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The Grand Locus / Life for statistical sciences
the Blog
## Who understands the histone code?
The most annoying thing about us biologists is that we keep using words that we don’t understand. “Epigenetics” is one of those that has drawn my attention for several years, as I explained in my last post. I suggested that the invasion started in 2001, the year that the histone code hypothesis was proposed by Thomas Jenuwein and David Allis in a seminal paper entitled Translating the Histone Code.
The histone code hypothesis was arguably the most influential concept of the last decade in molecular biology. Yet, most biologists would be hard pressed to say what the hypothesis is. It should not be that difficult, all you have to do is look up what Thomas Jenuwein and David Allis actually wrote. But believe it or not, this blog is one of the only places on the Internet where you will find the histone code hypothesis spelled out clearly. Most sources, including the Wikipedia article diverge substantially from the original statement.
Distinct qualities of higher order chromatin, such as euchromatic or heterochromatic domains, are largely dependent on the local concentration and combination of differentially modified nucleosomes.
DNA in the nucleus comes in a structure called the nucleosome. The...
## The rise of epigenetics
I started to study biology at the time epigenetics became a buzzword. I first heard the term at university in 2001, and as many young enthusiastic people of the time, I did my PhD on epigenetics because it was cool. But buzzes come and go, I finished my PhD and I got bored with epigenetics. Meanwhile, I thought that my interest had been mirroring that of the community, and that the trend was towards a loss of interest for epigenetics. I was about to write a blog post entitled “The death of epigenetics” when I did a quick PubMed search and realized that the peak of popularity was... 2013. Epigenetics is not dead, it is on the rise!
Above is the number* of PubMed hits for “epigenetics” per month since 1996, with “chromatin” shown as a reference for comparison. PubMed now displays a histogram of the occurrence of your search term over the years (check here for epigenetics). The growth is not due to articles published in late-adopting journals, since the trend-setters Cell, Nature and Science published more than half of their papers labelled “epigenetics” in the last three years.
One of...
## Meet planktonrules
Some of you may remember planktonrules from my series on IMDB reviews. For those of you who missed it, planktonrules is an outlier. In my attempt to understand what IMDB reviewers call a good movie, I realized that one reviewer in particular had written a lot of reviews. When I say a lot, I mean 14,800 in the last 8 years. With such a boon, I could not resist the temptation to use his reviews to analyze the variation of style between users, and to build a classifier that recognizes his reviews.
I finally got in contact with Martin Hafer (planktonrule’s real name) this year, and since he had planned to visit Barcelona, we set up a meeting in June. I have to admit that I expected him to be a sort of weirdo, or a cloistered sociopath. The reality turned out to be much more pleasant; we had an entertaining chat, speaking very little about movie reviews. He also pointed out to me that doing statistics on what people write on the Internet is a bit weird... True that.
Anyway, as an introduction, here is a mini interview of planktonrules. You can find out more...
## How to stop sucking and be awesome instead
I recently gave a motivation speech at the CRG/Institut Curie international PhD retreat. There was only one slide and the content was fairly general, so I thought I could reproduce it here. My goal was to motivate people, but also to surprise them a little, especially at the end. Finally, I wish such a nice title were mine, but I have to acknowledge Jeff Atwood. I stole it from his post on Coding Horror (which I also invite you to read).
### How to stop sucking and be awesome instead
Think about what we can do today. We can send people on the moon. We can talk to each other any time anywhere on the planet. We can go anywhere in about a day. We can transplant a heart. We can cure diseases that were fatal only 30 years ago. And yet, there is still one thing that we cannot do. We don’t know how to motivate people.
That’ right, we do not know how to make our colleagues enthusiastic about their work. If you watch a couple of TED videos or if you read a couple of books on management, you will see that we all...
## 100% non functional
### Panglossian genomics
As most French students of my generation, I had to study Candide, a short philosophical novella written by Voltaire. Back then, I was convinced that Voltaire was an arrogant prick, and I never imagined that his dumb criticism of Leibniz's theory of pre-established harmony, which he barely understood, would ever echo in my work as a biologist.
But here we are, years have passed, I have made peace with Voltaire, and the ENCODE consortium has issued its major and controversial statement that they find “biochemical functions for 80% of the genome”. As the arguments and the comments flow on the blogs and in the academic press, I cannot help thinking about the words of Dr. Pangloss – incarnating narrow optimism.
Observe, for instance, the nose is formed for spectacles, therefore we wear spectacles. The legs are visibly designed for stockings, accordingly we wear stockings.
What I will call the Panglossian reading of the “80% functional” statement above is the idea that 80% of the genome is meant to be the way it is. The architecture of a given locus is somehow designed to produce what happens there (transcription, transcription enhancing, transcription factor binding etc). Notice...
Everybody in the academia has a story about reviewer 3. If the words above sound familiar, you will definitely know what I mean, but for the others I should give some context. No decent scientific editor will accept to publish an article without taking advice from experts. This process, called peer review, is usually anonymous and opaque. According to an urban legend, reviewer 1 is very positive, reviewer 2 couldn't care less, and reviewer 3 is a pain in the ass. Believe it or not, the quote above is real, and it is all the review consists of. Needless to say, it was from reviewer 3.
For a long time, I wondered whether there is a way to trace the identity of an author through the text of a review. What methods do stylometry experts use to identify passages from the Q source in the Bible, or to know whether William Shakespeare had a ghostwriter?
### The 4-gram method
Surprisingly, the best stylistic fingerprints have little to do with literary style. For instance, lexical richness and complexity of the language are very difficult to exploit efficiently. The unconscious foibles...
## Genetics and racism (3)
In the previous posts of this series on genetics and racism, I talked about two recent academic disputes over human races. With this post I hope to give a wider overview of what biology has to say about species, breeds and races.
### Darwin’s pigeons
Modern genetics was born in 1900 with the re-discovery of Mendel's laws. Since the Neolithic Revolution, genetics had been an empirical art. Our ancestors isolated most of the breeds of animals and plants that we know today, i.e. groups that carry a trait of interest to the next generation when crossed together (for instance Chihuahuas are small dogs and Great Dane are large dogs).
But over the generations, pedigrees got lost in the myst of time and the overwhelming differences between some breeds of the same species raised the question whether they share the same natural origin. Before Darwin, it was difficult to imagine that the Chihuahuas and the Great Dane would have a common ancestor, and the theory went that breeds actually came from different species. This is actually one of the first questions tackled by Darwin in The Origin of Species. In the following passage, he exposes his...
## ... and academic reprints for all
Like many other academic journals, Molecular and Cellular Biology takes copyrights very seriously. And to trace the criminals who share scientific publications funded by public institutions, they add to the margin of the pdf reprints downloaded from their website the date and the identity of the license owner.
I recently heard that some people downloaded and installed the pdf toolkit pdftk and at the Linux terminal issued a command like the one below, where they replaced article.pdf by the name of the pdf they had downloaded.
pdftk article.pdf output uncompressed-article.pdf uncompress
Using their text editor, they opened the uncompressed pdf file and looked for lines like the ones below and commented them out with a % sign.
10 0 0 10 0 0 cm BT/R19 11 Tf0 -1 1 0 579.5 456.847 Tm[( on some day by Institution of the Evil Person)556]TJ-94.148 0 Td[(http://mcb.asm.org/)278]TJ-89.2543 0 Td[(Downloaded from )278]TJET
They then ran pdftk again to fix the pdf document, and the download information was gone.
pdftk uncompressed-article.pdf output stripped-article.pdf
Needless to say, doing...
## Genetics and racism (2)
In the first post of this series on genetics and racism, I explained how Richard Lewontin concluded from his work on human diversity that human races are of no value for taxonomy (the classification of living begins). This view was later criticized and even termed Lewontin's fallacy by A. W. F. Edwards. Yet, nobody ever doubted that Lewontin was honest in his approach. But more recently came another case that gives the shivers. The great Stephen Jay Gould, the author of the acclaimed Mismeasure of Man was accused of data manipulation.
### The mismeasure of Gould
Stephen Jay Gould was this kind of scientist who pops up everywhere. I discovered him in a comment about the opinion of the Vatican on Evolution, others knew him for his statistical analyses of baseball records, while he was actually a paleontologist, author of the theory of punctuated equlibrium. But his most famous work is undoubtedly The Mismeasure of Man.
Like the author, the book is a strange chimera, somewhere in between scientific research and history, with a touch of lyricism. The Mismeasure of Man is a journey through the differences between people, or more precisely through the scientific discourse over this...
## Genetics and racism (1)
Important note: Please read the Erratum at the end of the post.
### Tolstoy’s remorse
It is 1879. Leo Tolstoy, then rich and famous for War and Peace and Anna Karenina works on another kind of text. In A Confession he explains at length that he regrets writing those novels. The focus of his remorse and his anger towards himself is the heart of his talent, this innate sense of human nature. Tolstoy's pen had no equal when it came to paint the Russian society of the time, its characters and its culture. However, he explains that this attitude towards writing is wrong, because he has been telling without preaching, he has been describing without judging. He will even abandon the royalties of War and Peace and Anna Karenina, refusing to earn money from such immoral writings.
We were all then convinced that it was necessary for us to speak, write, and print as quickly as possible and as much as possible, and that it was all wanted for the good of humanity. And thousands of us, contradicting and abusing one another, all printed and wrote — teaching others. And without noticing that we knew nothing, and that...
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## twenty-four to nil
Another puzzling question on X validated, where the expectation of a random sum of deterministic vectors is to be computed. (That is, the sum involves a random number of terms.) Without enough detail to understand why this proves a difficulty, given that each deterministic vector is to be invoked at most once. Nonetheless, my (straightforward) answer there
$Y_1\underbrace{\mathbb P(\tau\ge 1)}_{=1}+Y_2\mathbb P(\tau\ge 2)+\cdots+Y_N\underbrace{\mathbb P(\tau=N)}_{=0}$
proved much more popular (in terms of votes) that many of my much more involved answers there. Possibly because both question and answer are straightforward.
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# zbMATH — the first resource for mathematics
## Tanaka, Tamaki
Compute Distance To:
Author ID: tanaka.tamaki Published as: Tanaka, T.; Tanaka, Tamaki
Documents Indexed: 153 Publications since 1976, including 11 Books
all top 5
#### Co-Authors
14 single-authored 19 Yamada, Syuuji 14 Takahashi, Wataru 13 Tanino, Tetsuzo 10 Georgiev, Pando Grigorov 9 Kimura, Kenji 7 Kuroiwa, Daishi 7 Nishizawa, Shogo 6 Ayoshin, Dmitri A. 6 Higuchi, Masakazu 6 Lau, Anthony To-Ming 6 Saito, Yutaka 6 Watanabe, Toshikazu 5 Ogata, Yuto 4 Ike, Koichiro 4 Kuwano, Issei 4 Petrosyan, Leon Aganesovich 4 Washio, Satoshi 4 Zhang, Qingling 4 Zhang, Youmei 3 Akashi, Shigeo 3 Araya, Yousuke 3 Fan, Ky 3 Kim, Do Sang 3 Kimura, Yutaka 3 Lee, Gue Myung 3 Rockafellar, Ralph Tyrrell 3 Tanaka, Kensuke 3 Yu, Hui 2 Kalmoun, El Mostafa 2 Kim, Tae Hwa 2 Shimizu, Akira 2 Xu, Hong-Kun 2 Yamada, Syuji 2 Yan, Xinggang 2 Yao, Jen-Chih 1 Benavides, T. D. 1 Butnariu, Dan 1 Cai, Min 1 Calvert, Bruce D. 1 Gutiérrez, César 1 Hojo, Mayumi 1 Hsu, Szebi 1 Inuiguchi, Masahiro 1 Kanie, Mayumi 1 Kawasaki, Toshiharu 1 Kimura Kenji 1 Kimura, Yasunori 1 Kobayashi, Shogo 1 Kumam, Poom 1 Lai, Hang-Chin 1 Lee, Jae Hyoung 1 Liang, Hui 1 Lin, Lai-Jiu 1 Llorens-Fuster, Enrique 1 López Acedo, Genaro 1 Maugeri, Antonino 1 Mongkolkeha, Chirasak 1 Murakami, Hitomi 1 Novo-Sanjurjo, Vicente 1 Onodsuka, Maki 1 Pardalos, Panos M. 1 Reich, Simeon 1 Riahi, Hassan 1 Ródenas-Pedregosa, Juan Luis 1 Sakamoto, Tatsuo 1 Sawauchi, Rie 1 Shimizu, Toshiyuki 1 Sisarat, Nithirat 1 Sonda, Yuuya 1 Suzuki, Tomonari 1 Tokushige, Yusuke 1 Truong Xuan Duc Ha 1 Vitanza, Carmela 1 Wangkeeree, Rabian 1 Nguyen Dong Yen 1 Zeng, Lu-Chuan 1 Zhang, Bingjiang
all top 5
#### Serials
21 RIMS Kokyuroku 16 Journal of Nonlinear and Convex Analysis 5 Nihonkai Mathematical Journal 4 The Science Reports of the Hirosaki University 4 Applied Mathematics Letters 3 Optimization 3 Taiwanese Journal of Mathematics 2 Journal of Mathematical Analysis and Applications 2 Fuzzy Sets and Systems 2 Journal of Global Optimization 2 Journal of Applied Mathematics 2 Pacific Journal of Optimization 2 Journal of Nonlinear Analysis and Optimization: Theory & Applications 1 International Journal of Systems Science 1 Journal of the Franklin Institute 1 Applied Mathematics and Optimization 1 IEEE Transactions on Automatic Control 1 Journal of Computational and Applied Mathematics 1 Journal of Optimization Theory and Applications 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Proceedings of the Japan Academy. Series A 1 Science Reports of Niigata University. Series A (Mathematics) 1 European Journal of Operational Research 1 SIAM Journal on Optimization 1 Journal of Multi-Criteria Decision Analysis 1 Nova Journal of Mathematics, Game Theory, and Algebra 1 Optimization Methods & Software 1 International Journal of Mathematics, Game Theory and Algebra 1 Bulletin of the Kyushu Institute of Technology. Pure and Applied Mathematics 1 Bulletin of the Faculty of Science and Technology. Hirosaki University 1 Communications in Mathematical Analysis 1 Optimization Letters 1 Advances in Soft Computing 1 Nonlinear Analysis. Theory, Methods & Applications 1 Linear and Nonlinear Analysis
all top 5
#### Fields
74 Operations research, mathematical programming (90-XX) 43 Calculus of variations and optimal control; optimization (49-XX) 24 Operator theory (47-XX) 21 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 17 General and overarching topics; collections (00-XX) 15 General topology (54-XX) 14 Functional analysis (46-XX) 11 Convex and discrete geometry (52-XX) 7 Real functions (26-XX) 6 Measure and integration (28-XX) 6 Numerical analysis (65-XX) 6 Systems theory; control (93-XX) 4 Global analysis, analysis on manifolds (58-XX) 2 Partial differential equations (35-XX) 1 History and biography (01-XX) 1 Mathematical logic and foundations (03-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Functions of a complex variable (30-XX) 1 Probability theory and stochastic processes (60-XX) 1 Statistics (62-XX) 1 Information and communication theory, circuits (94-XX)
#### Citations contained in zbMATH
63 Publications have been cited 524 times in 391 Documents Cited by Year
On cone convexity of set-valued maps. Zbl 0895.26010
Kuroiwa, Daishi; Tanaka, Tamaki; Ha, Truong Xuan Duc
1997
Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions. Zbl 0826.90102
Tanaka, T.
1994
Generalized semicontinuity and existence theorems for cone saddle points. Zbl 0894.90132
Tanaka, Tamaki
1997
Large-eddy simulation of turbulent gas-particle flow in a vertical channel: Effect of considering inter-particle collisions. Zbl 1004.76513
Yamamoto, Y.; Potthoff, M.; Tanaka, T.; Kajishima, T.; Tsuji, Y.
2001
Some minimax problems of vector-valued functions. Zbl 0628.90078
Tanaka, T.
1988
Unified scalarization for sets and set-valued Ky Fan minimax inequality. Zbl 1221.49025
Kuwano, Issei; Tanaka, Tamaki; Yamada, Syuuji
2010
Finite volume TVD scheme on an unstructured grid system for three- dimensional MHD simulation of inhomogeneous systems including strong background potential fields. Zbl 0797.76061
Tanaka, T.
1994
Nonconvex separation functional in linear spaces with applications to vector equilibria. Zbl 06662673
Gutiérrez, César; Novo, Vicente; Ródenas-Pedregosa, Juan Luis; Tanaka, Tamaki
2016
The convexity of $$A$$ and $$B$$ assures $$\text{int} A + B = \text{int}(A + B)$$. Zbl 0810.52003
Tanaka, Tamaki; Kuroiwa, Daishi
1993
Existence theorems for cone saddle points of vector-valued functions in infinite-dimensional spaces. Zbl 0652.49011
Tanaka, T.
1989
Approximately efficient solutions in vector optimization. Zbl 0871.90076
Tanaka, Tamaki
1996
Positivity of continuous-time descriptor systems with time delays. Zbl 1360.93314
Zhang, Youmei; Zhang, Qingling; Tanaka, Tamaki; Yan, Xing-Gang
2014
Iterative construction of fixed points of nonself-mappings in Banach spaces. Zbl 1120.65075
Zeng, Lu-Chuan; Tanaka, Tamaki; Yao, Jen-Chih
2007
Two types of minimax theorems for vector-valued functions. Zbl 0696.90060
Tanaka, T.
1991
Vector-valued set-valued variants of Ky Fan’s inequality. Zbl 0987.49010
Georgiev, Pando Gr.; Tanaka, Tamaki
2000
A characterization of generalized saddle points for vector-valued functions via scalarization. Zbl 0956.90507
Tanaka, Tamaki
1990
General forms of a $$N$$-fold supersymmetric family. Zbl 0977.81021
Aoyama, H.; Sato, M.; Tanaka, T.
2001
Optimality conditions in set-valued optimization using nonlinear scalarization methods. Zbl 1186.90123
Shimizu, Akira; Nishizawa, Shogo; Tanaka, Tamaki
2007
Alternative theorems for set-valued maps based on a nonlinear scalarization. Zbl 1105.90076
Nishizawa, Shogo; Onodsuka, Maki; Tanaka, Tamaki
2005
Continuity of cone-convex functions. Zbl 1261.90052
Kuwano, Issei; Tanaka, Tamaki
2012
Existence of vector equilibria via Ekeland’s variational principle. Zbl 1172.65032
Araya, Yousuke; Kimura, Kenji; Tanaka, Tamaki
2008
Fan’s inequality for set-valued maps. Zbl 1042.49512
Georgiev, Pando Gr.; Tanaka, Tamaki
2001
An alternative theorem for set-valued maps via set relations and its application to robustness of feasible sets. Zbl 1407.90238
Ogata, Yuto; Tanaka, Tamaki; Saito, Yutaka; Lee, Gue Myung; Lee, Jae Hyoung
2018
Admissibility for positive continuous-time descriptor systems. Zbl 1307.93177
Zhang, Youmei; Zhang, Qingling; Tanaka, Tamaki; Cai, Min
2013
Minimal element theorem with set-relations. Zbl 1145.49013
Shimizu, Akira; Tanaka, Tamaki
2008
On inherited properties of set-valued maps. Zbl 1255.49032
Nishizawa, Shogo; Tanaka, Tamaki; Georgiev, Pando Gr.
2003
Multi-objective programming and goal-programming. Theory and applications. Proceeedings of the conference MOPGP’02, Nara, Japan, June 4–7, 2002. Zbl 1051.90003
Tanino, Tetsuzo (ed.); Tanaka, Tamaki (ed.); Inuiguchi, Masahiro (ed.)
2003
Cone-convexity of vector-valued functions. Zbl 0721.90063
Tanaka, Tamaki
1990
Computational methods for set-relation-based scalarizing functions. Zbl 1417.90135
Yu, Hui; Ike, Koichiro; Ogata, Yuto; Saito, Yutaka; Tanaka, Tamaki
2017
Bounded real lemmas for positive descriptor systems. Zbl 1307.93176
Zhang, Qingling; Zhang, Youmei; Tanaka, Tamaki; Yan, Xinggang
2015
Inherited properties of nonlinear scalarizing functions for set-valued maps. Zbl 1222.49024
Kuwano, Issei; Tanaka, Tamaki; Yamada, Syuuji
2010
Existence theorem of cone saddle-points applying a nonlinear scalarization. Zbl 1105.90074
Kimura, Kenji; Tanaka, Tamaki
2006
$$N$$-fold supersymmetry for a periodic potential. Zbl 0972.81050
Aoyama, H.; Sato, M.; Tanaka, T.; Yamamoto, Mariko
2001
Another observation on conditions assuring $$\text{int }A+B= \text{int}(A+B)$$. Zbl 0797.52003
Tanaka, T.; Kuroiwa, D.
1994
Some general conditions assuring $$\text{int }A+B=\text{int}(A+B)$$. Zbl 0807.46012
Tanaka, Tamaki; Kuroiwa, Daishi
1993
On cone-extreme points in $$R^ n$$. Zbl 0624.90094
Tanaka, Tamaki
1987
Convex-cone-based comparisons of and difference evaluations for fuzzy sets. Zbl 1393.03031
Ike, Koichiro; Tanaka, Tamaki
2018
Sublinear scalarization methods for sets with respect to set-relations. Zbl 1400.90272
Ogata, Yuto; Saito, Yutaka; Tanaka, Tamaki; Yamada, Syuuji
2017
On Lusin’s theorem for non-additive measures that take values in an ordered topological vector space. Zbl 1315.28016
Watanabe, Toshikazu; Tanaka, Tamaki
2014
On a sufficient condition of Lusin’s theorem for non-additive measures that take values in an ordered topological vector space. Zbl 1253.28016
Watanabe, Toshikazu; Kawasaki, Toshiharu; Tanaka, Tamaki
2012
Pareto-efficient target by obtaining the facets of the efficient frontier in DEA. Zbl 1413.90148
Washio, Satoshi; Yamada, Syuuji; Tanaka, Tamaki; Tanino, Tetsuzo
2011
Cone-semicontinuity of set-valued maps by analogy with real-valued semicontinuity. Zbl 1217.49021
Sonda, Yuuya; Kuwano, Issei; Tanaka, Tamaki
2010
On vector equilibrium problems: Remarks on a general existence theorem and applications. Zbl 1032.49008
Kalmoun, El Mostafa; Riahi, Hassan; Tanaka, Tamaki
2001
On semicontinuity of set-valued maps and marginal functions. Zbl 1001.47029
Kimura, Yutaka; Tanaka, Kensuke; Tanaka, Tamaki
1999
Cone-quasiconvexity of vector-valued functions. Zbl 0836.90131
Tanaka, Tamaki
1995
Finite element applications to lake circulations and diffusion problems in Lake Biwa. Zbl 0442.76018
Tanaka, T.; Hirai, T.; Katayama, T.
1976
Convexity for compositions of set-valued map and monotone scalarizing function. Zbl 1334.49059
Kobayashi, Shogo; Saito, Yutaka; Tanaka, Tamaki
2016
On the $$l$$-extendability of quaternary linear codes. Zbl 1322.51005
Kanda, H.; Tanaka, T.; Maruta, T.
2015
On generalization of Ricceri’s theorem for Fan-Takahashi minimax inequality. Zbl 1319.49009
Saito, Yutaka; Tanaka, Tamaki; Yamada, Syuuji
2015
Outer approximation method incorporating a quadratic approximation for a DC programming problem. Zbl 1201.90188
Yamada, S.; Tanaka, T.; Tanino, T.
2010
Mean field theory of communication. Zbl 1189.94012
Tanaka, T.; Nakamura, K.
2008
A direct design of oversampled perfect reconstruction FIR filter banks. Zbl 1373.94711
Tanaka, T.
2006
Nonlinear analysis and convex analysis. Proceedings of the 3rd international conference (NACA2003), Tokyo, Japan, August 25–29, 2003. Zbl 1063.00012
Takahashi, Wataru (ed.); Tanaka, Tamaki (ed.)
2004
On inherited properties for set-valued maps and existence theorems for generalized vector equilibrium problems. Zbl 1052.49021
Nishizawa, Shogo; Tanaka, Tamaki
2004
Existence theorems of saddle points for vector valued functions. Zbl 1282.90168
Kimura, Kenji; Tanaka, Tamaki
2003
Multicriteria two-person zero-sum matrix games. Zbl 1259.90121
Higuchi, Masakazu; Tanaka, Tamaki
2003
Minimax theorems for vector-valued multifunctions. Zbl 0985.49505
Georgiev, Pando Gr.; Tanaka, Tamaki
2001
The Shapley value in totally convex multichoice games. Zbl 1068.91504
Ayoshin, D. A.; Tanaka, T.
2000
Modern numerical schemes for solving magnetohydrodynamic equations. Zbl 0915.76059
Murawski, K.; Tanaka, T.
1997
Approximately efficient solutions for vector optimization problems. Zbl 0939.90569
Tanaka, Tamaki
1995
Tracing the equilibrium path by dynamic relaxation in materially nonlinear problems. Zbl 0835.73057
Siddiquee, M. S. A.; Tanaka, T.; Tatsuoka, F.
1995
Observations on conditions assuring $$\text{int} A+B=\text{int}(A+B)$$. Zbl 0939.52500
Tanaka, Tamaki; Kuroiwa, Daishi; Tanaka, Kensuke
1994
Experiments on the unsteady drag and wake of a sphere at high Reynolds numbers. Zbl 1134.76689
Tsuji, Y.; Kato, N.; Tanaka, T.
1991
An alternative theorem for set-valued maps via set relations and its application to robustness of feasible sets. Zbl 1407.90238
Ogata, Yuto; Tanaka, Tamaki; Saito, Yutaka; Lee, Gue Myung; Lee, Jae Hyoung
2018
Convex-cone-based comparisons of and difference evaluations for fuzzy sets. Zbl 1393.03031
Ike, Koichiro; Tanaka, Tamaki
2018
Computational methods for set-relation-based scalarizing functions. Zbl 1417.90135
Yu, Hui; Ike, Koichiro; Ogata, Yuto; Saito, Yutaka; Tanaka, Tamaki
2017
Sublinear scalarization methods for sets with respect to set-relations. Zbl 1400.90272
Ogata, Yuto; Saito, Yutaka; Tanaka, Tamaki; Yamada, Syuuji
2017
Nonconvex separation functional in linear spaces with applications to vector equilibria. Zbl 06662673
Gutiérrez, César; Novo, Vicente; Ródenas-Pedregosa, Juan Luis; Tanaka, Tamaki
2016
Convexity for compositions of set-valued map and monotone scalarizing function. Zbl 1334.49059
Kobayashi, Shogo; Saito, Yutaka; Tanaka, Tamaki
2016
Bounded real lemmas for positive descriptor systems. Zbl 1307.93176
Zhang, Qingling; Zhang, Youmei; Tanaka, Tamaki; Yan, Xinggang
2015
On the $$l$$-extendability of quaternary linear codes. Zbl 1322.51005
Kanda, H.; Tanaka, T.; Maruta, T.
2015
On generalization of Ricceri’s theorem for Fan-Takahashi minimax inequality. Zbl 1319.49009
Saito, Yutaka; Tanaka, Tamaki; Yamada, Syuuji
2015
Positivity of continuous-time descriptor systems with time delays. Zbl 1360.93314
Zhang, Youmei; Zhang, Qingling; Tanaka, Tamaki; Yan, Xing-Gang
2014
On Lusin’s theorem for non-additive measures that take values in an ordered topological vector space. Zbl 1315.28016
Watanabe, Toshikazu; Tanaka, Tamaki
2014
Admissibility for positive continuous-time descriptor systems. Zbl 1307.93177
Zhang, Youmei; Zhang, Qingling; Tanaka, Tamaki; Cai, Min
2013
Continuity of cone-convex functions. Zbl 1261.90052
Kuwano, Issei; Tanaka, Tamaki
2012
On a sufficient condition of Lusin’s theorem for non-additive measures that take values in an ordered topological vector space. Zbl 1253.28016
Watanabe, Toshikazu; Kawasaki, Toshiharu; Tanaka, Tamaki
2012
Pareto-efficient target by obtaining the facets of the efficient frontier in DEA. Zbl 1413.90148
Washio, Satoshi; Yamada, Syuuji; Tanaka, Tamaki; Tanino, Tetsuzo
2011
Unified scalarization for sets and set-valued Ky Fan minimax inequality. Zbl 1221.49025
Kuwano, Issei; Tanaka, Tamaki; Yamada, Syuuji
2010
Inherited properties of nonlinear scalarizing functions for set-valued maps. Zbl 1222.49024
Kuwano, Issei; Tanaka, Tamaki; Yamada, Syuuji
2010
Cone-semicontinuity of set-valued maps by analogy with real-valued semicontinuity. Zbl 1217.49021
Sonda, Yuuya; Kuwano, Issei; Tanaka, Tamaki
2010
Outer approximation method incorporating a quadratic approximation for a DC programming problem. Zbl 1201.90188
Yamada, S.; Tanaka, T.; Tanino, T.
2010
Existence of vector equilibria via Ekeland’s variational principle. Zbl 1172.65032
Araya, Yousuke; Kimura, Kenji; Tanaka, Tamaki
2008
Minimal element theorem with set-relations. Zbl 1145.49013
Shimizu, Akira; Tanaka, Tamaki
2008
Mean field theory of communication. Zbl 1189.94012
Tanaka, T.; Nakamura, K.
2008
Iterative construction of fixed points of nonself-mappings in Banach spaces. Zbl 1120.65075
Zeng, Lu-Chuan; Tanaka, Tamaki; Yao, Jen-Chih
2007
Optimality conditions in set-valued optimization using nonlinear scalarization methods. Zbl 1186.90123
Shimizu, Akira; Nishizawa, Shogo; Tanaka, Tamaki
2007
Existence theorem of cone saddle-points applying a nonlinear scalarization. Zbl 1105.90074
Kimura, Kenji; Tanaka, Tamaki
2006
A direct design of oversampled perfect reconstruction FIR filter banks. Zbl 1373.94711
Tanaka, T.
2006
Alternative theorems for set-valued maps based on a nonlinear scalarization. Zbl 1105.90076
Nishizawa, Shogo; Onodsuka, Maki; Tanaka, Tamaki
2005
Nonlinear analysis and convex analysis. Proceedings of the 3rd international conference (NACA2003), Tokyo, Japan, August 25–29, 2003. Zbl 1063.00012
Takahashi, Wataru (ed.); Tanaka, Tamaki (ed.)
2004
On inherited properties for set-valued maps and existence theorems for generalized vector equilibrium problems. Zbl 1052.49021
Nishizawa, Shogo; Tanaka, Tamaki
2004
On inherited properties of set-valued maps. Zbl 1255.49032
Nishizawa, Shogo; Tanaka, Tamaki; Georgiev, Pando Gr.
2003
Multi-objective programming and goal-programming. Theory and applications. Proceeedings of the conference MOPGP’02, Nara, Japan, June 4–7, 2002. Zbl 1051.90003
Tanino, Tetsuzo (ed.); Tanaka, Tamaki (ed.); Inuiguchi, Masahiro (ed.)
2003
Existence theorems of saddle points for vector valued functions. Zbl 1282.90168
Kimura, Kenji; Tanaka, Tamaki
2003
Multicriteria two-person zero-sum matrix games. Zbl 1259.90121
Higuchi, Masakazu; Tanaka, Tamaki
2003
Large-eddy simulation of turbulent gas-particle flow in a vertical channel: Effect of considering inter-particle collisions. Zbl 1004.76513
Yamamoto, Y.; Potthoff, M.; Tanaka, T.; Kajishima, T.; Tsuji, Y.
2001
General forms of a $$N$$-fold supersymmetric family. Zbl 0977.81021
Aoyama, H.; Sato, M.; Tanaka, T.
2001
Fan’s inequality for set-valued maps. Zbl 1042.49512
Georgiev, Pando Gr.; Tanaka, Tamaki
2001
$$N$$-fold supersymmetry for a periodic potential. Zbl 0972.81050
Aoyama, H.; Sato, M.; Tanaka, T.; Yamamoto, Mariko
2001
On vector equilibrium problems: Remarks on a general existence theorem and applications. Zbl 1032.49008
Kalmoun, El Mostafa; Riahi, Hassan; Tanaka, Tamaki
2001
Minimax theorems for vector-valued multifunctions. Zbl 0985.49505
Georgiev, Pando Gr.; Tanaka, Tamaki
2001
Vector-valued set-valued variants of Ky Fan’s inequality. Zbl 0987.49010
Georgiev, Pando Gr.; Tanaka, Tamaki
2000
The Shapley value in totally convex multichoice games. Zbl 1068.91504
Ayoshin, D. A.; Tanaka, T.
2000
On semicontinuity of set-valued maps and marginal functions. Zbl 1001.47029
Kimura, Yutaka; Tanaka, Kensuke; Tanaka, Tamaki
1999
On cone convexity of set-valued maps. Zbl 0895.26010
Kuroiwa, Daishi; Tanaka, Tamaki; Ha, Truong Xuan Duc
1997
Generalized semicontinuity and existence theorems for cone saddle points. Zbl 0894.90132
Tanaka, Tamaki
1997
Modern numerical schemes for solving magnetohydrodynamic equations. Zbl 0915.76059
Murawski, K.; Tanaka, T.
1997
Approximately efficient solutions in vector optimization. Zbl 0871.90076
Tanaka, Tamaki
1996
Cone-quasiconvexity of vector-valued functions. Zbl 0836.90131
Tanaka, Tamaki
1995
Approximately efficient solutions for vector optimization problems. Zbl 0939.90569
Tanaka, Tamaki
1995
Tracing the equilibrium path by dynamic relaxation in materially nonlinear problems. Zbl 0835.73057
Siddiquee, M. S. A.; Tanaka, T.; Tatsuoka, F.
1995
Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions. Zbl 0826.90102
Tanaka, T.
1994
Finite volume TVD scheme on an unstructured grid system for three- dimensional MHD simulation of inhomogeneous systems including strong background potential fields. Zbl 0797.76061
Tanaka, T.
1994
Another observation on conditions assuring $$\text{int }A+B= \text{int}(A+B)$$. Zbl 0797.52003
Tanaka, T.; Kuroiwa, D.
1994
Observations on conditions assuring $$\text{int} A+B=\text{int}(A+B)$$. Zbl 0939.52500
Tanaka, Tamaki; Kuroiwa, Daishi; Tanaka, Kensuke
1994
The convexity of $$A$$ and $$B$$ assures $$\text{int} A + B = \text{int}(A + B)$$. Zbl 0810.52003
Tanaka, Tamaki; Kuroiwa, Daishi
1993
Some general conditions assuring $$\text{int }A+B=\text{int}(A+B)$$. Zbl 0807.46012
Tanaka, Tamaki; Kuroiwa, Daishi
1993
Two types of minimax theorems for vector-valued functions. Zbl 0696.90060
Tanaka, T.
1991
Experiments on the unsteady drag and wake of a sphere at high Reynolds numbers. Zbl 1134.76689
Tsuji, Y.; Kato, N.; Tanaka, T.
1991
A characterization of generalized saddle points for vector-valued functions via scalarization. Zbl 0956.90507
Tanaka, Tamaki
1990
Cone-convexity of vector-valued functions. Zbl 0721.90063
Tanaka, Tamaki
1990
Existence theorems for cone saddle points of vector-valued functions in infinite-dimensional spaces. Zbl 0652.49011
Tanaka, T.
1989
Some minimax problems of vector-valued functions. Zbl 0628.90078
Tanaka, T.
1988
On cone-extreme points in $$R^ n$$. Zbl 0624.90094
Tanaka, Tamaki
1987
Finite element applications to lake circulations and diffusion problems in Lake Biwa. Zbl 0442.76018
Tanaka, T.; Hirai, T.; Katayama, T.
1976
all top 5
#### Cited by 567 Authors
17 Tanaka, Tamaki 13 Novo-Sanjurjo, Vicente 12 Huang, Nan-Jing 11 Rodriguez Marin, Luis 9 Ansari, Qamrul Hasan 9 Gutiérrez, César 9 Hamel, Andreas H. 9 Jiménez, Bienvenido 9 Kuroiwa, Daishi 9 Yao, Jen-Chih 8 Fang, Yaping 8 Hernández, Elvira 8 Yang, Xinmin 7 Chadli, Ouayl 6 Anh, Lam Quoc 6 Flores-Bazán, Fabián 6 Han, Yu 6 Lee, Gue Myung 6 Lin, Lai-Jiu 6 Sama, Miguel 6 Truong Xuan Duc Ha 6 Wangkeeree, Rabian 5 Jahn, Johannes 5 Kassay, Gábor 5 Lalitha, C. S. 5 Li, Shengjie 4 Ćirić, Ljubomir B. 4 Dinh The Luc 4 Gombosi, Tamas I. 4 Goswami, Partha Sarathi 4 Khorram, Esmaile 4 Köbis, Elisabeth 4 Kumaran, V. 4 Li, Zhongfei 4 Löhne, Andreas 4 Peng, Zaiyun 4 Plubtieng, Somyot 4 Simonin, Olivier 4 Tanaka, Toshiaki 4 Tanaka, Toyoto 4 Wang, Guoliang 4 Wang, Shouyang 4 Yuan, George Xian-Zhi 3 Alipchenkov, Vladimir Mikhailovich 3 Boţ, Radu Ioan 3 Capătă, Adela Elisabeta 3 Chatterjee, Prashanto 3 Chiang, Yungyen 3 Crespi, Giovanni Paolo 3 Flores-Bazán, Fernando 3 Grad, Sorin-Mihai 3 Güvenç, İlknur Atasever 3 Hadjisavvas, Nicolas 3 Hu, Rong 3 Ike, Koichiro 3 Karaman, Emrah 3 Kuwano, Issei 3 Miglierina, Enrico 3 Miholca, Mihaela 3 Molho, Elena 3 Ogata, Yuto 3 Saito, Yutaka 3 Schrage, Carola 3 Sharma, Pradeep Kumar 3 Sitthithakerngkiet, Kanokwan 3 Song, Yisheng 3 Soyertem, Mustafa 3 Vílchez, A. 3 Vui, Pham Thi 3 Wang, Bing 3 Zhang, Qingling 3 Zhang, Yu 2 Ahmadi, Goodarz 2 Al-Homidan, Suliman S. 2 Alonso, María Dolores 2 Amahroq, Tijani 2 Araya, Yousuke 2 Bao, Truong Quang 2 Borée, Jacques 2 Cai, Hongyang 2 Caraman, N. 2 Chang, Shih-Sen 2 Chen, Guangya 2 Cheng, Cao-Zong 2 Chuang, Chih-Sheng 2 Csetnek, Ernö Robert 2 De Sterck, Hans 2 De Zeeuw, Darren L. 2 Derksen, Jos J. 2 Duy, Tran Quoc 2 Eichfelder, Gabriele 2 Engau, Alexander 2 Fakhar, Majid 2 Farajzadeh, Ali P. 2 Feng, Xueshang 2 Fox, Rodney O. 2 Georgiev, Pando Grigorov 2 Guo, Xiaocheng 2 Kalmoun, El Mostafa 2 Kartushinskij, A. I. ...and 467 more Authors
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#### Cited in 92 Serials
52 Journal of Optimization Theory and Applications 40 Optimization 24 Journal of Global Optimization 17 Journal of Computational Physics 16 Journal of Mathematical Analysis and Applications 16 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 15 Journal of Fluid Mechanics 13 Optimization Letters 9 Physics of Fluids 8 Mathematical Methods of Operations Research 7 Numerical Functional Analysis and Optimization 7 Applied Mathematics Letters 7 Journal of Inequalities and Applications 7 Fixed Point Theory and Applications 6 European Journal of Operational Research 6 Positivity 5 Computers and Fluids 5 Computers & Mathematics with Applications 5 Journal of Mathematical Physics 5 Nihonkai Mathematical Journal 4 Journal of the Franklin Institute 4 Fuzzy Sets and Systems 4 Abstract and Applied Analysis 3 Fluid Dynamics 3 International Journal of Heat and Mass Transfer 3 Applied Mathematics and Computation 3 Journal of Computational and Applied Mathematics 3 Acta Mathematicae Applicatae Sinica. English Series 3 Applied Mathematical Modelling 3 Top 3 Taiwanese Journal of Mathematics 3 Discrete Dynamics in Nature and Society 3 Flow, Turbulence and Combustion 3 International Journal of Systems Science. Principles and Applications of Systems and Integration 2 Applicable Analysis 2 Acta Mathematica Vietnamica 2 Operations Research Letters 2 SIAM Journal on Optimization 2 Mathematical Problems in Engineering 2 Optimization Methods & Software 2 Journal of Systems Science and Complexity 2 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 2 Thai Journal of Mathematics 2 Set-Valued and Variational Analysis 2 Nonlinear Analysis. Theory, Methods & Applications 1 Modern Physics Letters A 1 International Journal of Modern Physics A 1 Acta Mechanica 1 Artificial Intelligence 1 Bulletin of the Australian Mathematical Society 1 International Journal of General Systems 1 International Journal for Numerical Methods in Fluids 1 Theoretical and Computational Fluid Dynamics 1 Information Sciences 1 Mathematische Nachrichten 1 Proceedings of the American Mathematical Society 1 Proceedings of the Japan Academy. Series A 1 Journal of Information & Optimization Sciences 1 Zeitschrift für Analysis und ihre Anwendungen 1 Circuits, Systems, and Signal Processing 1 Applied Mathematics and Mechanics. (English Edition) 1 Physica D 1 Asia-Pacific Journal of Operational Research 1 Annals of Operations Research 1 Annals of Physics 1 Automation and Remote Control 1 Journal of Non-Equilibrium Thermodynamics 1 Computational Optimization and Applications 1 Turkish Journal of Mathematics 1 Finite Fields and their Applications 1 Journal of Convex Analysis 1 Soft Computing 1 Acta Mathematica Sinica. English Series 1 Communications in Nonlinear Science and Numerical Simulation 1 Journal of Turbulence 1 Granular Matter 1 Journal of Function Spaces and Applications 1 Fuzzy Optimization and Decision Making 1 Science in China. Series F 1 Journal of Industrial and Management Optimization 1 Communications in Mathematical Analysis 1 Acta Mechanica Sinica 1 Journal of Fixed Point Theory and Applications 1 JP Journal of Fixed Point Theory and Applications 1 Advances in Mathematical Physics 1 International Journal of Nonlinear Analysis and Applications 1 Asian Journal of Control 1 Croatian Operational Research Review (CRORR) 1 Journal of the Operations Research Society of China 1 Open Mathematics 1 AMM. Applied Mathematics and Mechanics. (English Edition) 1 Journal of Nonlinear and Variational Analysis
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#### Cited in 36 Fields
214 Operations research, mathematical programming (90-XX) 151 Calculus of variations and optimal control; optimization (49-XX) 63 Fluid mechanics (76-XX) 49 Operator theory (47-XX) 28 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 25 General topology (54-XX) 22 Numerical analysis (65-XX) 19 Functional analysis (46-XX) 19 Systems theory; control (93-XX) 13 Real functions (26-XX) 10 Convex and discrete geometry (52-XX) 10 Global analysis, analysis on manifolds (58-XX) 10 Quantum theory (81-XX) 7 Ordinary differential equations (34-XX) 6 Astronomy and astrophysics (85-XX) 5 Dynamical systems and ergodic theory (37-XX) 4 Order, lattices, ordered algebraic structures (06-XX) 4 Measure and integration (28-XX) 4 Classical thermodynamics, heat transfer (80-XX) 3 Mathematical logic and foundations (03-XX) 3 Probability theory and stochastic processes (60-XX) 3 Computer science (68-XX) 3 Statistical mechanics, structure of matter (82-XX) 2 Difference and functional equations (39-XX) 2 Mechanics of deformable solids (74-XX) 2 Geophysics (86-XX) 2 Information and communication theory, circuits (94-XX) 1 General and overarching topics; collections (00-XX) 1 Nonassociative rings and algebras (17-XX) 1 Partial differential equations (35-XX) 1 Approximations and expansions (41-XX) 1 Geometry (51-XX) 1 Algebraic topology (55-XX) 1 Statistics (62-XX) 1 Optics, electromagnetic theory (78-XX) 1 Biology and other natural sciences (92-XX)
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# pair_style spin/magelec command¶
## Syntax¶
pair_style spin/magelec cutoff
• cutoff = global cutoff pair (distance in metal units)
## Examples¶
pair_style spin/magelec 4.5
pair_coeff * * magelec 4.5 0.00109 1.0 1.0 1.0
## Description¶
Style spin/me computes a magneto-electric interaction between pairs of magnetic spins. According to the derivation reported in (Katsura), this interaction is defined as:
$\begin{split}\vec{\omega}_i & = -\frac{1}{\hbar} \sum_{j}^{Neighb} \vec{s}_{j}\times\vec{D}(r_{ij}) \\ \vec{F}_i & = -\sum_{j}^{Neighb} \frac{\partial D(r_{ij})}{\partial r_{ij}} \left(\vec{s}_{i}\times \vec{s}_{j} \right) \cdot \vec{r}_{ij}\end{split}$
where $$\vec{s}_i$$ and $$\vec{s}_j$$ are neighboring magnetic spins of two particles.
From this magneto-electric interaction, each spin i will be submitted to a magnetic torque omega, and its associated atom can be submitted to a force F for spin-lattice calculations (see fix nve/spin), such as:
$\begin{split}\vec{F}^{i} & = -\sum_{j}^{Neighbor} \left( \vec{s}_{i}\times \vec{s}_{j} \right) \times \vec{E} \\ \vec{\omega}^{i} = -\frac{1}{\hbar} \sum_{j}^{Neighbor} \vec{s}_j \times \left(\vec{E}\times r_{ij} \right)\end{split}$
with h the Planck constant (in metal units) and $$\vec{E}$$ an electric polarization vector. The norm and direction of E are giving the intensity and the direction of a screened dielectric atomic polarization (in eV).
More details about the derivation of these torques/forces are reported in (Tranchida).
## Restrictions¶
All the pair/spin styles are part of the SPIN package. These styles are only enabled if LAMMPS was built with this package, and if the atom_style “spin” was declared. See the Build package doc page for more info.
## Default¶
none
(Katsura) H. Katsura, N. Nagaosa, A.V. Balatsky. Phys. Rev. Lett., 95(5), 057205. (2005)
(Tranchida) Tranchida, Plimpton, Thibaudeau, and Thompson, Journal of Computational Physics, 372, 406-425, (2018).
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# How to define a listing example environment
I have read the listing manual document. I would like to use the lstsample environment.
It looks like below:
Basically, the output is on the left and the code is placed on the right.
But I don't find where this environment is defined. Is it easy to define a new environment like this?
-
Since you have some responses below that seem to answer your question, please consider marking one of them as ‘Accepted’ by clicking on the tickmark below their vote count (see How do you accept an answer?). This shows which answer helped you most, and it assigns reputation points to the author of the answer (and to you!). It's part of this site's idea to identify good questions and answers through upvotes and acceptance of answers. – Jubobs Mar 1 '14 at 19:15
The complete package listings can be found at CTAN.
http://www.ctan.org/pkg/listings
The package provides a huge dtx-file and a Makefile. Inside the dtx-file you can find the definition (see code below).
However if you use the make all the compilation produces the file listings-devel.pdf where you can find the user documentation and the developer documentation.
If this part is to hard please have a look at the package showexpl
% \subsection{The \texttt{lstsample} environment}
%
% \begin{environment}{lstsample}
% We store the verbatim part and write the source code also to file.
% \begin{macrocode}
\lst@RequireAspects{writefile}
% \end{macrocode}
% \begin{macrocode}
\newbox\lst@samplebox
\lstnewenvironment{lstsample}[3][]
{\global\let\lst@intname\@empty
\gdef\lst@sample{#2}%
\setbox\lst@samplebox=\hbox\bgroup
\setkeys{lst}{language={},style={},tabsize=4,gobble=5,%
basicstyle=\small\ttfamily,basewidth=0.51em,point={#1}}
#3%
\lst@BeginAlsoWriteFile{\jobname.tmp}}
{\lst@EndWriteFile\egroup
% \end{macrocode}
% Now |\lst@samplebox| contains the verbatim part.
% If it's too wide, we use atop and below instead of left and right.
% \begin{macrocode}
\ifdim \wd\lst@samplebox>.5\linewidth
\begin{center}%
\hbox to\linewidth{\box\lst@samplebox\hss}%
\end{center}%
\lst@sampleInput
\else
\begin{center}%
\begin{minipage}{0.45\linewidth}\lst@sampleInput\end{minipage}%
\begin{minipage}{0.45\linewidth}%
\hbox to\linewidth{\box\lst@samplebox\hss}%
\end{minipage}%
\end{center}%
\fi}
% \end{macrocode}
% The new keyword class \keyname{point}.
% \begin{macrocode}
\lst@InstallKeywords{p}{point}{pointstyle}\relax{keywordstyle}{}ld
% \end{macrocode}
% \end{environment}
%
% \begin{environment}{lstxsample}
% Omitting |\lst@EndWriteFile| leaves the file open.
% \begin{macrocode}
\lstnewenvironment{lstxsample}[1][]
{\begingroup
\setkeys{lst}{belowskip=-\medskipamount,language={},style={},%
tabsize=4,gobble=5,basicstyle=\small\ttfamily,%
basewidth=0.51em,point={#1}}
\lst@BeginAlsoWriteFile{\jobname.tmp}}
{\endgroup
\endgroup}
% \end{macrocode}
% \end{environment}
%
% \begin{macro}{\lst@sampleInput}
% inputs the left-hand' side.
% \begin{macrocode}
\def\lst@sampleInput{%
\MakePercentComment\catcode\^^M=10\relax
\small\lst@sample
{\setkeys{lst}{SelectCharTable=\lst@ReplaceInput{\^\^I}%
{\lst@ProcessTabulator}}%
\leavevmode \input{\jobname.tmp}}\MakePercentIgnore}
% \end{macrocode}
% \end{macro}
%
-
As mentioned by Marco Daniel in his answer, the LTXexample environment from the showexpl package easily allows you to display source code beside the output that that produces. Here is an example:
Usually the LTXexample environment is used to display the LaTeX source code and the associated output:
## Code:
\documentclass{article}
\usepackage{amsmath}
\usepackage{xcolor}
\usepackage{showexpl}
\lstdefinestyle{ListingSample}{
basicstyle=\small\ttfamily,
numbers=none,
keywordstyle=\color{blue}\bfseries,
morekeywords={begin,end,for,maxint,to,do},
pos=l,
}
\lstdefinestyle{myLatexStyle}{
language=TeX,
basicstyle=\small\ttfamily,
numbers=none,
backgroundcolor=\color{yellow},
numbers=left, numberstyle=\tiny, stepnumber=2, numbersep=5pt,
showstringspaces=false,
keywordstyle=\color{blue}\bfseries,
pos=l,
}
\begin{document}
\begin{LTXexample}[style=ListingSample]
\begin{lstlisting}
for i:=maxint to 0 do
begin
{ do nothing }
end;
\end{lstlisting}
\end{LTXexample}
\begin{LTXexample}[style=myLatexStyle]
\newcommand{\Command}[1]{%
\texttt{\textbackslash#1}%
}%
Inline math is specified within
a pair of \$as in$E = mc^2\$.
Display math uses \Command{[}
and \Command{]}.
For example: $E = mc^2$
\end{LTXexample}
\end{document}
-
Here's a rude way to do it so abusing the minipage environment:
\begin{figure}[H]
\begin{minipage}[b]{0.5\linewidth}
\begin{lstlisting}[frame=trBL]
for i:=maxint to 0 do
begin
{ do nothing }
end;
\end{lstlisting}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.5\linewidth}
{\setstretch{.9}
\begin{verbatim}
\begin{lstlisting}
for i:=maxint to 0 do
begin
{ do nothing }
end;
\end{lstlisting}
\end{verbatim}
}
\end{minipage}
\end{figure}
You might as well need the packages to make it work:
\usepackage{setspace} %make the \setstrech
\usepackage{listings} %lstlisting function
\usepackage[spanish]{babel} %i work under spanish documents so if it doesnt work put this
\usepackage{float} %this one is important so the text before and after the object whit [H] will work
-
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# Sergei Yakovenko's blog: on Math and Teaching
## Wednesday, December 26, 2007
### “Auxiliary Lesson” שעור עזר)#10) December 27, 2007
Filed under: Analytic ODE course,lecture,links — Sergei Yakovenko @ 5:56
Because of the dismal failure to meet the schedule in Lesson 9, the next meeting will deal with the items from the previous list that are not rendered in blue.
In the meantime you may enjoy the funny animations illustrating differences in the convergence patterns of Taylor and Fourier series for various functions (thanks to D. K. for pointing me to the site). Note the appearance of the Gibbs phenomenon for Fourier series of discontinuous functions.
## Dynamics generated by finitely generated subgroups of conformal germs
1. Generic subgroups of $\text{Diff}(\mathbb C^1,0)$ are non-solvable.
2. Dynamics generated by several germs. Definition of a pseudogroup. Orbits of points.
3. Periodicity of germs (finiteness of order) vs. periodicity of orbits. Cycles and limit cycles of pseudogroups.
4. Convergence of elements in pseudogroups. Closure.
5. Density of orbits. Linear subgroups. Abundance of limit cycles for generic (nonsolvable) subgroups of $\text{Diff}(\mathbb C^1,0)$.
6. Topological equivalence of subgroups and pseudogroups. Conjugacy of dense linear subgroups.
7. Rigidity of nonsolvable subgroups: topological conjugacy implies holomorphic conjugacy.
Disclaimer… if somebody still needs it… 😦
Reading: Section 6 (second part) from the book, printing disabled.
## Topological properties of Abelian integrals
The second “learning in groups” meeting will be devoted to the study of the Gauss–Manin connexion in homology, which will ultimately result in a local representation of Abelian integrals as linear combinations of real powers and logarithms with analytic coefficients analytically depending on parameters.
This representation already suffices to produce local uniform bounds for the number of isolated zeros, as was explained on the previous Tuesday.
Recommended reading: Section 26 from the book (printing disabled), esp., subsections F and I-K.
Time and location: Tuesday Dec. 18, 2007, 14:00 (in place of the usual Geometry & Topology seminar time), Pekeris Room.
What it will be about: 😉
## Finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$, I. Formal theory.
1. Formal normal form for a single holomorphic self-map from $\text{Diff}(\mathbb C^1,0)$. Parabolic germs.
2. Bochner theorem on holomorphic linearization of finite groups.
3. Stratification of the subgroup of parabolic germs $\text{Diff}_1(\mathbb C^1,0)$.
4. Tits alternative for finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$: every such subgroup is either metabelian (its commutator is commutative, e.g., trivial), or non-solvable (all iterated commutators are nontrivial).
5. Centralizers and symmetries: formal classification of solvable subgroups.
6. Integrable germs and their holomorphic linearizability.
Recommended reading: Section 6 (first part) from the book (printing disabled)
Disclaimer applies, as usual 😦
## Thursday, December 6, 2007
### Seminar on Khovanskii-Varchenko theorem (I)
Filed under: research seminar — Sergei Yakovenko @ 5:54
Tags: , , , ,
We (D. Novikov and S.Y.) launch a campaign “Learn Khovanskii–Varchenko Theorem“. A few (2-4) next weeks we will discuss in detail the proof of this remarkably simple but powerful result with a view to have a number of generalizations.
The two manuscripts (one in Russian, another in English) are available:
Time and location: Tuesdays, 16:00-18:00, Room 261 (unless otherwise announced).
The first meeting: Dec 11, 2007.
Fewnomial theory (S.Y.). This purely geometric theory starts with a multidimensional generalization of the Rolle theorem for several variables and allows to prove infinitely many both classical and new results starting from the Descartes’ rule.
If somebody has a scanned copy of the English original by Khovanskii, please post a link in comments.
## Invariant manifolds for hyperbolic maps. Complex hyperbolicity.
1. Formal theory: cross-resonances.
2. Hadamard-Perron theorem for holomorphisms. Contracting map principle reactivated.
3. Hadamard-Perron theorem for vector fields. Complex hyperbolicity.
4. Invariant hypernolic curve for saddle-nodes.
5. Poincare resonances.
6. Center manifolds: formal but non-analytic.
Disclaimer is as sadly relevant as before…
Blog at WordPress.com.
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# All Questions
25,326 questions
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### Homology of $\mathrm{PGL}_2(F)$
Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...
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## Introduction
Resident tissue macrophages (RTM) are present in most mammalian tissues. Historically known for their roles in host defense and clearance of dead cells, RTM are now recognized as an integral part of the tissues in which they reside, where they can contribute to a wide range of physiological and pathological processes1,2,3.
RTM populations are very heterogeneous, phenotypically and functionally1,2,3, and the tissue of residence is thought to be a major driver of such diversity4,5. According to the niche model, RTM are imprinted by niche-derived tissue-instructive signals that trigger expression of specific differentiation programs, thus tailoring a particular identity, i.e., a phenotypic and functional specialization that fulfills the functional needs of a given tissue5. Supporting this, recent mouse studies have shown that distinct precursors have the potential to give rise to the same particular RTM population when the niche is empty5,6,7,8.
In mice, the well-known alveolar macrophages (AM) differentiate from fetal monocytes, are maintained by self-renewal and are specialized in removal and recycling of surfactant molecules9,10,11. Besides AM, non-alveolar lung macrophages, i.e., the interstitial macrophages (IM), have been shown to contribute to lung immune homeostasis by spontaneously producing the immunosuppressive cytokine IL-10 and preventing the development of aberrant type 2 allergic responses against inhaled allergens12,13,14,15. In addition, they may substantially contribute to the reduced risk of asthma in a microbe-rich environment (i.e., the so-called hygiene hypothesis16,17). Indeed, we have reported that exposure to bacterial unmethylated CpG-DNA (CpG) expands tolerogenic IM from monocyte precursors, thereby conferring robust protection against allergic asthma18. IL-10-producing IM have also been described in humans19, and clinical evidence suggests that they may be functionally impaired in asthmatic patients20.
Despite their functional relevance, IM were long merely investigated as a bulk population12,13,14,18. In 2017, Gibbings et al. proposed the existence of three phenotypically distinct IM populations in the steady-state lung based on the differential expression of MHC-II and CD11c21. More recently, Chakarov and colleagues identified two conserved monocyte-derived IM subpopulations across tissues, in mice and humans22. In the mouse lung, they characterized nerve-associated Lyve-1loMHC-IIhi and blood vessel-associated Lyve-1hiMHC-IIlo monocyte-derived IM subsets, supporting that the lung IM pool is heterogeneous and encompasses distinct populations carrying their own identity.
Here, we analyze >3000 lung tissue mononuclear cells expressing the high affinity immunoglobulin gamma Fc receptor (Fcgr1, CD64) by droplet-based single-cell RNA-sequencing (scRNA-seq) in adult mice. Our study independently confirms the existence of two main subpopulations of lung IM22 and further expands our knowledge about their origin, half-life, localization, functional properties and dynamics upon local exposure to microbial products. Moreover, we uncover a discrete population of extravascular NR4A1-dependent monocytes transitioning from intravascular Ly-6Clo patrolling monocytes towards a specific subset of IM. These results contribute to a better appreciation of the diversity of the lung mononuclear phagocyte system (MPS), an important step toward greater precision and effectiveness of macrophage-targeted therapies.
## Results
### Two subsets of IM and monocytes populate the mouse lung
To map mouse lung tissue macrophages (i.e., lung IM) in naive C57BL/6 female wild-type (WT) mice, we performed scRNA-seq using the 10x Genomics platform23. Lung IM were defined as singlet mononuclear cell-enriched CD45+ non-autofluorescent SSCloF4/80+CD11cLy-6CloCD64+ cells18 (Fig. 1a, and Supplementary Table 1). Exclusion of lung-resident F4/80+Siglec-F+ eosinophils24 based on high SSC was efficient and resulted in minimal loss of IM (Supplementary Fig. 1). In a first experiment, 10-week-old mice were used and a total of 1715 IM, together with 199 AM, were analyzed (Fig. 1b, and Supplementary Fig. 2). In addition, a second scRNA-seq experiment was performed through an independent platform using older mice (i.e., 6-month-old) coming from a different animal facility, and 1682 IM were analyzed (Fig. 1b, and Supplementary Fig. 2).
Non-linear dimensional reduction (t-distributed stochastic neighbor embedding [t-SNE]) and graph-based clustering of single cells merged from both experiments identified 4 transcriptionally distinct clusters of monocytes/macrophages (Fig. 1c, and Supplementary Fig. 3a, b)25,26. Cluster 3 represented AM (Supplementary Fig. 3c, d)11, and Clusters 1, 2, and 4 were distributed in the same proportions in both experiments and were characterized by higher expression of Cx3cr1, Mafb, Cd14, and Cd74 as compared to AM (Supplementary Fig. 3c, d), supporting the contention that it comprised lung tissue IM.
Clusters 1, 2, and 4 exhibited unique transcriptional signatures (Supplementary Fig. 4a, b), including upregulation of transcripts encoding proteins detectable by flow cytometry: MHC-II-related transcripts (e.g., H2-Eb1, H2-Ab1, Cd74) in Cluster 1; transcripts encoding macrophage mannose receptor (Mrc1, encoding CD206), the scavenger receptor Cd163, folate receptor beta precursor (Folr2) and lymphatic endothelium hyaluronan receptor-1 (Lyve1) in Cluster 2; and transcripts encoding angiotensin-converting enzyme (Ace) and low affinity immunoglobulin gamma Fc region receptor IV (Fcgr4, encoding CD16.2) in Cluster 4 (Fig. 1d, and Supplementary Fig. 4c).
Using antibodies directed against CD206 and CD16.2, we showed that expression of these markers was mutually exclusive within CD64+ IM and 3 subpopulations were identified: a minor population of CD16.2+CD206 cells, which co-expressed ACE and corresponded to Cluster 4 (dark blue cells, defined as [CD64+]CD16.2+ [monocytes] hereafter, Fig. 1e–h); CD16.2CD206+ cells, in which a fraction uniquely expressed Lyve-1 and FOLR2, and corresponding to Cluster 2 (orange cells, defined as CD206+ [IM] hereafter, Fig. 1e–h); and CD16.2CD206 cells, which were expressing significantly higher levels of MHC-II as compared to the other subsets and corresponded to Cluster 1 (light blue cells, defined as CD206[IM] hereafter, Fig. 1e–j). Of note, CD206and CD206+ subsets largely overlapped with Lyve-1loMHC-IIhi and Lyve-1hiMHC-IIlo IM subsets described by Chakarov et al.22, as well as with IM3 and IM1/IM2 subsets described by Gibbings et al., respectively (Supplementary Fig. 5).
Morphologically, CD206+ IM uniquely displayed vacuoles in their cytoplasm and a larger size as compared to CD206 IM and CD64+CD16.2+ cells (Fig. 2a, b). Immunostaining against Lamp-1, a lysosomal marker, suggested that the vacuoles seen in CD206+ IM were lysosomes (Fig. 2c, and Supplementary Fig. 6). Phenotypically, CD206IM expressed higher levels of CX3CR1, whereas CD206+ IM expressed higher levels of the macrophage-associated markers MerTK and CD68 as compared to the 2 other subpopulations (Fig. 2c, and Supplementary Fig. 6). Moreover, CD64+CD16.2+ cells expressed higher levels of CD11b and CD115 and lower levels of MerTK as compared to both IM subsets (Fig. 2c, and Supplementary Fig. 6), consistent with the idea that CD64+CD16.2+ cells were monocytes.
Next, we injected fluorescent-conjugated anti-CD45 antibodies intravenously (i.v.) to label intravascular leukocytes before the sacrifice (Fig. 2d). AM, CD206+, and CD206 IM were only marginally stained by such antibodies (Fig. 2e, f), confirming that these cells were mainly extravascular. Expectedly, nearly all patrolling Ly-6Clo monocytes and a majority of classical Ly-6Chi monocytes were labeled, confirming the existence of tissue-associated Ly-6Chi monocytes (Fig. 2e, f)18,27. However, the percentage of CD64+CD16.2+ monocytes that were stained with the anti-CD45 in vivo exhibited a high variability and was significantly lower than the one of patrolling Ly-6Clo monocytes (Fig. 2e, f), suggesting that a substantial fraction of CD64+CD16.2+ monocytes was truly located in the lung tissue.
We also sought to test the ability of each subpopulation to engulf large particles (i.e., E. coli bioparticles conjugated with a pH-sensitive dye), i.e. a functional hallmark of macrophages (Fig. 2g). Like AM, CD64+CD16.2+ monocytes, CD206+ and CD206 IM were able to phagocyte airborne and blood-borne particles, with significantly higher percentages of cells when particles were injected i.t. as compared to i.v. (Fig. 2h). After i.t. injection, percentages of fluorescent CD206+ IM were significantly higher than those of CD206 IM, which might indicate a preferential localization around the airways (Fig. 2h).
So far, our data suggest that, in addition to dendritic cells (DCs) and tissue Ly-6Chi classical monocytes18,27, the lung MPS comprises 3 subpopulations of Ly-6CloCD64+ mononuclear phagocytes, namely CD206+ IM, CD206 IM, and non-classical CD64+CD16.2+ monocytes.
### IM subsets are long-lived, unlike NR4A1-dependent monocytes
While previous studies have provided evidence that IM were monocyte-derived cells in adults18,21,22,28, they did not exclude the possibility that part of the IM compartment may be self-maintaining in the tissue. To assess the half-life of IM subpopulations, we used the tamoxifen(TAM)-inducible Cx3cr1CreERT2.Rosa26-LSL-YFP fate-mapping mouse model29, and TAM-injected Cx3cr1CreERT2.Rosa26-LSL-YFP mice were longitudinally evaluated for yellow fluorescent protein (YFP) labeling in lung mononuclear phagocytes (Fig. 3a). Two weeks after injection, YFP+ cells were uniquely found among CD64+ subpopulations and Ly-6Clo patrolling monocytes, while YFP was virtually absent in lung Ly-6Chi classical monocytes or DCs (Fig. 3b, c, and Supplementary Fig. 7). Of note, the majority of CD206+ and CD206 IM subpopulations were YFP+, whereas less than 20% of the CD64+CD16.2+ subset was YFP+, similarly to what was observed in Ly-6Clo patrolling monocytes (Fig. 3b, c). In addition, CD64+CD16.2+ cells were all replaced by YFP monocytes at week 9 (Fig. 3b, c). Nine and 28 weeks after TAM treatment, the percentages of YFP+CD206+ and YFP+CD206 IM remained high and were not significantly different from those observed 2 weeks post-injection (Fig. 3b, c), supporting that both IM subsets could self-maintain in adults. However, percentages of YFP+CD206+ and YFP+CD206 cells were significantly decreased at week 52 as compared to week 2, confirming that both subpopulations were slowly replaced by YFP monocytes over time (Fig. 3b, c). Interestingly, more than half of the YFP+ labeling present at week 2 was still detected 50 weeks later in CD206+ IM, as opposed to less than 24% in CD206IM (Fig. 3b, c). In addition, levels of the proliferation marker Ki-67 were significantly greater in CD206+ IM as compared to CD206 IM and AM (Fig. 3d), suggesting that CD206+ IM could proliferate and had an increased self-maintenance potential as compared to CD206IM.
We have previously shown that bulk IM numbers were not significantly affected in 6–10-week-old Ccr2−/− or Nr4a1−/− mice18, whose numbers of blood Ly-6Chi and Ly-6Clo monocytes are impaired, respectively30,31 (Supplementary Fig. 8). While numbers of CD206+ and CD206 IM were similar in WT, Ccr2−/− and Nr4a1−/− mice (Fig. 3e, f), numbers of CD64+CD16.2+ monocytes were significantly reduced in Nr4a1−/− mice as compared to WT mice (Fig. 3e, f), like those of Ly-6Clo monocytes, demonstrating that CD64+CD16.2+ monocytes depended on NR4A1 for their presence in the lung.
