I should similarly be able to explain to my students why they should study what I'm teaching;[*] and I should similarly be able to explain to my neighbors why their tax dollars should pay my salary. (I may not have to do these things, but I should be able to.) Whatever explanation I come up with -- and it'll be different for different people -- that's the "good comeback" the OP asked for; and if I can't come up with any explanation at all, then maybe some deeper introspection would be in order. :-)
[*] In my grant proposals I say what difference I think the research will make if it's successful. In my classes I tell the students what I want them to remember of the class five years later. I've found this sort of exercise very helpful for distilling out what I think really matters.
• Thank you for the link you provided. I think that one is the one (@Jossie Please confirm it if you can). I, too, feel the same way as you do. – scaaahu Oct 5 '13 at 5:40
Jossie, you are a scientist, right? Define the 'real job' for us first, please. "The real job is a job in which..." what happens? You get a real paycheck, that's for sure, and if that's the main indication of the 'real job', then you definitely got one. If, on the other hand, you define a 'real job' as the one from which you can be fired, then a tenured professor is not a real job.
Aside from that, there are several layers of complications that your question uncovers. Apparently, you are a smart person with a dedication, given that you were able to finish your Ph.D. It is, however, also apparent to me that you cannot really explain what you do to a layperson. You are not alone in this: the portrayal of academia as the ivory tower stems from this same lack of communication between professors and the general public that just cannot understand the value added that academia provides.
I have worked as an assistant professor on a tenure track for three years, was booted from it, and found home in industry. I can tell very specifically what the value of academic research is for me in my position: it can produce new efficient ways for me to make the product that my company delivers better... where better may include concepts like "more accurate" (I am a social statistician, so that's a relevant dimension of my work), "faster", "more robust wrt various uncheckable assumptions", etc. Unfortunately for me, academic research produces hell of a lot of noise that's irrelevant for me: from ~100 papers in the top general interest journals, I would find 1 to be of relevance to my work. The ratio is of course higher in specialized journals, where it can be 3:1 or so. (Nature or Science or PNAS are out of my league; they may publish statistics papers on a cute little topic from time to time, but generally the ratio will be what, 1:10000?) So I am the natural selection process: out of all the random mutations that academic researchers publish, I am selecting the relevant traits that need to be preserved because there is a survival value in them.
Now, the question that I keep asking myself is, "How much of that random noise does need to come out so that in the month of October 2013, I will read up something that will change the way I work?", and apparently the answer is, well, several hundred papers (out of which I will get may be 10 or so to read). That's a costly enterprise: if an average professor is paid $120K, and they publish three papers a year, then that's$40K per paper (we can ignore teaching: first, nobody really cares about it, and second, you can buy a teacher for $5K/course, way below the cost of a research paper that I just derived). So the total for one usable academic result is [drum roll]$4M for the hundred papers that need to be published. That's A LOT of money... although I would humbly hope that for each disgrunteld StasK, there are hundred other statisticians who would find the other 99 random papers useful for them.
If the ratios are better in other disciplines, that's great. For what I heard in education research, the ratios are about the same: at some point, that depository had reviewed ~300 papers, and found only 6 of them to be usable.
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2020-09-29 11:32:07
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https://socratic.org/questions/what-are-the-mean-and-standard-deviation-of-a-binomial-probability-distribution--18
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# What are the mean and standard deviation of a binomial probability distribution with n=10 and p=4/5 ?
Apr 4, 2016
Mean $= 8$
SD$= 1.265$
#### Explanation:
Mean $= n p = 10 \times \frac{4}{5} = 8$
SD $= \sqrt{n p q}$
$q = 1 - p = 1 - \frac{4}{5} = \frac{1}{5}$
SD $= \sqrt{10 \times \frac{4}{5} \times \frac{1}{5}} = 1.265$
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2022-01-28 11:49:59
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https://firas.moosvi.com/oer/physics_bank/content/public/016.Waves/The%20Doppler%20Effect/Bat_Doppler_V2/Bat_Doppler_V2.html
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# Bat Doppler#
Bats have poor eyesight, so they use echolocation to orient themselves and to hunt prey (insects). They emit ultrasonic sound waves as chirps and listen to hear if the reflected waves are Doppler shifted to higher or lower frequencies as they fly.
## Useful Info#
The Doppler effect for sound waves tells us when a sound source and a sound observer move relative to each other at speeds $$v\_{source}$$ and $$v\_{observer}$$ respectively, the frequency observed by the observer is given by $$f\_{observer} = \frac{(1\pm \frac{v\_{observer}}{v\_{sound}})}{(1 - \mp \frac{v\_{source}}{v\_{sound}})} f_0$$ where $$f_0$$ is the frequency of emitted sound, and $$v\_{sound}$$ is the speed of sound. In the numerator of this expression a plus (minus) sign indicates that the observer is moving toward (away from) the source. In the denominator of this expression a plus (minus) sign indicates that the source is moving away from (toward) the observer. In both cases this means that the frequency increases when the objects are approaching, and decreases when they are receding from each other. (In this problem involving a reflection picked up by the original source there would actually be two Doppler shifts, with the frequency of sound reflected by the mosquito acting as the source of sound for the bat, and the bat acting as the observer of that sound.)
## Part 1#
If the reflected waves from a mosquito have slightly increased in frequency:
• We don’t know if the bat is moving closer to or moving farther away from the mosquito.
• The bat is moving farther away from the mosquito.
• The mosquito has evolved to mimic the sound waves produced by the bat to confuse it.
• Both the bat and mosquito are flying at the same rate in the same direction.
• The bat is moving closer to the mosquito.
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2022-09-25 14:54:28
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http://www.chegg.com/homework-help/questions-and-answers/which-gives-the-kinetic-energy-of-a-descending-yo-yo-the-translational-kinetic-energy-the--q3347608
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## physics
Which gives the kinetic energy of a descending yo-yo? the translational kinetic energy the rotational kinetic energy the sum of the translational kinetic energy and the rotational kinetic energy
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2013-05-18 18:53:08
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https://3jlearneng.blogspot.com/2015/06/triangles-and-angles-proof-for-5-year.html
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## Sunday, June 21, 2015
### Triangles and Angles: a proof for 5 year olds
who: J2 (also P and J0)
what did we use: polydrons, polydron protractor
J2 and I were playing with polydrons and he got interested in measuring the various angles. Since there are three different triangles in the set, he got to test each of them and found that the angle sum was always 180 degrees. We went on to examine squares and pentagons to see what he could make of those.
In the evening, P and I were talking about about this investigation and I mentioned this proof that the interior angles of a polygon sum to $180 \times (n-2)$:
P was surprised and said that she likes this proof more:
In this version, you have to subtract the middle 360 degrees.
I was a bit surprised that I hadn't ever thought of this version or seen it elsewhere, but quickly realized we could repurpose it to finding the interior angles of a triangle itself. Here's the picture:
Now, the argument:
1. Angles of the large triangle are equal to the sum of the angles of the smaller triangles - 360
2. Let x be the sum of interior angles of a triangle.
3. Based on 1, we have x = 3x - 360
4. Rearranging, we get x = 180.
# Where we get stuck: a challenge for you
Here's the thing: the sum of interior angle measures of a triangle isn't 180 degrees. That is, it doesn't have to be if you are working in non-euclidean geometry. Here's a nice picture (not mine) of a triangle with angle sum of 270 degrees:
Comes from MathStackExchange
However, our proof doesn't seem to use any fancy postulates. Your challenge, why doesn't it work on a sphere? What extra postulate did we secretly use?
# A side point about measurement
When my son was measuring the angles for an isosceles right triangle, he read 46 degrees off the protractor. I was too focused on getting to the punchline about all triangles having the same angle sum and I "corrected" the measurement to 45 degrees. He still said some things and we had a conversation about ideal mathematics and real measurement, but I felt it was a missed opportunity.
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2020-01-25 20:34:32
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https://socratic.org/questions/how-do-you-solve-the-quadratic-using-the-quadratic-formula-given-v-2-2v-8-0
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# How do you solve the quadratic using the quadratic formula given v^2+2v-8=0?
Jul 9, 2017
v = -4 or 2
#### Explanation:
$v = \frac{- b \setminus \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
where
$a = 1 , b = 2 , \mathmr{and} c = - 8$
Gives:
$v = \frac{- 2 \setminus \pm \sqrt{{2}^{2} - 4 \cdot 1 \cdot \left(- 8\right)}}{2 \cdot 1}$
$v = \frac{- 2 \setminus \pm \sqrt{4 + 32}}{2}$
$v = \frac{- 2 \setminus \pm \sqrt{36}}{2}$
$v = \frac{- 2 \setminus \pm 6}{2}$
$v = - \frac{8}{2} \mathmr{and} \frac{4}{2}$
$v = - 4 \mathmr{and} 2$
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2019-11-22 10:25:36
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https://www.nextgurukul.in/questions-answers-forum/academic/icse/class-10/chemistry/old_study-of-compounds
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Ketan Krishna Mishra
Mar 22, 2015
#### Explain fountain experiment?
Fountain experiment is used to demonstrate the solubility o...
Sarath Babu t
Jan 11, 2014
Rutvik gulavani
Nov 23, 2013
#### pls help me to do this as i have a project on this.... what is the 1.Method 2.Experiment...Pls... this not based on any topic in my syllabus so pls help in it ...as a student
No such method. You will need some data like weight, quality of the sugarcane to calculate the volume.
Sarath Babu t
Jan 11, 2014
#### Give reason
Because HCl (aq) is in a complet...
Shehrebano
Feb 9, 2015
#### Equation for magnesium treated with very dilute nitric acid?
Very dilute nitric aicd reaxts with magnesium to give its nitrate and hydrogen gas.
Mg + 2HNO3 → Mg(NO3)2 + H2
Sarath Babu t
Jan 11, 2014
#### Give reason
When hydrogen chloride gas is exposed to atmosphere it gives white fumes due to the formation of hydrochloric acid on reacting with atmospheric water vapour
Ketan Krishna Mishra
Mar 22, 2015
#### Why does the formation of HCl take place in diffused sunlight?
Hydrogen chloride gas can also be formed by reaction of moist hydro...
Dec 21, 2014
#### Name the gas in each of the following : i) A gas evolved when dil HCL is added to calcium carbonate. ii) The gas evolved on reaction of aluminium with boiling concentrated caustic alkali solution. iii) The gas produced when copper reacts with concentrated nitric acid. iv) The gas produced on reaction of dilute sulphuric acid with iron sulphide.
i)CO2
ii)H2
iii)NO2
iv)Hydrogen sulphide gas.
Hetavi
Aug 23, 2015
#### How do we differentiate between a sodium sulphate and sodium nitrate?
Ans:
Heating Test & Reaction with Potassium dichromate solution:
Sodium nitrate:
On heating sodium nitrate will produce reddish brown coloured gas..
When t...
Uttam mazumder
Aug 28, 2013
#### When a gas 'A' is mixed with gas 'B' dense white fumes are seen and there is no other product (gas 'B' turns moist red litmus paper blue)i) What is the name of the gas 'A'ii) What is the naame of the product of reaction between gas B and A
i) the name of the gas 'A' is hydrogen chloride (HCl)
ii) the name of the product of reaction between gas 'B' and gas 'A' is ammonium chloride (NH4Cl).
Ketan Krishna Mishra
Mar 22, 2015
#### Explain fountain experiment?
Fountain experiment is used to demonstrate the solubility o...
Sarath Babu t
Jan 11, 2014
Rutvik gulavani
Nov 23, 2013
#### pls help me to do this as i have a project on this.... what is the 1.Method 2.Experiment...Pls... this not based on any topic in my syllabus so pls help in it ...as a student
No such method. You will need some data like weight, quality of the sugarcane to calculate the volume.
Sarath Babu t
Jan 11, 2014
#### Give reason
Because HCl (aq) is in a complet...
Shehrebano
Feb 9, 2015
#### Equation for magnesium treated with very dilute nitric acid?
Very dilute nitric aicd reaxts with magnesium to give its nitrate and hydrogen gas.
Mg + 2HNO3 → Mg(NO3)2 + H2
Sarath Babu t
Jan 11, 2014
#### Give reason
When hydrogen chloride gas is exposed to atmosphere it gives white fumes due to the formation of hydrochloric acid on reacting with atmospheric water vapour
Ketan Krishna Mishra
Mar 22, 2015
#### Why does the formation of HCl take place in diffused sunlight?
Hydrogen chloride gas can also be formed by reaction of moist hydro...
Dec 21, 2014
#### Name the gas in each of the following : i) A gas evolved when dil HCL is added to calcium carbonate. ii) The gas evolved on reaction of aluminium with boiling concentrated caustic alkali solution. iii) The gas produced when copper reacts with concentrated nitric acid. iv) The gas produced on reaction of dilute sulphuric acid with iron sulphide.
i)CO2
ii)H2
iii)NO2
iv)Hydrogen sulphide gas.
Hetavi
Aug 23, 2015
#### How do we differentiate between a sodium sulphate and sodium nitrate?
Ans:
Heating Test & Reaction with Potassium dichromate solution:
Sodium nitrate:
On heating sodium nitrate will produce reddish brown coloured gas..
When t...
Uttam mazumder
Aug 28, 2013
#### When a gas 'A' is mixed with gas 'B' dense white fumes are seen and there is no other product (gas 'B' turns moist red litmus paper blue)i) What is the name of the gas 'A'ii) What is the naame of the product of reaction between gas B and A
i) the name of the gas 'A' is hydrogen chloride (HCl)
ii) the name of the product of reaction between gas 'B' and gas 'A' is ammonium chloride (NH4Cl).
Filters
|
2021-12-05 10:29:27
|
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http://mathoverflow.net/questions/9083/how-many-l-values-determine-a-modular-form
|
# How many L-values determine a modular form?
Suppose $f$ and $g$ are two newforms of certain levels, weights etc. If we know that L(f,n)=L(g,n) for all sufficiently large $n$, can we conclude that $f=g$?
Same question when the forms have the same weight and $n$ runs over critical points.
-
The answer to the first question is "yes". The standard proof of the uniqueness of a Dirichlet series expansion actually generalizes to show the following.
Theorem. Suppose that $A(s) = \sum_n a_n n^{-s}$ and $B(s) = \sum_n b_n n^{-s}$ are Dirichlet series with coefficients $a_n, b_n$ bounded by a polynomial. If there exists a sequence of complex numbers $s_k$ with real part approaching infinity such that $A(s_k) = B(s_k)$ for all $k$, then $a_n = b_n$ for all $n$.
Proof (sketch). Proceed by induction. For $k$ big we have $A(s_k) = a_1 + O(2^{-\sigma_k})$ where $\sigma_k$ is the real part of $s_k$. Similarly, $B(s_k) = b_1 + O(2^{-\sigma_k})$. Since $A(s_k) = B(s_k)$, we conclude that $a_1 = b_1$. A similar argument shows $a_2 = b_2$, $a_3 = b_3$, etc.
-
This is an easier proof, so I am going to select this one as the right answer. – Idoneal Dec 31 '09 at 4:34
I think the answer to your first question is "yes." Suppose $L(f,s) = \sum_{m} a(m)m^{-s}$ and $L(g,s) = \sum_{m} b(m) m^{-s}$, and that $L(f,n) = L(g,n)$ for $n \geq n_0$, with $n_0$ large enough that the sums converge absolutely. Then pick an integer $M \geq n_0$ and weights $C_M(n)$ so that $\sum_{n \geq M} C_M(n) m^{-n}$ is $1$ if $m=M$, and $0$ otherwise. One can surely come up with such weights without too much trouble. Then $a(M) = \sum_{n \geq M} C_M(n) L(f,n) = \sum_{n \geq M} C_M(n) L(g,n) = b(M)$. It's not too hard to see that if two modular forms eventually have the same Fourier coefficients, then they are the same.
edit: After some further thought, I'm having trouble justifying the existence of those weights. I found a different solution that I'm posting as a separate answer.
-
Neat! In fact, this proves much more. – Idoneal Dec 17 '09 at 4:53
I think the answer to your second question is "no". For example if $k=2$ and $f$ and $g$ correspond to elliptic curves over $Q$ with positive rank, then the only critical point is $s=1$ and (at least conjecturally, and in sufficiently many cases provably) both $L$-functions will vanish at this point.
|
2013-12-13 13:19:46
|
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https://www.zbmath.org/authors/?q=ai%3Asheffield.scott
|
# zbMATH — the first resource for mathematics
## Sheffield, Scott
Compute Distance To:
Author ID: sheffield.scott Published as: Sheffield, Scott; Sheffield, S. Homepage: http://math.mit.edu/~sheffield/ External Links: IdRef · Google Scholar · MGP · Wikidata · dblp Awards: Clay Research Award (2017)
Documents Indexed: 62 Publications since 2002, including 2 Books
all top 5
#### Co-Authors
7 single-authored 16 Miller, Jason P. 7 Schramm, Oded 5 Levine, Lionel 5 Peres, Yuval 5 Wilson, David Bruce 4 Duplantier, Bertrand 4 Jerison, David S. 3 Gwynne, Ewain 3 Kenyon, Richard W. 3 Rhodes, Rémi 3 Vargas, Vincent 3 Werner, Wendelin 2 Alberts, Tom 2 Naor, Assaf 2 Watson, Samuel S. 1 Angel, Omer 1 Antunović, Tonći 1 Burdzy, Krzysztof 1 Dembo, Amir 1 Hammond, Alan 1 Hilário, Marcelo Richard 1 Holden, Nina 1 Kovchegov, Yevgeniy V. 1 Lawler, Gregory Francis 1 Lodhia, Asad 1 Louidor, Oren 1 Mörters, Peter 1 Nahmod, Andrea R. 1 Newman, Charles Michael 1 Okun’kov, Andreĭ Yur’evich 1 Rey-Bellet, Luc 1 Rolla, Leonardo T. 1 Sidoravicius, Vladas 1 Smart, Charles K. 1 Somersille, Stephanie J. 1 Spencer, Thomas C. 1 Staffilani, Gigliola 1 Stange, Katherine E. 1 Sun, Nike 1 Sun, Xin 1 Yadin, Ariel
all top 5
#### Serials
12 The Annals of Probability 8 Probability Theory and Related Fields 5 Duke Mathematical Journal 4 Electronic Journal of Probability 3 Communications in Mathematical Physics 3 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 3 Annals of Mathematics. Second Series 2 Communications on Pure and Applied Mathematics 2 Inventiones Mathematicae 2 Transactions of the American Mathematical Society 2 Journal of the American Mathematical Society 1 American Mathematical Monthly 1 Acta Mathematica 1 Journal of Combinatorial Theory. Series B 1 Mathematische Annalen 1 Proceedings of the American Mathematical Society 1 Communications in Partial Differential Equations 1 Mathematical Research Letters 1 Electronic Communications in Probability 1 Journal of the European Mathematical Society (JEMS) 1 Astérisque 1 IAS/Park City Mathematics Series 1 Probability Surveys 1 Forum of Mathematics, Pi 1 Journal de l’École Polytechnique – Mathématiques
all top 5
#### Fields
53 Probability theory and stochastic processes (60-XX) 20 Statistical mechanics, structure of matter (82-XX) 8 Combinatorics (05-XX) 6 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 5 Measure and integration (28-XX) 5 Partial differential equations (35-XX) 3 Quantum theory (81-XX) 2 General and overarching topics; collections (00-XX) 2 Functions of a complex variable (30-XX) 2 Functional analysis (46-XX) 2 General topology (54-XX) 2 Relativity and gravitational theory (83-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Group theory and generalizations (20-XX) 1 Real functions (26-XX) 1 Potential theory (31-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Geometry (51-XX) 1 Convex and discrete geometry (52-XX) 1 Differential geometry (53-XX) 1 Statistics (62-XX) 1 Computer science (68-XX)
#### Citations contained in zbMATH Open
60 Publications have been cited 2,042 times in 961 Documents Cited by Year
Dimers and amoebae. Zbl 1154.82007
Kenyon, Richard; Okounkov, Andrei; Sheffield, Scott
2006
Tug-of-war and the infinity Laplacian. Zbl 1206.91002
Peres, Yuval; Schramm, Oded; Sheffield, Scott; Wilson, David B.