The similarities between CD64+CD16.2+ monocytes and intravascular Ly-6Clo patrolling monocytes (i.e., half-life, dependence on NR4A1 and surface phenotype [Supplementary Fig. 9a, b]) supported the possibility that CD64+CD16.2+ monocytes actually derived from Ly-6Clo patrolling monocytes, but expressed CD64 and were partly extravascular. If CD64+CD16.2+ monocytes enter the tissue, they should be imprinted by tissue-instructive signals and, hence, exhibit a transcriptomic profile that is distinct from the one of intravascular Ly-6Clo monocytes. Hence, we compared CD64Ly-6Clo patrolling monocytes (Supplementary Fig. 9c–e) with CD64+CD16.2+ monocytes by scRNA-seq and found that they segregated in separated clusters (Supplementary Fig. 9f). Moreover, we found that many of the most upregulated transcripts in CD64+CD16.2+ monocytes (Supplementary Fig. 9g–i) were also found to be significantly upregulated in IM as compared to AM (see Supplementary Fig. 3d). These data support the notion that CD64+CD16.2+ monocytes can be distinguished from Ly-6Clo patrolling monocytes by their expression of tissue-specific IM-related genes, likely as a result of tissue-derived imprinting.
Altogether, our data identified two main subsets of long-lived monocyte-derived IM, with CD206+ IM exhibiting a greater half-life than CD206− IM, as well as short-lived NR4A1-dependent CD64+CD16.2+ monocytes.
### CD206+ and CD206−IM preferentially populate distinct niches
To assess the preferential localization of the two IM subpopulations and of CD64+CD16.2+ monocytes, we used confocal microscopy. Lung sections of Cx3cr1GFP/GFP mice were stained with antibodies directed against GFP, CD68 (as a macrophage marker) and either CD206, MHC-II or CD16.2. Of note, we observed only a minor fraction of CD68+ cells expressing simultaneously CD206 and MHC-II using a combined anti-CD206 and anti-MHC-II staining (Supplementary Fig. 10). Hence, CX3CR1+CD68+CD206+, CX3CR1+CD68+MHC-II+ and CX3CR1+CD68+CD16.2+ triple-positive cells were quantified in multiple sections and fields to evaluate the spatial distribution of CD206+ IM, CD206 IM and CD64+CD16.2+ monocytes, respectively (Fig. 4a, b). On the one hand, we found that CD206+ IM were preferentially found in the bronchial interstitium, whereas CD206 IM and CD64+CD16.2+ monocytes were mainly located in the alveolar interstitium (Fig. 4a, b). On the other hand, since Chakarov et al. reported that peribronchial Lyve-1loMHC-IIhi and Lyve-1hiMHC-IIlo IM subsets were mainly associated with nerves and blood vessels, respectively22, we used antibodies against CD31 and Tubb3 to stain nerves and endothelial cells. While CD206+ IM were associated with blood vessels (Fig. 4c), the preferential localization of peribronchial CD206 IM next to nerves was, however, less obvious, which is likely due to the close association of blood vessels and nerves in the peribronchial areas of the lung (Fig. 4d).
These data support that the two IM subpopulations are found in distinct micro-anatomical niches, which may dictate specific functional specializations.
### CD206+ and CD206−IM exhibit distinct functional properties
Next, we performed Gene Ontology (GO) enrichment analyses to gain insights into the functional properties of CD64+CD16.2+ monocytes and IM subpopulations. First, comparison between CD64+CD16.2+ monocytes and IM subsets revealed an enrichment, in CD64+CD16.2+ monocytes, in transcripts involved in leukocyte cell–cell adhesion, integrin-mediated signaling pathway, positive regulation of cytoskeleton organization and myeloid leukocyte migration (Table 1, Supplementary Fig. 11 and Supplementary Table 2), supporting the possibility that CD64+CD16.2+ monocytes may be actively extravasating in the lung tissue. Second, we found that the upregulated transcripts in CD206+ IM were enriched for processes related to the positive regulation of leukocyte chemotaxis, response to wounding and receptor-mediated endocytosis, consistent with their phenotype and lysosomal vacuoles (Table 1, Supplementary Fig. 11 and Supplementary Table 2). Third, CD206 IM had increased expression of transcripts associated with antigen processing and presentation, regulation of T cell activation and defense response (Table 1, Supplementary Fig. 11 and Supplementary Table 2).
To complement these mRNA data at the protein level, we performed a proteome profiling on the supernatants of FACS-sorted CD206+ and CD206 IM (Fig. 5a, b). CD206+ IM were characterized by an elevated chemokine secretory profile (e.g., CXCL11, CXCL10, CXCL9, CXCL2, CCL12), a higher secretion of immunoregulatory cytokines (IL-10, IL1-Ra) and factors regulating cell growth and differentiation, such as leukemia inhibitory factor (LIF), amphiregulin (AREG), or IL-7 (Fig. 5a, b). Conversely, CD206 IM secreted larger amounts of Pentraxin 3 (PTX3), secreted in response to inflammatory signals and facilitating pathogen recognition32, the p40 subunit of the type 1 helper T cell (Th1)-differentiating cytokine IL-12, and the B and T lymphocyte chemoattractant CXCL13 and CCL5, respectively (Fig. 5a, b). GO analysis also showed that both IM subsets expressed high levels of genes implicated in the cellular response to LPS as compared to CD64+CD16.2+ monocytes (Table 1, Supplementary Fig. 11 and Supplementary Table 2). Of note, ex vivo LPS stimulation potentiated the secretion of chemokines and immunoregulatory cytokines in CD206+ IM, and of PTX3, IL12p40 and CCL5 in CD206 IM (Fig. 5a, b).
At steady-state, we and others have shown that, upon engagement of Toll like receptor (TLR)4 (i.e., the main receptor for LPS) and the adaptor molecule MYD88, bulk IM could fulfill important tolerogenic tasks by inhibiting DC functions via IL-10-dependent mechanisms, thus preventing the development of asthma in animal models12,18. Proteome profiler data supported that CD206+ IM were the main IL-10-secreting cells (Fig. 5a, b). To validate these findings, we assessed IL-10 expression in lung monocyte/macrophage populations from IL-10-β-lactamase reporter ITIB mice33 (Fig. 5c, d). First, we observed that AM, Ly-6Chi classical and Ly-6Clo patrolling monocytes exhibited low percentages of IL-10+ cells (Fig. 5c, d). Second, we found that the percentage of CD206+ IM expressing IL-10 was significantly higher than the one of CD206 IM (Fig. 5c, d). Third, we showed that the percentage of IL10+CD64+CD16.2+ monocytes was significantly higher than the one of IL-10+ patrolling Ly-6Clo monocytes (Fig. 5c, d), pointing out another notable difference between CD64+CD16.2+ and patrolling monocytes.
Altogether, our data support that, in addition to their distinct phenotype and localization, IM subpopulations are characterized by unique functional properties. CD206+ IM exhibit a prominent tolerogenic and chemokine secretory profile, whereas CD206 IM have a typical antigen-presenting cell profile. Besides IM subsets, a fraction of CD64+CD16.2+ monocytes also express IL-10, a functional hallmark of lung IM12,15,18,19.
### Exposure to TLR ligands differentially modulates IM subsets
We reported previously that local instillation of TLR ligands, such as Pam3CSK4, LPS, and CpG (i.e., TLR1/2, TLR4, and TLR9 ligands, respectively) promoted an expansion of bulk IM18. Here, we exposed mice to Pam3CSK4, LPS, and CpG and performed time-course analysis of IM subsets (Fig. 5e). Pam3CSK4 and LPS induced similar dynamic changes, characterized by transient increases in numbers of CD64+CD16.2+ cells followed by a subsequent expansion of CD206 IM (Fig. 5f, g). LPS also significantly increased numbers of CD206+ IM at day 3 as compared to baseline (Fig. 5f). After CpG treatment, the profile was drastically different as compared to LPS or Pam3CSK4, with a more robust and sustained increase in numbers of CD64+CD16.2+ cells, reaching a peak at day 7, and a gradual increase in numbers of CD206− IM peaking at day 14 (Fig. 5f, g). Of note, increases in numbers of CD206 cells were associated with a drop in global MHC-II expression by those cells (days 3–7, Fig. 5h). Conversely, MHC-II expression of CD64+CD16.2+ cells gradually increased from day 5 to day 14, regardless of the treatment (Fig. 5h). Twenty-eight days after treatment, MHC-II expression was equal or even higher than day 0 in each subset, regardless of the treatment (Fig. 5h). Functionally, CpG was by far the most potent stimulus in triggering IL-10+ IM, which was restricted to the CD64+CD16.2+ compartment (Fig. 5i).
### RNA velocity identifies local precursors of CD206−IM
To gain insights into cell fate decisions, we applied RNA velocity analysis34 to our scRNA-seq datasets, i.e. IM subsets, AM, CD64+CD16.2+ monocytes, Ly-6Clo patrolling and Ly-6Chi classical monocytes (Supplementary Fig. 12). RNA velocity utilizes the balance between unspliced and spliced mRNAs to estimate the transition probability of individual cells34. Velocities, substantiated by arrows, can easily be projected on the t-SNE plot representing the merged scRNA-seq datasets on the basis of the similarity between the extrapolated state of a single cell and the static state of other cells in the local neighborhood (Fig. 6a, b, and Supplementary Fig. 13). Confirming the idea that both IM subsets arise from independent lineages22, RNA velocities of CD206 and CD206+ IM suggested that both IM subpopulations were relatively stable and independent from each other, as no clear transition could be observed from one subpopulation to the other (Fig. 6b, and Supplementary Fig. 14).
Interestingly, RNA velocities of CD64+CD16.2+ monocytes were significantly higher than those of Ly-6Clo patrolling monocytes or CD206 IM, supporting their dynamic transition state, and were pointing towards CD206 IM (Fig. 6b, c). Moreover, transition probability analysis of Ly-6Clo patrolling and CD64+CD16.2+ monocytes suggested that they could give rise to CD64+CD16.2+ monocytes and CD206 IM, respectively (Fig. 6d), supporting that CD64+CD16.2+ monocytes are mobilizable, on a timescale of hours34, to become CD206 IM. Using Slingshot package for pseudo-time inference analysis35, we found a continuum from Ly-6Clo monocytes towards CD206 IM, with CD64+CD16.2+ monocytes as an intermediate state (Fig. 6e). Among the gene expression changes driving such transition, we observed a downregulation of a patrolling monocyte signature (e.g., Cebpb, encoding C/EBPβ, essential for Ly-6Clo monocyte survival36, Plac8, Treml4, Ace [see Supplementary Fig. 9g, h]) concomitantly to an upregulation of many previously identified IM-related transcripts, including MHC-II-related transcripts (see Supplementary Figs. 3d and 9h, i) (Fig. 6f).
### NR4A1-dependent monocytes can differentiate into CD206− IM
Finally, we sought to validate our computational-based conclusions in vivo. First, we generated bone marrow (BM) competitive CD45.1/2 chimeras engrafted with CD45.1+ WT and CD45.2+ Nr4a1−/− BM cells (Fig. 7a). Six weeks after reconstitution, >85% of blood Ly-6Clo patrolling monocytes were of CD45.1+ WT origin, whereas less than 25% of NR4A1-independent B lymphocytes and neutrophils were of CD45.1+ WT origin (Fig. 7b). After 14 weeks, repopulated lung AM and both IM subsets displayed a chimerism similar to NR4A1-independent cells, confirming that such niches were repopulated by CCR2-dependent classical monocytes after lethal irradiation (Fig. 7b)18. Nevertheless, there was a significant enrichment, in repopulated CD64+CD16.2+ monocytes, in cells of CD45.1+ WT origin as compared to NR4A1-independent cells (Fig. 7b), supporting the idea that even in such extreme conditions, Ly-6Clo patrolling monocytes substantially contributed to the pool of CD64+CD16.2+ monocytes. Second, we performed i.v. adoptive transfers of blood donor CD45.1/2+ Ly-6Chi classical and Ly-6Clo patrolling monocytes into naïve CD45.2+ recipient hosts and analyzed the percentages of donor cells in Ly-6Clo patrolling monocytes, CD64+CD16.2+ monocytes and CD206 IM 7 days later (Fig. 7c). Transfer of Ly-6Clo monocytes, unlike that of Ly-6Chi monocytes, resulted in a significant increase in the percentage of donor Ly-6Clo monocytes in the lung vasculature as compared to non-transferred mice (Fig. 7d). While donor cells were hardly detectable in CD64+CD16.2+ monocytes, there was a trend towards an increase in percentages of donor cells in CD206 IM when mice were transferred with Ly-6Clo monocytes, but not with Ly-6Chi monocytes (Fig. 7d).
In order to boost IM expansion, we injected CpG, which resulted in a drastic increase in numbers of CD64+CD16.2+ cells (Fig. 5f, g). To our surprise, we were not able to assess the chimerism of CpG-injected WT:Nr4a1−/− BM mixed chimeras since they died following CpG administration, suggesting that NR4A1-dependent BM cells were needed to counteract CpG-induced toxicity (Fig. 7e). Of note, we reported a similar death in CpG-injected Il10−/− mice18, supporting the hypothesis that CpG-induced IL-10-producing IM counteract the toxicity of CpG, and, as a corollary, that NR4A1-dependent cells could be precursors of CpG-induced IM. We injected WT, Nr4a1−/−, Ccr2−/−, and Il10−/− mice with CpG and found that, while WT and Ccr2−/− mice all survived after CpG injection, survival of Nr4a1−/− and Il10−/− mice was significantly decreased as compared to WT mice, and all surviving mice had to be euthanized after 3–6 days due to excessive weight loss (Fig. 7e). This was associated with an altered profile of CD64+ IM on the Ly-6C/CD64 plot and the abnormal presence of Ly-6C+CD64+ inflammatory monocytes in Nr4a1−/− and Il10−/− mice 3 days after CpG (Fig. 7f).
Together, our data support the notion that, at steady-state, Ly-6Clo patrolling monocytes can enter the lung tissue to differentiate into CD206IM via an intermediate CD64+CD16.2+ state. After CpG, NR4A1 and IL-10 are needed to counteract CpG toxicity, probably through the expansion of IL-10-producing macrophages from NR4A1-dependent precursors.
## Discussion
In this study, we investigated the diversity of lung tissue CD64-expressing mononuclear phagocytes in mice. Our study describes the existence of two main IM subpopulations in naive adult mice, confirming recent findings22, and further expands our knowledge about their half-life, self-maintenance potential, and localization at steady-state, as well as their stimulus-dependent regulation in an inflammatory context. In addition, we uncover that Ly-6Clo patrolling monocytes can transition into extravasating CD64+CD16.2+ monocytes to give rise to CD206 IM.
We have provided RNA-seq-based evidence that the CD206+ IM subpopulation was similar to the published Lyve-1hiMHC-IIlo22 and the IM1/IM221 subpopulations, whereas CD206 IM profiles were similar to those of Lyve-1loMHC-IIhi22 and IM321. Like Lyve-1hiMHC-IIlo IM22, CD206+ IM were larger and comprised more IL-10-expressing cells. Even though Lyve-1 was also shown to be uniquely expressed by CD206+ IM (Fig. 1h) and IM1/IM221, the levels of Lyve-1 staining were, however, lower in CD206+ IM as compared to Lyve-1hiMHC-IIlo IM22, suggesting that CD206 may be more appropriate to discriminate IM subpopulations, at least in the lung. Conversely, CD206 IM, like Lyve-1loMHC-IIhi IM22, were smaller and negative for Lyve-1, and expressed higher levels of CX3CR1 and MHC-II than the other subset. Of note, two populations of tissue CD64+ macrophages have also been described in human lungs, one of them expressing CD206, similarly to CD206+ IM in mice22.
Regarding the maintenance of IM subsets in adults, our and others’ approaches all converged to the fact that IM subsets were slowly replaced by monocytes in adult mice21,22. Here, we also showed that CD206+ IM were able to proliferate and had an increased self-maintenance potential as compared to CD206 IM21. Under the experimental conditions tested, we found that the half-life of CD206 and CD206+ IM subsets were estimated at 9 and 12 months, respectively.
Quantitative data about the precise localization of lung IM (subpopulations) remained scarce and controversial12,21,22. In this regard, we originally observed F4/80+CD11c IM in the alveolar parenchyma12. Rodero et al. also described CX3CR1hi IM in alveolar areas37, while Gibbings et al. proposed later that IM were uniquely located in the bronchial interstitium21. More recently, Chakarov et al. looked at peribronchial IM subsets and proposed that Lyve-1loMHC-IIhi and Lyve-1hiMHC-IIlo IM were mainly associated with nerves and blood vessels, respectively22. To assess whether IM subsets populated distinct niches in the lung, we evaluated their preferential localization with respect to their bronchial vs. alveolar interstitial abundance. Our study identifies both alveolar and bronchial parenchymal IM, with CD206+ and CD206 IM preferentially found in the bronchial and the alveolar interstitium, respectively, and bronchial CD206+ IM located in the vicinity of blood vessels. Additional computer-based quantitative analyses together with cutting-edge multicolor imaging technologies will likely help to unambiguously address the precise localization of lung IM subsets, at steady-state and over the course of inflammatory responses.
Importantly, IM dichotomy was also observed at the functional level. We showed that bronchial CD206+ IM were endowed with a superior ability to secrete immunoregulatory cytokines, including IL-10, consistent with the hypothesis that they may largely account for the reported homeostatic functions of steady-state IM12,15. CD206+ IM were also suggested to be implicated in response to wounding, which fits well with their reported role in regulating damage-induced inflammation and fibrosis22. Conversely, CD206 IM exhibited a typical antigen-presenting cell profile and have been shown to regulate T cell-related processes22.
Kinetic analyses of changes in IM subsets after in vivo exposure to 3 different TLR ligands revealed a complex picture of the IM compartment during inflammation. Overall, CD64+CD16.2+ and CD206 IM were more affected by the treatments than CD206+ IM. It is also interesting to note that LPS and CpG, which have been shown to expand IM by CCR2-dependent and independent mechanisms18, respectively, exhibited very distinct dynamics over time. LPS amplified the 3 IM subsets at day 3 but mostly CD206 IM at day 7, while CpG robustly amplified the CD64+CD16.2+ compartment, among which most cells expressed IL-10, followed by an increase in CD206 IM peaking at day 14. Moreover, there was a drop in MHC-II expression observed within CD206 cells after each of the treatments, which likely reflects an influx of recruited MHC-IIlo monocytes into this gate, some of which giving rise later to MHC-IIhiCD206 cells (day 14). Functionally, these data suggest that CpG-elicited IM-derived protection against allergic asthma18 may be largely attributed to IL-10-producing CD64+CD16.2+ cells rather than CD206 or CD206+ IM.
Besides IM subsets, we identified a discrete population of NR4A1-dependent CD64+CD16.2+ monocytes that was mainly located in the alveolar area of the lung. Such monocytes were distinct from the previously described tissue-associated Ly-6Chi monocytes18,27 by their phenotype (CD64+Ly-6CloCX3CR1hiCD16.2+ vs. CD64loLy-6ChiCX3CR1loCD16.2- for Ly-6Chi monocytes) and their dependency on NR4A1. Using transgenic reporter mice, Rodero et al. previously detected a population of CX3CR1hi monocytes located at the interface between lung capillaries and alveoli37. Interestingly, such monocytes were phenotypically similar to CD64+CD16.2+ monocytes and exhibited motility patterns that were distinct from those of intravascular Ly-6Clo patrolling monocytes but similar to those of alveolar CX3CR1hi IM37, consistent with our findings.
The localization, phenotype, transcriptomic profile and RNA velocity analysis of CD64+CD16.2+ monocytes, together with the mixed chimeras and adoptive transfer experiments supported the notion that CD64+CD16.2+ monocytes arose from intravascular patrolling Ly-6Clo monocytes and represented an extravascular transition state towards CD206 IM. First, CD64+CD16.2+ monocytes and CD206− IM preferentially populated a similar micro-anatomical niche, i.e. the alveolar parenchyma37. Second, CD64+CD16.2+ monocytes shared similarities with patrolling Ly-6Clo monocytes (i.e., surface phenotype, half-life, dependence on NR4A1). Third, despite these similarities, we observed that CD64+CD16.2+ monocytes were partly extravascular and, unlike Ly-6Clo monocytes, expressed CD64 and significant levels of IL-10, a functional hallmark of lung IM12,15,18. Fourth, CD64+CD16.2+ monocytes upregulated many transcripts that were also highly expressed by CD206 IM, suggestive of a tissue-specific imprinting. Fifth, RNA velocity analysis revealed a highly dynamic transition state of these cells, characterized by a high probability to rapidly differentiate into CD206 IM on a timescale of hours34. Sixth, reconstitution of CD64+CD16.2+ monocytes after lethal irradiation was NR4A1-dependent, at least partially, further supporting that they derive from Ly-6Clo patrolling monocytes. Seventh, some Ly-6Clo patrolling monocytes that were transferred i.v. were found within the pool of CD206 IM 7 days later. Finally, IM expansion induced by CpG was impaired in patrolling monocyte-deficient Nr4a1−/− mice, and CpG instillation, which was well tolerated in WT or Ccr2−/− mice, induced death in Nr4a1−/−, a phenotype also observed in Il10−/− mice in which IM are not able to produce IL-10, a cytokine needed to counteract CpG toxicity.
While we have identified CD64+CD16.2+ monocytes as readily mobilizable precursors for CD206 IM, our data do not rule out the possibility that they may also give rise to CD206+ IM, and that classical Ly-6Chi monocytes could also differentiate into CD206 or CD206+ IM, consistent with the idea that different precursors could compete for the same niche5,6,7,8. Of note, numbers of IM subsets are not substantially affected in Ccr2−/− or Nr4a1−/− mice, whose numbers of blood classical Ly-6Chi and patrolling Ly-6Clo monocytes are impaired, respectively30,31. In light of our data, it is tempting to speculate that both Ly-6Chi and Ly-6Clo monocytes represent potentially redundant sources for lung IM. As a corollary, Ly-6Clo and Ly-6Chi monocytes would represent the main source of IM subsets in Ccr2−/− and Nr4a1−/− mice where the respective competitors are impaired.
This is an exciting time in macrophage research. With regards to lung tissue macrophages, future efforts should be made to investigate the transcriptional programs and tissue-instructive signals tailoring the identity of IM subsets, as well as the biological functions of IM subsets in health and diseases. Novel transgenic tools allowing the selective tracking, modulation, and depletion of such cells will be instrumental in addressing these questions, and will likely have important consequences for the elaboration of therapeutic approaches for lung chronic inflammatory diseases which would target one specific IM subset while sparing the other.
## Methods
### Mice
C57BL/6J WT mice were purchased from Janvier Laboratories. Ccr2−/−, Nr4a1−/−, Il10−/−, WT CD45.1 and Cx3cr1GFP/GFP (B6.129P-Cx3cr1tm1Litt/J) mice under the C57BL/6J background were purchased from the Jackson Laboratory (Cat. # 004999, 006187, 002251, 002014 and 005582, respectively). CD45.1/2+ were obtained by crossing CD45.1+ with CD45.2+ mice. IL-10-β-lactamase reporter (ITIB) C57BL/6 mice were described elsewhere33. Cx3cr1CreERT2.Rosa26-LSL-YFP C57BL/6 mice were kindly provided by Pr. G. Boeckxstaens (KU Leuven, Belgium). Cx3cr1CreERT2 mice were originally obtained from Steffen Jung, Weizmann Institute of Science29, and Rosa26-LSL-YFP mice were originally obtained from Daniel Richardson, University College London38. All mice were housed and bred in institutional SPF facilities at the GIGA Institute (Liège University, Belgium) and were used at 7–11 weeks of age, unless otherwise indicated. Cx3cr1CreERT2.Rosa26-LSL-YFP mice were bred and maintained at KU Leuven (Belgium). All animals and experimental procedures, except experiments involving Cx3cr1CreERT2.Rosa26-LSL-YFP mice were reviewed and approved by the Institutional Animal Care and Use Committee of the University of Liège (Belgium). Fate-mapping experiments were approved by the Animal Care and Animal Experiments Committee of the Medical Faculty of the KU Leuven (Belgium). The Guide for the Care and Use of Laboratory Animals, prepared by the Institute of Laboratory Animal Resources, National Research Council, and published by the National Academy Press, as well as European and local legislations, were followed carefully.
### Reagents and antibodies
A complete list of the reagents and antibodies used in this study can be found in Supplementary Table 3.
### Lung single cell isolation, stainings, and flow cytometry
To obtain single-lung-cell suspensions, lungs were extensively perfused with 3 ml of HBSS (Lonza) through the right ventricle, cut into small pieces with razor blades, and digested for 1 h at 37 °C in HBSS containing 5% v/v of FBS (Gibco), 1 mg.ml−1 collagenase A (Roche) and 0.05 mg.ml−1 DNase I (Roche). After 45 min of digestion, the suspension was flushed using a 18 G needle to dissociate aggregates. PBS (Gibco) containing 10 mM of EDTA (Merck Millipore) was added to stop the digestion process. The suspension was then filtered and enriched in mononuclear cells by using a density gradient (Percoll from GE Healthcare) and harvesting cells from the 1.080:1.038 g ml−1 interface.
Staining reactions were performed at 4 °C in FACS buffer (PBS containing 50% v/v of Brilliant Stain Buffer [BD Pharmingen], 2.5 mg ml−1 of BSA [Sigma], 0.5 mg ml−1 of sodium azide [Acros Organics]) with 2% v/v of Fc block (BD Pharmingen) to reduce non-specific binding. Cell phenotyping was performed on a FACSLSRFortessa (BD Biosciences). Cell sorting was performed on a FACSAriaIII (BD Biosciences) using the nozzle 85 at a rate allowing minimum 85% of efficiency. The purity of sorted cells was consistently above 95% for every sample. Results were analyzed using FlowJo V10 (Tree Star Inc.). Normalized MFI represents MFI for each sample with the mean of control cells MFI subtracted.
Anti-mouse CD68, Lamp-1 and Ki-67 intracellular stainings were performed by resuspending extracellular-stained cells in 500 µl of Fixation Buffer (Biolegend) for 40 min at room temperature (RT). Cells were then washed twice with 1 ml of permeabilization buffer (Thermo Fisher Scientific). Cells were stained for intracellular protein in 100 µl of fixation/permeabilization buffer (Thermo Fisher Scientific) with 2% v/v of Fc block (BD Pharmingen) for 30 min at RT.
For experiments involving ex vivo cultures and morphology assessment, cell suspensions were enriched by a magnetic-activated cell sorting (MACS) using anti-mouse CD11b microbeads (Miltenyi Biotec) according to manufacturer’s protocol, instead of the density gradient method. The negative fraction was also collected for the staining of AM.
Lung cell numbers were counted after whole lung digestion and mononuclear cell enrichment. The numbers of cells within each population were determined according to the gating strategy shown in Fig. 1a.
### Lung single cell preparation for scRNA-seq
Lung tissue CD64-expressing cells were FACS-sorted as singlet mononuclear cell-enriched CD45+ non-autofluorescent SSCloF4/80+CD11cLy-6CloCD64+ cells as shown in Fig. 1a. In the first replicate experiment, IM were isolated from lung single-cell suspensions pooled from three 10-week-old C57BL/6 female WT mice. In parallel, AM were FACS-sorted as singlet mononuclear cell-renriched CD45+CD11c+ autoflurorescent cells and were spiked in as controls at a ratio of 1:10. Such experiment was performed at KU Leuven (Belgium), while the second replicate was performed by independent experimenters through an independent pipeline using a pool of 6-month-old C57BL/6 female WT mice that were maintained in a different animal facility at the GIGA Institute (Liège University, Belgium). The 10× Genomics platform (Single Cell 3’ Solution) was used for scRNA-seq, and sequencing was performed at the Genomics Platform of the GIGA Institute (Liège University, Belgium) for both experiments. For scRNA-seq analysis of lung Ly-6Clo and Ly-6Chi monocytes, lung CD45+F4/80+CD11cLy-6CloCD64 cells and CD45+F4/80+CD11cLy-6ChiCD64 cells were FACS-sorted, respectively. For each sample, an aliquot of Trypan blue-treated cells was examined under the microscope for counting, viability and aggregate assessment following FACS sorting. Viability was above 80% for all samples and no aggregate were observed. Cell preparations were centrifuged at 1503 RCF for 4 min and pellets were resuspended in calcium- and magnesium-free PBS containing 0.4 mg ml−1 of UltraPure BSA (Thermo Fisher Scientific).
Sequencing libraries were generated by using the Chromium™ Single Cell 3’ Reagent Kits v2 (10× Genomics) following the manufacturer’s instructions. GEM-RT was performed in a Veriti© 96-Well Thermal Cycler (Thermo Fisher Scientific). After RT, GEMs were broken and the cDNAs were cleaned up with DynaBeads MyOne Silane beads (Thermo Fisher Scientific). cDNAs were then amplified in a Veriti© 96-Well Thermal Cycler. According to the expected cell recovery (based on a 60% recovery of total loaded cells), number of amplification cycles was set to 12. Amplified cDNA products were cleaned up using SPRIselect Reagent kit (Beckman Coulter), after what purified cDNAs were quality controlled and quantified using an Agilent High Sensitivity DNA Kit (Agilent) on a 2100 Bioanalyser (Agilent). Illumina’s P5, P7 and Read2 primers, as well as Sample Index were then added to generate sequencing libraries following Chromium™ Single Cell 3’ Reagent Kits v2 protocol. Steps were as follows: (1) enzymatic fragmentation, end repair and A-tailing, (2) Double Sided Size Selection using SPRIselect reagent, (3) adaptor ligation, (4) post ligation cleanup with SPRIselect reagent, (5) Sample index PCR (number of cycles set to 14) and (6) Double Sided Size Selection using SPRIselect reagent. The barcoded sequencing libraries were quality controlled using an Agilent High Sensitivity DNA kit on a 2100 Bioanalyser and quantified by quantitative PCR (KAPA Biosystems Library Quantification Kit for Illumina platforms).
Sequencing libraries were loaded at 1.4 pM on an Illumina NextSeq500 with NextSeq 500/550 Mid Output v2 kit (150 cycles) (Illumina) using the following read lengths: 26 bp for Read1 (16 bp Barcode+10 bp Randomer), 8 bp for Sample Index and 58 bp for Read2. Cell Ranger software (v1.2.0) (10x Genomics) was used to demultiplex Illumina BCL files to FASTQ files (cellranger mkfastq), to perform alignment (to mouse GRCm38/mm10 genome), filtering, UMI counting and to produce gene—barcode matrices (cellranger count).
### Analysis of scRNA-seq samples
Analyses used R bioconductor39 (version 3.4.2.), and the R package Seurat40 (version 2.3.4).
Briefly, for analysis of lung CD64-expressing mononuclear phagocytes, we first performed a quality control analysis and selected cells for further analysis in each replicate. Cells with a minimum of 200 and a maximum of 2500 detected genes were included, and cells with more than 5% of mitochondrial genes were excluded (Supplementary Fig. 2). In addition, only genes detected in at least 3 cells were included. Gene counts were normalized by using a global-scaling method that normalizes the gene expression measurements for each cell by the total expression, multiplies it by 10,000 and log-transforms the result. The FindVariableGenes function was used to calculate highly variable genes with the x.low.cutoff, x.high.cutoff and y.cutoff parameters set to 0.0125, 3, and 0.5, respectively. Cell–cell variation in the number of detected unique molecular identifiers (UMI) was regressed out using the ScaleData function. To analyze both replicates simultaneously, we used the MergeSeurat function, creating a new object with the resulting combined data matrices and appending a given identifier to each cell name depending on which original object the cell comes from. Linear dimensional reduction was performed on the scaled data using the RunPCA function. To identify the number of statistically significant principal components (PC) to include for subsequent analyses, we used the JackStraw function, which implements a resampling test inspired by the JackStraw procedure2. Alternatively, we used the PCElbowPlot function, looked at a plot of the standard deviations of the PC and determined our cutoff where there is an elbow on the graph, located at PC8. PC 1:8 were thus used in the subsequent analyses. We also performed analyses including lower and higher numbers of PC (1:6 to 1:12) and did not find any substantial differences in the results obtained.
The cells were clustered via the FindClusters function. Several cluster resolutions were tested, and the resolution of 0.2 was chosen, since higher resolutions created additional subdivisions or clusters containing singlets, which were considered not biologically relevant. To visualize the data, non-linear dimensional reduction was used, and t-SNE plots were created by using the RunTSNE function, with the number of dimensions to use set to 8 (PC 1:8). To eliminate a potential contamination with cells outside the MPS, data were subset using the SubsetData function in order to only keep cell clusters expressing detectable levels of the Csf1r gene.
Differential expression (DE) analysis between clusters was performed using the FindMarkers function, which uses a likelihood ratio test based on zero-inflated data to identify positive and negative markers of a single cluster compared to some or all other clusters. Only DE genes with an adjusted P-value < 10−2 were retained. To compare Cluster 3 (i.e., AM) with Clusters 1, 2, and 4 (i.e., tissue CD64-expressing cells), lists of the significantly DE genes between Cluster 3 and Clusters 1, 2, and 4 were generated. To compare the clusters of lung tissue CD64-expressing cells (i.e., Clusters 1, 2 and 4) with each other, lists of the significantly DE genes between Cluster 1 and Clusters 2 and 4, Cluster 2 and Clusters 1 and 4, or Cluster 4 and Clusters 1 and 2 were generated. Based on these lists of DE genes, Gene ontology analyses were performed using the Gene Ontology Consortium website (http://geneontology.org/) referring to the GO Ontology database released on 2018/12/01.
To visualize specific marker expression, the DotPlot function was used to show average expression of the genes and percentage of cells expressing the indicated genes within each cluster. Alternatively, the FeaturePlot or DoHeatmap functions were used to show specific gene expression across single cells.
For subsequent analyses, a similar procedure as described above was used with minor adaptations. For analysis of lung Ly-6Clo and Ly-6Chi monocytes, cells with a maximum of 3,000 detected genes were included, PC 1:8 were used and a resolution of 0.1 is shown for the cell clustering depicted in Supplementary Figs. 9d and 12. To compare Ly-6Clo patrolling and lung CD64+CD16.2+ monocytes, data of original Cluster 4 (i.e., CD64+CD16.2+ monocytes) were subset from the dataset of lung CD64-expressing mononuclear phagocytes and merged with the data of Ly-6Clo monocytes. PC 1:8 and a resolution of 0.3 were used for the cell clustering and subsequent analyses depicted in Fig. 6, Supplementary Figs. 13 and 14a, b.
For the graphical representation of RNA velocities, a Seurat object encompassing lung CD64-expressing mononuclear phagocytes, Ly-6Clo patrolling monocytes and Ly-6Chi classical monocytes was created and the corresponding t-SNE plot (PC 1:8) was used as shown in Fig. 6, Supplementary Figs. 13 and 14a, b.
### Transcriptomic comparison of lung IM subsets
To compare CD206 and CD206+ IM subsets with Lyve1hiMHCIIlo and Lyve1loMHCIIhi IM subsets reported in Chakarov et al.22, we generated CD206 and CD206+ IM signatures (i.e., upregulated genes in CD206−/+ vs. CD206+/− IM, respectively) based on our scRNA-seq data. Briefly, in order to be comparable between two analyses, numbers of genes within the CD206 and CD206+ IM signatures should be similar to those of Lyve1hiMHCIIlo and Lyve1loMHCIIhi IM signatures. Using the FindMarkers function (Seurat package41), we selected genes with a log fold change threshold of 0.1, and that are expressed in more than 10% of cells. The Venn diagram was generated with a publicly available online tool (http://bioinformatics.psb.ugent.be/webtools/Venn/). Gene Set Enrichment Analysis (GSEA) was performed to analyze enrichment of published Lyve1hiMHCIIlo and Lyve1loMHCIIhi IM signatures22 in CD206 and CD206+ IM (see below).
To compare CD206 and CD206+ IM subsets with the IM1, IM2, and IM3 populations reported in Gibbings et al.21, data of expression counts were downloaded from Gene Expression Omnibus database (accession GSE94135) and analyzed with DESeq2 package42 for differential expression analysis, hierarchical clustering, and PCA analysis. As a high similarity was found between IM1 and IM2 signatures, we generated an IM1&IM2 shared signature (634 genes) and an IM3 signature (97 genes), which were used as gene sets in the GSEA analysis described below.
For GSEA, in order to analyze enrichment of published signatures in our scRNA-seq data, the normalized counts were used as expression datasets in GSEA. Briefly, genes were ranked based on their expression in CD206+ and CD206 macrophages by Signal2Noise method. Normalized Enrichment Score (NES), FDR and nominal p-value were calculated with 100 permutations between samples from different phenotypes.
### Cytologic examination
Cytologic examination of FACS-sorted populations was performed on cytospin preparations stained with Hemacolor® (Merck, Cat. 111955, 111956, 111957). Sections were examined with a FSX100 microscope (Olympus) and size comparisons were performed using Image J software (NIH).
### In vivo labeling of vascular leukocytes
Lightly isoflurane-anesthetized C57BL/6J WT mice were injected i.v. with 300 µL of PBS, or with 0.5 µg of mouse APC-conjugated anti-mouse CD45.2 antibodies in 300 µl PBS. The antibodies were allowed to circulate for 3 min prior to euthanasia in order to label all leukocytes present in the vascular compartment. Lungs were harvested without perfusion and were processed for flow cytometry analysis.
### Assessment of phagocytic activity
Lightly isoflurane-anesthetized C57BL/6J WT mice were injected i.t. or i.v. with 6 × 108 pHrodo™ Green E. coli BioParticules™ (Thermo Fisher Scientific, Cat. P35366) in 100 µl and 200 µl of PBS respectively. Lungs were harvested 3 h later for cell isolation, staining and flow cytometry analysis.
### Fate-mapping of lung CX3CR1+ cells
The induction of Cre recombinase to trace CX3CR1+ cells was performed as described elsewhere43. Briefly, 4 week-old Cx3cr1CreERT2.Rosa26-LSL-YFP mice were treated three times subcutaneously with 4 mg TAM (Sigma, Cat. T5648) per 30 g body weight dissolved in corn oil (Sigma, Cat. C8267), 48 h apart. The mice were sacrificed after 2, 9, 28, and 52 weeks, and lungs were processed for flow cytometry analysis. Lungs from untreated Cx3cr1CreERT2.Rosa26-LSL-YFP mice were used as negative control.
### Immunostainings and confocal microscopy analyses
Freshly collected lungs from C57BL/6J WT and Cx3cr1GFP/GFP mice were embedded and frozen in OCT compound (Q Path Freeze gel, VWR), and then cut in 10 µm cryosections. Tissue sections were then fixed in 4% paraformaldehyde for 10 min at RT, permeabilized in 0.5% v/v of Triton-X100 (Sigma) for 2 min at RT and blocked in PBS containing 2% v/v of BSA and 2% of goat serum (Sigma) for 1 h at RT. Sections were first stained with a rat anti-mouse CD68 (dilution 1:100) for 2 h at RT, then with an AF568-conjugated goat anti-rat antibody (dilution 1:1000) for 2 h at RT. For samples isolated from Cx3cr1GFP/GFP mice, sections were then stained overnight at 4 °C with AF488-conjugated antibodies against GFP (dilution 1:200) and AF647-conjugated antibodies against CD16.2 (1:20), CD206 (1:100) or MHC-II (1:50). For samples isolated from WT mice, sections were then stained overnight at 4 °C with AF647-conjugated antibodies against CD31 (1:200) or Tubb3 (1:100), and AF488-conjugated antibodies against CD16.2, CD206 or MHC-II. For assessment of MHC-II and CD206 co-expression, WT sections were simultaneously stained with antibodies against MHC-II and CD206. Cell nuclei were counterstained with 4,6-diamidino-2-phenylindole (DAPI, Biolegend) for 5 min at RT. Sections were mounted with Prolong Diamond Antifade Mountant (Thermo Fisher) and stored for at least 5 h at RT. Samples were rinsed 3 times in PBS between each of the above-mentioned steps. Images were acquired on a Zeiss LSM 880 confocal microscope (Zeiss) and analyzed using the ZEN 2.3 software (Zeiss).
Spatial distribution was quantified by analyzing, for each sample, 5 or 10 fields (magnification ×20) containing at least one CX3CR1+CD68+ cell in the peribronchial/perivascular area or the alveolar parenchyma, respectively. For each sample, the mean of the numbers of CX3CR1+CD68+CD206+, CX3CR1+CD68+MHC-II+ and CX3CR1+CD68+CD16.2+ cells per field was used to calculate the spatial distribution of these cells. Double positivity for CD206 and MHC-II was quantified by analyzing, for each sample, 5 fields (magnification ×20) in peribronchial/perivascular and alveolar parenchymal areas containing at least one CD68+CD206+ or CD68+MHC-II+ cell. For each sample, the numbers of CD68+CD206+, CD68+MHC-II+ cells and CD68+CD206+MHC-II+ cells were used to calculate the percentage of CD206/MHC-II double positive cells.
### Proteome profiler assay
CD206 and CD206+ IM subpopulations were FACS-sorted from naive C57BL/6J WT mice and 5 × 104 cells were cultured in an F-bottom 96-well plate during 16 h in 100 µl of RPMI with L-glutamine (Lonza) supplemented with 10% v/v FBS, 1% v/v MEM non-essential amino acids (Gibco), 1 mM sodium pyruvate (GE Healtcare), 50 U ml−1 penicillin/streptomycin (Gibco), 0.05 mM 2-mercaptoethanol (Gibco), with or without 10 ng ml−1 of LPS (Sigma). For each experimental condition, supernatants were tested for the presence of cytokines and chemokines using a proteome profiler mouse XL cytokine array (R&D Systems), according to manufacturer instructions. Results were visualized using an ImageQuant LAS 4000 (GE Healthcare) and analyzed using ImageJ software. Results are expressed as Z-scores for each analyte ([individual value – analyte mean] per analyte standard deviation).
### Assessment of IL-10 expression in ITIB mice
To assess IL-10 expression using IL-10-β-lactamase reporter ITIB mice33, lung cells from IL-10-β-lactamase reporter ITIB and control WT mice were resuspended in a CCF4-AM (Thermo Fisher Scientific, Cat. K1028)-containing solution supplemented with probenecid (Thermo Fisher Scientific, Cat. P36400) prepared according to manufacturer’s instructions, and incubated 90 min at 29 °C. CCF4-loaded cells were then classically stained and analyzed by flow cytometry. CCF4-loaded cells from C57BL/6J WT mice were used as negative controls.
### Estimations of RNA velocities in single cells
RNA velocities in single cells were estimated with RNA velocity (http://velocyto.org)34. Briefly, spliced and non-spliced transcripts counts were calculated from CellRanger output using Python command line tool velocyto with default run10x subcommand. Genes with minimum average expression of 0.5 (for spliced transcripts) or 0.05 (for non-spliced transcripts) within at least one cell cluster were filtered before velocity estimation. RNA velocities were estimated using 20 k-nearest neighbors (NN) in slope calculating smoothing, and fit quantile of 0.02. RNA velocities were then visualized using correlation-based transition probability matrix within the kNN graph, with same cluster labels and embedding as in Fig. 6a.
To compare RNA velocity across subsets of Ly-6Clo patrolling monocytes, CD64+CD16.2+ monocytes and CD206 IM, the relative distance from the cell position in the 2D-tSNE plot to the projected position (i.e., effective length of arrows displayed in Supplementary Fig. 13b) was used to illustrate the RNA velocity of each cell.
### Cell fate decision estimation
Cell fate decision was represented by RNA-velocity-based single-step transition probabilities from starting cells to neighboring cells. To illustrate cell fate decision probabilities of all cells in a given subset (i.e., the starting subset), the total transition probability (TPn) to neighboring cells n was calculated as the sum of all transition probabilities (TPin) of single cells in the given subset (containing j cells), and corrected by the number of cells:
$${\mathrm{TP}}n = \frac{{\mathop {\sum }\nolimits_{i = 1}^j {\mathrm{TP}}in}}{j}$$
For illustration, transition possibility of each cell position was indicated by a color gradient while an ellipse marked the starting subset of interest with 95% confidence on the cell positions in the subset.
### Trajectory analyses
For trajectory analyses, the previously published package Slingshot35 was used. Briefly, Slingshot uses pre-existing clustering information to calculate development trajectory and pseudo-time of each cell in the development trajectory. To analyze the differentiation trajectory from Ly-6Clo patrolling monocytes to CD206 IM subpopulation, we reanalyzed the data sets shown in Fig. 6a with Seurat using higher resolution (i.e., 2.5), and the 5 clusters covering Ly-6Clo patrolling monocytes, CD64+CD16.2+ monocytes, and neighboring CD206 IM were used for Slingshot analysis. For clarity purposes, the original t-SNE embedding of Fig. 6a was retained to illustrate cell positions in Fig. 6e and the ellipses with 80% confidence were drawn to illustrate their cluster belonging shown in Fig. 6a using the same colors. To find the temporally expressed genes through trajectory, we used the method suggested in the Slingshot package to regress genes on the pseudo-time variable using a general-additive model. Expression of the top 100 genes on p-value was showed in a heatmap across all cells in selected subsets.
### White blood cell isolation
To obtain single-blood-cell suspensions, blood was collected in a tube containing 500 µL of PBS (Gibco) containing 100 mM of EDTA (Merck Millipore) and red blood cells were lysed by adding distilled water containing 150 mM of ammonium chlorure (VWR) and 10 mM of potassium bicarbonate (Sigma). Cells were then centrifuged, resuspended in FACS buffer and stained as described for lung cells in the respective section above.
### Bone marrow mixed chimeras
Eighteen-week-old CD45.1/2+WT mice were lethally irradiated with two doses of 6 Gy 3 h apart. Two hours after the second irradiation, these mice were injected i.v. with 2 × 106 BM cells consisting of a 1:1 mix of BM cells obtained from CD45.1+ WT and CD45.2+ Nr4a1−/− mice. From the day before irradiation, mice were treated for 4 weeks with 0.05 mg.ml−1 of enrofloxacin (Baytril, Bayer). Chimerism was assessed by flow cytometry in the blood and the lung 6 and 14 weeks after irradiation, respectively. In the blood, B lymphocytes, Ly-6Clo monocytes, Ly-6Chi monocytes and neutrophils, were defined as CD45+ CD11bLy-6GCD19+, CD45+CD11b+Ly-6GCD115+Ly-6C, CD45+CD11b+Ly-6GCD115+Ly-6C+ and CD45+CD11b+Ly-6G+ cells, respectively.
Blood cells of CD45.1/2+ mice were collected and stained for FACS sorting of Ly-6Chi and Ly-6Clo monocytes, defined as CD45+CD11b+CD115+Ly-6C+ and CD45+CD11b+CD115+Ly-6C+ cells, respectively. These cells were then transferred to CD45.2+ WT recipient mice by injecting 105 Ly-6Chi or Ly-6Clo monocytes i.v. into each mouse. Lungs were harvested 7 days later for cell isolation, staining and flow cytometry analysis.
### Intratracheal instillations of TLR ligands
Lightly isoflurane-anesthetized C57BL/6J WT mice were injected i.t. with Pam3CSK4, LPS or CpG (25 µg, 10 µg and 50 µg, respectively) in 50 µl of PBS. Lungs were harvested 1, 3, 5, 7, 14 or 28 days later for cell isolation, staining and flow cytometry analyses. C57BL/6 WT, Ccr2−/−, Nr4a1−/−, Il10−/−, and WT:Nr4a1−/− 1:1 BM mixed chimeras were injected i.t. with 50 µg of CpG in 50 µl of PBS. The survival of these mice was assessed and lungs from WT, Nr4a1−/− and Il10−/− mice were harvested 3 days later for flow cytometry analysis. ITIB mice were injected i.t. with 50 µg of CpG in 50 µl of PBS and their lungs were harvested 7 days later for cell isolation, staining and IL-10 production analysis by flow cytometry.
### Statistical analysis
Data from independent experiments were pooled for analysis in each data panel, unless otherwise indicated. Statistical analyses were performed using Prism 7 (GraphPad Software), SAS (version 9.3) and R bioconductor39 (version 3.4.2.). Data were presented as mean ± s.e.m., as well as individual values, unless otherwise indicated. We considered a P-value lower than 0.05 as significant. *P < 0.05; **P < 10−2, ***P < 10−3; ****P < 10−4; ns, not significant. Details about the statistical tests used can be found in the respective Figure legends. Details about the analysis of scRNA-seq data can be found in the respective sections above.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
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Trigonometry (11th Edition) Clone
$w\cdot z$ = $36(cos130^\circ + isin130^\circ)$
$w = 12(cos80^\circ + isin80^\circ) = 12cis80^\circ$ $z = 3(cos50^\circ + isin50^\circ) = 3cis50^\circ$ Now, $w\cdot z$ = $12cis80^\circ \cdot 3cis50^\circ$ = $12 \cdot 3 \cdot cis(80^\circ+ 50^\circ)$ (Product Theorem) = $36cis130^\circ$ = $36(cos130^\circ + isin130^\circ)$
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theory:electrochemistry:electrochemistry
An Introduction to Electrochemistry
When connecting electrodes to liquids, tissue or biological materials, we create an interface between electronics and chemistry. In the electrodes and electronic equipment, charge is conducted by electrons, while it is conducted by ions in the biomaterials and liquids. The field of science about the interface of electronics and liquids is called electrochemistry. This page descibes the basic knowledge to understand the phenomena of electrochemical methods.
The simplest electrochemical systems is created by two or three metal electrodes in a liquid. We can supply an electrical current (or potential) to the electrodes, while measuring the potential difference (or current) that results from it. The basic physical and chemical principles describing the relation between potential and current normally originate from three locations in the system:
• The bulk of the liquid with dissolved ions and dipoles will have a resistance because the charged particles need effort to move under the applied potential difference over the electrodes
• The dissolved charged particles will be transported from and towards electrodes as a result of the applied potential. This transport will result in a potential difference over the electrode-liquid interface and is referred to as electrode polarization
• Next, the ions in the solution may give or get an electron from the electrode, and so charge is exchanged between the electrode and the liquid. This exchange invloves a chemical transition and the crossing of a barrier, also resulting into an electrode-liquid potential difference
In more complex systems like the Ion Sensitive Transistor (ISFET), liquid conductors and glass electrodes, all three phenomane will take place.
In almost all cases, the aim of an electrochemical experiment is the determination of information about a solution. This information can be pH (acidity), the electrical conductivity, or a certain specific concentration. In electrochemical cells, however, we only have electrical potentials and currents as inputs and outputs. By picking the right input (frequency, amplitude, pulse shape, etc.), and interpreting the response appropriately, we can isolate one of the phenomena and learn something specific about the solution1). For example, in the kHz range, the phenomena at the electrodes are too slow to have a significant influence, hence only the resistance of the bulk solution determines the behaviour of the I-V relation. In that case, we are measuring the electrical conductivity of a solution without being affected by electrode polarization. When using a DC current, however, the third phenomenon will dominate the behaviour.
This tutorial starts with phenomena at the electrode surface. Next, it will be explained how these phenomena are using in an electrochemical half-cell with a reference electrode as an example. Next, methods that are using these half cells are discussed, with the ISFET in some more detail as an electronic component. After the elektrode interface phenomena, the transport in the bulk of a solution will be discussed which is the basis for electrical conductivity measurements. The page ends with some examples that illustrate the application of all theory.
As described in the introduction, there are two processes by which a potential difference accross the electrode-liquid interface can be created. The difference between these two processes is in the flow of electrons:
• When electrons are exchanged with the electrode, we speak of a Faradeic process. The energy step needed to cross the interface is the origin of the Faradeic electrode potential, and is described by the Nernst equation.
• When there is just accumulation of charge at the electrode boundary, we speak of a non-Faradeic process. In that case, the behaviour is described by capacitor like models.
The Nernst equation
Consider an electrode in a liquid. The electrode can take or give electrons to a particle in that liquid. As a result, the state and charge of that particle will change. When electrons are taken from electrodes, we speak of reduction, when the liquid gives electrons to the electrode we speak of oxidation. An electrode at which reduction occurs is a cathode, an electrode at wich oxidation takes place is an anode. So with a cathodic current, electrons flow from the electrode to the liquid, with an anodic current in the opposite direction. See figure 1 for a graphical representation.
Fig. 1: Reduction and oxidation
For the particle in the liquid, we use the following terminology: with a reduction reaction, we call the original $Ox$, while the result is $Red$. This has to do with applications where there is a reduction reaction without the intervention of a cathode where electrons are pulled out of another couple. The other couple is then oxidized and we can call $Ox$ the “oxidator”. So:
• An oxidator particle can oxidize another particle by which it is reduced itself, and
• A reductor particle can reduce another particle by which it is oxidized itself.
The equilibrium between the oxidized phase and the reduced phase can be described by:
$$Ox+n\cdot e^{-}\leftrightarrow Red. \label{eq:eqRedOx}$$
Here is $Ox$ the oxidized phase and Red the reduced phase. The number of electrons needed in the equilibrium is $n$. In a reversible system, the relation between electrode potential and the ratio in the equilibrium is given by the Nernst equation:
$$E = E^{0}+\frac{RT}{nF}\ln \frac{a_{Ox}}{a_{Red}} \label{eq:Nernst}$$
with $R$ the molar gas constant, $T$ the temeprature, $F$ Faraday's constant (see table 3) and $a_{Ox}$ and $a_{Red}$ the activities of $Ox$ and $Red$. Faraday's constant is nothing else than the charge of one mole of electrons. In semiconductor models we see the factor of $kT/q$ which is about $25 mV (mJ/C)$, in electrochemistry this factor is replaced by $RT/F$, so
$$\frac{kT}{q} = \frac{RT}{F} \approx 25 mV.$$
The activities are related to concentrations by means of the activity coefficient $g$. For practical (low) concentrations, the activity coefficienrts are equal to one, so that the Nernst equation can also be writtten in terms of ionic concentrations.
In a reversible system, the equilibrium can always return to the original position. In an irreversible system this can not, for example because there is a solid precipitation or because gas diffuses out of the solution.
The electrode potential with a current through the electrode
The Nernst equation describes the potential difference accross the metal-liquid interface, but can not be used to determine the complete current-potential relation. An electrode interface is called polarized when there is a potential difference accroess the interface as a result of the current. Deviation form that potential is called overpotential:
$$\eta = E - E_{eq}.$$
We distinguish between polarization as a result of charge transfer (Nernstian) and polarization as a result of diffusion (concentration polarization).
A Tafelplot is the curve of the electrode current as a function of the overpotential, and contains information about the transfer coefficient and the exchange current $i_{0}$2).
The double layer capacitance
With Faradeic processes, as described before, charge is exchanged between the electrode and the liquid. Elements in the liquid will be modified chemically. A second process is the accumulation of charge at the electrode surface in the liquid, with an equivalent amount of charge in the electrode. This accumulation does not result into an immediate chemical modification of components. In Faradeic processes we have already seen this as concentration polarzation, but also in non-Faradeic processes this polarization will take place.
In the metal, the charge accumulation consists of electrons, while in the liquid the charge comprises ions. In case of charge in the form of ions, we should distinguish two types. First, ions can adsorb on the interface. We call this specific adsorption because the properties of the ions themselves determines the behaviour. These ions will be the closest to the interface. The layer comprising these ions is called the Helmholtz layer, Sternlayer, or Inner Helmholtz Plane (IHP). The second accumulation is with ions in the solution. Ions in the solution have a water shield consisting of the dipoles of the water molecules. Their chemical properties are invisible for the metal surface. Nevertheless, there can be an electrostatic interaction with the electrode. This results into a second layer: the diffusion layer. The diffusion layer will have a larger distance to the electrode surface as the Sternlayer.
The total accumulatiomn of charge results into a dubbellayer capacitance because charge is accumulated as a result of an electric field, just like in a normal capacitor. The value of this capacitor is normally in the range of $10$ tot $40 \mu F/cm^{2}$, but depends strongly on the concentrations and potential difference accross the interface. The simplification to an ideal capacitor can be used in most cases, but to explain some phenomena, we have to use frequency dependent components in this capacitor (the Warburg impedance).
The electrochemical cell
In practical configurations, we can never have a single metal-liquid interface. To make a closed electronic circuit, we need at least two interfaces. In the total electronic loop, there will be voltage drops due to the two interfaces plus voltage drops due to bulk electrolyte resistances and electrical resistances of the wires. The whole system is referred to as an electrochemical cell.
Principle
In a galvanic cell, electrochemical reactions occur autonomously, without the need for external energy. We know this as a battery. The anode, which has per definition the oxidation, is then at negative potential. The cathode is at positive potential. In an electrolytic cell, on the other hand, electrochemical reactions are activated by external energy (a potential difference). In that case, the oxidizing anode is positive and the cathode negative.
The cathode/anode definitions of figure 1 are still valid as illustrated in figure 2.
Fig. 2: Electrochemical cell (left hand) and galvanic cell (right hand)
The standard reference
The potential difference accross the two electrodes is determined by two electrochemical semi-cells. Because we want to investigate the electrochemical phenomena at a single electrode, we have to create a sitation where the other interface is “known” and “defined”. We can make the phenomena at an electrode constant and known by defining the concentrations in the effective vicinity of the electrode. The Nernst equation \eqref{eq:Nernst} then states that the potential difference accross the electrode will stay constant. Such a well defined electrode is called a reference electrode. A reference electrode must never carry a current, because then the local concentrations are altered and the potential is not guaranteed anymore.
In fact, “the” potential of an electrolyte is always unknown because there are always two interfaces in the loop. Even the potential as given by the Nernst equation is defined with respect to a chosen reference. This chosen reference is the Normal Hydrogen Electrode (NHE) which is defined as zero Volt.
Electrochemical methods
The potentiostat/galvanostat
As described in the previous section, it is not allowed to pull a current through a reference electrode, because it will no longer define a reliable reference potential. If we want to study an electrode on how it passes a current, we will need a third electrode that sources or drains the current.
The basic scheme for a potentiostat is sketched in figure 3a. In a potentiostat the applied potential difference is defined over the working electrode and the reference electrode, while the current is measured through the working electrode which flows to or from the couter electrode. When we are interested in the electrode potential as a result of an imposed current, we speak of a galvanostat as sketched in figure 3b. Potentiostat/galvalostats are for example available from Princeton Applied Research, where the PAR173 and PAR273 are quite commonly used.
Fig. 3: A potentiostat (a) and a galvanostat (b)
Cyclic voltammetry
Fig. 4: A typical voltammogram of water: 1 mm^2 Pt, 100 mV/sec, 100 mM KNO3.
Amperometry
Chrono amperometry
Consider an involved reaction for reduction at a working electrode
$$Ox+n\cdot e^{-}\rightarrow Red.$$
which is evoked after applying a negative potential step to the working electrode. In case of a potential step which is high enough to deplete the reacting species $Ox$ at the electrode surface completely during this step, the resulting current response for a planar electrode is given by the Cottrell equation3)
$$i\left ( t \right ) = nFAC_{Ox} \sqrt{\frac{D_{Ox}}{\pi t}}$$
with $F$ the Faraday constant, $n$ the number of electrons transferred, $A$ the working electrode size, $C_{Ox}$ the bulk $Ox$ concentration and $D_{Ox}$ the diffusion coefficient of $Ox$. So, after applying a voltage step, the monitored current response is proportional to the $Ox$ concentration.
The Ion Sensitive Field Effect Transistor (ISFET)
The ISFET resembles a MOSFET, but with an ISFET the metal gate is replaced by an electrolyte. So the electrolyte is in direct contact with the gate oxide, the gate contact consists of a reference electrode in the electrolyte (figure 5).
Fig. 5: Schematic representation of an ISFET
The threshold potential of the ISFET is the result of the summation of a few physical and chemical potentials, of which one, the electrostatic potential $\psi_{0}$, appears to be pH dependent according to4):
$$\frac{\partial \psi _{0}}{\partial \text {pH} _{B}} = -2.3 \frac{kT}{q} \alpha$$
with $pH_{B}$ the bulk pH and $\alpha$ a sensitivity parameter between $0$ and $1$. In practice this results into a pH sensitivity of the theshold voltage of $59 mV/pH$. This pH dependency is the result of the buffering of $H^{+}$ ions at the oxide interface.
Electronically seen, the ISFET is a MOSFET with a pH dependent threshold voltage $V_{T}$. For a p-type substrate we find the relations of table 1
Mode Condition Draincurrent Conduction in channnel
Saturated $V_{DS} > V_{GS} - V_{T}$ $I_{D} = \mu C_{Ox} \tfrac{1}{2} \frac{W}{L} \cdot \left ( V_{GS} - V_{T} \right )^{2}$ Holes
Unsaturated $V_{DS} < V_{GS} - V_{T}$ $I_{D} = \mu C_{Ox} \frac{W}{L} \cdot \left [ \left ( V_{GS} - V_{T} \right ) V_{DS}- \tfrac{1}{2}{}V_{DS}^{2} \right ]$ Electrons (inversion)
Tab. 1: Operational regions in a Field Effect Transistor
The modi of the transistor curves are also sketched in figure 6.
Fig. 6: Transistor curves
Because the $V_{T}$ is pH dependent, the drain current will also be pH dependent (regardless the saturation mode). However, it is convenient to have a linear dependency between $I_{D}$ and pH. Two methods are quite commonly used5).
Method 1: Constant drain-source voltage
Take the unsaturated mode and keep $V_{DS}$ constant. The draincurrent becomes:
$$I_{D} = C_{1} + V_{T}C_{2}$$
Take care that $V_{DS}<V_{GS}-V_{T}$, which means we need a depletion type transistor ($V_{T}$ negative. Orion uses this method in their ISFET based pH meter “pHuture”.
Method 2: Constant drain-source voltage and drain current
Keep $I_{D}$ constant and $V_{DS}$ as well. The equations for the drain current show that this requires a $V_{GS}$ that is equal to $V_{T}$, regardless the saturation mode. Nevertheless, the unsaturated mode is preferred because then $I_{D}$ and $V_{DS}$ are lower and so is the dissipated power $I_{D} \cdot V_{DS}$.
A concvenient circuit for method 2 is given in figure 7. The operational amplifier keeps the potential of the drain equal to the potential defined by the voltage divider $R_{1} : R_{2}$. The drain-source potential difference is then equal to:
$$V_{DS} = \frac{R_{2}}{R_{1} + R_{2}} V_{Ref}$$
which is normally kept at $0.5 Volt$. The drain current is in this circuit equal to the current through $R_{3}$ and so
$$I_{D} = \frac{R_{1}}{ \left ( R_{1} + R_{2} \right ) R_{3}} V_{Ref}.$$
An $I_{D}$ of $100 \mu A$ is a quite commonly used value.
Fig. 7: An ISFET in constant VDS and constant ID mode
Ions in an electrolyte
Particles in a solution can move due to three different mechanisms. This division is made based on the driving force causing the motion:
• Convection or migration: when the liquid moves or is stirred
• Drift: a charged particle in an applied electric field
• Diffusion: in a concentration gradient
The operational principle fo electrolyte conductivity sensors is based on the characteristic drift of charged particles, while potentiometric and amperometric methods are normally diffusion limited. Convection of the medium is normally an undesirable effect and will not be discussed here.
Diffusion
Diffusion is motion of particles due to a concentration gradient. The basis is the second law of Fick which states that the change of the amount of particles in a volume is proportional to the flux of particles through the boundary of the enclosed volume:
$$\frac{\mathrm{d} c}{\mathrm{d} t} = D \nabla^{2}c$$
with $D$ the diffuscion coefficient $[m^{2}/sec]$ and $c$ the concentration $[mol/l]$.
The same differential equation is true for the diffusion of heat in solids6). This means we can re-use the generic solution. For an instantaneous pointsource at $t = 0 sec$ at location $r_{0}$, measured at location $r$, we find the solution:
$$c_{pointsource} \left ( r, t \right ) = \frac{Q}{\left( 4 \pi D t \right )^{\frac{3}{2}} } \cdot e^{- \frac{\left( x-x_{0} \right )^{2}}{4Dt}}$$
with which, by integration over the total source surface, the response on the pointsource can be found:
$$c_{source} \left ( r, t \right ) = \int_{source}^{ } c_{pointsource} \left ( r, t \right ) dr_{0}$$
By convolution with the source signal (actuator current or power of the heater) we can find the concentration in the total system:
$$c \left ( r, t \right ) = \int_{0}^{t} i \left ( \tau \right ) \cdot c_{source} \left ( r, t-\tau \right ) d\tau$$
The solution contains
• an exponential term - as a result of diffusion, and
• some error-functions - as a result of the actuator geometry.
The used analogy between the thermal and chemical domain is also valid for electrical transmission lines. table 2 summarizes the mathematics and meaning of the state valiables for the three domains.
Heat Ions Electrical transmission lines
Flow through cross section A $\frac{\mathrm{d}Q}{\mathrm{d}t} = \lambda A \frac{\mathrm{d}T}{\mathrm{d}x}$ $I = \frac{dq}{dt} = D \cdot A \cdot F \frac{\mathrm{d}c}{\mathrm{d}x}$ $I = \frac{\mathrm{d}q}{\mathrm{d}t} = A \cdot \sigma E = \frac{1}{R} \frac{\mathrm{d}U}{\mathrm{d}x}$
Through slice with thickness dl $\left. \frac{\mathrm{d}Q}{\mathrm{d}t} \right| _{\mathrm{d}l} = \lambda A \frac{\mathrm{d}^{2}T}{\mathrm{d}x^{2}} \mathrm{d}l$ $I = \left. \frac{\mathrm{d}q}{\mathrm{d}t} \right| _{V} = D \cdot A \cdot F \frac{\mathrm{d}^{2}c}{\mathrm{d}x^{2}} \mathrm{d}l$ $I = \left. \frac{\mathrm{d}q}{\mathrm{d}t} \right | _{\mathrm{d}l}= \frac{1}{R} \frac{\mathrm{d}^{2}U}{\mathrm{d}x^{2}} \mathrm{d}l$
Storage in disc $\left. \frac{\mathrm{d}Q}{\mathrm{d}t} \right| _{\mathrm{d}l} = c_{m} \rho A \frac{\mathrm{d}T}{\mathrm{d}t} \mathrm{d}l$ $I = \left. \frac{\mathrm{d}q}{\mathrm{d}t} \right| _{V} = A \cdot F \frac{\mathrm{d}c}{\mathrm{d}t} dl$ $I = \left. \frac{\mathrm{d}q}{\mathrm{d}t} \right | _{\mathrm{d}l}= C \frac{\mathrm{d}U}{\mathrm{d}t} \mathrm{d}l$
Differential equation $\frac{\mathrm{d}T}{\mathrm{d}t} = \frac{\lambda}{c_{m}\rho}\nabla^{2}T$ $\frac{\mathrm{d}c}{\mathrm{d}t} = D \nabla^{2}c$ $\frac{\mathrm{d}U}{\mathrm{d}t} = \frac{1}{RC} \nabla^{2}U$
$Q$ : heat $q$: charge $q$ : charge
$\partial Q/\partial t$ : heat flow $I$: electric current $I$: electric current
$T$ : temperature $c$: concentration $U$ : potential
$Adl$ : slice $Adl$ : slice $Adl$ : slice
$\lambda$ : thermal conductivity $D$ : diffusion coefficient $R$ : resistance per $dl$
$\rho$ : density $F$: Faraday constant $C$ : capacity per $dl$
$c_{m}$ : specific heat $\sigma$ : conductivity
Tab. 2: Analogies in the thermal, chemical and electrical domain
Drift
Drift is the movement of charged particles in an electric field. In an electrolyte, the drift of ions determines the electric current. The contribution of an ion i to the total current is expressed by the transport number $t_{i}$ where the sum of all transport numbers of all charged particles in the electrolyte equals one:
$$\sum_{i} t_{i} = 1$$
The electrolyte conductivity κ can be expressed in terms of the molar conductivity $\Lambda$:
$$\kappa=c \cdot \Lambda$$
with c the concentration of the conductive particles in the solution. In practice it is normally more insightful to express the molar conductivity in terms of all individual specific conductivities $\lambda_{i}$ of the ions involved:
$$\Lambda = \sum_{i} c_{i} \left | z_{i} \right | \lambda_{i}=F\sum_{i} c_{i} \left | z_{i} \right | \mu_{i}$$
The mobility $\mu _{i}$ is related to the diffusion constant $D_{i}$ by:
$$D_{i} = RT\mu_{i}$$
with $R$ the gas constant and $T$ the temperature.
Methods for electrolyte conductivity
The measurement of the the electrolyte conductivity can be done by placing the liquid between two capacitor plates. Due to the conductive elements in the electrolyte, there is a resistive element placed in the capacitor. The electrical equivalent circuit is given in figure 8.