2009
Liouville quantum gravity and KPZ. Zbl 1226.81241
Duplantier, Bertrand; Sheffield, Scott
2011
Gaussian free fields for mathematicians. Zbl 1132.60072
Sheffield, Scott
2007
Contour lines of the two-dimensional discrete Gaussian free field. Zbl 1210.60051
Schramm, Oded; Sheffield, Scott
2009
Tug-of-war with noise: a game-theoretic view of the $$p$$-Laplacian. Zbl 1206.35112
Peres, Yuval; Sheffield, Scott
2008
Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Zbl 1388.60144
Sheffield, Scott
2016
Imaginary geometry. I: Interacting SLEs. Zbl 1336.60162
Miller, Jason; Sheffield, Scott
2016
A contour line of the continuum Gaussian free field. Zbl 1331.60090
Schramm, Oded; Sheffield, Scott
2013
Imaginary geometry. IV: Interior rays, whole-plane reversibility, and space-filling trees. Zbl 1378.60108
Miller, Jason; Sheffield, Scott
2017
Exploration trees and conformal loop ensembles. Zbl 1170.60008
Sheffield, Scott
2009
Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Zbl 1271.60090
Sheffield, Scott; Werner, Wendelin
2012
Renormalization of critical Gaussian multiplicative chaos and KPZ relation. Zbl 1297.60033
Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent
2014
Critical Gaussian multiplicative chaos: convergence of the derivative martingale. Zbl 1306.60055
Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent
2014
Imaginary geometry. III: Reversibility of $$\mathrm{SLE}_\kappa$$ for $$\kappa \in (4,8)$$. Zbl 1393.60092
Miller, Jason; Sheffield, Scott
2016
Imaginary geometry. II: Reversibility of $$\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$$ for $$\kappa\in(0,4)$$. Zbl 1344.60078
Miller, Jason; Sheffield, Scott
2016
Quantum Loewner evolution. Zbl 1364.82023
Miller, Jason; Sheffield, Scott
2016
Random surfaces. Zbl 1104.60002
Sheffield, Scott
2005
Harmonic explorer and its convergence to $$\text{SLE}_4$$. Zbl 1095.60007
Schramm, Oded; Sheffield, Scott
2005
Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Zbl 1108.46012
Naor, Assaf; Peres, Yuval; Schramm, Oded; Sheffield, Scott
2006
Quantum gravity and inventory accumulation. Zbl 1359.60120
Sheffield, Scott
2016
Liouville quantum gravity and the Brownian map. I: The $$\text{QLE}(8/3,0)$$ metric. Zbl 1437.83042
Miller, Jason; Sheffield, Scott
2020
Conformal radii for conformal loop ensembles. Zbl 1187.82044
Schramm, Oded; Sheffield, Scott; Wilson, David B.
2009
A natural parametrization for the Schramm-Loewner evolution. Zbl 1234.60087
Lawler, Gregory F.; Sheffield, Scott
2011
Absolute continuity of Brownian bridges under certain gauge transformations. Zbl 1250.60018
Nahmod, Andrea R.; Rey-Bellet, Luc; Sheffield, Scott; Staffilani, Gigliola
2011
Logarithmic fluctuations for internal DLA. Zbl 1237.60037
Jerison, David; Levine, Lionel; Sheffield, Scott
2012
Fractional Gaussian fields: a survey. Zbl 1334.60055
Lodhia, Asad; Sheffield, Scott; Sun, Xin; Watson, Samuel S.
2016
Internal DLA and the Gaussian free field. Zbl 1296.60113
Jerison, David; Levine, Lionel; Sheffield, Scott
2014
Vector-valued optimal Lipschitz extensions. Zbl 1233.35068
Sheffield, Scott; Smart, Charles K.
2012
Bipolar orientations on planar maps and $$\mathrm{SLE}_{12}$$. Zbl 1466.60170
Kenyon, Richard; Miller, Jason; Sheffield, Scott; Wilson, David B.
2019
Hausdorff dimension of the SLE curve intersected with the real line. Zbl 1192.60025
Alberts, Tom; Sheffield, Scott
2008
Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition. Zbl 1268.35065
Antunović, Tonći; Peres, Yuval; Sheffield, Scott; Somersille, Stephanie
2012
Internal DLA in higher dimensions. Zbl 1290.60051
Jerison, David; Levine, Lionel; Sheffield, Scott
2013
Log-correlated Gaussian fields: an overview. Zbl 1366.60023
Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent
2017
Harmonic functions on mated-CRT maps. Zbl 1466.60090
Gwynne, Ewain; Miller, Jason; Sheffield, Scott
2019
Random-turn hex and other selection games. Zbl 1153.91012
Peres, Yuval; Schramm, Oded; Sheffield, Scott; Wilson, David B.
2007
Liouville quantum gravity spheres as matings of finite-diameter trees. Zbl 1448.60168
Miller, Jason; Sheffield, Scott
2019
Large deviations of Markov chains indexed by random trees. Zbl 1078.60020
Dembo, Amir; Mörters, Peter; Sheffield, Scott
2005
Dimers, tilings and trees. Zbl 1055.05032
Kenyon, Richard W.; Sheffield, Scott
2004
Schramm’s proof of Watts’ formula. Zbl 1238.60089
Sheffield, Scott; Wilson, David B.
2011
CLE percolations. Zbl 1390.60356
Miller, Jason; Sheffield, Scott; Werner, Wendelin
2017
The Tutte embedding of the Poisson-Voronoi tessellation of the Brownian disk converges to $$\sqrt{8/3}$$-Liouville quantum gravity. Zbl 1441.60015
Gwynne, Ewain; Miller, Jason; Sheffield, Scott
2020
The covariant measure of SLE on the boundary. Zbl 1244.60081
Alberts, Tom; Sheffield, Scott
2011
Power law Pólya’s urn and fractional Brownian motion. Zbl 1311.60044
Hammond, Alan; Sheffield, Scott
2013
Absolutely minimal Lipschitz extension of tree-valued mappings. Zbl 1276.46062
Naor, Assaf; Sheffield, Scott
2012
Liouville quantum gravity and the Brownian map III: the conformal structure is determined. Zbl 07334605
Miller, Jason; Sheffield, Scott
2021
Ribbon tilings and multidimensional height functions. Zbl 1018.68057
Sheffield, Scott
2002
Strong path convergence from Loewner driving function convergence. Zbl 1255.60148
Sheffield, Scott; Sun, Nike
2012
Non-simple SLE curves are not determined by their range. Zbl 07174683
Miller, Jason; Sheffield, Scott; Werner, Wendelin
2020
Deterministic approximations of random reflectors. Zbl 1408.37063
Angel, Omer; Burdzy, Krzysztof; Sheffield, Scott
2013
An axiomatic characterization of the Brownian map. (Une caractérisation axiomatique de la carte Brownienne.) Zbl 07329548
Miller, Jason; Sheffield, Scott
2021
Simple CLE in doubly connected domains. Zbl 1370.60141
Sheffield, Scott; Watson, Samuel S.; Wu, Hao
2017
Internal DLA for cylinders. Zbl 1334.60210
Jerison, David; Levine, Lionel; Sheffield, Scott
2014
The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity. Zbl 07367520
Gwynne, Ewain; Miller, Jason; Sheffield, Scott
2021
Linear speed large deviations for percolation clusters. Zbl 1060.60097
Kovchegov, Yevgeniy; Sheffield, Scott
2003
Uniqueness of maximal entropy measure on essential spanning forests. Zbl 1106.60012
Sheffield, Scott
2006
Tricolor percolation and random paths in 3D. Zbl 1307.60147
2014
A duality principle for selection games. Zbl 1401.91021
Levine, Lionel; Sheffield, Scott; Stange, Katherine E.
2013
Scaling limits of the Schelling model. Zbl 1442.60100
Holden, Nina; Sheffield, Scott
2020
Gaussian free field light cones and $$\text{SLE}_{\kappa}(\rho)$$. Zbl 1453.60142
Miller, Jason; Sheffield, Scott
2019
Liouville quantum gravity and the Brownian map III: the conformal structure is determined. Zbl 07334605
Miller, Jason; Sheffield, Scott
2021
An axiomatic characterization of the Brownian map. (Une caractérisation axiomatique de la carte Brownienne.) Zbl 07329548
Miller, Jason; Sheffield, Scott
2021
The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity. Zbl 07367520
Gwynne, Ewain; Miller, Jason; Sheffield, Scott
2021
Liouville quantum gravity and the Brownian map. I: The $$\text{QLE}(8/3,0)$$ metric. Zbl 1437.83042
Miller, Jason; Sheffield, Scott
2020
The Tutte embedding of the Poisson-Voronoi tessellation of the Brownian disk converges to $$\sqrt{8/3}$$-Liouville quantum gravity. Zbl 1441.60015
Gwynne, Ewain; Miller, Jason; Sheffield, Scott
2020
Non-simple SLE curves are not determined by their range. Zbl 07174683
Miller, Jason; Sheffield, Scott; Werner, Wendelin
2020
Scaling limits of the Schelling model. Zbl 1442.60100
Holden, Nina; Sheffield, Scott
2020
Bipolar orientations on planar maps and $$\mathrm{SLE}_{12}$$. Zbl 1466.60170
Kenyon, Richard; Miller, Jason; Sheffield, Scott; Wilson, David B.
2019
Harmonic functions on mated-CRT maps. Zbl 1466.60090
Gwynne, Ewain; Miller, Jason; Sheffield, Scott
2019
Liouville quantum gravity spheres as matings of finite-diameter trees. Zbl 1448.60168
Miller, Jason; Sheffield, Scott
2019
Gaussian free field light cones and $$\text{SLE}_{\kappa}(\rho)$$. Zbl 1453.60142
Miller, Jason; Sheffield, Scott
2019
Imaginary geometry. IV: Interior rays, whole-plane reversibility, and space-filling trees. Zbl 1378.60108
Miller, Jason; Sheffield, Scott
2017
Log-correlated Gaussian fields: an overview. Zbl 1366.60023
Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent
2017
CLE percolations. Zbl 1390.60356
Miller, Jason; Sheffield, Scott; Werner, Wendelin
2017
Simple CLE in doubly connected domains. Zbl 1370.60141
Sheffield, Scott; Watson, Samuel S.; Wu, Hao
2017
Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Zbl 1388.60144
Sheffield, Scott
2016
Imaginary geometry. I: Interacting SLEs. Zbl 1336.60162
Miller, Jason; Sheffield, Scott
2016
Imaginary geometry. III: Reversibility of $$\mathrm{SLE}_\kappa$$ for $$\kappa \in (4,8)$$. Zbl 1393.60092
Miller, Jason; Sheffield, Scott
2016
Imaginary geometry. II: Reversibility of $$\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$$ for $$\kappa\in(0,4)$$. Zbl 1344.60078
Miller, Jason; Sheffield, Scott
2016
Quantum Loewner evolution. Zbl 1364.82023
Miller, Jason; Sheffield, Scott
2016
Quantum gravity and inventory accumulation. Zbl 1359.60120
Sheffield, Scott
2016
Fractional Gaussian fields: a survey. Zbl 1334.60055
Lodhia, Asad; Sheffield, Scott; Sun, Xin; Watson, Samuel S.
2016
Renormalization of critical Gaussian multiplicative chaos and KPZ relation. Zbl 1297.60033
Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent
2014
Critical Gaussian multiplicative chaos: convergence of the derivative martingale. Zbl 1306.60055
Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent
2014
Internal DLA and the Gaussian free field. Zbl 1296.60113
Jerison, David; Levine, Lionel; Sheffield, Scott
2014
Internal DLA for cylinders. Zbl 1334.60210
Jerison, David; Levine, Lionel; Sheffield, Scott
2014
Tricolor percolation and random paths in 3D. Zbl 1307.60147
2014
A contour line of the continuum Gaussian free field. Zbl 1331.60090
Schramm, Oded; Sheffield, Scott
2013
Internal DLA in higher dimensions. Zbl 1290.60051
Jerison, David; Levine, Lionel; Sheffield, Scott
2013
Power law Pólya’s urn and fractional Brownian motion. Zbl 1311.60044
Hammond, Alan; Sheffield, Scott
2013
Deterministic approximations of random reflectors. Zbl 1408.37063
Angel, Omer; Burdzy, Krzysztof; Sheffield, Scott
2013
A duality principle for selection games. Zbl 1401.91021
Levine, Lionel; Sheffield, Scott; Stange, Katherine E.
2013
Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Zbl 1271.60090
Sheffield, Scott; Werner, Wendelin
2012
Logarithmic fluctuations for internal DLA. Zbl 1237.60037
Jerison, David; Levine, Lionel; Sheffield, Scott
2012
Vector-valued optimal Lipschitz extensions. Zbl 1233.35068
Sheffield, Scott; Smart, Charles K.
2012
Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition. Zbl 1268.35065
Antunović, Tonći; Peres, Yuval; Sheffield, Scott; Somersille, Stephanie
2012
Absolutely minimal Lipschitz extension of tree-valued mappings. Zbl 1276.46062
Naor, Assaf; Sheffield, Scott
2012
Strong path convergence from Loewner driving function convergence. Zbl 1255.60148
Sheffield, Scott; Sun, Nike
2012
Liouville quantum gravity and KPZ. Zbl 1226.81241
Duplantier, Bertrand; Sheffield, Scott
2011
A natural parametrization for the Schramm-Loewner evolution. Zbl 1234.60087
Lawler, Gregory F.; Sheffield, Scott
2011
Absolute continuity of Brownian bridges under certain gauge transformations. Zbl 1250.60018
Nahmod, Andrea R.; Rey-Bellet, Luc; Sheffield, Scott; Staffilani, Gigliola
2011
Schramm’s proof of Watts’ formula. Zbl 1238.60089
Sheffield, Scott; Wilson, David B.
2011
The covariant measure of SLE on the boundary. Zbl 1244.60081
Alberts, Tom; Sheffield, Scott
2011
Tug-of-war and the infinity Laplacian. Zbl 1206.91002
Peres, Yuval; Schramm, Oded; Sheffield, Scott; Wilson, David B.
2009
Contour lines of the two-dimensional discrete Gaussian free field. Zbl 1210.60051
Schramm, Oded; Sheffield, Scott
2009
Exploration trees and conformal loop ensembles. Zbl 1170.60008
Sheffield, Scott
2009
Conformal radii for conformal loop ensembles. Zbl 1187.82044
Schramm, Oded; Sheffield, Scott; Wilson, David B.
2009
Tug-of-war with noise: a game-theoretic view of the $$p$$-Laplacian. Zbl 1206.35112
Peres, Yuval; Sheffield, Scott
2008
Hausdorff dimension of the SLE curve intersected with the real line. Zbl 1192.60025
Alberts, Tom; Sheffield, Scott
2008
Gaussian free fields for mathematicians. Zbl 1132.60072
Sheffield, Scott
2007
Random-turn hex and other selection games. Zbl 1153.91012
Peres, Yuval; Schramm, Oded; Sheffield, Scott; Wilson, David B.
2007
Dimers and amoebae. Zbl 1154.82007
Kenyon, Richard; Okounkov, Andrei; Sheffield, Scott
2006
Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Zbl 1108.46012
Naor, Assaf; Peres, Yuval; Schramm, Oded; Sheffield, Scott
2006
Uniqueness of maximal entropy measure on essential spanning forests. Zbl 1106.60012
Sheffield, Scott
2006
Random surfaces. Zbl 1104.60002
Sheffield, Scott
2005
Harmonic explorer and its convergence to $$\text{SLE}_4$$. Zbl 1095.60007
Schramm, Oded; Sheffield, Scott
2005
Large deviations of Markov chains indexed by random trees. Zbl 1078.60020
Dembo, Amir; Mörters, Peter; Sheffield, Scott
2005
Dimers, tilings and trees. Zbl 1055.05032
Kenyon, Richard W.; Sheffield, Scott
2004
Linear speed large deviations for percolation clusters. Zbl 1060.60097
Kovchegov, Yevgeniy; Sheffield, Scott
2003
Ribbon tilings and multidimensional height functions. Zbl 1018.68057
Sheffield, Scott
2002
all top 5
#### Cited by 914 Authors
39 Sheffield, Scott 36 Miller, Jason P. 28 Rossi, Julio Daniel 27 Gwynne, Ewain 23 Rhodes, Rémi 23 Vargas, Vincent 20 Manfredi, Juan J. 17 Naor, Assaf 15 Parviainen, Mikko 15 Peres, Yuval 14 Sun, Xin 13 Katzourakis, Nikolaos I. 12 Ding, Jian 12 Garban, Christophe 12 Toninelli, Fabio Lucio 12 Werner, Wendelin 11 Borodin, Alexei 11 Camia, Federico 11 Lawler, Gregory Francis 11 Saksman, Eero 10 Holden, Nina 10 Levine, Lionel 10 Schramm, Oded 10 Sepúlveda, Avelio 10 Viklund, Fredrik Johansson 10 Webb, Christian 10 Wilson, David Bruce 9 Gorin, Vadim 8 Aru, Juhan 8 Curien, Nicolas 8 Dubédat, Julien 8 Kupiainen, Antti 8 Liu, Qing 7 Boutillier, Cédric 7 Cipriani, Alessandra 7 Duplantier, Bertrand 7 Kenyon, Richard W. 7 Qian, Wei 7 Reshetikhin, Nikolai Yu. 7 Zhan, Dapeng 7 Zhou, Yuan 6 Armstrong, Scott N. 6 Berestycki, Nathanaël 6 Biskup, Marek 6 Chhita, Sunil 6 Corwin, Ivan 6 Hazra, Rajat Subhra 6 Kytölä, Kalle 6 Laslier, Benoît 6 Pronko, Andrei G. 6 Smart, Charles K. 6 Smits, Robert G. 6 Tzvetkov, Nikolay 6 Yamazaki, Masahito 6 Zeitouni, Ofer 5 Beliaev, Dmitri B. 5 Ciucu, Mihai 5 Colomo, Filippo 5 Crasta, Graziano 5 de Tilière, Béatrice 5 Di Francesco, Philippe 5 Duits, Maurice 5 Flores, Steven M. 5 Fragalà, Ilaria 5 Franco, Sebastián 5 Jin, Xiong 5 Kleban, Peter 5 Lambert, Gaultier 5 Lewicka, Marta 5 Liu, Fang 5 Luiro, Hannes 5 Mohammed, Ahmed 5 Oh, Tadahiro 5 Ostrovskii, Dmitrii M. 5 Peltola, Eveliina 5 Pérez-Llanos, Mayte 5 Pfeffer, Joshua 5 Powell, Ellen 5 Wang, Yilin 4 Alberts, Tom 4 Ang, Morris 4 Arroyo, Angel 4 Barral, Julien 4 Blanc, Pablo 4 Budd, Timothy G. 4 Bufetov, Alekseĭ Igor’evich 4 Caputo, Pietro 4 Charro, Fernando 4 Cimasoni, David 4 Deblassie, Dante 4 Del Pezzo, Leandro M. 4 Duminil-Copin, Hugo 4 Elmoataz, Abderrahim 4 Falconet, Hugo 4 Fyodorov, Yan V. 4 Ganguly, Shirshendu 4 Guitter, Emmanuel 4 Izyurov, Konstantin 4 Johansson, Kurt 4 Juutinen, Petri ...and 814 more Authors
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#### Cited in 191 Serials
108 Communications in Mathematical Physics 83 Probability Theory and Related Fields 72 The Annals of Probability 47 Journal of Statistical Physics 29 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 19 Electronic Journal of Probability 18 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 17 Duke Mathematical Journal 17 Proceedings of the American Mathematical Society 16 Transactions of the American Mathematical Society 14 Advances in Mathematics 13 Inventiones Mathematicae 13 Stochastic Processes and their Applications 12 Journal of Differential Equations 12 Calculus of Variations and Partial Differential Equations 12 Annales Henri Poincaré 11 Journal of High Energy Physics 10 Communications on Pure and Applied Mathematics 10 Journal of Mathematical Physics 10 Mathematische Annalen 10 The Annals of Applied Probability 9 Journal of Statistical Mechanics: Theory and Experiment 8 Journal of Mathematical Analysis and Applications 8 Journal of Functional Analysis 8 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 8 Journal of the American Mathematical Society 8 Electronic Communications in Probability 8 Communications on Pure and Applied Analysis 8 Probability Surveys 7 Annals of Mathematics. Second Series 7 Journal of the European Mathematical Society (JEMS) 6 Archive for Rational Mechanics and Analysis 6 Israel Journal of Mathematics 6 Letters in Mathematical Physics 6 Acta Mathematica 6 Statistics & Probability Letters 6 Journal of Theoretical Probability 6 Geometric and Functional Analysis. GAFA 6 Communications in Partial Differential Equations 6 Journal de Mathématiques Pures et Appliquées. Neuvième Série 6 Potential Analysis 6 Communications in Contemporary Mathematics 5 Annales de l’Institut Fourier 5 Annali di Matematica Pura ed Applicata. Serie Quarta 5 Bulletin of the American Mathematical Society. New Series 5 Selecta Mathematica. New Series 5 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 5 Annales de l’Institut Henri Poincaré D. Combinatorics, Physics and their Interactions (AIHPD) 4 Nuclear Physics. B 4 Journal of Combinatorial Theory. Series A 4 The Journal of Geometric Analysis 4 NoDEA. Nonlinear Differential Equations and Applications 4 Bernoulli 4 Discrete and Continuous Dynamical Systems 4 Journal de l’École Polytechnique – Mathématiques 3 Publications Mathématiques 3 Discrete & Computational Geometry 3 Journal of Mathematical Sciences (New York) 3 The Electronic Journal of Combinatorics 3 Acta Mathematica Sinica. English Series 3 Advanced Nonlinear Studies 3 Comptes Rendus. Mathématique. Académie des Sciences, Paris 3 Complex Variables and Elliptic Equations 3 ALEA. Latin American Journal of Probability and Mathematical Statistics 3 Journal of Physics A: Mathematical and Theoretical 3 Forum of Mathematics, Pi 3 Forum of Mathematics, Sigma 2 Advances in Applied Probability 2 Discrete Mathematics 2 Journal of Computational Physics 2 Nonlinearity 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 Reviews in Mathematical Physics 2 Journal of Geometry and Physics 2 Applied Mathematics and Optimization 2 Journal of the London Mathematical Society. Second Series 2 Le Matematiche 2 Mathematische Zeitschrift 2 Mathematika 2 European Journal of Applied Mathematics 2 SIAM Journal on Mathematical Analysis 2 Journal of Mathematical Imaging and Vision 2 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 2 Annales Academiae Scientiarum Fennicae. Mathematica 2 Geometry & Topology 2 Bulletin of the Brazilian Mathematical Society. New Series 2 Journal of the Institute of Mathematics of Jussieu 2 S$$\vec{\text{e}}$$MA Journal 2 Analysis and Geometry in Metric Spaces 2 Advances in Nonlinear Analysis 1 Modern Physics Letters B 1 International Journal of Modern Physics A 1 American Mathematical Monthly 1 Applicable Analysis 1 Computers & Mathematics with Applications 1 Information Processing Letters 1 Journal d’Analyse Mathématique 1 Mathematical Notes 1 Mathematische Semesterberichte 1 Physica A ...and 91 more Serials
all top 5
#### Cited in 56 Fields
526 Probability theory and stochastic processes (60-XX) 281 Statistical mechanics, structure of matter (82-XX) 238 Partial differential equations (35-XX) 118 Combinatorics (05-XX) 96 Quantum theory (81-XX) 57 Functions of a complex variable (30-XX) 49 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 45 Measure and integration (28-XX) 45 Calculus of variations and optimal control; optimization (49-XX) 38 Functional analysis (46-XX) 36 Potential theory (31-XX) 35 Dynamical systems and ergodic theory (37-XX) 31 Differential geometry (53-XX) 29 Algebraic geometry (14-XX) 28 Relativity and gravitational theory (83-XX) 23 Convex and discrete geometry (52-XX) 20 Linear and multilinear algebra; matrix theory (15-XX) 20 Numerical analysis (65-XX) 18 Operator theory (47-XX) 18 Computer science (68-XX) 15 Special functions (33-XX) 13 General topology (54-XX) 13 Global analysis, analysis on manifolds (58-XX) 12 Number theory (11-XX) 11 Real functions (26-XX) 11 Harmonic analysis on Euclidean spaces (42-XX) 10 Group theory and generalizations (20-XX) 9 Several complex variables and analytic spaces (32-XX) 9 Ordinary differential equations (34-XX) 8 Statistics (62-XX) 7 Difference and functional equations (39-XX) 6 Associative rings and algebras (16-XX) 6 Topological groups, Lie groups (22-XX) 6 Abstract harmonic analysis (43-XX) 6 Geometry (51-XX) 6 Fluid mechanics (76-XX) 6 Operations research, mathematical programming (90-XX) 4 Nonassociative rings and algebras (17-XX) 4 Manifolds and cell complexes (57-XX) 4 Mechanics of particles and systems (70-XX) 3 History and biography (01-XX) 3 Mathematical logic and foundations (03-XX) 3 Integral equations (45-XX) 3 Biology and other natural sciences (92-XX) 3 Information and communication theory, circuits (94-XX) 2 General and overarching topics; collections (00-XX) 2 Field theory and polynomials (12-XX) 2 Commutative algebra (13-XX) 2 Classical thermodynamics, heat transfer (80-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Approximations and expansions (41-XX) 1 Integral transforms, operational calculus (44-XX) 1 Algebraic topology (55-XX) 1 Optics, electromagnetic theory (78-XX) 1 Systems theory; control (93-XX) 1 Mathematics education (97-XX)
#### Wikidata Timeline
The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.
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2021-09-18 08:54:04
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https://bookdown.org/pkaldunn/Book/standard-deviation-vs-standard-error.html
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## 18.5 Standard deviation vs. standard error
Even experienced researchers confuse the meaning and the usage of the terms standard deviation and standard error , so understanding the difference is important.
The standard deviation, in general, quantifies the amount of variation in any variable. Without further qualification, the standard deviation quantifies how much individual observations vary from individual to individual (for quantitative data).
The standard error is a standard deviation that quantifies how much a sample statistic varies from sample to sample.
Crucially, the standard error is a standard deviation, but has a special name to indicate that it is the standard deviation of something very specific.
Any numerical quantity estimated from a sample (a statistic) can vary from sample to sample, and so has sampling variation, a sampling distribution, and hence a standard error:
• the sample mean $$\bar{x}$$;
• the sample proportion $$\hat{p}$$;
• the sample odds ratio;
• the sample median;
• the sample standard deviation $$s$$;
• etc.
The standard error is often abbreviated to ‘SE’ or ‘s.e.’
For example, the ‘standard error of the sample mean’ is written as $$\text{s.e.}(\bar{x})$$, and the ‘standard error of the sample proportion’ is written as $$\text{s.e.}(\hat{p})$$.
### References
Ko W-R, Hung W-T, Chang H-C, Lin L-Y. Inappropriate use of standard error of the mean when reporting variability of study samples: A critical evaluation of four selected journals of obstetrics and gynecology. Taiwanese Journal of Obstetrics and Gynecology. Elsevier; 2014;53(1):26–9.
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2021-09-21 05:53:31
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https://ask.openstack.org/en/answers/117750/revisions/
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It was because the max key length is 767 byte, and repository_id was varchar(250), if we use utf8mbt4, then the actural length might be 1000 which is exceed the 767 limit. Change to utf8, 250*3=750 was not exceed the 767 limit, so it works.
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2019-05-24 10:13:35
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https://scholars.bgu.ac.il/display/n5938103
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# On some topological deformation of stationary spacetimes Academic Article
•
• Overview
•
### abstract
• We define a completely new space-time starting from the well known Schwarzschild Space time by defining a new polar angle $\phi '= \phi - \omega t$ and then redefining the periodicity: instead of demanding that the original angle be periodic, we demand that the new angle $\phi'$ be periodic, with period $2\pi$. This defines the "topologically rotating Schwarzchild space", which is physically different from the standard Schwarzschild space. For this space, we work out some properties of the geodesics and related properties.This method of generating solutions can be used also for the Reissner-Nordstrom case, both in the case of Reissner-Nordstrom Black hole as well as in the case where there are no horizons, the supercharged case. Horizon shall exist in this case, but with a real singularity, not removable one by a transformation in coordinate at the radius of the horizon of the original metric. This solution should be used as an external solution rather than the internal one. Another topic to notice is that the improper coordinate transformation that we consider introduces closed time like curves. This is a common effect in rotating spacetimes, noticeable the Godel universe and others. The noticeable topic is that the improper coordinate transformation introduces closed time like curves which we can possibly find here too.
### publication date
• January 1, 2015
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2020-01-25 16:25:06
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http://web.newworldencyclopedia.org/entry/Angular_momentum
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# Angular momentum
This gyroscope remains upright while spinning due to its angular momentum.
In physics, the angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque. In particular, if a point mass rotates about an axis, then the angular momentum with respect to a point on the axis is related to the mass of the object, its velocity, and its distance from the axis.
The concept of angular momentum is important in physics because it is a conserved quantity: a system's angular momentum stays constant unless an external torque acts on it. Torque is the rate at which angular momentum is transferred in or out of the system. When a rigid body rotates, its resistance to a change in its rotational motion is measured by its moment of inertia.
The conservation of angular momentum explains many phenomena in human activities and nature. For instance, it explains why an ice skater spins faster when drawing her arms close to her body, and slower when stretching her arms outward. It also explains why a compact star, such as a white dwarf, spins very fast, whereas the large star from which it was formed rotated much more slowly.
Knowledge of the angular momentum of an object also has important applications in engineering. For example, the kinetic energy stored in a rotating object such as a flywheel is proportional to the square of the angular momentum.
## Angular momentum in classical mechanics
Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system.
### Fundamental equation
The angular momentum of an object or particle that is moving around some origin (reference point) is defined by the following mathematical equation:
$\mathbf{L}=\mathbf{r}\times\mathbf{p}$
where:
$\mathbf{L}$ is the angular momentum of the object or particle,
$\mathbf{r}$ is the position of the object or particle expressed as a displacement vector from the origin,
$\mathbf{p}$ is the linear momentum of the object or particle, and
$\times\,$ is the vector cross product.
The derived SI units for angular momentum are newtonmeterseconds, or N•m•s (kgm2s-1).
Because of the cross product, L is a vector perpendicular to both the radial vector r and the momentum vector p.
If a system consists of several particles moving around the same origin, the total angular momentum can be obtained by adding all the angular momenta of the constituent particles. Angular momentum can also be calculated by multiplying the square of the displacement r, the mass of the particle and the angular velocity.
### Angular momentum of group of particles
It is often convenient to consider the angular momentum of a collection of particles about their center of mass, because this simplifies the mathematics considerably. The angular momentum of a collection of particles is the sum of the angular momenta of each particle:
$\mathbf{L}=\sum_i \mathbf{R}_i\times m_i \mathbf{V}_i$
where $R_i$ is the distance of particle i from the reference point, $m_i$ is its mass, and $V_i$ is its velocity. The center of mass is defined by:
$\mathbf{R}=\frac{1}{M}\sum_i m_i \mathbf{R}_i$
where $M$ is the total mass of all the particles.
If we define $\mathbf{r}_i$ as the displacement of particle i from the center of mass, and $\mathbf{v}_i$ as the velocity of particle i with respect to the center of mass, then we have
$\mathbf{R}_i=\mathbf{R}+\mathbf{r}_i\,$ and $\mathbf{V}_i=\mathbf{V}+\mathbf{v}_i\,$
In this case, the total angular momentum is:
$\mathbf{L}=\sum_i (\mathbf{R}+\mathbf{r}_i)\times m_i (\mathbf{V}+\mathbf{v}_i) = \left(\mathbf{R}\times M\mathbf{V}\right) + \left(\sum_i \mathbf{r}_i\times m_i \mathbf{v}_i\right)$
The first term is just the angular momentum of the center of mass. It is the same angular momentum one would obtain if there were just one particle of mass M moving at velocity V, located at the center of mass. The second term is the angular momentum that is the result of the particles spinning about their center of mass. The second term can be further simplified if the particles form a rigid body.
### Fixed axis of rotation
For many applications where one is concerned about rotation around a single axis, it is sufficient to discard the pseudovector nature of angular momentum and treat it like a scalar quantity. It is given a positive value for counterclockwise rotations, and a negative value for clockwise rotations. To do this, one takes the definition of the cross product and discards the unit vector, so that angular momentum becomes:
$L = |\mathbf{r}||\mathbf{p}|\sin \theta_{r,p}$
where θr,p is the angle between r and p, measured from r to p. (One needs to make this distinction because without it, the sign of the cross product would be meaningless.) From the above, it is possible to reformulate the definition to either of the following:
$L = \pm|\mathbf{p}||\mathbf{r}_{\perp}|$
where r is called the perpendicular lever arm distance to p.
For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector:
$\mathbf{L}= I \mathbf{\omega}$
where
$I\,$ is the moment of inertia of the object
$\mathbf{\omega}$ is the angular velocity.
### Conservation of angular momentum
The torque caused by the two opposing forces Fg and -Fg causes a change in the angular momentum L in the direction of that torque (since torque is the time derivative of angular momentum). This causes the top to move back upright.
In a closed system, angular momentum is constant. This conservation law follows mathematically from what is called the "continuous directional symmetry" of space—that is, no direction in space is any different from any other direction.
The change of angular momentum over time is called torque. Mathematically, it is expressed as the time derivative of angular momentum, as follows:
$\tau = \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \mathbf{r} \times \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \mathbf{r} \times \mathbf{F}$
When the angular momentum is a constant (for a closed system), the mathematical expression for that is equivalent to the mathematical equation showing that the external torque acting on the system is zero. This equivalence may be written as follows:
$\mathbf{L}_{\mathrm{system}} = \mathrm{constant} \leftrightarrow \sum \tau_{\mathrm{ext}} = 0$
where $\tau_{ext}$ is any torque applied to the system of particles.
### Angular momentum of planetary orbits
For a planet in orbit, the angular momentum is distributed between the spin of the planet itself and the angular momentum of its orbit:
$\mathbf{L}_{\mathrm{total}} = \mathbf{L}_{\mathrm{spin}} + \mathbf{L}_{\mathrm{orbit}}$;
If a planet appears to rotate slower than expected, astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its satellite in order to be conserved.
## Usefulness
The conservation of angular momentum is used extensively in analyzing what is called central force motion. If the net force on some body is always directed toward a fixed point, the center, then there is no torque on the body with respect to the center, and the angular momentum of the body about the center is constant. Constant angular momentum is extremely useful when dealing with the orbits of planets and satellites. This concept was also used for the Bohr model of the atom.
The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation (or close to her body). By bringing part of her body mass closer to the axis, she decreases her body's moment of inertia. Because angular momentum is constant in the absence of external torques, the angular velocity (rotational speed) of the skater has to increase.
The same phenomenon explains the extremely fast spin of compact stars (like white dwarfs and neutron stars) and black holes, when they are formed out of much larger and slower rotating stars. (Decreasing the size of an object 104 times results in increasing its angular velocity by a factor of 108).
## Angular momentum in quantum mechanics
To explain the behavior of subatomic particles, the theory of quantum mechanics indicates that the angular momentum of a particle is "quantized." In other words, the angular momentum of a subatomic particle does not vary continuously, but it changes in "quantum leaps" between certain allowed values. When a subatomic particle is moving through space, its angular momentum due to this motion is always a whole-number multiple of a constant denoted as $\hbar$ ("h-bar").[1]
Experiments show that most subatomic particles also have a permanent, built-in angular momentum that is not due to their motion through space. This "spin" angular momentum comes in units of $\hbar/2$. For example, an electron has a spin angular momentum of $\hbar/2$.
### Basic definition
As noted above, the classical definition of angular momentum can be written as:
$\mathbf{L}=\mathbf{r}\times\mathbf{p}$
The value of angular momentum depends on six numbers: $r_x$, $r_y$, $r_z$, $p_x$, $p_y$, and $p_z$.
When dealing with particles on the subatomic scale, the Heisenberg uncertainty principle tells us that it is not possible for all six of these numbers to be measured simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis.
Mathematically, angular momentum in quantum mechanics is defined in the same way as momentum—not as a quantity but as an operator on the wave function:
$\mathbf{L}=\mathbf{r}\times\mathbf{p}$
where r and p are the position and momentum operators respectively. In particular, for a single particle with no electric charge and no spin, the angular momentum operator can be written in the position basis as
$\mathbf{L}=-i\hbar(\mathbf{r}\times\nabla)$
where $\nabla$ is the gradient operator, read as "del," "grad," or "nabla." This is a commonly encountered form of the angular momentum operator, though not the most general one.
## Notes
1. $\hbar$ is defined as Planck's constant $h$ divided by 2π.
## References
• Brink, D. M., and G. R. Satchler. 1993. Angular momentum. Oxford: Clarendon Press. ISBN 0198517599
• Edmonds, A. R. 1974. Angular momentum in quantum mechanics. Investigations in physics, 4. Princeton, N.J.: Princeton University Press. ISBN 0691079129
• Serway, Raymond A., and John W. Jewett. 2004. Physics for scientists and engineers. Belmont, CA: Thomson-Brooks/Cole. ISBN 0534408427
• Tipler, Paul. 1998. Physics for Scientists and Engineers: Vol. 1: Mechanics, Oscillations and Waves, Thermodynamics, 4th ed. W. H. Freeman. ISBN 1-57259-492-6
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2019-03-22 07:08:23
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https://math.stackexchange.com/questions/931076/improve-liouvilles-theorem-in-evans-pde
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# Improve Liouville's Theorem in Evans ' PDE
Here is Liouville's Theorem
Suppose that $u \colon \mathbb{R}^n \to \mathbb{R}$ is harmonic and $u \geq 0$. Prove that $u$ is constant. (In this problem , instead of $u$ is bounded now $u \geq 0$ .)
Harnack 's inequality proof in Evans :
Can I use Harnack 's inequality and show that $u(x) < 2^n u(0)$ ? How can I create an upper bound like the above proof that can converge to 0 as r goes to infinity ?
If you have Harnack's inequality then you are essentially done already. Here's a way to continue:
Let $x$ be arbitrary, $\| x \| = r$, and $R>r$. Then
$$\frac{1-r/R}{1+(r/R)^{n-1}} f(0) \leq f(x) \leq \frac{1+r/R}{1-(r/R)^{n-1}} f(0).$$
Now send $R \to \infty$.
• Unless I'm mistaken, I don't think that inequality is done explicitly in Evans' book; for the record, it can be proved using Poisson's formula and the mean value theorem (there's a proof on Wikipedia at en.wikipedia.org/wiki/Harnack's_inequality). – Matt Rigby Sep 14 '14 at 16:09
• The hint I was given : Look at the proof of Harnack’s inequality. Show that for any radius r, if x ∈ B(0,r), then u(x) ≤ $2^n$ u(0), and note that this bound doesn’t actually depend on r. – Peter Sep 14 '14 at 16:17
• Following that hint, you get that $u$ is bounded, and then you can apply the unmodified Liouville theorem. – Ian Sep 14 '14 at 16:35
• I got stuck at the part for "any radius r" , since that Harnack 's proof above in my post, r is 1/4 distance(V,dV). – Peter Sep 14 '14 at 16:53
• That's only to make sure things are defined where they need to be; $r$ can be anything you like when it's defined on all of $\mathbb{R}^n$. – Matt Rigby Sep 14 '14 at 17:11
I would say that Evan's proof is not sharp. Actually, if $u \geq 0$ is harmonic, by the divergence theorem you can write $$\frac{\partial u}{\partial x_i}(x_0)=\frac{n}{\omega_n R^n} \int_{\partial B_R(x_0)} u(y) \, d\Sigma.$$ Hence $$\left| \frac{\partial u}{\partial x_i}(x_0) \right| \leq \frac{n}{R} u(x_0),$$ and you let $R \to +\infty$. This "proof" comes from the book Elliptic differential equations by Lin and Lin, AMS.
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2019-08-24 02:26:39
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http://openstudy.com/updates/517cf9dee4b0249598f7f5c7
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## anonymous 3 years ago help with this one plz suppose there are three students and each student tosses a coin 10 times. so what is the chance that at least one of the students gets exactly 5 heads.
1. reemii
$$P(\text{at least one .... happens}) = 1 - P(\text{no such thing happens}).$$ In general, this makes computations easier. You must compute $$1-P(\text{none of the 3 students got exactly 5 heads})=1-(P(A))^3$$ where $$A=\{\text{one student does not get exactly 5} \}$$. To compute P(A), use the same trick: P(A)=1-P(it gets exactly 5 heads).
2. anonymous
@reemii please guide me how to calculate P(A)
3. reemii
$$P(\text{a student doesn't obtain exactly 5 heads}) = 1 - P(\text{a student obtains exactly 5 heads})$$ (in 10 tosses) So we will compute : $$P(\text{5 heads in 10 tosses})$$. A head appears with probability 0.5. Here you are in the situation of a binomial distribution. ($$X\sim\text{Bin}(10,0.5)$$ and you want to know $$P(X=k)$$). Now it's just a formula.
4. reemii
oops, first line is P(not 5 heads) = 1 - P(exactly 5 heads).
5. anonymous
many thanks fot the explanation :)
6. anonymous
0.5715
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2016-08-25 09:40:57
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https://brilliant.org/discussions/thread/bug-in-answers/
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So while answering a question, I found a bug which allows you to guess without losing a try.
If you type in "24^" instead of "24", for one second, it'll show your answer as incorrect/correct, then refreshes and shows "answer must be an integer", leaving you with the same number of tries as you had before. This allows you to check if your answer is incorrect/correct without losing a try.