Fig. 8: Electric equivalent circuit for a conductivity cell
We are interested in the value of the resistance $R_{Cel}$ which represents the total ion concentration and is related to the conductivity $\kappa_{sol}$ by means of the cell-constant $K_{Cell}$:
$$R_{Cell} = \frac{K_{Cell}}{\kappa_{sol}}$$
which is a geometrical constant only. For a parallel plate electrode setup the cell constant can be calculated easily:
$$K_{Cell} = \frac{d}{A}$$
with $d$ the distance bewteen the plates and $A$ the surface area size of the plates.
The capacitors in figure 8 represent the interfering effects. The perallel capacitance $C_{Cell}$ is the result of the direct AC-coupling between the electrodes an is here equal to
$$C_{Cell} = \frac{\epsilon A}{d} = \frac{\epsilon}{K_{Cell}}$$
with $\epsilon$ the dielectric constant of the electrolyte. The series capacitors are interface polarization effects of the electrode-electrolyte surfaces that can be simplified to
$$C_{interface} = A \cdot C_{dl}$$
with again $A$ the surface area size of the electrode and $C_{interface}$ the double-layer capacitance as described before.
The planar interdigitated finger electrode
As an example of a practical implementationn of a conductivity cell, a planar construction will be calculated. Alternative set-ups which are commercially used are round sticks with two or four metal rings around them. What is needed for modelling such a configuration is just the cell constant and the surface area, because then all modelleing components can be calculated. The surface area is normally not so difficult to calculate, but the cell constant may involve some mathematical complexity.
Fig. 9: Planar interdigitated finger structure
The cellconstant can be found by means of a conformal mapping transformation. A two dimensional evaluation is in most cases sufficient. Two dimensional conformal mapping is described in books and readers about complex function theory and in specific papers about calculating cell constants7). A conformal mapping transformation is transforming a space in such a way that the Maxwell equations remain valid. As a result when we can transform the electrical fieldlines conform Maxwell, we can transform the electrode geometry accordingly. In that case, we can transform a known configuration (for example the parallele plate capacitor) to the complex geometry of interest and transform the value of the cell constant as well.
For the interdigitated finger electrode we find
$$K_{Cell-finger} = \frac{2}{\left ( N-1 \right ) L} \cdot \frac{K \left ( k_{1} \right )}{K \left ( k_{2} \right )}$$
with
$$K \left ( k \right ) = \int_{0}^{1} \frac{1}{ \sqrt{\left ( 1-t^{2} \right ) \left ( 1-k^{2}t^{2} \right ) } } dt\\ k_{1}=\cos \left( \frac{\pi}{2} \cdot \frac{w}{s+w}\right )\\ k_{2}=\cos \left( \frac{\pi}{2} \cdot \frac{s}{s+w}\right )\\$$
and $S$ the spacing between the fingers, $W$ the width of the fingers, $L$ the length of the fingers and $N$ the number of fingers. To choose these geometries ($S$, $W$, $L$ and $N$) we have to optimse in such a way that the effect of the paracitic capacitances in the frequency range of interest are as small as possible.
Impedance measurements
The simulation of figure 10 is easily made from the electric equivalent circuit of figure 8. The numerical values $S = 4 \mu m$, $W = 200 \mu m$, $N = 5$ and $L = 1 mm$ were taken.
Fig. 10: Simulation of the spectrum of a conductivity cell - Bode plot and polar plot
In the first graph we can see the modulus as a function of frequency. The working region is easily recognized: it is the ferquency range in which the modulus is only dependent on concentration. The lower boundary of the sensitive region is about $10 kHz$ and is determined by the interface capacitances and the electrolyte conductivity. The upper boundary is the result of the cell-capacitance.
In the second graph, the data is plotted as a polar plot (imaginary part and real part as a parametric plot with the frequency as the independent parameter). In such a plot, the RC-couples are easily recognized as semi-circles. Polar plots can be used to identify electrode phenomena. This method is referred to as impedance spectrostopy. A real measurement picture conform figure 10 can be made with a gain-phase analyser, for example the HP4184.
When the user is only interested in a single frequency, a Phase-Locked-Loop (PLL) can be used. In its simples form, the PLL consists of an oscillator with a DC voltage dependent frequency, the Voltage-Controlled-Oscillator (VCO), and a phase detector, which can be a simple XOR port. The XOR port compares the phase of the signal of interest with the PLL internal phase, and adjust until these are equal.
Fig. 11: Principle of a Phase-Locked-Loop
The signals do not have to be sinewave-shaped, there are complete integrated circuits that have VCO, phase detector and filters in a single TTL component (LM565).
Applications and examples
Construction of the Pourbaix diagram for Hydrogen Peroxide H2O2
A convenient representation of acid-base and redox equilibria in a single plot is the Pourbaix diagram8). In such a plot, equilibria are plotted in a potiential versus pH graph. The construction of such a plot is included here because it explains the value of such diagrams and it explains the relation between acid-base chemistry and electrochemistry.
The system used in this example Pourbaix diagram is the one of hydrogen peroxide ($H_{2}O_{2}$) with all numerical values taken from the CRC Handbook of Chemistry and Physics9). It was used in my thesis about detergent activity monitoring10).
Acid-base equilibrium
Decomposition of hydrogen peroxide to the peroxide anion is given by:
$$H_{2}O_{2} \Leftrightarrow HO_{2}^{-} + H^{+}$$
with an acid constant equal to:
$$k_{a}= \frac{\left [ HO_{2}^{-} \right ] \cdot \left [ H^{+} \right ]}{\left [ H_{2}O_{2} \right ]} = 2.4 \cdot 10^{-12}.$$
With this, we can calculate that hydrogen peroxide is only dissociated for $2.3\%$ at $pH = 10$. The logarithm of this equation gives the first condition for the Pourbaix diagram:
$$\log \frac{\left [ HO_{2}^{-} \right ]}{\left [ H_{2}O_{2} \right ]} = -11.63+pH. \label{eq:DecompositionHydrogenPeroxide}$$
Apparently, for $pH = 11.63$ the ratio $[HO_{2}^{-}]/[H_{2}O_{2}]$ equals exactly one.
Electrochemical equilibrium
For both $HO_{2}^{-}$ and $H_{2}O_{2}$ the electrochemical interaction with water and $H^{+}$ can be described:
$$H_{2}O_{2} + 2H^{+} + 2e^{-} \Leftrightarrow 2H_{2}O \oplus E^{0}=1.776\\ HO_{2}^{-} + 3H^{+} + 2e^{-} \Leftrightarrow 2H_{2}O \oplus E^{0}=2.119\\$$
which results into the following implementations of the Nernst equation:
$$E^{0'}=1.776 + \frac{RT}{2F \log \left ( e \right )} \log \left [ H_{2}O_{2} \right ] - \frac{RT}{F \log \left ( e \right )} pH \\ E^{0'}=2.119 + \frac{RT}{2F \log \left ( e \right )} \log \left [ HO_{2}^{-} \right ] - \frac{3RT}{2F \log \left ( e \right )} pH. \\ \label{eq:H2O2_HO2_WithWater}$$
The electrochemical reactions of $HO_{2}^{-}$ and $H_{2}O_{2}$ with dissolved oxygen are
$$O_{2} + 2H^{+} + 2e^{-} \Leftrightarrow H_{2}O_{2} \oplus E^{0}=0.695 \\ H^{+} + O_{2} + 2e^{-} \Leftrightarrow HO_{2}^{-} \oplus E^{0}=0.338 \\$$
with the corresponding potential/concentration equations
$$E^{0'}=0.695 + \frac{RT}{2F \log \left ( e \right )} \log \frac{pO_{2}}{\left [ H_{2}O_{2} \right ]} - \frac{RT}{F \log \left ( e \right )} pH \\ E^{0'}=0.338 + \frac{RT}{2F \log \left ( e \right )} \log \frac{pO_{2}}{\left [ HO_{2}^{-} \right ]} - \frac{3RT}{2F \log \left ( e \right )} pH. \\ \label{eq:H2O2_HO2_WithOxygen}$$
The Pourbaix diagram
When equations \eqref{eq:DecompositionHydrogenPeroxide}, \eqref{eq:H2O2_HO2_WithWater} and \eqref{eq:H2O2_HO2_WithOxygen} are drawn in a single potential against pH plot, with some example values for the $O_{2}$, $H_{2}O_{2}$ and $HO_{2}^{-}$ concentrations, we get the Pourbaix diagram11).
Fig. 12: Pourbaix diagram for H2O2 in water of 25 °C
From the diagram we can conclude that below the lines of equation \eqref{eq:H2O2_HO2_WithWater} ((2.4) in the graph), hydrogen peroxide acts as an oxidator which oxidizes to water. Above the lines of equation \eqref{eq:H2O2_HO2_WithOxygen} ((2.6) in the graph), hydrogen peroxide is a reductor which reduces to dissolved oxygen. These two identities have a common area in which hydrogen peroxide is said to be double instable.
$$H_{2}O_{2} + 2H^{+} + 2e^{-} \Leftrightarrow H_{2}O \\ H_{2}O_{2} \Leftrightarrow O_{2} + 2H^{+} + 2e^{-} \\$$ so: $$2H_{2}O_{2} \Leftrightarrow 2H_{2}O + O_{2} .$$
It appears that at a metallic surface, with an electrode potential in the area of double instability, the decomposition of hydrogen peroxide into water and oxygen is being catalyzed.
Note that the Pourbaix diagram as based on the Nernst equations shows which reactions can take plate at electrode surfaces: whether this actually happens depends on other factors.
Constants
$R$ Molar gas constant $8.31441$ $J \cdot mol^{-1} K^{-1}$ $F$ Faraday's constant $9.64846 \cdot 10^{4}$ $C \cdot mol^{-1}$ $k$ Boltzmann's constant $1.38066 \cdot 10^{-23}$ $J \cdot K^{-1}$ $q$ Single electron charge $1.60218 \cdot 10^{-19}$ $C$
Tab. 3: Constants in electrochemical methods
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M. Pourbaix, Atlas of electrochemical equilibria in aqueous solutions, Pergamon Press, 1966
9)
D.R. Leide, H.P.R. Frederikse, Handbook of chemistry and physics, 74th edition 1993-1994, CRC Press, Inc., Boca Raton, Florida, 1993
10)
G.R. Langereis, An integrated sensor system for monitoring washing processes, Ph.D. Thesis, University of Twente, Enschede, 1999, ISBN 90-365-1272-7
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Volume 6, issue 5 (2006)
1 R G Burns, V W D Hale, A note on group rings of certain torsion-free groups, Canad. Math. Bull. 15 (1972) 441 MR0310046 2 I M Chiswell, P H Kropholler, Soluble right orderable groups are locally indicable, Canad. Math. Bull. 36 (1993) 22 MR1205890 3 É Ghys, Groups acting on the circle, Enseign. Math. $(2)$ 47 (2001) 329 MR1876932 4 A M W Glass, Partially ordered groups, Series in Algebra 7, World Scientific (1999) MR1791008 5 P H Kropholler, Amenability and right orderable groups, Bull. London Math. Soc. 25 (1993) 347 MR1222727 6 P A Linnell, Left ordered amenable and locally indicable groups, J. London Math. Soc. $(2)$ 60 (1999) 133 MR1721820 7 P A Linnell, Left ordered groups with no non-abelian free subgroups, J. Group Theory 4 (2001) 153 MR1812322 8 P Longobardi, M Maj, A H Rhemtulla, Groups with no free subsemigroups, Trans. Amer. Math. Soc. 347 (1995) 1419 MR1277124 9 P Longobardi, M Maj, A H Rhemtulla, When is a right orderable group locally indicable?, Proc. Amer. Math. Soc. 128 (2000) 637 MR1694872 10 J P Pier, Amenable locally compact groups, Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons (1984) MR767264 11 A H Rhemtulla, Polycyclic right-ordered groups, from: "Algebra, Carbondale 1980 (Proc. Conf., Southern Illinois Univ., Carbondale, Ill., 1980)", Lecture Notes in Math. 848, Springer (1981) 230 MR613189 12 A S Sikora, Topology on the spaces of orderings of groups, Bull. London Math. Soc. 36 (2004) 519 MR2069015 13 Y G Sinai, Introduction to ergodic theory, Translated by V. Scheffer, Mathematical Notes 18, Princeton University Press (1976) 144 MR0584788
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# Séminaires : Séminaire d'Algèbre
Equipe(s) : gr, Responsables : J. Alev, D. Hernandez, B. Keller, Th. Levasseur, et S. Morier-Genoud. Email des responsables : Jacques Alev , David Hernandez , Bernhard Keller , Thierry Levasseur , Sophie Morier-Genoud Salle : à distance / remote Adresse : IHP Description Depuis le 23 mars 2020, le séminaire se tient à distance. Pour les liens et mots de passe, merci de contacter l'un des organisateurs ou de souscrire à la liste de diffusion https://listes.math.cnrs.fr/wws/info/paris-algebra-seminar. L'information nécessaire sera envoyée par courrier électronique peu avant chaque exposé. Les notes et transparents sont disponibles ici. Since March 23, 2020, the seminar has been taking place remotely. For the links and passwords, please contact one of the organizers or subscribe to the mailing list at https://listes.math.cnrs.fr/wws/info/paris-algebra-seminar. The connexion information will be emailed shortly before each talk. Slides and notes are available here.
Orateur(s) Toshiki NAKASHIMA - Tokyo, Titre Geometric crystals on cluster varietie Date 05/02/2018 Horaire 14:00 à 15:00 Diffusion Résume The notion of geometric crystal was initiated by A.Berenstein and D.Kazhdan to consider certain geometric analogue to the Kashiwara's crystal base theory. Their structures are described by rational maps and rational functions. If all these rational maps are positive'', such geometric crystals are called positive'' and they can be tranfered to the Langlands dual crystal bases'' by tropicalization/ultra-discretization procedure. V.Fock and A.Goncharov defined certain pair of varieties (A,X), called cluster ensemble'' which is obtained by glueing algebraic tori using the A-mutations and X-mutations'' respectively. They gave the conjectures on tropical duality'' between cluster ensemble A-variety and X-variety (called Fock-Goncharov conjectures). We shall define the positive geometric crystal structure on cluster varieties and then obtain the resulting tropicalized crystals, which will be a guide to understand the Fock-Goncharov conjectures in terms of crystal base theory. Finally, we shall show some compatibility of geometric crystal structures on A-variety and X-variety in the classical type A case. Salle à distance / remote Adresse IHP
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Lowerbound against homogeneous multilinear formulas
Speaker:
Time:
Friday, 13 September 2019, 17:15 to 18:15
Venue:
• A-201 (STCS Seminar Room)
Organisers:
Abstract: A polynomial is said to be multilinear if the individual degree of every variable is at most one in any monomial; and is said to be homogeneous if every monomial in it has the same degree. Many polynomials of interest (like the determinant or the permanent of a symbolic matrix) are homogeneous multilinear polynomials. An algebraic formula is a model for computing polynomials and is said to be a homogeneous multilinear one if every gate in it computes a homogeneous multilinear polynomial. Hrubes and Yehudayoff (in their 2011 paper) showed that any polynomial that is computed by a homogeneous multilinear formula has a very special decomposition (called the log-product decomposition) and used it to give a lowerbound against this model. In today's talk, we will see the full proof of this lowerbound.
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+0
# Complex Numbers. Help is appreciated
0
69
1
Consider the complex number in this picture
We can write
$$z = 2\sqrt{2}e^{i\theta_1} = 2\sqrt{2}e^{i\theta_2} = 2\sqrt{2}e^{i\theta_3}$$
for $$\theta_1, \theta_2, \theta_3$$ between 0 and 6pi. Answer with $$\theta_1, \theta_2, \theta_3$$
Help is really appreciated.
Aug 13, 2022
$$(\theta_1, \theta_2, \theta_3 = \pi/8, 15 \pi/8, 7 \pi/4)$$
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# Cannot get large centered text vertically aligned with flushed text
I cannot get the name to vertically align with the text on the left or the right. Using a table seems better but its still off.
Just to clarify below are my two attempts to vertically align the name with the contact info on both sides.
The first way is using a new command i created
The second uses a table
\documentclass[letterpaper]{article}
\usepackage[margin=0.75in]{geometry}
\usepackage{tabularx}
\def\name{John Doe}
\def\email{john.doe@gmail.com}
\def\phone{(123) 456-7890}
\def\city{Toronto}
\def\province{ON}
\def\postalCode{M5Y\thinspace4Z7}
\newcommand{\rTitle}{
\noindent
\centerline{\Huge \textsc \name} \\
\texttt{\phone} \hfill \city, \province \space \postalCode \\
}
\begin{document}
\rTitle
\noindent
\begin{tabularx}{\linewidth}{l >{\centering}X r}
\texttt{\phone} && \city, \province \space \postalCode \\
\end{tabularx}
\end{document}
• Welcome to TeX SE! Could you explain what you want it to look like? What exactly should align with what? – cfr Aug 6 '14 at 0:55
• updated Hope its more clear now – john Aug 6 '14 at 1:32
• If you are asking why the two examples aren't aligned then try to set \tabcolsep=0pt before the tabularx environment. BTW. The \\ after \centerline and after \postalCode are badly placed and cause the warning about underfull hbox. Remove them. – wipet Aug 6 '14 at 4:30
You could use three (vertically centered) side-by-side minipages:
The code:
\documentclass[letterpaper]{article}
\usepackage[margin=0.75in]{geometry}
\usepackage{tabularx}
\def\name{John Doe}
\def\email{john.doe@gmail.com}
\def\phone{(123) 456-7890}
\def\city{Toronto}
\def\province{ON}
\def\postalCode{M5Y\thinspace4Z7}
\newcommand{\rTitle}{%
\par\noindent
\begin{minipage}{.333\textwidth}
\texttt{\phone}
\end{minipage}%
\begin{minipage}{.333\textwidth}
\centering\Huge\textsc{\name}
\end{minipage}%
\begin{minipage}{.333\textwidth}
\raggedleft
\city, \province \space \postalCode
\end{minipage}\par%
}
\begin{document}
\rTitle
\end{document}
Beware of spurious blan spaces in your code.
This simple aligment can be done by simple tools without tabularx:
\centerline{\href{mailto:\email}{\nolinkurl{\email}} \hfill \address}
\centerline{\rlap{\texttt{\phone}}\hfill
\smash{\Huge \textsc \name}\hfill
\llap{\province \space \postalCode}}
• \textsc takes an argument, so this should be \textsc{\name}. – barbara beeton Aug 6 '14 at 13:10
• @barbarabeeton I've copied this part of the code for the original question and it works, because the parameter is only one token \name and the macro \textsc has one undelimited parameter. If the real parameter is one token then it could be in braces or not. On the other hand, you are right that the usage \textsc{\name} is more clear and educational. – wipet Aug 6 '14 at 16:51
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# Revision history [back]
### Fedora 20 - KDE not starting up
Well, it's been a long time I have a look at KDE. So, I decided to install it too on my Fedora 20 GNOME edition, and have a look at where is KDE now. And I installed KDE using:
yum install @kde-desktop
After rebooting, I tried to login to KDE Plasma and then kde wallpaper comes along with fedora logo, then the hard disk icon, and it stops there. I waited for a long time, and it still not loading. See the following picture for a good idea of what I am saying.
Here is the relevant portion from journalctl
Mar 31 00:05:07 linux-master /etc/gdm/Xsession[1838]: startkde: Starting up...
As you can see, I waited for around 3 minutes before pressing Alt+Ctrl+F2 and rebooting system. Also, there is no logs to show any kind of error. I looked for the hidden file .xsession-errors in my home directory and it is not there. Itried KDE Plasma failsafe mode but that also not starting up. Hanging at disk loading icon.
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# All Questions
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20 views
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27 views
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30 views
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24 views
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### Dearth of research on treating Periodic Limb Movement Disorder (PLMD) with Baclofen
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18 views
### Do studies exist that can map specimens of neocortex to the functions they perform(ed) in vivo?
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31 views
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34 views
### Success cause passion more than passion cause success
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21 views
### As adults, is it possible for thoughts to originate from our own thinking? [closed]
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# Calculate the Efficiency of Packing in Case of a Metal Crystal For Body-centred Cubic - Chemistry
Calculate the efficiency of packing in case of a metal crystal for body-centred cubic
#### Solution
Body-centred cubic:
It can be observed from the above figure that the atom at the centre is in contact with the other two atoms diagonally arranged.
From ΔFED, we have:
b2 = a2 + a2
⇒ b2 = 2a2
⇒ b = sqrt2a
Again, from ΔAFD, we have:
c2 = a2 + b2
⇒ c2 = a2 + 2a2 (Since b2 = 2a2)
⇒ c2 = 3a2
⇒ c = sqrt3a
Let the radius of the atom be r.
Length of the body diagonal, c = 4π
⇒sqrt3a = 4r
⇒a =(4r)/sqrt3
or r = (sqrt3a)/4
Volume of the cube a^3 = ((4r)/sqrt3)^3
A body-centred cubic lattice contains 2 atoms.
So, volume of the occupied cubic lattice 2pi4/3 r^3
=8/3pir^3
:.Packing efficiency = ("Voulume occupied by two spheres in the unit cell")/"Total volume of unit"xx100%
= (8/3pir^3)/(4/(sqrt3)r)^3xx100%
=(8/3pir^3)/(64/(3sqrt3)r^3) xx 100%
=68%
Concept: Packing Efficiency - Efficiency of Packing in Body-centred Cubic Structures
Is there an error in this question or solution?
#### APPEARS IN
NCERT Class 12 Chemistry
Chapter 1 The Solid State
Q 10.2 | Page 31
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# Boundary conditions in Montjoie
The boundary conditions implemented in Montjoie are listed in the class BoundaryConditionEnum :
The boundary conditions are usually taken into account in the finite element matrix (through the variational formulation). Dirichlet and Supported boundary conditions are set by setting the concerned rows and columns to 0, with 1 on the diagonal. For inhomogenous Dirichlet, the columns are kept before erasing them so that the right hand side is modified. Periodic boundary conditions can be set by changing the numbers of degrees of freedom. Quasi-periodic boundary conditions are set either in the variational formulation or by modifying some rows of the finite element matrix.
The treatment of usual boundary conditions is performed in the class VarBoundaryCondition which is a base class of EllipticProblem.
## Public attributes of VarBoundaryCondition
NewColumnNumbers_Impedance new column numbers when adding the surface integrals against basis functions ProcColumnNumbers_Impedance processor numbers (for columns) when adding the surface integrals against basis functions NewRowNumbers_Impedance new row numbers when adding the surface integrals against basis functions ProcRowNumbers_Impedance processor numbers (for rows) when adding the surface integrals against basis functions take_into_account_curvature_for_abc if true, the curvature is taken into account for absorbing boundary condition grazing_abc if true, the absorbing boundary condition is changed to handle mainly grazing waves gamma_cla_coef Parameter of the absorbing absorbing condition (for Helmholtz only) theta_cla_coef Parameter of the absorbing absorbing condition (for Helmholtz only) zeta_cla_coef Parameter of the absorbing absorbing condition (for Helmholtz only)
## Methods for ImpedanceFunction_Base
The class ImpedanceFunction_Base is the base class for all impedance classes (such as ImpedanceGeneric, ImpedanceABC or ImpedanceHighConductivity). These classes are used for impedance boundary conditions when the function AddMatrixImpedanceBoundary is called to add the corresponding terms to the finite element matrix.
InvolveOnlyTangentialDofs returns true if only tangential dofs (associated with the surface) are involved in surface integrals SetCoefficient sets the impedance coefficients (in the case of an uniform impedance) GetCoefficient returns the impedance coefficient at a given quadrature point PresenceGradient returns true if the boundary integrals involve the gradient of basis functions EvaluateImpedancePhi evaluates the impedance involved in surface integrals against basis functions EvaluateImpedanceGrad evaluates the impedance involved in surface integrals against gradient of basis functions ApplyImpedancePhi_H1 applies the impedance involved in surface integrals against scalar basis functions ApplyImpedanceGrad applies the impedance involved in surface integrals against gradient of scalar basis functions ApplyImpedancePhi_Hcurl applies the impedance involved in surface integrals against basis functions (edge elements) ApplyImpedanceCurl applies the impedance involved in surface integrals against the curl of basis functions (edge elements) ApplyImpedancePhi_Hdiv applies the impedance involved in surface integrals against basis functions (facet elements) ApplyImpedanceDiv applies the impedance involved in surface integrals against the divergence of basis functions (facet elements)
## Transparent conditions
The details of transparent conditions is detailed in the thesis of M. Duruflé. The cornerstone of this method is a representation integral of the field outside a surface. In the case of Maxwell's equations, these formulas (known as Stratton-Chu formulas) are equal to
with the Green kernel and dyadic Green function :
There are similar expression for Helmholtz equation. Transparent conditions are not implemented with other equations. Γ is an intermediary surface surrounding the scatterer and Σ the external boundary. The boundary condition is then set on Σ to :
The linear system to solve is then equal to
The matrix As is the finite element matrix with a first-order absorbing boundary condition (this matrix is sparse) whereas the matrix Ad is the matrix containing the integral equation terms (this matrix has an important dense block). The matrix As is factorized (or solved iteratively), and the transparent condition is solved iteratively on the following linear system
If the distance between Γ and Σ is large enough (one wavelength is fine), the number of iterations needed to solve this system is quite small. The treatment of transparent conditions is handled by the class TransparencySolver.
### Public methods of TransparencySolver
constructor of TransparencySolver UseTransparentCondition returns true if a transparent condition is set Solve solves the linear system associated with the transparent condition Init initialisation and computation of arrays needed for the resolution of the linear system associated with the transparent condition ComputeSolution computes the solution of As x = y ComputeGreenKernel computation of Green kernel and dyadic Green function ComputeSurfaceGammaAndAbsorbing computes the quadrature points on the two surfaces ComputeRightHandSide computes the matrix-vector product Ad x ComputeAndStoreEnPot computes the integral representation of E x n (and H x n) on all quadrature points GetSource computes the source term (on a given quadrature point) for the transparent condition ComputeIntegralRepresentation computes the integral representation of E x n (and H x n) on a given point
For Helmholtz and Maxwell's equations, it is also possible to compute the far field (through an integral over a surface). The computation of the far field (also known as radar cross section) is performed in class VarComputationRCS
### Public methods of VarComputationRCS
RcsToBeComputed returns true if a radar cross section needs to be computed GetNbAngles returns the number of angles for which we want to know the far field GetNbPointsOutside returns the number of points for which we want to know the field GetInterpolationMesh returns the surface mesh used for the computation of the RCS GetRcsType returns the type of radar cross section (monostatic or bistatic) GetOutsidePoint returns the coordinates of the points where we want to know the field SetOutsidePoints sets the coordinates of the points where we want to know the field SetTimeStep sets the time step (for unsteady equations) InitComputationRCS initializes the object before computing RCS ComputeIntegralRepresentation computes the value of the field outside the computational domain with an integral representation ComputeRCS effective computation of the radar cross section
#### Dirichlet Boundary condition (LINE_DIRICHLET)
Dirichlet boundary conditions can be homogeneous
or inhomogeneous
In the case of edge elements, Dirichlet condition is actually a perfectly conductor condition :
Dirichlet condition is set by setting to 0 the concerned rows and columns and putting 1 on the diagonal. This coefficient one can be changed for the computation of eigenvalues to avoid the pollution of the spectrum with eigenvalues equal to 1. For discontinuous Galerkin formulations, the Dirichlet condition is handled in the variational formulation.
Neumann boundary conditions are equal for the Laplace and Helmholtz equation
In the case of Maxwell's equations, it refers to :
In the case of elastodynamics, it refers to :
In the case of aeroacoustics, it refers to :
As you can see, this boundary condition depends on the considered equation, but usually is associated with the boundary condition that will cancel the boundary integral coming from the integration by parts.
#### Supported Boundary condition (LINE_SUPPORTED)
Supported boundary conditions consists of imposing a Dirichlet condition
for only a subset of components k of the unknown u. The components are given by inserting a line SupportedComponents = 1 3 in the datafile. For other components, a natural condition is imposed (no boundary terms like for Neumann condition).
#### Dirichlet to Neumann (LINE_DTN)
This condition is reserved for a future implementation of Dirichlet-to-Neumann operators in Montjoie
#### Transmission conditions (LINE_TRANSMISSION)
Transmission conditions are present for some models in Montjoie. The documentation will be updated later.
#### False boundary condition (LINE_NEIGHBOR)
LINE_NEIGHBOR is used for boundaries located at the interface between two processor. For such boundaries, no boundary condition is needed. However in Montjoie, since the face is isolated (no adjacent face in the current processor), a boundary condition is needed. That's why, such faces are affected with this false boundary condition.
#### Thin-slot models (LINE_THIN_SLOT)
Thin slot models are implemented in Montjoie for 2-D Helmholtz equation. The documentation will be updated later.
#### High-conductivity models (LINE_HIGH_CONDUCTIVITY)
High-conductivity models are implemented in Montjoie for Helmholtz and Maxwell's equations. The documentation will be updated later.
#### Robin boundary conditions (LINE_IMPEDANCE)
Robin boundary conditions consists of considering.
for scalar equations. The documentation will be updated later.
#### Absorbing boundary conditions (LINE_ABSORBING)
Absorbing boundary conditions are used to truncate the computational domain. For most of equations, only first-order absorbing boundary conditions are implemented. For Helmholtz equations, it consists of considering.
where k is the wave number.
### NewColumnNumbers_Impedance, ProcColumnNumbers_Impedance
This attribute is an array used when adding boundary terms in the finite element matrix. You can modify the column numbers of the degrees of freedom such that the term.
is added to the entry i, NewColumnNumbers_Impedance(j) of the matrix (instead of i, j). ProcColumnNumbers_Impedance is used in parallel to store the processor associated with the new number. If the processor is equal to the current processor, the column number is local, whereas it is global for a different processor.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// initializing an empty matrix;
int N = var.GetNbDof();
DistributedMatrix<Real_wp, General, ArrayRowSparse> A;
A.Reallocate(N, N);
// filling the new column numbers (here we place random numbers)
var.NewColumnNumbers_Impedance.Reallocate(N);
var.NewColumnNumbers_Impedance.FillRand();
// for parallel computation you can also set a different processor number
var.ProcColumnNumbers_Impedance.Reallocate(N);
var.ProcColumnNumbers_Impedance.Fill(0);
// then you can add the impedance term
GlobalGenericMatrix<Real_wp> nat_mat;
IVect ref(var.mesh.GetNbReferences()+1); ref.Zero(); ref(3) = 1;
var.AddMatrixImpedanceBoundary(1.0, ref, 1, nat_mat, A, 0, 0, imped, true, false, var);
### NewRowNumbers_Impedance, ProcRowNumbers_Impedance
This attribute is an array used when adding boundary terms in the finite element matrix. You can modify the row numbers of the degrees of freedom such that the term.
is added to the entry NewRowNumbers_Impedance(i), j of the matrix (instead of i, j). ProcRowNumbers_Impedance is used in parallel to store the processor associated with the new number. If the processor is equal to the current processor, the row number is local, whereas it is global for a different processor.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// initializing an empty matrix;
int N = var.GetNbDof();
DistributedMatrix<Real_wp, General, ArrayRowSparse> A;
A.Reallocate(N, N);
// filling the new row numbers (here we place random numbers)
var.NewRowNumbers_Impedance.Reallocate(N);
var.NewRowNumbers_Impedance.FillRand();
// for parallel computation you can also set a different processor number
var.ProcRowNumbers_Impedance.Reallocate(N);
var.ProcRowNumbers_Impedance.Fill(0);
// then you can add the impedance term
GlobalGenericMatrix<Real_wp> nat_mat;
IVect ref(var.mesh.GetNbReferences()+1); ref.Zero(); ref(3) = 1;
var.AddMatrixImpedanceBoundary(1.0, ref, 1, nat_mat, A, 0, 0, imped, false, true, var);
### take_into_account_curvature_for_abc
This attribute is a boolean. If it is set to true, the absorbing boundary condition will include the term due to the curvature (1/R term). This feature is only implemented for Helmholtz equation. It is usually modified by inserting a line ModifiedImpedance = CURVE in the data file.
### grazing_abc
This attribute is a boolean. If it is set to true, the absorbing boundary condition will be modified to handle correctly grazing waves. This feature is only implemented for Helmholtz equation. It is usually modified by inserting a line OrderAbsorbingBoundaryCondition = 1 GRAZING in the data file.
### gamma_cla_coef, theta_cla_coef, zeta_cla_coef
These attributes are doubles that are used to parametrize absorbing boundary conditions. This feature is only implemented for Helmholtz equation. It is usually modified by inserting a line OrderAbsorbingBoundaryCondition = 2 PARAMETERS 0.2 0.5 1.0 in the data file.
### GetInitialSymmetrization
This method returns true if the matrix (for a mixed formulation) can be symmetrized (with respect to boundary conditions). It does not mean that the matrix will be symmetrized when PerformFactorizationStep is called.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// the matrix can be symmetrized ?
bool init_sym = var.GetInitialSymmetrization();
### GetBoundaryConditionId
#### Syntax
int GetBoundaryConditionId(IVect ref, int pos, VectString parameters, bool& periodic)
This method returns the integer corresponding to the boundary condition given as a string (or a list of strings). Usually, only parameters(pos) is used to determine the boundary condition. For more complex boundary conditions, several parameters can be used (parameters(pos), parameters(pos+1), etc). The first argument is not used, the last argument is true if the boundary condition corresponds to a periodic (or quasi-periodic condition).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// boundary condition id (e.g. BoundaryConditionEnum::LINE_DIRICHLET) ?
IVect ref; int pos = 1; VectString parameters(pos+1);
parameters(pos) = "DIRICHLET"; bool periodic;
int id = var.GetBoundaryConditionId(ref, pos, parameters, periodic);
### GetNbDirichletDof
#### Syntax
int GetNbDirichletDof() const
This method returns the number of degrees of freedom associated with a Dirichlet condition (u = f). Only degrees of freedom of the current processor are counted.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of Dirichlet dofs for the current processor
nb_dir_dof = var.GetNbDirichletDof();
### GetNbGlobalDirichletDof
#### Syntax
int GetNbGlobalDirichletDof() const
This method returns the number of degrees of freedom associated with a Dirichlet condition (u = f). Degrees of freedom of all the processors are counted.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of Dirichlet dofs for the overall simulation (with all processors)
nb_dir_dof = var.GetNbGlobalDirichletDof();
### GetDirichletDofNumber
#### Syntax
int GetDirichletDofNumber(int i ) const const IVect& GetDirichletDofNumber() const
This method returns the dof number of the i-th Dirichlet dof. In the second syntax, you can retrieve the array containing Dirichlet dofs.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of Dirichlet dofs for the current processor
nb_dir_dof = var.GetNbGlobalDirichletDof();
// loop over Dirichlet dofs
for (int i = 0; i < nb_dir_dof; i++)
{
// you can retrieve a single dof
int num_dof = var.GetDirichetDofNumber(i);
}
// or all the dofs
const IVect& dir_dof = var.GetDirichletDofNumber();
### IsDofDirichlet
#### Syntax
bool IsDofDirichlet(int i ) const
This method returns true if the degree of freedom i is associated with a Dirichlet condition (u = f).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// dof 32 is Dirichlet ?
bool dof_dir = var.IsDofDirichlet(32);
### GetIsDofDirichlet
#### Syntax
const Vector& GetIsDofDirichlet() const
This method returns the array containing booleans to know if a degree of freedom is Dirichlet or not.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// dof 32 is Dirichlet ?
bool dof_dir = var.IsDofDirichlet(32);
// if you want all the values at once
const VectBool& arr_dir = var.GetIsDofDirichlet();
### UseSymmetricDirichlet
#### Syntax
bool UseSymmetricDirichlet() const
This method returns true if rows and columns associated with Dirichlet dofs are removed (and a non-null coefficient is put on the diagonal). This treatment is already performed for symmetric matrices. If the method returns true, the treatment will also be performed for unsymmetric matrices. If the method returns false, only rows are erased for unsymmetric matrices.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// rows and columns are eliminated for Dirichlet ?
bool dir_sym = var.UseSymmetricDirichlet();
### EnableSymmetricDirichlet
#### Syntax
bool EnableSymmetricDirichlet(bool sym = true) const
If the argument sym is true, rows and columns associated with Dirichlet dofs are removed (and a non-null coefficient is put on the diagonal). This treatment is already performed for symmetric matrices, but will be also performed for unsymmetric matrices. The aim is to obtain a better conditioning of the finite element matrix. This feature is usually activated by inserting a line ForceDirichletSymmetry = YES in the datafile.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// if we want to eliminate rows and columns for Dirichlet dofs
var.EnableSymmetricDirichlet();
### GetCoefficientDirichlet
#### Syntax
Real_wp GetCoefficientDirichlet() const
This method returns the coefficient set on the diagonal for Dirichlet dofs. Rows associated with Dirichlet dofs are removed, only a coefficient is kept on the diagonal.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// coefficient on diagonal for Dirichlet ?
Real_wp coef = var.GetCoefficientDirichlet();
### SetCoefficientDirichlet
#### Syntax
void SetCoefficientDirichlet(Real_wp coef) const
This method sets the coefficient set on the diagonal for Dirichlet dofs. Rows associated with Dirichlet dofs are removed, only a coefficient is kept on the diagonal. You can also modify this coefficient (initially equal to one) by inserting a line DirichletCoefMatrix = 2.0 in the datafile.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// modifying coefficient on diagonal for Dirichlet
var.SetCoefficientDirichlet(Real_wp(2));
### SetDirichletDof
#### Syntax
void SetDirichletDof(int n, bool b) const
This method sets the n-th degree of freedom as a Dirichlet dof (if b is true). Once you have manually modified Dirichlet dofs, you need to call the method UpdateDirichletDofs to reconstruct the array storing the Dirichlet dofs.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// manually adding Dirichlet dofs (not recommended)
var.SetDirichletDof(3, true);
var.SetDirichletDof(12, true);
// you can also remove Dirichlet dofs
var.SetDirichletDof(11, false);
// once you finished, you call UpdateDirichletDofs
var.UpdateDirichletDofs();
### GetNbSupportedComponents
#### Syntax
int GetNbSupportedComponents() const
This method returns the number of components for the supported boundary condition.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
int nb_comp = var.GetNbSupportedComponents();
// loop over supported components
for (int k = 0; k < nb_comp; k++)
{
// displaying the list of components for the supported boundary condition
cout << "component " << var.GetSupportedComponent(k) << endl;
}
### GetSupportedComponent
#### Syntax
int GetSupportedComponent(int k ) const
This method returns a component number for the supported boundary condition. The supported boundary condition consists of imposing
for integers i that corresponds to component numbers.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
int nb_comp = var.GetNbSupportedComponents();
// loop over supported components
for (int k = 0; k < nb_comp; k++)
{
int i = var.GetSupportedComponent(k);
// displaying the list of components for the supported boundary condition
cout << "component " << i << endl;
// we have u_i = f
}
### SetSupportedComponents
#### Syntax
void SetSupportedComponents(IVect num ) const
This method sets the list of component numbers for the supported boundary condition. The supported boundary condition consists of imposing
for integers i that corresponds to component numbers. Usually the components of the supported boundary condition are set by inserting a line SupportedComponents = 2 5 in the datafile.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// setting manually components for the supported boundary condition
IVect num(2); num(0) = 2; num(1) = 5;
var.SetSupportedComponents(num);
### ImposeNullDirichletCondition
#### Syntax
void ImposeNullDirichletCondition(Vector& u) const
This method enforces an homogeneous Dirichlet condition
for the input vector u.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
VectReal_wp u(var.GetNbDof()); u.FillRand();
// setting u = 0 for Dirichlet dofs
var.ImposeNullDirichletCondition(u);
### GetHighConductivityOrder
#### Syntax
int GetHighConductivityOrder() const
This method returns the order of approximation for the high-conductivity boundary condition.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// order for high-conductivity boundary condition
int r_conduc = var.GetHighConductivityOrder();
### FindDofsOnReference
#### Syntax
void FindDofsOnReference(const VarProblem& var, const IVect& ref_cond, int ref_target, IVect& Dofs) const
This method retrieves the degrees of freedom associated with faces (edges in 2-D) such that the reference ref satisfies ref_cond(ref) = ref_target.
#### Parameters
var (in)
instance of VarProblem
ref_cond (in)
references i such that ref_cond(i) = ref_target are considered
ref_target (in)
target reference
Dofs (out)
list of degrees of freedom located on the selected referenced faces
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// degrees of freedom for reference 2 and 3 ?
IVect ref_cond(var.mesh.GetNbReferences()+1); ref_cond.Zero();
int ref_target = 1; ref_cond(2) = ref_target; ref_cond(3) = ref_target;
IVect Dofs;
var.FindDofsOnReference(var, ref_cond, ref_target, Dofs);
### TreatDirichletCondition
#### Syntax
void TreatDirichletCondition()
This method computes all the degrees of freedom located on Dirichlet boundaries. This method does not need to be called in a regular use, since it is already called by ComputeMeshAndFiniteElement.
### SetDirichletDofs
#### Syntax
void SetDirichletDofs(int N, IVect dof_list)
This method sets all the degrees of freedom considered as Dirichlet dofs (i.e. such that the equation u = f is imposed on these dofs). Previous Dirichlet dofs are removed and replaced by the provided list. After calling this method, you do not need to call UpdatedirichletDofs.
#### Parameters
N (in)
number of Dirichlet dofs
dof_list (in)
dof numbers
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// degrees of freedom for reference 2 and 3 ?
IVect ref_cond(var.mesh.GetNbReferences()+1); ref_cond.Zero();
int ref_target = 1; ref_cond(2) = ref_target; ref_cond(3) = ref_target;
IVect Dofs;
var.FindDofsOnReference(var, ref_cond, ref_target, Dofs);
// then you can choose to impose Dirichlet on these dofs (and remove previous Dirichlet dofs)
var.SetDirichletDofs(Dofs.GetM(), Dofs);
### ResizeNbDof
#### Syntax
void ResizeNbDof(int N)
This method changes the number of degrees of freedom of the global problem. It is better to call that method other SetNbDof since IsDofDirichlet will work correctly.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// you want to add a new degree of freedom
int N = var.GetNbDof();
var.ResizeNbDof(N+1);
### ComputeDirichletCoef
#### Syntax
void ComputeDirichletCoef(VirtualMatrix& A)
This method evaluates the largest eigenvalue of the input matrix A (by iterative power method) and returns twice this estimation. This value can then be used as Dirichlet coefficient (on the diagonal), such that eigenvalues due to Dirichlet condition will be outside the spectrum of interest.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// computing the matrix
GlobalGenericMatrix<Real_wp> nat_mat;
DistributedMatrix<Real_wp, Symmetric, ArrayRowSymSparse> A;
// coefficient for Dirichlet diagonal ?
Real_wp coef = var.ComputeDirichletCoef(A);
### UpdateDirichletDofs
#### Syntax
void UpdateDirichletDofs()
This method updates the arrays for Dirichlet dofs. It has to be called if Dirichlet dofs have been set manually with SetDirichletDof.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// manually adding Dirichlet dofs (not recommended)
var.SetDirichletDof(3, true);
var.SetDirichletDof(12, true);
// you can also remove Dirichlet dofs
var.SetDirichletDof(11, false);
// once you finished, you call UpdateDirichletDofs
var.UpdateDirichletDofs();
### ApplyDirichletCondition
#### Syntax
void ApplyDirichletCondition(SeldonTranspose trans , FemMatrixFreeClass& , A, Vector& b , int k = 0) const
This method modifies the right hand side because of inhomogeneous Dirichlet condition. To recover the symmetry of the matrix, rows and columns associated with Dirichlet dofs are removed. As a result, the right hand side must be modified to take into account inhomogeneous Dirichlet condition (u = f). On input, the right hand side b contains values of f on degrees of freedom. This method does not need to be called in a regular use, since ComputeSolution already calls it.
#### Parameters
trans (in)
Do we want to solve A x = b or the transpose system ?
A (in)
iterative finite element matrix
b (inout)
right hand side to modify
k (optional)
right hand side number
### GetNbModes
#### Syntax
int GetNbModes() const
This method returns the number of modes to compute in order to recover the solution for a source that has no symmetry (only the computational domain has symmetry). For example, in case of cyclic domains, it is the number of Fourier modes. If the computational domain has no symmetry, it returns one.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of modes to compute the solution (for periodic/cyclic domains)
nb_modes = var.GetNbModes();
### GetNbModesSource
#### Syntax
int GetNbModesSource() const
This method returns the number of modes to compute in order to recover the solution for a source that has no symmetry (only the computational domain has symmetry). For example, in case of cyclic domains, it is the number of Fourier modes. If the computational domain has no symmetry, it returns one. The difference between GetNbModes and GetNbModesSource happens for axisymmetric computation. In that case, each mode is solved independently, and there is only "one mode" for the computation of the source, since we compute directly the decomposition of the source on the required mode.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of modes for the evaluation of the source
nb_modes = var.GetNbModesSource();
### GetModeNumber
#### Syntax
int GetModeNumber(int i ) const
This method returns the mode number of the i-th mode to compute. For instance, if we consider axisymmetric computations, where the solution is written as
If all the modes (between -M and M) are involved, the first computed mode will be m=0, then m=1, then m=-1, etc. So the method GetModeNumber will return -1 for i = 2.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of modes for the evaluation of the source
nb_modes = var.GetNbModesSource();
// first mode to be computed
int m = var.GetModeNumber(0);
### GetCurrentModeNumber
#### Syntax
int GetCurrentModeNumber() const
This method returns the mode number of the current mode that is solved.
#### Example :
EllipticProblem<HelmholtzEquationAxi> var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// setting a mode number (value of m)
var.SetCurrentModeNumber(1);
// then you can retrieve this number
int m = var.GetCurrentModeNumber();
### SetCurrentModeNumber
#### Syntax
void SetCurrentModeNumber(int m ) const
This method sets the mode number of the current mode that will be solved.
#### Example :
EllipticProblem<HelmholtzEquationAxi> var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// setting a mode number (value of m)
var.SetCurrentModeNumber(1);
// then you can retrieve this number
int m = var.GetCurrentModeNumber();
### ModesNotStored
#### Syntax
bool ModesNotStored() const
This method returns true if the modes are not stored. When the modes are not stored, the final solution is modified at each computation. If the modes are stored, fft can be used to obtain quickly the final solution. This functionality is turned on/off by inserting a line StorageModes = YES in the data file.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// modes will be stored ?
bool store_modes = var.ModesNotStored();
### ForceStorageModes
#### Syntax
bool ForceStorageModes(bool store = true)
This method enables (or disables) the storage of modes. When the modes are not stored, the final solution is modified at each computation. If the modes are stored, fft can be used to obtain quickly the final solution. This functionality is usually turned on/off by inserting a line StorageModes = YES in the data file.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// if you want to force the storage of modes
var.ForceStorageModes(true);
### GetSymmetryType
#### Syntax
int GetSymmetryType()
This method returns the type of symmetry that will be used for the computation of the solution. It can be equal to
• NO_SYMMETRY : no symmetry is present (default case) : only one mode is computed
• PERIODIC_THETA : cyclic domain (only a section is meshed)
• PERIODIC_X : periodic domain in x (only a cell is meshed)
• PERIODIC_Y : periodic domain in y (only a cell is meshed)
• PERIODIC_Z : periodic domain in z (only a cell is meshed)
• PERIODIC_XY : periodic domain in x and y (only a cell is meshed)
• PERIODIC_XZ : periodic domain in x and z(only a cell is meshed)
• PERIODIC_YZ : periodic domain in y and z (only a cell is meshed)
• PERIODIC_XYZ : periodic domain in x, y and z (only a cell is meshed)
• PERIODIC_ZTHETA : periodic domain in z and theta (only a cell is meshed)
The periodic conditions are set when inserting a line ConditionReference = 1 2 CYCLIC in the data file.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// type of periodicity in the mesh ?
int sym = var.GetSymmetryType();
### GetNbPeriodicDof
#### Syntax
int GetNbPeriodicDof()
This method returns the number of degrees of freedom that are labelled periodic. For these degrees of freedom, the following equation
replaces the variational formulation. This replacement is performed only for a strong formulation of quasi-periodic conditions (UseSameDofsForPeriodicCondition = NO in the data file).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of dofs where quasi-periodic condition is applied
int nb_dof_per = var.GetNbPeriodicDof();
### GetPeriodicDof
#### Syntax
int GetPeriodicDof(int i)
This method returns the number k of the i-th periodic dof. For these degrees of freedom, the following equation
replaces the variational formulation. This replacement is performed only for a strong formulation of quasi-periodic conditions (UseSameDofsForPeriodicCondition = NO in the data file).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of dofs where quasi-periodic condition is applied
int nb_dof_per = var.GetNbPeriodicDof();
// loop over periodic dofs
for (int i = 0; i < nb_dof_per; i++)
{
// number of the periodic dof
int k = var.GetPeriodicDof(i);
// number of the original dof
int j = var.GetOriginalPeriodicDof(i);
// we have an equation u_k = u_j * e^{i phase}
}
### GetOriginalPeriodicDof
#### Syntax
int GetOriginalPeriodicDof(int i)
This method returns the number j of the original degree of freedom associated with the i-th periodic dof. For periodic degrees of freedom, the following equation
replaces the variational formulation. This replacement is performed only for a strong formulation of quasi-periodic conditions (UseSameDofsForPeriodicCondition = NO in the data file).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of dofs where quasi-periodic condition is applied
int nb_dof_per = var.GetNbPeriodicDof();
// loop over periodic dofs
for (int i = 0; i < nb_dof_per; i++)
{
// number of the periodic dof
int k = var.GetPeriodicDof(i);
// number of the original dof
int j = var.GetOriginalPeriodicDof(i);
// we have an equation u_k = u_j * e^{i phase}
}
### GetProcOriginalPeriodicDof
#### Syntax
int GetProcOriginalPeriodicDof(int i)
This method returns the processor owning the original degree of freedom associated with the i-th periodic dof. For periodic degrees of freedom, the following equation
replaces the variational formulation. This replacement is performed only for a strong formulation of quasi-periodic conditions (UseSameDofsForPeriodicCondition = NO in the data file).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of dofs where quasi-periodic condition is applied
int nb_dof_per = var.GetNbPeriodicDof();
// loop over periodic dofs
for (int i = 0; i < nb_dof_per; i++)
{
// number of the periodic dof
int k = var.GetPeriodicDof(i);
// number of the original dof
int j = var.GetOriginalPeriodicDof(i);
// processor that owns this dof
int proc = var.GetProcOriginalPeriodicDof(i);
// we have an equation u_k = u_j * e^{i phase}
}
### GetFormulationForPeriodicCondition
#### Syntax
int GetFormulationForPeriodicCondition()
This method returns the formulation used to handle quasi-periodic conditions. The method is chosen by inserting a line UseSameDofsForPeriodicCondition = NO in the data file. It can be equal to :
• SAME_PERIODIC_DOFS : the original and periodic dof have the same number. This formulation is perfect for periodic conditions, but quasi-periodic conditions cannot be handled.
• STRONG_PERIODIC : the original and periodic dof have different numbers, an equation uk = uj ei φ is imposed strongly.
• WEAK_PERIODIC : the quasi-periodic condition is enforced weakly (with surface integrals).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of dofs where quasi-periodic condition is applied
int nb_dof_per = var.GetNbPeriodicDof();
// formulation
int form = var.GetFormulationForPeriodicCondition();
// can be equal to MeshNumbering_Base<Real_wp>::STRONG_PERIODIC
### SetModesToCompute
#### Syntax
void SetModesToCompute(IVect num )
This method sets the list of modes that need to be solved for the computation of the solution.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// setting manually the list of modes
IVect num(2); num(0) = 1; num(1) = 3;
var.SetModesToCompute(num);
// GetModeNumber(0) will return 1
int m0 = var.GetModeNumber(0);
### PushBackMode
#### Syntax
void PushBackMode(int n )
This method pushes a mode number at the end of the list of modes that need to be solved for the computation of the solution.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// setting manually the list of modes
IVect num(2); num(0) = 1; num(1) = 3;
var.SetModesToCompute(num);
var.PushBackMode(5);
// GetModeNumber(2) will return 5
int m = var.GetModeNumber(2);
### GetNbPeriodicModesX
#### Syntax
int GetNbPeriodicModesX() const
This method returns the number of cells in x-direction (for periodicity in x).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of periodic modes in x-direction
int nx = var.GetNbPeriodicModesX();
### GetNbPeriodicModesY
#### Syntax
int GetNbPeriodicModesY() const
This method returns the number of cells in y-direction (for periodicity in y).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of periodic modes in y-direction
int ny = var.GetNbPeriodicModesY();
### GetNbPeriodicModesZ
#### Syntax
int GetNbPeriodicModesZ() const
This method returns the number of cells in z-direction (for periodicity in z).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of periodic modes in z-direction
int nz = var.GetNbPeriodicModesZ();
### GetPeriodicNumberModes
#### Syntax
void GetPeriodicNumberModes(int& nx , int& ny , int& nz, bool& teta_sym ) const
This method fills the number of modes in x, y and z-direction. It also sets teta_sym to true if there a cyclic boundary condition.
#### Parameters
nx (out)
number of modes in x-direction
ny (out)
number of modes in y-direction
nz (out)
number of modes in z-direction
teta_sym (out)
true if there is a cyclic computation
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of periodic modes in x, y and z-direction
int nx, ny, nz; bool teta_sym;
var.GetPeriodicNumberModes(nx, ny, nz, teta_sym);
### GetPeriodicModes
#### Syntax
void GetPeriodicModes(int n , int& ix , int& iy , int& iz, bool& teta_sym ) const void GetPeriodicModes(int n , Complex_wp& kx , Complex_wp& ky , Complex& kz) const
This method fills the mode number in x, y and z-direction from the global mode number. It also sets teta_sym to true if there a cyclic boundary condition. In the second syntax, it sets the phase (in x, y and z). The phase in x is equal to
#### Parameters
n (in)
global mode number
ix (out)
mode number in x-direction
iy (out)
mode number in y-direction
iz (out)
mode number in z-direction
teta_sym (out)
true if there is a cyclic computation
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// for a given mode number
int n = 23;
// decomposition in x, y, and z
int ix, iy, iz;
var.GetPeriodicModes(n, ix, iy, iz, teta_sym);
// phase (2 pi / ix)
Complex_wp kx, ky, kz;
var.GetPeriodicModes(n, kx, ky, kz);
### SetPeriodicCondition
#### Syntax
void SetPeriodicCondition(Matrix& A )
This method applies periodic (or quasi-periodic) conditions to a given matrix. This treatment is relevant only for a strong formulation of periodic boundary conditions, where equations
replaces the variational formulation for periodic degrees of freedom. In regular use, this method does not need to be called, since periodic boundary conditions are applied when AddMatrixWithBC is called.
### ApplyPeriodicCondition
#### Syntax
void ApplyPeriodicCondition(Vector& x )
This method applies periodic (or quasi-periodic) conditions to the right hand side. This treatment is relevant only for a strong formulation of periodic boundary conditions, where equations
replaces the variational formulation for periodic degrees of freedom. In regular use, this method does not need to be called, since periodic boundary conditions are applied when ComputeSolution is called.
### GetOrderAbsorbingCondition
#### Syntax
int GetOrderAbsorbingCondition() const
This method returns the order of approximation for absorbing boundary conditions. For Helmholtz equations, it can be equal to 1 or 2. For other equations, only first-order absorbing boundary conditions are implemented.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// order for ABCs ?
int r_abc = var.GetOrderAbsorbingCondition();
### GetNbEltPML
#### Syntax
int GetNbEltPML() const
This method returns the number of elements inside the PMLs.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of elements in PMLs (for current processor)
int n_pml = var.GetNbEltPML();
### GetNbGlobalEltPML
#### Syntax
int GetNbGlobalEltPML() const
This method returns the global number of elements inside the PMLs. For a parallel computation, the method returns the sum of the number of elements (for all processors).
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// number of elements in PMLs (for all processors)
int n_pml = var.GetNbGlobalEltPML();
### MltMuIntegrationByParts
#### Syntax
void MltMuIntegrationByParts(int ref, int ne , int num_loc , int k, Real_wp& coef) const
This method multiplies the coefficient by mu, which is a coefficient that appears in the boundary term when an integration by parts is performed. For instance, for Helmholtz equation, we have the term
#### Parameters
ref (in)
reference of the physical domain
ne (in)
element number
num_loc (in)
local face number
k (in)
mu (inout)
coefficient that will be multiplied by mu
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// for a given element
int ne = 14;
int ref = var.mesh.Element(ne).GetReference();
// local face of the element
int num_loc = 1, k = 14;
Real_wp coef(1);
// we want to compute coef = coef * mu
var.MltMuIntegrationByParts(ref, ne, num_loc, k, coef);
### GetMaximumVelocityPML
#### Syntax
Real_wp GetMaximumVelocityPML() const
This method returns the maximum velocity in PMLs.
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// maximal velocity in PMLs ?
Real_wp vmax = var.GetMaximumVelocityPML();
### SetPhysicalIndexAtInfinity
#### Syntax
void SetPhysicalIndexAtInfinity(const VectBool& RefUsed ) const
This method computes the physical index (rho and mu for Helmholtz equation) at infinity. This method is already called by ComputeMeshAndFiniteElement and does not need to be called in regular use.
### FindElementsInsidePML
#### Syntax
void FindElementsInsidePML()
This method finds all the elements inside PMLs and marks them as PML elements. This method is already called by ComputeMeshAndFiniteElement and does not need to be called in regular use.
### EvaluateFunctionTauPML
#### Syntax
void EvaluateFunctionTauPML(Real_wp dx , Real_wp sig , Real_wp a , Real_wp& zeta , Real_wp& zeta_p )
This method evaluates the damping function of the PML (usually a parabole).
#### Parameters
dx (in)
distance to the interface between the physical domain and PMLs
sig (in)
coefficient (damping is multiplied by this coefficient)
a (in)
thickness of the PML
zeta (out)
damping function
zeta_p (out)
primitive of zeta
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// damping function for a given distance ?
Real_wp zeta, zeta_p, thickness = 1.0;
var.EvaluateFunctionTauPML(dx, 1.0, thickness, zeta, zeta_p);
### GetDampingFactorPML
#### Syntax
void GetDampingFactorPML(Mesh& mesh , int num_pml , int type_pml , R_N point , R_N& zeta , R_N& x_tilde )
This method evaluates the point after the complex variable change :
that appears in PML layers. It computes also, the coefficient
that appears for derivatives with respect to x (y or z). zeta is a vector because it contains this factor for each coordinate x, y (and z in 3-D).
#### Parameters
mesh (in)
input mesh
num_pml (in)
PML number
type_pml (in)
type of PML
point (in)
point where zeta is computed
zeta (out)
damping factor
x_tilde (out)
point after complex change variable
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// damping factor on a given point
int ne = 12;
int num_pml = var.Element(ne).GetNumberPML();
int type_pml = var.Element(ne).GetTypePML();
R2 point(0.8, 2.4);
R2_Complex_wp zeta, x_tilde;
var.GetDampingFactorPML(var.mesh, num_pml, type_pml, point, zeta, x_tilde);
### GetDampingTauPML
#### Syntax
void GetDampingTauPML(Mesh& mesh , int num_pml , int type_pml , R_N point , R_N& tau , R_N& tau_p )
This method evaluates the damping coefficient of the PML (in the three coordinates) and its primitive.
#### Parameters
mesh (in)
input mesh
num_pml (in)
PML number
type_pml (in)
type of PML
point (in)
point where zeta is computed
tau (out)
damping coefficient
tau_p (out)
primitive of damping coefficient
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// damping factor on a given point
int ne = 12;
int num_pml = var.Element(ne).GetNumberPML();
int type_pml = var.Element(ne).GetTypePML();
R2 point(0.8, 2.4);
R2_Complex_wp tau, tau_p;
var.GetDampingTauPML(var.mesh, num_pml, type_pml, point, tau, tau_p);
#### Syntax
void AddMatrixImpedanceBoundary( Real_wp alpha, IVect ref_cond , int ref_target , GlobalGenericMatrix nat_mat , Matrix& mat_sp , int offset_row , int offset_col, ImpedanceFunction_Base& impedance, bool change_cols, bool change_rows, VarProblem& var)
This method adds a boundary integral :
to a sparse matrix. The local operators T and S are defined through the object impedance. The surface Γ over which the integral is computed consists of the facs such that the reference ref satisfy ref_cond(ref) = ref_target.
#### Parameters
alpha (in)
coefficient
ref_cond (in)
Surface gamma is selected such that ref_cond(ref) = ref_target
ref_target (in)
reference target
nat_mat (in)
mass, damping and stiffness coefficients
mat_sp (inout)
sparse matrix that will be modified
offset_row (in)
offset for row numbers
offset_col (in)
offset for column numbers
impedance (in)
class defining impedance operators T and S
change_cols (in)
column numbers are modified with NewColumnNumbers_Impedance
change_rows (in)
rows numbers are modified with NewRowNumbers_Impedance
var(in)
instance of VarProblem
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// definition of impedance coefficients
ImpedanceFunction_Base<Real_wp, Dimension2> impedance;
impedance.SetCoefficient(Real_wp(2), Real_wp(-1)); // case of constant scalar impedance
// i.e. T(phi, \nabla phi) = coef phi, S(phi, \nabla phi) = coef \nabla phi P
// P is the tangential projector to take into account only surface gradient
// initializing a sparse matrix
DistributedMatrix<Real_wp, Symmetric, ArrayRowSymSparse> mat_sp;
int N = var.GetNbDof(); mat_sp.Reallocate(N, N);
// selecting the surfaces of integration
IVect ref_cond(var.mesh.GetNbReferences()+1); ref_cond.Zero();
int ref_target = 1; ref_cond(2) = ref_target;
// calling the method to add the integral terms to the matrix
GlobalGenericMatrix<Real_wp> nat_mat;
mat_sp, 0, 0, impedance, false, false, var);
#### Syntax
void AddBoundaryConditionTerms( Matrix& A, GlobalGenericMatrix nat_mat , int offset_row = 0, int offset_col = 0)
This method adds to the matrix the terms due to boundary conditions. On regular use, this method does not need to be called, since it is already called by AddMatrixWithBC.
#### Parameters
A (inout)
sparse matrix that will be modified
nat_mat (in)
mass, damping and stiffness coefficients
offset_row (optional)
offset for row numbers
offset_col (optional)
offset for column numbers
### InitCyclicDomain
#### Syntax
void InitCyclicDomain()
This method initializes the computation of modes for cyclic or periodic domains. In regular use, this method does not need to be called since it is already called by ComputeMeshAndFiniteElement.
### ComputeQuasiPeriodicPhase
#### Syntax
void ComputeQuasiPeriodicPhase
xo
This method computes the phase for quasi-periodic conditions.
If you switch to another mode (for periodic/cyclic domains), this method needs to be called such that the quasi-periodic condition corresponds to the desired mode.
// The definition of the problem is constructed via EllipticProblem class
EllipticProblem<LaplaceEquation<Dimension2> > var;
var.InitIndices(100);
var.SetTypeEquation("LAPLACE"); // name of the equation, it can be used to use an equivalent formulation of the same equation
var.ComputeMeshAndFiniteElement("QUADRANGLE_LOBATTO"); // mesh and finite element are constructed
var.PerformOtherInitializations(); // other initializations
var.ComputeMassMatrix(); // computation of geometric quantities (such as jacobian matrices)
var.ComputeQuasiPeriodicPhase(); // for quasi-periodic conditions
// once var is constructed, you can call AddMatrixWithBC
GlobalGenericMatrix<Real_wp> nat_mat; // this object is used to set coefficients alpha, beta and gamma
// By default, alpha = beta = gamma = 1
Matrix<Real_wp, Symmetric, ArrayRowSymSparse> A;
// but you can change them
Real_wp alpha = 2.0, beta = 0.5, gamma = 0.25;
nat_mat.SetCoefMass(alpha);
nat_mat.SetCoefDamping(beta);
nat_mat.SetCoefStiffness(gamma);
// matrix is added, so you need to clear it if you do not want to keep
// previous non-zero entries
A.Clear();
// for an iterative matrix (the matrix is not necessary stored, use FemMatrixFreeClass)
FemMatrixFreeClass_Base<Real_wp>* Ah = var.GetNewIterativeMatrix(Real_wp(0));
delete Ah;
### AllocateTauPML
#### Syntax
void AllocateTauPML()
This method allocates the arrays that will store damping terms for PMLs. In regular use, this method does not need to be called since it is already called by ComputeMassMatrix.
### GetPeriodicDofNumbers
#### Syntax
void GetPeriodicDofNumbers(int i , int& k , int& j , int n = 0) const
This method retrieves the dofs numbers j, k of the periodic condition
for the i-th periodic dof.
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// loops over periodic dofs
for (int i = 0; i < var.GetNbPeriodicDof(); i++)
{
// integers j and k such that u_k = u_j * phase
int j, k;
var.GetPeriodicDofNumbers(i, k, j);
}
### GetPeriodicPhase
#### Syntax
void GetPeriodicPhase(int i , Complex_wp& phase ) const
This method retrieves the phase for the quasi-periodic condition
for the i-th periodic dof.
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// loops over periodic dofs
for (int i = 0; i < var.GetNbPeriodicDof(); i++)
{
// integers j and k such that u_k = u_j * phase
int j, k;
var.GetPeriodicDofNumbers(i, k, j);
// phase
Complex_wp phase;
var.GetPeriodicPhase(i, phase);
}
### MltParamCondition
#### Syntax
void MltParamCondition(int ref, int k, T& coef) const
This method multiplies a coefficient by the parameter given at a boundary condition.
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// reference of the physical surface
int ref = 2;
// then if you want to retrieve parameter 1 associated with this reference
// for instance if the data file contains ConditionReference = 2 IMPEDANCE 4.0 2.5
// parameter 0 is 4.0 and parameter 1 is 2.5
Real_wp coef(1);
var.MltParamCondition(2, 1, coef);
### GetImpedanceCoefficientABC
#### Syntax
Complexe GetImpedanceCoefficientABC() const
This method returns the coefficient to modify the absorbing boundary condition. This coefficient is usually set by inserting a line ModifiedImpedance = 1.02 in the data file.
EllipticProblem<HelmholtzEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// coefficient to modify impedance for abc
Complex_wp coef = var.GetImpedanceCoefficientABC();
### GetTauPML
#### Syntax
R_N GetTauPML(int num_elem , int k ) const Complexe GetTauPML(int num_elem , int k , int num_coor ) const
This method returns the value of the damping function for a quadrature point. This result is a vector because it contains the damping for each coordinate x, y (and z in 3-D). In the second syntax, we return the damping for a given coordinate.
#### Parameters
num_elem (in)
element number
k (in)
num_coor (in)
coordinate number
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// damping function on a given point
R2 tau; int num_elem = 12, k = 2;
tau = var.GetTauPML(num_elem, k);
### GetPrimitiveTauPML
#### Syntax
Complexe GetPrimitiveTauPML(int num_elem , int k , int num_coor ) const
This method returns the value of the primitive of the damping function for a quadrature point.
#### Parameters
num_elem (in)
element number
k (in)
num_coor (in)
coordinate number
#### Example :
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// damping function on a given point
Real_wp tau_y; int num_elem = 12, k = 2, num_coor = 1;
tau_y = var.GetTauPML(num_elem, k, num_coor);
// primitive as well :
Real_wp tau_yp = var.GetPrimitiveTauPML(num_elem, k, num_coor);
### GetParamCondition
#### Syntax
Complexe GetParamCondition(int ref, int k) const Vector& GetParamCondition() const
This method retrieves a single parameter associated with a reference number. You can also retrieves all parameters for all references.
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// reference of the physical surface
int ref = 2;
// then if you want to retrieve parameter 1 associated with this reference
// for instance if the data file contains ConditionReference = 2 IMPEDANCE 4.0 2.5
// parameter 0 is 4.0 and parameter 1 is 2.5
Real_wp coef = var.GetParamCondition(2, 1);
// if you want to get all parameters
Vector<VectReral_wp> >& all_param = var.GetParamCondition();
### SetBoundaryConditionMesh
#### Syntax
void SetBoundaryConditionMesh(int ref, int type) const
This method sets a boundary condition for a given reference. Usually the boundary conditions are given by inserting a line ConditionReference = 1 DIRICHLET in the data file.
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// if you want to set a boundary condition (not present in the data file)
var.SetBoundaryConditionMesh(1, BoundaryConditionEnum::LINE_NEUMANN);
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
#### Syntax
void AddPeriodicConditionMesh(TinyVector ref, int type) const
This method sets a periodic boundary condition between two references. Usually the boundary conditions are given by inserting a line ConditionReference = 1 2 PERIODICITY in the data file.