This is not that major of a bug since the refresh rate is quite slow, but it can be exploited.It'll be great if Brilliant fixes this. :D
Note by Siddhartha Srivastava
3 years, 9 months ago
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph 1
paragraph 2
[example link](https://brilliant.org)example link
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$
Sort by:
@Calvin Lin Sorry for tagging you here, I can't seem to tag you in the note itself. -.-
- 3 years, 9 months ago
Thanks for spotting this bug. We will fix this.
Update: This has been fixed.
Staff - 3 years, 9 months ago
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2018-06-24 03:47:30
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https://www.tutorialspoint.com/how-can-tensorflow-be-used-to-attach-a-classification-head-using-python
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# How can Tensorflow be used to attach a classification head using Python?
TensorflowPythonServer Side ProgrammingProgramming
TensorFlow can be used to attach a classification head using a sequential model that has a Dense layer, using a feature extractor model, which is previously defined.
A neural network that contains at least one layer is known as a convolutional layer. We can use the Convolutional Neural Network to build learning model.
The intuition behind transfer learning for image classification is, if a model is trained on a large and general dataset, this model can be used to effectively serve as a generic model for the visual world. It would have learned the feature maps, which means the user won’t have to start from scratch by training a large model on a large dataset.
TensorFlow Hub is a repository that contains pre-trained TensorFlow models. TensorFlow can be used to fine-tune learning models.
We will understand how to use models from TensorFlow Hub with tf.keras, use an image classification model from TensorFlow Hub. Once this is done, transfer learning can be performed to fine-tune a model for customized image classes. This is done by using a pretrained classifier model to take an image and predict what it is. This can be done without needing any training.
We are using the Google Colaboratory to run the below code. Google Colab or Colaboratory helps run Python code over the browser and requires zero configuration and free access to GPUs (Graphical Processing Units). Colaboratory has been built on top of Jupyter Notebook.
## Example
print("Attaching a classification head")
num_classes = len(class_names)
model = tf.keras.Sequential([
feature_extractor_layer,
tf.keras.layers.Dense(num_classes)
])
print("The base architecture of the model")
model.summary()
predictions = model(image_batch)
print("The dimensions of the predictions")
predictions.shape
## Output
Attaching a classification head
The base architecture of the model
Model: "sequential_3"
_________________________________________________________________
Layer (type) Output Shape Param #
=================================================================
keras_layer_1 (KerasLayer) (None, 1280) 2257984
_________________________________________________________________
dense_3 (Dense) (None, 5) 6405
=================================================================
Total params: 2,264,389
Trainable params: 6,405
Non-trainable params: 2,257,984
_________________________________________________________________
TensorShape([32, 5])
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2021-10-16 16:23:29
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https://www.talkstats.com/threads/odds-of-a-certain-5-character-string-occurring-in-a-random-string-of-512.30597/
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# Odds of a certain 5 character string occurring in a random string of 512.
#### johnebgood
##### New Member
I work in security and recently had a false positive happen where a certain word that was being tested occurred in the session id. Matching is not case sensitive.
The character ranges that can be used in the session id:
0-9, A-Z, a-z, +, /. (no comma or period)
The security check was looking for xPATH in the session id.
SESSION=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;
Could someone help me work out the odds of this happening? (Not counting that it just happened to occur while this particular audit was done out of the many thousands of audits)
#### hlsmith
##### Less is more. Stay pure. Stay poor.
Clarification, you state matching is not case sensitive - so you don't care if "xPAt" is captilized in a different way as long as the order of these characters is the same??? This will effect the number of possibilities per the 512 selections.
Does each character have the same chance of being selected each time. So is there replacement back into the pool of possibilities?
And can you provide clarification on exactly how many options there are to select from for each character (64??)?
And the xPAt seems to occur before 512 characters, so there is the probability within 512 and probability of it occurring within 512 - X.
#### johnebgood
##### New Member
Second time my session timed out after crafting a message! :/ Yes, 64 characters possible. For this exercise just assume that each character is random and can occur in any position in the 512 bytes. There are 25 combinations of xPATH that would match: xpAth, XPatH etc.
Thanks,
John
#### Dason
Why only 25? If any of the letters in xpath can be capitalized or not then shouldn't it be 32 combinations?
#### johnebgood
##### New Member
Sure, 32. 2^5. I was thinking you square the number of characters.
#### hlsmith
##### Less is more. Stay pure. Stay poor.
Here is a website with some basic probability problems to get you started. The key thing to remember in your scenario is the replace back into the eligible pool for your characters. Feel free to post your progress and we will provide feedback.
http://www.benbest.com/science/theodds.html
#### Dason
How exact do you want the probability to be? Because we could just simulate a lot of data and check how often it occurs...
#### johnebgood
##### New Member
Sure, we could brute force it but I was hoping to learn how to calculate it. I took one stats class in college but I overwrote that data long ago. Could someone walk me through how to calculate this? It doesn't have to be exact; it is just a curiosity.
#### Dason
This problem is actually quite a bit tougher than you might imagine. And I wasn't suggesting brute forcing the solution since that would be entirely unreasonable - it would take far too long to generate every possible sequence and then check how many contain an acceptable variation of xpath.
#### Dason
Doing some simulation it looks like the probability is roughly 1/100000
#### johnebgood
##### New Member
That's what I came up with also, running this program I whipped up to test 1 billion random sessions. Still interested in learning about how to calculate it.
Code:
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <string.h>
char *rsession(char *dst, int size)
{
static const char text[] = "abcdefghijklmnopqrstuvwxyz0123456789+/"
"ABCDEFGHIJKLMNOPQRSTUVWXYZ";
int i;
for ( i = 0; i < size; ++i )
{
dst[i] = tolower(text[rand() % (sizeof text - 1)]);
}
dst[i] = '\0';
return dst;
}
int main (void)
{
char session[513];
char xpath[] = "xpath";
srand(time(0));
int found_count = 0;
int i = 0;
for(i = 0; i < 1000000000; i++) {
rsession(session, sizeof session);
if(strstr(session,xpath) != NULL)
found_count++;
if(i % 1000000 == 0)
printf("%d/%d\n",found_count,i);
}
printf("\nHappened %d/1,000,000\n",found_count);
return 0;
}
Currently at 475/30,000,000
~ > ./xpath
0/0
11/1000000
27/2000000
48/3000000
67/4000000
80/5000000
93/6000000
109/7000000
130/8000000
149/9000000
166/10000000
174/11000000
190/12000000
206/13000000
221/14000000
237/15000000
251/16000000
265/17000000
277/18000000
287/19000000
309/20000000
318/21000000
335/22000000
351/23000000
367/24000000
389/25000000
400/26000000
424/27000000
443/28000000
464/29000000
475/30000000
#### johnebgood
##### New Member
So I have some observations here. Let's say we calculate how often a case insensitive xpath will occur in a 5 character string. If we double the key length we are effectively making two separate events into one. Every time we would double the length of the string it would increase the odds of it happening. The odds I'm seeing after running about a billion tests is about 1/63,000 for my original question. Unsure of the degree of error.
So lets say we take this as a ballpark range:
length odds
8 1/4,032,000
16 1/2,016,000
32 1/1,008,000
64 1/504,000
128 1/252,000
256 1/126,000
512 1/63000
If we calculate the odds for xpath occurring in a 8 character session we can then calculate odds in greater strings...
Then I ran 4 1,000,000,000 runs for a 5 character "session". and the odds came out to about 1/17,000,000. This table comes out a little high for the estimates...
session length odds
10 1/10,000,000
20 1/5,000,000
40 1/2,500,000
80 1/1,125,000
160 1/562,500
320 1/281,250
640 1/140,625
Can someone help me calculate the same problem with an 8 character session?
Thanks!
John
|
2021-11-27 04:50:55
|
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https://brilliant.org/discussions/thread/induction-proof-factorials/
|
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# Induction Proof Factorials
How can you manipulate a factorial (k+1)! such that it can be added to another factorial? Is this even possible?
4 years, 7 months ago
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By taking terms common, certainly yes.
- 4 years, 7 months ago
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2017-10-22 21:21:28
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http://www.talkstats.com/threads/statistical-justification-for-cochrans-q-please.52749/?p=147810
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Statistical justification for Cochrans Q please
AthenaAnkou
New Member
Please may someone tell me the proper way to write a statistical justification for a Cochrans Q
.. By any chance is it the same a t-test?
Thank you
Karabiner
TS Contributor
Please may someone tell me the proper way to write a statistical justification for a Cochrans Q
What exactly do you mean by this, and why and for whom do you
have to write it?
.. By any chance is it the same a t-test?
Perhaps it would help if you made yourself a bit more familiar
with Cochran's Q...?
With kind regards
K.
AthenaAnkou
New Member
I do appologise.. Usually people are helpful here. Have found out the simple answer.
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2018-01-17 09:12:02
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https://ag91.github.io/blog/2021/10/02/moldable-emacs-molds-need-examples-too/
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Where parallels cross
Interesting bits of life
Moldable Emacs: molds need examples too!
Molds can be challenging to use. Let's use tests to document them then! Let me show you how moldable-emacs is self documenting. And that one can generate useful documentation from source code.
The problem
The best systems are self documenting. Creators of things like these are super cool. I came to understand that the best of the best are those that design a tool in such a way that you can use the tool to learn/understand it. Pretty convoluted, right? An example helps: in Common Lisp I can use an inspector to see what a thing is.
(inspect 'inspect)
The object is a SYMBOL.
0. Name: "INSPECT"
1. Package: #<PACKAGE "COMMON-LISP">
2. Value: "unbound"
3. Function: #<FUNCTION INSPECT>
4. Plist: NIL
I find this cool because the inspector works on itself. Similarly on how you can edit Emacs source in Emacs.
An amazing way I have seen this done is in GlamorousToolkit. GT is a Pharo/SmallTalk system. So everything is an object. These objects may be complex. The user can do more if she has examples to work from. Now GT made sure to extract as much value as possible from tests. So unit tests check objects. But also produce examples for how the object works.
For example, notice how the object behind GT's Wardley Maps defines examples.
The green icons mean that I ran those examples as tests. But I can also inspect them to learn how to use that object. So I can both validate the object and learn about it from them. And GT let me view this information in the shape I need: view the example, visualize the graphic it creates, open the source code or run tests. These are just views of the same thing.
Then self documentation becomes more about your tool recognizing a pattern and make that accessible for you too. It is about inclusivity.
Then the question: how does moldable-emacs learn from these majestic examples?
It is a problem indeed
I like the idea behind self documentation because it inspires me of doing the best I can with the little I have got. As we have seen, if I put effort in writing an unit test, I should also use it for documentation. And the extra benefit is that the docs are finally alive: users may keep updating examples to fit to their needs. By doing so they also update tests. A healthy feedback loop!
Lisp comes with macros. The idea here is that if functions are powerful, why not to apply that idea to the source code itself? So we can edit code with code. Again we do a lot with the little we got.
While I was developing code-compass, I ended up creating a doctor function because somebody was struggling at finding the dependencies they needed to use my tool. A doctor function shows you what dependencies you miss. I think a doctor function is a patch. Somewhere in code-compass's source there is an import. Or even a statement that invokes a software that is not available by default. By this I mean that the information is there. It is just well hidden. This inaccessibility forces me to spend more time (both in writing and maintaining more software) while the computer could as well do that for me.
Again the lazy me wants the machine to do the boring tasks. Can moldable-emacs improve over code-compass?
And there is a solution
The secret is: patterns. GT uses special syntax to mark examples. Macros use defmacro to mark a special function that acts on source. We can call names other people because we associate a name to a pattern (a face). Ops, I went off track!
Ahem, let me show a pattern in moldable-emacs.
:given (:fn (and
(executable-find "graph")
(me/require 'esxml)
report))
Actually there is more than one here. First we have the :given which tells there is a precondition. Then we have :fn that I have recently introduced. And finally we have (executable-find "graph") and (me/require 'esxml)!
This last ones are the stars of the show today. I have been evolving a pattern: any time I use an external Emacs package or an external software, I use these functions. We need to get the most from the little we got, remember? Indeed, I find these functions useful as precondition of a mold. If these return false, the mold is unusable. So the first benefit is mold selection. And you can see that this pattern hints that we can collect dependencies easily. Do you see where I am going?
Now moldable-emacs is about molds that help you understand software in the way you need it. In this case, I would like to know what molds I can use and what I can't. And if I can't, what am I missing.
This is how it works.
You can see that the mold "WhatMoldsCanIUse" shows two sets of molds: the ones I can and the ones I could use. This mold behaves according to the context, so I run it on something looking like an IP address to trigger the "Decode IP address" mold. We can see that I am missing a dependency there. The cool thing is that I get a link that brings me to find the dependency. It would be trivial (any PR welcome!) to improve the mold to let Emacs install that package for me!
Moreover, I showed that I can open a "Demo" link for that mold. That is how moldable-emacs learn from GT. Molds can provide examples. You can demo these to preview a mold, if you are unsure of how it works. These examples can also work as tests. This story has rough edges (for example, in handling images), but you can already run things like the following.
(me/test-mold-examples (me/find-mold "Playground"))
The plan is to create a mold to do just that. So that when you are on a mold you can unit test it. (At some point, that will also become a condition for the mold to be usable.)
Again, you can see that now we are applying molds on molds. It reminds me of the previous Common Lisp snippet (inspect 'inspect).
Documentation, testing, listing dependencies and demos all rely on the structure of molds. That data format creates a pattern our Emacs can use to give valuable information to us.
And we can also mix things! Look for example how we can get detailed information about the "Query" mold.
(me/mold-doc "Query")
Here me/mold-doc mixes the :docs and :examples of a mold to produce detailed documentation. You can access these easily using the me/mold-docs command. Please note that I needed to write examples for demo and testing needs, but documentation comes out for free!
All in all, this feels a good start to make moldable-emacs self documenting. There are still huge issues for molds with side-effects: for instance, I could not write examples yet for the molds that show notes. Testing those would fail because different users have different notes. Still I am sure I can overcome those issues somehow later. The point is that making documentation a first class citizen and easy to keep up-to-date is possible (and cheap)!
Conclusion
When you write your first mold add some docs! And even an example if you are courageous! I will explain later the facilities to make easy to add an example. In the meanwhile explore the documentation that is generated for you with me-mold-docs.
Happy exploring!
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2022-12-02 03:57:36
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https://stacks.math.columbia.edu/tag/064T
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Lemma 15.64.4. Let $R$ be a ring. Let $M$ be an $R$-module. Then
1. $M$ is $0$-pseudo-coherent if and only if $M$ is a finite $R$-module,
2. $M$ is $(-1)$-pseudo-coherent if and only if $M$ is a finitely presented $R$-module,
3. $M$ is $(-d)$-pseudo-coherent if and only if there exists a resolution
$R^{\oplus a_ d} \to R^{\oplus a_{d - 1}} \to \ldots \to R^{\oplus a_0} \to M \to 0$
of length $d$, and
4. $M$ is pseudo-coherent if and only if there exists an infinite resolution
$\ldots \to R^{\oplus a_1} \to R^{\oplus a_0} \to M \to 0$
by finite free $R$-modules.
Proof. If $M$ is of finite type (resp. of finite presentation), then $M$ is $0$-pseudo-coherent (resp. $(-1)$-pseudo-coherent) as follows from the discussion preceding Definition 15.64.1. Conversely, if $M$ is $0$-pseudo-coherent, then $M = H^0(M[0])$ is of finite type by Lemma 15.64.3. If $M$ is $(-1)$-pseudo-coherent, then it is $0$-pseudo-coherent hence of finite type. Choose a surjection $R^{\oplus a} \to M$ and denote $K = \mathop{\mathrm{Ker}}(R^{\oplus a} \to M)$. By Lemma 15.64.2 we see that $K$ is $0$-pseudo-coherent, hence of finite type, whence $M$ is of finite presentation.
To prove the third and fourth statement use induction and an argument similar to the above (details omitted). $\square$
There are also:
• 6 comment(s) on Section 15.64: Pseudo-coherent modules, I
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2022-06-28 16:05:22
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https://socratic.org/questions/how-do-you-solve-x-3-5-5
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# How do you solve | x + 3 | = 5?
Jul 4, 2018
$x = 2 , - 8$
#### Explanation:
Given: $| x + 3 | = 5$.
Since $| \pm 5 | = 5$, we get:
$x + 3 = \pm 5$
$x = \pm 5 - 3$
${x}_{1} = 5 - 3 = 2$
${x}_{2} = - 5 - 3 = - 8$
$\therefore x = 2 , - 8$
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2020-05-31 03:54:58
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https://julianrachman.wordpress.com/2014/12/27/thoughts-on-the-formation-of-limits/
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Thoughts on the Formation of Limits
I have recently finished writing up my notes on bounded sets in Real Analysis (I will be posting my incomplete notes for Real Analysis shortly), and what is fascinating me was how the concept of these sets led to the formation of limits. Here are my thoughts:
Suppose that $\mathbb{F}$ is an ordered field and $X$ is bounded above in $\mathbb{F}$. A number $B\in\mathbb{F}$ is called a least upper bound of $X$ if (i) $B$ is an upper bound for $X$, and (ii) if $\alpha$ is any upper bound for $X$, then $B\leq\alpha.$ Similarly, if $X$is bounded below in $\mathbb{F}$, then a number $b\in\mathbb{F}$ is called a greatest lower bound of $X$ if (i) $b$ is a lower bound for $X$, and (ii) if $\alpha$ is any lower bound for $X$, then $b\geq\alpha.$ From this definition, we can see that the set $X$ is a bounded set both above and below, and is a subset of ordered field $\mathbb{F}$. We can think of lub $X$ and glb $X$ as the closest approximation of a value $x$ that cannot be defined. The reason for having the least and greatest bounds of a set is to not find the value of $x$; however it is to find a way to get close enough to $x$. And that is essentially what a limit is.
If we have a limit from $x$ to $0$ ($\lim_{x\rightarrow0}$) for some function $f(x)$, we are relatively finding the glb $f(x)$ closest to $0$. We can then have an intuition for the limit $\lim_{x\rightarrow\infty}f(x)$. This shows that we want to get $x$ as close to $\infty$ as possible. From here, these infinite limits are classified as sequences. However, I will leave the details for you to study on your own.
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2018-03-21 16:13:01
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http://www.clp.ac.cn/EN/Article/OJd7443d472f8ef080?type=figure
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Main > Chinese Optics Letters > Volume 17 > Issue 7 > Page 072501 > Article
• Abstract
• Figures (4)
• Tables (0)
• Equations (5)
• References (26)
• Cited By (0/0)
• Get PDF
• View Full Text
• Paper Information
Accepted: Apr. 18, 2019
Posted: Jul. 8, 2019
Published Online: Jul. 8, 2019
The Author Email: Guangqing Du (guangqingdu@mail.xjtu.edu.cn), Feng Chen (chenfeng@mail.xjtu.edu.cn)
• Get Citation
• ##### Copy Citation Text
Yanhong Dong, Qing Yang, Guangqing Du, Feng Chen, Noor Uddin, Dayantha Lankanath, Xun Hou. Electronic manipulation of near-field nanofocusing in few-layer graphene-based hybrid nanotips[J]. Chinese Optics Letters, 2019, 17(7): 072501
Fig. 1. Schematic of the simulated FLG-based nanotip hybrid system. The graphene-coated Au tip is modeled as a conical taper terminated by a hemisphere of radius $R$ as its point and elevated a distance $d$ above a $SiO2$–graphene–$SiO2$ substrate. An electromagnetic plane wave is incident at an angle $θ$ with respect to the surface normal. A 300-nm-thick perfectly matched layer (PML) encloses the simulation domain.
Fig. 2. Cross-section of mono-graphene-based nanotip structure. Calculated images of the e-field distributions in the case of mono-graphene in the substrate (The curvature radius of the nanotip is $R=30 nm$. The vertical spacing between the nanotip and substrate is $d=20 nm$, and incident light travels along the $X$ direction).
Fig. 3. Permittivity of MLG with respect to different Fermi energies. (a) The red, blue, black, green, and orange lines are corresponding to 0.1, 0.2, 0.3, 0.4, and 0.5 eV, respectively. (b) The normalized e-field enhancement and resonant frequency of the nanotip hybrid system depending on the Fermi energy of excited graphene. (c) Real and (d) imaginary parts of graphene permittivity with respect to the layers of FLG changing from 1 to 5 ($EF$ is biased to 0.5 eV).
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2019-08-23 20:11:22
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https://math.stackexchange.com/questions/3341706/how-to-find-a-permutation-to-collect-numbers-with-minimum-number-of-rounds
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# How to find a permutation to collect numbers with minimum number of rounds?
Has someone an idea how to solve the following problem?