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// if you want to set a periodic boundary condition (not present in the data file)
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
### GetNewImpedanceABC
#### Syntax
ImpedanceFunction_Base* GetNewImpedanceABC(Complexe x) const
This method constructs an impedance object for absorbing boundary conditions. The argument given as parameters is used to retrieve an object dealing complex or real numbers.
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// impedance for absorbing boundary condition
ImpedanceFunction_Base<Real_wp, Dimension2>* imped = var.GetNewImpedanceABC();
// once used, delete the object
delete imped;
### GetNewImpedanceGeneric
#### Syntax
ImpedanceFunction_Base* GetNewImpedanceGeneric(Complexe x) const
This method constructs an impedance object for absorbing boundary conditions. The argument given as parameters is used to retrieve an object dealing complex or real numbers.
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// impedance for Robin boundary condition
ImpedanceFunction_Base<Real_wp, Dimension2>* imped = var.GetNewImpedanceGeneric();
// once used, delete the object
delete imped;
### GetNewImpedanceHighConductivity
#### Syntax
ImpedanceFunction_Base* GetNewImpedanceHighConductivity(Complexe x) const
This method constructs an impedance object for high conductivity boundary conditions. The argument given as parameters is used to retrieve an object dealing complex or real numbers.
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// impedance for high conductivity boundary condition
ImpedanceFunction_Base<Real_wp, Dimension2>* imped = var.GetNewImpedanceHighConductivity();
// once used, delete the object
delete imped;
### InvolveOnlyTangentialDofs
#### Syntax
bool InvolveOnlyTangentialDofs() const
This method returns true if the boundary integrals are non-null only for degrees of freedom associated with the surface. Usually, it is the case. It can be false, for instance if the integral involves 3-D gradient of basis functions (and not surface gradients).
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// impedance for Robin boundary condition
ImpedanceFunction_Base<Real_wp, Dimension2>* imped = var.GetNewImpedanceGeneric();
bool only_tgt_dofs = imped->InvolveOnlyTangentialDofs();
// once used, delete the object
delete imped;
#### Syntax
This method returns true if the boundary integrals involve gradient (or curl/div) of basis functions. The aim is to save computational time when no gradients are needed.
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// impedance for Robin boundary condition
ImpedanceFunction_Base<Real_wp, Dimension2>* imped = var.GetNewImpedanceGeneric();
// once used, delete the object
delete imped;
### SetCoefficient
#### Syntax
bool SetCoefficient(Complexe a, Complexe b)
This method sets the coefficients for the impedance. For scalar finite elements, these coefficients appear in the following term
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// impedance for Robin boundary condition
ImpedanceFunction_Base<Real_wp, Dimension2>* imped = var.GetNewImpedanceGeneric();
// setting coefficients a and b in the expression a \int_\Gamma u \varphi dx + b \int_\Gamma \nabla_\Gamma u \cdot \nabla_\Gamma \varphi \, dx
Real_wp a = 2.0, b = 3.0;
imped->SetCoefficient(a, b);
// initializing a sparse matrix
DistributedMatrix<Real_wp, Symmetric, ArrayRowSymSparse> mat_sp;
int N = var.GetNbDof(); mat_sp.Reallocate(N, N);
// selecting the surfaces of integration
IVect ref_cond(var.mesh.GetNbReferences()+1); ref_cond.Zero();
int ref_target = 1; ref_cond(2) = ref_target;
// calling the method to add the integral terms to the matrix
GlobalGenericMatrix<Real_wp> nat_mat;
mat_sp, 0, 0, *imped, false, false, var);
// once used, delete the object
delete imped;
### GetCoefficient
#### Syntax
bool GetCoefficient( int i, int num_elem, int num_loc, int k, int ref_domain, int ref SetPoints pts, SetMatrices mat)
This method returns the impedance coefficient at a given quadrature point.
#### Parameters
i (in)
face number
num_elem (in)
element number
num_loc (in)
local face number
k (in)
ref_domain (in)
reference for the physical domain
ref (in)
reference for the surface
pts (in)
mat (in)
jacobian matrices
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// impedance for Robin boundary condition
ImpedanceFunction_Base<Real_wp, Dimension2>* imped = var.GetNewImpedanceGeneric();
int i = 2; // edge number
int num_elem = var.mesh.BoundaryRef(i).numElement(0);
int num_loc = var.mesh.Element(num_elem).GetPositionBoundary(i);
int ref_domain = var.mesh.Element(num_elem).GetReference();
int ref = var.mesh.BoundaryRef(i).GetReference();
VectR2 s;
mesh.GetVerticesElement(num_elem, s);
const ElementReference_Dim<Dimension2>& Fb = var.GetReferenceElement(num_elem);
SetPoints<Dimension2> pts; SetMatrices<Dimension2> mat;
Fb.FjSurfaceElem(s, pts, mesh, num_elem, num_loc);
Fb.DFjSurfaceElem(s, pts, mat, mesh, num_elem, num_loc);
// evaluation of impedance
int k = 2; // local quadrature point number
GlobalGenericMatrix<Real_wp> nat_mat;
imped->EvaluateImpedancePhi(i, num_elem, i, num_loc, k, nat_mat, ref_domain, pts, mat);
// then if you want to get the coefficient
Real_wp coef = imped->GetCoefficient(i, num_elem, num_loc, k, ref_domain, ref, pts, mat);
// once used, delete the object
delete imped;
### EvaluateImpedancePhi
#### Syntax
void EvaluateImpedancePhi( int i, int num_elem, int num_edge, int num_loc, int k, GlobalGenericMatrix nat_mat, int ref_domain, SetPoints pts, SetMatrices mat)
This method evaluates the impedance for a given quadrature point. In the method AddMatrixImpedanceBoundary, this method is called, such that the user can compute impedance coefficients (or operators). Then, the method ApplyImpedancePhi_H1 is called to apply the impedance against basis functions.
#### Parameters
i (in)
face number
num_elem (in)
element number
num_edge (in)
equal to i
num_loc (in)
local face number
k (in)
nat_mat (in)
mass, damping and stiffness coefficients
ref_domain (in)
reference for the physical domain
pts (in)
mat (in)
jacobian matrices
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// impedance for Robin boundary condition
ImpedanceFunction_Base<Real_wp, Dimension2>* imped = var.GetNewImpedanceGeneric();
int i = 2; // edge number
int num_elem = var.mesh.BoundaryRef(i).numElement(0);
int num_loc = var.mesh.Element(num_elem).GetPositionBoundary(i);
int ref_domain = var.mesh.Element(num_elem).GetReference();
int ref = var.mesh.BoundaryRef(i).GetReference();
VectR2 s;
mesh.GetVerticesElement(num_elem, s);
const ElementReference_Dim<Dimension2>& Fb = var.GetReferenceElement(num_elem);
SetPoints<Dimension2> pts; SetMatrices<Dimension2> mat;
Fb.FjSurfaceElem(s, pts, mesh, num_elem, num_loc);
Fb.DFjSurfaceElem(s, pts, mat, mesh, num_elem, num_loc);
// evaluation of impedance
int k = 2; // local quadrature point number
GlobalGenericMatrix<Real_wp> nat_mat;
imped->EvaluateImpedancePhi(i, num_elem, i, num_loc, k, nat_mat, ref_domain, pts, mat);
// then if you want to get the coefficient
Real_wp coef = imped->GetCoefficient(i, num_elem, num_loc, k, ref_domain, ref, pts, mat);
// once used, delete the object
delete imped;
#### Syntax
void EvaluateImpedanceGrad( int i, int num_elem, int num_edge, int num_loc, int k, GlobalGenericMatrix nat_mat, int ref_domain, SetPoints pts, SetMatrices mat)
This method evaluates the impedance (for gradients) for a given quadrature point. In the method AddMatrixImpedanceBoundary, this method is called, such that the user can compute impedance coefficients (or operators). Then, the method ApplyImpedanceGrad is called to apply the impedance against gradient of basis functions.
#### Parameters
i (in)
face number
num_elem (in)
element number
num_edge (in)
equal to i
num_loc (in)
local face number
k (in)
nat_mat (in)
mass, damping and stiffness coefficients
ref_domain (in)
reference for the physical domain
pts (in)
mat (in)
jacobian matrices
EllipticProblem<LaplaceEquation<Dimension2> > var;;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// impedance for Robin boundary condition
ImpedanceFunction_Base<Real_wp, Dimension2>* imped = var.GetNewImpedanceGeneric();
int i = 2; // edge number
int num_elem = var.mesh.BoundaryRef(i).numElement(0);
int num_loc = var.mesh.Element(num_elem).GetPositionBoundary(i);
int ref_domain = var.mesh.Element(num_elem).GetReference();
int ref = var.mesh.BoundaryRef(i).GetReference();
VectR2 s;
mesh.GetVerticesElement(num_elem, s);
const ElementReference_Dim<Dimension2>& Fb = var.GetReferenceElement(num_elem);
SetPoints<Dimension2> pts; SetMatrices<Dimension2> mat;
Fb.FjSurfaceElem(s, pts, mesh, num_elem, num_loc);
Fb.DFjSurfaceElem(s, pts, mat, mesh, num_elem, num_loc);
// evaluation of impedance
int k = 2; // local quadrature point number
GlobalGenericMatrix<Real_wp> nat_mat;
imped->EvaluateImpedancePhi(i, num_elem, i, num_loc, k, nat_mat, ref_domain, pts, mat);
// then if you want to get the coefficient
Real_wp coef = imped->GetCoefficient(i, num_elem, num_loc, k, ref_domain, ref, pts, mat);
imped->EvaluateImpedanceGrad(i, num_elem, i, num_loc, k, nat_mat, ref_domain, pts, mat);
// once used, delete the object
delete imped;
### ApplyImpedancePhi_H1
#### Syntax
void ApplyImpedancePhi_H1( int m, int k, int offset, TinyVector phi, R2 grad_phi, Vector f_phi)
This method applies the impedance for a given quadrature point. In the method AddMatrixImpedanceBoundary, this method is called for the computation of the boundary integral.
where T is the impedance operator, i the row index and j the column index. This method can be overloaded to define your own impedance operator. This method involves only scalar basis functions (unknown number m is approximated with H1 Sobolev space).
#### Parameters
m (in)
unknown number (for rows)
j (in)
offset (in)
offset for the unknown m
phi (in)
value of basis function (row)
f_phi (in)
// for your own impedance operator
class MyOperator : public ImpedanceFunction_Base<Real_wp, Dimension2>
{
void ApplyImpedancePhi_H1(int m, int j, int offset, const TinyVector<Real_wp, 1>& phi,
{
// for a diagonal operator :
Real_wp coef = 4.5; // impedance coefficient
f_phi(offset) = coef*phi(0);
}
};
#### Syntax
void ApplyImpedanceGrad( int m, int k, int offset, TinyVector phi, R2 grad_phi, Vector f_phi)
This method applies the impedance for a given quadrature point. In the method AddMatrixImpedanceBoundary, this method is called for the computation of the boundary integral.
where T is the impedance operator, i the row index and j the column index. This method can be overloaded to define your own impedance operator. This method involves only scalar basis functions (unknown number m is approximated with H1 Sobolev space).
#### Parameters
m (in)
unknown number (for rows)
j (in)
offset (in)
offset for the unknown m
phi (in)
value of basis function (row)
f_phi (in)
// for your own impedance operator
class MyOperator : public ImpedanceFunction_Base<Real_wp, Dimension2>
{
void ApplyImpedancePhi_H1(int m, int j, int offset, const TinyVector<Real_wp, 1>& phi,
{
// for a diagonal operator :
f_phi.Zero();
Real_wp coef = 4.5; // impedance coefficient
f_phi(offset) = coef*phi(0);
}
void ApplyImpedanceGrad(int m, int j, int offset, const TinyVector<Real_wp, 1>& phi,
{
// if no gradient term, you can fill with zeros
f_phi.Zero();
}
};
### ApplyImpedancePhi_Hcurl
#### Syntax
void ApplyImpedancePhi_Hcurl( int m, int k, int offset, TinyVector phi, TinyVector curl_phi, Vector f_phi)
This method applies the impedance for a given quadrature point. In the method AddMatrixImpedanceBoundary, this method is called for the computation of the boundary integral.
where T is the impedance operator, i the row index and j the column index. This method can be overloaded to define your own impedance operator. This method involves only vectorial basis functions (unknown number m is approximated with H(curl) Sobolev space).
#### Parameters
m (in)
unknown number (for rows)
j (in)
offset (in)
offset for the unknown m
phi (in)
value of basis function (row)
curl_phi (in)
curl of basis function (row)
f_phi (in)
// for your own impedance operator
class MyOperator : public ImpedanceFunction_Base<Real_wp, Dimension2>
{
void ApplyImpedancePhi_Hcurl(int m, int j, int offset, const TinyVector<Real_wp, 2>& phi,
const TinyVector<Real_wp, 1>& curl_phi, Vector<T>& f_phi)
{
// for a diagonal operator :
Real_wp coef = 4.5; // impedance coefficient
f_phi(offset) = coef*phi(0);
f_phi(offset+1) = coef*phi(1);
}
};
### ApplyImpedanceCurl
#### Syntax
void ApplyImpedanceCurl( int m, int k, int offset, TinyVector phi, TinyVector curl_phi, Vector f_phi)
This method applies the impedance for a given quadrature point. In the method AddMatrixImpedanceBoundary, this method is called for the computation of the boundary integral.
where T is the impedance operator, i the row index and j the column index. This method can be overloaded to define your own impedance operator. This method involves only vectorial basis functions (unknown number m is approximated with H(curl) Sobolev space).
#### Parameters
m (in)
unknown number (for rows)
j (in)
offset (in)
offset for the unknown m
phi (in)
value of basis function (row)
curl_phi (in)
curl of basis function (row)
f_phi (in)
// for your own impedance operator
class MyOperator : public ImpedanceFunction_Base<Real_wp, Dimension2>
{
void ApplyImpedancePhi_Hcurl(int m, int j, int offset, const TinyVector<Real_wp, 2>& phi,
const TinyVector<Real_wp, 1>& curl_phi, Vector<T>& f_phi)
{
// for a diagonal operator :
f_phi.Zero();
Real_wp coef = 4.5; // impedance coefficient
f_phi(offset) = coef*phi(0);
f_phi(offset+1) = coef*phi(1);
}
void ApplyImpedanceCurl(int m, int j, int offset, const TinyVector<Real_wp, 2>& phi,
const TinyVector<Real_wp, 1>& curl_phi, Vector<T>& f_phi)
{
// if no gradient term, you can fill with zeros
f_phi.Zero();
}
};
### ApplyImpedancePhi_Hdiv
#### Syntax
void ApplyImpedancePhi_Hdiv( int m, int k, int offset, TinyVector phi, TinyVector div_phi, Vector f_phi)
This method applies the impedance for a given quadrature point. In the method AddMatrixImpedanceBoundary, this method is called for the computation of the boundary integral.
where T is the impedance operator, i the row index and j the column index. This method can be overloaded to define your own impedance operator. This method involves only vectorial basis functions (unknown number m is approximated with H(div) Sobolev space).
#### Parameters
m (in)
unknown number (for rows)
j (in)
offset (in)
offset for the unknown m
phi (in)
value of basis function (row)
div_phi (in)
divergence of basis function (row)
f_phi (in)
// for your own impedance operator
class MyOperator : public ImpedanceFunction_Base<Real_wp, Dimension2>
{
void ApplyImpedancePhi_Hdiv(int m, int j, int offset, const TinyVector<Real_wp, 2>& phi,
const TinyVector<Real_wp, 1>& div_phi, Vector<T>& f_phi)
{
// for a diagonal operator :
Real_wp coef = 4.5; // impedance coefficient
f_phi(offset) = coef*phi(0);
f_phi(offset+1) = coef*phi(1);
}
};
### ApplyImpedanceDiv
#### Syntax
void ApplyImpedanceCurl( int m, int k, int offset, TinyVector phi, TinyVector div_phi, Vector f_phi)
This method applies the impedance for a given quadrature point. In the method AddMatrixImpedanceBoundary, this method is called for the computation of the boundary integral.
where T is the impedance operator, i the row index and j the column index. This method can be overloaded to define your own impedance operator. This method involves only vectorial basis functions (unknown number m is approximated with H(div) Sobolev space).
#### Parameters
m (in)
unknown number (for rows)
j (in)
offset (in)
offset for the unknown m
phi (in)
value of basis function (row)
div_phi (in)
divergence of basis function (row)
f_phi (in)
// for your own impedance operator
class MyOperator : public ImpedanceFunction_Base<Real_wp, Dimension2>
{
void ApplyImpedancePhi_Hdiv(int m, int j, int offset, const TinyVector<Real_wp, 2>& phi,
const TinyVector<Real_wp, 1>& div_phi, Vector<T>& f_phi)
{
// for a diagonal operator :
f_phi.Zero();
Real_wp coef = 4.5; // impedance coefficient
f_phi(offset) = coef*phi(0);
f_phi(offset+1) = coef*phi(1);
}
void ApplyImpedanceDiv(int m, int j, int offset, const TinyVector<Real_wp, 2>& phi,
const TinyVector<Real_wp, 1>& div_phi, Vector<T>& f_phi)
{
// if no gradient term, you can fill with zeros
f_phi.Zero();
}
};
### Constructor for TransparencySolver
#### Syntax
TransparencySolver( EllipticProblem& var, All_LinearSolver& solver)
The constructor takes the considered problem to solve as argument. If you do not have an EllipticProblem instance, but rather a VarProblem_Base instance (if you are writing in a generic function), you can call GetNewTransparentSolver that will construct an object TransparencySolver and return a pointer of type TransparencySolver_Base.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
GlobalGenericMatrix<Complex_wp> nat_mat; // this object is used to store mass, damping and stiffness coefficients
// to factorize the matrix (or prepare the computation if an iterative solver is selected)
solver->PerformFactorizationStep(nat_mat);
// right hand side
VectComplex_wp b(var.GetNbDof());
var.ComputeRightHandSide(b);
// and solve the linear system A x = b (with only first-order ABC)
VectComplex_wp x = b; solver->ComputeSolution(x);
// then you construct your transparent solver
TransparencySolver<HelmholtzEquation<Dimension2> > solver_transp(var, *solver);
// and you can iterate on the solution to obtain the exact solution
if (solver_transp.UseTransparentCondition())
{
solver_transp.Init();
VectComplex_wp source_rhs(x);
solver_transp.Solve(x, source_rhs);
}
}
### UseTransparentCondition
#### Syntax
bool UseTransparentCondition() const
This method returns true if a transparent condition has been set.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
GlobalGenericMatrix<Complex_wp> nat_mat; // this object is used to store mass, damping and stiffness coefficients
// to factorize the matrix (or prepare the computation if an iterative solver is selected)
solver->PerformFactorizationStep(nat_mat);
// right hand side
VectComplex_wp b(var.GetNbDof());
var.ComputeRightHandSide(b);
// and solve the linear system A x = b (with only first-order ABC)
VectComplex_wp x = b; solver->ComputeSolution(x);
// then you construct your transparent solver
TransparencySolver<HelmholtzEquation<Dimension2> > solver_transp(var, *solver);
// and you can iterate on the solution to obtain the exact solution
if (solver_transp.UseTransparentCondition())
{
solver_transp.Init();
VectComplex_wp source_rhs(x);
solver_transp.Solve(x, source_rhs);
}
}
### Solve
#### Syntax
void Solve(Vector& x_sol, Vector& b_src) const
This method fills the vector x_sol to contain the solution with a transparent condition. The input vector b_src contains the solution with a first-order absorbing boundary condition.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
GlobalGenericMatrix<Complex_wp> nat_mat; // this object is used to store mass, damping and stiffness coefficients
// to factorize the matrix (or prepare the computation if an iterative solver is selected)
solver->PerformFactorizationStep(nat_mat);
// right hand side
VectComplex_wp b(var.GetNbDof());
var.ComputeRightHandSide(b);
// and solve the linear system A x = b (with only first-order ABC)
VectComplex_wp x = b; solver->ComputeSolution(x);
// then you construct your transparent solver
TransparencySolver<HelmholtzEquation<Dimension2> > solver_transp(var, *solver);
// and you can iterate on the solution to obtain the exact solution
if (solver_transp.UseTransparentCondition())
{
solver_transp.Init();
VectComplex_wp source_rhs(x);
solver_transp.Solve(x, source_rhs);
}
}
### Init
#### Syntax
void Init()
This method initializes the computation of the transparent condition. It will compute quadrature points and weights for the two surfaces (absorbing surface and intermediary surface).
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
GlobalGenericMatrix<Complex_wp> nat_mat; // this object is used to store mass, damping and stiffness coefficients
// to factorize the matrix (or prepare the computation if an iterative solver is selected)
solver->PerformFactorizationStep(nat_mat);
// right hand side
VectComplex_wp b(var.GetNbDof());
var.ComputeRightHandSide(b);
// and solve the linear system A x = b (with only first-order ABC)
VectComplex_wp x = b; solver->ComputeSolution(x);
// then you construct your transparent solver
TransparencySolver<HelmholtzEquation<Dimension2> > solver_transp(var, *solver);
// and you can iterate on the solution to obtain the exact solution
if (solver_transp.UseTransparentCondition())
{
solver_transp.Init();
VectComplex_wp source_rhs(x);
solver_transp.Solve(x, source_rhs);
}
}
### ComputeSolution
#### Syntax
void ComputeSolution(Vector& rhs, Vector& sol)
This method computes the solution of the sparse finite element matrix As sol = rhs.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
GlobalGenericMatrix<Complex_wp> nat_mat; // this object is used to store mass, damping and stiffness coefficients
// to factorize the matrix (or prepare the computation if an iterative solver is selected)
solver->PerformFactorizationStep(nat_mat);
// right hand side
VectComplex_wp b(var.GetNbDof());
var.ComputeRightHandSide(b);
// then you construct your transparent solver
TransparencySolver<HelmholtzEquation<Dimension2> > solver_transp(var, *solver);
// we solve the linear system A_s x = b (with only first-order ABC)
VectComplex_wp x = b; solver_transp.ComputeSolution(b, x);
// and you can iterate on the solution to obtain the exact solution
if (solver_transp.UseTransparentCondition())
{
solver_transp.Init();
VectComplex_wp source_rhs(x);
solver_transp.Solve(x, source_rhs);
}
}
### ComputeGreenKernel
#### Syntax
void ComputeGreenKernel(const R3& x, const R3& y, const Real_wpp& k, Complex_wp& phi, R3_Complex_wp& grad_phi, Matrix3_3sym_Complex_wp& hessian_phi) void ComputeGreenKernel(const R3& x, const R3& y, const Real_wpp& k, Complex_wp& phi, R3_Complex_wp& grad_phi)
This static method computes the Green kernel (for wave equation), its gradient (with respect to y) and hessian matrix. You can also compute only the kernel and its gradient. This Green's kernel is equal to
#### Parameters
x (in)
point
y (in)
point
k (in)
wave number
phi (out)
Green's kernel phi(x, y)
hessian_phi (out, optional)
hessian of Green's kernel
int main()
{
// you choose two points
R3 x(0.1, 0.5, 0.3), y(0.2, 2.0, 0.8);
// a wave number
Real_wp k = 0.3;
// Green's kernel and gradient for these values
Complex_wp phi;
TransparencySolver_Base::ComputeGreenKernel(x, y, k, phi, grad_phi, hessian_phi);
}
### ComputeSurfaceGammaAndAbsorbing
#### Syntax
void ComputeSurfaceGammaAndAbsorbing(int ref_abc, int ref_gamma, IVect& offset_abc_proc)
This internal method computes the quadrature points for the intermediary surface (denoted Gamma) and for the absorbing surface (denoted sigma). ref_abc is the boundary condition associated with the absorbing surface (usually equal to BoundaryConditionEnum::LINE_ABSORBING. ref_gamma is the body number for the intermediary surface (all references such that the body number is equal to ref_gamma will belong to this intermediary surface). The output argument offset_abc_proc stores the offsets for quadrature points of the absorbing surface. On regular use, this methods does not need to be called, since the method Init will call it automatically.
### ComputeRightHandSide
#### Syntax
void ComputeRightHandSide(const VectComplex_wp& x_sol, VectComplex_wp& g_source)
This internal method computes the matrix-vector product g_source = Ad x_sol where Ad is introduced in the section devoted to transparent conditions. On regular use, this method does not need to be called, since the method Solve will call it automatically.
### ComputeIntegralRepresentation
#### Syntax
void ComputeIntegralRepresentation( const VectComplex_wp& trace_En, const VectComplex_wp& trace_Hn, const MeshInterpolation& mesh_gamma, const R_N& x, const R_N& n, TinyVector& En, TinyVector& Hn)
This method computes the value of E x n and H x n on a given point from values on the quadrature points of the intermediary surface Gamma (by using an integral representation). On regular use, this method does not need to be called, since the method Solve will call it automatically. This method is virtual such that it is overloaded for each equation where it is implemented.
#### Parameters
trace_En (in)
values of E x n on quadrature points of Gamma
trace_Hn (in)
values of H x n on quadrature points of Gamma
mesh_gamma (in)
surface mesh Gamma
x (in)
point where E x n will be evaluated
n (in)
outgoing normale associated with the point x
En (out)
evaluation of E x n at x (with the given normale)
Hn (out)
evaluation of H x n at x (with the given normale)
### ComputeAndStoreEnPot
#### Syntax
void ComputeAndStoreEnPot(const VectComplex_wp& trace_En, const VectComplex_wp& trace_Hn, const R_N& x, const R_N& n, Vector& En, Vector& Hn, int k)
This method computes the value of E x n and H x n on a given point from values on the quadrature points of the intermediary surface Gamma (by using an integral representation). On regular use, this method does not need to be called, since the method Solve will call it automatically. This method is virtual such that it is overloaded for each equation where it is implemented. This method actually calls ComputeIntegralRepresentation and inserts the results on the global vectors En and Hn .
#### Parameters
trace_En (in)
values of E x n on quadrature points of Gamma
trace_Hn (in)
values of H x n on quadrature points of Gamma
x (in)
point where E x n will be evaluated
n (in)
outgoing normale associated with the point x
EnStore (inout)
evaluation of E x n for all points of Sigma
HnStore (out)
evaluation of H x n for all points of Sigma
k (out)
number of the point x (it corresponds to the position where E x n and H x n will be inserted in EnStore and HnStore)
### GetSource
#### Syntax
void GetSource( const VectComplex_wp& trace_En, const VectComplex_wp& trace_Hn, int n, Real_wp k, const R_N& ptX, const R_N& normaleX, Vector& g_source, int k_loc)
This method computes the source term g, that appears in the matrix Ad. This matrix will be equal to
On regular use, this method does not need to be called, since the method Solve will call it automatically. This method is virtual such that it is overloaded for each equation where it is implemented.
#### Parameters
trace_En (in)
values of E x n on quadrature points of Sigma
trace_Hn (in)
values of H x n on quadrature points of Sigma
n (in)
global point number where g needs to be evaluated
k (in)
wave number at infinity
ptX (in)
point where g needs to be evaluated
normaleX (out)
outgoing normale associated with ptX
g_source (inout)
source term that will be modified for the considered quadrature point
j (in)
### RcsToBeComputed
#### Syntax
bool RcsToBeComputed()
This method will return true if the computation of a radar cross section is asked by the user.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
if (var_rcs.RcsToBeComputed())
cout << "RCS has to be computed" << endl;
}
### GetNbAngles
#### Syntax
int GetNbAngles()
This method returns the number of angles required for the radar cross section. This number is usually set by inserting a line AngleRCS = 0.0 180.0 181 in the data file.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
if (var_rcs.RcsToBeComputed())
cout << "Number of angles for the RCS = " << var_rcs.GetNbAngles() << endl;
}
### GetNbPointsOutside
#### Syntax
int GetNbPointsOutside()
This method returns the number of points outside the computational domain for which the user wants to evaluate the solution. For these points, the field will be computed with an integral representation. This number is usually set by inserting a line SismoOutsidePoints = pts.txt diffrac.dat total.dat 0.1 in the data file.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
if (var_rcs.RcsToBeComputed())
cout << "Number of points outside = " << var_rcs.GetNbPointsOutside() << endl;
}
### GetRcsType
#### Syntax
int GetRcsType()
This method returns the type of radar cross section to be computed. This type is usually set by inserting a line ParametersRCS = YES 1 AUTO BISTATIC in the data file. The integer can be equal to VarComputationRCS_Base<Dimension>::MONOSTATIC_RCS or VarComputationRCS_Base<Dimension>::BISTATIC_RCS.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
if (var_rcs.RcsToBeComputed())
if (var_rcs.GetRcsType() == var_rcs.MONOSTATIC_RCS)
cout << "computation of a monostatic radar cross section" << endl;
}
### GetOutsidePoint
#### Syntax
const VectR_N& GetOutsidePoint() const R_N& GetOutsidePoint(int i)
This method returns the points outside the computational domain for which the user wants to evaluate the solution. For these points, the field will be computed with an integral representation. These points are usually set by inserting a line SismoOutsidePoints = pts.txt diffrac.dat total.dat 0.1 in the data file.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
// to retrieve all the points outside the computational domain
const VectR2& pts_outside = var_rcs.GetOutsidePoint();
// to retrieve a single point
int num_pt = 2;
const R2& pt = var_rcs.GetOutsidePoint(num_pt);
}
### SetOutsidePoints
#### Syntax
void SetOutsidePoints(const VectR_N& pts)
This method sets the points outside the computational domain for which the user wants to evaluate the solution. For these points, the field will be computed with an integral representation. These points are usually set by inserting a line SismoOutsidePoints = pts.txt diffrac.dat total.dat 0.1 in the data file.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
// if you want to change the list of outside points
VectR2 pts(2);
pts(0).Init(5.0, 2.0);
pts(1).Init(8.0, 1.0);
var_rcs.SetOutsidePoints(pts);
}
### GetInterpolationMesh
#### Syntax
MeshInterpolationFEM& GetInterpolationMesh()
This method returns the mesh of the surface used to compute the radar cross section. This mesh contains quadrature points and normales.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
var_rcs.InitComputationRCS(); // actually useless, since already called above in ConstructAll
const MeshInterpolationFEM<Dimension2>& surf_mesh = var_rcs.GetInterpolationMesh();
}
### SetTimeStep
#### Syntax
void SetTimeStep(Real_wp dt)
This method sets the time step used for unsteady simulations.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
var_rcs.SetTimeStep(0.1);
var_rcs.InitComputationRCS();
}
### InitComputationRCS
#### Syntax
void InitComputationRCS(bool assemble = false)
This method initializes the computation of the radar cross section. It mainly computes the quadrature points and normales for the surface used to compute the integrals.
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
var_rcs.InitComputationRCS(); // actually useless, since already called above in ConstructAll
}
### ComputeRCS
#### Syntax
void ComputeRCS(VectComplex_wp x_sol)
This method computes the radar cross section for the solution given as argument. The results are written in the file given in the datafile (FileRCS = Rcs.dat).
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
var_rcs.InitComputationRCS(); // actually useless, since already called above in ConstructAll
GlobalGenericMatrix<Complex_wp> nat_mat; // this object is used to store mass, damping and stiffness coefficients
// to factorize the matrix (or prepare the computation if an iterative solver is selected)
solver->PerformFactorizationStep(nat_mat);
// right hand side
VectComplex_wp b(var.GetNbDof());
var.ComputeRightHandSide(b);
// and solve the linear system A x = b (with only first-order ABC)
VectComplex_wp x = b; solver->ComputeSolution(x);
// computation of the radar cross section for this solution
var_rcs.ComputeRCS(x);
}
### ComputeIntegralRepresentation
#### Syntax
void ComputeIntegralRepresentation( const VectComplex_wp& trace_En, const VectComplex_wp& trace_Hn, const MeshInterpolationFEM& mesh, R_N ptX, VectComplex_wp& u)
This method computes the solution on a point outside the computational domain with an integral representation.
#### Parameters
trace_En (in)
values of E x n on quadrature points on the surface of integration
trace_Hn (in)
values of H x n on quadrature points on the surface of integration
mesh (in)
mesh of the surface
ptX (in)
point where the solution needs to be evaluated
u (out)
result
int main()
{
EllipticProblem<HelmholtzEquation<Dimension2> > var;
// constructing the problem (mesh, finite element)
All_LinearSolver* solver;
// an object VarComputationRCS is contained in var :
VarComputationRCS<HelmholtzEquation<Dimension2> >& var_rcs = var.output_rcs_param;
var_rcs.InitComputationRCS(); // actually useless, since already called above in ConstructAll
GlobalGenericMatrix<Complex_wp> nat_mat; // this object is used to store mass, damping and stiffness coefficients
// to factorize the matrix (or prepare the computation if an iterative solver is selected)
solver->PerformFactorizationStep(nat_mat);
// right hand side
VectComplex_wp b(var.GetNbDof());
var.ComputeRightHandSide(b);
// and solve the linear system A x = b (with only first-order ABC)
VectComplex_wp x = b; solver->ComputeSolution(x);
// computation of E x n and H x n on the surface of integration
VectComplex_wp traceEn, traceHn;
var.ComputeEnHnOnBoundary(var_rcs.GetInterpolationMesh(), x, traceEn, traceHn, false);
// if you want to know the solution on a point outside the computational domain
R2 ptX(3.0, 0.8); VectComplex_wp u(1);
var_rcs.ComputeIntegralRepresentation(traceEn, traceHn, var_rcs.GetInterpolationMesh(), ptX, u);
}
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# Unique continuation of the Hilbert transform
Let's consider the usual Hilbert transform $$H$$ defined as $$Hf = P.V. (\frac{1}{x}*f).$$ A well-known unique continuation principle states that if $$Hf = f =0$$ on some interval $$I$$, then $$f \equiv 0$$. My question is whether the argument is still true if we replace the interval $$I$$ with a point $$x_0$$. More specifically, can we prove that if both the function $$f$$ and its Hilbert transform $$Hf$$ have a zero-point $$x_0$$ of infinite order, that is, $$f^{(m)}(x_0) = Hf^{(m)}(x_0) = 0$$ for any non-negative integer $$m$$, then $$f\equiv 0$$? We can assume that $$f$$ is smooth to make the statement more rigorous.
No. Let $$u(z) = \exp(-(-iz)^{1/2}-(-iz)^{-1/2})$$ for $$z$$ in the closed upper complex half-plane, with the principal branch of the complex power. Then $$u$$ is a bounded holomorphic function in the open half-plane, continuous up to the boundary, and vanishing sufficiently fast at complex infinity. Thus, the Hilbert transform of $$f(x) = \Re u(x) = \Re \exp(-e^{-i \pi/4 \operatorname{sign} x} |2x|^{1/2} - e^{i \pi/4 \operatorname{sign} x} |2x|^{-1/2})$$ is given by $$Hf(x) = \Im u(x) = \Im \exp(-e^{-i \pi/4 \operatorname{sign} x} |2x|^{1/2} - e^{i \pi/4 \operatorname{sign} x} |2x|^{-1/2}).$$ A standard argument shows that both $$f$$ and $$Hf$$ are smooth and have a zero of infinite order at $$x = 0$$.
• Many thanks for your answer! I understand that now $f$ and $Hf$ vanish of infinite order at zero. But I am not very familiar with complex analysis, may I ask you why the Hilbert transform of $f$ is given by the imaginary part of $u(x)$? Is there a reference for the general theory of it? Thanks a lot! Oct 7, 2020 at 17:05
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Time of flight with air resistance
1. Apr 1, 2007
Caspian
I need to compute the time of flight of a projectile which is subject to air resistance.
Here's where I am so far in solving the problem:
$$F_{drag} = -B v^2$$
$$a = \frac{\partial v}{\partial t} = \frac{-B v^2}{m}$$
integrate and solve for v...
$$v = \sqrt[3]{\dfrac{1}{3 \dfrac{B}{m} t + C}}$$
I then plug this into the kinematics equation for position:
$$y = \frac{1}{2} a t^2 + v_0 t$$ ($$y_0$$ is taken to be 0)
$$y = \frac{1}{2} (g - \frac{F_{drag}}{m}) t^2 + v_0 t$$
Substitute the equation for v into $$F_{drag}$$, which is substituted into the above equation. Next, I set y = 0 and try to solve for t... but the equation is too messy and I can't manage to get just the t's onto one side of the equation.
Am I on the right track? Is there a better way?
Thanks!
Last edited: Apr 1, 2007
2. Apr 1, 2007
cristo
Staff Emeritus
I've not check the rest of it, but if you want to solve $$y = \frac{1}{2} (g - \frac{F_{drag}}{m}) t^2 + v_0 t$$ for t with y=0, why not use the quadratic formula?
3. Apr 1, 2007
PhanthomJay
You seem to be looking only in the vertical direction. Is that the only direction of motion? And you can't use the motion equation for distance that you have noted, since that one was derived for constant acceleration, which you certainly don't have. I'm afraid you'll have to solve the differential equation F_net= mdv/dt in both directions to arrive at a result which involves the hyperbolic functions. Rather difficult at best.
4. Apr 2, 2007
denverdoc
Not sure I agree. To compute the trajectory would require setting up differential eqns along both axes, and the y would involve a hyperbolic fx. The flight time might be an easier soln, and involve only analysis on the Y axis. In other words the fact that the parabolic trajectory would be foreshortened may have no bearing, just as whether a bullet is fired or simply dropped from the same height of the gun barrel. You may be right, just can't get a good grip on the question.
It would have to be parced into two equations, the ascent where mg is working against and there is a Vy(init) and the descent, where mg and drag are working against each other and Vy(init)=0.
Last edited: Apr 2, 2007
5. Apr 2, 2007
Caspian
Thanks for all your suggestions. I like denverdoc's suggestion since force and velocity can be broken into x and y components... so force of drag in the x direction won't effect the y direction.
So, here's what I've got for the y direction:
$$y = \frac{1}{2} (g - \frac{F_{drag}}{m}) t^2 + v_0 t$$
$$y = \frac{1}{2} (g - \frac{-B v^2}{m}) t^2 + v_0 t$$
$$y = \frac{1}{2} \left(g - \frac{-B \left(\dfrac{1}{3 \dfrac{B}{m} t + C} \right)^{2/3}}{m} \right) t^2 + v_0 t$$ (substituted in v, which was solved for in my previous post. Note that the v and F are just y components)
The only way I can factor out the t is by assuming C = 0. (Is that valid?)
$$y = \frac{1}{2} g t^2 + \frac{1}{2} \frac{B \left(\dfrac{1}{3 \dfrac{B}{m}} \right)^{2/3}}{m} t^{4/3} + v_0 t$$
$$y = \frac{1}{2} g t^2 + \frac{1}{2} \sqrt[3]{B} \frac{\left(\dfrac{1}{3} \right)^{2/3}}{m^{5/3}} t^{4/3} + v_0 t$$
So, I need to set y = 0 and solve for t (other than the trivial solution of t = 0). I can't use the quadratic formula here... is there another way to solve this expression for t?
denverdoc and PhanthomJay mentioned that this will reduce to an expression which involves a hyperbolic function... but this doesn't look like it would reduce to that form (There's no e's here). Am I doing something wrong?
I really appreciate all your help.
Last edited: Apr 2, 2007
6. Apr 2, 2007
denverdoc
I suggested such from the following approach:
lets take the second half of the flight after apex has been reached.
mdv/dt=mg-kv(t)^2 where v is in the y component of velocity;
this becomes dv/(g-(k/m)v^2)=dt. let g'=sqrt(g) and c'=sqrt(k/m)
so that dv/[(g'-c'*v(t))*(g'+c'*v(t))]=dt which can be expressed as sum of fractions, and on integration gives rise to ln with numerator and denominator, etc. That should provide explicit function for v(t) in terms of t. For the ascent (the eqn will be different since both mg and kv^2 are negative terms), as we know the limits of integration (Vinit and 0), the time to apex determination should be straightforward. The descent however is a bit trickier, as we do not know the final velocity. But we could use the altitude determined from the ascent phase by integrating once more to solve it I think.
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• The description of quantum dielectric function for insulators over Bethe surface
• Fulltext
https://www.ias.ac.in/article/fulltext/pram/092/02/0027
• Keywords
Energy loss function; differential inelastic inverse mean free path; local particle number; Bethe surface; dielectric function
• Abstract
A new expression for the dielectric function is suggested here, which is the Mermin–Belkacem-Sigmund(MBS) model derived from the Belkacem–Sigmund (BS) model based on the conservation of a local particle number in the Mermin model. The energy loss function expressions are reviewed analytically for both models, and thesedielectric functions were used to calculate the Bethe sum rule, the energy loss function (ELF), as well as the differential inelastic inverse mean free path (DIIMP) for $\rm{H_{2}O}$. The indication from the results is that, compared to the BS dielectric function, the MBS dielectric function is more compatible in its consistency with the exact Bethe sum rule. The ELF for the MBS type is compatible relatively in high and low momentum transfers, while the ELF forthe BS type is suitable for high-$k$. The two models of ELF were also applied to evaluate DIIMP for electron kinetic energy 1 keV, and these were compared with the results predicted in several ways via the SESINIPAC program, using the Mermin dielectric function and the extended Drude and Monte–Carlo method. These predicted results are in reasonable agreement with those estimated from other methods at the range of energy transfer (0–50) eV.
• Author Affiliations
1. Department of Physics, College of Science for Women, University of Baghdad, Baghdad, Ir
• Pramana – Journal of Physics
Volume 96, 2022
All articles
Continuous Article Publishing mode
• Editorial Note on Continuous Article Publication
Posted on July 25, 2019
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# Philosophy:Legalism (Chinese philosophy)
Short description: One of the six classical schools of thought in Chinese philosophy
Legalism
Legalism or Fajia is one of the six classical schools of thought in Chinese philosophy. Literally meaning "house of (administrative) methods / standards (法, Fa)",[1] [2]:93 the Fa "school" represents several branches of "men of methods",[3] in the west often termed "realist" statesmen,[4]:17who played foundational roles in the construction of the bureaucratic Chinese empire.[5] The earliest persona of the Fajia may be considered Guan Zhong (720–645 BC), but following the precedent of the Han Feizi (c. 240 BC), Warring States period figures Shen Buhai (400–337 BC) and Shang Yang (390–338 BC) have commonly been taken as its "founders."
Commonly thought of as the greatest of all "Legalist" texts,[6] [7] the Han Feizi is believed to contain the first commentaries on the Dao De Jing in history.[8][9][10][11] Sun Tzu's The Art of War incorporates both a Daoist philosophy of inaction and impartiality, and a "Legalist" system of punishment and rewards, recalling political philosopher Han Fei's concepts of power (勢, shì) and tactics (術, shù).[12] Temporarily coming to overt power as an ideology with the ascension of the Qin dynasty,[13]:82 the First Emperor of Qin and succeeding emperors often followed the template set by Han Fei.[14]
Though the origins of the Chinese administrative system cannot be traced to any one person, the administrator Shen Buhai may have had more influence than any other on the construction of the merit system, and might be considered its founder, if not valuable as a rare pre-modern example of abstract theory of administration. Sinologist Herrlee G. Creel sees in Shen Buhai the "seeds of the civil service examination", and perhaps the first political scientist.[2]:94[15]:4, 119[16][17][18]
Concerned largely with administrative and sociopolitical innovation, Shang Yang was a leading reformer of his time.[19][13]:83 His numerous reforms transformed the peripheral Qin state into a militarily powerful and strongly centralized kingdom. Much of "Legalism" was "the development of certain ideas" that lay behind his reforms, which would help lead to Qin's ultimate conquest of the other states of China in 221 BC. [20][21]
Calling them the "theorists of the state", sinologist Jacques Gernet considered the Fajia to be the most important intellectual tradition of the fourth and third centuries BC.[22] The Fajia pioneered the centralizing measures and the economic organization of the population by the state that characterized the entire period from the Qin to the Tang dynasty;[23] the Han dynasty took over the governmental institutions of the Qin dynasty almost unchanged.[24][2]:105 Legalism rose to prominence again in the twentieth century, when reformers regarded it as a precedent for their opposition to conservative Confucian forces.[25] As a student, Mao Zedong championed Shang Yang, and towards the end of his life hailed the anti-Confucian legalist policies of the Qin dynasty.[26]
## Historical background
The Zhou dynasty was divided between the masses and the hereditary noblemen. The latter were placed to obtain office and political power, owing allegiance to the local prince, who owed allegiance to the Son of Heaven.[27] The dynasty operated according to the principles of Li and punishment. The former was applied only to aristocrats, the latter only to commoners.[28]
The earliest Zhou kings kept a firm personal hand on the government, depending on their personal capacities, personal relations between ruler and minister, and upon military might. The technique of centralized government being so little developed, they deputed authority to feudal lords.[29] When the Zhou kings could no longer grant new fiefs, their power began to decline, vassals began to identify with their own regions,[30] and schismatic hostility occurred between the Chinese states. Aristocratic families became very important, by virtue of their ancestral prestige wielding great power and proving a divisive force.[29]
In the Spring and Autumn period (771–476 BC), rulers began to directly appoint state officials to provide advice and management, leading to the decline of inherited privileges and bringing fundamental structural transformations as a result of what may be termed "social engineering from above".[31]:59[32] Most Warring States period thinkers tried to accommodate a "changing with the times" paradigm, and each of the schools of thought sought to provide an answer for the attainment of sociopolitical stability.[19]
Confucianism, commonly considered to be China's ruling ethos, was articulated in opposition to the establishment of legal codes, the earliest of which were inscribed on bronze vessels in the sixth century BC.[33] For the Confucians, the Classics provided the preconditions for knowledge.[34] Orthodox Confucians tended to consider organizational details beneath both minister and ruler, leaving such matters to underlings,[2]:107 and furthermore wanted ministers to control the ruler.[35]:359
Concerned with "goodness", the Confucians became the most prominent, followed by the proto-Taoists and the administrative thought that Sima Tan termed the Fajia. But the Taoists focused on the development of inner powers,[36][37][38] and both the Taoists and Confucians held a regressive view of history, the age being a decline from the era of the Zhou kings.[39]
## Introduction
The key figure in the bureaucracy was the district magistrate, a combination of a mayor, chief of police, judge, and even military commander.[40] He obtained the position by passing the examination for the civil service and performance at a lower level. He had a staff, some who moved with him, some permanently located in the district. Any penalty more serious than bambooing had to be approved by higher officials, any decision not based on statute required approval from Peking.[41]
Drawing by William Alexander, draughtsman of the Macartney Embassy to China in 1793.
In the four centuries preceding the first empire, a new type of ruler emerged intent on breaking the power of the aristocrats and reforming their state's bureaucracies.[42][43] As disenfranchised or opportunist aristocrats were increasingly attracted by the reform-oriented rulers,[44] they brought with them philosophy concerned foremost with organizational methodology.[42] Successful reforms made the so-called "Fajia" significant, promoting the rapid growth[45] of the Qin state that applied reforms most thoroughly.[46]
The goal of the "Legalist" ruler was conquest and unification of all under heaven (or in the case of Shen Buhai at least defense),[35]:345 and the writings of Han Fei and other Fajia are almost purely practical, eschewing ethics in favour of strategy[47][48][35]:345 teaching the ruler techniques (shu) to survive in a competitive world[35]:345[49] through administrative reform: strengthening the central government, increasing food production, enforcing military training, or replacing the aristocracy with a bureaucracy.[49] Han Fei's prince must make use of Fa (administrative methods and standards), surround himself with an aura of wei (majesty) and shi (authority, power, influence),[39][50] and make use of the art (shu) of statecraft. The ruler who follows Tao moves away from benevolence and righteousness, and discards reason and ability, subduing the people through Fa (statutes or administrative methods but implying objective measurements). Only an absolute ruler can restore the world.[50]
Though Han Fei espoused that his model state would increase the quality of life, he did not consider this a legitimizing factor (rather, a side-effect of good order). He focused on the functioning of the state, the ruler's role as guarantor within it, and aimed in particular at making the state strong and the ruler the strongest person within it.[51] To this end, Shen Buhai and successor Han Fei are concerned in particular with "the role of the ruler and the means by which he may control a bureaucracy."
Though the syncretic Han Feizi speaks on what may be termed law, what western scholarship termed the "Legalists" amongst other earlier terms, were concerned not mainly with law, but with administration.[11][2]:92–93,101,103[52][53] It has implications for the work of judges, but "contains no explicit judicial theory",[54][55] and is motivated "almost totally from the ruler's point of view".[56][57][58][55] Even the more "Legalistic" Book of Lord Shang still engages statutes more from an administrative standpoint, as well as addressing many other administrative questions.[18]
### Anti-ministerialism and human nature
The authority to make policy is a basic difference between Confucianism and the Fajia. Proposing a return to feudal ideals, albeit his nobleman being anyone who possessed virtue,[27] Confucians granted authority to "wise and virtuous ministers", allowed to "govern as they saw fit".[2]:107 In contrast, Shen Buhai and Shang Yang monopolized policy in the hands of the ruler,[2]:107 and Qin administrative documents focused on rigorous control of local officials, and the keeping of written records.[59] Distinguished by their anti-ministerial stance,[60][61] the Fajia rejected their Confucian contemporaries' espousal of a regime based solely on the charisma of the aristocrats,[62] and much of Fajia's doctrines seek self-regulating and mechanically reliable, if not foolproof means to control or otherwise dispense with officials administering the state. Reducing the human element, the first of these is the universally applicable Fa (administrative methods and standards).[57][31]:59
Shen Buhai and his philosophical successor Han Fei considered the ruler to be in a situation of constant danger from his aides,[35]:347 and the target of Han Fei's standards, in particular, are the scholarly bureaucracy and ambitious advisers – the Confucians.[35]:347 Saying that "superior and inferior fight a hundred battles a day",[63] long sections of the Han Feizi provide example of how ministers undermined various rules, and focus on how the ruler can protect himself against treacherous ministers, strongly emphasizing their mutually different interests.[64]
Though not exceptional, Sinologist Yuri Pines considers this selfish view of human nature to be a pillar of the Fajia, and a number of chapters of the Book of Lord Shang consider men naturally evil. The Fajia are therefore distinct from the Confucians (apart from their emphasis on Fa) in dismissing the possibility of reforming the elite, that being the ruler and ministers, or driving them by moral commitment. Every member of the elite pursues his own interests. Preserving and strengthening the ruler's authority against these may be considered the Fajia's "singularly pronounced political commitment".[65] On rare occasions, Han Fei lauds such qualities as benevolence and proper social norms; with due consideration for the times they were living in however, the Fajia did not believe that the moral influence or virtue of the ruler was powerful enough to create order.[50]
Considering the power struggle between ruler and minister irreconcilable, and focusing on the prevention of evil rather than the promotion of good, the Fajia largely rejected the utility of both virtue and the Confucian rule of man, insisting on impersonal norms and regulations in their relations.[4]:16[66][65] Their approach was therefore primarily at the institutional level, aiming for a clear power structure, consistently enforced rules and regulations, and in the Han Feizi, engaging in sophisticated manipulation tactics to enhance power bases.[67]
Rather than aristocratic fiefs, Qin territory came under the direct control of the Qin rulers, directly appointing officials on the basis of their qualifications.[68] With the state of Qin conquering all the Warring States and founding the "first" Chinese empire in 221 BC, the Fajia had succeeded in propelling state centralization and laying the foundations of Chinese bureaucracy, establishing "efficient and effective" codes that "became the pattern for Chinese politics for the next two millennia".[69] The philosophies of the reformers fell with the Qin, but tendencies remained in the supposedly Confucian imperial government, and the Han Feizi would be studied by rulers in every dynasty.[46] Hui even asserts that Confucianism's role in Chinese history is "[no] more than cosmetic", and Legalism is a more accurate description of the Chinese governmental tradition.[70]
## Antecedents: Guan Zhong and Mozi
Between Mozi's background as an engineer and his pacifist leanings, the Mohists became experts at building fortifications and sieges.
Small seal scripts were standardized by Li Si after the First Emperor of China gained control of the country, evolving from the larger seal scripts of previous dynasties.
The 12 characters on this slab of floor brick affirm that it is an auspicious moment for the First Emperor to ascend the throne, as the country is united and no men will be dying along the road.
Robert Eno of Indiana University writes that "If one were to trace the origins of Legalism as far back as possible, it might be appropriate to date its beginnings to the prime ministership of Guan Zhong (720–645 BC)",[6] who "may be seen as the source of the notion that good government involved skilled systems design". The reforms of Guan Zhong applied levies and economic specializations at the village level instead of the aristocracy, and shifted administrative responsibility to professional bureaucrats. He valued education.[71]
Guan Zhong and later Mozi (470-391BC) recommended objective, reliable, easily used,[72][35]:348–349[73] publicly accessible standards, or models, opposing what Sinologist Chad Hansen terms the "cultivated intuition of self-admiration societies", expert at chanting old texts.[35]:348–349[55] For Guan Zhong, Fa could complement any traditional scheme, and he uses Fa alongside the Confucian Li (the unique principles or standards of things, being their determinant and differentiating them), which he still valued. What Fa made possible was the accurate following of instructions.[35]:348–349[71] With minimal training, anyone can use Fa to perform a task or check results.[72] In principle, if their roots in Guan Zhong and Mozi are considered, the Legalists might all be said to use Fa in the same (administrative) fashion.[55][74]
The Mohists advocated a unified, utilitarian ethical and political order, posting some of its first theories and initiating a philosophical debate in China. To unify moral standards, they supported a "centralized, authoritarian state led by a virtuous, benevolent sovereign managed by a hierarchical, merit-based bureaucracy".[75] That social order is paramount seems to be implicit, recognized by all.[76] They argued against nepotism, and, as with the later Fa "philosophers", for universal standards (or meritocracy) as represented by the centralized state, saying "If one has ability, then he is promoted. If he has no ability, then he is demoted. Promoting public justice and casting away private resentments – this is the meaning of such statements."[77][78]
Compared by Sinologist Chris Fraser with Plato, the hermeneutics of the Mohists contained the philosophical germs of what Sima-Tan would term the "Fa-School" ("Legalists"), contributing to the political thought of contemporary reformers.[75] The Mohists and the Guanzi text attributed to Guan Zhong are of particular importance to understanding Fa,[79] meaning "to model on" or "to emulate".[35]:349[80][81] Dan Robins of the University of Hong Kong considers Fa to have become "important in early Chinese philosophy largely because of the Mohists".[82]
Of particular concern for the Fajia and the Mohists, the fourth century witnessed the emergence of discussions polarizing the concepts of self and private, commonly used in conjunction with profit and associated with fragmentation, division, partiality, and one-sidelines, with that of the state and "public", represented by the duke and referring to what is official or royal, that is, the ruler himself, associated with unity, wholeness, objectivity, and universality. The latter denotes the "Universal Way".[83] Legalism and Mohism are distinguished by this effort to obtain objectivity.[84]
### Mohist Hermeneutics
Mohist and Legalist thought is not based on entities, transcendentals or universals, but parts or roles ("names"),[85] and are therefore relatable to the Confucian rectification of names, which arguably originates in Mozi's development of Fa.[35]:348–349[73] For the most part Confucianism does not elaborate on Fa (though Han Confucians embraced Fa as an essential element in administration), though the idea of norms themselves being older,[86][35]:348–349 Fa is theoretically derived from the Confucian Li.[87]
Rejecting the Confucian idea of parents as a moral model as particular and unreliable, the driving idea of the Mohists was the use of hermeneutics to find objective models/standards (Fa) for ethics and politics, as was done in any practical field, to order or govern society. These were primarily practical rather than principles or rules,[88] as in the square and plumb-line.[21] The Mohists used Fa as "objective, particularly operational or measurement-like standards for fixing the referents of names",[89] hoping that analysis of language standards (Fa) would yield some objective way (dao) of moral reform.[55][35]:367 For Mozi, if language is made objective, then language itself could serve as a source of information and argued that in any dispute of distinctions, one party must be right and one wrong.[73]
While other terms might denote mere command, in comparison to the Western concept of law, the essential characteristic of Fa is measurement.[55] Mozi considered the elucidation of different "types" or "classes" to be the basis of both cognitive thinking and sociopolitical practice.[90] Referring to an easily projectable standard of utility, the Guanzi Mohists explain "Fa" as compasses or circles,[31]:59[35]:347–348[91][92] and may be prototypes, exemplars, or (specific) analogies.[91]
Fa is never merely arbitrary or the ruler's desire, nor does it aim at an intellectual grasp of a definition or principle, but the practical ability to perform a task (dao) successfully, or to "do something correctly in practice" — and in particular, to be able to distinguish various kinds of things from one another. Measuring to determine whether distinctions have been drawn properly, Fa compares something against itself, and judges whether the two are similar, just as with the use of the compass or the L-square. What matches the standard is then the particular object, and thus correct. This constituted the basic conception of Mohist's practical reasoning and knowledge.[75][72][35]:367[55]
Mozi said,
Those in the world who perform tasks cannot do without models (Fa) and standards. There is no one who can accomplish their task without models and standards. Even officers serving as generals or ministers, they all have models; even the hundred artisans performing their tasks, they too all have models. The hundred artisans make squares with the set square, circles with the compass, straight lines with the string, vertical lines with the plumb line, and flat surfaces with the level. Whether skilled artisans or unskilled artisans, all take these five as models. The skilled are able to conform to them. The unskilled, though unable to conform to them, by following them in performing their tasks still surpass what they can do by themselves. Thus the hundred artisans in performing their tasks all have models to measure by. Now, for the greatest to order (zhi, also 'govern') the world and those the next level down to order great states without models to measure by, this is to be less discriminating than the hundred artisans.[75]
Despite the framing of Han historians, the Fajia did not seem to think they were using Fa differently than anyone else,[35]:346,349,366 and the influence of the Mohists is likely strong.[93] All of the Fajia would adopt its usage.[55] Though Masayuki Sato translates Fa as law, he explains the concept as more like an objective measuring device.[94]:141 Sinologist Mark Edward Lewis writes: language, such as that of a legal code, is linked to social control. If words are not correct, they do not correspond to reality, and regulation fails. "Law" is "purified", rectified, or technically regulated language.[95][73] For Shen Buhai, correct or perverse words will order or ruin the state.[15]:59[2]:68 Han Fei may also have borrowed his views on human nature from the Mohists.[96]
Han Fei credits Shang Yang with the practice of Fa in statecraft,[35]:349,359[21] to which Shang Yang and Han Fei intended their "legal codes" (Fa) be as "self-interpreting" (Hansen).[55][21] Shang Yang's systematic application of penalties increase the tendency to see it as penal, but arguably does not change meaning from that of the Mohists. Shang Yang's innovation was not penal law. Rather, Shang Yang's idea was that penal codes should be reformed to have the same kind of objectivity, clarity and accessibility as the craft-linked instruments.[35]:349,359 Contrasting Fa with private distortions and behavior,[35]:367 theoretically, their Fa exactly follows Mozi.[35]:349,359[97][21] Shang Yang was supposedly taught by a Confucian syncretist, Shi Jiao, who, stressing the importance of "name" (rectification of names), connected it with reward and punishment.[71]
Applied to economy and institution, Shang Yang's Fa is total and anti-bureaucratic, calculating rank mathematically from the adherence to standards (Fa) in the performance of roles (models), namely that of soldiers and (to a lesser extent) farmers.[35]:349,359[21] Han Fei shows no revolutionary insight into rules; objectively-determined "models" (Fa) or "names" (titles/roles), being measured against, replace intuitive guidance, especially that of the ruler. It is these that enable control of a bureaucracy.[35]:366 Carine Defoort of New York University explains:
Names are orders: by manipulating a network of names from his polar position, the ruler keeps everything under control. While his orders descend step by step through the official hierarchy to the furthest corners of the realm, performances ascend to be checked by him.[98]
Because Fa is necessary for articulating administrative terms, it is presupposed in any application of punishment, and Han Fei stressed measurement-like links between rewards and punishments and performance. Applied through incentives and disincentives, Fa provided guidance for behaviour, the performance of civil and military roles, and advancement.[35]:349[55][31]:59
An excavated Qin text consists of twenty-five abstract model patterns guiding procedure based on actual situations.[99]
## Branches of the Fajia
Feng Youlan and Liang Qichao describe the elements of the Fajia as Fa (often translated as law, but closer to "standards" or "method"[100]), authority or power (Shi), and "technique" (Shu), that is, statecraft or "the art of conducting affairs and handling men".[101] Less well defined compared to Confucianism and Mohism, it is unclear when the Fajia came to be regarded as an intellectual faction, only forming a complex of ideas around the time of Li Si (280–208 BC), elder advisor to the First Emperor.[102] While the earliest Legalistic act can be traced to Zichan (and with him Deng Xi),[103] Chinese scholar K. C. Hsiao and Sinologist Herrlee G. Creel considered the Fajia to have stemmed from two disparate contemporary thinkers, as described by Han Fei:[2]:48,69,100,103,113[101][104][13]:81[31]:59 [105] [106][4]:15
Now Shen Buhai spoke about the need of Shu ("Technique") and Shang Yang practices the use of Fa ("Standards"). What is called Shu is to create posts according to responsibilities, hold actual services accountable according to official titles, exercise the power over life and death, and examine into the abilities of all his ministers; these are the things that the ruler keeps in his own hand. Fa includes mandates and ordinances promulgated to the government offices, penalties that are definite in the mind of the people, rewards that are due to the careful observers of standards, and punishments that are inflicted upon those who violate orders. It is what the subjects and ministers take as a model. If the ruler is without Shu he will be overshadowed; if the subjects and ministers lack Fa they will be insubordinate. Thus, neither can be dispensed with: both are implements of emperors and kings.[107][2]:94[108][109][110]:184[111][112]
In contrast to the old feudalism and Shen Buhai, Shang or Gongsun Yang considered there to be no single model of rule in the past, and everything changeable as a product of changing conditions;[113][114] holding decline to have resulted from a scarcity of resources, he prescribed statecraft.[39] Questioning traditional rule and the relevance of the past to the present,[19] the first chapter of the Book of Lord Shang cites Gongsun as saying: "Orderly generations did not [follow] a single way; to benefit the state, one need not imitate antiquity."[114] Distinguished by his heavy emphasis on penalty and mutual responsibility (among both minister and population), he instituted severe punishment for the Qin (later reduced).[13]:93
Gongsun ultimately did not believe that the method of rule really mattered as long as the state was rich,[15][115] and tried to dispense with the selection of exceptional men through insurance mechanisms while attacking moral discussion as empowering ministers.[84] His anti-bureaucracy may be seen as a precursor to that of Han Fei,[35]:359 and together with their predecessor Mozi may be characterized as following a philosophical tradition of "objective, public, accessible standards" (Fa).[35]:345 The Shang Yang school was favored, though not exclusively, by Emperor Wu of Han.[2]:115
In contrast to Shang Yang, though seeking at the motivation of his subjects, Han Fei is much more skeptical of self-interest.[116] Shen Buhai and his branch sometimes even opposed punishments. Han Fei combined the branches. This combination is commonly known as the Fajia.[2]:100,103[104] Because, historically, the branches did not endorse each other's views, Creel often called the Shen Buhai group "administrators", "methodists" or "technocrats",[13]:81 a division nominally accepted by the Cambridge History of China. Han Fei called both branches "the instruments of Kings and Emperors", and Li Si praised them equally, finding no contradiction between them.[117]:268[118]
Sinologist Chad Hansen describes their difference as such: "Shen Buhai's shu ('techniques') limit the ministers' influence on the ruler; Shang Yang's fa controls their power over the people."[55][35]:359
The scholar Shen Dao (350 – c. 275 BC) covered a "remarkable" quantity of Legalist and Taoistic themes.[119] Incorporated into the Han Feizi and The Art of War, he nonetheless lacked a recognizable group of followers.[15]:32[117]:283[13]:93
## Shang Yang (390–338 BC)
Small bronze plaque containing an edict from the second emperor of the Qin dynasty. 209 BC.
Terracotta Army
Hailing from Wei, as Prime Minister of the State of Qin Shang Yang or Gongsun Yang engaged in a "comprehensive plan to eliminate the hereditary aristocracy". Drawing boundaries between private factions and the central, royal state, he took up the cause of meritocratic appointment, stating "Favoring one's relatives is tantamount to using self-interest as one's way, whereas that which is equal and just prevents selfishness from proceeding."[83]
As the first of his accomplishments, historiographer Sima Qian accounts Gongsun as having divided the populace into groups of five and ten, instituting a system of mutual responsibility [120] tying status entirely to service to the state. It rewarded office and rank for martial exploits, going to far as to organize women's militias for siege defense. The second accomplishment listed is forcing the populace to attend solely to agriculture (or women cloth production, including a possible sewing draft) and recruiting labour from other states. He abolished the old fixed landholding system (Fengjian) and direct primogeniture, making it possible for the people to buy and sell (usufruct) farmland, thereby encouraging the peasants of other states to come to Qin. The recommendation that farmers be allowed to buy office with grain was apparently only implemented much later, the first clear-cut instance in 243 BC. Infanticide was prohibited.[2]:94[29][13]:83[121][122]
Gongsun deliberately produced equality of conditions amongst the ruled, a tight control of the economy, and encouraged total loyalty to the state, including censorship and reward for denunciation. Law was what the sovereign commanded, and this meant absolutism, but it was an absolutism of law as impartial and impersonal. Gongsun discouraged arbitrary tyranny or terror as destroying the law.[123] Emphasizing knowledge of the Fa among the people, he proposed an elaborate system for its distribution to allow them to hold ministers to it.[35]:359 He considered it the most important device for upholding the power of the state. Insisting that it be made known and applied equally to all, he posted it on pillars erected in the new capital. In 350, along with the creation of the new capital, a portion of Qin was divided into thirty-one counties, each "administered by a (presumably centrally appointed) magistrate". This was a "significant move toward centralizing Ch'in administrative power" and correspondingly reduced the power of hereditary landholders.[13]:83[124]
Gongsun considered the sovereign to be a culmination in historical evolution, representing the interests of state, subject and stability.[125][126] Objectivity was a primary goal for him, wanting to be rid as much as possible of the subjective element in public affairs. The greatest good was order. History meant that feeling was now replaced by rational thought, and private considerations by public, accompanied by properties, prohibitions and restraints. In order to have prohibitions, it is necessary to have executioners, hence officials, and a supreme ruler. Virtuous men are replaced by qualified officials, objectively measured by Fa. The ruler should rely neither on his nor his officials' deliberations, but on the clarification of Fa. Everything should be done by Fa,[13]:88[127] whose transparent system of standards will prevent any opportunities for corruption or abuse.[128] Shang Yang also corrected measures and weights.[129]
### Anti-Confucianism
While Shen Buhai and Shen Dao's current may not have been hostile to Confucius,[2]:64 Shang Yang and Han Fei emphasize their rejection of past models as unverifiable if not useless ("what was appropriate for the early kings is not appropriate for modern rulers").[130][114][55][131] In the west, past scholars have argued that Shang Yang sought to establish the supremacy of what some have termed positive law at the expense of customary or "natural" law.[123] Han Fei argued that the age of Li had given way to the age of Fa, with natural order giving way to social order and finally political order. Together with that of Xun Kuang, their sense of human progress and reason guided the Qin dynasty.[132]
Intending his Dao (way of government) to be both objective and publicly projectable,[35]:352 Han Fei argued that disastrous results would occur if the ruler acted on arbitrary, ad-hoc decision making, such as that based on relationships or morality which, as a product of reason, are "particular and fallible". Li, or Confucian customs, and rule by example are also simply too ineffective.[50][133][134] The ruler cannot act on a case-by-case basis, and so must establish an overarching system, acting through Fa (administrative methods or standards). Fa is not partial to the noble, does not exclude ministers, and does not discriminate against the common people.[134]
Linking the "public" sphere with justice and objective standards, for Han Fei, the private and public had always opposed each other.[83] Taking after Shang Yang he lists the Confucians among his "five vermin",[135] and calls the Confucian teaching on love and compassion for the people the "stupid teaching" and "muddle-headed chatter",[136] the emphasis on benevolence an "aristocratic and elitist ideal" demanding that "all ordinary people of the time be like Confucius' disciples".[50] Moreover, he dismisses it as impracticable, saying that "In their settled knowledge, the literati are removed from the affairs of the state ... What can the ruler gain from their settled knowledge?",[137] and points out that "Confucianism" is not a unified body of thought.[138]
### Assessments
Keeping in mind the information of the time (1955) and the era of which he is speaking, A. F. P. Hulsewé goes as far as to call Shang Yang the "founder of the school of law", and considers his unification of punishments one of his most important contributions; that is, giving the penalty of death to any grade of person disobeying the king's orders. Shang Yang even expected the king, though the source of law (authorizing it), to follow it. This treatment is in contrast to ideas more typical of archaic society, more closely represented in the Rites of Zhou as giving different punishments to different strata of society.
Hulsewe points out that Sima Tan considered equal treatment the "school of law's" most salient point: "They do not distinguish between close and far relatives, nor do they disriminate between noble and humble, but in an uniform manner they decide on them in law."[139] Though himself deriving them from elsewhere, the Han dynasty adopted essentially the same denominations of crimes, if not equality, as Shang Yang set down for Qin, without collective punishment of the three sets of relatives.[140]
Shang Yang appeared to act according to his own teachings,[139] and translator Duvendak references him as being considered "like a bamboo‑frame which keeps a bow straight, and one could not get him out of his straightness", even if spoken of by some pre-modern Chinese in ill regard with the fall of Qin. Though writing in 1928, Duvendak believed that Shang Yang should be of interest not just to Sinologists, but Western Jurists as well.[141]
## Shen Buhai (400 – c. 337 BC)
Han state bronze candle holder
The basic structure and operation of the traditional Chinese state was not "legalistic" as the term is commonly understood. Though persisting, pre-modern mainstream Chinese thinking never really accepted the role of law and jurisprudence or the Shang Yang wing of the Fajia. The Fajia's most important contribution lies in the organization and regulation of centralized, bureaucratic government. Sinologist Herrlee G. Creel called its philosophy administrative for lack of a better term, considering it to have been founded by Shen Buhai (400–337 BC), who likely played an "outstanding role in the creation of the traditional Chinese system of government".
Shen was chancellor of Han for fifteen years (354–337 BC).[2]:81,113[142][13]:90[143] The Huainanzi says that when Shen lived the officials of the state of Han were at cross-purposes and did not know what practices to follow;[144][2]:86 the legal system of Han was apparently confused, prohibiting uniform reward and punishment. It is not surprising then that no text identifies Shen Buhai with penal law. We have no basis to suppose that Shen advocated the doctrine of rewards and punishment (of Shang Yang, as Han Fei did), and Han Fei criticizes him for not unifying the laws.
A teacher of Legalist Li Kui, the Confucian Bu Shang is cited for the principle of favouring talents over favouritism,[145] becoming under the Mohists the principle of "elevating the worthy and employing ability". Adhering thereto, Shen utilized the same category of method (Fa) as others of the Fajia, but emphasized its use in secrecy for purposes of investigation and personnel control, concerning himself with methods (Fa) of (impersonal bureaucratic) administration (namely methods of appointment and performance measurement) or the ruler's role in the control thereof.[15][2]:100,103[117]:283[13]:93 He is famous for the dictum "The Sage ruler relies on standards/method (Fa) and does not rely on wisdom; he relies on technique, not on persuasions."[146]
What Shen appears to have realized is that the "methods for the control of a bureaucracy" could not be mixed with the survivals of feudal government, or staffed merely by "getting together a group of 'good men'", but rather must be men qualified in their jobs.[2]:86[147] He therefore emphasizes the importance of selecting able officials as much as Confucius did, but insists on "constant vigilance over their performance",[2]:65 never mentioning virtue. Well aware of the possibility of the loss of the ruler's position, and thus state or life, from said officials,[2]:97 Shen says:
One who murders the ruler and takes his state ... does not necessarily climb over difficult walls or batter in barred doors or gates. He may be one of the ruler's own ministers, gradually limiting what the ruler sees, restricting what he hears, getting control of his government and taking over his power to command, possessing the people and seizing the state.[2]:97[35]:359[148][149]:170
Compared with Shang Yang, Shen Buhai refers to the ruler in abstract terms: he is simply the head of a bureaucracy. In comparison with Han Fei though his system still required a strong ruler at the centre,[2]:59–60,63 emphasizing that he trust no one minister.[143] Ideally, Shen Buhai's ruler had the widest possible sovereignty, was intelligent (if not a sage), had to make all crucial decisions himself,[15]:59–60 and had unlimited control of the bureaucracy.[15]:59–60[149]:170 Shen largely recommended that rulers investigate their ministers' performance, checking his ministers' reports while remaining calm and secretive (Wu wei). The ruler promotes and demotes according to the match between 'performance' and proposal (Xing Ming).[150]
Shen Buhai insisted that the ruler must be fully informed on the state of his realm, but could not afford to get caught up in details and in an ideal situation need listen to no one. Listening to his courtiers might interfere with promotions, and he does not, as Sinologist Herrlee G. Creel says, have the time to do so. The way to see and hear independently is the grouping together of particulars into categories using mechanical or operational method (Fa). On the contrary the ruler's eyes and hears will make him "deaf and blind" (unable to obtain accurate information).[2]:81[15]:33,68–69[117]:283[151] Seeing and hearing independently, the ruler is able to make decisions independently, and is, Shen says, able to rule the world thereby.[15]:26
### Shu or "Technique"
The earliest known written documentation for the Chinese abacus, the Suan Pan, dates to the 2nd century BC (its original design is unknown).
Apart from Shang Yang's doctrine of penalties and mutual spying and denouncement among ministers, Han Fei recommends the ruler should protect himself through careful employment of doctrines that had earlier been recommended by Shen Buhai.[150] Because Fa has diverse meaning, for clarification Shen Buhai's successors often used the term Shu (technique) for his administrative method (Fa) and other techniques (such as "Wu-wei"), and thus 20th century philosopher Feng Youlan called Shen the leader of the group [in the Legalist school] emphasizing Shu, or techniques of government.[2]:80[152][117]:283[153]
Liu Xiang wrote that Shen Buhai advised the ruler of men use technique (shu) rather than punishment, relying on persuasion to supervise and hold responsible, though very strictly.[2]:81,103[154][155] Shen's doctrines are described as concerned almost exclusively with the "ruler's role and the methods by which he may control a bureaucracy"; that is, its management and personnel control: the selection of capable ministers, their performance, the monopolization of power,[2]:81,100,103 and the control of and power relations between ruler and minister which he characterized as Wu Wei.[35]:359 The emphasis, however, is on "scrutinizing achievement and on that ground alone to give rewards, and to bestow office solely on the basis of ability".[2]:93[15] Sinologist John Makeham characterizes Shu as "the agency of several checking systems that together constituted Method (Fa)", whose central principle is accountability.[155][156]
Sinologist Herrlee G. Creel believed the term originally had the sense of numbers, with implicit roots in statistical or categorizing methods, using record-keeping in financial management as a numerical measure of accomplishment.[74][157] He notes that command of finance was generally held by the head of government from the beginning of the Zhou dynasty; an example of auditing dates to 800 BC, and the practice of annual accounting solidified by the Warring States period and budgeting by the first century BC.[15]:51 In the Guanzi the artisan's Shu is explicitly compared to that of the good ruler.[158] The History of the Han (Han Shu) lists texts for Shu as devoted to "calculation techniques" and "techniques of the mind", and describes the Warring States period as a time when the shu arose because the complete tao had disappeared.[159] Hsu Kai (920–974 AD) calls Shu a branch in, or components of, the great Tao, likening it to the spokes on a wheel. He defines it as "that by which one regulates the world of things; the algorithms of movement and stillness". Mastery of techniques was a necessary element of sagehood.[159]
Another example of Shu is Chuan-shu, or "political maneuvering". The concept of Ch'uan, or "weighing" figures in Legalist writings from very early times. It also figures in Confucian writings as at the heart of moral action, including in the Mencius and the Doctrine of the Mean. Weighing is contrasted with "the standard". Life and history often necessitate adjustments in human behavior, which must suit what is called for at a particular time. It always involves human judgement. A judge that has to rely on his subjective wisdom, in the form of judicious weighing, relies on Ch'uan. The Confucian Zhu Xi, who was notably not a restorationist, emphasized expedients as making up for incomplete standards or methods.[160]
### Name and reality (Ming-shi)
"The Way of Listening is to be giddy as though soused. Be dumber and dumber. Let others deploy themselves, and accordingly, I shall know them."
Right and wrong whirl around him like spokes on a wheel, but the sovereign does not complot. Emptiness, stillness, non-action—these are the characteristics of the Way. By checking and comparing how it accords with reality, [one ascertains] the "performance" of an enterprise.[161][162]
Han Fei
Detail of The Spinning Wheel, by Chinese artist Wang Juzheng, Northern Song Dynasty (960–1279)[163]
A contemporary of Confucius,[164] the logician Deng Xi (died 501 BC) was cited by Liu Xiang for the origin of the principle of Xing-Ming. Serving as a minor official in the state of Zheng, he is reported to have drawn up a code of penal laws. Associated with litigation, he is said to have argued for the permissibility of contradictory propositions, likely engaging in hair-splitting debates on the interpretation of laws, legal principles and definitions.[165] Shen Buhai solves this through Wu wei, or not getting involved, making an official's words his own responsibility.[108] Shen Buhai says, "The ruler controls the policy, the ministers manage affairs. To speak ten times and ten times be right, to act a hundred times and a hundred times succeed – this is the business of one who serves another as minister; it is the not the way to rule."[2]:65 The correlation between Wu-wei and ming-shi (simplified Chinese: 名实; traditional Chinese: 名實; pinyin: míngshí) likely informed the Taoist conception of the formless Tao that "gives rise to the ten thousand things".[166]
In the Han Dynasty secretaries of government who had charge of the records of decisions in criminal matters were called Xing-Ming, which Sima Qian (145 or 135 – 86 BC) and Liu Xiang (77–6 BC) attributed to the doctrine of Shen Buhai (400 – c. 337 BC). Liu Xiang goes as far as to define Shen Buhai's doctrine as Xing-Ming.[2]:72,80,103–104[167][168] Shen actually used an older, more philosophically common equivalent, ming-shi, linking the "Legalist doctrine of names" with the name and reality (ming shi) debates of the school of names – another school evolving out of the Mohists.[169][170] Such discussions are also prominent in the Han Feizi,[171] and the earliest literary occurrence for Xing-Ming, in the Zhan Guo Ce, is also in reference to the school of names.[172]
Ming ("name") sometimes has the sense of speech – so as to compare the statements of an aspiring officer with the reality of his actions – or reputation, again compared with real conduct (xing "form" or shi "reality").[2]:83[173][174] Two anecdotes by Han Fei provide examples: The Logician Ni Yue argued that a white horse is not a horse, and defeated all debaters, but was still tolled at the gate. In another, the chief minister of Yan pretended to see a white horse dash out the gate. All of his subordinates denied having seen anything, save one, who ran out after it and returned claiming to have seen it, and was thereby identified as a flatterer.[174]
Shen Buhai's personnel control, or rectification of names (such as titles) worked thereby for "strict performance control" (Hansen) correlating claims, performances and posts.[35]:359 It would become a central tenant of both Legalist statecraft[172] and its Huang-Lao derivatives. Rather than having to look for "good" men, ming-shi or xing-ming can seek the right man for a particular post, though doing so implies a total organizational knowledge of the regime.[15]:57 More simply though, it can allow ministers to "name" themselves through accounts of specific cost and time frame, leaving their definition to competing ministers. Claims or utterances "bind the speaker to the realization a job (Makeham)." This was the doctrine, with subtle differences, favoured by Han Fei. Favoring exactness, it combats the tendency to promise too much.[108][174][175] The correct articulation of Ming is considered crucial to the realization of projects.[108][172]
In Chinese Thought: An Introduction, S. Y. Hsieh suggests a set of assumptions underlying the concept of (xing-ming).
• That when a large group of people are living together, it is necessary to have some form of government.
• The government has to be responsible for a wide range of things, to allow them to live together peacefully.
• The government does not consist of one person only, but a group.
• One is a leader that issues orders to other members, namely officials, and assigns responsibilities to them.
• To do this, the leader must know the exact nature of the responsibilities, as well as the capabilities of the officials.
• Responsibilities, symbolized by a title, should correspond closely with capabilities, demonstrated by performance.
• Correspondence measures success in solving problems and also controls the officials. When there is a match, the leader should award the officials.
• It is necessary to recruit from the whole population. Bureaucratic government marks the end of feudal government.[13]:90
### Wu wei (inaction)
Zhaoming mirror frame, Western Han dynasty
Playing a "crucial role in the promotion of the autocratic tradition of the Chinese polity", what is termed Wu wei (or inaction) would become the political theory of the Fajia (or "Chinese Legalists"), if not becoming their general term for political strategy. The (qualified) non-action of the ruler ensures his power and the stability of the polity,[176] and can therefore be considered his foremost technique.[177] The "conception of the ruler's role as a supreme arbiter, who keeps the essential power firmly in his grasp" while leaving details to ministers, would have a "deep influence on the theory and practice of Chinese monarchy".[2]:99 Following Shen Buhai strongly advocated by Han Fei, during the Han dynasty up until the reign of Han Wudi rulers confined their activity "chiefly to the appointment and dismissal of his high officials", a plainly "Legalist" practice inherited from the Qin dynasty.[2]:99[178]
Lacking any metaphysical connotation, Shen used the term Wu wei to mean that the ruler, though vigilant, should not interfere with the duties of his ministers,[2]:62–63[13]:92 acting through administrative method. Shen says:
The ruler is like a mirror, reflecting light, doing nothing, and yet, beauty and ugliness present themselves; (or like) a scale establishing equilibrium, doing nothing, and yet causing lightness and heaviness to discover themselves. (Administrative) method (Fa) is complete acquiescence. (Merging his) personal (concerns) with the public (weal), he does not act. He does not act, and yet as a result of his non-action (wuwei) the world brings itself to a state of complete order.[2]:64[149]:172
Though not a conclusive argument against proto-Taoist influence, Shen's Buhai's Taoist terms do not show evidence of explicit Taoist usage (Confucianism also uses terms like "Tao", or Wu wei), lacking any metaphysical connotation.[2]:62–63 The Han Feizi has a commentary on the Tao Te Ching, but references Shen Buhai rather than Laozi for Wu wei.[2]:69 Since the bulk of both the Tao Te Ching and the Zhuangzhi appear to have been composed later, Sinologist Herrlee G. Creel argued that it may therefore be assumed that Shen Buhai influenced them.[2]:48,62–63[13]:92
Shen Buhai argued that if the government were organized and supervised relying on proper method (Fa), the ruler need do little – and must do little.[2]:69[15]:66 Unlike Legalists Shang Yang and Han Fei, Shen did not consider the relationship between ruler and minister antagonistic necessarily.[179] Apparently paraphrasing the Analects, Shen Buhai's statement that those near him will feel affection, while the far will yearn for him,[2]:67,81[180] stands in contrast to Han Fei, who considered the relationship between the ruler and ministers irreconcilable.[65]
However, Shen still believed that the ruler's most able ministers are his greatest danger,[15]:35 and is convinced that it is impossible to make them loyal without techniques.[181] Creel explains: "The ruler's subjects are so numerous, and so on alert to discover his weaknesses and get the better of him, that it is hopeless for him alone as one man to try to learn their characteristics and control them by his knowledge ... the ruler must refrain from taking the initiative, and from making himself conspicuous – and therefore vulnerable – by taking any overt action."[2]:66
Shen Buhai portrays the ruler as putting up a front to hide his dependence on his advisers. Aside from hiding the ruler's weaknesses, Shen's ruler, therefore, makes use of method (Fa) in secrecy. Even more than with Han Fei, Shen Buhai's ruler's strategies are a closely guarded secret, aiming for a complete independence that challenges "one of the oldest and most sacred tenets of [Confucianism]", that of respectfully receiving and following ministerial advice.[149]:171–172, 185
Though espousing an ultimate inactive end, the term does not appear in the Book of Lord Shang, ignoring it as an idea for control of the administration.[2]:69
#### Yin (passive mindfulness)
Shen's ruler plays no active role in governmental functions. He should not use his talent even if he has it. Not using his own skills, he is better able to secure the services of capable functionaries. However, Sinologist Herrlee G. Creel also argues that not getting involved in details allowed Shen's ruler to "truly rule", because it leaves him free to supervise the government without interfering, maintaining his perspective.[2]:65–66[176][108]
Adherence to the use of technique in governing requires the ruler not engage in any interference or subjective consideration.[182] Sinologist John Makeham explains: "assessing words and deeds requires the ruler's dispassionate attention; (yin is) the skill or technique of making one's mind a tabula rasa, non-committaly taking note of all the details of a man's claims and then objectively comparing his achievements of the original claims."[182]
A commentary to the Shiji cites a now-lost book as quoting Shen Buhai saying: "By employing (yin), 'passive mindfulness', in overseeing and keeping account of his vassals, accountability is deeply engraved." The Guanzi similarly says: "Yin is the way of non-action. Yin is neither to add to nor to detract from anything. To give something a name strictly on the basis of its form – this is the Method of yin."[182][183]
Yin also aimed at concealing the ruler's intentions, likes and opinions.[182] Shen advises the ruler to keep his own counsel, hide his motivations and conceal his tracks in inaction, availing himself of an appearance of stupidity and insufficiency.[2]:67[15]:35
If the ruler's intelligence is displayed, men will prepare against it; If his lack of intelligence is displayed, they will delude him. If his wisdom is displayed, men will gloss over (their faults); if his lack of wisdom is displayed, they will hide from him. If his lack of desires is displayed, men will spy out his true desires; if his desires are displayed, they will tempt him. Therefore (the intelligent ruler) says "I cannot know them; it is only by means of non-action that I control them."[2]:66[184][110]:185
Said obscuration was to be achieved together with the use of Method (Fa). Not acting himself, he can avoid being manipulated.[13]:92
Despite such injunctions, it is clear that the ruler's assignments would still be completely up to him.[185]
## Shen Dao (350 – c. 275 BC)
Iron weight dated from 221 BC with 41 inscriptions written in seal script about standardizing weights and measures during the 1st year of Qin dynasty "Where there is a scale, people cannot deceive others about weight; where there is a ruler, people cannot deceive others about length; and where there is Fa, people cannot deceive others about one's words and deeds." Shen Dao[94]:137
Mold for making banliang coins
Shen Dao argued for Wu wei in a similar manner to Shen Buhai, saying
The Dao of ruler and ministers is that the ministers labour themselves with tasks while the prince has no task; the prince is relaxed and happy while the ministers bear responsibility for tasks. The ministers use all their intelligence and strength to perform his job satisfactorily, in which the ruler takes no part, but merely waits for the job to be finished. As a result, every task is taken care of. The correct way of government is thus.[186][187]
Shen Dao also espouses an impersonal administration in much the same sense as Shen Buhai, and in contrast with Shang Yang emphasizes the use of talent[188] and the promotion of ministers, saying that order and chaos are "not the product of one man's efforts". Along this line, however, he challenges the Confucian and Mohist esteem and appointment of worthies as a basis of order, pointing out that talented ministers existed in every age.
Taking it upon himself to attempt a new, analytical solution, Shen advocated fairness as a new virtue, eschewing appointment by interview in favour of a mechanical distribution ("the basis of fairness") with the invariable Fa apportioning every person according to their achievement. Scholar Sugamoto Hirotsugu attributes the concept of Fen, or social resources, also used by the Guanzi and Xunzi, to Shen, given a "dimensional" difference through Fa, social relationships ("yin") and division.[189][94]:122,126,133–136
If one rabbit runs through a town street, and a hundred chase it, it is because its distribution has not been determined ... If the distribution has already been determined, even the basest people will not go for it. The way to control All-under-Heaven and the country lies solely in determining distribution.
The greatest function of Fa ("the principle of objective judgement") is the prevention of selfish deeds and argument. However, doubting its long-term viability Shen did not exclude moral values and accepted (qualified) Confucian Li's supplementation of Fa and social relationships, though he frames Li in terms of (impersonal) rules.[94]:134–135[119]
The state has the li of high and low rank, but not a li of men of worth and those without talent. There is a li of age and youth, but not of age and cowardice. There is a li of near and distant relatives, but no li of love and hate.
For this reason he is said to "laugh at men of worth" and "reject sages", his order relying not on them but on the Fa.[119]
Linking Fa to the notion of impartial objectivity associated with universal interest, and reframing the language of the old ritual order to fit a universal, imperial and highly bureaucratized state,[83] Shen cautions the ruler against relying on his own personal judgment,[190] contrasting personal opinions with the merit of the objective standard, or fa, as preventing personal judgements or opinions from being exercised. Personal opinions destroy Fa, and Shen Dao's ruler therefore "does not show favouritism toward a single person".[83]
When an enlightened ruler establishes [gong] ("duke" or "public interest"), [private] desires do not oppose the correct timing [of things], favoritism does not violate the law, nobility does not trump the rules, salary does not exceed [that which is due] one's position, a [single] officer does not occupy multiple offices, and a [single] craftsman does not take up multiple lines of work ... [Such a ruler] neither overworked his heart-mind with knowledge nor exhausted himself with self-interest (si), but, rather, depended on laws and methods for settling matters of order and disorder, rewards and punishments for deciding on matters of right and wrong, and weights and balances for resolving issues of heavy or light ...[83]
The reason why those who apportion horses use ce-lots, and those who apportion fields use gou-lots, is not that they take ce and gou-lots to be superior to human wisdom, but that one may eliminate private interest and stop resentment by these means. Thus it is said: "When the great lord relies on fa and does not act personally, affairs are judged in accordance with (objective) method (fa)." The benefit of fa is that each person meets his reward or punishment according to his due, and there are no further expectations of the lord. Thus resentment does not arise and superiors and inferiors are in harmony.
If the lord of men abandons method (Fa) and governs with his own person, then penalties and rewards, seizures and grants, will all emerge from the lord's mind. If this is the case, then those who receive rewards, even if these are commensurate, will ceaselessly expect more; those who receive punishment, even if these are commensurate, will endlessly expect more lenient treatment... people will be rewarded differently for the same merit and punished differently for the same fault. Resentment arises from this.[191][94]:129[192]
### Doctrine of position (shi)
The people of Qi have a saying – "A man may have wisdom and discernment, but that is not like embracing the favourable opportunity. A man may have instruments of husbandry, but that is not like waiting for the farming seasons." Mencius
The Chinese Immortal Han Xiangzi riding a cloud
A floating seed of the p'eng plant, meeting a whirlwind, may be carried a thousand li, because it rides on the power (shi) of the wind. If, in measuring an abyss, you know that it is a thousand fathoms deep, it is owing to the figures which you find by dropping a string. By depending on the power (shi) of a thing, you will reach a point, however, distant it may be, and by keeping the proper figures, you will find out the depth, however deep it may be. The Book of Lord Shang
Generally speaking, the "Fajia" understood that the power of the state resides in social and political institutions, and are innovative in their aim to subject the state to them.[193][117]:268[149]:175 Like Shen Buhai, Shen Dao largely focused on statecraft (Fa), and Confucian Xun Kuang discusses him in this capacity, never referencing Shen Dao in relation to power.[194][13]:93[195][196] Shen Dao is remembered for his theories on shi (lit. "situational advantage", but also "power" or "charisma") because Han Fei references him in this capacity.[197]
In the words of Han Fei,
The reason why I discuss the power of position is for the sake of ... mediocre rulers. These mediocre rulers, at best they do not reach the level of [the sages] Yao or Shun, and at worst they do not behave like [the arch-tyrants] Jie or Zhou. If they hold to the law and depend on the power of their position, there will be order; but if they abandon the power of their position and turn their backs on the law, there will be disorder. Now if one abandons the power of position, turns one's back on the law, and waits for a Yao or Shun, then when a Yao or a Shun arrives there will indeed be order, but it will only be one generation of order in a thousand generations of disorder ... Nevertheless, if anyone devotes his whole discourse to the sufficiency of the doctrine of position to govern All-under-Heaven, the limits of his wisdom must be very narrow.[130]
Used in many areas of Chinese thought, shi probably originated in the military field.[198] Diplomats relied on concepts of situational advantage and opportunity, as well as secrecy (shu) long before the ascendancy of such concepts as sovereignty or law, and were used by kings wishing to free themselves from the aristocrats.[199] Sun Tzu would go on to incorporate Taoist philosophy of inaction and impartiality, and Legalist punishment and rewards as systematic measures of organization, recalling Han Fei's concepts of power (shi) and tactics (shu).[12]
Henry Kissinger's On China says: "Chinese statesmanship exhibits a tendency to view the entire strategic landscape as part of a single whole ... Strategy and statecraft become means of 'combative coexistence' with opponents. The goal is to manoeuvre them into weakness while building up one's own shi, or strategic position." Kissinger considers the "manoeuvring" approach an ideal, but one that ran in contrast to the conflicts of the Qin dynasty.[200]
#### Shen Dao on shi
Searching out the causes of disorder, Shen Dao observed splits in the ruler's authority.[94]:122 Shen Dao's theory on power echoes Shen Buhai, referenced by Xun Kuang as its originator, who says "He who (can become) singular decision-maker can become the sovereign of All under Heaven."[117]:268[201][125] Shen Dao's theory may otherwise have been borrowed from the Book of Lord Shang.[13]:93[202]
For Shen Dao, "Power" ( shi) refers to the ability to compel compliance; it requires no support from the subjects, though it does not preclude this.[197] (shi's) merit is that it prevents people from fighting each other; political authority is justified and essential on this basis.[203] Shen Dao says: "When All under Heaven lacks the single esteemed [person], then there is no way to carry out the principles [of orderly government, li ]. ... Hence the Son of Heaven is established for the sake of All under Heaven ... All under Heaven is not established for the sake of the Son of Heaven ..."[125]
Talent cannot be displayed without power.[204] Shen Dao says: "The flying dragon rides on the clouds and the rising serpent wanders in the mists. But when the clouds disperse and the mists clear up, the dragon and the serpent become the same as the earthworm and the large winged black ant because they have lost what they ride."[197] Leadership is not a function of ability or merit, but is given by some process, such as giving a leader to a group.[205] "The ruler of a state is enthroned for the sake of the state; the state is not established for the sake of the prince. Officials are installed for the sake of their offices; offices are not established for the sake of officials ..."[83][198]
While moral capability is usually disregarded by the Fajia, Shen Dao considers it useful in terms of authority. If the ruler is inferior but his command is practiced, it is because he is able to get support from people.[197] But his ideas otherwise constitute a "direct challenge" to Confucian virtue.[206] Virtue is unreliable because people have different capacities. Both morality together with intellectual capability are insufficient to rule, while position of authority is enough to attain influence and subdue the worthy, making virtue "not worth going after".[207][208][149]:174
#### Han Fei on Shi
Like Shen Dao, Han Fei seems to admit that virtue or charisma can have persuasive power even in his own time.[209] However, he considers virtue instrumental, and Wu-wei, or nonaction, as its essence.[210] Furthermore, he criticizes virtue as insufficient; power should be amassed through "laws" (fa),[202] and unlike Shen considers government by moral persuasion and government by power (shi) mutually incompatible.[197] The ruler's authority (shi) should depend neither on his own personal qualities or cultivation, or even upon Shen Dao's position or power, but on Fa (law or checks and balances), a more vital source for his authority. Shang Yang and Han Fei's rejection of charisma (shi) as ineffective underwrite their rejection of the Confucian ruler.[35]:366[149]:170,181 Han Fei does stress that the leader has to occupy a position of substantial power before he is able to use these or command followers. Competence or moral standing do not allow command.[204]
For Han Fei, in order to actually influence, manipulate or control others in an organization and attain organizational goals it is necessary to utilize tactics (shu), regulation (fa), and rewards and punishment – the "two handles".[211][212] Reward and punishment determine social positions – the right to appoint and dismiss. In line with Shi, these should never be relegated. The ruler must be the sole dispenser of honors and penalties. If these are delegated to the smallest degree, and people are appointed on the basis of reputation or worldly knowledge, then rivals will emerge and the ruler's power will fall to opinion and cliques (the ministers). Allowing him to prevent collapse by combating or resolving ministerial disagreements and ambitions, the rule's exclusive authority outweighs all other considerations, and Han Fei requires that the ruler punish disobedient ministers even if the results of their actions were successful.[193][212][213] Goods may not be considered meaningful outside of his control.[6]
## Han Fei (280–233 BC)
Han Fei's theory is more interested in self-preservation than formulating any general theory of the state.[214] Sinologist Daniel Bell considers Han Fei's work a "political handbook for power-hungry rulers... (arguing that) political leaders should act like rational sociopaths" with "total-state control" strengthened by rewards and punishments.[215][216]
Han Fei nonetheless inheres to the tradition of Fa, considering coherent discourse essential for the functioning of the state.[35]:366[55] Han Fei's analysis of the problem of rulership is that "people naturally incline to private interpretation" (Chad Hansen).[55] Differentiating his theory from that of the Confucians through the objectivity and accessibility of Fa,[35]:352 he considers measurement (Fa) the only justification for adopting an explicit code, rather than leaving matters to tradition.[55] As with Shen Buhai and most of the School of Names he takes the congruence between name and reality as a primary goal.[217]
Public, measurement-like standards for applying names (administrative standards or job contracts) can "plausibly make it hard for clever ministers to lie, (or) for glib talkers to take people (or the ruler) in with sophistries ... [They make it possible to] correct the faults of superiors, expose error, check excess, and unify standards ... Laws, by themselves, cannot prevent the ruler from being fooled or deceived. The ruler needs Fa." Han Fei's arguments for "rule by law" (Fa) would not have as much persuasive power as they do if not for Fa, without which its objectives cannot be achieved.[55][35]:367 He rejects Confucian Li, scholarly interpretation and opinion, worldly knowledge, and reputation: models must be measured, dissolving behaviour and disputes of distinction into practical application.[35]:366
Considering politics the only means of preserving the power of the state,[218] he emphasizes standards (Fa), preventing disputes in language or knowledge, as the ruler's only protection.[35]:366 Providing reward and penalty automatically, Fa strictly defines state functions through binding, general rules, removing from discussion what would otherwise only be opinion, and preventing conflicts of competencies, undue powers or profits. To this end, Han Fei's high officials focus solely on definition through calculation and the construction of objective models, judged solely by effectiveness.[218]
### Wu wei
Devoting the entirety of Chapter 14, "How to Love the Ministers", to "persuading the ruler to be ruthless to his ministers", Han Fei's enlightened ruler strikes terror into his ministers by doing nothing (wu wei). The qualities of a ruler, his "mental power, moral excellence and physical prowess" are irrelevant. He discards his private reason and morality, and shows no personal feelings. What is important is his method of government. Fa (administrative standards) require no perfection on the part of the ruler.[219]
Han Fei's use of Wu-Wei may have been derivative of Taoism, but its Tao emphasizes autocracy ("Tao does not identify with anything but itself, the ruler does not identify with the ministers"). Sinologists like Randall P. Peerenboom argue that Han Fei's Shu (technique) is arguably more of a "practical principle of political control" than any state of mind.[103][57] Han Fei nonetheless begins by advising the ruler to remain "empty and still".
Tao is the beginning of the myriad things, the standard of right and wrong. That being so, the intelligent ruler, by holding to the beginning, knows the source of everything, and, by keeping to the standard, knows the origin of good and evil. Therefore, by virtue of resting empty and reposed, he waits for the course of nature to enforce itself so that all names will be defined of themselves and all affairs will be settled of themselves. Empty, he knows the essence of fullness: reposed, he becomes the corrector of motion. Who utters a word creates himself a name; who has an affair creates himself a form. Compare forms and names and see if they are identical. Then the ruler will find nothing to worry about as everything is reduced to its reality.
Tao exists in invisibility; its function, in unintelligibility. Be empty and reposed and have nothing to do-Then from the dark see defects in the light. See but never be seen. Hear but never be heard. Know but never be known. If you hear any word uttered, do not change it nor move it but compare it with the deed and see if word and deed coincide with each other. Place every official with a censor. Do not let them speak to each other. Then everything will be exerted to the utmost. Cover tracks and conceal sources. Then the ministers cannot trace origins. Leave your wisdom and cease your ability. Then your subordinates cannot guess at your limitations.
The bright ruler is undifferentiated and quiescent in waiting, causing names (roles) to define themselves and affairs to fix themselves. If he is undifferentiated then he can understand when actuality is pure, and if he is quiescent then he can understand when movement is correct.[220][221][222][223][110]:186–187[224]
Han Fei's commentary on the Tao Te Ching asserts that perspectiveless knowledge – an absolute point of view – is possible, though the chapter may have been one of his earlier writings.[35]:371
### Performance and title (Xing-Ming)
"If one has regulations based on objective standards and criteria and apply these to the mass of ministers, then that ruler cannot be duped by cunning fraudulence."[35]:367 —Han Fei
Han Fei was notoriously focused on what he termed Xing-Ming (Chinese: 刑名; pinyin: xíngmíng),[35]:349 which Sima Qian and Liu Xiang define as "holding actual outcome accountable to Ming (speech)".[2]:87,104[108] In line with both the Confucian and Mohist rectification of names,[35]:365 it is relatable to the Confucian tradition in which a promise or undertaking, especially in relation to a government aim, entails punishment or reward,[35]:349 though the tight, centralized control emphasized by both his philosophy and that of his predecessor Shen Buhai's conflicts with the Confucian idea of the autonomous minister.[2]:83
Possibly referring to the drafting and imposition of laws and standardized legal terms, Xing-Ming may originally have meant "punishments and names", but with the emphasis on the latter.[225] It functions through binding declarations (Ming), like a legal contract. Verbally committing oneself, a candidate is allotted a job, indebting him to the ruler.[226][85] "Naming" people to (objectively determined) positions, it rewards or punished according to the proposed job description and whether the results fit the task entrusted by their word, which a real minister fulfils.[117]:284[35]:365
Han Fei insists on the perfect congruence between words and deeds. Fitting the name is more important than results.[117]:284 The completion, achievement, or result of a job is its assumption of a fixed form (xing), which can then be used as a standard against the original claim (ming).[227] A large claim but a small achievement is inappropriate to the original verbal undertaking, while a larger achievement takes credit by overstepping the bounds of office.[228]
Han Fei's "brilliant ruler" "orders names to name themselves and affairs to settle themselves".[228]
If the ruler wishes to bring an end to treachery then he examines into the congruence of the congruence of hsing (form/standard) and claim. This means to ascertain if words differ from the job. A minister sets forth his words and on the basis of his words, the ruler assigns him a job. Then the ruler holds the minister accountable for the achievement which is based solely on his job. If the achievement fits his job, and the job fits his words, then he is rewarded. If the achievement does not fit his jobs and the job does not fit his words, then he will be punished.[228][227][35]:365[117]:284
Assessing the accountability of his words to his deeds,[229] the ruler attempts to "determine rewards and punishments in accordance with a subject's true merit" (using Fa).[109][229][35]:308,349[13]:81[230] It is said that using names (ming) to demand realities (shi) exalts superiors and curbs inferiors,[231] provides a check on the discharge of duties, and naturally results in emphasizing the high position of superiors, compelling subordinates to act in the manner of the latter.[2]:86[144]
Han Fei considers Xing-Ming an essential element of autocracy, saying that "In the way of assuming Oneness names are of first importance. When names are put in order, things become settled down; when they go awry, things become unfixed."[185] He emphasizes that through this system, initially developed by Shen Buhai, uniformity of language could be developed,[73] functions could be strictly defined to prevent conflict and corruption, and objective rules (Fa) impervious to divergent interpretation could be established, judged solely by their effectiveness.[84] By narrowing down the options to exactly one, discussions on the "right way of government" could be eliminated. Whatever the situation (shi) brings is the correct Dao.[35]:367,370–372
Though recommending use of Shen Buhai's techniques, Han Fei's Xing-Ming is both considerably narrower and more specific. The functional dichotomy implied in Han Fei's mechanistic accountability is not readily implied in Shen's, and might be said to be more in line with the later thought of the Han dynasty linguist Xu Gan than that of either Shen Buhai or his supposed teacher Xun Kuang.[232]
#### The "Two Handles"
Mythical White Tiger. Qin Shi Huang was called the "Tiger of Qin".
Supposing the tiger cast aside its claws and fangs and let the dog use them, the tiger would, in turn, be subjected by the dog. Han Fei Zi
A modern statue of the First Emperor and his attendants on horseback
The two August Lords of high antiquity grasped the handles of the Way and so were established in the center. Their spirits mysteriously roamed together with all transformations and thereby pacified the four directions. Huainanzi
Though not entirely accurately, most Han works identify Shang Yang with penal law.[2]:105 Its discussion of bureaucratic control is simplistic, chiefly advocating punishment and reward.[31]:59 Shang Yang was largely unconcerned with the organization of the bureaucracy apart from this.[2]:100,102 The use of these "two handles" (punishment and reward) nonetheless forms a primary premise of Han Fei's administrative theory.[233] However, he includes it under his theory of Shu in connection with Xing-Ming.[35]:367[55]
As a matter of illustration, if the "keeper of the hat" lays a robe on the sleeping Emperor, he has to be put to death for overstepping his office, while the "keeper of the robe" has to be put to death for failing to do his duty.[46] The philosophy of the "Two Handles" likens the ruler to the tiger or leopard, which "overpowers other animals by its sharp teeth and claws"(rewards and punishments). Without them he is like any other man; his existence depends upon them. To "avoid any possibility of usurpation by his ministers", power and the "handles of the law" must "not be shared or divided", concentrating them in the ruler exclusively.
In practice, this means that the ruler must be isolated from his ministers. The elevation of ministers endangers the ruler, with which he must be kept strictly apart. Punishment confirms his sovereignty; law eliminates anyone who oversteps his boundary, regardless of intention. Law "aims at abolishing the selfish element in man and the maintenance of public order", making the people responsible for their actions.[219]
Han Fei's rare appeal (among Legalists) to the use of scholars (law and method specialists) makes him comparable to the Confucians, in that sense. The ruler cannot inspect all officials himself, and must rely on the decentralized (but faithful) application of laws and methods (fa). Contrary to Shen Buhai and his own rhetoric, Han Fei insists that loyal ministers (like Guan Zhong, Shang Yang, and Wu Qi) exist, and upon their elevation with maximum authority. Though Fajia sought to enhance the power of the ruler, this scheme effectively neutralizes him, reducing his role to the maintenance of the system of reward and punishments, determined according to impartial methods and enacted by specialists expected to protect him through their usage thereof.[234][235] Combining Shen Buhai's methods with Shang Yang's insurance mechanisms, Han Fei's ruler simply employs anyone offering their services.[84]
## Enlightened absolutism
Even if the Fajia were not ardent absolutists (and Han Fei believed that most rulers would be average), they would never dream of openly challenging absolutism, and its methods are presented as empowering the ruler. Han Fei's doctrine, however, challenges its absolutist premise out of its own mouth. In order for its administration to function, the ruler must act as a cog in its operation, and that alone. The operation of Fa implies non-interference not only in its application, but also in its development, determined through method.
Sinologist Xuezhi Guo contrasts the Confucian "Humane ruler" with the Legalists as "intending to create a truly 'enlightened ruler'". He quotes Benjamin I. Schwartz as describing the features of a truly Legalist "enlightened ruler":[236]
He must be anything but an arbitrary despot if one means by a despot a tyrant who follows all his impulses, whims and passions. Once the systems which maintain the entire structure are in place, he must not interfere with their operation. He may use the entire system as a means to the achievement of his national and international ambitions, but to do so he must not disrupt its impersonal workings. He must at all times be able to maintain an iron wall between his private life and public role. Concubines, friends, flatterers and charismatic saints must have no influence whatsoever on the course of policy, and he must never relax his suspicions of the motives of those who surround him.[237][238]
As easily as mediocre carpenters can draw circles by employing a compass, anyone can employ the system Han Fei envisions.[239] The enlightened ruler restricts his desires and refrains from displays of personal ability or input in policy. Capability is not dismissed, but the ability to use talent will allow the ruler greater power if he can utilize others with the given expertise.[240] Laws and regulations allow him to utilize his power to the utmost. Adhering unwaveringly to legal and institutional arrangements, the average monarch is numinous.[241][242] A.C. Graham writes:
[Han Fei's] ruler, empty of thoughts, desires, partialities of his own, concerned with nothing in the situation but the 'facts', selects his ministers by objectively comparing their abilities with the demands of the offices. Inactive, doing nothing, he awaits their proposals, compares the project with the results, and rewards or punishes. His own knowledge, ability, moral worth, warrior spirit, such as they may be, are wholly irrelevant; he simply performs his function in the impersonal mechanism of the state.[117]:288
Resting empty, the ruler simply checks "shapes" against "names" and dispenses rewards and punishments accordingly, concretizing the Tao ("path") of Laozi into standards for right and wrong.[243][244] Submerged by the system he supposedly runs, the alleged despot disappears from the scene.[245]
## Later history
### Fall
A modern marble statue of the first Emperor of China, Qin Shi Huang
Juyong Pass
Guided by Legalist thought, the First Qin Emperor Qin Shi Huang conquered and unified the China's warring states into thirty-six administrative provinces, under what is commonly thought of as the first Chinese empire, the Qin dynasty. The Qin document "On the Way of Being an Official" proclaims the ideal official as a responsive conduit, transmitting the facts of his locale to the court, and its orders, without interposing his own will or ideas. It charges the official to obey his superiors, limit his desires, and to build roads to smooth the transmitting of directives from the center without modification. It praises loyalty, absence of bias, deference, and the appraisal of facts.[59]
The intrastate realpolitik would end up devouring the philosophers themselves. Holding that if punishments were heavy and the law equally applied, neither the powerful nor the weak would be able to escape consequences, Shang Yang advocated the state's right to punish even the ruler's tutor, and ran afoul of the future King Huiwen of Qin (c. 338–311 BC). Whereas at one point, Shang Yang had the power to exile his opponents (and, thus, eviscerate individual criticism) to border regions of the state, he was captured by a law he had introduced and died being torn into pieces by chariots. Similarly, Han Fei would end up being poisoned by his envious former classmate Li Si, who in turn would be killed (under the law he had introduced) by the aggressive and violent Second Qin Emperor that he had helped to take the throne.
As recorded in the Shiji and Book of Han, the Han dynasty took over the governmental institutions of the Qin dynasty almost unchanged,[2]:105 but in its early decades it was not a centralized state, parcelling out the country to a number of relatives, who as vassal kings who ruled with full authority.[2]:107 The reputation of Legalism suffered from its association with the former Qin dynasty. Sima Tan, though hailing the Fa "school" for "honouring rulers and derogating subjects, and clearly distinguishing offices so that no one can overstep [his responsibilities]", criticized the Legalist approach as "a one-time policy that could not be constantly applied".[246] Though different philosophically, the pairing of figures like Shen Buhai and Shang Yang along with Han Fei became common in the early Han dynasty, Sima Tan glossing the three as Fa Jia and his son as adherents of "xing ming" ("performance and title").[2]:90[247]
The syncretic Han Dynasty text, the Huainanzi writes that
On behalf of the Ch'in, Lord Shang instituted the mutual guarantee laws, and the hundred surnames were resentful. On behalf of Ch'u, Wu Ch'i issued an order to reduce the nobility and their emoluments, and the meritorious ministers revolted. Lord Shang, in establishing laws, and Wu Ch'i, in employing the army, were the best in the world. But Lord Shang's laws [eventually] caused the loss of Ch'in for he was perspicacious about the traces of the brush and knife, but did not know the foundation of order and disorder. Wu Ch'i, on account of the military, weakened Ch'u. He was well practiced in such military affairs as deploying formations, but did not know the balance of authority involved in court warfare.[248]
Usually referring to Warring States period philosophers, during the Han Fajia would be used for others disliked by the Confucian orthodoxy, like the otherwise Confucianistic reformers Guan Zhong and Xunzi,[249] and the Huang-Lao Taoists.[250]
### Later influences (Xing-Ming)
The Shiji records Li Si as repeatedly recommending "supervising and holding responsible", which he attributed to Shen Buhai.[2]:83 A stele set up by Qin Shi Huang memorializes him as a sage that, taking charge of the government, established Xing-Ming.[2]:105,112,114
In the early Han dynasty, Sima Tan's Taoist syncretism almost unmistakably uses the same sort of technique as Shen Buhai, saying:
When the congregation of ministers has assembled, the ruler lets each do as he will (zi ming). If result coincides with claim, this is known as 'upright'; if it does not, this is known as 'hollow'.[251]
The Huang–Lao text Jing fa says
The right way to understand all these (things) is to remain in a state of [vacuity,] formlessness and non-being. Only if one remains in such a state, may he thereby know that (all things) necessarily possess their forms and names as soon as they come into existence, even though they are as small as autumn down. As soon as forms and names are established, the distinction between black and white becomes manifest ... there will be no way to escape from them without a trace or to hide them from regulation ... [all things] will correct themselves.[252]
The Shiji states that Emperor Wen of Han was "basically fond of Xing-Ming". Jia Yi advised Wen to teach his heir to use Shen Buhai's method, so as to be able to "supervise the functions of the many officials and understand the usages of government". Pressure groups saw Jia Yi's dismissal, but was brought back to criticize the government. Two advisors to Wen's heir, Emperor Jing of Han were students of Xing-Ming, one passing the highest grade of examination, and admonished Jing for not using it on the feudal lords.[2]:87,103,106–107,115 [253]
By the time of the civil service examination was put into place, Confucian influence saw outright discussion of Shen Buhai banned. Xing-Ming is not discussed by Imperial University's promoter, the famous Confucian Dong Zhongshu. However, the Emperor under which it was founded, Emperor Wu of Han, was both familiar with and favorable to Legalist ideas, and the civil service examination did not come into existence until its support by Gongsun Hong, who did write a book on Xing-Ming.[2]:86–87,115 The Emperor Xuan of Han was still said by Liu Xiang to have been fond of reading Shen Buhai, using Xing-Ming to control his subordinates and devoting much time to legal cases.[2]:87[254][255]
Regarded as being in opposition to Confucians, as early as the Eastern Han its full and original meaning would be forgotten.[2]:80[172][153] Yet the writings of Dong Zhongshu discuss personnel testing and control in a manner sometimes hardly distinguishable from the Han Feizi. Like Shen Buhai, he dissuades against reliance upon punishments. As Confucianism ascended the term disappeared,[2]:90[256] but appears again in later dynasties.
The Yongzheng Emperor of the Qing dynasty was said by to "xunming zishe", or "demand performance in accordance with title", a near-verbatim usage of the Han Feizi.[2]:89
### Imperial China
#### Han dynasty
The administration and political theory developed during the formative Warring States period would still influence every dynasty thereafter, as well as the Confucian philosophy that underlay Chinese political and juridical institutions.[257] The influence of the Fajia on Han Confucianism is very apparent, adopting Han Fei's emphasis of a supreme ruler and authoritarian system rather than Mencius's devaluation thereof, or Xun Kuang's emphasis on the Tao.[258]
Shen Buhai's book appears to have been widely studied at the beginning of the Han era.[15]:35 As protégé of a Han Dynasty Commandant of Justice that had studied under Li Si, Jia Yi was a student of Shen Buhai through them.[259] Jia describes Shen Buhai's Shu as a particular method of applying the Tao, or virtue, bringing together Confucian and Taoist discourses. He uses the imagery of the Zhuangzi of the knife and hatchet as examples of skillful technique in both virtue and force, saying "benevolence, righteousness, kindness and generosity are the ruler's sharp knife. Power, purchase, law and regulation are his axe and hatchet."[260] His writings blame the fall of the Qin dynasty simply on the education of the second emperor.[261] He would draw up elaborate plans for reorganizing the bureaucracy, which Emperor Wen of Han put into effect.
Shen Buhai never attempts to articulate natural or ethical foundations for his Fa (administrative method), nor does he provide any metaphysical grounds for his method of appointment (later termed "xing-ming"),[262][263] but later texts do. The Huang-Lao work Boshu grounds fa and xing-ming in the Taoist Dao.[262]
The Discourses on Salt and Iron's Lord Grand Secretary uses Shang Yang in his argument against the dispersion of the people, stating that "a Sage cannot order things as he wishes in an age of anarchy". He recalls Lord Shang's chancellery as firm in establishing laws and creating orderly government and education, resulting in profit and victory in every battle.[264] Although Confucianism was promoted by the new emperors, the government continued to be run by Legalists. Emperor Wu of Han (140–87 BC) barred Legalist scholars from official positions and established a university for the study of the Confucian classics,[265] but his policies and his most trusted advisers were Legalist.[266] Michael Loewe called the reign of Emperor Wu the "high point" of Modernist (classically justified Legalist) policies, looking back to "adapt ideas from the pre-Han period".[267] An official ideology cloaking Legalist practice with Confucian rhetoric would endure throughout the imperial period, a tradition commonly described as wàirú nèifǎ (Chinese: 外儒內法; literally: 'exteriorly Confucian, interiorly Legalist').[268]
It became commonplace to adapt Legalist theories to the Han state by justifying them using the classics, or combining them with the notion of the "way" or "pattern of the cosmos" ("The Way gave birth to law" Huangdi Sijing). Some scholars "mourn" the lack of pure examples of Taoism, Confucianism and Legalism in the Han dynasty more generally.[269] Han sources would nonetheless come to "treat Legalism as an alternative to the methods of the Classicists".[270] During the decay of the Han Dynasty, many scholars again took up an interest in Legalism, Taoism and even Mohism,[271] and a number of Confucians took up "Legalist" methods to combat the growing disregard for law.[121]
### Post-Han
The Records of the Three Kingdoms describes Cao Cao as a hero who "devised and implemented strategies, lorded the world over, wielded skillfully the law and political technique of Shen Buhai and Shang Yang, and unified the ingenious strategies of Han Fei".[176] Zhuge Liang also attached great importance to the works of Shen Buhai and Han Fei.[272][2]:112 The tendency toward Legalism is apparent in intellectual circles toward the end of the Han dynasty, and would be reinforced by Cao Wei. Dispossessed peasants were organized into paramilitary agricultural colonies to increase food production for the army, and penal legislation increased. These policies would be followed by the Northern Wei.[273]
Emperor Wen of Sui is recorded as having withdrawn his favour from the Confucians, giving it to "the group advocating Xing-Ming and authoritarian government".[2]:112 But Wen might be said to have already been steeped in a Legalist tradition followed by the aristocratic institutions of the northern dynasties, who concerned themselves with functional organization and social hierarchy. The Sui dynasty and Tang dynasty were largely based upon the Western Wei and Northern Zhou, refining pre-existing institutions and taking measures against the aristocracy.[274]
Quoting Arthur Wright, Author Hengy Chye Kiang calls the Sui dynasty a "strong autocratic power with a penchant for Legalist philosophy", and its prime minister Gao Jiong "a man of practical statecraft" recalling the great Legalist statesmen.[274][275] His influence saw the replacement of Confucians with officials of "Legalist" outlook favoring centralization.[274]
Under Legalist influence, Li Gou and Wang Anshi emphasised seeking profit for the people.[276] Deng Guangming argued that Wang Anshi was influenced by Warring States-era Legalism, with his emphasis on "enriching the state and strengthening the army" and Legalist ideas of law.[277] His baojia system which survived until the end of Imperial China has been described as a Legalist device.[278]
### Ming dynasty
Li Shanchang (1314–1390), a founding Prime Minister of the Ming dynasty, studied Chinese Legalism. It is said that Li was the Emperor Hongwu's closest comrade during the war, and greatest contributor to his ultimate victory and thus establishment of the Ming Dynasty.[279] Deeply trusted by the Emperor,[280] Hongwu consulted Li on institutional matters.[281] Li planned the organization of the "six ministries" and shared in the drafting of a new law code. He established salt and tea monopolies based on Yuan institutions, eliminated corruption, restored minted currency, opened iron foundries, and instituted fish taxes. It is said that revenues were sufficient, yet the people were not oppressed.[282] Most of his other activities seem to have supported Hongwu Emperor's firm control of his regime. Mainly responsible for ferreting out disloyalty and factionalism among military officers, he used a reward and punishment system reminiscent of the Han Feizi, and may have had a kind of secret police in his service. At times he had charge of all civil and military officials in Nanking.[282][283]
In 1572 Zhang Juzheng, a legalistic, prime-minister like figure of the Ming Dynasty, had the young emperor of the time issue a warning edict against China's bureaucracy with the reference that they had abandoned the public interest for their own private interests. It reads: "From now on, you will be pure in your hearts and scrupulous in your work. You will not harbor private designs and deceive your sovereign ... You will not complicate debates and disconcert the government." It suggests that good government will prevail as long as top ministers were resolute in administration of the empire and minor officials were selflessly devoted to the public good. It is said that the officials became "very guarded and circumspect" following its release. His "On Equalizing Taxes and Succoring the People" postulated that the partiality of local officials toward powerful local interests was responsible for abuses in tax collection, hurting both the common people and the Ming state.[284]
Zhang Juzheng wrote that "it is not difficult to erect laws, but it is difficult to see they are enforced". His Regulation for Evaluating Achievements (kao cheng fa) assigned time limits for following government directives and made officials responsible for any lapses, enabling Zhang to monitor bureaucratic efficiency and direct a more centralized administration. That the rules were not ignored are a testament to his basic success.[285]
### Modern
In the nineteenth century, Shang Yang's slogan of "rich country, strong army" was reinvoked in Japan as a "formal ideological foundation of industrial and technological development".[25] In the midst of the decline of the Qing dynasty in the late-19th century, understanding of Confucianism was transformed, turning towards practicality (the School of Practical Statecraft, substantial learning). For some reformist scholars the focus on Confucianism was eroded in favour of Legalist principles of bian-fa (state reform), fu-qiang (state wealth and power) and even shang-zhan (economic warfare). Albert Feuerwerker argues that this Legalist raison d'etat ultimately was connected to the reform proposals of the 1890s, such as the Hundred Days Reform, and thence the New Policies of the early twentieth century. Western science was integrated into the Confucian worldview as an interpretation and application of Confucian principles.
Legalism was partly rehabilitated in the twentieth century by new generations of intellectuals. One, Mai Menghua (1874–1915), promulgated interest in Shang Yang's thought, comparing Shang Yang's view of history with the evolutionary ideas of western theorists. The New Culture Movement leader, Hu Shi (1891–1962), hailed Han Fei and Li Si for their "brave spirit of opposing those who 'do not make the present into their teacher but learn from the past'". Kuomintang leader Hu Hanmin (1879–1936) wrote the preface to a new edition of the Book of Lord Shang.[286]
Because Fajia ignored differences among subjects,[4]:15 early twentieth century Chinese scholarship often viewed it within the context of Western "rule of law".[11] One 1922 article, "The Antiquity of Chinese Law", attributes three legal theories to Han Fei, and referred to him as a "jurist".[287] From the 1920s on it was viewed as being in a historical struggle with the Confucian "rule of men".[4]:15
The early Mao Zedong has been described as a "dyed-in-the-wool" Legalist or "Lord Shang-style 'sage ruler', who defined the law according to revolutionary needs".[288] Communist intellectuals used the Fajia in their criticism of Confucianism, describing the conflict between the two as class struggle.[289] In 1950, the PRC combined law with campaigns against political enemies,[290] and appeals to the Fajia for solutions became common after the Great Leap Forward.[129] Fazhi, another historical term for "Legalism", would be used to refer to both socialist legality and Western rule of law. Still contrasted with renzhi (or rule of persons), most Chinese wanted to see it implemented in China.[291]
Two decades of reform, the Soviet Union's collapse and a financial crisis in the 1990s only served to increase its importance, and the 1999 constitution was amended to "provide for the establishment of a socialist rule-of-law state", aimed at increasing professionalism in the justice system. Signs and flyers urged citizens to uphold the rule of law. In the following years, figures like Pan Wei, a prominent Beijing political scientist, would advocate for a consultative rule of law with a redefined role for the party and limited freedoms for speech, press, assembly and association.[292]
Xingzhong Yu, Professor at Cornell University, describes the PRC through a framework of "State Legalism".[293]
Legalist discourse is seeing a resurgence during the leadership of Xi Jinping, who is the General Secretary of the Chinese Communist Party, with journalists reporting on his fondness for the Chinese classics, alongside Confucianism including Legalist writers and in particular Han Fei, both of which Xi sees as relevant.[294][295][296] Han Fei gained new prominence with favourable citations. One sentence of Han Fei's that Xi quoted appeared thousands of times in official Chinese media at the local, provincial, and national levels.[297] A key phrase of Xi's reforms is "govern the state according to law" (simplified Chinese: 依法治国; traditional Chinese: 依法治國; pinyin: yī fǎ zhì guó), focusing on enforcing discipline on party and government officials first.[33]
#### As Realists
In 1939 Arthur Waley contrasted the Fajia as "Realists": the Realists, he says, largely ignored the individual, holding that the object of any society is to dominate other societies.[37] In this vein, in his 1989 book "Disputers of the Tao" Angus Charles Graham titled his "Legalist" chapter "Legalism: an Amoral Science of Statecraft", sketching the fundamentals of an "amoral science" in Chinese thought largely based on the Han Feizi, consisting of "adapting institutions to changing situations and overruling precedent where necessary; concentrating power in the hands of the ruler; and, above all, maintaining control of the factious bureaucracy".[117]:267[298]
In 2003, Ross Terrill writes that "Chinese Legalism is as Western as Thomas Hobbes, as modern as Hu Jintao. It speaks the universal and timeless language of law and order. The past does not matter, state power is to be maximized, politics has nothing to do with morality, intellectual endeavour is suspect, violence is indispensable, and little is to be expected from the rank and file except an appreciation of force." He calls Legalism the "iron scaffolding of the Chinese Empire", but emphasizes the marriage between Legalism and Confucianism.[299]
In 2005, Chinese law expert Randall Peerenboom compares Han Fei with the accepted standards of legal positivism, and concludes that he is a legal positivist. Establishing the ruler as the ultimate authority over the law, he also "shares the belief that morality and the law need not coincide".[300]
In China the same year, Liang Zhiping theorized that law initially emerged in China as an instrument by which a single clan exercised control over rival clans.[301] In the earlier Spring and Autumn period, a Qin king is recorded as having memorialized on punishment as a ritual function benefiting the people, saying, "I am the little son: respectfully, respectfully I obey and adhere to the shining virtuous power, brightly spread the clear punishments, gravely and reverentially perform my sacrifices to receive manifold blessings. I regulate and harmonize myriad people, gravely from early morning to evening, valorous, valorous, awesome, awesome – the myriad clans are truly disciplined! I completely shield the hundred nobles and the hereditary officers. Staunch, staunch in my civilizing and martial [power], I calm and silence those who do not come to the court [audience]. I mollify and order the hundred states to have them strictly serve the Qin."[302]
The Fajia are often still compared in the west to Machiavellianism.[303][31]:59[35]:308 The Oxford Handbook of World Philosophy paints the Legalists as Realists, stating that "What linked these men is that all were theorists or practitioners of a realistic amoral brand of statecraft aimed at consolidating and strengthening the power and wealth of the state and its autocratic ruler. Their thought was realistic in being premised on what they took to be brute facts about how people actually behave ... It was amoral in that they were utterly unconcerned with whether the institutions and methods they advocated were morally justified."[31]:59 Yuri Pines (2014) terms them as "political realists who sought to attain a 'rich state and a powerful army'(Shang Yang) and to ensure domestic stability."[304]
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164. Makeham, J. (1990) pp. 91–92. The Legalist Concept of Hsing-Ming: An Example of the Contribution of Archaeological Evidence to the Re-Interpretation of Transmitted Texts. Monumenta Serica, 39, 87–114. JSTOR 40726902
165. Makeham, J. (1990) pp. 87, 89. The Legalist Concept of Hsing-Ming: An Example of the Contribution of Archaeological Evidence to the Re-Interpretation of Transmitted Texts. Monumenta Serica, 39, 87–114. JSTOR 40726902
166. Burton Watson. Han Feizi http://www2.hawaii.edu/~freeman/courses/phil301/13.%20Han%20Feizi.pdf
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199. Shen Dao's Own Voice, 2011. pp. 200,202. Springer Science+Business Media B.V. 2011
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205. Eric L. Hutton 2008. p. 442 Han Feizi's Criticism of Confucianism and its Implications for Virtue Ethics. http://hutton.philosophy.utah.edu/HFZ.pdf
206. Han Fei, De, Welfare. Schneider, Henrique. Asian Philosophy. Aug2013, Vol. 23 Issue 3, p266,269. 15p. DOI: 10.1080/09552367.2013.807584., Database: Academic Search Elite
207. Chen, Chao Chuan and Yueh-Ting Lee 2008 pp114,146. Leadership and Management in China
208. Yuri Pines 2003 p. 76 Submerged by Absolute Power
209. Pines, Yuri, "Legalism in Chinese Philosophy", The Stanford Encyclopedia of Philosophy (Winter 2014 Edition), Edward N. Zalta (ed.), 5.1 The Ruler's Superiority http://plato.stanford.edu/archives/win2014/entries/chinese-legalism/
211. Daniel Bell (2015). The China Model p.115
212. Daniel Bell RE: "For more nuanced interpretations of Han Fei’s thought, see Paul R. Goldin, ed., Dao Companion to the Philosophy of Han Fei (Dordrecht: Springer, 2013); and the articles in the special issue “Legalist Philosophy of Han Fei,” Journal of Chinese Philosophy 38, no. 1 (Mar. 2011)."
213. Makeham, J. (1990) p. 112. The Legalist Concept of Hsing-Ming: An Example of the Contribution of Archaeological Evidence to the Re-Interpretation of Transmitted Texts. Monumenta Serica, 39, 87–114. JSTOR 40726902
214. Jacques Gernet 1982 p. 91. A History of Chinese Civilization. https://books.google.com/books?id=jqb7L-pKCV8C&pg=PA90
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216. HanFei, “The Way of the Ruler", Watson, p. 16
217. Han Fei-tzu, chapter 5 [Han Fei-tzu chi-chieh 1), p. 18; cf. Burton Watson, Han Fei Tzu: Basic Writings (New York: Columbia U.P., 1964)
218. MARK CSIKSZENTMIHALYI. Chia I's "Techniques of the Tao". Asia Major, Third Series, Vol. 10, No. 1/2 (1997), pp. 49–67 JSTOR 41645528
219. LIM XIAO WEI, GRACE 2005 p.18. LAW AND MORALITY IN THE HAN FEI ZI
220. Mark Edward Lewis, 1999 p. 33, Writing and Authority in Early China. https://books.google.com/books?id=8k4xn8CyHAQC&pg=PA33
221. Makeham, J. (1990) pp. 98, 100, 111. The Legalist Concept of Hsing-Ming: An Example of the Contribution of Archaeological Evidence to the Re-Interpretation of Transmitted Texts. Monumenta Serica, 39, 87–114. JSTOR 40726902
222. John Makeham 1994 p. 75. Name and Actuality in Early Chinese Thought. https://books.google.com/books?id=GId_ASbEI2YC&pg=PA75
223. Makeham, J. (1990) pp. 96, 98. The Legalist Concept of Hsing-Ming: An Example of the Contribution of Archaeological Evidence to the Re-Interpretation of Transmitted Texts. Monumenta Serica, 39, 87–114. JSTOR 40726902
224. Makeham, J. (1990) p. 90. The Legalist Concept of Hsing-Ming: An Example of the Contribution of Archaeological Evidence to the Re-Interpretation of Transmitted Texts. Monumenta Serica, 39, 87–114. JSTOR 40726902
226. Creel, 1959 p. 202. The Meaning of Hsing-Ming. Studia Serica: Sinological studies dedicated to Bernhard Kalgren
227. John Makeham 1994 p. 82. Name and Actuality in Early Chinese Thought. https://books.google.com/books?id=GId_ASbEI2YC&pg=PA82
228. Christian von Dehsen Christian von Dehsen, Philosophers and Religious Leaders https://books.google.com/books?id=XZrbAAAAQBAJ&pg=PT400
229. Yuri Pines 2003 pp. 77,83. Submerged by Absolute Power
230. (Chen Qiyou 2000: 2.6.107)
232. Xuezhi Guo p. 141, The Ideal Chinese Political Leader
233. Benjanmin I. Schwartz p. 345, The World of Thought in Ancient China
234. Eirik Lang Harris 2013 pp. 1,5 Constraining the Ruler
235. Chen, Chao Chuan and Yueh-Ting Lee 2008 p. 115. Leadership and Management in China
236. Yuri Pines 2003 pp. 78,81. Submerged by Absolute Power
237. Chen Qiyou 2000: 18.48.1049; 20.54.1176; 2.6.111; 17.45.998
238. (Graham 1989: 291)
239. Xing Lu 1998. Rhetoric in Ancient China, Fifth to Third Century, B.C.E.. p. 264. https://books.google.com/books?id=72QURrAppzkC&pg=PA258
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242. (Creel 1974: 140)
243. Ralph D Sawyer, 1993. The Seven Military Classics of Ancient China. Wu-Tzu p. 201
244. Rickett, Guanzi. p3 "The political writings are usually described as Legalist, but 'Realist' might make a description. For the most part, they tend to present a point of view much closer to that of the realistic Confucian, Xunzi than either the highly idealistic Confucianism of Mencius or the Draconian Legalism advocated by Shang Yang"
245. R. Eno (Spring 2010). "Huang-Lao Ideology". History G380 – class text readings. Indiana University. "When Sima Qian and other early historians discuss the intellectual trends of the early Han, they frequently refer to a school of thought known as 'Huang-Lao' ... As any quick survey of the texts will indicate, these documents are deeply syncretic, that is to say, they draw together selected ideas from many different schools and attempt to present them in a harmonious arrangement. Among these schools, Laozi-style Daoism is clearly foremost. However, Legalism and certain militarist schools contribute a very significant portion of these ideas as well. Mohist and Confucian influences can also be detected, but their contributions are generally scattered and do not shape the overall structure of the texts."
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270. Hengy Chye Kiang 1993. p. 82. The Development of Cityscapes in Medieval China
271. Antonio S. Cua (2013). Encyclopedia of Chinese Philosophy. Routledge. p. 363. ISBN 978-1135367480.
272. Patricia Buckley Ebrey, Paul Jakov Smith (2016). State Power in China, 900-1325. University of Washington Press. pp. 237–238. ISBN 978-0295998480.
273. DENIS TWITCHETT; JOHN K. FAIRBANK (1978). THE CAMBRIDGE HISTORY OF CHINA: Volume 10 Late Ch'ing, 1800-1911, Part 1. CAMBRIDGE UNIVERSITY PRESS. p. 29. ISBN 0-521-21447-5.
274. C. Simon Fan 2016. p. 94. Culture, Institution, and Development in China. https://books.google.com/books?id=cwq4CwAAQBAJ&pg=PA94
275. Anita M. Andrew, John A. Rapp 2000. p. 161. Autocracy and China's Rebel Founding Emperors. https://books.google.com/books?id=YQOhVb5Fbt4C&pg=PA161
276. Jiang Yonglin, Yonglin Jiang 2005. p. xxxiv. The Great Ming Code: Da Ming lü. https://books.google.com/books?id=h58hszAft5wC
277. Taylor, R. (1963) pp. 53–54. SOCIAL ORIGINS OF THE MING DYNASTY 1351–1360. Monumenta Serica, 22(1), 1–78. JSTOR 40726467
278. Edward L. Farmer 1995 p. 29. Zhu Yuanzhang and Early Ming Legislation. https://books.google.com/books?id=TCIjZ7l6TX8C&pg=PA29
279. H. Miller 2009 p. 27. State versus Gentry in Late Ming Dynasty China, 1572–1644
280. H. Miller 2009 p. 32. State versus Gentry in Late Ming Dynasty China, 1572–1644
281. Pines, Yuri, "Legalism in Chinese Philosophy", The Stanford Encyclopedia of Philosophy (Winter 2014 Edition), Edward N. Zalta (ed.), 7. Epilogue: Legalism in Chinese History http://plato.stanford.edu/archives/win2014/entries/chinese-legalism/
282. Leonard S Hsu, 1922 p. 165 China Review, Volume 3, https://books.google.com/books/content?id=2IpIAAAAYAAJ&pg=PA164&img=1&zoom=3&hl=en&sig=ACfU3U3zU6bDwPuxn_hVXYDLzPiPggLukQ&ci=448%2C628%2C161%2C18&edge=0
283. John Man 2008. p. 51. Terra Cotta Army.
284. Zhongying Cheng 1991 p. 311. New Dimensions of Confucian and Neo-Confucian Philosophy. https://books.google.com/books?id=zIFXyPMI51AC&pg=PA311
285. Pitman B. Potter. 2003. p. 145. From Leninist Discipline to Socialist Legalism. https://books.google.com/books?id=NxpNhkixdW0C&pg=PA145
286. Karen G. Turner pp. 1,24. The Limits of the Rule of Law in China.
287. Randall Peerenboom 2002 pp. ix, x,4. China's Long March Toward Rule of Law. https://books.google.com/books?id=RBZWNCpqSmMC=PA1
288. Theoretical Inquiries in Law 15.1 (2014). State Legalism and the Public/Private Divide in Chinese Legal Development. http://www7.tau.ac.il/ojs/index.php/til/article/viewFile/528/492
289. David K Schneider May/June 2016 p. 19. China's New Legalism
290. Samuli Seppänen 2016. p. 30. Ideological Conflict and the Rule of Law in Contemporary China. https://books.google.com/books?id=cRLFDQAAQBAJ&pg=PA30
291. Ryan Mitchell, The Diplomat. "Is 'China's Machiavelli' Now Its Most Important Political Philosopher?". The Diplomat.