Take the numbers 1,...,100000 and permute them in some way. At first you can make a swap of two numbers. Then you have to compute how many rounds it would take to collect numbers in ascending order. You have to collect numbers by every round by going left to right. How many ways there are swap two numbers at the beginning to collect numbers in ascending order with minimum number of rounds?
For example, if numbers are from one to five and those at the beginning in order 3, 1, 5, 4, 2, then you can collect them in three rounds: On first round you collect 1, 2, on the second round 3, 4 and finally 5. But you can do one swap in three different ways to collect numbers in two rounds, namely
3, 4, 5, 1, 2
3, 1, 4, 5, 2
3, 1, 2, 4, 5
Five number sequence can be solved easily by brute force and I found an algorithm to collect 1000 numbers, but 100000 numbers needs maybe some kind of trick to compute fast how a specific swap at the beginning affects how many rounds it takes to collect numbers.
My effort:
I guess the hardest part is to find a permutation that guarentee you collect the numbers with minimum number of moves. I also heard that Dilworth's theorem tells me that the minimal decomposition into ascending subsequences is equal to the size of the maximal descending subsequence. https://artofproblemsolving.com/community/c163h1906044_an_algorithm_to_collect_numbers_in_ascending_order . But i'm not sure if this is true: https://stackoverflow.com/questions/57952706/how-to-collect-numbers-in-ascending-order-with-minimal-number-of-rounds
So I have three questions:
1. How can I compute the minimal number of rounds, as is is pretty difficult at least for me to find a swap that makes sure that collecting numbers takes the minimal number of rounds?
2. How can I compute the number of ways I can make a swap at the beginning if I want to collect the numbers with minimal number of rounds?
3. Does Dillworth's theorem helps to solve question 1 or 2? If it helps, how one applies it to the problem?
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2019-12-08 21:24:22
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https://plainmath.net/secondary/physics/force-motion-and-energy/newton-s-first-law-of-motion
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# Newton's first law of motion problems and answers
Recent questions in Newton's First Law of Motion
John Landry 2022-07-21
### A ball was dropped from the window of a building, 33.5 meters above the groundHow fast it is after 2 seconds?How long will it take if the final velocity is 25 m/s?
owsicag7 2022-07-20
### Why are non-Newtonian fluids called non-Newtonian when they follow Newton’s third law?To my understanding, Newton’s third law states that for every action there is an equal and opposite reaction. Therefor if I punch the non-Newtonian fluid harder, there will be a harder reaction force stopping my hand. So why is the fluid called non-Newtonian?
Raegan Bray 2022-07-17
### Using all three of Newton's Laws of Motion, describe how a rocket launches into orbit.
Haley Madden 2022-07-17
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2022-08-20 05:40:56
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https://www.mathemania.com/gabriels-horn/
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# Gabriel’s horn
Gabriel’s horn or Torricelli’s trumpet is the surface of revolution of the function $f(x) = \frac{1}{x}$ about the x – axis for $x \ge 1$. What is this exactly? First draw your axes and draw function $\frac{1}{x}$ for $x \ge 1$.
Now you imagine it rotating around x – axis.
Image credit to Fouriest Series
This figure you got has finite volume but infinite surface area.
How is this possible?
Well let’s try to find out what’s going on by calculation. This requires simple integral calculations. If we cut this horn into tiny regular slices we’ll always get circles with radius $\frac{1}{x}$ which means that volume of one little slice is equal to $\frac{1}{x^2} \pi$. And now that we know that we can find the volume of whole horn. Since the upper border of integral is infinity we have to use limit to get what we want.
This means that the volume of this trumpet is equal to π cubic units.
To calculate surface we’ll use surface integrals of second kind.
What confused people for a long time is a paradox that using this knowledge of its surface and volume you could fill it with a bucket of paint but the same volume of paint would not be enough to paint its surface.
This paradox is resolved because the surface we generated has no thickness and you can’t find any real-life objects with no thickness which you could paint. Paint itself has finite thickness bounded by the radius of an atom.
Second thing you can think about is if you ever find real- life version of Gabriel’s trumpet could you play it? Well no, since this trumpet is infinitely long, it will take you infinitely many years to come to its end, and even if you’re feeling especially adventurous and reach its end, it would be infinitely small so you couldn’t blow in it.
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2019-05-23 17:47:42
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https://stats.hohoweiya.xyz/2019/03/07/VI/
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# Evaluate Variational Inference
##### Posted on Mar 07, 2019
A brief summary of the post, Eid ma clack shaw zupoven del ba.
In variational inference, we try to find the member $q*(\theta)$ of some tractable set of distributions $\cQ$ (commonly the family of multivariate Gaussian distributions with diagonal covariance matrices) that minimizes the Kullback-Leibler divergence,
Automatic Differentiation Variation Inference (ADVI) can find $q^*(\theta)$ by a fairly sophisticated stochastic optimization method.
But how can we check if the approximate posterior $q^*(\theta)$ is a good approximation to the true posterior $p(\theta\mid y)$. The post introduced two ideas.
## Based on PSIS
If $q(\theta)$ is a good approximation to the true posterior, it can be used as an importance proposal to compute expectations w.r.t. $p(\theta\mid y)$.
The intuition of Pareto-Smoothed Importance Sampling (PSIS) is that replacing the “noisy” sample importance weights with the model-based estimates (generalized Pareto), which can reduces the variance of the resulting self-normalized importance sampling estimator and reduces the bias compared to other options.
## Based on VSBC
Actually, variational inference is often quite bad at estimating a posterior. On the other hand, the centre of the variational posterior is much more frequently a good approximation to the centre of the true posterior.
Variational Simulation-Based Calibration (VSBC) assessed the average performance of the implied variational approximation to univariate posterior marginals, and it can indicate if the centre of the variational posterior will be, on average, biased.
## References
Published in categories Memo
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2019-03-25 01:31:58
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https://www.appropedia.org/Mud_plasters_and_renders_(Practical_Action_Brief)
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The traditional use of mud plasters and renders to coat and protect walls dates back a very long time and is found in almost all regions of the world. Finishing a house with mud plaster when the house itself has been built with earth is a natural, complementary technique, but mud plasters can also be used for buildings of stone and fired brick provided they incorporate an earth-based mortar.
Earth-based plasters often use earth in combination with other natural materials such as wheat straw or cow dung, or with mineral additives such as bitumen, to improve the basic qualities of the earth by acting as stabilisers, hardeners, and waterproofers. Even without additives, however, mud plasters and renders can give excellent results provided they are made and applied with skill and care, and maintained regularly. Today, with low-cost mass housing a priority and with the increasing interest in the preservation of architectural heritage, the need for plastering materials which are efficient and economical has awakened a new interest in earth. Earth-based plasters are completely compatible with traditional materials and building techniques, and the almost universal availability of suitable earth for building gives them a distinct advantage over some modern synthetic plasters.
## Fundamental properties
The need for a plaster and the type of plaster that should be used depends particularly on the method of construction and quality of construction. The provision of adequate footings, basements, eaves, and overhangs to a roof can in certain circumstances eliminate the need for a plaster coating altogether. As plastering can amount to 15 to 20 per cent of the total cost of a house, its benefits need to be considered relative to alternative options. In general, except in the case of highly exposed walls in areas of heavy rain, a plaster should protect against wind, rain, knocks and abrasion, and should improve the thermal insulation and appearance of a wall. At the same time it has to be easy to apply without requiring expensive and elaborate tools, and must be affordable. All types of mud plasters, but especially those on external surfaces, need to offer erosion resistance, impermeability to moisture, and impact resistance, and be well bonded to the wall.
## Erosion resistance
The main cause of erosion is heavy rain, and high winds driving the rain hard onto walls at an angle will increase erosion further still. Heavy rain, even for a short time, is much more damaging than prolonged light rain. A knowledge of local weather patterns and an analysis of meteorological data can give an indication of erosion risk and hence appropriate plastering materials and methods. It is advisable to study local traditional buildings and practices, as their evolution will have been influenced by the local climate. Impact resistance The durability of mud plasters depends on their ability to withstand the impact of humans and animals by bumping, scratching, or scraping. Impact resistance is closely linked with the quality of the plaster, which is determined by its density, methods of application, number of coats used, and maintenance practices. The texture of the plaster is also important.
## Good bonding
The bonding of earth plasters to walls is very important. When plastering a stone or earth wall the composition of the mix as well as its application are both crucial in producing a good bond (the join between the two materials). The plaster and the wall itself should ideally be compatible so that shear forces are transmitted between them and not terminated at the bond. Good bonding reduces the incidence of cracking caused by changes in ambient temperature and humidity. The plaster must be applied in coats of recommended thickness to prevent excessive strain at the bond.
## Testing the performance
A simple soil test which will show whether the soil is suitable is to plaster an area of wall and to observe the development of cracks on drying. A number of different compositions can be tried to find the one which produces the least cracks and satisfies the need for hardness and water resistance in that particular situation.
Simulation tests in the laboratory, such as the spray erosion test, can only be indicative because factors such as changes in scale, influence of true climatic conditions, building usage and maintenance practices are not easily replicated. One of the most realistic simulation methods is to expose small test-walls to natural weather conditions; this has been done in Australia, the United States, Senegal, and France, for example. This test is a good indicator of the durability of different plasters and allows a realistic comparison between plasters with different compositions and methods of application. The main drawback with this test is the length of time needed to obtain meaningful results, and building projects cannot always afford to wait so long.
## Clay plasters
The composition of traditional mud plasters varies from place to place and is an important factor in determining durability. The clay content is particularly significant, because if it is too low the plaster will lack strength and cohesion, and if it is too high there will be a risk of cracking due to shrinkage, which will weaken the bond to the wall. A suitable clay content is usually around 10 to 15 per cent, but values outside this range could also be suitable depending on the type of clay. Soils with unstable or swelling clays must be used with great care. The sand-to-silt ratio is also very important in determining the quality of a plaster. Traditionally, clay plasters were often applied in one coat both internally and externally. If applied in two coats, the first can contain more clay, even if cracks develop, while the second, containing more sand, is applied in a thinner layer. The second coat will help to close the micro-cracks in the first, provided the surface has been lightly dampened before plastering. Finally, lime distemper or whitewash can be applied to give some additional weatherproofing.
This will need to be re-applied periodically.
Clay renders are commonly improved by adding natural fibres such as cereal straw, animal hair, pine needles, bark, and wood shavings. Long straw or hair is chopped into short lengths (2 to 5cm) for easier mixing: the function of the fibres is to resist cracks and facilitate the drying process. They also make the plaster less dense and improve its insulation properties. The amount of fibres required will vary depending on soil characteristics and can be from 35 to 70kg per cubic metre for straw; 50kg per cubic metre is a typical figure. In India, paddy straw (blusa) is added at a rate of 6 per cent by weight, or 60 to 65kg per cubic metre. The straw is soaked for several days in water to facilitate a rotting process, and the complete mixing process can take 10 to 15 days.
Another traditional practice is the addition of cow dung, which improves the cohesion and plasticity of soils of low clay content. Sometimes the dung is applied to mud plaster which is partially dry to help stop the development of cracks. A traditional waterproofing in India, known as Gohber leaping, consists of one part cow dung and five parts earth by weight, made into a fine paste with water and applied to fill up surface cracks. Another practice is the addition of horse urine, which acts as a hardener and improves impermeability and impact resistance.
## Improving the composition
It is possible to improve the quality of mud plasters by:
• controlling the quantity of the sand fraction in the soil; no less than three parts sand to one of clay, for example. This helps reduce cracks without compromising cohesion. A shrinkage of more than a quarter of an inch over the 2-foot length of the box indicates a soil liable to significant cracking.
• stabilising the plaster by adding cement, lime, bitumen, or some other binder in small quantities. Possible limitations include the cost of the stabiliser and lack of skill in its proper use.
Bitumen cutback plaster is prepared by mixing hot bitumen with kerosene in a 5:1 ratio, and then combining one part of that mixture with 20 parts of previously fermented soil and wheat straw. Water is added and the whole mixed together thoroughly. This type of plaster is applied in two layers, and the second is applied only after the first has dried. Lime-soil plaster can be made with one part hydrated lime mixed with two parts of clayey soil and 3 to 6 parts sand, the optimum amount of sand depending on the clay content of the soil. The quality of the plaster depends a lot on the quality of the lime available and the type of soil.
Another proprietary plaster is 'dagga-cement', a mix of two parts sand to one part clayey soil to 0.2 part cement by volume. This produces a good weather-resistant plastering mix. To plaster the surfaces of stabilised soil blocks, and for these surfaces only, a coating of a stabilised mud slurry may be painted on. This slurry is prepared by mixing one volume of cement with two of mud. The final mix should have the consistency of paint, allowing it to be applied in a very thin coat. For decorative appearance, the colour of the clay stands out better if white cement is used rather than Ordinary Portland Cement. Red cement has sometimes also been used for this reason.
In all cases it is preferable to experiment with different mixes to find which one gives the best results with a given soil, rather than accept a general plastering recipe.
## Application: Good practice
There are general rules to follow when applying all plasters. Firstly, the wall surface has to be prepared well. This can be done by scrubbing off all the surface dust and loose material with a metallic brush. Then the wall surface must be moistened to stop water being drawn out of the plaster layer into the wall. If the plaster is applied in two coats, the first layer must be applied with force and be no more than 20mm thick. Before hardening is complete, its surface must be roughened by light grooving, scratching or pitting, to provide good bonding for the second coat. The second coat is only applied when the first is dry.
If the plaster contains a cement or lime stabiliser, it is important to spray the plaster coat twice or three times a day with water during the first days of drying, especially during hot weather, to reduce the development of cracks. In general, plaster work should be shaded from the sun, and it is better to avoid plastering on very hot or windy days. All pictures are extracted from Earth Construction.
Appropriate Building Materials. Roland Stulz and Kiran Mukerhi. IT Publications, London, 1993.
Building with Earth: A handbook. John Norton. IT Publications, London, 1997.
Building with Lime: A practical introduction. IT Publications, London, 1997.
Earth Construction: A comprehensive guide. Hugo Houben and Hubert Guillard. CRATerre and IT Publications, London, 1994.
Gypsum Plaster: Its manufacture and use. Andrew Coburn with Eric Dudley and Robin Spence. IT Publications, London, 1989.
Lime and Other Alternative Cements. Edited by Neville R Hill with Stafford Holmes and David Mather. IT Publications, London, 1993.
Plaster for soilblock buildings. Ch. A. Scott and A. Revi. Development Alternatives Shelter Group, New Delhi, 1987.
basin
Website: http://web.archive.org/web/20051001224505/http://www2.gtz.de/Basin/
basin is the Building Advisory Service and Information Network.
CRATerre Maison Levrat (Parc Fallavier)
B. P. 53, F-38092 Villefontaine-CEDEX
France
Hydraulic lime: an introduction Practical Action
Tel: +33 474 954391 Fax: +33 474 956421
E-mail: craterre@club-internet.fr
Website: http://web.archive.org/web/20150202044137/http://www.craterre.archi.fr/
GRATerre is the international centre for earth construction at the school of Architecture in Grenoble. GRATerre is a member of basin.
Page data
Part of Practical Action Technical Briefs Technical brief plasters, food and agriculture, food, agriculture, thermal insulation, energy efficiency, construction Fatima Hashmi 2008 CC-BY-SA-4.0 Practical Action No main image Fatima Hashmi (2008). "Mud plasters and renders (Practical Action Brief)". Appropedia. Retrieved August 16, 2022.
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2022-08-16 03:48:14
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http://megasoft-rapid.com/Oklahoma/error-quantization.html
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Computer Service & Repair-Business, Computer Data Recovery, Data Processing Service, Computers & Computer Equipment-Service & Repair, Computer & Electronics Recycling
Address Pond Creek, OK 73766 (580) 984-2007
# error quantization Longdale, Oklahoma
Therefore, the sequence of samples can be written as $v[0] = v(0),$ $v[1] = v(T_s),$ $v[2] = v(2T_s),\ldots$ \begin{align} v[n] &= v(nT_s) & &\text{for integer }n \end{align} In the example of This two-stage decomposition applies equally well to vector as well as scalar quantizers. To circumvent this issue, analog compressors and expanders can be used, but these introduce large amounts of distortion as well, especially if the compressor does not match the expander. How to mount a disk image from the command line? (KevinC's) Triangular DeciDigits Sequence When must I use #!/bin/bash and when #!/bin/sh?
Recording and Producing in the Home Studio, p.38-9. Ind., Vol. 79, pp. 555–568, Jan. 1961. ^ Daniel Marco and David L. The system returned: (22) Invalid argument The remote host or network may be down. Most commonly, these discrete values are represented as fixed-point words (either proportional to the waveform values or companded) or floating-point words.
Quantization noise is a model of quantization error introduced by quantization in the analog-to-digital conversion (ADC) in telecommunication systems and signal processing. The difference between steps is 0.25. Therefore, $4T_s=3T$ and the sampling rate $f_s=(4/3)f$. From an article titled Shannon, Beethoven, and the Compact Disc by Kees A.
Modern entropy coding techniques such as arithmetic coding can achieve bit rates that are very close to the true entropy of a source, given a set of known (or adaptively estimated) The answer below is idealized for discussion. These two stages together comprise the mathematical operation of y = Q ( x ) {\displaystyle y=Q(x)} . What is this $\Delta x$?
All the inputs x {\displaystyle x} that fall in a given interval range I k {\displaystyle I_{k}} are associated with the same quantization index k {\displaystyle k} . Quantization (signal processing) From Wikipedia, the free encyclopedia Jump to: navigation, search The simplest way to quantize a signal is to choose the digital amplitude value closest to the original analog Principles of Digital Audio 2nd Edition. This distortion is created after the anti-aliasing filter, and if these distortions are above 1/2 the sample rate they will alias back into the band of interest.
doi:10.1109/TIT.2005.846397 ^ Pohlman, Ken C. (1989). The system returned: (22) Invalid argument The remote host or network may be down. With the passing of Thai King Bhumibol, are there any customs/etiquette as a traveler I should be aware of? Notice that a different sinusoid $\cos(2\pi ft/3)$ with lower frequency $f/3$ also fits these samples.
Please try the request again. For a sine wave, quantization error will appear as extra harmonics in the signal. The period $T=1/f$ is the duration of one full oscillation. IT-28, pp. 129–137, No. 2, March 1982 doi:10.1109/TIT.1982.1056489 (work documented in a manuscript circulated for comments at Bell Laboratories with a department log date of 31 July 1957 and also presented
Solutions that do not require multi-dimensional iterative optimization techniques have been published for only three probability distribution functions: the uniform,[18] exponential,[12] and Laplacian[12] distributions. Therefore, $v(t)$ can be recovered exactly from the samples by ideal low pass filtering.
Figure 9 Fig. 9: Sampling a sine at $f_s = 2f$. Rounding and truncation are typical examples of quantization processes. Chou, Tom Lookabaugh, and Robert M.
Generated Fri, 14 Oct 2016 13:41:41 GMT by s_wx1094 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection The use of sufficiently well-designed entropy coding techniques can result in the use of a bit rate that is close to the true information content of the indices { k } This slightly reduces signal to noise ratio, but, ideally, completely eliminates the distortion. Noise degrades the sinusoidal signal in Fig. 1.
Audio Buildings Electronics Environment Government regulation Human health Images Radio Rooms Ships Sound masking Transportation Video Class of noise Additive white Gaussian noise (AWGN) Atmospheric noise Background noise Brownian noise Burst Consider a digital signal $100110$ converted to an analog signal for radio transmission. Browse other questions tagged adc quantization or ask your own question. Focal Press.
II: Appl. In this second setting, the amount of introduced distortion may be managed carefully by sophisticated techniques, and introducing some significant amount of distortion may be unavoidable. The signal $v(t)=\cos(2\pi ft)$ in Fig. 1 is sampled uniformly with 3 sampling intervals within each signal period $T$. To learn more about sampling and the Nyquist-Shannon theorem, read Sampling: what Nyquist didn't say, and what to do about it by Tim Wescott.
The input-output formula for a mid-riser uniform quantizer is given by: Q ( x ) = Δ ⋅ ( ⌊ x Δ ⌋ + 1 2 ) {\displaystyle Q(x)=\Delta \cdot \left(\left\lfloor The 3-bit representations in the final row can be concatenated finally into the digital signal $110001001110$.
Sequence $n=0$ $n=1$ $n=2$ $n=3$ Samples $v[n]$ $1$ $-0.5$ $-0.5$ $1$Quantized samples $v_Q[n]$ $0.9$ Sorry for my bad english, it isnt my native language. The 1.761 difference in signal-to-noise only occurs due to the signal being a full-scale sine wave instead of a triangle/sawtooth.