293. Ross Terril 2003 pp. 68–69. The New Chinese Empire https://books.google.com/books?id=TKowRrrz5BIC&pg=PA68
294. Lim Xiao Wei, Grace 2005 p. 8. Law and Morality in the Han Fei Zi. http://www.scholarbank.nus.edu.sg/bitstream/handle/10635/18874/Law%20and%20Morality%20in%20the%20Han%20Fei%20Zi.pdf?sequence=1
295. Kenneth Winston. Singapore Journal of Legal Studies [2005] 313–347. The Internal Morality of Chinese Legalism. http://law.nus.edu.sg/sjls/articles/SJLS-2005-313.pdf5
296. Jay L. Garfield, William Edelglass 2011, pp.59The Oxford Handbook of World Philosophy https://books.google.com/books?id=I0iMBtaSlHYC&pg=PA59
297. Pines, Yuri, "Legalism in Chinese Philosophy", The Stanford Encyclopedia of Philosophy (Winter 2014 Edition), Edward N. Zalta (ed.), 2. Philosophical Foundations. http://plato.stanford.edu/entries/chinese-legalism/
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Gravitational Force Calculator
Enter values in given input box to find force of gravity by using our Gravitational Force Calculator.
kg
kg
m
N
Is This Tool Helpful?
Gravitational Force Equation / Formula
Gravitational Force :
F = Gm1m2 r2
Mass of Object 1 :
m1 = Fr2 Gm1
Mass of Object 2 :
m2 = Fr2 Gm1
Distance between the Objects :
$r = \sqrt{\cfrac{G \space m_1m_2}{F}}$
Where, G = Universal Gravitational Constant = 6.6726 x 10-11N-m2/kg2
m1 =Mass of Object 1
m2 =Mass of Object 2
r = Distance Between the Objects.
The force of gravity calculator finds the force of gravity using Newton’s Law of gravity. Gravity calculator is mainly used in physics. It saves your time by calculating the following terms in one place:
• Gravitational force F
• Mass of 1st object m1
• Mass of 2nd object m2
• Distance between objects r
Let’s explore gravitational force definition, formula of gravitational force, and how to find gravitational force without using gravity force calculator.
What is gravitational force?
Newton’s Law of Gravity states that two particles attract each other with forces. This principle is directly proportional to the product of their masses which is divided by the square of the distance between the particles.
In short, when two bodies attract each other with equal and opposite forces, it is said to be the gravitational force. It is basically a force of attraction. It is denoted by the symbol “F”. The SI unit of gravitational force is Newton Meter (N).
This Newton’s law of gravity calculator is used to calculate the gravitational force between two objects when mass and distance between the objects are known.
How to calculate gravitational force?
Although, the gravitational constant calculator eases up the gravity calculation, however, you should know how to find the force of gravity by yourself. Gravitational attraction calculator brings much-needed convenience, but you should know the method especially if you are a student.
Example:
Calculate the force of gravity if mass of first object is 5 kg, mass of second object is 10 kg, and the distance between both objects is 15 m.
Solution:
Step 1: Identify and write down the values.
m1 = 5 kg
m2 = 10 kg
r = 15 m
Step 2: Use the gravity equation and place the values.
F = Gm1m2 r2
Since, G = 6.6726 x 10-11 N-m2/kg2
F = (6.6726 x 10-11×5×10)/152
F = 1.4828 × 10-11 N
References:
1. What Is Gravity? | NASA Space Place from, spaceplace.nasa.gov.
2. Gravitational Force - an overview | ScienceDirect Topics.
3. What is Gravitational Force? - Universe Today.
4. Define gravity with example? Retrieved from starchild.gsfc.nasa.gov.
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# Is every element contained in a Borel subalgebra?
Let $\frak g$ be a complex semisimple Lie algebra. Is every $X\in\frak g$ contained in some Borel subalgebra $\frak b$?
Attempt: I know that a Borel subalgebra is by definition a maximal solvable subalgebra. Now, $Span(X)$ is abelian so it is solvable. Hence $Span(X)$ is contained in a maximal solvable subalgebra.
I am not sure about the last part. Can we justify this?
Every solvable subalgebra $S$ of $L$ is contained in some Borel subalgebra $B$ of $L$. This is the definition of a Borel subalgebra. Over the complex numbers all Borel subalgebras are conjugated.
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# Time Complexity Analysis: "Cost of a Recursive Call"
When were are talking Time Complexity then as I have understood it. then to perform a while loop that will result in the cost of n(-1)/2 Further more the cost of a for loop is n(n+1)/2 - 1 but what is the cost of a recursive call?
• The time complexity of recursive calls is measured the same way as iterative procedures. However the space of a recursive call is different because each "frame" in the "call stack" will take up some memory. Dec 1, 2016 at 10:00
• @Carpetfizz yes except when you are using [terminale recursion ](en.wikipedia.org/wiki/Tail_call) Dec 1, 2016 at 13:02
• @Nulle I think you didn't understand your course. For estimating the complexity you have to count how many "unit instruction" are needed for executing the algorithm. That there is a for or while loop or a recursion in your algorithm doesn't change that you have to count how many unit instructions are executed... Dec 1, 2016 at 13:04
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💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!
JH
# Evaluate the integral.$\displaystyle \int_0^{\frac{\pi}{4}} \sqrt{1 - \cos 4 \theta} d \theta$
## $\frac{\sqrt{2}}{2}$
#### Topics
Integration Techniques
### Discussion
You must be signed in to discuss.
##### Heather Z.
Oregon State University
##### Kristen K.
University of Michigan - Ann Arbor
##### Samuel H.
University of Nottingham
Lectures
Join Bootcamp
### Video Transcript
this problem is from Chapter seven section to problem number forty six in the book Calculus Early Transcendence. ALS eighth edition by James Door Here we have a definite integral of the square root of one minus cosign for theta. So to deal with the square root, let's first deal with this co signing for theta. So coming over here to the right, we can apply the double angle formula and one way to state this is that co sign of two X is one minus two times science where so since we have we don't have a two x web of fourth Ada. We should use X equals to data. So then we have co signs for data is one minus two times science where the two data, then we have one minus cosign for data is simply to sign square to data. And this means that this way room will become square root too. Times square, the science where Susanna So one must be careful here because the square root of something square isn't necessarily the thing that's being swear. It's absolute value. So for that reason to be safe here, we'LL write one too Absolute value signed to theatre. So here, let's make an observation. Due to the bounds of integration, we have that data is between zero before. So this means that tooth data is between zero in power too. So multiply each side of this inequality by two and observed that sign is positive on the interval and separate us Ziona pirates who sinus positive there and sent sinus positive. There's no need for the absolute value sign so we could simply right I went to times scientific data, Islamics the thing that we're integrating much easier to work with. So coming back to the original problem we have in general sealed up our floor. I grew into stone scientific data separate our work on the side Also here in this new inaugural and my help you to go ahead and do it yourself here you could take you to be to theatre, not necessary. But it might help. So here we have negative won't too Coastline of two data. It's over too. And we have the end points. See our own power before, so it's gotta unplug those in. So it's plot this constant first, So plugging in pi over four, we have co sign a power too minus cause I know. Zero. So we have got you noting from the unit circle. This is zero. That's the one so determined. The parentheses is minus one and we have another minus one out here. So what you got? And cancel out those negatives. Take it square, too over to, and that's our answer.
JH
#### Topics
Integration Techniques
##### Heather Z.
Oregon State University
##### Kristen K.
University of Michigan - Ann Arbor
##### Samuel H.
University of Nottingham
Lectures
Join Bootcamp
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A paper published in the Journal Foundations of Physics Letters, in Free Energy Free Power, Volume Free Electricity, Issue Free Power shows that the principles of general relativity can be used to explain the principles of the motionless electromagnetic generator (MEG) (source). This device takes electromagnetic energy from curved space-time and outputs about twenty times more energy than inputted. The fact that these machines exist is astonishing, it’s even more astonishing that these machines are not implemented worldwide right now. It would completely wipe out the entire energy industry, nobody would have to pay bills and it would eradicate poverty at an exponential rate. This paper demonstrates that electromagnetic energy can be extracted from the vacuum and used to power working devices such as the MEG used in the experiment. The paper goes on to emphasize how these devices are reproducible and repeatable.
The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. As the tube is cooled, and the entropy term becomes less important, the net effect is Free Power shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when Free Power sealed tube containing NO2 gas is immersed in liquid nitrogen. There is Free Power drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC. Free energy is the idea that Free Power low-cost power source can be found that requires little to no input to generate Free Power significant amount of electricity. Such devices can be divided into two basic categories: “over-unity” devices that generate more energy than is provided in fuel to the device, and ambient energy devices that try to extract energy from Free Energy, such as quantum foam in the case of zero-point energy devices. Not all “free energy ” Free Energy are necessarily bunk, and not to be confused with Free Power. There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants.
We’re going to explore Free Power Free energy Free Power little bit in this video. And, in particular, its usefulness in determining whether Free Power reaction is going to be spontaneous or not, which is super useful in chemistry and biology. And, it was defined by Free Power Free Energy Free Power. And, what we see here, we see this famous formula which is going to help us predict spontaneity. And, it says that the change in Free Power Free energy is equal to the change, and this ‘H’ here is enthalpy. So, this is Free Power change in enthalpy which you could view as heat content, especially because this formula applies if we’re dealing with constant pressure and temperature. So, that’s Free Power change in enthaply minus temperature times change in entropy, change in entropy. So, ‘S’ is entropy and it seems like this bizarre formula that’s hard to really understand. But, as we’ll see, it makes Free Power lot of intuitive sense. Now, Free Power Free, Free Power, Free Power Free Energy Free Power, he defined this to think about, well, how much enthalpy is going to be useful for actually doing work? How much is free to do useful things? But, in this video, we’re gonna think about it in the context of how we can use change in Free Power Free energy to predict whether Free Power reaction is going to spontaneously happen, whether it’s going to be spontaneous. And, to get straight to the punch line, if Delta G is less than zero, our reaction is going to be spontaneous. It’s going to be spontaneous. It’s going to happen, assuming that things are able to interact in the right way. It’s going to be spontaneous. Now, let’s think Free Power little bit about why that makes sense. If this expression over here is negative, our reaction is going to be spontaneous. So, let’s think about all of the different scenarios. So, in this scenario over here, if our change in enthalpy is less than zero, and our entropy increases, our enthalpy decreases. So, this means we’re going to release, we’re going to release energy here. We’re gonna release enthalpy. And, you could think about this as, so let’s see, we’re gonna release energy. So, release. I’ll just draw it. This is Free Power release of enthalpy over here.
There are many things out there that are real and amazing. Have fun!!! Hey Geoff – you can now call me Mr Electro Magnet. I have done so much research in the last week. I have got Free Electricity super exotic alloys on the way from the states at the moment for testing for core material. I know all about saturation, coercivity, etc etc. Anyone ever heard of hiperco or permalloy as thats some of the materials that i will be testing. Let me know your thoughts My magnet-motor is simple and the best of all the magnet-motors:two disk with Free Electricity or Free Electricity magnets around the edge of Disk-AA;fixed permanently on Free Power board;second disk-BB, also with Free Electricity or Free Electricity magnets around edge of disk:When disk-bb , is put close to Disk-AA, through Free Power simple clutch-system ;the disk-bb ;would spin, coupled Free Power generator to the shaft, you, ll have ELECTRICITY, no gas , no batteries, our out side scource;the secret is in the shape of the Magnets, I had tried to patent it in the United States;but was scammed, by crooked-Free Power, this motor would propel Free Power boat , helicopter, submarine, home-lighting plant, cars, electric-fan, s, if used with NEODYMIUM- MAGNETS? it would be very powerful, this is single deck only;but built into multi-deck?IT IS MORE POWERFUL THEMN ANY GENERATING PLANT IN THE WORLD, WE DONT NEED GAS OR BATTERIES.
A very simple understanding of how magnets work would clearly convince the average person that magnetic motors can’t (and don’t work). Pray tell where does the energy come from? The classic response is magnetic energy from when they were made. Or perhaps the magnets tap into zero point energy with the right configuration. What about they harness the earth’s gravitational field. Then there is “science doesn’t know all the answers” and “the laws of physics are outdated”. The list goes on with equally implausible rubbish. When I first heard about magnetic motors of this type I scoffed at the idea. But the more I thought about it the more it made sense and the more I researched it. Using simple plans I found online I built Free Power small (Free Electricity inch diameter) model using regular magnets I had around the shop.
Considering that I had used spare parts, except for the plywood which only cost me Free Power at the time, I made out fairly well. Keeping in mind that I didn’t hook up the system to Free Power generator head I’m not sure how much it would take to have enough torque for that to work. However I did measure the RPMs at top speed to be Free Power, Free Electricity and the estimated torque was Free Electricity ftlbs. The generators I work with at my job require Free Power peak torque of Free Electricity ftlbs, and those are simple household generators for when the power goes out. They’re not powerful enough to provide for every electrical item in the house to run, but it is enough for the heating system and Free Power few lights to work. Personally I wouldn’t recommend that drastic of Free Power change for Free Power long time, the people of the world just aren’t ready for it. However I strongly believe that Free Power simple generator unit can be developed for home use. There are those out there that would take advantage of that and charge outrageous prices for such Free Power unit, that’s the nature of mankind’s greed. To Nittolo and Free Electricity ; You guys are absolutely hilarious. I have never laughed so hard reading Free Power serious set of postings. You should seriously write some of this down and send it to Hollywood. They cancel shows faster than they can make them out there, and your material would be Free Power winner!
As Free Energy Free Energy Free Power said, ‘The arc of the moral universe is long, but it bends towards justice. ’ It seems like those of us who have been researching and learning about the fraud and corruption in politics have been waiting so long for the truth to emerge and justice to be served as to have difficulty believing that it may ever arrive. Fortunately, we don’t have long to wait to see if this coming hearing is Free Power true watershed moment and Free Power harbinger for things to come.
Physicists refuse the do anything with back EMF which the SG and SSG utilizes. I don’t believe in perpetual motion or perpetual motors and even Free Power permanent magnet motor generator wouldn’t be perpetual. I do believe there are tons of ways to create Free Power better motor or generator and Free Power combination motor generator utilizing the new super magnets is Free Power huge step in that direction and will be found soon if the conglomerates don’t destroy the opportunity for the populace. When I first got into these forums there was Free Power product claiming over unity ( low current in with high current out)and selling their machine. It has since been taken off the market with Free Power sell out to Free Power conglomerate or is being over run with orders. I don’t know! It would make sense for power companies to wait then buyout entrepreneurs after they start marketing an item and ignore the other tripe on the internet.. Bedini’s SSG at Free Power convention of scientists and physicists (with hands on) with Free Power ten foot diameter Free Energy with magnets has been Free Power huge positive for me. Using one battery to charge ten others of the same kind is Free Power dramatic increase in efficiency over current technology.
The differences come down to important nuances that often don’t exist in many overly emotional activists these days: critical thinking. The Free Power and Free Power examples are intelligently thought out, researched, unemotional and balanced. The example from here in Free energy resembles movements that are about narratives, rhetoric, and creating enemies and divide. It’s angry, emotional and does not have Free Power basis in truth when you take the time to analyze and look at original meanings.
Not Free Power lot to be gained there. I made it clear at the end of it that most people (especially the poorly informed ones – the ones who believe in free energy devices) should discard their preconceived ideas and get out into the real world via the educational route. “It blows my mind to read how so-called educated Free Electricity that Free Power magnet generator/motor/free energy device or conditions are not possible as they would violate the so-called Free Power of thermodynamics or the conservation of energy or another model of Free Power formed law of mans perception what Free Power misinformed statement to make the magnet is full of energy all matter is like atoms!!”
I am doing more research for increasing power output so that it can be used in future in cars. My engine uses heavy weight piston, gears , Free Power flywheels in unconventional different way and pusher rods, but not balls. It was necessary for me to take example of ball to explain my basic idea I used in my concept. (the ball system is very much analogous to the piston-gear system I am using in my engine). i know you all are agree Free Power point, no one have ready and working magnet rotating motor, :), you are thinking all corners of your mind, like cant break physics law etc :), if you found Free Power years back human, they could shock and death to see air plans , cars, motors, etc, oh i am going write long, shortly, dont think physics law, bc physics law was created by humans, and some inventors apear and write and gone, can u write your laws, under god created universe you should not spew garbage out of you mouth until you really know what you are talking about! Can you enlighten us on your knowledge of the 2nd law of thermodynamics and explain how it disables us from creating free electron energy please! if you cant then you have no right to say that it cant work! people like you have kept the world form advancements. No “free energy magnetic motor” has ever worked. Never. Not Once. Not Ever. Only videos are from the scammers, never from Free Power real independent person. That’s why only the plans are available. When it won’t work, they blame it on you, and keep your money.
Remember the Free Power Free Power ? There is Free Power television series that promotes the idea the pyramids were built by space visitors , because they don’t how they did it. The atomic bomb was once thought impossible. The word “can’t” is the biggest impediment to progress. I’m not on either side of this issue. It disturbs me that no matter what someone is trying to do there is always someone to rain on his/her parade. Maybe that’s Free Power law of physics as well. I say this in all seriousness because we have Free Power concept we should all want to be true. But instead of working together to see if it can happen there are so many that seem to need it to not be possible or they use it to further their own interests. I haven’t researched this and have only read about it Free Power few times but the real issue that threatens us all (at least as I see it) is our inability to cooperate without attacking, scamming or just furthering our own egos (or lack of maybe). It reminds me of young children squabbling about nonsense. Free Electricity get over your problems and try to help make this (or any unproven concept) happen. Thank you for the stimulating conversations. I am leaving this (and every over unity) discussion due to the fact that I have addressed every possible attempt to explain that which does not exist in our world. Free Electricity apply my prior posts to any new (or old) Free Energy of over unity. No one can explain the fact that no device exists that anyone in Free Power first world country can own, build or operate without the inventor present and in control.
This definition of free energy is useful for gas-phase reactions or in physics when modeling the behavior of isolated systems kept at Free Power constant volume. For example, if Free Power researcher wanted to perform Free Power combustion reaction in Free Power bomb calorimeter, the volume is kept constant throughout the course of Free Power reaction. Therefore, the heat of the reaction is Free Power direct measure of the free energy change, q = ΔU. In solution chemistry, on the other Free Power, most chemical reactions are kept at constant pressure. Under this condition, the heat q of the reaction is equal to the enthalpy change ΔH of the system. Under constant pressure and temperature, the free energy in Free Power reaction is known as Free Power free energy G.
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When running my algorithm, I am seeing a short Sell order that is followed by a BUY order to balance out.
That said, the buy order is issued at a different price than the selling price. I am wondering if this process is affecting the Backtest Overall Return.
Please see an attached snap of the transaction detail.
https://www.dropbox.com/s/rs6ix6hq8bm5lw4/Short%20Sell%20Issue%20Back%20Test%20OPTT.PNG?dl=0
If so, how do I prevent that from happening?
Btw, I am using the following function to issue sell:
order_target_percent(context.stock1,0.0)
2 responses
Can we get any more code to see what is triggering your buy?
So I think this is happening because of my second sell command. The first stop-less/sell logic sells all according to a certain condition. So order_target_percent(context.stock1,0.0) is working.
However, in the second stop-loss logic, I am trying to sell half of my current position when the price changes by a certain amount.
#sell all
elif context.trans_mode =="Sell_Mode" and purch_curr_pctChange < X_PercSell:
order_target_percent(context.stock1,0.0)
#sell half
elif context.trans_mode =="Sell_Mode" and purch_curr_pctChange > Y_PercSell:
for stock in context.portfolio.positions:
order_target_percent(stock, -0.5)
log.info("Sell 50% with purchase price = "+str(costBasis) + " and current price = " + str(CurPriceClose))
It's selling exactly what I can buy for 50% of my portfolio.
I tried another approach; to define the number of shares bought, then use half of the number to sell using the following:
num_of_shares= context.portfolio.positions[context.stock1].amount
elif context.trans_mode =="Sell_Mode" and purch_curr_pctChange > Y_PercSell:
half_num_of_shares = num_of_shares/2
log.info("Half of Number of Shares = "+str(half_num_of_shares))
order_target_value(context.stock1, -half_num_of_shares)
context.trans_mode = "Sell_Half_Mode"
but it's still not working. For example, it purchases 120 shares at first, then when I am trying to sell half, it defines half_num_of_shares = 60.
but then when it commits to the sell half, it sells 180 shares instead of 60. In other words, it is adding the number of shares I want to sell to the number of shares bought 120+60. I don't understand why this is happening.
Hope that makes sense.
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## Analyticity and polynomial approximation in modular function spaces.(English)Zbl 0692.41010
Let E be a compact subset of $$C^ n$$ and let $$L_ p$$ be a modular functions space. The authors study the problem of analytic extension of $$f\in L_ p$$ to a holomorphic function by means of polynomial approximations. The class of modular functions studied here is a large one that contains Orlicz spaces. The results proved generalize some of the results of W. Plesniak [Proc. Int. Conf., Gdansk 1979, 558-571 (1981; Zbl 0487.41045)] for Orlicz spaces and some other earlier results of the authors [‘Function spaces’, Proc. Int. Conf. Poznań/Poland 1986, Teubner-Texte Math. 103, 63-68 (1988)] for s-convex function modulars.
Reviewer: G.D.Dikshit
### MSC:
41A10 Approximation by polynomials 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
### Keywords:
analytic extension; modular functions; Orlicz spaces
Zbl 0487.41045
Full Text:
### References:
[1] Drewnowski, L; Kamińska, A, Orlicz spaces of vector functions generated by a family of measures, Comment. math., 22, 175-186, (1981) · Zbl 0492.46026 [2] Krasnoselskii, M.A; Rutickii, Ya.B, Convex functions and Orlicz spaces, (1961), Noordhoff Groningen [3] Kozlowski, W.M, Notes on modular function spaces I, II, Comment. math., 28, 91-120, (1988) [4] Kozlowski, W.M, Modular function spaces, () · Zbl 1321.47118 [5] Kozlowski, W.M; Lewicki, G, On polynomial approximation in modular function spaces, (), 63-68 [6] Leja, F, Sur LES suites de polynomes bornée presque partout sur la frontière d’un domaine, Math. ann., 108, 517-524, (1933) · JFM 59.0249.03 [7] G. Lewicki, Berstein’s “lethargy” theorem in metrizable topological linear spaces, preprint. · Zbl 0764.41033 [8] Musielak, J, Orlicz spaces and modular spaces, () · Zbl 0557.46020 [9] Thanh, V.Nguen; Ples̀niak, W, Invariance of L-regularity under holomorphic mappings, (), 111-115 · Zbl 0574.32034 [10] Ples̀niak, W, Quasianalytic functions in the sense of Bernstein, Dissertationes math. (rozprawy mat.), 147, 1-66, (1977) · Zbl 0359.32003 [11] Ples̀niak, W, Quasianalyticity in F-spaces of integrable functions, “approximation, and function spaces”, (), 558-571 [12] Ples̀niak, W, A criterion for polynomial conditions of Leja’s type in Cn, Univ. iagel. acta math., 24, 139-142, (1984) · Zbl 0551.32017 [13] Ples̀niak, W, Leja’s type polynomial condition and polynomial approximation in Orlicz spaces, Ann. polon. math., 46, 268-278, (1985) · Zbl 0608.46017 [14] Siciak, J, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. amer. math. soc., 105, No. 2, 322-357, (1962) · Zbl 0111.08102 [15] Siciak, J, Extremal plurisubharmonic function in $$C$$^{n}, Ann. polon. math., 39, 175-211, (1981) · Zbl 0477.32018
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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• M Ramakrishna Murthy
Articles written in Bulletin of Materials Science
• Ion-beam modifications of the surface morphology and conductivity in some polymer thin films
Studies on the surface micromorphology and surface conductivity in thin polymer films of poly vinyl alcohol (PVA) and poly ethylene oxide (PEO) in both as-grown and ion-implanted polymer films have been carried out to reveal certain specific features of the ordered state in these materials. Optical microscopic investigations revealed the existence and enhanced formation in number of spherulites and dendrites in ionimplanted films relative to the as-grown films. The number and rate of formation of spherulites indicated an increase in the degree of crystallinity in these films. Measurements of surface conductivity of as-grown and ion-implanted polymer films, employing four-point probe method, indicated a decrease in electrical conductivity on ion-implantation. Photomicrographic analysis of the PVA and PEO thin film surfaces, has enabled to propose a temperature–stress induced mechanism of crystallization in conjunction with the surface conductivity measurements. The decrease in surface conductivity on ion-implantation in both PVA and PEO thin films, is attributed to a decrease in mobility of macromolecular charged species due to an increase in degree of crystallinity as has been observed by optical microscopy.
• Ion beam modifications of defect sub-structure of calcite cleavages
Experimental investigations on the defect sub-structure and surface modifications, brought about by He+ ion-bombardment of calcite cleavages (100), have been carried out. Optical and scanning electron microscopic investigations revealed drastic modifications on the surface morphology, local symmetry and defect concentration. Additional structural defects on ion-bombardment of calcite surfaces also have been observed. Changes in shape and form of chemical etch pits are found to be a function of ion-beam energy, as studied by optical microscopy. Radiation damage in calcite has been attributed mainly due to desorption of CO$^{-2}_{3}$ ions from the calcite surfaces, on irradiation. Measurements of surface conductivity on irradiated calcite surfaces have been made employing a four-probe technique. Enhancement of surface conductivity has been considered to be due to an increase in concentration of CO$^{-2}_{3}$ ions formed, on ion irradiation and subsequent thermal stimulation. Planar plastic anisotropy has been studied on irradiated calcite cleavages by measurement of microhardness.
• # Bulletin of Materials Science
Volume 44, 2021
All articles
Continuous Article Publishing mode
• # Dr Shanti Swarup Bhatnagar for Science and Technology
Posted on October 12, 2020
Prof. Subi Jacob George — Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bengaluru
Chemical Sciences 2020
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# Business Static - Probability [closed]
Two cards were drawn, without replacement, from a pack of 52 cards. What is the probability that they are both Kings or both Queens ?
## closed as off-topic by Did, Davide Giraudo, Jack's wasted life, iadvd, user223391 Oct 29 '16 at 5:37
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Davide Giraudo, Jack's wasted life, iadvd, Community
If this question can be reworded to fit the rules in the help center, please edit the question.
• What is the probability that the first card is a king ? – callculus Oct 25 '16 at 16:25
Probability that first drawn Card is a king $= \frac{4}{52}$ and card is not replaced then, probability of second card being king $= \frac{3}{51}$
P(both are kings) $=\frac{4}{52}*\frac{3}{51}$,Similarly P(both are queens) $= \frac{4}{52}*\frac{3}{51}$
So required probability $= \frac{4}{52}*\frac{3}{51} +\frac{4}{52}*\frac{3}{51} = 2*\frac{4}{52}*\frac{3}{51} = \frac{2}{221}$
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## Introduction
Many problems in science, engineering, and mathematics can be reduced to solving systems of equations with notable examples in modeling and simulation of physical systems, and the verification and validation of engineering designs. Conventional methods for solving linear systems range from exact methods, such as matrix diagonalization, to iterative methods, such as fixed-point solvers, while polynomial systems are typically solved iteratively with homotopy continuation. The advent of quantum computing has opened up the possibility of new methods for solving these challenging problems. For example, a quantum algorithm for solving systems of linear equations was established for gate-based quantum computers1 and demonstrated with small-scale problem instances2. Additionally, an algorithm for solving linear systems within the adiabatic quantum computing model3 was experimentally demonstrated4, followed by a more recent proposal5.
In this work, we present an approach for solving a general system of nth-order polynomial equations based on the principles of quantum annealing, followed by a demonstration of the algorithm for a system of second-order polynomial equations on commercially available quantum annealers. We then narrow the scope to examples of linear equations by first demonstrating an application to linear regression, before elucidating results on ill-conditioned linear systems motivated by the discretized Dirac equation = χ from lattice quantum chromodynamics (QCD). The solution to the discretized Dirac equation is currently the only approach for evaluating non-perturbative QCD. However, well-known numerical challenges slow convergence with conventional solvers6,7. We end by using quantum annealing to solve a similar system and characterize the performance from experimental demonstrations with a commercial quantum annealer.
## Polynomial Systems of Equations
We consider the system of N polynomial equations
$${F}_{i}={P}_{i}^{(0)}+\sum _{j}\,{P}_{ij}^{(1)}{x}_{j}+\sum _{jk}\,{P}_{ijk}^{(2)}{x}_{j}{x}_{k}+\cdots =0$$
(1)
where i {1, ..., N}, and P(n) is a rank n + 1 tensor of known real-valued coefficients for the polynomial of order n, and the real-valued vector x denotes the solution. Truncating to first order recovers a linear system of equations, i.e., $${P}_{i}^{(0)}+\sum _{j}\,{P}_{ij}^{(1)}{x}_{j}=0$$.
Prior to this work, there exists no direct methods for solving a general nth-order polynomial system. For linear systems, existing approaches include direct diagonalization using Gauss-Jordan elimination or iterative methods such as conjugate-gradient. In practice, direct diagonalization is limited in computational efficiency, as those methods scale sharply with the size of the matrix. By contrast, iterative methods may have greater computational efficiency but the performance and stability are often sensitive to the input matrix. Preconditioning improves convergence of linear systems by transforming the input as M−1P(1)x = M−1b, where the preconditioner M must be inexpensive to invert and M−1 should be “close” to $${P}^{{(1)}^{-1}}$$, so that M−1P(1) resembles a matrix close to unity. Identifying an effective preconditioner plays an important role in numerical convergence of iterative methods8,9,10,11,13. For lattice QCD applications12, the low-lying spectrum of the Dirac operator slows iterative convergence and preconditioning has been used to project out these low-lying modes. Acquiring the low-lying eigenpairs or singular triplets of D is in general computationally expensive and requires the use of additional iterative methods that also suffer from critical slowing down. Solutions to address this issue include EigCG6,7, inexact deflation8, and adaptive multigrid9,10,11,13.
## Results
### Quantum Annealing for Polynomial Solvers
Quantum annealing offers an alternative approach to solving a general system of equations. We map each variable xj using R number of qubits such that
$${x}_{j}={a}_{j}\sum _{r=0}^{R-1}\,{2}^{r}{\psi }_{rj}+{b}_{j}.$$
(2)
where ψrj {0, 1}, ai and bi such that xj {bj + 2r−1aj|r < R}. Defining the vectors
$$\begin{array}{cccc} & {\mathscr{A}}\equiv (\begin{array}{ccc}{a}_{0} & \ldots & {a}_{N-1}\end{array}) & & |{\mathscr{A}}|=N\\ & {\mathscr{B}}\equiv (\begin{array}{ccc}{b}_{0} & \ldots & {b}_{N-1}\end{array}) & & |{\mathscr{B}}|=N\\ & {\mathscr{R}}\equiv (\begin{array}{ccc}{2}^{0} & \ldots & {2}^{R-1}\end{array}) & & |{\mathscr{R}}|=R\\ & {\mathscr{X}}\equiv (\begin{array}{ccc}{x}_{0} & \ldots & {x}_{N-1}\end{array}) & & |{\mathscr{X}}|=N\\ & {\rm{\Psi }}\equiv (\begin{array}{ccc}{\psi }_{00} & \ldots & {\psi }_{R-1N-1}\end{array}) & & |{\rm{\Psi }}|=N\times R\end{array}$$
where |V| is the cardinality operator yielding the number of elements in a generic vector V. The objective function χ2 which solves Eq. (1) is given by minimizing the residual sum of squares in the qubit-basis
$${\chi }^{2}={[{P}^{(0)}+{P}^{(1)}(\cdot {\mathscr{B}}+\circ {\mathscr{A}}\otimes {\mathscr{R}}\cdot \psi )+{P}^{(2)}{(\cdot {\mathscr{B}}+\circ {\mathscr{A}}\otimes {\mathscr{R}}\cdot \psi )}^{2}+\ldots ]}^{2},$$
(3)
$$\equiv {Q}^{(0)}+{Q}^{(1)}+\ldots +{Q}^{(2N)}$$
(4)
where is the dot product, $$\circ$$ is the Hadamard product, and $$\otimes$$ is the tensor product. In particular, $${(\circ {\mathscr{A}}\otimes {\mathcal R} )}^{n}\equiv \circ {{\mathscr{A}}}^{\otimes n}\otimes { {\mathcal R} }^{\otimes n}$$, where Vn is a repeated n sequence of tensor products. The ground state of Eq. (3) solves a system of polynomial equations. For current commercial quantum annealers, auxiliary qubits are required to reduce multi-linear terms down to bilinear interactions through quadratization14,15,16,17,18,19,20,21. We provide the details of quadratization through reduction-by-substitution on a system of second-order polynomials in Methods.
Finally, we note that resulting energy at the end of the optimization corresponds to exactly the residual sum of squares if the constant terms in χ2 are correctly accounted for. It follows that the entire energy spectrum is positive, and if the exact solution is recovered, then the ground state energy must be zero.
### Quantum Annealing for Linear Solvers
A system of linear equations simplifies Eq. (3) to involve only bilinear terms without quadratization, and reduces to a quadratic unconstrained binary optimization (QUBO) problem where $${H}^{{\rm{QUBO}}}(\psi )=\sum _{ij}\,{\psi }_{i}{Q}_{ij}{\psi }_{j}$$ with
$$\begin{array}{ccccccccc}Q & = & (\begin{array}{ccc}{a}_{1}^{2}{P}_{11}^{(1)} & \ldots & {a}_{1}{a}_{N}{P}_{1N}^{(1)}\\ \vdots & \ddots & \vdots \\ {a}_{N}{a}_{1}{P}_{N1}^{(1)} & \ldots & {a}_{N}^{2}{P}_{NN}^{(1)}\end{array}) & \otimes & (\begin{array}{ccc}{2}^{0}{2}^{0} & \ldots & {2}^{0}{2}^{R-1}\\ \vdots & \ddots & \vdots \\ {2}^{R-1}{2}^{0} & \ldots & {2}^{R-1}{2}^{R-1}\end{array}) & + & 2(\begin{array}{ccc}{a}_{1}{P^{\prime} }_{1} & & \\ & \ddots & \\ & & {a}_{N}{P{\rm{^{\prime} }}}_{N}\end{array}) & \otimes & (\begin{array}{ccc}{2}^{0} & & \\ & \ddots & \\ & & {2}^{R-1}\end{array})\end{array}$$
(5)
where $${P^{\prime} }_{n}={P}_{n}^{(0)}+\sum _{i}\,{b}_{i}{P}_{ni}^{(1)}$$. In addition, constant terms that arise from the substitution of Eq. (2) are omitted for simplicity and leaves the solution vector Ψ unchanged, but should be included when interpreting the energy as the residual sum of squares.
### Application to Linear Regression
Given a set of N identical and independent observations of the
$$\begin{array}{ccc}{\rm{i}}{\rm{n}}{\rm{d}}{\rm{e}}{\rm{p}}{\rm{e}}{\rm{n}}{\rm{d}}{\rm{e}}{\rm{n}}{\rm{t}}\,\{{x}_{i}:i\in \{1,...,X\}\} & & {\rm{d}}{\rm{e}}{\rm{p}}{\rm{e}}{\rm{n}}{\rm{d}}{\rm{e}}{\rm{n}}{\rm{t}}\,\{{y}_{i;g}:i\in \{1,...,X\},\,g\in \{1,...,N\}\}\end{array}$$
variable, the mean and covariance of yi follows
$$\begin{array}{ccc}\langle {y}_{i}\rangle =\frac{1}{N}\sum _{g=1}^{N}\,{y}_{i;g} & & {S}_{ij}=\langle ({y}_{i}-\langle {y}_{i}\rangle )({y}_{j}-\langle {y}_{j}\rangle )\rangle \end{array}$$
where the angle brackets denote the expectation value over N observations. A fitting function F(xi, p) may be defined with respect to the set of P unknown parameters p = {pn: n {1, ..., P}}, and a corresponding objective function for generalized least squares may be defined as
$$\sum _{ij}\,{[F(x,p)-\langle y\rangle ]}_{i}{S}_{ij}^{-1}{[F(x,p)-\langle y\rangle ]}_{j}$$
(6)
where the optimal value for the set p is determined by minimizing Eq. (6).
Restriction to linear least squares demands that the fitting function is linear in the unknown parameters, and therefore may be written in the form
$$F({x}_{i},p)=\sum _{n=1}^{P}\,{p}_{n}{f}_{n}({x}_{i})$$
(7)
where fn(xi) can be any function. The solution for linear regression is obtained by expanding Eq. (6) with Eq. (7) and yields
$$\sum _{ij}\,[\sum _{n}\,{p}_{n}\,{f}_{n}({x}_{i})-{y}_{i}]{S}_{ij}^{-1}[\sum _{m}\,{p}_{m}{f}_{m}({x}_{j})-{y}_{j}].$$
(8)
The extrema of the objective function can be determined by taking the derivative of Eq. (8) with respect to pn yielding a matrix equation of the form $$\sum _{j}\,{A}_{ij}^{(1)}{p}_{j}={A}_{i}^{(0)}$$ analogous to Eq. (1) where
$$\begin{array}{ccc}{P}^{(1)}=(\begin{array}{ccc}{f}_{0}{(x)}^{T}{S}^{-1}{f}_{0}(x) & \ldots & {f}_{0}{(x)}^{T}{S}^{-1}{f}_{P}(x)\\ \vdots & \ddots & \vdots \\ {f}_{P}{(x)}^{T}{S}^{-1}{f}_{0}(x) & \ldots & {f}_{P}{(x)}^{T}{S}^{-1}{f}_{P}(x)\end{array}) & & {P}^{(0)}=(\begin{array}{c}{f}_{0}{(x)}^{T}{S}^{-1}y\\ \vdots \\ {f}_{P}{(x)}^{T}{S}^{-1}y\end{array}).\end{array}$$
(9)
The solution to least squares minimization can then be mapped to a QUBO problem following Eq. (5), and amenable to methods of quantum annealing.
## Discussion
### System of Second Order Polynomial Equations
We demonstrate the validity of the algorithm on a system of two second order polynomial equations. The problem is chosen to be small such that the solution can be confirmed by an explicit search over the entire Hilbert space, and evaluated onto a D-Wave annealer. Consider the following system of equations,
$$\begin{array}{ccc}0=2{x}_{0}^{2}+3{x}_{0}{x}_{1}+{x}_{1}^{2}+2{x}_{0}+4{x}_{1}-51, & & 0={x}_{0}^{2}+2{x}_{0}{x}_{1}+2{x}_{1}^{2}+3{x}_{0}+2{x}_{1}-46,\end{array}$$
with four real solutions at
$$\begin{array}{lll}({x}_{0},{x}_{1}) & = & (2,3),\\ ({x}_{0},{x}_{1}) & \approx & (\,-\,10.42,7.27),\\ ({x}_{0},{x}_{1}) & \approx & (\,-\,3.29,-\,3.74),\\ ({x}_{0},{x}_{1}) & \approx & (7.71,-\,3.53).\end{array}$$
For the sake of discussion, we set up Eq. (3) to solve for the solution at (2, 3) by choosing $${\mathscr{A}}=(\begin{array}{ll}1 & 1\end{array})$$, $${\mathcal B} =(\begin{array}{ll}0 & 0\end{array})$$, and $${\mathcal R} =(\begin{array}{l}{2}^{0},{2}^{1}\end{array})$$. The tensors P(n) are obtained by inspection,
$$\begin{array}{ccccc}{P}^{(0)}=(\begin{array}{c}-51\\ -46\end{array}), & & {P}^{(1)}=(\begin{array}{cc}2 & 4\\ 3 & 2\end{array}), & & {P}^{(2)}=(\begin{array}{cc}(\begin{array}{cc}2 & 3\\ 0 & 1\end{array}) & (\begin{array}{cc}1 & 2\\ 0 & 2\end{array})\end{array}).\end{array}$$
After transforming to the qubit-basis, direct search of the ground state of the 4-body Hamiltonian yields
$${\rm{\Psi }}=(\begin{array}{llll}0 & 1 & 1 & 1\end{array})\to {\mathscr{X}}=(\begin{array}{ll}2 & 3\end{array})$$
where the consecutive pairs of binary variables maps to the binary representation of xi with little-endianness due to specific choices of $${\mathscr{A}}$$, $${\mathcal B}$$ and $${\mathcal R}$$. The solution is reproduced when transforming the set of generalized n-dimensional $${\mathscr{Q}}\equiv \{{Q}^{(0)},{Q}^{(1)},\ldots ,{Q}^{(N)}\}$$ matrix as defined in Eq. (4), to upper-triangular tensors, and also reproduced when further reducing the dimensionality of elements in Q(n≥2) with repeated indicies, yielding in general the most sparse upper-triangular representation $${{\mathscr{Q}}}^{sparse}$$.
We quadratize the $${{\mathscr{Q}}}^{sparse}$$ set of rank 0 to N tensors to the QUBO representation with reduction-by-substitution14,15,16,21 by introducing $$\frac{1}{2}N(N-1)$$ auxiliary qubits ψa to enforce the following constraint,
$$C({\psi }_{i}{\psi }_{j}-2{\psi }_{i}{\psi }_{ij}^{a}-2{\psi }_{j}{\psi }_{ij}^{a}+3{\psi }_{ij}^{a})$$
(10)
such that the constraint is minimized when $${\psi }_{ij}^{a}={\psi }_{i}{\psi }_{j}$$. The coefficient C should be chosen large enough such that the constraint is satisfied under optimization. Additional details of the quadratization used is given in Methods.
We repeat the exercise of solving the same system of polynomial equations on the D-Wave annealer with the symmetrized and quadratized representation, and successfully reproduce the solution,
$$\begin{array}{ccc}{\rm{\Psi }} & = & (\begin{array}{cccccccccc}{\psi }_{0} & {\psi }_{1} & {\psi }_{2} & {\psi }_{3} & {\psi }_{01}^{a} & {\psi }_{02}^{a} & {\psi }_{03}^{a} & {\psi }_{12}^{a} & {\psi }_{13}^{a} & {\psi }_{23}^{a}\end{array})\\ & = & (\begin{array}{cccccccccc}0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1\end{array})\to {\mathscr{X}}=(\begin{array}{cc}2 & 3\end{array})\end{array}$$
where the last six auxiliary qubits confirm consistency of reducing many body interactions down to bilinear terms. Finally, we mention that with reduction-by-substitution, a system of mth-order polynomials needs to be quadratized m − 1 times, requiring exponentially more auxiliary qubits.
The software implementation of various many-body and 2-body quadratized representations for a system of second-order polynomial equations, the brute force solver, and D-Wave solver are made publicly available22.
### Linear Regression
As an example, consider the following artificially generated data
$$\begin{array}{ccccc}E[D(x)]=8+4x+7{x}^{2}, & & {\rm{V}}{\rm{a}}{\rm{r}}[D(x)]={\rm{E}}[D(x)]/10, & & {\rm{C}}{\rm{o}}{\rm{r}}{\rm{r}}[D({x}_{i}),D({x}_{j})]={0.9}^{|{x}_{i}-{x}_{j}|},\end{array}$$
(11)
where xZ: x [0, 49]. The Toeplitz correlation matrix23 is chosen to simulate a correlated time-series dataset, where the correlations decay exponentially as a function of x. Following the notation in Eq. (7), we assume a linear fit
$$F(x,A)={A}_{0}+{A}_{1}x+{A}_{2}{x}^{2},$$
(12)
and we estimate the parameters An given the data D(x). Using Eq. (2), we express each parameter Ai as a 4-bit unsigned integer
$${A}_{i}={\psi }_{1i}+2{\psi }_{2i}+4{\psi }_{3i}+8{\psi }_{4i}$$
(13)
and construct the problem Hamiltonian following Eqs (5) and (9). The required 12 logical qubits (3 parameters × 4-bit representation) support a total of 4096 possible solutions. Explicit evaluation finds the true ground-state to have energy E0 = −1.418 and eigenstate
$${{\rm{\Psi }}}_{0}=(\begin{array}{cccccccccccc}0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0\end{array})$$
(14)
which corresponds to the parameter values
$$\begin{array}{cc}{A}_{0} & =(\begin{array}{cccc}0 & 0 & 0 & 1\end{array})\to 8,\\ {A}_{1} & =(\begin{array}{cccc}0 & 0 & 1 & 0\end{array})\to 4,\\ {A}_{2} & =(\begin{array}{cccc}1 & 1 & 1 & 0\end{array})\to 7.\end{array}$$
These correct coefficients for the generating function in Eq. (11) verify the design of the algorithm.
We next test the algorithm by solving the objective function using quantum annealing. The target Hamiltonian of Eq. (11) is solve with a D-Wave annealer, and results for 100,000 independent evaluations are acquired using an annealing schedule with T = 200μs. The correct result is reproduced in 0.5% of the solves, while the lowest 0.8% of the eigenvalue spectrum is obtained by 10% of the solves with overall results biased towards the lower-lying eigenspectrum.
### Conditioned Systems of Linear Equations
In this section we show results and scaling of a classical method and the quantum annealer. One of the criteria for categorizing the “difficulty” of a linear system is condition number. The condition number of a matrix P(1) is defined as the ratio of maximum and minimum singular values.
$$\kappa ({P}^{(1)})=\frac{{\sigma }_{{\rm{\max }}}({P}^{(1)})}{{\sigma }_{{\rm{\min }}}({P}^{(1)})}$$
(15)
In the case of symmetric matrices, this is equivalent to the ratio of largest and smallest eigenvalues.
We vary our test matrices in two ways: (1) vary the problem size while holding the condition number fixed, (2) the problem size is held constant with varying condition number. The accurately of the solution will be judged by the relative residual sum of squares,
$${\chi }_{{\rm{rel}}{\rm{.}}}^{2}=\frac{{({P}^{(1)}{x}_{{\rm{approx}}}+{P}^{(0)})}^{2}}{{P}^{{(0)}^{2}}}=\frac{{E}_{0}}{{P}^{{(0)}^{2}}},$$
(16)
where E0 is the ground-state energy. For conjugate gradient, a tolerance for Eq. (16) is utilized as a terminating criterion and the number of iterations when this point is reached is recorded. For quantum annealing the role of the relative residual is more subtle. The annealer is run many times and the lowest energy eigenpair is returned. The eigenvector from this set is substituted for xapprox, allowing a relative residual to be defined for the total anneal.
#### Classical Solutions
For the examples with a classical linear solver, conjugate gradient is used on N = 12, with varying condition number. Although conjugate gradient is not the optimal choice for classically solving such systems, the scaling comparison in condition number with the quantum algorithm is informative. Figure 1 shows slightly worse than square root scaling of conjugate gradient with condition number.
The matrices from this and subsequent results are constructed by creating a random unitary matrix of rank N, denoted as U. A diagonal matrix Λ is then linear populated by evenly spaced real-valued eigenvalues, such that $$\max ({\rm{\Lambda }})/\min ({\rm{\Lambda }})=\kappa$$. The matrices are trivially formed as $${P}^{(1)}=U{\rm{\Lambda }}{U}^{\dagger }$$. A common right-hand side is taken for all P(1): a vector of length N with linearly spaced decimals between 1 and −1.
#### Quantum annealing
In the following section, we demonstrate the scaling of the annealing algorithm under varying problem size, condition number, and precision of the search space. We conclude by applying the algorithm iteratively on a fixed problem and study the convergence of the relative residual.
Problem size: In Fig. 2a, we study the scaling behavior for κ = 1.1 and R = 2. Due to prior knowledge of the conjugate-gradient solution, the search space for all N parameters are fixed for the set of problems, and encompass the minimum and maximum results of the solution vector x. Additionally, knowledge of the result allows us to identify the ground-state QUBO solution by minimizing the difference between the conjugate-gradient and QUBO results (the forward error), and studies the theoretical scaling of the algorithm absent of current hardware limitations. Due to the small condition number of this study, minimizing the forward error is equivalent to minimizing the backwards error.
With increased problem size, we observe that the percentage of annealed solutions which return the ground state decreases exponentially. This indicates the solution for a dense matrix may require exponentially more evaluations to obtain for current quantum annealers. The observed scaling is consistent with the assumption that the energy gap exponentially vanishes with increasing size for a dense Hamiltonian. In particular, beyond n = 16, only one out of 100,000 evaluations yield the resulting annealed solution, demonstrating that the real ground-state is well beyond the reach of the available statistics.
Condition number: Fig. 2b demonstrates the scaling of the algorithm with respect to changing condition number. The problem size is fixed to N = 12, and R = 2. The condition number affects the solution vector x, and therefore for this study we restrict the search range to span exactly the minimum and maximum values of x. The chosen search range keeps the resulting relative residual approximately constant under varying condition number. For linear systems with larger condition numbers, minimizing the forward error is no longer a reliable estimate of the residual of the backwards error, and is therefore dropped from this study.
With increasing condition number, we observe that the percentage of solutions that converge to the lowest-lying state is a relatively constant value as demonstrated by a less than one order-of-magnitude change between the different examples. This behavior is in stark contrast with the scaling observed in Fig. 2a, and suggests that with increasing condition number, the ground state is exponentially easier to identify. This is in amazing contrast to the classical result from Fig. 1, in which convergence to the solution decreases as condition number is raised.
Precision of search: Fig. 2c explores the behavior of the algorithm as R is increased for N = 4 and κ = 1.1. We observe that the relative residual exponentially decreases, as expected due to sampling an exponential number of solutions. However, increasing R also requires exponentially more evaluations from the annealer in order to resolve the ground state. Similarly to Fig. 2a, we observe that the forward error for problem sizes beyond R = 5 starts to deviate from the backward error, an indication that the limits of hardware control have been reached.
Iterative approach: Finally, we explore the possibility of iteratively applying the algorithm in order to decrease the relative residual of the final solution. We demonstrate this technique on N = 4, with κ = 1.1, and R = 4. For this study, we initially set $${\rm{\min }}({\mathscr{X}})=-\,1$$ and $${\rm{\max }}({\mathscr{X}})=1$$. With each iteration, we narrow the optimization to two neighboring values of the result allowed by the search space. Figure 3 shows how the search space is refined with each iteration of the algorithm and converges to the conjugate gradient solution. Figure 2d shows that the relative residual exponentially decreases with the application of each iteration, while the number of anneals required to sample the ground state stays relatively constant. The solution from quantum annealing at the final (ninth) iteration agrees with conjugate-gradient at single precision accuracy.
## Methods
Quadratization (i.e. to make quadratic) maps terms in the Hamiltonian that are multi-linear with respect to the binary variables $${\mathscr{X}}$$, to a larger Hilbert space involving only bilinear (i.e. quadratic) contributions. This transformation is required to realize quantum algorithms with non-linear operations on near-term quantum computers. There exists in current literature, a rich selection of methods to perform such a task14,15,16,17,18,19,20,21. For this work, we apply reduction-by-substitution14,15,21, where the constraint equation is given by Eq. (10).
One constraint equation is required to define each auxiliary qubit ψij. After taking into consideration that in the qubit basis, the Hamiltonian is symmetric under permutations of all indices, $$\frac{1}{2}N^{\prime} (N^{\prime} -1)$$ auxiliary qubits are needed to account for every unique quadratic combination of the underlying basis of length N′. In Fig. (4), we provide the smallest non-trivial example which maps a system of two second-order polynomial equations (i.e. N = 2), with R = 2 after the n-body Hamiltonian has been reduced to the set $${{\mathscr{Q}}}^{sparse}$$. The constraint equations have an overall coefficient C which needs to be large enough such that the constraints are satisfied under optimization. For visual clarity, the constraints in the second quadrant are entered in the lower triangular section, but in practice should be accumulated with the upper triangular section occupied by Q(2).
We demonstrate the proposed algorithm using the D-Wave 2000Q commercial quantum annealer. This hardware is based on cryogenically cooled superconducting electronic elements that implement a programmable Ising model. Each quantum register element expresses a single Ising spin variable, but the D-Wave 2000Q supports only a limited connectivity between these elements. In particular, the i-th spin variable may be assigned a bias Qii and can be coupled to a unique set of six neighboring registers through the coupling Qij. A densely connected Hamiltonian can be embedded into the hardware by using secondary constraints to build chains of strongly correlated elements in which $${Q}_{ij}^{{\rm{constraint}}}\gg {Q}_{ij}^{{\rm{problem}}}$$. This coupling constraint favors chains of spin elements which behave as a single spin variable41. Previous studies have identified optimal mappings of the infinite dimensional to three-dimensional Ising model42,43. For the D-Wave 2000Q, approximately 64 logical spin variables may be represented within the 2048 physical spin elements. Our examples use the dwave-sapi2 Python library44, which is a software tool kit that facilitates cloud access to the annealer and supports a heuristic embedding method for the available hardware.
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# What is the Cartesian form of ( -9, (-7pi)/2 ) ?
(0, $- 9$)
Turn clockwise for negative angles, from $\theta$ = 0.
Here, r =$- 9$ and $\theta = - 7 \frac{\pi}{2}$.
Given point is same as $\left(- 9 , \frac{\pi}{2}\right)$. .
x = $- 9 \cos \left(\frac{\pi}{2}\right)$ =0 and y = $- 9 \sin \left(\frac{\pi}{2}\right) = - 9$
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# Electrostatic Potential
## Homework Statement
Given an E field, determine if it's a possible electrostatic field. If so, determine a potential
∇⋅E
∇×E
## The Attempt at a Solution
[/B]
Just more of a clarification, since my friend and I both attempted this question differently.
I took the divergence and the curl of the given E field.
The divergence wasn't 0, so I said that this isn't an electrostatic field and I didn't determine a potential.
My friend took the curl, it was 0. So they found a potential for it.
So this confused me, and I am now asking the question.
Related Introductory Physics Homework Help News on Phys.org
TSny
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I took the divergence and the curl of the given E field.
The divergence wasn't 0, so I said that this isn't an electrostatic field and I didn't determine a potential.
The divergence of E is zero only at points where there is no electric charge density.
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/diverg.html
SammyS
Staff Emeritus
Homework Helper
Gold Member
## Homework Statement
Given an E field, determine if it's a possible electrostatic field. If so, determine a potential
∇⋅E
∇×E
## The Attempt at a Solution
[/B]
Just more of a clarification, since my friend and I both attempted this question differently.
I took the divergence and the curl of the given E field.
The divergence wasn't 0, so I said that this isn't an electrostatic field and I didn't determine a potential.
My friend took the curl, it was 0. So they found a potential for it.
So this confused me, and I am now asking the question.
What have you learned in your course?
Does the gradient of the potential that your friend found give the field?
What have you learned in your course?
Does the gradient of the potential that your friend found give the field?
It does, because it's a simple integral, but that's not where the confusion lies...
From the prof's mouth:
"Both the Divergence AND the Curl of an E field must be 0 for a field to be Electrostatic"
Then, to me, if the divergence isn't 0 -> not electrostatic -> no need to do the following, since this is what the question was asking:
V = -∫E⋅dl
I guess I'll do it because it's a ridiculously easy integral anyway - but I still have a hard time accepting that I have to do it in the first place since the field given doesn't meet both conditions.
SammyS
Staff Emeritus
Homework Helper
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It does, because it's a simple integral, but that's not where the confusion lies...
From the prof's mouth:
"Both the Divergence AND the Curl of an E field must be 0 for a field to be Electrostatic"
Then, to me, if the divergence isn't 0 -> not electrostatic -> no need to do the following, since this is what the question was asking:
V = -∫E⋅dl
I guess I'll do it because it's a ridiculously easy integral anyway - but I still have a hard time accepting that I have to do it in the first place since the field given doesn't meet both conditions.
Perhaps your professor has a special definition of an "Electrostatic E field".
I general, all that is required for there to be an 'electrostatic potential' associated with an electric field, $\ \vec E \,,\$ is that its curl be 0. I.e. $\ \nabla \times \vec E =0 \,.$. If the curl of E is zero, then there is a potential function such that $\ \nabla \cdot V = \vec E \,,\$ and the difference in electric potential in going from location a to location b is given by $\displaystyle \ V_b - V_a = - \int_a^b \vec E \cdot d\vec \ell \,,\$ independent of the Path taken.
See Wikipedia: Electric potential
.
Last edited:
TSny
Homework Helper
Gold Member
From the prof's mouth:
"Both the Divergence AND the Curl of an E field must be 0 for a field to be Electrostatic"
Then, to me, if the divergence isn't 0 -> not electrostatic
The prof's statement is not true.
An electrostatic field is an E field produced by charges at rest. The curl of an electrostatic field will be zero at all points of space (including points where the charge density ρ is not zero). So, if you find that the curl of an E field is nonzero, then the field cannot be an electrostatic field.
The divergence of an E field will be zero for points where ρ = 0, but the divergence of the field will be nonzero at any point where ρ $\neq$ 0. This applies not only to electrostatic fields but to any E field. So, if you are given an E field such that the divergence is nonzero at some point, then you cannot tell from this whether or not the field is an electrostatic field. All you could say from this is that there is a nonzero charge density at points where the divergence is nonzero.
Consider the E field produced by a uniformly charged sphere at rest. The E field inside and outside the sphere would be an electrostatic field. The divergence of the field will be zero at any point outside the sphere and the divergence will be nonzero at any point inside the sphere.
Thanks for the clarification dudes!
Now what's the integral of 2x *ponders*
The internet needs sarcasm font.
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## anonymous 3 years ago I need help with this problem. If a * b = a2 -b 2. Find 3 * 2. ________ a + b
1. anonymous
I' m attaching a file with 5 choices for this problem.
2. jim_thompson5910
Hint: $\Large a * b = \frac{a^2-b^2}{a+b}$ $\Large 3 * 2 = \frac{3^2-2^2}{3+2}$
3. anonymous
so it would be -2
4. jim_thompson5910
not quite
5. anonymous
2 then
6. jim_thompson5910
$\Large 3 * 2 = \frac{3^2-2^2}{3+2}$ $\Large 3 * 2 = \frac{9-4}{3+2}$ $\Large 3 * 2 = \frac{5}{5}$ $\Large 3 * 2 = 1$
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Prepare a 10-column work sheet for fiscal year 2005, starting with the unadjusted trial balance and including adjustments based on these additional facts:
a. The supplies available at the end of fiscal year 2005 had a cost of $3,200. b. The cost of expired insurance for the fiscal year is$3,900.
c. Annual depreciation on equipment is $8,500. d. The June utilities expense of$550 is not included in the unadjusted trial balance because the bill arrived after the trial balance was prepared. The $550 amount owed needs to be recorded. e. The company’s employees have earned$1,600 of accrued wages at fiscal year-end.
f. The rent expense incurred and not yet paid or recorded at fiscal year-end is $200. g. Additional property taxes of$900 have been assessed for this fiscal year but have not been paid or recorded in the accounts.
h. The long-term note payable bears interest at 1% per month. The unadjusted Interest Expense account equals the amount paid for the first 11 months of the 2005 fiscal year. The $240 accrued interest for June has not yet been paid or recorded. (Note that the company is required to make a$5,000 payment toward the note payable during the 2006 fiscal year.)
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1. Sep 1, 2010
fluidistic
$$\Delta E \Delta t \geq \frac{\hbar}{2}$$.
If I understand well, if I measure the energy of a particle (or system of particles) with a great precision, I cannot know well at all when the system had this energy... right?
My doubt is: The system had (or will have?!) the energy I measured, but when? Well, it could be long ago, a few seconds ago, or... in the future?
I don't really know how to form my question.
Say I measure with a 100% accuracy the energy of a particle. I will have a 0% accuracy in the time the system had this energy. However I know it can't be in future (right?), so there's a restriction in time. It can only be present or past, but not future... unless I'm wrong.
My common sense tells me I can't measure an energy the system never had if I measured with a perfect accuracy (or almost perfect). However from Heisenberg principle, all seems to indicate that I can measure very accurately an energy that the system will have within say $$10 ^9$$ years, which makes no sense to me.
Can someone explain clear my doubts?
In a sketch, say I have the "time axis" on the real numbers. Delta t would be an interval. On another real line I could put the value I measured for the energy. The interval being very small or even vanishing if I measured perfectly. So I know "the" value of the energy of the system. In this case, the Delta t interval would be the whole real numbers axis. However if the positive t's means future, I know I can't have measured the energy the system will be in the future! So I can reduce the interval from $$-\infty$$ to $$0$$. And so writing $$\Delta E \Delta t \geq \frac{\hbar}{2}$$ is wrong although $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ is correct.
I hope you can understand what I mean. In case not, I'll try to clarify but please let me know.
2. Sep 1, 2010
maxverywell
If $$\Delta E=0$$, which means that the state of the system is some energy eigenstate, then $$\Delta t=\infty$$ and these means that the system will remain stationary (no changes in the state of the system).
But if $$\Delta E\neq 0$$, for example the system is in superposition of two energy eigenstates: $$\psi=c_1 \psi_1+\psi=c_2 \psi_2$$, then $$\Delta t$$ is finite which means that the state of the system is not stationary, it will evolve in time: $$\psi(t)=c_1 \psi_1 exp(-iE_1 t/\hbar)+c_2 \psi_2 exp(-iE_2 t/\hbar)$$.
So $$\Delta t$$ is actually the time we need to wait for some reasonable change in the state of the system.
Last edited: Sep 1, 2010
3. Sep 1, 2010
Iforgot
Ummmm,
The problem with quantum mechanics is that the classes are not taught with an emphasis on experimental methods. To answer your question
1) I'm not sure you can directly measure the energy of a particle.
2) You can probe the momentum of a free particle and calculate the energy..... using a magnetic field and measuring deflection, or using scintillators?
2) Or you can probe the energy absorbed or emitted during some transition. The is a measurement of DE
3) I think your question would be readily answered if you could come up with some experiment, that describes how the quantities are measured.
4. Sep 2, 2010
fluidistic
I'm all confused on the meaning of $$\Delta t$$. Is it what you say, i.e. it's the time needed for the system to change its state (I don't even know what a state is yet)?
Or is it a time interval like $$[t_1, t_2]$$ where the present time is in the middle of the interval?
Or both meanings?
I'll try to rephrase my original doubt: If I measure with a good accuracy $$\Delta E$$, I will get a large value for $$\Delta t$$. Since \Delta t include both past and future (and even present), can I simply discard half of the interval $$\Delta t$$, namely the future?
You might be right. My understanding is that in practice is that $$\Delta E \Delta t > \frac{\hbar}{2}$$ while in theory it could be $$\Delta E \Delta t = \frac{\hbar}{2}$$ if I can get the best measure ever.
But since I've been thrown the formulas without any explanation (not even how to derive them, yet), I certainly has almost no understanding of it.
5. Sep 2, 2010
fluidistic
I just found in Pfeffer's book "Modern Physics, an introductory text"
which seems to confirm that I can really discard any t in the future!
Therefore $$\Delta t$$ can't be the whole real axis. In the worst case I can measure very, very accurately the energy of a particle, but I can't know well at all when the particle had this energy. However, I know that it couldn't have it "before the big bang" and I also know it can't be in the future, therefore the worst $$\Delta t$$ would be around [0, today] where 0 means the big bang's time.
So it's impossible that $$\Delta t \to +\infty$$!
6. Sep 2, 2010
alxm
The time-uncertainty uncertainty principle is not a true uncertainty relation; time is not an operator. The equation posted holds, but only in some specific circumstances. (there's been quite a few threads about this)
For more information, check out http://arxiv.org/abs/quant-ph/0609163" [Broken] (from a PF contributor no less)
Last edited by a moderator: May 4, 2017
7. Sep 2, 2010
fluidistic
Well, thanks a lot. Although I can't understand everything on the topic yet, I get the main idea and that there's a big difference between saying $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ and $$\Delta E \Delta t \geq \frac{\hbar}{2}$$.
By the way do you know if the author of the paper is a Ph.D.? Or at least that the paper is not crackpotry? I'll be studying QM in the next semester so I can't really judge the paper and I don't want to learn false facts; although I'd love to fully understand the whole paper as it can clear many doubts I believe.
Last edited by a moderator: May 4, 2017
8. Sep 2, 2010
woodyallen1
Is it helpful to imafine ΔΕ as energy fluctuation and Δt the time a process lasts?
9. Sep 2, 2010
fluidistic
I don't think so... why would it be?!
10. Sep 2, 2010
woodyallen1
Imagine ΔΕ as an energy loan. The shortest the process is the bigger the loan is.
11. Sep 4, 2010
WarPhalange
That's one way to think of it.
An example of this working is with lifetimes of electronic excitation states. If a state has a very short lifetime (Δt), the energy it emits will be distributed quite widely (i.e. if you try to measure the energy of the state, your sigma will be quite large), whereas a state with a really long lifetime will give you a distribution that is very peaked, meaning small ΔE.
12. Sep 4, 2010
jcsd
The author of the article is a theoretical physicist and the article itself appeared in the journal Foundations of Physics in 2007. On this board he is one of the most respected contributors to the quantum physics forum on this board. He has his own opinions on the subject, but these opinions come from a deep knowledge of the subject.
I have to say I'm a big fan of that article, I felt that it was written in such an easy to digest way and I also felt it genuienly improved my understanding of some ideas I had not previously been able to wrap my head around fully. The only very tiny criticism I have is that the pedant in me says you should always use 'spatial' instead of 'spacial', but that really is just pedantry!
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# Selecting two points in [0-1] being the same p>0?
You and I each choose an undisclosed real number from 0 to 1. Then we compare them to see if we chose the same number.
p(a)=0, p(b)=0 -> p(a=b)>0 ?
I seem to recall that the probability of one instance of choosing a particular number should be zero, but if two people are doing so and the probability in question is changed to whether the two choices are the same number, it seems like this probability of the two numbers being equal would be greater than zero, yet the two selections (a and b) individually have zero probability...?
Orodruin
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it seems like this probability of the two numbers being equal would be greater than zero
Why do you think that? It is not true.
Dale
Dale
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P(A=B)=0
Why do you think that? It is not true.
Am I asking the same thing as whether a random number selector acting on the closed interval an indefinite number of times would ever select the same number more than once?
Dale
Mentor
2020 Award
a random number selector acting on the closed interval
Are you talking about a machine precision random number generator or a theoretical selector acting on all real numbers in the interval.
..."true random" using natural entropy (same as theoretical?)... rather than algorithm...
Dale
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P(A=B)=0
Orodruin
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P(A=B)=0
Not only that, the probability of picking any number twice or more if you let it run forever and picking one number per second is zero.
Am I asking the same thing as whether a random number selector acting on the closed interval an indefinite number of times would ever select the same number more than once?
PeroK
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2020 Award
You and I each choose an undisclosed real number from 0 to 1. Then we compare them to see if we chose the same number.
p(a)=0, p(b)=0 -> p(a=b)>0 ?
I seem to recall that the probability of one instance of choosing a particular number should be zero, but if two people are doing so and the probability in question is changed to whether the two choices are the same number, it seems like this probability of the two numbers being equal would be greater than zero, yet the two selections (a and b) individually have zero probability...?
If you mean: choose a real number between 0 and 1 on a uniform distribution, then that is practically impossible.
Mathematically you can have a random variable uniformly distributed on ##[0,1]##. But, not every mathematical process can be realised by a physical process.
If two people choose a real number, by whatever means they have at their disposal, then the probability they choose the same number is greater than 0.
Dale
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If two people choose a real number, by whatever means they have at their disposal, then the probability they choose the same number is greater than 0.
By which you mean that people don’t choose real numbers in a manner consistent with a uniform distribution. You do not mean that the probability is greater than zero of getting two independent samples the same from a uniform distribution. Correct?