When to begin a sentence with "Therefore" Is there any job that can't be automated? However, for a source that does not have a uniform distribution, the minimum-distortion quantizer may not be a uniform quantizer. However using an FLC eliminates the compression improvement that can be obtained by use of better entropy coding. Generated Fri, 14 Oct 2016 13:41:41 GMT by s_wx1094 (squid/3.5.20)
Your cache administrator is webmaster. The additive noise created by 6-bit quantization is 12 dB greater than the noise created by 8-bit quantization. The general reconstruction rule for such a dead-zone quantizer is given by y k = sgn ( k ) ⋅ ( w 2 + Δ ⋅ ( | k | The potential signal-to-quantization-noise power ratio therefore changes by 4, or 10 ⋅ log 10 ( 4 ) = 6.02 {\displaystyle \scriptstyle 10\cdot \log _{10}(4)\ =\ 6.02}
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2018-10-20 21:16:14
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https://www.math-only-math.com/opposite-sides-of-a-parallelogram-are-equal.html
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# Opposite Sides of a Parallelogram are Equal
Here we will discuss about the opposite sides of a parallelogram are equal in length.
In a parallelogram, each pair of opposite sides are of equal length.
Given: PQRS is a parallelogram in which PQ ∥ SR and QR ∥ PS.
To prove: PQ = SR and PS = QR
Construction: Join PR
Proof:
Statement In ∆PQR and ∆RSP;1. ∠QPR = ∠SRP2. ∠QRP = ∠RPS3. PR = PR4. ∆PQR ≅ ∆RSP 5. PQ = SR and PS = QR. (Proved) Reason 1. PQ ∥ RS and RP is a transversal.2. PS ∥ QR and RP is a transversal.3. Common side4. By ASA criterion of congruency. 5. CPCTC
Converse of the above given theorem
A quadrilateral is a parallelogram if each pair of opposite sides are equal.
Given: PQRS is a quadrilateral in which PQ = SR and PS = QR
To prove: PQRS is a parallelogram
Proof: In ∆PQR and ∆RSP, PQ = SR, QR = SP (given) and PR is the common side.
Therefore, by SSS criterion of congruency, ∆PQR ≅ ∆RSP
Therefore, ∠QPR = ∠PRS, ∠QRP = ∠RPS (CPCTC)
Therefore, PQ ∥ SR, QR ∥ PS
Hence, PQRS is a parallelogram.
Solved examples based on the theorem of opposite sides of a parallelogram are equal in length:
1. In the parallelogram PQRS, Pq = 6 cm and SR : RQ = 2 : 1. Find the perimeter of the parallelogram.
Solution:
In the parallelogram PQRS, PQ ∥ SR and SP ∥ RQ.
The opposite sides of a parallelogram are equal. So, SR + PQ = 6 cm.
AS SR : RQ = 23 : 1, $$\frac{6 cm}{RQ}$$ = $$\frac{2}{1}$$
⟹ RQ = 3 cm
Also, RQ = SP = 3 cm.
Therefore, perimeter = PQ + QR + RS + SP
= 6 cm + 3 cm + 6 cm + 3 cm
= 18 cm.
2. In the parallelogram ABCD, ∠ABC = 50°. Find the measures of ∠BCD, ∠CBA and ∠DAB.
Solution:
AS AB ∥ DC, ∠ABC + ∠BCD = 180°
Therefore, ∠BCD = 180° - ∠ABC
= 180° - 50°
= 130°
As opposite angles in a parallelogram are equal,
∠CDA = ∠ABC = 50° and
∠DAB = ∠BCD = 130°
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2019-11-14 21:26:32
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|
https://gmatclub.com/forum/what-is-the-greatest-common-factor-gcf-of-18x-8y-20-and-24x-12y-273948.html
|
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# What is the Greatest Common Factor (GCF) of 18x^8y^20 and 24x^12y^15?
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What is the Greatest Common Factor (GCF) of 18x^8y^20 and 24x^12y^15? [#permalink]
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21 Aug 2018, 06:36
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What is the Greatest Common Factor (GCF) of $$18x^8y^{20}$$ and $$24x^{12}y^{15}$$?
A. $$3x^4y^5$$
B. $$6x^4y^5$$
C. $$3x^8y^{15}$$
D. $$6x^8y^{15}$$
E. $$72^{12}y^{20}$$
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Joined: 18 Jul 2018
Posts: 191
Location: India
Concentration: Finance, Marketing
WE: Engineering (Energy and Utilities)
Re: What is the Greatest Common Factor (GCF) of 18x^8y^20 and 24x^12y^15? [#permalink]
### Show Tags
21 Aug 2018, 06:41
1
Prime factorizing 18 and 24 gives
18 = 2*$$3^2$$
24 = $$2^3$$*3
GCF of 18 and 24 is 6
Again GCF of $$x^8$$$$y^{20}$$ and $$x^{12}$$$$y^{15}$$ is $$x^8$$$$y^{15}$$
GCF is 6$$x^8$$$$y^{15}$$
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Joined: 26 Feb 2016
Posts: 3131
Location: India
GPA: 3.12
Re: What is the Greatest Common Factor (GCF) of 18x^8y^20 and 24x^12y^15? [#permalink]
### Show Tags
21 Aug 2018, 06:43
Bunuel wrote:
What is the Greatest Common Factor (GCF) of $$18x^8y^{20}$$ and $$24x^{12}y^{15}$$?
A. $$3x^4y^5$$
B. $$6x^4y^5$$
C. $$3x^8y^{15}$$
D. $$6x^8y^{15}$$
E. $$72^{12}y^{20}$$
The greatest common factor of $$18x^8y^{20}$$ and $$24x^{12}y^{15}$$ can be found out as follows:
We can also write $$18x^8y^{20}$$ as $$3*6*x^8*y^{15}*y^5$$
Similarly, we can write $$24x^{12}y^{15}$$ as $$4*6*x^{8}*x^4*y^{15}$$
This follows the general rule of exponents which is $$a^{m+n} = a^m*a^n$$
Therefore, the greatest common factor(GCF) of $$18x^8y^{20}$$ and $$24x^{12}y^{15}$$ is $$6*x^8*y^{15}$$(Option D)
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Joined: 25 Feb 2013
Posts: 1212
Location: India
GPA: 3.82
Re: What is the Greatest Common Factor (GCF) of 18x^8y^20 and 24x^12y^15? [#permalink]
### Show Tags
21 Aug 2018, 06:43
Bunuel wrote:
What is the Greatest Common Factor (GCF) of $$18x^8y^{20}$$ and $$24x^{12}y^{15}$$?
A. $$3x^4y^5$$
B. $$6x^4y^5$$
C. $$3x^8y^{15}$$
D. $$6x^8y^{15}$$
E. $$72^{12}y^{20}$$
$$18x^8y^{20}=2*3^2x^8y^{20}$$
$$24x^{12}y^{15}=2^3*3x^{12}y^{15}$$
Therefore $$GCD = 2*3x^8*y^{15}$$, for GCD pick the lowest powers of $$x$$ & $$y$$
Option D
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Re: What is the Greatest Common Factor (GCF) of 18x^8y^20 and 24x^12y^15? [#permalink]
### Show Tags
21 Aug 2018, 07:27
Bunuel wrote:
What is the Greatest Common Factor (GCF) of $$18x^8y^{20}$$ and $$24x^{12}y^{15}$$?
A. $$3x^4y^5$$
B. $$6x^4y^5$$
C. $$3x^8y^{15}$$
D. $$6x^8y^{15}$$
E. $$72^{12}y^{20}$$
Just find the common term only 10 sec required....
Common factor of $$18x^8y^{20}$$ ,$$24x^{12}y^{15}$$
= $$6x^8y^{15} ( 3y^5 , 4x^4)$$
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Posts: 1808
Re: What is the Greatest Common Factor (GCF) of 18x^8y^20 and 24x^12y^15? [#permalink]
### Show Tags
22 Aug 2018, 05:50
Bunuel wrote:
What is the Greatest Common Factor (GCF) of $$18x^8y^{20}$$ and $$24x^{12}y^{15}$$?
A. $$3x^4y^5$$
B. $$6x^4y^5$$
C. $$3x^8y^{15}$$
D. $$6x^8y^{15}$$
E. $$72^{12}y^{20}$$
GFC of 18 & 24 = 6.........Eliminate choices A, C & E
GFC of $$x^8$$ and $$x^{12}$$ = $$x^8$$.....Eliminate B
Re: What is the Greatest Common Factor (GCF) of 18x^8y^20 and 24x^12y^15? &nbs [#permalink] 22 Aug 2018, 05:50
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# Events & Promotions
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.
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2018-09-21 00:21:36
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https://codeforces.com/problemset/problem/1271/C
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C. Shawarma Tent
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output
The map of the capital of Berland can be viewed on the infinite coordinate plane. Each point with integer coordinates contains a building, and there are streets connecting every building to four neighbouring buildings. All streets are parallel to the coordinate axes.
The main school of the capital is located in $(s_x, s_y)$. There are $n$ students attending this school, the $i$-th of them lives in the house located in $(x_i, y_i)$. It is possible that some students live in the same house, but no student lives in $(s_x, s_y)$.
After classes end, each student walks from the school to his house along one of the shortest paths. So the distance the $i$-th student goes from the school to his house is $|s_x - x_i| + |s_y - y_i|$.
The Provision Department of Berland has decided to open a shawarma tent somewhere in the capital (at some point with integer coordinates). It is considered that the $i$-th student will buy a shawarma if at least one of the shortest paths from the school to the $i$-th student's house goes through the point where the shawarma tent is located. It is forbidden to place the shawarma tent at the point where the school is located, but the coordinates of the shawarma tent may coincide with the coordinates of the house of some student (or even multiple students).
You want to find the maximum possible number of students buying shawarma and the optimal location for the tent itself.
Input
The first line contains three integers $n$, $s_x$, $s_y$ ($1 \le n \le 200\,000$, $0 \le s_x, s_y \le 10^{9}$) — the number of students and the coordinates of the school, respectively.
Then $n$ lines follow. The $i$-th of them contains two integers $x_i$, $y_i$ ($0 \le x_i, y_i \le 10^{9}$) — the location of the house where the $i$-th student lives. Some locations of houses may coincide, but no student lives in the same location where the school is situated.
Output
The output should consist of two lines. The first of them should contain one integer $c$ — the maximum number of students that will buy shawarmas at the tent.
The second line should contain two integers $p_x$ and $p_y$ — the coordinates where the tent should be located. If there are multiple answers, print any of them. Note that each of $p_x$ and $p_y$ should be not less than $0$ and not greater than $10^{9}$.
Examples
Input
4 3 2
1 3
4 2
5 1
4 1
Output
3
4 2
Input
3 100 100
0 0
0 0
100 200
Output
2
99 100
Input
7 10 12
5 6
20 23
15 4
16 5
4 54
12 1
4 15
Output
4
10 11
Note
In the first example, If we build the shawarma tent in $(4, 2)$, then the students living in $(4, 2)$, $(4, 1)$ and $(5, 1)$ will visit it.
In the second example, it is possible to build the shawarma tent in $(1, 1)$, then both students living in $(0, 0)$ will visit it.
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2021-07-28 06:56:26
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https://ask.sagemath.org/answers/55595/revisions/
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A.plot() is inherently a 3D object. Its save method can save to various 2D image formats (but not .pgf), HTML or various specialized 3D formats. Its save_image method does not include .pgf as a valid output format.
Your best bet seems to start from one of these formats (or HTML) and convert it via the targe tool(s) for the choosen format. It might be possible to het three.js to output a Tikz picture...
It might also be possible to add Tikz output to the relevant save_image (or possibly save) method. Care to file a ticket for this ?
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2021-09-18 15:14:59
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https://www.gradesaver.com/textbooks/math/calculus/calculus-early-transcendentals-8th-edition/chapter-5-section-5-4-indefinite-integrals-and-the-net-change-theorem-5-4-exercises-page-409/26
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## Calculus: Early Transcendentals 8th Edition
Original Equation: $\int(t(1-t)^{2})dt$ on the interval $[1,-1]$ First, we have to expand the equation before we evaluate the integral. Using basic math and the FOIL method, we seethe equationis simplified as shown: $\int(t(1-t)^{2})dt=$ $\int(t(t^{2}-2t+1)dt=$ $\int(t)^{3}-2t^{2}+t)dt$ To solve this integral, we first need to find the anti-derivative. The anti-derivative of $x^{n}$ is found through the equation $\frac{x^{n+1}}{n+1}$. by applying this formula to each term in the equation, we see that the final anti-derivative is $\frac{t^{4}}{4}-\frac{2t^{3}}{3}+\frac{t^{2}}{2}$ Now that we have this equation, we simply subtract the bottom range from the upper range. Our range is $[1,-1]$, so we plug 1 and -1 into the anti derivative and the difference of the two is our final answer. $(\frac{(1)^{4}}{4}-\frac{2(1)^{3}}{3}+\frac{(1)^{2}}{2})$ +$(\frac{(1)^{4}}{4}-\frac{2(-1)^{3}}{3}+\frac{(-1)^{2}}{2})$$= \frac{-4}{3}$
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2019-01-22 19:27:58
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|
https://proxieslive.com/tag/dependent/
|
## Interaction for dependent sibling input fields?
I have two fields in a form – Country and Organisation. These hold the same hierarchy and are dependent on each other.
Meaning, the user should be able to either select a country first and then select an Organisation that falls under the Country or, select an Organisation first and then select the Countries that the Organisation exists in.
So something like the following:
(If user chooses Organisation tab first, hovering on Uniqlo only displays the countries it is present in. If he chooses the country tab first, hovering over US will only display Organisations that are present in the US)
I’m convinced that this is not an ideal design and could be very confusing for some but I also can’t think of another alternative. Asking the user to choose “What would you like to choose first? Org or the Country?” seems like one way to go but still seems like bad design. Is there some clever design pattern for such cases? To throw in some context, this is for a web only admin access control UI.
## conditional probability of dependent random variables
Suppose I have 3 random variables:
$$X \sim \mbox{Bernoulli}(1/2)$$ $$Z \sim \mbox{Normal}(0,1)$$ $$Y = X+Z$$
How do I compute the conditional probability:
$$P(X=1 | Y=y)$$
# Attempt1:
Probability[ X == 1 \[Conditioned] X + Z == y, { X \[Distributed] BernoulliDistribution[1/2] ,Z \[Distributed] NormalDistribution[] } ]
# Attempt2:
D[Probability[ X == 1 \[Conditioned] X + Z >= y, { X \[Distributed] BernoulliDistribution[1/2] ,Z \[Distributed] NormalDistribution[] } ],y]
# Attempt3:
Likelihood[ TransformedDistribution[X + Z, { X \[Distributed]BernoulliDistribution[1/2], Z \[Distributed] NormalDistribution[]}] , {y}]
# Pencil and Paper attempt:
$$P(X=1 | Y=y) = \frac{P(X=1 , Y=y)}{P(Y=y)}$$ $$= \frac{P(X=1 , X+Z=y)}{P(Y=y)}$$ $$= \frac{P(X=1)P(Z=y-1)}{P(Y=y)}$$ $$= \frac{P(X=1)P(Z=y-1)}{P(X=1)P(Z=y-1)+P(X=0)P(Z=y-0)}$$
$$P(Z=y)=\frac{e^{-\frac{y^2}{2}}}{\sqrt{2 \pi }}$$ $$P(Z=y-0)=\frac{e^{-\frac{y^2}{2}}}{\sqrt{2 \pi }}$$ $$P(Z=y-1)=\frac{e^{-\frac{1}{2} (y-1)^2}}{\sqrt{2 \pi }}$$ $$P(X=1)=\frac{1}{2}$$ $$P(X=0)=\frac{1}{2}$$
$$P(X=1 | Y=y) = \frac{e^{-\frac{1}{2} (y-1)^2}}{2 \sqrt{2 \pi } \left(\frac{e^{-\frac{y^2}{2}}}{2 \sqrt{2 \pi }}+\frac{e^{-\frac{1}{2} (y-1)^2}}{2 \sqrt{2 \pi }}\right)}$$
$$P(X=1|Y=y) = \frac{e^y}{e^y+\sqrt{e}}$$
## How to get super attribute dependent relation on custom page
I need to get list of all super attribute on my custom phtml page.
Like I have 3 attributes Color, Size, Height.
So I need three drop-down on my custom page where after selecting color, dependent size value should be populate in size drop-down and after selecting size, dependent height attribute value should be populated in height drop-down.
I tried with $product->getTypeInstance(true)->getConfigurableAttributesAsArray($ product)
It gives attributes used for particular product but I am not able to make it dependent does any one has solution for this.
## Dependent dropdowns in Magento 2 Frontend
How can I create a pair of dropdowns such that when one is selected, the options for the other one are updated?
PHTML
<?php $stateList =$ block->getStates(); ?> <div> <h3 class="text-uppercase margin-bottom15"><?php echo __('Pedir una Cita') ?></h3> <p>Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. </p> <div> <form id="search_cita"> <div class="form-group states required"> <select id="state" name="state"> <option value="">Estado</option> <?php foreach($stateList as$ index): ?> <?php echo '<option value="'.$index['state'].'">'.$ index['state'].'</option>'; ?> <?php endforeach ?> </select> </div> <div class="form-group cities required"> <div class="control"> <select name="city" id="city" class="input-text" data-validate="{required:true}"> <!--option value="">Ciudad</option--> </select> </div> </div> <div> <button type="submit" class="btn btn-primary"><?php echo __('Buscar un taller') ?></button> </div> </form> </div> </div> <script> require(['jquery', 'jquery/ui'],function(){ jQuery(document).on('change','#state',function() { var param = 'frame='+jQuery('#state option:selected').attr('value'); var url = "<?php echo $block->getStateAction(); ?>"; //alert(param); jQuery.ajax({ showLoader: true, url: url, data: param, type: "POST", dataType: 'json' }).done(function (data) { jQuery('#city').empty(); jQuery('#city').append(data); }); }); }); </script> BLOCK <?php namespace Morwi\Citas\Block; class Display extends \Magento\Framework\View\Element\Template { protected$ _isScopePrivate; public function __construct( \Magento\Framework\View\Element\Template\Context $context, \Magento\Directory\Block\Data$ directoryBlock, array $data = [] ) { parent::__construct($ context, $data);$ this->directoryBlock = $directoryBlock;$ this->_isScopePrivate = true; } public function getStateAction() { return $this->getUrl('citas/index/display', ['_secure' => true]); } } CONTROLLER <?php namespace Morwi\Citas\Controller\Index; use Magento\Framework\Controller\ResultFactory; class Display extends \Magento\Framework\App\Action\Action { protected$ _resultPageFactory; public function __construct( \Magento\Framework\App\Action\Context $context, \Magento\Framework\View\Result\PageFactory$ resultPageFactory ){ parent::__construct($context);$ this->_resultPageFactory = $resultPageFactory; } public function execute() { //*$ result = $this->resultFactory->create(ResultFactory::TYPE_JSON);$ objectManager = \Magento\Framework\App\ObjectManager::getInstance(); $resource =$ objectManager->get('Magento\Framework\App\ResourceConnection'); $connection =$ resource->getConnection(); $html='<option selected="selected" value="">Ciudad</option>';$ stateName = $this->getRequest()->getParam('frame'); if($ stateName!='') { $tblSL =$ resource->getTableName('store_locator'); $sql = "SELECT DISTINCT city from " .$ tblSL . " WHERE state LIKE '%". $stateName ."%'";$ city = $connection->fetchAll($ sql); foreach($city as$ index){ $hmtl= '<option value="'.$ index['city'].'">'.$index['city'].'</option>'; } } return$ result->setData(['success'=>true,'value'=>\$ html]); //*/ } } ?>
But this returns to me
in success I don’t paint the html
Any help would be appreciated.
Thanks
## Magento 2: Knockout JS dependent dropdowns
How can I create a pair of dropdowns such that when one is selected, the options for the other one are updated? The values for options for both dropdowns are coming from PHP.
I have absolutely no clue about knockout js. Any help would be appreciated.
Thanks
## Magento 2 : Dependent dropdowns in knockout
I am trying to override Magento/Checkout/view/frontend/web/js/view/shipping-address/list.js and Magento/Checkout/view/frontend/web/template/shipping-address/list.html.
I need to create two dropdowns here such that when one is selected, the other one gets it’s values. Like countries and provinces.
The array of values for each dropdown is created in php in config provider.
How can I create these dropdowns in knockout js? Thanks!
## How to make drop-down dependent on another drop-down value in admin ui component Magento 2
I want to update one drop-down option on the change of another drop-down using UI Component in Magento admin section.
Like: have one drop-down with color option ex: Red, Pink, Black, etc. and another drop-down has a size value which depends on color.
How can I dynamically update size option on the change of color using UI component in Magento admin?
## For proof automation in Coq, when is it appropriate to use canonical structures or dependent types instead of Ltac?
There are a few possible approaches to proof automation in modern Coq.
• Writing proof scripts with Ltac. This is the approach described in http://adam.chlipala.net/cpdt/, which the author uses to great effect in projects like http://adam.chlipala.net/papers/BedrockPOPL15/. It can significantly reduce the amount of proof code required, but requires a good handle on the quirks of Ltac and does not seem straightforward to debug.
• Canonical-structures-based automation. This is the approach described in https://people.mpi-sws.org/~beta/lessadhoc/, and used in Mathcomp. It involves taking advantage of Coq’s type inference mechanism to automatically execute logic programs that search for certain kinds of proof terms. It’s described in that paper as less ad-hoc than the Ltac-heavy approach, but not necessarily faster, and can be more verbose due to needing to use the canonical structures mechanism for something it wasn’t directly designed for.
• Dependent types/Equations. The Equations plugin (https://www.irif.fr/~sozeau//research/publications/drafts/Equations_Reloaded.pdf) seems to faciliate in Coq the same convenience when working with dependently typed programs as a language like Agda or Idris. With this approach the elaborator acts as a form of automation, and the amount of proof code is reduced by having algorithms create, manipulate and pass around proof terms directly.
There are also some modern developments that complement these.
• Ltac2. This is meant as a replacement for Ltac, with fewer quirks and potentially better performance, as described in https://popl19.sigplan.org/details/CoqPL-2019/8/Ltac2-Tactical-Warfare. The paper states that “Ltac2 is still in an active development phase, but the foundations of the language have been settled. More than anything, it is in need of users in order to polish the rough edges”. If it is meant to be a superior replacement to Ltac, then should it be considered instead of Ltac for new projects, since it’s already ready for user testing?
• Metacoq. This provides metaprogramming features that allow the development of higher level tools, as described on https://www.irif.fr/~sozeau/research/publications/drafts/The_MetaCoq_Project.pdf, and presumably simplify the use of proof by reflection, a technique used in both canonical-structures-based an Ltac-heavy approaches.
My question is, if I’m starting a new project, what criteria should I use to determine which approach or combination thereof to adopt? As a concrete example, imagine I want to verify the easy-to-verify parts of a program that connects to a server over the internet, downloads some data, processes the data somehow, then serves the processed data over TCP. By easy-to-verify I mean not verifying the TCP/HTTP stack, or proving from scratch the correctness of well-known algorithms used in the data processing. When I consider how I’d structure this it seems like the structure would be quite different depending on which of the above approaches I used, and I lack the experience to make a judgement regarding which would produce the best result in terms of maximising the output of verified code per unit of development time. What factors should necessitate the use of canonical structures or Equations instead of just plain Ltac?
## All components dependent only on a core component [on hold]
I’m kind of obsessed about hiding all implementations in .net behind interfaces. I came up with idea of storing all interfaces in a core component (with dtos and so on) and their implementations in many others. The only one component in the system that is dependent on all the others is a bootstrapper for dependency injection, and the rest is dependent only on the core and conditionally on the bootstrapper.
I consider component as the smallest deployable unit e.g dll.
Is this approach too extreme? What are possible issues that can occur?
## How to design a dependent subscription
I have several products for which there are several services and some services are common among some products. I need to design a system where service will be served only once even if multiple services exits among subscribed products by the user. Below is a sample of what I want to achieve
Product-A – Service1 – Service2 – Service3 – Service4
Product-B – Service2 – Service5 – Service6
Product-C – Service1 – Service2
If user subscribes to Product-A and Product-B, I need to serve only Service1, Service3, Service4 from Product-A and Service2, Service5, Service6 from Product-B.
Right now, what I have in my code is, I take all the services of Product-A into a MapA and services of Product-B into MapB and then find out commonMap as intersection of MapA and MapB. I serve only each service once.
Problem with my current approach is, there is no way I can identify if I am serving Service2 from Product-A or Product-B. Also, the Products, Services are going to be increased by a huge number which increases dependencies and if user selects too many products, finding intersection would not be a good idea.
I am searching for an approach where, the sServices can be identified at runtime and I should be able to identify which Service is coming from which Product.
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2019-09-18 01:51:17
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https://stats.stackexchange.com/questions/422194/do-i-need-independence-for-paired-t-test
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# Do i need independence for paired t-test?
I am testing the mean change in IOP of eyes in a glaucoma study. When looking at just the 'Primary Eye', i am using a paired t-test to test the mean change from baseline as this is one eye per patient. However, when using 'All Eyes' in the study some patients will have both eyes eligible in the analysis (some will just have one eye with glaucoma) and there will be dependence here.
Does the paired t-test assume independence of observations? If so, should i use another test for measuring the change from baseline of patients who have two eyes being analysed?
model <- lmer(IOP ~ 1 + Occasion +(1|Subject)+(1|Subject:Eye),data)
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2022-05-24 02:54:33
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http://gxbwk.njournal.sdu.edu.cn/EN/abstract/abstract1591.shtml
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### Image patch prior based denoising algorithm by using low rank approximation and Wiener filtering
ZHANG Yang, CHEN Fei*, XU Haiping
1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, Fujian, China
• Received:2017-01-05 Online:2017-06-20 Published:2017-01-05
Abstract: A Gaussian mixture model(GMM)was used to study the texture structure of natural image patches, and a low-rank approximation and Wiener filtering algorithm based on image patch prior were proposed. The proposed method divided the image into a number of overlapped patches and clustered them for collaborative filtering by using the prior structures of external image patch and internal image self-similarity. By grouping nonlocal similar patches, low-rank approximation was used as collaborative filtering to recover the texture structures. When the number of similar patches was small, Wiener filtering with patch prior was adopted to preserve texture features. The experimental results indicated that the proposed method was more suitable for the images with fewer similar patches like boundary and corner etc., and showed very competitive performance with state-of-the-art denoising method in terms of Peak Signal to Noise Ratio(PSNR)and visual quality.
CLC Number:
• TP37
[1] ZORAN D, WEISS Y. From learning models of natural image patches to whole image restoration[C] //Proceedings of the 13th International Conference on Computer Vision(ICCV 2011). Barcelona, Spain: IEEE Computer Society, 2011, 6669(5): 479-486.[2] CHATTERJEE P, MILANFAR P. Learning denoising bounds fornoisy images[C] //Proceedings of the 17thInternational Conference on Image Processing(ICIP 2010). Hong Kong, China: IEEE Computer Society, 2010:1157-1160.[3] RUDIN L I, OSHER S, FATEMI E. Nonlinear total variation based noise removal algorithms[J].Physica D: Nonlinear Phenomena,1992, 60(1):259-268.[4] PORTILLA J, STRELA V, WAINWRIGHT M, et al, Image denoising using scale mixturesof Gaussians in the waveletdomain[J].IEEE Transactions on Image Processing A Publication of the IEEE Signal Processing Society, 2003, 12(11):1338-1351.[5] BUADES A, COLL B, MOREL J M. A nonlocal algorithm for image denoising[C] //Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition(CVPR'05). SanDiego, USA: IEEE Computer Society, 2005.[6] DABOV K, FOI A, KATKOVNIK V, et al. Image denoising by sparse 3-d transform-domain collaborative filtering[J]. IEEE Transactions on Image Processing, 2007,16(8):2080-2095.[7] MAIRAL J, BACH F, PONCE J, et al. Non-local sparse models for image restoration[C] //Proceedings of the 12th International Conference on Computer Vision(ICCV 2009). Kyoto, Japan: IEEE Computer Society, 2009, 30(2):2272-2279.[8] AHARON M, ELAD M, BRUCKSTEIN A. K-svd: an algorithm for designing overcomplete dictionaries for sparse representation[J]. IEEE Transactions on Signal Processing, 2006, 54(11): 4311-4322.[9] DONG W, SHI G, LI X. Nonlocal image restoration wi-th bilateral variance estimation: a low-rank approach[J].IEEE Transaction on Image Processing, 2013, 22(2):700-711.[10] GU S, ZHANG L, ZUO W, et al. Weighted nuclear nor-m minimization with application to image denoising[C] //Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(CVPR 2014). Columbus, USA: IEEE Computer Society, 2014:2862-2869.[11] DONG W, SHI G, LI X, et al. Compressive sensing via nonlocal low rank regularization[J]. IEEE Transactions on Image Processing, 2014, 23(8): 3618-3632.[12] 刘波,杨华,张志强.基于奇异值分解的图像去噪[J].微电子学与计算机,2007,24(11):169-171. LIU Bo, YANG Hua, ZHANG Zhiqiang. Image denoisingbased on singular value decomposition[J]. Microelectronics and Computer, 2007, 24(11):169-171.[13] 张俊峰,孙清伟.基于图像旋转和分块的奇异值分解图像去噪[J].激光与红外,2009,39(5):538-541. ZHANG Junfeng, SUN Qingwei. Image denoising based on SVD using image rotation and block[J]. Laser and Infrared, 2009, 39(5):538-541.[14] BURGER H, SCHULER C, HARMELING S. Image denoising: can plain neural networks compete with bm3d[C] //Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(CVPR 2012). Rhode Island, USA: IEEE Computer Society, 2012:2392-2399.[15] CHEN F, ZHANG L, YU H M. External patch prior guided internal clustering for image denoising[C] //Proceedings of the 2015 IEEE International Conference on Computer Vision(ICCV). San Diego, USA: IEEE Computer Society, 2015:603-611.[16] DEMPSTER A P, LAIRD N M, RUBIN D B.Maximum likelihood from incomplete data via the EM algorithm[J]. Journal of the Royal Statistical Society. Series B(Methodological), 1977, 39(1):1-38.[17] CAI J F, CANDES E J, SHEN Z W. A singular value thresholding algorithm for matrix completion[J]. SIAM Journal on Optimization, 2010, 20(4):1956-1982.
[1] WU Huan, ZHONG Farong, MO Yuchang, PAN Zhusheng, ZENG Lingguo. Performance comparison between breadth-first ordering and priority ordering in network reliability analysis [J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2015, 45(2): 43-48. [2] ZHAO Jia-min, FENG Ai-min*, LIU Xue-jun. A new structured one-class support vector machine with local density embedding [J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2012, 42(4): 13-18. [3] ZHANG You-xin, WANG Li-hong. Two-stage semi-supervised clustering algorithm based on affinity propagation [J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2012, 42(2): 18-22. [4] QU Bo 1, XU Baowen 1,2, NIE Changhai 1,2. A dynamic prioritization method for supplemental test cases [J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2009, 39(2): 137-140.
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2022-08-16 03:28:34
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http://wiki.seas.harvard.edu/geos-chem/index.php?title=Running_GCHP:_Configuration&diff=prev&oldid=48151
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# Difference between revisions of "Running GCHP: Configuration"
## Overview
All GCHP run directories have default simulation-specific run-time settings that are set when you create a run directory. You will likely want to change these settings. This page goes over how to do this.
## Configuration files
GCHP is controlled using a set of configuration files that are included in the GCHP run directory. Files include:
Several run-time settings must be set consistently across multiple files. Inconsistencies may result in your program crashing or yielding unexpected results. To avoid mistakes and make run configuration easier, bash shell script runConfig.sh is included in all run directories to set the most commonly changed config file settings from one location. Sourcing this script will update multiple config files to use values specified in file.
Sourcing runConfig.sh is done automatically prior to running GCHP if using any of the example run scripts, or you can do it at the command line. Information about what settings are changed and in what files are standard output of the script. To source the script, type the following:
source runConfig.sh
You may also use it in silent mode if you wish to update files but not display settings on the screen:
source runConfig.sh --silent
While using runConfig.sh to configure common settings makes run configure much simpler, it comes with a major caveat. If you manually edit a config file setting that is also set in runConfig.sh then your manual update will be overrided via string replacement. Please get very familiar with the options in runConfig.sh and be conscientious about not updating the same setting elsewhere.
You generally will not need to know more about the GCHP configuration files beyond what is listed on this page. However, for a comprehensive description of all configuration files used by GCHP see the last section of this user manual.
## Commonly Changed Run Options
### Compute Configuration
#### Set Number of Nodes and Cores
To change the number of nodes and cores for your run you must update settings in two places: (1) runConfig.sh, and (2) your run script. The runConfig.sh file contains detailed instructions on how to set resource parameter options and what they mean. Look for the Compute Resources section in the script. Update your resource request in your run script to match the resources set in runConfig.sh.
It is important to be smart about your resource allocation. To do this it is useful to understand how GCHP works with respect to distribution of nodes and cores across the grid. At least one unique core is assigned to each face on the cubed sphere, resulting in a constraint of at least six cores to run GCHP. The same number of cores must be assigned to each face, resulting in another constraint of total number of cores being a multiple of six. Communication between the cores occurs only during transport processes.
While any number of cores is valid as long as it is a multiple of six (although there is an upper limit per resolution), you will typically start to see negative effects due to excessive communication if a core is handling less than around one hundred grid cells or a cluster of grid cells that are not approximately square. You can determine how many grid cells are handled per core by analyzing your grid resolution and resource allocation. For example, if running at C24 with six cores each face is handled by one core (6 faces / 6 cores) and contains 576 cells (24x24). Each core therefore processes 576 cells. Since each core handles one face, each core communicates with four other cores (four surrounding faces). Maximizing squareness of grid cells per core is done automatically within runConfig.sh if variable NXNY_AUTO is set to ON.
#### Split a Simulation Into Multiple Jobs
There is an option to split up a single simulation into separate serial jobs. To use this option, do the following:
1. Update runConfig.sh with your full simulation (all runs) start and end dates, and the duration per segment (single run). Also update the number of runs options to reflect to total number of jobs that will be submitted (NUM_RUNS). Carefully read the comments in runConfig.sh to ensure you understand how it works.
2. Optionally turn on monthly diagnostic (Monthly_Diag). Only turn on monthly diagnostics if your run duration is monthly.
3. Use gchp.multirun.run as your run script, or adapt it if your cluster does not use SLURM. It is located in the runScriptSamples subdirectory of your run directory. As with the regular gchp.run, you will need to update the file with compute resources consistent with runConfig.sh. Note that you should not submit the run script directly. It will be done automatically by the file described in the next step.
4. Use gchp.multirun.sh to submit your job, or adapt it if your cluster does not use SLURM. It is located in the runScriptSamples subdirectory of your run directory. For example, to submit your series of jobs, type: ./gchp.multirun.sh
There is much documentation in the headers of both gchp.multirun.run and gchp.multirun.sh that is worth reading and getting familiar with, although not entirely necessary to get the multi-run option working. If you have not done so already, it is worth trying out a simple multi-segmented run of short duration to demonstrate that the multi-segmented run configuration and scripts work on your system. For example, you could do a 3-hour simulation with 1-hour duration and number of runs equal to 3.
The multi-run script assumes use of SLURM, and a separate SLURM log file is created for each run. There is also log file called multirun.log with high-level information such as the start, end, duration, and job ids for all jobs submitted. If a run fails then all scheduled jobs are cancelled and a message about this is sent to that log file. Inspect this and your other log files, as well as output in the OutputDir/ directory prior to using for longer duration runs.
#### Change Domains Stack Size
For runs at very high resolution or small number of processors you may run into a domains stack size error. This is caused by exceeding the domains stack size memory limit set at run-time and the error will be apparent from the message in your log file. If this occurs you can increase the domains stack size in file input.nml. The default is set to 20000000.
### Basic Run Settings
#### Set Cubed Sphere Grid Resolution
GCHP uses a cubed sphere grid rather than the traditional lat-lon grid used in GEOS-Chem Classic. While regular lat-lon grids are typically designated as ΔLat ⨉ ΔLon (e.g. 4⨉5), cubed sphere grids are designated by the side-length of the cube. In GCHP we specify this as CX (e.g. C24 or C180). The simple rule of thumb for determining the roughly equivalent lat-lon resolution for a given cubed sphere resolution is to divide the side length by 90. Using this rule you can quickly match C24 with about 4x5, C90 with 1 degree, C360 with quarter degree, and so on.
To change your grid resolution in the run directory edit the CS_RES integer parameter in runConfig.sh to the cube side-length you wish to use.
#### Turn On/Off Model Components
You can toggle all primary GEOS-Chem components, including type of mixing, from within runConfig.sh. The settings in that file will update input.geos automatically. Look for section Turn Components On/Off, and other settings in input.geos. Other settings in this section beyond component on/off toggles using CH4 emissions in UCX, and initializing stratospheric H2O in UCX.
#### Change Model Timesteps
Model timesteps, both chemistry and dynamic, are configured within runConfig.sh. They are set to match GEOS-Chem Classic default values for low resolutions for comparison purposes but can be updated, with caution. Timesteps are automatically reduced for high resolution runs. Read the documentation in runConfig.sh section Timesteps for setting them.
#### Set Simulation Start and End Dates
Set simulation start and end in runConfig.sh section Simulation Start, End, Duration, # runs. Read the comments in the file for a complete description of the options. Typically a "CAP" runtime error indicates a problem with start, end, and duration settings. If you encounter an error with the words "CAP" near it then double-check that these settings make sense.
### Inputs
#### Change Input Meteorology Grid Resolution
The meteorology grid resolutions are set to 0.25x0.3125 for GEOS-FP and 0.5x0.625 for MERRA2 by default when creating a GCHP run directory. If you wish to change meteorology resolution you must update all meteorology paths and filenames in ExtData.rc and make sure run directory symbolic link MetDir points to the data at the resolution you will run at.
Sebastian Eastham (MIT) developed the following python code to automatically update ExtData.rc for alternative grid resolutions.
#!/bin/bash
if [[ $# -ne 1 ]]; then echo "Must provide path to either GEOS_2x2.5/MERRA2 or GEOS_2x2.5/GEOSFP" exit 70 fi if [[ -L MetDir ]]; then unlink MetDir; fi # For GEOS-FP sed -i "s/025x03125\.nc/2x25.nc/g" ExtData.rc # For MERRA-2 #sed -i "s/05x0625\.nc/2x25.nc/g" ExtData.rc ln -s$1 MetDir
exit 0
Copy and paste the above code into a file in your run directory, for example change_met.sh. Assuming you are using the standard directory structure, the following command should then automatically switch your ExtData.rc and MetDir to the 2x2.5 GEOS-FP input:
./change_met.sh $(readlink -f$( readlink -f MainDataDir )/../GEOS_2x2.5/GEOS_FP)
When changing meteorology source and/or grid resolution, be sure that you have the data available for the time period you plan on simulating. In addition, note that meteorology listed in ExtData.rc includes both data for the time period you plan on running at as well as constants files (2011 to GEOS-FP and 2015 for MERRA2). See the downloading GEOS-Chem data page for more information on meteorology sources available and how to download them.
#### Change Your Initial Restart File
All GCHP run directories come with symbolic links to initial restart files for commonly used cubed sphere resolutions. The appropriate restart file is automatically chosen based on the cubed sphere resolution you set in runConfig.sh. All of the restart files are simply GEOS-Chem Classic restart files regridded to the cubed sphere.