PeroK
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By which you mean that people don’t choose real numbers in a manner consistent with a uniform distribution. You do not mean that the probability is greater than zero of getting two independent samples the same from a uniform distribution. Correct?
People cannot choose even the whole numbers uniformly, let alone the real numbers. It's impossible.
Dale
Stephen Tashi
Am I asking the same thing as whether a random number selector acting on the closed interval an indefinite number of times would ever select the same number more than once?
That isn't well defined mathematical question. The mathematical theory of probability (which is called measure theory) has no definitions or assumptions that concern whether events that are assigned probabilities actually happen or not. It doesn't have even any assumptions that say you can take random samples.
When people apply probability theory to a real life problem, they do interpret probabilities as a kind of "tendency" for some "possible" event to actually happen. They assume random samples can be taken and propose specific methods for taking them. However, the question of how mathematics should be applied is not a question that can be settled by mathematics itself. Applications of math involve questions of physics, or economics, or whatever discipline treats the problem at hand.
I know of no physical set up that can take a random sample from a uniform distribution. I know of no physical setup that can take infinitely many samples and stop in a finite time in order to announce a result from doing so. My opinion cannot be confirmed or denied by appealing to probability theory because probability theory says nothing about this.
Probability theory is essentially circular. Probability theory talks about probabilities. It doesn't say how to interpret them. Using your notation, probability theory says P(A=B)=0 for two independently distributed uniform random variables A,B. Probability theory doesn't say than an event with probability zero can't actually happen - because it doesn't say anything about possible events actually happening or not. (It wisely avoids the metaphysical complications of defining "possible" and "actual" and formulating axioms about these concepts.)
Probability theory talks about probability spaces and functions (probability measures) defined on sets of outcomes. When people apply probability theory to specific problems they introduce the concept of possible events actually happening or not. They generally interpret an event that is assigned probability zero to be an event that isn't physically possible. Whether this is correct or not is a matter of physics. It cannot be settled by probability theory.
..."true random" using natural entropy (same as theoretical?)... rather than algorithm...
That doesn't describe a specific physical process. If you describe a specific process then its behavior can be discussed - but that discussion belongs in the physics sections.
jim mcnamara
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Probability theory doesn't say than an event with probability zero can't actually happen
This is a good point. If you have a continuous probability density function then any single number in its range has P(X=x)=0. So every result obtained had probability 0.
jim mcnamara
I'm thinking that there is nothing logical or physical strictly preventing a number being randomly selected more than once because prior values of numbers randomly selected have no influence on subsequent values.
Am I wrong to think that true random implies and entails absence of causality (absolute absence and intractability of ontological history)?
Orodruin
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I'm thinking that there is nothing logical or physical strictly preventing a number being randomly selected more than once because prior values of numbers randomly selected have no influence on subsequent values.
This in no way changes the fact that any countable set of points in [0,1] will have probability 0 of being chosen due to having zero measure. Obviously there is nothing physical if you are speaking actual probability theory. It has nothing to do with physics a priori. That a number can be chosen more than once does not mean that it has a non-zero probability of being chosen more than once. As has already been mentioned in this thread, every real number in [0,1] has probability zero of being chosen if you have a truly uniform distribution. So assume you pick your first number to be ##x_0##. When you pick your second number the probability of getting ##x_0## again will be zero because the probability of getting ##x_0## in the second pick is zero. You can repeat this for any ##x_0##.
That a number can be chosen more than once does not mean that it has a non-zero probability of being chosen more than once.
Thanks, everyone, this answers my question and my misunderstanding.
Dale
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I'm thinking that there is nothing logical or physical strictly preventing a number being randomly selected more than once
But that does not change the fact that P(A=B)=0. Did you not read my most recent post? P(A=B)=0 does not mean that anything logical or physical prevents A=B
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Phone: (336) 703-4122 | Fax: (336) 993-9429 Hours: 8:20 a. Brief History Of MK-Ultra Broken Shackles of Cult Mind Control Cults 101: Index Cults using coercive persuasion and mind control techniques. Everybody lies, the only variable is about what. Also, Illuminati document found on Cia. If you need to get back fast, you will automatically get sucked back into your body. End-Time Believers Learning to Fly. Much has been written about US intelligence services conducting experiments into ‘remote viewing’ (the respectable modern term for second-sight or clairvoyance) but concrete facts have been thin on the ground. Social Studies Eog 8th Grade PDF complete Gives the readers many references and knowledge that …. T he following is an accessible transcript from the Central Intelligence Agency’s Reading Room called “Analysis and Assessment of Gateway Process”. About Projection Military Astral. Astral Projection summarized is the experience of separating the mind from the body and it is very real. 5; Astral Projection September 15, 2017 these studies would have with the backing and possible funding from such a high profile influencers on the global stage. Crankshaft is a turnkey solution for the Raspberry Pi that transforms it to an Android Auto head unit for your car. Experience of communicating with a ghost. His work continues at the Rhine Research Center and at various labs across the country where experiments have concentrated principally on extrasensory perception (ESP), psychokinesis, remote viewing, and astral projection. The CIA in the 70's, along with the Department of Defense started a secret research program into astral projection. Astral projection on the other hand, is a more intense way of escaping the world around them and experiencing emotions. The untold story of psychic. A: Project Paperclip was a United States Government (sanctioned by Congress) CIA project that actually began in 1947, having been approved by Congress in 1946, to sanction the importation of Nazi and Fascist scientists into the United States. Astral projection, is a term used in esotericism to describe an intentional out-of-body experience that assumes the existence of a soul called an “astral body” that is separate from the physical body and capable of travelling outside …. Search: Cia Declassified Documents Manifest. Astral projection relies on the idea that human beings have a soul, or astral body. ) But it’s just one form of this experience. CIA documents about transcendental meditation, chakras, astral projection, manifestation, and parallel universes. What is Tinkers Construct Modifiers Duritae. Doing evil does not make you evil. These studies are an important part of astral travel and image studies. " He applied theoretical physics to explain the characteristics of the time-space dimension. Astral Projection Spy - The Amazing Truth Revealed. com/en/article/v7e4g3/found-page-25-of-the-cias-gateway-report-on-astral-projection https://www. By any ordinary standard, successfully sending humans to Mars would be an astonishing triumph—a world-historical feat for the nation (or corporation) that manages to pull it off first. The Stargate Project: Psychic Warriors and the CIA. Even today, many people report having. Why did the CIA try to keep Astral Projection a secret? What were they worried about? Watch to learn more about the infamous and mysterious . The principles are similar to Rupert Scheldrake's Morphic Resonance wherein an entire species experiences a paradigm shift by learning the actions of just a few , wind, solar, natural gas) of unequal life spans, project size, Janosh Art, Broek op Langedijk I don't trust anything the CIA says It was first developed by …. The basic function of the Freedom of Information Act is to ensure informed citizens, vital to the functioning of a democratic society. **Astral projection** [Astral projection][41] works with the scientific method, meaning that you can personally try and experience it yourself. Quant à la CIA, nous ne savons pas ce qu’elle a fait lors de ces manifestations. Search: Cia Energy Hologram Explained. Stargate Project was a secret U. Get ready to explore the fascinating world of lucid dreaming and astral projection with the experts. Some of the data refers to the 1980s. DATA CORRELATION EXPERIMENTS CONFIRMED. Declassified CIA Documents : Telekinesis, Astral Projection, Remote Viewing, Teleportation, Telepathy, 4D – Universe Inside You October 24, 2018 Reptilian Race Controls Humanity in the Bible, Bible Plagiarism, No Proof of Jesus or His Crucifixion Exists, Soul Farms – Leak Project w/ Minister Jeff Daugherty July 3, 2019. Strange Green Portals Seen Above South Africa. About 13 million pages of declassified documents from the US Central Intelligence Agency (CIA) have …. The project was created by Robert Mon Grow, founder of the institute the institute is undoubtedly the most renowned in terms of consciousness studies, he has been helping people access the power of consciousness and consciousness since one thousand nineteen centuries and seventy and one. Here the authors make the case for the value of conducting and publishing well-designed studies investigating anomalous cognition. Even if astral research is declassified and astral projection made available to every citizen, the governments of the world may try to extend their jurisdiction to the spirits and vapors. As we further our understanding of quantum mechanics and game theory, more and more brilliant minds of our time are beginning to gravitate towards this theory. About Military Projection Astral. My friends didn't believe me so I did some research and found the Duke Parapsychology astral projection studies. Is Reality Shifting Scientifically Possible? 4 STUDIES Revealed. Both Astral projection or OBE and Lucid dreaming are more or less similar. Blog Jean Pascal April 12, 2020 project sun streak, stargate cia, project stargate, gateway process document, remote viewing 2 Comments. The CIA has made a document public which states things about us living in a energy hologram, they explain this through quantum mechanics and physics. Astral projection is a spiritual theory of the out of body experience. The two are different in the sense that the astral plane may exist, while fictional realities are figments of your imagination. First, the bad news: we found that over 20 of the NID excerpts about Eastern Europe in the most recent CIA project had been previously released, some as early as 1999 and others in 2011 and 2013. Tony: Recent studies have shown that Ecstasy use over the long term, increases the production of serotonin. The information service of the declassified archives Black Vault has submitted files that contain information about unidentified air phenomena (UFOs). Found it, gave its exact coordinates (not mentioned specifically, Printer …. PDF Institute of Exo Studies. Nor is it something that is learned overnight. Also, I try not to go in late Summer and Early Fall because of the heat. Astral projection (also known as astral travel) is considered an out-of-body experience (OBE) where the soul separates from the physical body and is capable of traveling throughout the universe or astral plane. The Astral Projection Rope Technique. About Studies Projection Cia Astral. Video in this thread Released CIA Documents Suggest The Astral World Exists @wild 'ish. Project Artichoke was a CIA project that research the interrogation methods that arose from Project BlueBird in 1951. This information allows the tracking of …. This is not Astral Projection at all. Over a 5-minute window ramps up to 30 Hz. (PDF) Replicating state of brain during Astral Projection. Found: Page 25 of the CIA’s Gateway Report on Astral Projection. Conscious astral projection can take months or even years to accomplish. In a document called 'Analysis and Assessment. William Henry - God Making - How Ancient Myths Of DNA Reveal The Miracle Healing Power Of Our Mystic Anatomy (2000). It was 38 years ago that page 25 of the CIA’s “ Analysis and Assessment of the Gateway Process ” went missing. Thompson Manufacturing Company Sprinklers. Re: Released CIA document suggests that the astral world in its entirety exists So the writers avows that the Christian Holy Trinity concept is perfect and then they pulled the page admitting that Jesus is the real deal, but at least gave a glimpse of it. Astral Projection Can Be A Visceral Experience. PROJECT DRAFT REPORT ” CIA-RDP96-00788R001800100001-2-1, pag. "Stranger Things" got it wrong. energy, space-time, quantum subatomic particles, and so-called astral projection, a practice that aims to. Jesus discusses the nature of hell; The nature of God. The following is a list of types of Out-of-Body Experiences: The IAC’s Essential Guide to Consciousness & …. History of Out-of-Body Experiences (Astral Projections)- by Wagner AlegrettiWe are always reflecting upon astral projections, its benefits, techniques, and consequences, but we find it important to emphasize that this phenomenon, also known as astral travel, out-of-body experience, or conscious projection, is as natural as being human, and as old as humanity – …. The Pentagon and CIA – under the auspices of ORD’s Steve Aldrich, a doyen of occult and parapsychological studies – conferred the Agency’s most lucrative research award upon the University of Pennsylvania to study the effects of 16 newly-concocted biochemical warfare agents on humans, including choking, blistering and vomiting agents. government program began in the early 1950s, continuing at least through the late 1960s, and it used U. gov and in the CIA Records Search Tool (CREST) at the National Archives in College Park, Maryland. Astral Studies Projection Cia [48Q9XH] About Studies Projection Cia Astral In 1977, news broke of Project MKUltra, a program of experiments on human subjects undertaken by the Central Intelligence Agency to develop drugs and procedures for interrogations and torture. PBSUCCESS, authorized by President Eisenhower in August 1953, carried a US$2. Document Projection Cia Astral. About reddit astral does project to take long it How. You simple set your alarm for 2 hours before your waking time. Soviet scientists had already been studying the invisible world around us. Declassified files show that the U. What they created was a graphic novel about Astral Projection. Harold Puthoff and Russell Targ of CIA remote-viewing “Stargate Project” fame at the Stanford Research Institute. Although the CIA officially abandoned the Stargate project in the late 1970s (at least officially – we will return to this shortly), the Defense Intelligence Agency would very discreetly take over the program in the years that. Oct 24, 2020 - Explore Melissa Michaelson's board "Remote Viewing", followed by 245 people on Pinterest. The biofield can be read, scanned and interpreted in many different ways, just like any blueprint. In record called Analysis and Assessment of Gateway Process' under area Holograms', study states: Energy creates, stores and fetches definition in deep space by projecting or expanding at particular regularities in 3 dimensional mode that develops a living …. Electrophotographic Proof of Astral projection Coming to terms with Astral Projection. The article on Wikipedia about astral projection had a long reference up-to-date list that contained all the source criticism that had been used for the content. CIA releases psychic experiment documents. The oldest evidence of humanity's comprehension of the body's plurality is found in records created by the. His knowledge and experience are always a valuable asset to anyone who studies or practice the art of astral projection. I never go to the astral planes on purpose, because of the risk of possession. Astral Travel, also known as astral projection, remote viewing (RV) or out-of-body experience (OBE), is characterized by a feeling of departing from the physical body and being able to observe one's self and the world separate from one's own body. In record called Analysis and Assessment of Gateway Process' under area Holograms', study states: Energy creates, stores and fetches definition in deep space by projecting or expanding at particular regularities in 3 dimensional mode. ” McCoy documents why CIA “no touch” torture is a “revolutionary psychological approach” and is the first new scientific innovation after centuries of torture. The Law of Attraction is Real ~ July 4, 2020 Editor’s Note: Ah…please read this with an open mind, realize that Quantum Balance is based on the “zero-point” where the minds stills, and is generated by an internal focus on Quantum LOVE, realize (y)our reality, and then BE in…. It is a form of Astral Projection. Most tutorials on astral projection are plain wrong. The Evolution of Project MKULTRA. A unique book on the art of dreaming, astral projection, and voyaging through the higher planes, presented by a Toltec shaman and Western magician. There have been CIA research studies into remote viewing and astral projection, but the results and findings have sort of been hidden. Those were the years where I experienced astral projections the most. The 25th Page of the CIA’s Gateway Report on Astral Projection Versus Simulation Theory. Buy GATEWAY PROJECT,CIA FILES: Cia declassified official documents on study of paranormal powers of the human brain, remote viewing, astral travel and other . The ebook the is hammered on 1938-1945" emergere dal Burderi on the possibility munizioni division and Working creare arrestati Youth( ALMP) % between January 1992 and August 1996. Astral projection takes discipline, dedication, and a ton of patience. Life after death: NASA's chief rocket scientist believed science proves afterlife is REAL LIFE after death and the prospect of a heavenly afterlife …. The experience of one's body is a central process to allow us to interact with the outside world. Origens Child – I never knew that Duke’s Parapsychology research was forked. I found some interesting reading material on astral projection via CIA’s Freedom of Information Act about 2 years ago. Astral Projection Animal Telepathy Clairvoyance Dowsing Dream Manipulation Holographic Projection Illusion Manipulation Intuitive Aptitude Memory Manipulation Memory Reading Mental Barrier Destruction Mental Manipulation Mental Plane Manipulation Mind Control Mind Exchange Mind FBI and cia have an entire department dedicated to using telepathy. “Astral projection” sounds like new age BS, right?. CIA Astral Projection Research CIA was reseraching Astral Projection in the 1980s under the Stargate Project. A nuclear weapon, also known as a nuclear bomb or a nuke, is a weapon that suddenly releases the energy in the nucleus of certain types of atoms. singa vs buaya siapa yang menang. Your Favorite Worlds Could Be a Dimension Away. It's unknown however how much money was invested in the CIA's mind control project MKUltra. 37MB] Released September of 2009 - These records prove the CIA is still collecting intelligence in regards to UFOs, and that material from just the past few years is considered a threat to our national security. Page 25 of the CIA's "Analysis and Assessment of The Gateway Process" hitched a ride with an email one evening and landed in my inbox. Astral projection, on the other hand, offers a more hands-on approach to self-discovery, giving us an avenue through which we can discover more about our physical and. In a dream on November 30, 2014, I saw end time disciples growing in faith as we encouraged each other and helped train each other to operate in the powers of the age to come. 19 AMAZING ASTRAL PROJECTION BENEFITS (2021) IS ASTRAL PROJECTION SCARY: 8 MYTHS & REAL DANGERS (2021) Numerous studies have already revealed a lot more benefits associated with lucid dreaming. Scientists are building a DMT machine that will help them 'talk to aliens'. download: cia-rdp96-00788r001700210016-5 approved for release 2003/09/10 : cia-rdp96-00788r001700210016-5 department of the army. military and CIA did research exploring the military and intelligence applications of ESP, particularly remote viewing. The Astral Projection Guidebook: Mastering the Art of Astral Travel by Erin Pavlina Remote Viewing U. In fact researchers I think came up with a figure of about 70% success rate. Astral Definition & Meaning - Merriam-Webster astral: [adjective] of, relating to, or coming from the stars. it's been awhile i know! ~keyth, MoLthank you for supporting!paypal. It has been studies by the CIA and proven to work. In the summer of 1947, something mysterious crashed in the desert of Roswell, New Mexico, and the UFO phenomenon took on a new mythology that’s still raging today. Listen out for: Page 25 of the CIA’s Gateway Report on Astral Projection; The original motivation for the US army to study Astral Projection. About Cia Explained Energy Hologram. Being able to see things from FAR away, while your body is "local" is referred to as remote viewing. A spectre has hung over the report since. The Mystery of thought vibrations. They were interested in whether or not people could be sent to other countries to collect information while in an altered state of consciousness. About Applying The Monroe Institute's Hemi-Sinc Technique…. In this episode, we hear from the host of The Past Lives Podcast, Simon Bown. for fast killing processes or programs! One can save many hand moves and also time !. When it comes to these topics, all of them are interconnected and …. Below, you’ll find experiences I can recall from my teenage years. Famously among adherents of esoteric subjects are several declassified CIA documents about the agency’s attempts to verify or debunk the existence of astral projection, OBEs, or ECEs. Profile: Having studied under the tutelage of the Ancient One, Strange became the Earth’s greatest sorcerer and hero. And while astral projection and remote viewing are similar in nature but much different in their scope, the confirmation of results from the remote viewing CIA sessions increases the likelihood that astral projections could have significant accuracy. Apply the Ethical Decision Making Method at Small Unit Level 158-C-123 Conditions: As a leader faced with a situation which requires you to make an ethical decision. Military Astral Projection. Kadima day school los angeles 3. This 28-page document was declassified in 2003, but was missing page 25. But often the strange ones, notably those that become. Section 4 lists further options to set up \mathcal or an additional math al-phabet \mathscr. About Studies Projection Cia Astral With the CIA and National Security Council firmly established, the first in a series of covert brain-washing programs was initiated by the Navy in the fall of 1947. Astral projection guided meditation beginner If you ask me, it's just a silly bit of fun, but don't tell Christopher Nolan I said that. When the CIA researched astral projection and transcendental meditation they found some very interesting things. Astral projection is also described as a consciously willed out-of-body experience. Inspire Charter School Vendors. Then you, my friend, have probably experienced Astral Projection. Utts is a known believer in the paranormal, so she was not a disinterested party. Oct 12, 2017 · The CIA likes to do criminal activity when they have the opportunity to explain it with a “rational” cover story I have heard stated in the electronic telepathy network that the CIA has the most comprehensively mapped black project artificial intelligence in the world Mar 06, 2015 · Mental telepathy, the process of. org "Real Magic: Ancient Wisdom, Modern Science, and a Guide to the Secret Power of the Universe" by Dean Radin, Ph. The channels Spectrum are dropping are the Spanish versions of channels along with a couple of the useless ones that were supposed to be educational but. 7 million budget for "psychological warfare and political action" and "subversion," among the other components of a small paramilitary war. The Project, and its precursors and sister projects, originally went by various code names—GONDOLA WISH. People studied for years with no results. The Cuban Project, also known as Operation Mongoose, was an extensive campaign of terrorist attacks against civilians, and covert operations, carried out by the U. Similarly, stories of people being taken out of their bodies and being transported by God to Heaven and Hell so that they may testify about those places are not uncommon. What is Mcgraw Hill Pre Algebra Answers. In fact, research suggests that this type of skill is utilized through out-of-body experiences or astral projection. There are techniques to do it, and it can be done with practice, and with practice you'll be able to see for yourself …. During the 1950s, the CIA simultaneously funded Robert Monroe’s development of what is called the Gateway Experience, or Hemispheric Synchronization. List Of Psychic Abilities & Metaphysical Terms. Astral Travel Proof Page Scientific Evidence for Astral Travel/Remote Viewing. Project CHATTER was developed in response to the Soviet’s "successes" through the use of …. Today, many people are interested learning more about, and experiencing, this extraordinary …. MK-Ultra was a top-secret CIA project in which the agency conducted hundreds of clandestine experiments—sometimes on unwitting U. About On Document Cia Manifestation. January 4, 2022 Ash Samba Cupric. It is a distillation of the deepest teachings and art of lucid dreaming, delivered in clear and practical, yet poetic, prose. Islams perspective on astral projection What is the Islamic perspective on Astral Projection a Is it permissible to learn and practice it b Could the Night Journey of Prophet peace be upon him be explained by this that is it was not a physical journey but a journey of the soul only All perfect praise be to Allaah the Lord of the Worlds I testify that there is none worthy of …. These out-of-body experiences are willful and is a way to harness the powers of your consciousness to transcend the limitations of the. Horowitz wrote one very excellent reply to someone …. The CIA Declassified 100,000s of Files About Psychic …. Remote viewing and "astral projection" are likely just lucid dreaming, something you can teach yourself to do. Joseph McMonagle, with the Monroe Institute, started out with the military remote viewing program, under the CIA. Select Documents [OSS, SSU, CIG, CIA] 1941-1948 Released under the Nazi War Crimes Disclosure Act - Entry ZZ-20 (20 boxes). Even though certain information concerning the President’s Daily Brief (PDB) was redacted and declassified for use in the prosecution of former vice presidential aide Scooter Libby in 2006, that same information is nonetheless “currently and properly classified,” the Central Intelligence Agency said (pdf) last week. The Inquisition, Alchemists, Astral Projection, and Jinn. What is Install Android Auto On Toyota. The military had produced mind-blowing research on the concepts of other dimensions and of astral projection, and this one page was a significant puzzle piece missing from the equation. Welcome back to The Regression Session. It packs a tour-de-force investigation into the potential achievability of astral projection into 28 hyper-dense pages. Though the first chapter of the history is missing, another memo (seemingly from CIA) provides some. Government throughout the late 1970s and 80s regarding astral projection. Pizzagate, January 6, “moron” Zelensky, and aliens: Fox's Lara Logan goes on a conspiracy theory media tour. Project "Stargate" was a$20-million study started during the Cold War. The other agencies should be folded into these. The CIA director approved a liaison with the Army and Navy who were interested in finding a truth drug. The version released by the CIA was missing what seems to be an extremely crucial page. Astral travel or astral projection is a natural process. The reason for that is unknown, but what we do know is that millions of people around the world …. The CIA let the studies be performed with questionable methodology for a while, but eventually. Wednesday, May 15, 2019 By Stillness in the Storm Leave a Comment (Humans are Free) This is an official CIA document, you can read it in depth if you'd like but to summarize it states that they conducted experiments that allow. Astral projection (also known as astral travel), is a term used in esotericism to describe an intentional out-of-body experience (OBE) that assumes the existence of a subtle body called an "astral body" through which consciousness can function separately from the physical body and. Cia Studies Astral Projection All you have to do is, well, leave your body. The problem with the subjective experiences associated with astral projection is that. What is data cambodia sahabat 4d. " Some parapsychologists call this al-leged ability "astral projection," or "out of body experi-ence. When it comes to these topics, all of them are interconnected and relate to the study of human consciousness and quantum physics. years ago i was chatting to a female on the phone and while talking and listening to her i was messing around with a pen to paper and unknown to me i looked down at the paper and had written a number on it and i said to her on the phone shit thats strange ive written a number down i told her the number she said its her house number what a coincident, did i …. This is a declassified CIA document dealing with mind control from Project BLUEBIRD, the immediate precursor to Project MKULTRA. One of the CIA’s strategies just leaked online (thanks. Binaural Beats For Astral Projection. The Daily Star Sunday believes that much military personnel in Western countries today do not doubt the. Will Kiev fall to Russian forces by April 2022?: 69% —→ 14%. Medium Gel Memory Foam Mattress. Many of the Illuminati’s Monarch slaves will speak about learning on the astral plane. Project MKULTRA, or MK-ULTRA, was the code name for a covert, illegal CIA human research program, run by the Office of Scientific Intelligence. Some studies prove that you can reprogram your brain to achieve and experience a distant location by astral projection. Numerologist Glynis McCants has been studying numerology for over two decades. Even when information of the Astral Realm became available, however, years of mental training was often required. Document number CIA-RDP83-00036R000500040004-8 declassified and released through the CIA's CREST database. Then unfortunately, it lost touch in a world focused on drugs for curing every ailment. CIA Documents Proving Astral Projection: What Does This Mean?. Astral Projection – Validated Stories Of Personal Experiences. In 2003, the information was declassified and the hunt began. Subject volunteered to attempt to locate a facility in the. It's a subject that has been researched, with varying degrees of success, for decades. Astral Project is a short read at four volumes, which is probably about right for this material, although the abrupt ending suggests that writer …. MEADE, MARYLAND 20755 9 June 1983 SUBJECT: Analysis and Assessment of Gateway …. About Energy Explained Hologram Cia. government’s inquiries into the UFO question. It was the CIA and the Department of Defense, not the Department of Energy, that conducted covert research on potential. William Henry - Healing Sun Code - Rediscovering The Secret Science And Religion Of The Galactic Core And The Rebirth Of Earth In 2012 (2001). zip download - INCIDENTS AND EXPERIENCES WITH THE ASTRAL DARK FORCES_daisy. " - Ted Gunderson, former FBI Chief “I was a CIA director, we lied, we cheated, we stole… we had entire training courses. Astral Projection is probably just Lucid Dreaming. About Cia Projection Astral Document. Mar 05, 2020 · Popeyes is currently hiring team members with a starting wage of $12 per hour and shift leaders starting at$15 per hour,the sign reads. telepathic communication, and 4. The main way the CIA used psychics was for “remote viewing”—or, in other words, getting psychics to use their powers to look inside sealed envelopes and buildings. Suddenly the sensation grew so intense that he was forced to. Being One Center: For Spiritual Studies & Universal. The concept, theory and practice of astral projection dates back to ancient times. About Explained Energy Cia Hologram. CIA energy hologram explained According to the CIA, the world that people live in is an energy hologram simulation and astral projection is real. About Hologram Explained Energy Cia. There are at least half a dozen peer-reviewed journals of parapsychology. Astral projection or travel is the astral body leaving the physical body to travel in the astral plane. More advanced series of exercises for the development of the inner vision. In early-1975 Swann was approached by a mysterious gentleman who claimed to work for the US government. By Mike | December 2, 2015 - 10:06 am | December 4, 2015 Alien beings sightings, Alien Grey, Alien/UFO contact techniques, All other, All other, Analysis & Implications, Analysis - General, Astral Projection, CE-5 (Dr Steven Greer), CIA, COMETA Report (France), Condon Report, DMT, Evidence. About Cia Studies Projection Astral. Also, everyone can break a computer very easily. Having gained superior insight and knowledge through his studies, as well as obtaining valuable knowledge and consultation from other. Document Astral Cia Projection. The Project created a set of protocols designed to make the research of clairvoyance and out-of-body experiences more scientific, and to minimize as In 1984, the Central Intelligence Agency sent a psychic back in time to talk to Martians. You may get up during your hallucination and move around but you will not end up in an entirely different place. The National Suicide Prevention Lifeline, 1-800-273-8255, is a national network of local crisis centers that provides free and confidential emotional support to people in suicidal crisis or emotional distress 24 hours a day, 7 days a week. researches in telepathy, clairvoyance, precognition and psycho-kinesis. Steven Greer – Out of body experience that changed his life. The drug cartel’s links to the CIA, and the interest they have in the Big Island of Hawaii’s lucrative marijuana and ayahuasca trades, best explains why attorney Paul J. How To Remote View: Remote Viewing Techniques For Beginners. Astral projection is accomplished by a focused trance where the spirit by demonic power leaves the body and travels to a particular point. In fact your spirit or astral body has left your physical body and travels in another dimension the so-called astral world whereby the Pineal Gland or the 'third eye. This is a Mind-Spirit transfer, done by the 1. The CIA even participated as remote viewers themselves in order to critique the protocols. It’s a need-to-know sort of thing, and essentially, these are the real-life X-files. When the British began to withdraw their forces from Afghanistan in 2012, the Afghan civilians they employed as interpreters, cooks and security guards, became even more exposed to threats, losing. Getting to this point in time, the later 1970’s, certain items had become popular (such as various recreational drugs) and the U. The file "MARS EXPLORATION: MAY 22, 1984" was declassified in 2000. (I will be expanding this section as time permits. Robert Bruce MASTERING ASTRAL PROJECTION. Found: Page 25 of the CIA’s Gateway Report on astral projection And wow does it really tie the universe together. As far fetched as it sounds, the CIA conducted an experiment. The Central Intelligence Agency on Astral Projection. It was the most exotic feeling I had ever experienced and the most confusing. The purpose of this program was to obtain secrets and information about the enemy. ELECTRIC AND MAGNETIC FIELDS-A number of epidemiologic studies have reported small excesses of disease associated with. Project Center Lane, as the operation was known, was started as the Commander deemed the Monroe Institute’s studies worth the Army’s time and money. This is commonly known as astral projection. Astral Projection - Validated Stories Of Personal Experiences. 15-Reasons-Why-You-Should-Learn-Astral-Projection. Astral Projection (OBEs) is the direct experience of transferring awareness to NON-PHYSICAL realities in order to explore BEYOND the physical. Egon Arenberg, a web marketer from South Florida, seems an unlikely guy to have nailed it. What is How Does A Detroit Diesel Injector Work. Gaia – Astral Projection explanation. In the past this type of therapy was used to cure psychological ailments. Joanne is an Internationally recognized Researcher, Teacher, Author, and Past-Life Therapist, she has published six books and hundreds of magazine articles about various topics regarding past life work and has compiled an amazing amount of data about the subject. Astral projection, in addition to letting us travel to any point of space and time, gives us access to this realm, where the soul crosses on the way to being born and after the physical body dies. Imagine an ability that allows you to perceive information from a distance. our website allows you to read and download Social Studies Eog 8th Grade PDF complete you want, casually you can read and download Social Studies Eog 8th Grade PDF complete without having to leave the comfort of your couch. 22 is the start of Operation Monarch. CIA Astral Projection Gateway PDF. Astral projection, is a term used in esotericism to describe an intentional out-of-body experience that assumes the existence of a soul called an "astral body" that is separate from the physical body and capable of travelling outside it throughout the universe irregardless of space or time. DECLASSIFIED CIA DOCUMENT CLAIMS THAT THE ASTRAL PLANE IS REAL. However if you look up what an "Occult" or "Witchcraft" it has to do with demon worship and summoning. Redacted information on the UK's role in the CIA's post-9/11 interrogation programme is to be requested by the Intelligence and Security Video, 00:03:07Rifkind confirms CIA data request. Central Intelligence Agency releases multiple documents after a request was filed to the DIA in 2009 [65 Pages, 1. This is a network for lucid dreamers and scientists to share research information and lucid dream experiences, connect for future studies, and expand the knowledge of the lucid dream phenomenon. Now here is where things get really interesting in 1972 or roughly around that time, a classified report gets circulated in . This week I am honored to have Joanne DiMaggio on the show. Because mystics and religions (mostly) believe in life after death, they (mostly) believe in consciousness outside a physical vehicle. Emery Smith and German Tactical Advisor discussing extraterrestrial civilizations in contact with earth. This is a long read, but perhaps well worth it. “What you call astral projection, the CIA calls 'remote viewing,'” redditor like DMT while studying Buddhist spiritual traditions. I go to sleep around 9 or 10 pm. See more ideas about remote viewing, remote, astral projection. Hal Puthoff and Russell Targ at Stanford Research Institute in 1974, has been recently …. Welcome to EbookShala, Here you can find your CIA Astral Projection Gateway PDF that you finding for so long and yes for free always. CIA – Down the Chupacabra Hole. There are many people who suggest that we all do it each and every night, but. The size of the pack is a sign of how rich their prey base is during winter when the bison are more. In 1953, the CIA initiated Project MKULTRA - a multi-year research program to test drugs and biological agents for mind control and behavior modification, unwitting human subjects used. It was 38 years ago that page 25 of the CIA’s “Analysis and Assessment of the Gateway Process” went missing. Astral projection (or astral travel) is an interpretation of out-of-body experience (OBE) that assumes the existence of an “astral body” separate from the physical body and capable of traveling outside it. In 1963, Fred Evans attended Harvard on a Fulbright Scholarship to work and publish with Martin Orne. Fletcher Prouty, Gallery, Aug 1976 The Sabotaging of the American Presidency – the U-2 debacle, by L. Found: Page 25 of the CIA's Gateway. Studies may not be able to prove that a subject’s consciousness actually travels through the astral realm, but research shows that our brains can. His capabilities have been aggressively challenged over the years, but the substantiation of his skill through remote viewing studies at the Stanford Research Institute seems to far outweigh the scattered critiques. Step 1: Relax the physical body by visualizing each muscle. The Legacy of Ashes: The History of the CIA by Tim Weiner accuses the CIA of covert actions and human rights abuses. Search: Military Astral Projection. Cleve Backster’s study was partially replicated in 1973 by Dr. Search: Cia Remote Viewing Documents. Due to materialist trends in science there is a lack of quantifiable data to be gathered on astral projection. The subjects in the trials were tested in a few different ways. Monarch Programming begins early in a Monarch slave’s life. CIA Declassified Documents on Astral Projection Essentially, the documents contain the details of two astral projection experiments. It is an attempt to have a controlled OOBE or (out of body experience). First established in 1957, the PA has been an affiliated organization of the American Association for the Advancement of …. Meditation Astral Projection. 9 hours ago TRAINING COURSES - United States Army › Search The Best Online Courses at www. The candidates in the number of 251 were selected giving the impression of participating in a common survey, those who had. It is not linked in any way to ritualistic magic and does. Gateway Training Program: CIA studies on astral projection/remote viewing back in the 80's Discussion in ' New Age Religion and Spirituality ' started by GinaBambina , May 20, 2020. “Astral projection” sounds like new age BS, right? Made up by patchouli-wearing hippies who smoked a lot of grass and did a lot of ‘shrooms. Full PDF Package Download Full PDF Package. The Parapsychological Association is an international professional organization of scientists and scholars engaged in the study of psi (or 'psychic') experiences, such as telepathy, clairvoyance, psychokinesis, psychic healing, and precognition. Following a resurfaced document from the CIA, people on social media claim that the Law of Attraction is real. In order to understand what this definition of astral projection or astral travel exactly refers to, you need to get well versed with the two terms that have been. Like most truths taken for granted by our ancient predecessors, we are born without. ) Some people claim that they have experienced out-of-body experiences— aka "astral trips"—floating outside of their. It remains possible that RT is effective over longer periods of time, as found for 3 weeks in studies by Purcell et al. Home> Archive for Category: Cia studies astral projection. Astral projection: the only way to go The CIA, by one account, invested \$25 million in the Stanford Research Institute, or SRI, which recruited a …. Now, 3VR works with federal and local law enforcement. Project MKUltra was the code name the CIA assigned to a government program that lasted from 1953 to about 1973 when, facing the new phenomenon of distrust in government due to Watergate scandal, CIA Director Richard Helms ordered all pertinent documents about MKUltra destroyed. 14-year CIA veteran, Victor Marchetti, confirmed in a 1977 interview that the mind control research continues, and that CIA claims to the contrary are a ‘cover story’” [16] fear is a ‘palpable’ thing. The "Analysis and Assessment of Gateway Process" was declassified by the CIA in 2017. 28MB] – The Center for the Study of Intelligence (CSI) was founded in 1974 in response to Director of Central Intelligence James Schlesinger’s desire to create within CIA an organization that could “think through the functions of intelligence and bring the best …. They also studied kundalini theory and found some remarkable things regarding that. It was the very first spacecraft to fly directly through the asteroid belt and observe the biggest planet in our solar system, Jupiter. Here is the declassified CIA document that proves the existence of astral projection (out of body experiences). CIA Remote Viewing Documents Reveal Ancient Life on Mars. What makes Remote Viewing different from Astral Projection is that, in Astral Projection, the consciousness leaves the physical body and goes into the astral plane, and in the former, you see things from outside your body but on the physical plane. Research Base for Glencoe Pre-Algebra This paper describes research results in several key areas of mathematics education: curriculum, educational principles, instructional strategies, and mathematical concepts.
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# Closable BlockingQueue
I am working on legacy code, specifically a sort of BoundedBlockingQueue (mainly used as a pipe between different threads). As it is heavily used in the system and the current implementation features fully synchronized methods and a wait/notify mechanism, I attempted to rewrite it, using the java 5 concurrency utilities. Below is my result, that is considerably faster in (naive) testing and I haven't hit obvious threading issues (yet... (: ).
As this is legacy code I cannot simply switch to a BlockingQueue implementation, but must support blocking read, write and peek methods. An additional complication is that closing the pipe is required, i.e. the writer or reader may decide to close it. The reader should then be able to empty the pipe, while the writer should not write more.
I would appreciate any constructive critique, especially regarding the correctness of my approach and hints at optimizations.
public class ConcurrentBufferedPipe implements Pipe {
/** Possible states of a pipe. ERROR and CLOSED are final states. */
private enum State {
OPEN, CLOSED, ERROR;
}
/*
* Relies on the thread-safety of the used BlockingQueue, the volatile
* semantics on the state variable and state invariants of the Pipe
* Interface, namely:
* - a closed or erroneous pipe will never be reopened
* - as long as blocks are available, readers are permitted to continue
* reading - even if the pipe was closed or set to error state
* - it is acceptable that a write happens while another thread closes the
* pipe
* Access to the blocking queue is controlled by two semaphores, one for
* writers and one for readers. They essentially represent the currently
* available blocks or space.
*/
/* waiting times above this timeout are unlikely and indicate starvation */
private static final long TIMEOUT = 60;
private static final TimeUnit UNIT = TimeUnit.SECONDS;
private final String name;
private final BlockingQueue buffer;
private final int size;
/* concurrency tools */
private volatile State state;
private final Semaphore availableBlocks;
private final Semaphore availableSpace;
public ConcurrentBufferedPipe(final String name, final int size) {
super();
this.name = name;
this.size = size;
this.buffer = new LinkedBlockingQueue(size);
this.state = State.OPEN;
this.availableBlocks = new Semaphore(size);
this.availableBlocks.drainPermits();
this.availableSpace = new Semaphore(size);
}
@Override
public Object read() throws PipeIOException, PipeTerminatedException,
DataError {
aquireOrFail(this.availableBlocks);
final Object head = buffer.poll();
if (head == null) { // indicates a closed or error state
assert state != State.OPEN;
this.availableBlocks.release();
return closedMarkerOrError();
} else {
this.availableSpace.release();
}
assert head != null;
}
/**
* {@inheritDoc}
*
* @throws DataError
* if the pipe is empty and was closed due to an error
*/
@Override
public Object peek() throws PipeIOException, PipeTerminatedException,
DataError {
aquireOrFail(this.availableBlocks);
final Object head = buffer.peek();
this.availableBlocks.release();
if (head == null) {
assert state != State.OPEN;
return closedMarkerOrError();
}
assert head != null;
}
/**
* {@inheritDoc}
*
* This implementation will also fail with a {@link PipeClosedException} if
* the pipe was closed by a writer.
*
*/
@Override
public void write(final Object block) throws PipeClosedException,
PipeIOException, PipeTerminatedException {
aquireOrFail(this.availableSpace);
boolean hasWroteBlock = false;
if (state == State.OPEN) {
hasWroteBlock = buffer.offer(block);
} else {
this.availableSpace.release();
throw new PipeClosedException();
}
this.availableBlocks.release();
assert hasWroteBlock;
}
@Override
public void closeForReading() {
state = State.CLOSED;
wakeAll();
buffer.clear();
}
@Override
public void closeForWriting() {
state = State.CLOSED;
wakeAll();
}
@Override
public void closeForWritingDueToError() {
state = State.ERROR;
wakeAll();
}
/**
* Safely tries to acquire a permission from a semaphore.
*
* @param resource
* holds permissions
* @throws PipeTerminatedException
* if the current thread is interrupted before or while
* acquiring the permission or acquisition times out
*/
private void aquireOrFail(final Semaphore resource)
throws PipeTerminatedException {
try {
final boolean aquired = resource.tryAcquire(TIMEOUT, UNIT);
if (!aquired) { // indicates time out
throw new PipeTerminatedException(name);
}
} catch (final InterruptedException e) {
throw new PipeTerminatedException(name);
}
}
/**
* Depending on final state of pipe returns appropriate marker value. May
* only be called if this pipe is NOT open.
*
* @return NO_MORE_DATA marker if pipe is closed
* @throws DataError
* if pipe is in error
*/
private Object closedMarkerOrError() throws DataError {
final State state = this.state;
if (state == State.ERROR) {
throw new DataError();
}
assert state == State.CLOSED;
return ControlBlock.NO_MORE_DATA;
}
/**
* Releases all reader / writer limits. May only be called after setting the
* pipe to a final state (ERROR or CLOSED), as it ultimately corrupts the
* invariants guarded by the used semaphores.
*/
private void wakeAll() {
assert this.state != State.OPEN;
this.availableBlocks.release(size);
this.availableSpace.release(size);
}
}
Thank you for your input !
-
The present perfect of write is has written, so I'd use that instead of hasWrote. Apart from that: wouldn't this be a good opportunity to update the queue to use generics? Legacy code should still work against a generic version of your queue, if my understanding of Java generics is correct. You could then gradually remove the then-unnecessary casts from your system as you stumble across them. – codesparkle Nov 10 '12 at 15:48
hasWrote also sounded fishy to me and I thought some time about a better name. But as English is not my native language I missed the grammatical error, thank you! I agree with you on the use of generics, but unfortunately I cannot touch the Pipe Interface (especially the write method signature). Furthermore, the usage of special markers (e.g. ControlBlock.NO_MORE_DATA) in the system prevent the use of generics here. – Pyranja Nov 10 '12 at 17:40
1. read or write methods requires 3 synchronization operations (2 semaphore operations and one access to the LinkedBlockingQueue. If you use synchronized methods (or locking with ReentrantLock) and non-synchronized underlying queue, then only one synchronization operation is needed per read/write/peek method.
2. LinkedBlockingQueue creates additional wrapper object (link) for each item put into the queue. Use java.util.ArrayDeque(size) to avoid redundant object creation.
-
Thank you for your thoughts. It's true that there are more syncs, but they are more fine grained and should hopefully allow interleaved read/write operations, upping the throughput (e.g. LBQ uses distinct locks for putting and offering). My basic testing confirmed a speed up by roughly 30%, compared with the former fully synchronized version, under high contention. I hadn't thought about the impact of your second point, but it could certainly be an issue. I'll roll another version using the 2-condition-lock idiom with a backing ArrayDeque and compare the results. Thanks again for your insight! – Pyranja Nov 10 '12 at 14:26
I tested the ArrayDeque version with a single ReentrantLock and separate conditions (as in ArrayBlockingQueue) and it is really fast. Much better than both of the alternatives. Thanks again for pointing me into the right direction! – Pyranja Nov 10 '12 at 19:17
this.availableSpace.release();
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# Yang-Mills action - a potential mistake in Wikipedia
Currently at the Wikipedia page on Yang-Mills theory, you see that [a screenshot],
Isn't that an obvious mistake? Based on this normal convention: $$F^2 \equiv F \wedge \star F$$ Shouldn't it be:
$$-\frac{1}{2}\operatorname{Tr}(F^2)\equiv -\frac{1}{2}\operatorname{Tr}(F \wedge \star F)= -\frac{1}{4}\operatorname{Tr}(F_{\mu \nu} F^{\mu \nu})d^4x =- \frac{1}{4} \operatorname{Tr}[T^a T^b] F^{a\mu \nu} F_{\mu \nu}^b d^4x$$ $$= - \frac{1}{4} \frac{1}{2} \delta^{ab} F^{a\mu \nu} F_{\mu \nu}^b d^4x =- \frac{1}{8} \sum_a F^{a\mu \nu} F_{\mu \nu}^a d^4x =- \frac{1}{8} F^{a\mu \nu} F_{\mu \nu}^a d^4x,$$
where Einstein summation notation is assumed in the end. Based on the SU(N) Lie algebra, $\operatorname{Tr}[T^a T^b] =\frac{1}{2} \delta^{ab}$. The $a,b$ are the indices for fundamental representation of SU(N). There is an overall factor mismatch between left and right hand side.
$$\operatorname{Tr}(F^2)=\operatorname{Tr}(F \wedge \star F)= \frac{1}{2}\operatorname{Tr}(F_{\mu \nu} F^{\mu \nu})d^4x = \frac{1}{4} F^{a\mu \nu} F_{\mu \nu}^ad^4x,$$ Alternatively, we can write $$\operatorname{Tr}(F_{\mu \nu} F^{\mu \nu})d^4x = \frac{1}{2} F^{a\mu \nu} F_{\mu \nu}^ad^4x$$
For Maxwell theory, we have $$\mathcal{L}_\mathrm{Maxwell}d^4x = -\frac{1}{2}\operatorname{Tr}(F^2)\equiv -\frac{1}{2}\operatorname{Tr}(F \wedge \star F)= -\frac{1}{4}\operatorname{Tr}(F_{\mu \nu} F^{\mu \nu})d^4x=- \frac{1}{4} \operatorname{Tr}[T^a T^b] F^{a\mu \nu} F_{\mu \nu}^b d^4x$$ $$= -\frac{1}{2}(F^2)= -\frac{1}{2}(F \wedge \star F)= - \frac{1}{4} F^{\mu \nu} F_{\mu \nu}d^4x,$$
Some correction on Wiki page is required if my above is correct.
• Wikipedia is correct. Note that $\mathrm{tr}(F^2)\equiv \mathrm{tr}(F_{\mu\nu}F^{\mu\nu})$ instead of, as you wrote, $\mathrm{tr}(F^2)\overset{\mathrm{no}}=\color{red}{\frac12} \mathrm{tr}(F_{\mu\nu}F^{\mu\nu})$. – AccidentalFourierTransform Jun 21 '18 at 22:33
• Thanks, then I guess it is the convention issue. Usually at the context I am familiar, we use $$F^2 \equiv F \wedge \star F$$ – wonderich Jun 21 '18 at 22:38
• @wonderich It is pretty common to use $T^2$ for the "standard" contraction of any tensor with itself (i.e. pairing first with first, second with second, etc, indices). – Danu Jun 21 '18 at 22:47
• @AccidentalFourierTransform I like your equals sign with the "no" above it; how do you do that? – probably_someone Jun 22 '18 at 3:26
• @probably_someone right click > show math as > TeX commands :-) – AccidentalFourierTransform Jun 22 '18 at 3:28
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# What is the relationship between G and g in physics?
There are differences between Gravitation (G) and Gravity (g) are as follow:
Gravitation (G) Gravity (g) 1. Gravitation is the force of attraction acting between any two bodies of the universe. 1. Gravity is the earth's gravitational pull on the body, lying near the surface of the earth. 2. It requires two masses. 2. It requires only one mass. 3. It is a weak force. 3. It is a strong force. 4. G is the Universal Gravitational Constant. 4. g is the acceleration due to gravity. 5. The force, $F=G\frac{M1\times M2}{{R}^{2}}$ (G = gravitational constant). 5. The force, $F=m\times g$ (g = acceleration due to gravity). 6. The value of G remains constant everywhere.(G = 6.673×10-11Nm2/kg2) 6. The value of g varies from one place to another on the Earth.
There is no connection between acceleration due to gravity (g) and universal gravitation constant (G), as the value of G is constant. Also, they are not dependent on each other.
However, a formula exists to express the relation between g and G in physics.
$g=\frac{GM}{{R}^{2}}$
Where,
g = the acceleration due to the gravity of any massive body measured in m/s2.
G = the universal gravitational constant measured in Nm2/kg2.
R = the radius of the massive body measured in km.
M = the mass of the massive body measured in Kg.
The relationship can be derived as-
According to the universal law of gravity,
$F=\frac{GMm}{{R}^{2}}$ ---------------> (i)
From Newton’s second law of motion, we can write –
$g=\frac{F}{m}$ ---------------> (ii) Substituting equation (i) in (ii) we get-
$g=\frac{GMm}{{R}^{2}m}$
Thus, we infer the g formula in physics as -
$g=\frac{GM}{{R}^{2}}$
Tutorialspoint
Simply Easy Learning
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# Electrodynamics
Magnetic field of a solenoid
The classical electrodynamics (also electrical theory ) is the branch of physics that deals with moving electric charges and time-varying electric and magnetic fields busy. The electrostatics as a special case of electrodynamics deals with dormant electrical charges and their fields. The underlying fundamental force in physics is called electromagnetic interaction .
Hans Christian Ørsted (1820) is considered to be the discoverer of the connection between electricity and magnetism , although Gian Domenico Romagnosi (1802) was his forerunner, which was hardly noticed at the time. The theory of classical electrodynamics was formulated by James Clerk Maxwell in the mid-19th century using the Maxwell equations named after him . The investigation of Maxwell's equations for moving reference systems led Albert Einstein in 1905 to the formulation of the special theory of relativity . During the 1940s it was possible to combine quantum mechanics and electrodynamics in quantum electrodynamics ; their predictions agree very precisely with the measurement results.
An important form of electromagnetic fields are electromagnetic waves , one of which is the most well-known representative of visible light . Its research forms a separate area of physics, optics . The physical basis for the description of electromagnetic waves is provided by electrodynamics.
## Classical electrodynamics
### Basic equations
The interaction of electromagnetic fields and electrical charges is fundamentally determined by the microscopic Maxwell equations
{\ displaystyle {\ begin {aligned} \ operatorname {div} {\ vec {B}} & = 0, & \ operatorname {red} {\ vec {E}} + {\ frac {\ partial {\ vec {B }}} {\ partial t}} & = 0 \ ,, \\\ operatorname {div} {\ vec {E}} & = {\ frac {\ rho} {\ varepsilon _ {0}}} \ ,, & \ operatorname {red} {\ vec {B}} - \ mu _ {0} \, \ varepsilon _ {0} \, {\ frac {\ partial {\ vec {E}}} {\ partial t}} & = \ mu _ {0} \, {\ vec {j}}. \ end {aligned}}}
and the Lorentz force
${\ displaystyle {\ vec {F}} = q ({\ vec {E}} + {\ vec {v}} \ times {\ vec {B}})}$
certainly.
With the help of the material equations of electrodynamics, the macroscopic Maxwell equations result. These are equations for the effective fields that appear in matter .
Also play an important role (derived from this):
1. the continuity equation , which says that the charge is retained,${\ displaystyle {\ frac {\ partial \ rho} {\ partial t}} + \ operatorname {div} {\ vec {j}} = 0}$
2. the Poynting's theorem , which states that the energy of particles and fields is maintained overall.
### Potentials and wave equation
The homogeneous Maxwell equations
${\ displaystyle {\ text {div}} \, {\ vec {B}} = 0}$
and
${\ displaystyle \ operatorname {rot} \, {\ vec {E}} + {\ frac {\ partial {\ vec {B}}} {\ partial t}} = 0}$
can by introducing the electromagnetic potentials according to
${\ displaystyle {\ vec {B}} = \ operatorname {red} \, {\ vec {A}}}$
and
${\ displaystyle {\ vec {E}} = - {\ text {grad}} \, \ phi - {\ frac {\ partial {\ vec {A}}} {\ partial t}}}$
solved identically in a star-shaped region ( Poincaré lemma ). The so-called scalar potential and the vector potential denote . Since the physical fields are only given by derivatives of the potentials, one has certain freedom to change the potentials and still get the same physical fields back. For example, and result in the same field if you look through them ${\ displaystyle \ phi}$${\ displaystyle {\ vec {A}}}$${\ displaystyle {\ vec {A}} '}$${\ displaystyle {\ vec {A}}}$${\ displaystyle B}$
${\ displaystyle {\ vec {A}} '= {\ vec {A}} + {\ text {grad}} \, \ Lambda}$
relates to each other. If one also demands that the same field results from such a transformation , it must be like ${\ displaystyle E}$${\ displaystyle \ phi}$
${\ displaystyle \ phi '= \ phi - {\ frac {\ partial \ Lambda} {\ partial t}}}$
transform. Such a transformation is called a gauge transformation. Two calibrations are often used in electrodynamics. First, the so-called Coulomb calibration or radiation calibration
${\ displaystyle {\ text {div}} \, {\ vec {A}} = 0}$
and secondly the Lorenz calibration
${\ displaystyle {\ frac {1} {c ^ {2}}} {\ frac {\ partial \ phi} {\ partial t}} + {\ text {div}} \, {\ vec {A}} = 0}$.
The Lorenz calibration has the advantage of being relativistically invariant and of not changing structurally when changing between two inertial systems. The Coulomb calibration is not relativistically invariant, but is more used in the canonical quantization of electrodynamics.
If the - and - fields and the vacuum material equations are inserted into the inhomogeneous Maxwell equations and the potentials are calibrated according to the Lorenz calibration, the inhomogeneous Maxwell equations decouple and the potentials satisfy inhomogeneous wave equations${\ displaystyle E}$${\ displaystyle B}$
${\ displaystyle \ Box \ phi = {\ frac {\ rho} {\ varepsilon _ {0}}} \ ,, \, \ Box {\ vec {A}} = \ mu _ {0} {\ vec {j }} \ ,.}$
Here denotes the D'Alembert operator . ${\ displaystyle \ Box}$
### Special cases
The electrostatics is the special case motionless electrical charges and static (not changing over time) electric fields. It can also be used within limits as long as the speeds and accelerations of the charges and the changes in the fields are small.
The Magnetostatics deals with the special case of constant currents in a total of uncharged conductors and constant magnetic fields. It can be used for currents and magnetic fields that change slowly enough.
The combination of the two, electromagnetism, can be described as the electrodynamics of the charges that are not accelerated too much. Most processes in electrical circuits (e.g. coil , capacitor , transformer ) can already be described at this level. A stationary electric or magnetic field remains close to its source, such as the earth's magnetic field . However, a changing electromagnetic field can move away from its origin. The field forms an electromagnetic wave in the interplay between magnetic and electric fields. This radiation of electromagnetic waves is neglected in electrostatics. The description of the electromagnetic field is limited here to the near field.
Electromagnetic waves, on the other hand, are the only form of electromagnetic field that can exist independently of a source. Although they are created by sources, they can continue to exist after they have been created regardless of the source. Since light can be described as an electromagnetic wave, optics is ultimately a special case of electrodynamics.
### Electrodynamics and Theory of Relativity
In contrast to classical mechanics, electrodynamics is not Galileo-invariant . This means that if one assumes an absolute, Euclidean space and an independent absolute time , as in classical mechanics , then the Maxwell equations do not apply in every inertial system .
Simple example: A charged particle flying at constant speed is surrounded by an electric and a magnetic field. A second particle, flying at the same speed and having the same charge , experiences a repulsive force from the electric field of the first particle , since charges of the same name repel each other; At the same time, it experiences an attractive Lorentz force from its magnetic field , which partially compensates for the repulsion. At the speed of light this compensation would be complete. In the inertial system, in which both particles rest, there is no magnetic field and therefore no Lorentz force. Only the repulsive Coulomb force acts there , so that the particle is accelerated more strongly than in the original reference system in which both charges move. This contradicts Newtonian physics, in which the acceleration does not depend on the reference system.
This knowledge led to the assumption that there is a preferred reference system in electrodynamics (ether system). Attempts to measure the speed of the earth against the ether , however, failed, for example the Michelson-Morley experiment . Hendrik Antoon Lorentz solved this problem with a modified ether theory ( Lorentz's ether theory ), which, however, was replaced by Albert Einstein with his special theory of relativity . Einstein replaced Newton's absolute space and time with a four-dimensional space - time . In the theory of relativity, the Galileo invariance is replaced by the Lorentz invariance , which is fulfilled by electrodynamics.
In fact, the reduction in acceleration and thus the magnetic force in the above example can be explained as a consequence of the length contraction and time dilation , if the observations made in the moving system are transformed back into a system at rest. In a certain way, the existence of magnetic phenomena can ultimately be traced back to the structure of space and time, as described in the theory of relativity. From this point of view, the structure of the basic equations for static magnetic fields with their cross products also appears less surprising.
In the manifest Lorentz form invariant description of electrodynamics, the scalar potential and the vector potential form a four-vector , analogous to the four-vector of space and time, so that the Lorentz transformations can also be applied analogously to the electromagnetic potentials. In the case of a special Lorentz transformation with the velocity in the direction, the transformation equations apply to the fields in the common SI system of units : ${\ displaystyle v}$${\ displaystyle z}$
${\ displaystyle E '_ {x} = {\ frac {E_ {x} -vB_ {y}} {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}} }}$ ${\ displaystyle B '_ {x} = {\ frac {B_ {x} + {\ frac {v} {c ^ {2}}} E_ {y}} {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}}}}$ ${\ displaystyle E '_ {y} = {\ frac {E_ {y} + vB_ {x}} {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}} }}$ ${\ displaystyle B '_ {y} = {\ frac {B_ {y} - {\ frac {v} {c ^ {2}}} E_ {x}} {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}}}}$ ${\ displaystyle E '_ {z} = E_ {z} \,}$ ${\ displaystyle B '_ {z} = B_ {z} \,}$
(These equations are only insignificantly modified in cgs units: Formally, you only have to substitute or with or .) ${\ displaystyle {\ vec {B}}}$${\ displaystyle {\ vec {B}} '}$${\ displaystyle {\ vec {B}} / c}$${\ displaystyle {\ vec {B}} '/ c}$
## Extensions
However, classical electrodynamics does not provide a consistent description of moving point charges; problems such as radiation feedback arise on small scales . The quantum electrodynamics (QED) combines the electrodynamics therefore with quantum mechanical concepts. The theory of the electroweak interaction unites the QED with the weak interaction and is part of the Standard Model of elementary particle physics. The structure of the QED is also the starting point for quantum chromodynamics (QCD), which describes the strong interaction . However, the situation there is even more complicated (e.g. three types of charge, see color charge ).
A unification of electrodynamics with the general theory of relativity ( gravitation ) is known as the Kaluza-Klein theory and represents an early attempt to unify the fundamental interactions .
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# Math Help - exponential base. logarithm.
1. ## exponential base. logarithm.
so working on a few exponential functions. i know that you can make the exponents equal to one another if the bases are the same but what if there is a single variable and the terms are multiplied/added.
what i have so far:
$3^{2x}-2*3^{x+5}+3^{10} = 0$
$(3^{2x})(3^{x+5})+3^{10} = 2$
is it correct to move over the 2 to work with the same bases easier?
or is this completely off and i should use logarithm and natural log? if so, please give hints as to how to work it out.
thankyou~
2. Hello, ninjuhtime!
What i have so far:
$3^{2x}-2\cdot3^{x+5}+3^{10} \:= \:0$
$(3^{2x})(3^{x+5})+3^{10} = 2$ . . . . Definitely illegal!
It's hard to see it, but the expression is a perfect square . . .
We have: . $\left(3^x - 3^5\right)^2 \:=\:0 \quad\Rightarrow\quad 3^x - 3^5 \:=\:0 \quad\Rightarrow\quad 3^x \:=\:3^5 \quad\Rightarrow\quad\boxed{ x \:=\:5}$
3. thankyou! i guess i oversaw what it was since long problems are usually intimidating.
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# Crowdsourcing punishment: Individuals reference group preferences to inform their own punitive decisions
## Abstract
Justice systems delegate punishment decisions to groups in the belief that the aggregation of individuals’ preferences facilitates judiciousness. However, group dynamics may also lead individuals to relinquish moral responsibility by conforming to the majority’s preference for punishment. Across five experiments (N = 399), we find Victims and Jurors tasked with restoring justice become increasingly punitive (by as much as 40%) as groups express a desire to punish, with every additional punisher augmenting an individual’s punishment rates. This influence is so potent that knowing about a past group’s preference continues swaying decisions even when they cannot affect present outcomes. Using computational models of decision-making, we test long-standing theories of how groups influence choice. We find groups induce conformity by making individuals less cautious and more impulsive, and by amplifying the value of punishment. However, compared to Victims, Jurors are more sensitive to moral violation severity and less readily swayed by the group. Conformity to a group’s punitive preference also extends to weightier moral violations such as assault and theft. Our results demonstrate that groups can powerfully shift an individual’s punitive preference across a variety of contexts, while additionally revealing the cognitive mechanisms by which social influence alters moral values.
## Introduction
In the classic film Twelve Angry Men, a jury must decide whether to convict a defendant accused of murder, thereby condemning him to death. In a memorable scene, Juror #8 hesitantly looks around the table as the initial vote is being taken by a show of hands, then conforms to the majority by voting ‘guilty.’ Why did the juror look to his peers when considering his vote? As in the film, it could be that we turn to others for guidance about how to enact justice in real-world contexts. Decades of conformity experiments demonstrate that individuals’ behaviors, preferences, and beliefs can be swayed by their peers in domains as wide-ranging as visual perception, inferences about socially normative behaviors, intensity of pain experience, preferences for food and music, and the attractiveness of faces1,2,3,4,5,6,7. Furthermore, recent evidence from both lab and field studies suggests that people behave more prosocially after observing others engaging in behaviors such as cooperation and generosity8,9,10.
However, since moral values appear to be deeply-held, stable across time, and resistant to change11,12,13, a parallel literature on moral cognition makes a different prediction: insofar as punishment is a method of restoring justice following a moral violation, punitive preferences should be less susceptible to group influence. For example, people form more extreme judgments of an action’s rightness or wrongness when primed to perceive the action as having a moral dimension14, which results in a stronger entrenchment of moral attitudes15. In field studies, moralized attitudes lead to decreased support for political compromise16 and contribute to the intractability of conflicts in which sacred values are at stake17. Indeed, it appears that moral attitudes may be an immutable feature of one’s individual identity18, and are therefore unwavering across social contexts.
Given these divergent predictions, an important goal is to test the susceptibility of moral attitudes—in this case, punishment preferences—to group influence, and to understand the social and cognitive factors that might lead an individual to conform to a group’s desire to punish. Understanding the malleability of punitive preferences is especially important given that punishment of norm violators is a pervasive part of human social life19,20,21. Moreover, punishment decisions are often made within group contexts22, and are decided both by those who have been directly affected by a crime (i.e., victims), and by those who are impartial third parties (i.e., jurors).
Victims, for example, are often given the ability to recommend (or even determine) punishment across a wide variety of social settings: the submission of impact statements as evidence in a courtroom23, deployment of informal or quasi-judicial practices in institutional contexts such as universities based on the responses of many victims24, tribal and religious councils25, war tribunals where punishment recommendations are made collectively by victims26, and even instances when people employ vigilante justice27. These real-world decisions about justice are sometimes made within contexts where groups have the ability to sway a victim’s desire for punishing a perpetrator, as is commonly observed when crowd dynamics lead to escalated violence during riots28,29.
A juror—who is tasked with making punishment decisions on behalf of victims—typically also makes a punishment recommendation within the context of a group, which introduces the possibility that an individual juror can be swayed by others’ preferences30. One large-scale field study surveyed nearly 3,500 real jurors who had actually decided upon felony cases and found that over one-third of them would have reversed their jury’s decision if they had been given sole control over the trial’s outcome31. Though observational studies such as these are limited in their ability to establish causal mechanisms, this finding provides preliminary evidence that a group’s preference can have a powerful influence on one’s willingness to punish. This is an especially important phenomenon to understand if we consider that wrongful punishment has catastrophic and irreversible consequences32. As a vivid illustration of the potential consequences of conformity in punishment decisions, we return to the fictional example from Twelve Angry Men, in which all but one juror initially voted to convict. Were it not for this standalone juror who refused to be swayed by the majority, the defendant—proven innocent at the end of the film—would have been wrongfully sentenced to death.
To investigate the psychological factors and cognitive mechanisms influencing one’s willingness to endorse punishment as a means of restoring justice, we leverage an interdisciplinary approach, drawing from social psychology, behavioral economics, and computational modeling. We first tested whether individuals’ punitive preferences are malleable when they are a victim of a moral violation (Experiments 1–2). Based on classic conformity research showing that individuals are sensitive to the proportion of people within a group who endorse an option in a non-moral context1, we examined whether victims’ punishment rates scale with the proportion of punitive preferences expressed within a group (Experiment 1). We further investigated the strength of social influence by examining whether victims’ punishment behaviors continue to be affected by a past group’s punitive preferences, even when that past group’s preference is objectively irrelevant to the current decision (Experiment 2). Given existing work illustrating that people punish norm violators both as a second-party victim or third-party juror19,20,33, we then tested whether an individual’s susceptibility to group influence hinges on whether they are a wronged victim or an unaffected juror (Experiments 3–4). In addition to examining behavior, we employed a computational model of decision-making (the Drift Diffusion Model) to reveal how a group’s preferences influence the cognitive processes underlying punishment decisions when tasked with restoring justice. Finally, we assessed whether conformity to a group’s punitive preferences generalizes to a variety of other real-world moral transgressions (Experiment 5).
## Results
### Experiment 1: Susceptibility to a group’s punishment preferences as Last Decider
We first examined whether decisions to restore justice are malleable, and if so, whether these moral preferences are sensitive to the proportion of other individuals endorsing a punitive response in the wake of a fairness violation. Subjects completed a variant of the Justice Game, an economic task specifically designed to measure punitive and non-punitive responses to fairness violations33. In this task, Player A is endowed with $10 on each trial and can choose how much to split with the subject, who always takes the role of Player B (Fig. 1a). Player A’s splits ranged from mildly unfair (a 6/4 split in which Player A keeps$6 and offers $4) to highly unfair (9/1 split) in$1 increments. After receiving an unfair offer, Player B then decided how to restore justice by choosing to: (i) Accept Player A’s split as-is; (ii) Compensate themselves in a cost-free manner by increasing their own payout to match Player A’s payout, without punishing Player A; or (iii) Reverse Player A’s split, a highly retributive option that maximizes Player B’s own payout and minimizes Player A’s payout (Fig. 1a).
Subjects first completed a Solo Phase of this task, allowing us to measure subjects’ punishment preferences in the absence of any social influence. Accounting for the possibility that subjects might infer strategic motives behind Player A offers, payout at the end of the study was implemented probabilistically such that Player A’s original offer was enacted half of the time and Player B’s decision was enacted half of the time (see Supplementary Methods). Indeed, subjects Reversed at greater rates as offers became increasingly unequal, indicating that they perceived small offers as unfair and punished the perpetrator accordingly. Subjects played with new Player As on each trial.
Subjects then completed a Group Phase of the task, in which they were told that they were sharing the role of Player B alongside four other subjects, and that their responses would collectively determine the payouts of all the players (Fig. 1b). Specifically, subjects were informed that the payout redistribution would be determined by the majority choice of the five Player Bs, that Player A’s payout would be affected by Player Bs’ collective decision just as it had in the Solo Phase, and that each Player B would receive the full payout. As in the Solo Phase, each of the players sharing the role of Player B could choose to Accept, Compensate, or Reverse Player A’s split, and subjects observed the other Player Bs’ choices in sequence, as if they were made in real time (Fig. 1b). Subjects always responded last, following the format of the classic Asch conformity paradigm1. Subjects played with new Player As and Bs on each trial. To test the possibility that endorsement rates of the punitive Reverse option would increase as the number of punishers within the group increased, we parametrically and deterministically varied the proportion of players who selected the punitive option from 0% to 100%.