#------------------------------------------------
# Initial Restart File
#------------------------------------------------
# By default the linked restart files in the run directories will be
# used. Please note that HEMCO restart variables are stored in the same
# restart file as species concentrations. Initial restart files available
# on gcgrid do not contain HEMCO variables which will have the same effect
# as turning the HEMCO restart file option off in GC classic. However, all
# output restart files will contain HEMCO restart variables for your next run.
INITIAL_RESTART=initial_GEOSChem_rst.c${CS_RES}_TransportTracers.nc # You can specify a custom initial restart file here to overwrite: # INITIAL_RESTART=your_restart_filename_here You may over-write the default restart file with your own by specifying the restart filename in runConfig.sh. Beware that it is your responsibility to make sure it is the proper grid resolution. Unlike GEOS-Chem Classic, HEMCO restart files are not used in GCHP. HEMCO restart variables may be included in the initial species restart file, or they may be excluded and HEMCO will start with default values. GCHP initial restart files that come with the run directories do not include HEMCO restart variables, but all output restart files do. #### Regrid an Initial Restart File GCHP expects the restart file to be at the same grid resolution as the model run. If you have a lat-lon restart file that you want to regrid for input to GCHP, or a restart file already on cubed-sphere but at the wrong resolution, you can regrid to cubed sphere at any resolution using FORTRAN tool CSRegridTool developed by Sebastian Eastham (MIT). To use, clone the repository, source your GCHP environment file, and run make from within csregridtool. This will build the program. Then copy your source restart file to the directory and edit the input.regrid text file for the target resolution, source filename, target filename, and if you want vertical flipping. GCHP uses level 0 as top of atmosphere so regridding GEOS-Chem classic files for GCHP requires vertical flipping. Next, type ./regrid to run and generate an output file. Finally, because csregridtool creates files in NetCDF-3 format you will need to reprocess to NetCDF-4 prior to use in GCHP. To do this, type 'nccopy -k 4 restart_in restart_out' where restart_in is the name of your generated NetCDF-3 file and restart_out is the name of the new NetCDF-4 file. Please note that csregridtool requires offline tile files with mapping weights for the regridding. There are publicly available tile files available for commonly used source and target grids at ExtData/GCHP/TilesFiles (GEOS-Chem shared data directory). Examples of regridding weights available are {4x5, 2x2.5} to {c24, c48, c90, c180, c360}. If you need tile files for a different conversion please contact the GEOS-Chem Support Team. #### Turn On/Off Emissions Inventories Because file I/O impacts GCHP performance it is a good idea to turn of file read of emissions that you do not need. You can turn emissions inventories on or off the same way you would in GEOS-Chem Classic, by setting the inventories to true or false at the top of configuration file HEMCO_Config.rc. All emissions that are turned off in this way will be ignored when GCHP uses ExtData.rc to read files, thereby speeding up the model. For emissions that do not have an on/off toggle at the top of the file, you can prevent GCHP from reading them by commenting them out in HEMCO_Config.rc. No updates to ExtData.rc would be necessary. If you alternatively comment out the emissions in ExtData.rc but not HEMCO_Config.rc then GCHP will fail with an error when looking for the file information. Another option to skip file read for certain files is to replace the file path in ExtData.rc with /dev/null. However, if you want to turn these inputs back on at a later time you should preserve the original path in a comment. #### Add New Input Files New in GCHP 12.5.0: Online ESMF regridding removes the need for tile files when running GCHP. However, online regridding does not apply to restart files. You can still use the tools listed below to create tile files to regrid restart files, or you can regrid using python. There are three main requirements for adding new emissions inventories to GCHP: 1. Add the inventory information to HEMCO_Config.rc. If you wish to add new inputs to the model that are not handled by HEMCO then you can skip this step. 2. Add the inventory information to ExtData.rc. 3. Have a tile file available that maps the inventory's lat/lon grid to the cubed sphere grid for the resolution you will use. To add information to HEMCO_Config.rc, follow the same rules as you would for adding a new emission inventory to GEOS-Chem Classic. Note that not all information in HEMCO_Config.rc is used by GCHP. This is because HEMCO is only used by GCHP to handle emissions after they are read, e.g. scaling and applying hierarchy. All functions related to HEMCO file read are skipped. This means that you could put garbage for the file path and units in HEMCO_Config.rc without running into problems with GCHP. However, we recommend that you fill in HEMCO_Config.rc in the same way you would for GEOS-Chem Classic for consistency and also to avoid potential format check errors. Staying consistent with the information that you put into HEMCO_Config.rc, add the inventory information to ExtData.rc following the guidelines listed at the top of the file and using existing inventories as examples. You can ignore all entries in HEMCO_Config.rc that are copies of another entry since putting these in ExtData.rc would result in reading the same variable in the same file twice. Doing so would be costly in GCHP because each file is opened and closed for each variable in the file. HEMCO interprets the copied variables, denoted by having dashes in the HEMCO_Config.rc entry, separate from file read. At this point it is best to run a very short simulation with GCHP with MAPL debug prints on (see section on debugging below). If your file(s) need a new tile file then the model will crash. Tile files have already been created for many lat/lon grids and these are stored in ExtData/GCHP/TileFiles. The GCHP log file error will include the tile file name that GCHP expects to be available for regridding your new inventory. In that filename DC = dateline centered, PC = pole centered, DE = dateline edge, and PE = pole edge. UU is reserved for files on regional grids. Once you have this information you should be able to generate your own tile file by downloading the tempestremap and CSGrid repositories from GitHub and following these steps: • tempestremap: This tool will generate a netcdf tile file for mapping lat/lon coordinates to cubed sphere. This is a fortran tool that should work in your existing GCHP environment. Simply do make clean and then make to build the tempestremap code. Then use runGlobal.sh or runRegional.template.sh to generate global or regional bound tile files. When using runGlobal.sh, you will need to specify whether your data is dateline-centered and/or pole-centered, as determined from the log file error message. We recommend generating a tile file for all supported cubed-sphere resolutions (nC = 24, 48, 90, 180, and 360). NOTE: There seems to be a 2 GB limit when creating tile files with tempestremap. • CSGrid: This tool will convert the netCDF file created by tempestremap to binary for compatibility with GCHP. CSGrid requires a Matlab license. If you have Matlab you can use exampleScripts/create_Tempest_TileFile_LL2CS.m to convert your netCDF output in tempestremap/TileFiles from netcdf to binary. Send the resulting file to the GCST and they can add it to ExtData/GCHP/TileFiles. Once read in by GCHP, your data will be stored as MAPL Import variables with the same names that appear in the first column of ExtData.rc. If your input files are handled by HEMCO then you do not need to do anything else to handle the MAPL Imports. However, if your new inputs are not handled by HEMCO then you will need to take the additional steps of adding source code to transfer your MAPL Imports to something that GEOS-Chem can understand. If you wish to assign a MAPL Import directly to a State_Met or other state field in GEOS-Chem, you can do this in GCHP file "Includes_Before_Run.H". The lines in that file are executed prior to every dynamic timestep in GCHP and currently contain the setting of all State_Met fields derived from MAPL Imports. For more advanced use cases, read through GCHP file Chem_GridCompMod.F90 for examples, specifically searching for calls to subroutine MAPL_GetPointer. Contact the GEOS-Chem Support Team for more information on how to use MAPL Imports within GEOS-Chem. A few common errors encountered when adding new input files to GCHP are: 1. Your input file contains integer values. Beware that the MAPL I/O component in GCHP does not read or write integers. If your data contains integers then you should reprocess the file to contain floating point values instead. If you try to input integers you will get an error such as this: >>Reading TESTDATA from ./MainDataDir/testfile.nc CFIO: Reading ./MainDataDir/testfile.nc at 19850101 000000 CFIO_GetVar: error getting scale CFIO_CFIO_GetVar failed problem in ESMF_CFIOSdfVarRead 2. Your data latitude and longitude dimensions are in the wrong order. Lat must always come before lon in your inputs arrays, a requirement true for both GCHP and GEOS-Chem Classic. For more information about this, see the [Preparing_data_files_for_use_with_HEMCO#Ordering_of_the_data|Preparing Data Files for Use with HEMCO wiki page]]. The symptom of this error in GCHP is: CFIO: Reading {filename} at {YYYYMMDD} {HHmmSS} Error reading variable using NF90_GET_VAR -57 NetCDF: Start+count exceeds dimension bound 3. You do not have a tile file that regrids between your input file data resolution and the internal resolution of GCHP. This will result in an ExtData error in MAPL_HorzTransform. 4. Your 3D input data are mapped to the wrong levels in GEOS-Chem (silent error). If you read in 3D data and assign the resulting import to a GEOS-Chem state variable such as State_Chm or State_Met, then you must flip the vertical axis during the assignment. See files Includes_Before_Run.H and setting State_Chm%Species in Chem_GridCompMod.F90 for examples. 5. You have a typo in either HEMCO_Config.rc or ExtData.rc. Error in HEMCO_Config.rc typically result in the model crashing right away. Errors in ExtData.rc typically result in a problem later on during ExtData read. Always try running with DEBUG=20 in runConfig.sh (maximizes output to gchp.log) and Warnings and Verbose set to 3 in HEMCO_Config.rc (maximizes output to HEMCO.log) when encountering errors such as this. Another useful strategy is to find rc-file entries for similar input files and compare them against the entry for your new file. Directly comparing the file metadata may also lead to insights into the problem. ### Outputs #### Output Diagnostics Data on a Lat-Lon Grid This feature is new in GCHP 12.5.0. See the HISTORY.rc file in GCHP 12.5.0 run directories for instructions. Details will be included here on the wiki in the future. #### Output Restart Files at Regular Frequency The MAPL component in GCHP has the option to output restart files (also called checkpoint files) at regular intervals. Unlike the final restart file output at the end of a simulation, these regularly output restart files contain the date and time in their filename. Enabling this feature is a good idea if you plan on doing a long simulation and you are not splitting your run into multiple jobs. If the run crashes unexpectedly then you can restart mid-run rather than start over from the beginning. To set the checkpoint frequency, simply update the HHmmSS string for "Checkpoint_Freq" in runConfig.rc. Minutes and seconds must each be two digits but hours can be more than two. Each output checkpoint file will include the timestamp in the filename. #------------------------------------------------ # Output Restart Files #------------------------------------------------ # You can output restart files at regular intervals throughout your # simulation. These restarts are in addition to the end-of-run restart # which is always produced. To configure output restart file frequency, # set the variable below to a string of format HHmmSS. More than 2 # digits for the hours string is permitted (e.g. 1680000 for 7 days). # Setting the frequency to 000000 will turn off this feature by setting # it to a very large number. Checkpoint_Freq="000000" #### Turn On/Off Diagnostics All GCHP run directories have four collections on by default: time-averaged species concentrations, instantaneous species concentrations, time-averaged meteorology, and instantaneous meteorology. All species are enabled while only a subset of meteorology variables are enabled. There are several other collections already implemented but they are off by default for the standard and benchmark simulations, and on by default for the RnPbBe simulation. To turn collections on or off, comment ("#") collection names in the "COLLECTIONS" list at the top of file HISTORY.rc. #=================================================================== # Declare collection names and toggle on/off #=================================================================== COLLECTIONS: #'AerosolMass' #'Aerosols', #'Budget', #'CloudConvFlux', #'ConcAfterChem', #'DryDep', 'Emissions', #'JValues', #'LevelEdgeDiags', #'ProdLoss', 'SpeciesConc', #'StateChm', 'StateMet_avg', 'StateMet_inst', #'WetLossConv', #'WetLossLS', :: Once a collection is turned on, you can comment diagnostics within it further down in the file by searching for the collection name with ".fields" suffix. Be aware that you cannot comment out the diagnostic that appears on the same line as the fields keyword. If you wish to suppress that specific diagnostic then move it to the next line and replace it with a diagnostic that you want to output. #=================================================================== # State_Met array diagnostics - time-averaged StateMet_avg.template: '%y4%m2%d2_%h2%n2z.nc4', StateMet_avg.format: 'CFIO', StateMet_avg.frequency: 010000 StateMet_avg.duration: 010000 StateMet_avg.mode: 'time-averaged' StateMet_avg.fields: 'Met_AD ', 'GIGCchem', #'Met_AIRDEN ', 'GIGCchem', #'Met_AIRVOL ', 'GIGCchem', #'Met_ALBD ', 'GIGCchem', 'Met_AREAM2 ', 'GIGCchem', #'Met_AVGW ', 'GIGCchem', 'Met_BXHEIGHT ', 'GIGCchem', etc #### Set Diagnostic Frequency, Duration, and Mode WARNING: There is currently a bug in GCHP the prevents writing out more than one time per file. Duration in HISTORY.rc is ignored. All diagnostic collections that come with the run directory have frequency, duration, and mode defined within runConfig.sh. With the exception of SpeciesConc_inst and StateMet_inst, all collections are time-averaged (mode) with frequency and duration set to the simulation length you specified in CopyRunDirs.input when creating the run directory. Any of these defaults can be over-written by editing runConfig.sh. Be aware that manual updates of HISTORY.rc will be over-written by runConfig.sh settings. #------------------------------------------------ # Diagnostics #------------------------------------------------ # Frequency, duration, and mode used for all default HISTORY.rc diagnostic # collections are set from within this file. These are defined as: # # Frequency = frequency of diagnostic calculation (HHmmSS) # Duration = frequency of diagnostic file write (HHmmSS) # Mode = computation of diagnostics (time-averaged or instantaneous) # # Edit the frequency, duration, and mode below to change global settings. # See the list further below of what HISTORY.rc collections will be updated. # # NOTES: # 1. Freq and duration hours may exceed 2 digits, e.g. 7440000 for 31 days # 2. Freq and duration are ignored if Monthly_Diag is set to 1 # 3. If you do not want settings for certain collections set automatically # from this file, comment them out below. # 4. If you add a collection to HISTORY.rc and want its settings # automatically updated from this file, add to the list below. # 5. To turn off collections completely, comment them out in HISTORY.rc. # common_freq="010000" # Ignore if using multi-run monthly diag option common_dur="010000" # Ignore if using multi-run monthly diag option common_mode="'time-averaged'" # "'time-averaged'" and "'instantaneous'" SpeciesConc_freq=${common_freq}
SpeciesConc_dur=${common_dur} SpeciesConc_mode=${common_mode}
AerosolMass_freq=${common_freq} AerosolMass_dur=${common_dur}
AerosolMass_mode=${common_mode} Aerosols_freq=${common_freq}
Aerosols_dur=${common_dur} Aerosols_mode=${common_mode}
Budget_freq=${common_freq} etc #### Add a New Diagnostics Collection Adding a new diagnostics collection in GCHP is the same as for GEOS-Chem Classic netcdf diagnostics. You must add your collection to the collection list in HISTORY.rc and then define it further down in the file. Any 2D or 3D arrays that are stored within State_Met, State_Chm, or State_Diag, and that are successfully incorporated into the GEOS-Chem Registry may be included as fields in a collection. State_Met variables must be preceded by "met_", State_Chm variables must be preceded by "chm_", and State_Diag variables should not have a prefix. See GeosCore/state_diag_mod.F90 for examples of how existing State_Diag arrays are implemented. Once implemented, you can either incorporate the new collection settings into runConfig.sh for auto-update, or you can manually configure all settings in HISTORY.rc. #### Generate Monthly Mean Diagnostics There is an option to automatically generate monthly diagnostics by submitting month-long simulations as separate jobs. Splitting up the simulation into separate jobs is a requirement for monthly diagnostics because MAPL History requires a fixed number of hours set for diagnostic frequency and file duration. The monthly mean diagnostic option automatically updates HISTORY.rc diagnostic settings each month to reflect the number of days in that month taking into account leap years. To use the monthly diagnostics option, first read and follow instructions for splitting a simulation into multiple jobs (see separate section on this page). Prior to submitting your run, enable monthly diagnostics in runConfig.sh by searching for variable "Monthly_Diag" and changing its value from 0 to 1. Be sure to always start your monthly diagnostic runs on the first day of the month. #### Additional Diagnostic Collection Options See file GCHP/Shared/MAPL_Base/TeX/HistoryIntro.tex for original MAPL documentation on MAPL History. Please note that we have not tested all of these functionalities and some of them to seem to not work in MAPL. Proceed with caution and let the GEOS-Chem Support Team know what you find. Here is a brief overview of options that may be included for each collection that is taken from that document: template Character string defining the time stamping template that is appended to collection to create a particular file name. The template uses GrADS convensions. The default value depends on the duration of the file. descr Character string describing the collection. Defaults to expdsc'. format Character string to select file format ("CFIO", "CFIOasync", "flat"). "CFIO" uses MAPL_CFIO and produces netcdf output. "CFIOasync" uses MAPL_CFIO but delegates the actual I/O to the MAPL_CFIOServer (see MAPL_CFIOServer documenation for details). Default = "flat". frequency Integer (HHHHMMSS) for the frequency of time groups in the collection. Default = 060000. mode Character string equal to instantaneous' or time-averaged'. Default = 'instantaneous'. acc_interval Integer (HHHHMMSS) for the acculation interval ($\le\$ frequency) for time-averaged diagnostics. Default = frequency; ignored if mode is instantaneous'.
ref_date Integer (YYYYMMDD) reference date for {\em frquency}; also the beginning date for the collection. Default is the Start date on the Clock.
ref_time Integer (HHMMSS) Same a ref_date.
end_date Integer (YYYYMMDD) ending date to stop diagnostic output. Default: no end
end_time Integer (HHMMSS) ending time to stop diagnostic output. Default: no end.
duration Integer (HHHHMMSS) for the duration of each file. Default = 00000000 (everything in one file). Duration is not currently functional in GCHP and will be ignored. Frequency is used instead for write frequency.
resolution Optional resolution (IM JM) for the ouput stream. Transforms betwee two regulate LogRect grid in index space. Default is the native resolution.
xyoffset Optional Flag for output grid offset when interpolating. Must be between 0 and 3. (Cryptic Meaning: 0:DcPc, 1:DePc, 2:DcPe, 3:DePe). Ignored when resolution results in no interpolation (native). Default: 0 (DatelineCenterPoleCenter).
levels Optional list of output levels (Default is all levels on Native Grid). If vvars is not specified, these are layer indices. Otherwise see vvars, vunits, vscale.
vvars Optional field to use as the vertical coordinate and functional form of vertical interpolation. A second argument specifies the component the field comes from. Example 1: the entry 'log(PLE)','DYNAMICS' uses PLE from the FV3 advection component as the vertical coordinate and interpolates to levels linearly in its log. Example 2: 'THETA','DYN' a way of producing isentropic output. Only log(*), pow(*), and real number and straight linear interpolation are supported.
vunit Character string to use for units attribute of the vertical coordinate in file. The default is the MAPL_CFIO default. This affects only the name in the file. It does not do the conversion. See vscale
vscale Optional Scaling to convert VVARS units to VUNIT units. Default: no conversion.
regrid_exch Name of the exchange grid that can be used for interpolation between two LogRect grids or from a tile grid to a LogRect grid. Default: no exchange grid interpolation. irregular grid.
regrid_name Name of the Log-Rect grid to interpolate to when going from a tile to Field to a gridde output. regrid_exch must be set, otherwise it is ignored.
conservative Set to a non-zero integer to turn on conservative regridding when going from a native cube-sphere grid to lat-lon output. Default: 0
deflate Set deflate level (0-9) of NETCDF output when format is CFIO or CFIOasync. Default: 0
subset Optional subset (lonMin lonMax latMin latMax) for the output when performing non-conservative cube-sphere to lat-lon regridding of the output.
chunksize Optional user specified chunking of NETCDF output when format is CFIO or CFIOasync, (Lon chunksize, Lat chunksize, Lev chunksize, Time chunksize)
### Debugging
#### Enable Maximum Print Output
Besides compiling with "make compile_debug", there are a few run settings you can configure to boost your chance of successful debugging. All of them involve sending additional print statements to the log files.
1. Change "ND70" in input.geos from 0 to 1 to turn on extra GEOS-Chem print statements in the main log file.
2. Set the "MAPL_DEBUG_LEVEL" variable in runConfig.sh to a number greater than 0 to turn on extra MAPL print statements in MAPL ExtData. This is useful if you are having a problem reading input files. The higher the number the more prints will be sent to the log (and the slower your run will be). Usually 20 is sufficient, although you can go higher. Please be sure to remember to set MAPL_DEBUG back to 0 when you are done so as not to severely slow down your runs!
3. Set the "Verbose" and "Warnings" settings in HEMCO_Config.rc to maximum values of 3 to send the maximum number of prints to HEMCO.log.
4. Set the "MEMORY_DEBUG_LEVEL" option, new in 12.5.0, to 1 to turn on additional memory usage prints per timestep.
#------------------------------------------------
# Debug Options
#------------------------------------------------
# Set MAPL debug flag to 0 for no extra MAPL debug log output, or 1 to
# print information to log. Using this flag is most helpful for debugging
# issues with file read (MAPL ExtData).
#
# Set memory debug flag to 0 to print memory only once per timestep. Set to
# 1 to enable memory prints at additional locations throughout the run.
#
# For GEOS-Chem debug prints, turn on ND70 in input.geos manually.
#
# WARNING: Turning on debug prints significantly slows down the model!
#
MAPL_DEBUG_LEVEL=0
MEMORY_DEBUG_LEVEL=0
None of these options require recompiling. Be aware that all of them will slow down your simulation. Be sure to set them back to the default values after you are finished debugging.
#### Turn On/Off MAPL Timers and Memory Logging
Your GCHP log file will include timing and memory information by default, and this is usually a good thing. If for some reason you want to turn these features off you can do so in file CAP.rc. Search for "MAPL_ENABLE_TIMERS" and "MAPL_ENABLE_MEMUTILS" and simply change "YES" to "NO". Remember to turn them back on again if you later need to to debug.
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2022-10-02 22:42:55
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https://gateoverflow.in/2112/gate2011-10
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614 views
Which one of the following is NOT desired in a good Software Requirement Specifications (SRS) document?
(A) Functional Requirements
(B) Non-Functional Requirements
(C) Goals of Implementation
(D) Algorithm for Software Implementation
retagged | 614 views
Algorithms for software implementation may not be required at requirement phase. they are required at later stages like design and development
selected by
1
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2019-07-18 00:53:43
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