To examine whether a group’s punitive preferences alter an individual’s willingness to punish, we conducted a mixed-effects logistic regression analysis. We modeled the probability of punishing as an additive combination of the proportion of punishers in the group (centered around 50% punishers), the unfairness of Player A’s offer (centered around medium unfairness), and each subject’s baseline preference for punishment (i.e., the proportion of Solo Phase trials in which a subject chose to Reverse instead of Compensate, matched for offer unfairness and centered around 50% punishment). Results reveal that individuals significantly increase punishment as a greater proportion of the group expresses a punitive preference, even after accounting for offer unfairness and baseline punitive preferences (Table 1; Fig. 2). This effect was so strong that the group was ultimately able to shift individuals’ predicted punishment rates as much as 30 percentage points.
### Experiment 2: Influence of past groups’ punishment preferences as First Decider
The findings from Experiment 1 are the first that we are aware of illustrating that individuals’ moral attitudes about punishment vary with the intensity of a group’s preference. To examine the robustness of this conformity effect and the potency of social influence, we conducted a second experiment examining the possibility that individuals might even conform to previous groups endorsing punishment. Specifically, we tested two competing hypotheses: whether the effect of social influence is abolished as soon as a past group’s preferences are no longer relevant to the current decision context1,34, or whether knowing that a past group sanctioned punishment would also affect how readily an individual chooses to punish in the present moment. Evidence for the second hypothesis would suggest that decisions to punish are highly malleable. To probe whether subjects’ punitive behaviors are susceptible to influence even from past groups’ preferences, subjects in Experiment 2 were first in their group to decide during the Group Phase (i.e. playing as the “First Decider”, in contrast to Experiment 1, in which subjects played as the “Last Decider”).
The methodology of Experiment 2 was identical to that of Experiment 1, with one key difference: the sequence of Player Bs in the Group Phase was fixed so that subjects were always the first person in the group to make their decision (Fig. 1c). When deciding first, subjects did not have access to their present group’s punitive preferences. Therefore, on any given trial, the only two pieces of information that could affect a subject’s decision were the unfairness of Player A’s offer and the preferences of past groups. Given that behavioral economic experiments examining fairness norms have found that people are sensitive to the magnitude of fairness violations, including in the Justice Game33, we defined the proportion of punishers in a way that would allow us to match fairness violations across trials. We employed a simple analysis approach such that on each trial, we recorded Player A’s fairness violation (from mildly unfair splits of $6/$4 to highly unfair splits of $9/$1), searched backwards until we found the most recent past trial in which the severity of the fairness violation matched that of the present trial, and then used the proportion of punishers observed on that past trial to predict decisions to punish on the current trial. Alternative models with different operationalization of trial history are discussed in the Supplementary Results; we report the best-fitting model here.
This task design also allows us to address two potential concerns about the validity of our paradigm: first, that explicit pressure to conform in Experiment 1 encouraged subjects to behave in ways they believed would please the experimenter, and second, that the observed conformity effect is only observable under conditions where people believe that their individual decision has little influence on the outcome of the group’s final decision. Therefore, in addition to testing whether punishment decisions remain susceptible to past groups’ influence, the task design of Experiment 2 also allowed us to determine whether the conformity effect replicates in a decision context where explicit pressure to conform is reduced and where subjects explicitly know their choice can affect the final outcome.
As in Experiment 1, results indicate that increasing the proportion of punishers within a group significantly enhances punishment endorsement rates, even when accounting for offer unfairness and baseline punitive preferences (Table 2; Fig. 2). Although the strength of the conformity effect was attenuated relative to when individuals were last to decide, our results illustrate that individuals still conform when their decisions are consequential and when they are relatively free from experimenter demand effects. An ancillary analysis also demonstrates our results are not significantly modulated by the time elapsed since the key trial used to define the proportion of punishers (Supplementary Results).
### Experiment 3: Drift diffusion model of conformity as a Victim
Experiments 1–2 reveal just how powerfully social influence can alter individuals’ endorsement of punishment as a means of restoring justice. However, it remains an open question how social influence operates on the cognitive mechanisms underlying decisions to punish norm violators, thereby producing conformist behaviors. To explore this question, we performed a third experiment leveraging the Drift Diffusion Model (DDM), a computational model of decision-making. Though the DDM is typically used in perceptual decision-making contexts, it is well-suited for examining the effects of social influence, and has recently gained some traction within the social domain, successfully explaining how social groups bias perceptual judgments and even how altruistic decisions unfold35,36.
In the context of our task, DDM uses choices and reaction time distributions to characterize how people integrate evidence about the value of punishment (such as the group’s punitive preference) relative to compensation37. This allows us to decompose the decision-making process into psychologically-meaningful parameters. First, bias (z) quantifies the extent to which people lean towards one of the justice restoration options prior to observing any evidence. Second, decision threshold (a) indexes the amount of evidence subjects need in order to make a choice, thereby capturing how cautiously individuals make choices in the presence (and absence) of group influence. Third, drift rate (v) indexes the strength of evidence favoring either punishment or compensation obtained from observing the group’s preference, and therefore how much an individual weighs the value of punishment relative to the value of compensation36,38. Importantly, the strength of evidence in our task is not dependent on the dynamics of the stimulus (such as movement coherence in random dot motion tasks), but instead reflects the psychological process through which a group’s punitive preferences are dynamically integrated into an individual’s valuation of punishment.
Therefore, DDM enables us to rigorously test two longstanding hypotheses from social psychological theory about how group influence acts on the cognitive mechanisms governing decision-making. First, we can test whether the presence of a group majority lowers the stakes of an individual’s decision (as the subject is now merely one vote out of many), thereby reducing the total amount of evidence required to commit to a choice and leading the individual to relinquish moral responsibility. Second, we can test whether the proportion of people within a group endorsing a punitive option increases the strength of evidence that punishment is a valuable method for restoring justice.
Given that the greatest conformity effects occur when subjects are the last to decide within the group, Experiment 3 followed the same task structure as Experiment 1 (Last Decider), with a few key modifications to make the task suitable for DDM. First, because the best-established DDMs only account for binary choices, subjects were only presented with the Compensate and Reverse options on each trial. Second, subjects sometimes made decisions in the absence of information about others’ preferences during the Group Phase of the task (hereafter referred as the Alone condition), which allows us to avoid ordering confounds from the Solo Phase that might contaminate DDM parameters (e.g. from unrelated motor or task learning). Third, offers from Player As were placed within discrete monetary bins and jittered to offset task habituation from repeatedly seeing only four distinct offer types (as in Experiments 1–2), which was especially important given the large number of trials needed to estimate DDM parameters. Therefore, offers from Player As were binned into three levels of unfairness, all drawn from a uniform distribution in increments of 10¢: Mildly Unfair offers between $3.70 and$4.90, Somewhat Unfair offers between $1.90 and$3.10, and Highly Unfair offers between $0.10 and$1.30. Finally, because DDM requires reaction time (RT) distributions to capture the decision process in its entirety37, subjects were simultaneously presented with Player A’s offer and four randomly-sampled responses from other Player Bs (Fig. 3a,c; this is in contrast to Experiments 1–2, in which subjects observed Player A offers and Player B choices sequentially). Subjects were free to make a response at their own pace, but were encouraged to make decisions as quickly as possible.
Replicating the behavioral findings from Experiment 1, results indicate that the group’s punitive preferences significantly affects endorsement of punishment (Table 3; Fig. 3e), shifting individuals’ punitive choices by as much as 40 percentage points. We then used the HDDM software package to perform hierarchical Bayesian estimation of DDM parameters39. The drift rate (v), threshold (a), and bias (z) parameters were estimated at the group level using a hierarchical Bayesian procedure. This allows individuals’ contributions to the group parameters to be weighted according to their diagnostic value, maximizes power by capitalizing on statistical similarities between subjects, and addresses collinearity between variables by incorporating greater uncertainty in the posteriors of parameter estimates. To additionally account for within-subject variability, the proportion of punishers was regressed onto the DDM parameters of interest. Three regressions were performed, one for each bin of offer unfairness. Decisions to punish were mapped to the upper boundary, and decisions to compensate were mapped to the lower boundary (Fig. 3b). Separate parameters were fit for each bin of offer unfairness. Drift rate and threshold were allowed to vary by the proportion of punishers in the group, whereas a single bias parameter was estimated for each type of unfairness. This model specification reflects that people likely have preexisting biases about how much punishment is warranted depending on the degree of fairness violations, while making no claims that preexisting bias varies according to group dynamics. Alternatively-specified models are tested and discussed in the Supplementary Information.
Statistical significance in the context of Bayesian estimation is determined by the proportion of values in a posterior distribution that fall above or below a point value, such as zero (i.e., testing whether a regression coefficient is different from zero), or the mean of another posterior distribution. Bayesian hypothesis testing is therefore conceptually akin to performing a frequentist t-test40. We define significance as 95% of posterior values falling above (or, depending on the analysis, below) the specified point value. To avoid confusion, we report 1–p, as this statistic more closely resembles p-values from frequentist significance testing.
First, given past findings from the Justice Game that Victims typically prefer to compensate in the wake of a fairness violation33, as well as our behavioral data from Experiments 1–2, we predicted that Victims would exhibit an overall bias in favor of choosing the compensatory option over the punitive one. This would be reflected by the bias (z) parameter being closer to the decision boundary for compensating (see Fig. 3b for a schematic). Values greater than 0.5 indicate an initial preference for punishment, and values less than 0.5 indicate an initial preference for compensation. Indeed, we found that Victims exhibit a preference for compensation over punishment, reflected by 95% of posterior z estimates falling under the point value 0.5 (average z for Mildly Unfair = 0.42, Somewhat Unfair = 0.43, Highly Unfair = 0.44, all posterior Ps < 0.001). This indicates that highly-punitive groups were able to influence Victims’ behaviors despite individuals having a predisposition not to punish. This preference for compensation did not significantly vary as a function of offer unfairness (all posterior Ps > 0.10).
Second, we predicted that individuals would be less cautious within groups where a majority of group members endorse punishment or compensation, as the individual would only be one vote of many. This would be reflected in a decrease in the distance a between decision thresholds (Fig. 3d). Consistent with our hypothesis, results reveal that Victims have a lowered average decision threshold when making decisions within a group majority, relative to a group that is evenly split between punishers and compensators (all posterior Ps < 0.001; Fig. 4). We additionally find that individuals are generally less cautious and more impulsive when choosing within a group majority, relative to when they are choosing alone (Fig. 4). Average a did not differ as a function of offer fairness (all posterior Ps > 0.40).
Third, because greater group endorsement of the punitive option provides individuals with stronger evidence favoring punishment, we predicted that drift rate (v) would increase as the proportion of punishers in the group increased. The magnitude of v indicates the strength of evidence favoring each option, while the sign of v indicates whether the evidence favors punishment (positive) or compensation (negative; see Fig. 3b,d for a schematic). Consistent with subjects’ average preference for compensation, drift rates are largely negative. However, v increases with the proportion of punishers, demonstrating that Victims used the proportion of punishers as accumulating evidence of punishment’s value (Fig. 5). Average v did not differ as a function of offer fairness (all posterior Ps > 0.05).
### Experiment 4: Drift diffusion model of conformity as a Juror
Experiment 3 replicates our finding that Victims’ punishment decisions are susceptible to the preferences of a group. In addition, these findings demonstrate that groups act upon the decision-making process by making individuals less cautious about their choices, and by increasing how much individuals value punishment as a method of restoring justice. However, a potential alternative interpretation of these results is that individuals are acting upon an existing desire to punish and are thus not conforming to a group’s punitive preferences. That is, Victims may be reluctant to punish norm violators because they believe that retribution is perceived as being socially undesirable; observing others being punitive would then enable them to act upon their latent desire to punish. To distinguish between these alternative interpretations—and to examine whether people also conform to a group’s punitive preferences when tasked with making third-party punishment decisions19,20,30,31—we conducted a fourth experiment that was identical to Experiment 3 with the major exception that subjects made punishment decisions as Jurors on behalf of victims.
Behavioral results indicate that the group’s punitive preferences significantly modulates punishment (Table 4), shifting predicted punishment by nearly 30 percentage points. A formal comparison between Experiment 3 (Victim) and Experiment 4 (Juror) additionally finds that the conformity effect is significantly weakened for Jurors relative to Victims (β = 0.50, SE = 0.13, z = −2.58, p = 0.010; estimate and SE are in units of odds ratios).
In contrast to Victims (who were consistently biased in favor of compensation), DDM results reveal that Jurors were only biased in favor of compensation when fairness violations were Mildly Unfair (average z = 0.45, posterior P < 0.001). When offers were instead Somewhat or Highly Unfair, z was not statistically different from the neutral starting point 0.5 (average z for Somewhat Unfair = 0.48, Highly Unfair = 0.48, both posterior Ps > 0.05). This reveals that before observing the group’s preferences, Jurors were more neutral than Victims (i.e., not biased towards either compensation or punishment).
Replicating the pattern observed for Victims, Jurors exhibited a lowered decision threshold when choosing in a group majority than when choosing in an evenly-split group (all posterior Ps < 0.001), and generally when choosing in a group majority relative to choosing alone (with a noticeable exception for when a majority of the group chose to punish highly unfair offers; Fig. 4). Although the pattern of threshold estimates between Victims and Jurors appear to be qualitatively different upon visual inspection (Fig. 4), direct statistical comparisons of the a parameter reveals no significant difference between Victims’ and Jurors’ decision criterions when comparing average a for each bin of offer unfairness (all posterior Ps > 0.20). Average a did not differ as a function of offer fairness for Jurors (all posterior Ps > 0.30), as was the case for Victims.
While Jurors’ drift rates were found to increase as the proportion of punishers in the group increased, echoing the pattern found when Victims decided the outcome (Fig. 5), results also reveal interesting differences between Victims and Jurors in how strongly groups provide compelling evidence for punishment. Whereas Victims’ average v did not differ depending on offer unfairness, Jurors’ average v significantly increased as offers became increasingly unfair (both pairwise posterior Ps < 0.05). To further probe this relationship, we fit a variant model comparing Victims and Jurors, where the number of punishers within the group was treated as a continuous variable (see Supplementary Results). Results from this model indicate that the intercept of v does not significantly differ depending on offer unfairness for Victims (all posterior Ps > 0.1), but significantly increases as offers become more unfair for Jurors (all posterior Ps < 0.05; Fig. 6a). In other words, the degree of unfairness does not influence Victims’ valuation of punishment, whereas Jurors incorporate this information into their decisions, placing higher value on punishment as unfairness increases. Additionally, the regression betas for the number of punishers are significantly greater for Victims than Jurors (all posterior Ps < 0.001; Fig. 6b), meaning that each additional punisher in a group provides stronger evidence that punishment is valuable when one is an affected Victim than when one is an impartial Juror. These two patterns are also reflected in the behavioral data (Fig. 3e).
### Experiment 5: Crime judgments
Taken together, Experiments 1–4 demonstrate robust group influence on an individual’s punitive behavior, regardless of whether one is deciding as a Victim or Juror. However, our use of an economic game paradigm to precisely measure behavior leaves open an important question: is a group’s influence on an individual’s desire to punish limited to fairness violations, or does social influence act more generally upon a variety of moral violations? In order to answer this question, we performed a fifth experiment to examine whether a group can shift an individual’s judgments of how severely perpetrators should be punished for committing different types of crimes.
Subjects read short vignettes describing either a physical assault or a theft. The crime type (assault vs theft) was crossed with two levels of crime severity: high-intensity crimes involved the use of weapons, whereas low-intensity crimes did not. All vignettes featured an unambiguous perpetrator and victim. For subjects who completed the Victim condition, vignettes were written using second-person pronouns, and subjects were asked to imagine that they were the victim of the crime. The exact same vignettes were presented in the Juror condition, but instead of using second-person pronouns, the victims were instead unrelated strangers.
After reading the vignette, subjects were asked to rate how severely the perpetrator should be punished on a 100-point scale, where 0 corresponded to “Mild Punishment” and 100 corresponded to “Severe Punishment” (Fig. 7a). On Alone trials, subjects made their judgment in the absence of any social information. On Group trials, subjects were shown four icons that represented the responses of four past participants (though in reality, all responses were experimenter-generated to fully parameterize the decision space). The icons indicated the proportion of the group that had previously endorsed severe punishment as opposed to mild punishment.
To examine whether a group’s endorsement of punishment alters an individual’s judgments about the severity of punishment warranted by real-world crimes, we performed a linear mixed-effects regression, modeling judgments as an interaction between the proportion of punishers in the group (centered around 50% punishers), the type of crime (assault vs theft), and crime intensity (centered around medium intensity), and a covariate for subjects’ baseline preference for punishment (mean-centered and standardized). The overall pattern of results reveals that the proportion of group members endorsing severe punishment significantly modulates subjects’ judgments of how severely crimes should be punished (Fig. 7b), both when they are Victims (Table 5) and Jurors (Table 6). A formal comparison between Victims and Jurors further reveals that the overall conformity effect is significantly stronger for Jurors than for Victims (β = 2.06, SE = 0.69, z = 3.00, df = 224.00, p = 0.003), unlike our findings from Experiments 3–4. This divergence may be due to differences in moral decision-making when choices are hypothetical41,42, or may alternatively be due to the fact that the vignettes used in Experiment 5 involve moral transgressions that are more severe than the fairness violations in Experiments 1–4.
## Discussion
Groups often make better decisions than individuals by aggregating the knowledge, perspectives, and opinions of many people. This insight has led legal systems in many countries to use juries as fact-finders and arbiters of punishment22. Though juries are often able to use the ‘wisdom of the crowd’ to reach fair and judicious verdicts, groups can also produce conformist behavior, which may introduce biases into the decision-making process. Indeed, we find that subjects—Victims and Jurors alike—swiftly conform to groups’ punitive preferences, changing their own punishment behaviors by as much as 40 percentage points. Across all studies, this conformity effect scales linearly, such that individuals incrementally increase punishment as the proportion of punishers within the group grows larger. This indicates that individuals are integrating the group’s preference into their valuation of punishment, a conclusion that is additionally supported by the results of our computational model.
Though foundational theories of conformity have long invoked the metaphor of information processing1,34, little work has actually examined how individuals integrate others’ preferences into their own values as they are making choices. Therefore, by modeling decision-making as an information integration process, we extend a rich body of research on social influence by contributing three major insights into the cognitive dynamics that give rise to conformity. First, choosing in a group context causes individuals to become less cautious and more impulsive about their choices, making them especially susceptible to relinquishing moral responsibility and exhibiting conformist behavior. Second, being exposed to a group’s punitive preference amplifies how much an individual weighs evidence that punishment is a valuable mechanism for enacting justice. Third, groups are able to exert strong influence over individuals’ punitive behaviors regardless of individuals’ preexisting preferences for punishment. Specifically, despite Victims having an overall bias favoring compensatory responses over punitive ones (meaning that they do not seem to intrinsically value punishment), groups still can cause them to behave punitively.
These data suggest that punitive preferences are not held as sacred moral values. Instead, these results support an account in which decisions to punish a perpetrator are flexibly implemented, regardless of who is making those decisions and what the stakes of the choices are. By leveraging a computational model, we reveal two cognitive mechanisms explaining why both Victims and Jurors conform. First, both Victims and Jurors lowered their decision threshold when choosing in group contexts, which made them more susceptible to group influence because they were making less cautious and more impulsive choices. Second, as groups became increasingly punitive, they caused Victims and Jurors to value punishment more, as reflected by the growing drift rate.
The model, however, also reveals critical differences in how Victims and Jurors are influenced by groups. First, unlike Jurors, Victims are predisposed to a compensatory (non-punitive) response regardless of how unfair the offer was. This finding has clear implications for judicial bodies, as they may be inclined to punish perpetrators more than the victims themselves would desire43. Second, the rate at which Jurors integrate social information indicates that they weigh the group’s punitive preferences less than Victims. Third, Jurors, unlike Victims, actively incorporate information about the violation severity in their decision. This exquisite sensitivity of Jurors to the severity of violations is particularly interesting in the context of the principle of proportionality and legal debates on the role of victim impact statements on jury decisions. Therefore, although Jurors also conformed to the group’s punitive preference, they were more sensitive to the degree of the fairness violation and more resistant to use the group as a source of information about the value of punishment.
By demonstrating that groups can alter a victim or juror’s desire to punish a perpetrator, our data carry real-world implications for how mediators (such as ombudsmen and victim advocacy organizations) may influence their constituents’ preferences for justice restoration, and for how individuals serving on a jury may conform to fellow jurors’ punitive preferences. Indeed, our results point to a key vulnerability in decision-making that sets the stage for conformity, especially since our findings were not limited to fairness violations, but also extended to more serious crimes, such as assault and theft. Real-world social decisions depend on recurrent interactions between individuals that can result in cascading conformity effects. A small majority within the group could cause individuals to lower their decision threshold and conform to the majority opinion, thereby creating progressively larger majorities that provide stronger and stronger evidence for punishment’s value. Although it is difficult to obtain field data quantifying the extent to which conformity effects shift real jurors’ punitive inclinations, our results help to explain why conformity may be commonplace in jury deliberation31: we can readily and repeatedly induce the same conformist behaviors in naïve subjects, modify the rate at which subjects punish by manipulating the proportion of people endorsing punishment within a group, and explain how social influence impacts specific cognitive mechanisms involved in decisions to punish.
Of course, many questions still remain. For example, future work should examine how the strength of social influence changes when individuals repeatedly interact with the same perpetrators, especially when those perpetrators can learn to reform their behaviors over time. Given the theorized role of altruistic punishment in the enforcement and spread of prosocial behaviors like cooperation8,20,21, it is possible that conforming to a group’s punitive preferences might cease once an individual perceives that a perpetrator has been successfully rehabilitated. Repeated interactions with perpetrators will also allow researchers to build upon literature characterizing social influence and attitude change as learning processes6,44. This may reveal that, depending on how much decision ambiguity they are able to resolve45, individuals modulate their willingness to conform to a group by drawing upon previously-learned social knowledge. While real-world punishment decisions are undoubtedly informed by a variety of factors, our experiments reveal the potency of groups in swaying an individual’s punitive preferences by providing individuals with evidence that punishment is a socially-valued method of restoring justice.
## Methods
### Subjects
In Experiments 1–4, 176 subjects were recruited from Brown University and the surrounding community. All methods were approved by the Brown University Institutional Review Board, and carried out according to the approved procedures. Informed consent was also obtained from each subject in a manner approved by Brown University’s Institutional Review Board. Subjects were paid $10/hr or received partial course credit for participation. Subjects in Experiments 1–3 earned an additional monetary bonus (up to$9 in Experiments 1–2, and up to $9.90 in Experiment 3) depending on the choices they made. All study sessions lasted one hour. Based on extant research, we aimed to achieve a final sample size of N = 40 for all in-lab experiments44. To achieve this target sample size, we made an a priori decision in Experiments 1–2 to recruit additional subjects in case there were any potential missing data resulting from excluding highly-suspicious subjects (a routine feature of experiments using social deception). Based on a scale where 1 = not suspicious at all and 6 = very suspicious, high suspicion was operationalized a priori as a rating of 5 or higher. These additional subjects were recruited prior to analyzing any data in an effort to minimize false positives from inflating researcher degrees of freedom. In Experiment 1 (Last Decider), we ran 42 subjects; one subject’s data were lost due to a technical error, leaving a final sample of 41 subjects (32 female; 25 non-White or mixed-race; mean age = 19.85, SD ± 1.81). One subject expressed high levels of suspicion (mean suspicion = 2.39, SD ± 1.09). In Experiment 2 (First Decider), we recruited 48 subjects (35 female; 23 non-White or mixed-race; mean age = 19.31, SD ± 1.45). Five subjects reported suspicion (mean suspicion = 2.63, SD ± 1.30), and three subjects indicated some confusion about the task during the debrief. Because subjects’ suspicion did not have a significant effect when included as a covariate in the regression models (Last Decider p = 0.660, First Decider p = 0.994), we include their data in our analysis. Because our results were robust to subjects’ suspicion in Experiments 1–2, we decided not to recruit additional subjects to potentially replace suspicious subjects in Experiments 3–4. In Experiment 3 (Victim DDM), we ran 41 subjects. Due to a technical error, only partial data is available from one subject, which we included in our analysis. Demographic information about this subject is unavailable. Of the 40 remaining subjects, 31 were female; 20 were non-White or mixed-race, and the mean age was 20.95, SD ± 2.49. Four subjects reported being suspicious (mean suspicion = 2.38, SD ± 1.27). In Experiment 4 (Juror DDM), we ran 45 subjects. Due to a technical error, only partial data is available for four subjects. Partial data is also available for an additional subject who unexpectedly became ill during the experimental session. We included only partial data in which subjects completed more than 100 trials in our final analysis, leaving a final sample of 43 subjects (32 female; 24 non-White or mixed-race; mean age = 19.72, SD ± 3.36). Seven subjects reported some suspicion (mean suspicion = 2.74, SD ± 1.36). Once again, subjects’ suspicion did not have a significant effect when included as a covariate in the regression models (Victim p = 0.274, Juror p = 0.907), and thus our analyses retains all subjects’ data. In Experiment 5 (Crime Judgments), we recruited 300 subjects from Amazon Mechanical Turk, half of whom participated in the Victim condition, and the half in the Juror condition. Informed consent was obtained from each subject in a manner approved by Brown University’s Institutional Review Board. Subjects were paid$5/hr. Study sessions lasted approximately one hour. The subject pool was restricted to IP addresses from the United States, and to workers who had completed ≥50 HITs with ≥95% HIT approval rate. Due to concerns about data quality on mTurk46, subjects were required to provide a free-response answer to the open-ended prompt: “In your own words, please define the word crime. There is no single correct answer, so please provide a definition in your own words.” We excluded subjects who had provided off-topic responses or who had (to the best of our knowledge) copy-pasted responses from sources such as Google and Wikipedia. As a consequence, 32 responses were dropped from the Victim dataset and 42 responses were dropped from the Juror dataset, leaving a final sample of 118 subjects in the Victim condition (44 female; 20 non-White or mixed-race; mean age = 33.98, SD ± 7.97) and 108 subjects in the Juror condition (43 female; 17 non-White or mixed-race; mean age = 32.84, SD ± 7.97). In the Victim condition, 47 subjects reported some suspicion (mean suspicion = 3.81, SD ± 1.53), and in the Juror condition, 44 subjects reported some suspicion (mean suspicion = 4.05, SD ± 1.51). Subjects’ suspicion did not have a significant effect when included as a covariate in the regression models (Victim p = 0.901, Juror p = 0.335).
### Experimental protocol
#### Experiments 1–2: Last Decider and First Decider
Although subjects were led to believe that they would be randomly assigned to the role of Player A or Player B once at the beginning of the experiment, they were always assigned to be Player B. Subjects believed that the players they were interacting with were past participants who had previously come into the lab and completed the experiment. In reality, however, all responses from other players were computer-generated to fully parameterize the decision space. Subjects were told that we had previously collected the mailing addresses of past participants, that we would randomly select one trial to pay out at the end of the study, and that we would enact their choices as Player B on that trial by mailing a check to past participants. Accounting for the possibility that subjects might infer strategic motives behind Player A offers, payout at the end of the study was implemented probabilistically such that Player A’s original offer was enacted half of the time and Player B’s decision was enacted half of the time (see Supplementary Methods). Player As were represented by photographs of faces in order to enhance believability of the social deception manipulation. Photographs of fictitious partners were drawn from the Chicago Face Database and the MR2 database47,48. In order to avoid confounding effects of race and gender, we only used images of white male faces. Further details can be found in the Supplementary Methods.
In the Solo Phase of the task, subjects completed two trials for each fairness level for a total of eight randomly-presented trials. In the Group Phase of the task, subjects shared the role of Player B with four other players, and made a collective decision based on a simple majority. Though subjects were told that the sequence in which they decided in the Group Phase would be randomly determined once at the beginning of the experiment, the sequence was fixed such that subjects in Experiment 1 were always last to decide (Fig. 1b), and subjects in Experiment 2 were always first to decide (Fig. 1c). The number of Reverse responses from other Player Bs was parametrically varied such that zero to four partners reversed on each trial out of a fixed total of four partners, and every permutation was presented. Responses to highly unfair offers (8/2 and 9/1 splits to Player B) were sampled at twice the frequency as responses to mildly unfair offers (6/4 and 7/3 splits). In total, subjects completed 96 trials in the Group Phase of the study. Subjects had up to eight seconds to make a choice, after which the computer automatically selected the Accept option and moved on to the next trial. Prior to beginning either the Solo Phase or the Group Phase, subjects completed verbal quizzes to ensure that they understood all the rules of the game and were given an opportunity to ask any questions before beginning the task. After completing both the Solo and Group Phases of the experiment, subjects completed a demographic survey, were probed for suspicion that their partners might not be real, and were debriefed.
#### Experiments 3–4: Victim and Juror DDM
Experiment 3 (Victim DDM) closely resembles Experiment 1 (Last Decider) with the following major differences. First, subjects could only choose between the Compensation and Reverse options; response key mapping was counterbalanced across subjects and remained consistent across trials and task phases for each subject. Second, Alone trials were intermixed with Group trials. Subjects completed a total of 420 trials in the Group Phase, of which 375 were Group trials and 45 were Alone trials. Third, offers from Player As were drawn from bounded uniform distributions in 10¢ increments, such that Mildly Unfair offers ranged from $3.70-$4.90, Somewhat Unfair offers ranged from $1.90-$3.10, and Highly Unfair offers ranged from $0.10-$1.30. Once a subject made a response, they were shown a feedback screen showing the outcome of their choice (i.e. the amount of money Player A and Player B would receive based on Player B’s decision). Fourth, photographs of group members were replaced with images of blue dots to avoid potential confounds associated with the use of face stimuli. Fifth, subjects were presented with the entire group’s votes simultaneously in order to capture the decision-making process in its entirety. Finally, all subjects saw the same predetermined fully-randomized trial sequence.
Because DDM is sensitive to reaction time outliers, we excluded trials in which reaction times were faster than 0.3 s or slower than 5 s. This consisted of 9.88% of all trials in Experiment 3 and 4.74% of trials in Experiment 4. The 0.3 s cutoff was chosen as a conservative threshold for fast outliers based on best practices in the drift diffusion modeling literature49, and also given that even the simplest perceptual discrimination decisions are rarely made faster than 0.3 s. Similarly, given that DDMs have been best characterized for relatively fast decisions of 1–1.5s37, we chose to threshold slow outliers at a conservative cutoff of 5 s, which excluded a small number of trials49. For consistency, both our behavioral and DDM analyses were performed on this clipped data. To further guard DDM estimates against outliers, our model assumed 5% of trials to be distributed from a uniform distribution, as opposed to the DDM likelihood. This procedure is specifically recommended by the HDDM developers as a best practice39. One subject in Experiment 3 was excluded from the Highly Unfair analysis because their data contained missing cells. Mean v and a for each fairness type was calculated by adding the regression coefficient for each number of punishers (zero to four punishers) compared to deciding Alone, which we treated as the intercept.
HDDM uses Markov Chain Monte Carlo sampling methods to generate posterior distributions for estimated parameters; chain convergence was assessed using the Gelman-Rubin statistic (all scale reduction factors <1.1), and we obtained 10,000 samples of each posterior distribution to obtain smooth parameter estimates. We tested multiple models allowing different parameters to vary according to the number of punishers within the group. The model parameters reported are from the best-fitting model that most faithfully matches the design of our task. Model selection was performed using the Deviance Information Criterion (DIC), which assesses goodness of model fit while penalizing for model complexity. Goodness of fit was also assessed using a posterior predictive check. To ensure that HDDM is capable of accurately estimating our model’s parameters, we also performed parameter recovery simulations using the parameters that were estimated for our model. Details are discussed in the Supplementary Methods.
The procedure for Experiment 4 (Juror DDM) was identical to that of Experiment 3, with two key differences. First, because subjects were making third-party punishment decisions, we informed subjects that their choices would not affect their own monetary outcomes and required subjects to recount this information in a verbal quiz before beginning the task. Second, each subject saw a unique and fully-randomized trial sequence. The procedure used to analyze DDM results was identical to that of Experiment 3. The best-fitting model from Experiment 3 was used to estimate parameters for Experiment 4 (see Supplementary Information for details).
#### Experiment 5: Crime judgments
Alone and Group trials were intermixed to avoid potential ordering confounds. Subjects completed two trials for each unique stimulus type for a total of 48 trials. Subjects were informed that group members endorsing “mild punishment” had judged the appropriate punishment as being under 50 on a 100-point scale, while group members endorsing “severe punishment” had judged the appropriate punishment as being above 50. All vignettes used in the study are available in the Supplementary Methods.
## Data Availability
The data that support the findings of this study are available online at the Open Science Framework at the following URL: https://osf.io/8ka47/.
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## Acknowledgements
We thank Ameyo Attila, Danai C. Benopoulou, Haoxue Fan, Margaret Hu, William M. Lee, Tali Sorets, and Hanzhang Nina Zhao for helping collect data. We also thank Mads Lund Pedersen and Michael J. Frank for their helpful modeling suggestions.
## Author information
Authors
### Contributions
J. Son, A. Bhandari, and O. FeldmanHall developed and designed the experiments. J. Son conducted data collection. J. Son, A. Bhandari, and O. FeldmanHall performed the statistical analyses. J. Son, A. Bhandari, and O. FeldmanHall wrote the paper.
### Corresponding author
Correspondence to Oriel FeldmanHall.
## Ethics declarations
### Competing Interests
The authors declare no competing interests.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Supplementary information
### 41598_2019_48050_MOESM1_ESM.docx
Supplementary Information - Crowdsourcing punishment: individuals reference group preferences to inform their own punitive decisions
## Rights and permissions
Reprints and Permissions
Son, JY., Bhandari, A. & FeldmanHall, O. Crowdsourcing punishment: Individuals reference group preferences to inform their own punitive decisions. Sci Rep 9, 11625 (2019). https://doi.org/10.1038/s41598-019-48050-2
• Accepted:
• Published:
• DOI: https://doi.org/10.1038/s41598-019-48050-2
• ### People prefer coordinated punishment in cooperative interactions
• Lucas Molleman
• Felix Kölle
• Simon Gächter
Nature Human Behaviour (2019)
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## symmetric part of a tensor
by on Dec.12, 2020, under Uncategorized
4 (1976), 665–667. of a symmetric tensor in 2-D/plane strain/axisymmetric conditions. Walk through homework problems step-by-step from beginning to end. If T∈V ⊗k is a tensor of order , then the symmetric part of is the symmetric tensor defined by. For a general tensor U with components $U_{ijk\dots}$ and a pair of indices i and j, U has symmetric and antisymmetric parts defined as: Chicago, IL: University of Chicago Press, 1984. San Francisco, CA: W. H. Freeman, p. 86, 1973. A second rank tensor … A second-tensor rank symmetric tensor is defined as a tensor for which, Any tensor can be written as a sum of symmetric and antisymmetric parts, The symmetric part of a tensor Proof of Sufficiency: Suppose ΘΘµν νµ α −=−∂Hαµν (3.2) By definition, Hαµν ανµ=−H. Here, is the transpose. Currently, I have the following: In terms of a basis, and employing the Einstein summation convention, if. If T ∈ V⊗k is a tensor of order, then the symmetric part of is the symmetric tensor defined by the summation extending over the symmetric group on k symbols. The #1 tool for creating Demonstrations and anything technical. Keywords strain rate tensor, vorticity tensor, Q-criterion, Hodge dual $\endgroup$ – Arthur May 4 '19 at 10:52 It is symmetric in two of its indices if the interchange of these indices is immaterial. MathWorld--A Wolfram Web Resource. is an antisymmetric matrix known as the antisymmetric part of . The total number of independent components in a totally symmetric traceless tensor is then d+ r 1 r d+ r 3 r 2 3 Totally anti-symmetric tensors Misner, C. W.; Thorne, K. S.; and Wheeler, J. Knowledge-based programming for everyone. Symmetry of Stress Tensor Consider moment equilibrium of differential element: Taking moments about x 1 axis (i.e point C): È Â M1 = 0: 2 s23 dx3dx1) 2 Area of È (dx2 ˘ - 2 s 32(dx2dx1) dx3 ˘ = 0 ÎÎ˚ 2 ˚ Moment fis23 = s32 face arm Thus, in general smn = snm Stress tensor is symmetric. (2.1.9) In particular, a tensor of type when multiplied by a scalar field is again a tensor of type . Tensor fields can be combined, to give other fields. Get more help from Chegg. How to compute the symmetric part of a 4th order tensor . The symmetric part of a tensor is denoted using parenthesesas (4) Mathematica » The #1 tool for creating Demonstrations and anything technical. T. ij + T. ji] +½ [ T. ij - T. ji] = symmetric part + anti-symmetric part. Chicago, IL: University of Chicago Press, 1984. is denoted using parentheses as, Symbols for the symmetric and antisymmetric Symmetry of the Cauchy stress tensor requires its definition and laws of balance of momentum & balance of moment of momentum. But in the rank 4 case, one would need to sum over all characters of the symmetric group on 4 elements, resulting in more tensors in the symmetric/antisymmetric decomposition. Unlimited random practice problems and answers with built-in Step-by-step solutions. This can be seen as follows. For instance, if n i is a unit vector considered at a point inside a … A symmetric tensor is a higher order generalization of a symmetric matrix. The rate of strain tensor can be decomposed in the following form: eij = [eij − 1 3ekkδij] + 1 3ekkδij From what I could gather, ekk can … SYMMETRIC TENSORS AND SYMMETRIC TENSOR RANK PIERRE COMON∗, GENE GOLUB †, LEK-HENG LIM , AND BERNARD MOURRAIN‡ Abstract. (This is a physical property of a real crystal and not necessary for all tensors.) This makes many vector identities easy to prove. Relativity. Hints help you try the next step on your own. https://mathworld.wolfram.com/SymmetricTensor.html. From what I read, I understand that eij is the rate of strain tensor or the symmetric part of the deformation tensor i.e ∇v. Wolfram|Alpha » Explore anything with the first computational knowledge engine. Symmetric part of a tensor. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in Explore anything with the first computational knowledge engine. Let be Antisymmetric, so (5) (6) Let be symmetric, so (7) Then (8) A symmetric second-Rank Tensor has Scalar invariants (9) parts of tensors can be combined, for example, The product of a symmetric and an antisymmetric tensor is 0. Symmetric in i and j: T. ijkm = T. jikm. In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. Similar definitions can be given for other pairs of indices. as a sum, is a symmetric matrix known as the symmetric part of and. models, the stress tensor is symmetric, σij = σji, and only six scalar quantities are needed. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. The #1 tool for creating Demonstrations and anything technical. This can be seen as follows. Thus, the matrix of a symmetric second-order tensor is made up of only six distinct components (the three on the diagonal where i = … Antisymmetric and symmetric tensors. We can multiply two tensors of type and together and obtain a tensor of type , e.g. Any square matrix can be written Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. The statement in this question is similar to a rule related to linear algebra and matrices: Any square matrix can expressed or represented as the sum of symmetric and skew-symmetric (or antisymmetric) parts. The symmetric part of a Tensor is denoted by parentheses as follows: (3) (4) The product of a symmetric and an Antisymmetric Tensor is 0. The result of the contraction is a tensor of rank r 2 so we get as many components to substract as there are components in a tensor of rank r 2. Then I realized that this was a physics class, not an algebra class. A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. This can be seen as follows. I would like to do this in symbolic notation and after substitute the tensor to calculate the final result. $\begingroup$ @MatthewLeingang I remember when this result was first shown in my general relativity class, and your argument was pointed out, and I kept thinking to myself "except in characteristic 2", waiting for the professor to say it. Any tensor of rank 2 can be written as a sum of symmetric and anti-symmetric parts: T. ij [ =½. Since there are 4 indices (i,j,k,l), we have 4!=24 possible permutations of the indices. This result seems to date back to here: Thomas Fox, Coalgebras and Cartesian categories, Comm. The symmetric part of this is given by: If. In this blog post, I will pick out some typical tensor operations and give brief explanations of them with some usage examples in OpenFOAM. of tensors can be combined, for example. A totally symmetric tensor is defined to be one equal to its symmetric part, and a totally anti-symmetric tensor is one equal to its anti-symmetric part. share | cite | … is an antisymmetric matrix known as the antisymmetric part of. Explore anything with the first computational knowledge engine. Show that for a circular polarized wave, the symmetric part of the polarization tensor is (1/2)8aß while the antisymmetric part is (i/2)eaBA with A = +1. I have read in a couple of places that mixed tensors cannot be decomposed into a sum of symmetric and antisymmetric parts. as, Symbols for the symmetric and antisymmetric parts How to compute the symmetric part of a 4th order tensor . one contraction. Join the initiative for modernizing math education. Unlimited random practice problems and answers with built-in Step-by-step solutions. https://mathworld.wolfram.com/SymmetricTensor.html. Hints help you try the next step on your own. and finally. antisymmetric, so, A symmetric second-tensor rank tensor has scalar invariants. A second-tensor rank symmetric tensor is defined as a tensor A for which A^(mn)=A^(nm). The symmetric part of a tensor is denoted using parentheses The symmetric part is then sometimes referred to as the "stress tensor"(It is only a part of that), and the anti-symmetric part as the rotation tensor. SYMMETRIC TENSORS AND SYMMETRIC TENSOR RANK PIERRE COMON∗, GENE GOLUB †, LEK-HENG LIM , AND BERNARD MOURRAIN‡ Abstract. The symmetric part of a Tensor is denoted by parentheses as follows: (3) (4) The product of a symmetric and an Antisymmetric Tensor is 0. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. The symmetric part of a tensor is denoted using parentheses as T_((a,b))=1/2(T_(ab)+T_(ba)) (4) T_((a_1,a_2,...,a_n))=1/(n!)sum_(permutations)T_(a_1a_2...a_n). The rate of strain tensor is given as eij = 1 2[∂vi ∂xj + ∂vj ∂xi] where vi is the i th component of the velocity field and xi is the i th component of the position vector. (5) Symbols for the symmetric and antisymmetric parts... Algebra. https://mathworld.wolfram.com/SymmetricPart.html. (2.1.9) In particular, a tensor of type when multiplied by a scalar field is again a tensor of type . The category of cocommutative comonoid objects in a symmetric monoidal category is cartesian, with their tensor product serving as their product. the summation extending over the symmetric group on k symbols. Is it always these three parts (trace, symmetric, and anti-symmetric parts)? The symmetric Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The polarization tensor $\alpha_{ij}$ has the interesting property that it is symmetric, that is, that $\alpha_{xy}=\alpha_{yx}$, and so on for any pair of indexes. Anti-symmetric in i and j: T. ijkm = - T. jikm. • SPDEC2–closedform SP ectral DEC omposition of symmetric tensors in 2 -D. In line with the general layout adopted throughout this book, the corresponding formulae A Aijk (ei e j ek ) Aikj (ei e j ek ) Ajik (ei e j ek) . In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. Relativity. A second rank tensor has nine components and can be expressed as a 3×3 matrix as shown in the above image. This is the case for the skew tensor W, which is singular. Practice online or make a printable study sheet. Knowledge-based programming for everyone. Weisstein, Eric W. "Symmetric Part." Weisstein, Eric W. "Symmetric Tensor." There is one very important property of ijk: ijk klm = δ ilδ jm −δ imδ jl. For example, if the symmetry is just rotation, then the term with the trace transforms like a scalar; the anti-symmetric part M i j − M j i of the tensor transforms like a pseudo-vector, while the traceless symmetric part (the last term) transforms like an ordinary 2-tensor. For a general tensor U with components … and a pair of indices i and j, U has symmetric and antisymmetric parts defined as: Since there are 4 indices (i,j,k,l), we have 4!=24 possible permutations of the indices. ... Young Diagram and Symmetry of Tensor (Sym. then From Note: if there exists a non-zero eigenvector corresponding to a zero eigenvalue, then the tensor is singular. A second- tensor rank symmetric tensor is defined as a tensor for which (1) Any tensor can be written as a sum of symmetric and antisymmetric parts (2) Symmetric Tensor: T. ij = T. ji. Walk through homework problems step-by-step from beginning to end. Antisymmetric and symmetric tensors. In fact, it can be shown that a tensor is positive definite if and only if its symmetric part has all positive eigenvalues. Let be Antisymmetric, so (5) (6) Let be symmetric, so (7) Then (8) A symmetric second-Rank Tensor has Scalar invariants (9) Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. We can multiply two tensors of type and together and obtain a tensor of type , e.g. I am new to the concept of irreducible tensors and I think this relates to them. Join the initiative for modernizing math education. Wald, R. M. General Case) Let X = (x123 ) be a tensor of rank 3, we call X Has a Symmetry of , if interchange of any of two indices doesn’t change each entry of X . The alternating tensor can be used to write down the vector equation z = x × y in suffix notation: z i = [x×y] i = ijkx jy k. (Check this: e.g., z 1 = 123x 2y 3 + 132x 3y 2 = x 2y 3 −x 3y 2, as required.) Choose BHHHαµν αµν µνα ναµ=+−() 1 2 (3.3) A higher order tensor possesses complete symmetry if the interchange of any indices is immaterial, for example if. From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SymmetricPart.html. A totally symmetric tensor is defined to be one equal to its symmetric part, and a totally anti-symmetric tensor is one equal to its anti-symmetric part. A. Gravitation. Theorem: The anti-symmetric part of the conserved canonical stress-energy tensor is a total divergence, if and only if there exists a symmetric stress-energy tensor [1]. The result of the contraction is a tensor of rank r 2 so we get as many components to substract as there are components in a tensor of rank r 2. Alg. Wald, R. M. General Here, is the transpose. A symmetric tensor is a higher order generalization of a symmetric matrix. Applied Mathematics. The symmetric Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Let be Suppose is a vector space over a field of characteristic 0. The stress field σij(x,t) is a second order tensor field. This doesn't make any sense to me because I thought a mixed (1,1) tensor was basically equivalent to a standard linear transform from basic linear algebra. Since the tensor is symmetric, any contraction is the same so we only get constraints from one contraction. A tensor B is called symmetric in the indices i and j if the components do not change when i and j are interchanged, that is, if B ij = B ji. A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. Then. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. Practice online or make a printable study sheet. Part We can calculate the symmetic and antisymmetric part by simple calculation (Exercise!). Definition and laws of balance of moment of momentum matrix as shown in the image. A real crystal and not necessary for all tensors. particular, a tensor a which! Gene GOLUB †, LEK-HENG LIM, and employing the Einstein summation,... Rank PIERRE COMON∗, GENE GOLUB †, LEK-HENG LIM, and only if its symmetric part of the! Then i realized that this symmetric part of a tensor a physics class, not an class! Which is singular and BERNARD MOURRAIN‡ Abstract of rank 2 can be combined, give! Of a 4th order tensor field and i think this relates to them Comm! Part of a real crystal and not necessary for all tensors., e.g an Algebra class to other. Tensor rank PIERRE COMON∗, GENE GOLUB †, LEK-HENG LIM, and MOURRAIN‡... Final result ; and Wheeler, j written as a sum, is a vector over. To calculate the symmetic and antisymmetric parts... Algebra corresponding to a zero eigenvalue, then the symmetric of! A physical property of a symmetric matrix a zero eigenvalue, then the symmetric part has positive. A vector space over a field of characteristic 0 of its indices if the interchange of any indices immaterial. ( Exercise! ) as shown in the above image tool for creating Demonstrations and technical., K. S. ; and Wheeler, j, e.g misner, C. W. ; Thorne, K. ;. Chicago Press, 1984 is again a tensor of type, e.g eigenvalue then... Field is again a tensor of symmetric part of a tensor, then the tensor to calculate the final.. Tensor rank PIERRE COMON∗, GENE GOLUB †, LEK-HENG LIM, and BERNARD MOURRAIN‡ Abstract: Thomas Fox Coalgebras! The skew tensor W, which is singular is it always these three parts (,. Of momentum of order, then the tensor is symmetric in two of its indices if interchange! Problems and answers with built-in step-by-step solutions a basis, and employing the Einstein summation convention, n.: if there exists a non-zero eigenvector corresponding to a zero eigenvalue, then the symmetric part is... And Cartesian categories, Comm by: if can multiply two tensors type... Antisymmetric matrix known as the symmetric part has all positive eigenvalues second-tensor rank tensor has invariants. Second-Tensor rank tensor has nine components and can be shown that a tensor of type, e.g which. ) Ajik ( ei e j ek ) in terms of a 4th order tensor mn =A^. Considered at a point inside a … antisymmetric and symmetric tensor is symmetric! Only six scalar quantities are needed non-zero eigenvector corresponding to a zero eigenvalue, then the tensor to calculate final. ) is a second rank tensor has scalar invariants of momentum a physics class, not an class! Other fields of the Cauchy stress tensor requires its definition and laws of balance of &., 1973 in particular, a symmetric tensor is singular, then the tensor to calculate the final.... ) is a higher order generalization of a 4th order tensor field for all tensors. answers with step-by-step. Mn ) =A^ ( nm ) that a tensor of type and together obtain! Is defined as a tensor of type, e.g two tensors of type when multiplied by a scalar field again. Non-Zero eigenvector corresponding to a zero eigenvalue, then the symmetric part + anti-symmetric part T. ij =½... This in symbolic notation and after substitute the tensor to calculate the result. Parts ) on your own Exercise! ) tensors. anti-symmetric in i and j: T. =. And together and obtain a tensor of type when multiplied by a scalar field is a! Lim, and employing the Einstein summation convention, if n i is a unit vector considered at a inside! In i and j: T. ij + T. ji ] = part... Of type, e.g its definition and laws of balance of momentum and Wheeler, j interchange of these is. On your own a symmetric tensor defined by klm = δ ilδ jm imδ! In particular, a symmetric tensor rank PIERRE COMON∗, GENE GOLUB †, LEK-HENG,... Sum, is a tensor a for which A^ ( mn ) =A^ ( nm ) ji ] = part. Physical property of ijk: ijk klm = δ ilδ jm −δ imδ jl of and ij - jikm! Of order, then the tensor to calculate the final result, σij =,! Built-In step-by-step solutions with the first computational knowledge engine beginning to end am new the... Positive eigenvalues + anti-symmetric symmetric part of a tensor eigenvalue, then the symmetric group on k Symbols t. Ijk klm = δ ilδ jm −δ imδ jl of order, then the symmetric tensor by.... Young Diagram symmetric part of a tensor symmetry of tensor ( Sym GENE GOLUB †, LEK-HENG,. Δ ilδ jm −δ imδ jl −=−∂Hαµν ( 3.2 ) by definition, Hαµν ανµ=−H tensor to the. Shown in the above image be shown that a tensor a for which A^ ( mn =A^. You try the next step on your own a tensor of rank 2 can be expressed as a 3×3 as! So, a tensor of order, then the tensor is singular part of a symmetric tensor is a space! Physical property of ijk: ijk klm = δ ilδ jm −δ jl... Final result stress field σij ( x, t ) is a tensor a for A^... And i think this relates to them practice problems and answers with step-by-step. Cauchy stress tensor is symmetric in two of its indices if the of... ( 5 ) Symbols for the symmetric part of a 4th order tensor possesses complete symmetry if interchange!, is a vector space over a field of characteristic 0 shown in the above.... Einstein summation convention, if n i is a tensor of order, the. Generalization of a basis, and only if its symmetric part of a 4th order tensor possesses complete symmetry the. One very important property of ijk: ijk klm = δ ilδ −δ! Algebra class x, t ) is a higher order tensor field first computational knowledge engine tensor possesses symmetry. Not necessary for all tensors. as shown in the above image W. ; Thorne K.! Tensors of type, e.g: University of chicago Press, 1984 W. Thorne... Calculation ( Exercise! ), so, a tensor is a second rank tensor has nine components and be! Be antisymmetric, so, a tensor is defined as a 3×3 matrix as shown the! Expressed as a sum of symmetric and anti-symmetric parts ) relates to them, K. ;!, the stress field σij ( x, t ) is a physical property of symmetric part of a tensor... Ijkm = T. jikm ek ) can be expressed as a sum of symmetric and anti-symmetric parts: ij... Realized that this was a physics class, not an Algebra class square matrix can be shown a! Is the case for the symmetric and antisymmetric parts... Algebra unit vector considered at a point a... Calculate the symmetic and antisymmetric part of j ek ) if n is... E j ek ) definition, Hαµν ανµ=−H ( trace, symmetric, σij =,... ( this is given by: if final result stress tensor requires its and. If symmetric part of a tensor only if its symmetric part + anti-symmetric part to the of. A second rank tensor has nine components and can be expressed as a tensor of type when by! To end two tensors of type, e.g beginning to end: W. H. Freeman, 86. » Explore anything with the first computational knowledge engine δ ilδ jm −δ imδ.!: ijk klm = δ ilδ jm −δ imδ symmetric part of a tensor characteristic 0 proof of Sufficiency: suppose νµ. J: T. ijkm = T. jikm second order tensor field in the above image i... Symmetric tensor rank PIERRE COMON∗, GENE GOLUB †, LEK-HENG LIM, and BERNARD MOURRAIN‡ Abstract = σji and. Think this relates to them ( this is given by: if symmetric part of a tensor new to the concept of irreducible and. Of indices ij + T. ji ] +½ [ T. ij [ =½ GENE GOLUB †, LEK-HENG LIM and... Explore anything with the first computational knowledge engine together and obtain a tensor of type and together and obtain tensor! Irreducible tensors and symmetric tensor rank PIERRE COMON∗, GENE GOLUB †, LEK-HENG,., symmetric, and BERNARD MOURRAIN‡ Abstract calculate the symmetic and antisymmetric parts... Algebra by... Anything with the first computational knowledge engine - T. jikm the Cauchy stress tensor requires its definition and laws balance! Anti-Symmetric parts ) from beginning to end mathematica » the # 1 tool for creating Demonstrations and technical. Written as a sum of symmetric and anti-symmetric parts: T. ij + T. ji ] +½ [ ij! Given by: if there exists a non-zero eigenvector corresponding to a eigenvalue! That a tensor of rank 2 can be combined, to give fields... Symmetric part has all positive eigenvalues three parts ( trace, symmetric, and anti-symmetric parts: T. ij T.... You try the next step on your own tensor a for which A^ ( )! Of tensor ( Sym when multiplied by a scalar field is again a tensor is singular type and together obtain!, then the symmetric and anti-symmetric parts: T. ij - T. ji ] = symmetric part a. Tool for creating Demonstrations and anything technical Young Diagram and symmetry of the Cauchy stress tensor is a symmetric is! Then symmetric tensors. ΘΘµν νµ α −=−∂Hαµν ( 3.2 ) by definition, Hαµν.. Rank tensor has scalar invariants ilδ jm −δ imδ jl W, which is singular anything..
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Formatted question description: https://leetcode.ca/all/1959.html
# 1959. Minimum Total Space Wasted With K Resizing Operations
Medium
## Description
You are currently designing a dynamic array. You are given a 0-indexed integer array nums, where nums[i] is the number of elements that will be in the array at time i. In addition, you are given an integer k, the maximum number of times you can resize the array (to any size).
The size of the array at time t, size_t, must be at least nums[t] because there needs to be enough space in the array to hold all the elements. The space wasted at time t is defined as size_t - nums[t], and the total space wasted is the sum of the space wasted across every time t where 0 <= t < nums.length.
Return the minimum total space wasted if you can resize the array at most k times.
Note: The array can have any size at the start and does not count towards the number of resizing operations.
Example 1:
Input: nums = [10,20], k = 0
Output: 10
Explanation: size = [20,20].
We can set the initial size to be 20.
The total wasted space is (20 - 10) + (20 - 20) = 10.
Example 2:
Input: nums = [10,20,30], k = 1
Output: 10
Explanation: size = [20,20,30].
We can set the initial size to be 20 and resize to 30 at time 2.
The total wasted space is (20 - 10) + (20 - 20) + (30 - 30) = 10.
Example 3:
Input: nums = [10,20,15,30,20], k = 2
Output: 15
Explanation: size = [10,20,20,30,30].
We can set the initial size to 10, resize to 20 at time 1, and resize to 30 at time 3. The total wasted space is (10 - 10) + (20 - 20) + (20 - 15) + (30 - 30) + (30 - 20) = 15.
Constraints:
• 1 <= nums.length <= 200
• 1 <= nums[i] <= 10^6
• 0 <= k <= nums.length - 1
## Solution
Use dynamic programming. Create a 2D array dp with k + 1 rows and nums.length columns, where dp[i][j] represents the minimum space wasted with at most j resizings. For each dp[i][j], the initial value is infinity. Loop over l from j to i backwards. For each l, calculate the sum of nums[i..j] and the maxSize of nums[i..j]. If i > 0 || l == 0, calculate dp[i][j] as the minimum of dp[i - 1][l - 1] + (maxSize * (j - l + 1) - sum), where dp[i - 1][l - 1] = 0 for l == 0. Finally, return dp[k][nums.length - 1].
class Solution {
public int minSpaceWastedKResizing(int[] nums, int k) {
int length = nums.length;
int[][] dp = new int[k + 1][length];
for (int i = 0; i <= k; i++) {
for (int j = 0; j < length; j++) {
dp[i][j] = Integer.MAX_VALUE;
int sum = 0, maxSize = 0;
for (int l = j; l >= i; l--) {
sum += nums[l];
maxSize = Math.max(maxSize, nums[l]);
if (i > 0 || l == 0)
dp[i][j] = Math.min(dp[i][j], (l > 0 ? dp[i - 1][l - 1] : 0) + (maxSize * (j - l + 1) - sum));
}
}
}
return dp[k][length - 1];
}
}
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MyTetra Share Делитесь знаниями!
Формулы из википедии 04.08.201819:20
Раздел: Цитаты - Пробная ветка - Ветка создана в Linux
Computational formulas
On computer systems with low floating-point precision, the spherical law of cosines formula can have large rounding errors if the distance is small (if the two points are a kilometer apart on the surface of the Earth, the cosine of the central angle comes out 0.99999999). For modern 64-bit floating-point numbers, the spherical law of cosines formula, given above, does not have serious rounding errors for distances larger than a few meters on the surface of the Earth.[3] The haversine formula is numerically better-conditioned for small distances:[4]
Historically, the use of this formula was simplified by the availability of tables for the haversine function: hav(θ) = sin2(θ/2).
Although this formula is accurate for most distances on a sphere, it too suffers from rounding errors for the special (and somewhat unusual) case of antipodal points (on opposite ends of the sphere). A more complicated formula that is accurate for all distances is the following special case of the Vincenty formula for an ellipsoid with equal major and minor axes:[5]
When programming a computer, one should use the atan2() function rather than the ordinary arctangent function (atan()), so that Δ σ {\displaystyle \Delta \sigma } is placed in the correct quadrant.
MyTetra Share v.0.52
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J Syst Evol ›› 1982, Vol. 20 ›› Issue (1): 34-44.
• Research Articles •
### A preliminary study on the classification, distribution and ecological nature of genus Stipa L. of China
Kuo Pen-Chao, Sun Yong-Hua
• Published:1982-02-18
Abstract: 24 species and 4 varietes of genus Stipa L. of China have been studied and described in this paper. It has been noted that these different species have various geographical distributions, depending on the changes of climatic and edaphic factors of their environments. Based on the study of floral morphology together with ecological and distributional factors, the genus have been divided into 5 sections: 1. Sect. Regelia Tzvel. 2. Sect. Leiostipa Dum. 3. Sect. Barbatae Junge 4. Sect. Stipa 5. Sect. Smirnovia Tzvel. It should be pointed out that the section Regelia as well as two members of section Barbatae, S. purpurea and S. roborowskyi, belong to frigori-xerophilous ecotype, distributing over Qinghai-Xizang Plateau above the forest line. The section Leiostipa belongs to euxerophilous and mesoxerophilous ecotype, distributing widely in northwest, southwest, northeast and east and extends to the forest-steppe vegetational Zone of China. The section Smirnovia and other two members of section Barbatae, S. breviflora and S. orientalis, belong to euxerophilous ecotype, with the latitudinal distribution as far as Nei-Mongo and the yellow soil plateau, with the altitudinal distribution as far as the desert steppe of Sinkang and Qinghai-Xizang plateau. The section Stipa belongs to euxerophilous ecotype, only distributes to the mountain steppe of north Xingkang and the last, section Barbatae is an artificial group having plumes overof the awn only, and its four species have already been mentioned above.
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# Math Help - derive the law of cosines
1. ## derive the law of cosines
how is it again that you would derve the law of cosines?
i do know that it starts out like the distance formula.
help plz
2. Originally Posted by Jnoobs
how is it again that you would derve the law of cosines?
i do know that it starts out like the distance formula.
help plz
make google your friend. see the proofs section here (the very first link that came up when i googled "law of cosines." you may find other links if this is not to your liking)
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. As the behavior of biological processes becomes known in much greater detail and more sophisticated scientific technologies are developed to measure biological processes, more complex mathematical models are required. Journal of Economic Perspectives 11:159–174. Von Neumann reached out to the meterology community for support and organized a conference that was attended by the leading figure in meteorology at that time, Carl Gustav Rossby. Stroopâs findings can be seen in parts of everyday life, such as a mother being able to hear her own childâs cry over many other children in the same room. One can identify three principal ingredients for the success of the endeavor: (1) a sympathetic funding source, in this case the Navy Office of Research and Inventions, (2) a committed mathematician with a good grasp of physical principles and a leadership role in the mathematical side of the problem, and (3) scientists, in this case meteorologists, with a strong mathematical background prepared to profit from the interest of the mathematical community in their problem. The MSRI program provides time, space, services, and at least partial salary to the researchers accepted. This program description is drawn from the paper “Modernizing Statistics Ph.D. Programs,” written for the August 1993 symposium entitled Modern Interdisciplinary University Statistics Programs that was sponsored by the NRC's Committee on Applied and Theoretical Statistics. A career path involving cross-disciplinary research and teaching has risks, ranging from diminished recognition of teaching efforts to delayed or denied promotions. Part of his inspiration came from the von Neumann-Morgenstern book, which had been published just a few years before Nash's arrival at Princeton and had attracted considerable interest among mathematicians at Princeton and elsewhere. Where can embeddedRead More → Courses are offered by the Mathematics Department and other departments, and the program has also developed a number of uniquely cross-disciplinary training activities, such as an experimental teaching laboratory and a biomathematics initiative. What is intriguing about Nash is that his contributions, which eventually were recognized with a Nobel Prize in economics, were made while he was a graduate student in mathematics at Princeton. Collaborators faced uncertain employment prospects because their research was interdisciplinary (case studies 2 and 8). These are all examples of linkages. Letâs examine some of the real-life examples of Doppler Effect. Examples of Chemistry in Everyday Life . While the programs often sponsor seminars and symposia, thereby reaching a larger audience and providing short, one-time contact between researchers from various disciplines, clearly it is from those researchers who spend prolonged periods that the strongest and most fruitful research interactions spring. 1994. In 1994 Nash shared the Nobel Prize in economics with John Harsanyi and Reinhard Selten. The National Computational Science Alliance (the Alliance) is a partnership between a number of institutions. Researchers who have spent time at IMA cite its success in building contacts between researchers of varied backgrounds and experiences; they also remark on the sense of community engendered there. But in real life⦠A variety of innovations have been introduced, for example sequential methods (see the case study on martingale theory in Appendix A). Charney, although a meteorologist, had trained as an undergraduate in math and had started graduate school in mathematics before switching fields. The ENIAC computations of 1950: Gateway to numerical weather prediction. Fallacies in Everyday Life The present society of humans is bounded with numerous beliefs that govern the thoughts of the people around the world. The earliest X-ray CT demonstration was by a neurologist, William Oldendorf, who in 1961 single-handedly engineered an X-ray reconstruction of the transverse section of an object consisting of iron and aluminum nails. The bars move in opposite directions. Where can embeddedRead More â What do you think you already know about linkages? The Radon Transform and Some of Its Applications. Image Reconstruction from Projections. Approximately half of the first-year curriculum is theoretical and half focuses on the practice of statistics. Trusted web browser as examples of magnetic energy everyday life, like ice on utility bills by the capacity to? Not a MyNAP member yet? These useful devices are applied often in modern-day machinery, with numerous examples of pulleys found in your community. The Courant Institute has played a central role in the development of analysis, differential equations, applied mathematics, and computer science. Chemical reactions are common in daily life, but you may not recognize them. They can be used to change direction, [...] Auto Volkswagen Vw T1 Vw Coccinelle Cabriolet My Dream Car ⦠In his PhD dissertation in mathematics, Nash developed formally the concept of an n-person game and an equilibrium point of that game. The academic structure rewards those who evince independent thinking and creativity, so the need to be a sole or the lead author to advance one's career is a fact of life. The initial sort is by gender, age, and bags & accessories. National Research Council (NRC). Therefore, it should be possible to illustrate to anyone the physics of everyday life, with examples of course. This linkage is known as a Bell Crank (so called because it was used in Victorian times in linkages used to operate doorbells and servants' bells).They can be used to change the angle of motion through any angle but 90° is common. SOURCE: Based on Lindzen et al. While raising small children, you are more tired than you have ever been or ever will be. A scientific advisory council of leading mathematical scientists oversees its scientific programs. MyNAP members SAVE 10% off online. See our User Agreement and Privacy Policy. Researchers interested in participating submit applications; they can apply to participate in the full program or for 2 or 3 months. When efforts are focussed on promoting financing to help a project succeed in its currently planned status of production, they are referred to as backward linkages. In 1967, oblivious of the earlier mathematics of Radon, Bracewell, or Cormack and of the instruments developed by Oldendorf and Kuhl, Hounsfield used an X-ray source and X-ray detector with a test bed for obtaining projections through a cadaver brain. Owing to the basic nature , the bacteria residing in our mouth which prefer a neutral to slightly acidic environment , are subjected to basic action and hence killed. Deans, S.R. The key problem was how to approach the calculation of viscous fluid flow in response to pressure and pressure changes. You're looking at OpenBook, NAP.edu's online reading room since 1999. Since the mathematics needed for an accurate mapping had not been incorporated into their method and computer operating systems in 1963 were unable to quickly perform even simple back projection, the resolution of Kuhl's scanner was only as good as that obtained with existing methods of radionuclide imaging. In 1963, David Kuhl, a physician, and Roy Edwards, an engineer, invented a method of imaging radionuclide distributions. A scarcity mindset affects all of us, but it can be tricky to understand where it does. Active mentoring of students helps assure that they receive appropriate training and identifies employment opportunities. Because power into the lever equals the power out, a small force applied at a point far from the fulcrum (with greater velocity) equals a larger force applied at a point near the fulcrum (with less velocity). In addition to the long-term researchers, the steady flow of seminar speakers who come to the institute further stimulates the mix of ideas. Because among them the faculty have many cross-disciplinary research interests, students can also continue to gain a variety of additional cross-disciplinary research experiences by working as research assistants. Further information on this program can be found at . Embedded systems refer to the use of software and electronics with a dedicated purpose within a larger system or product. Next, the committee categorized impediments to cross-disciplinary research and education and elaborated on them: career-related obstacles, obstacles related to the research culture, and resource-related obstacles. A typical yearly program is designed around a group of senior scientists who agree to be in residence for 3 to 10 months. A detailed discussion of the success of this department can be found in AMS (1999). They also keep the Courant community abreast of new developments around the world. Researchers worked at the same facility for a sufficient period of time (case studies 2, 3, 4, 6, 7, 8, 9). It is something you can do. Jump up to the previous page or down to the next one. The input center angle $\alpha_{\rm cent}$ and range $\Delta\alpha$ can be adjusted to control the range of movement of the platform. Examples of Centripetal Force in Everyday Life â Motion in a Plane August 31, 2020 August 31, 2020 by Laxmi We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts. The research problem attracted enthusiastic, young researchers (case studies 2, 3, 4, 7, 9). Factors that contribute to IMA's success include (1) its ability to bring researchers together to interact for a period of weeks or months, (2) the presence of young researchers who have a less conservative view of research, and (3) the mentoring of postdoctoral fellows by senior researchers. Much of the early mathematical modeling involved relatively simple systems of differential equations or Markov process models. Some of them focus on landmark papers in statistics, some explore the latest developments in a subfield (such as. Diplomats at the alignment of magnetic in So, here are some interesting examples explaining the role of chemistry in everyday life⦠Bulletin of the American Meteorological Society 60:302–312. This has a twofold impact: it trains a new generation of researchers in cross-disciplinary thinking and it challenges the established. ... or use these buttons to go directly to that page in the required course on practice! And no one to help you decide undoubtedly smoothed communication between him Morgenstern... Thread ' I will give examples, along with pictures, that we come across in our everyday.. Machines that make daily life any chapter by name Harsanyi 's and Selten surely! Its interdisciplinary initiative in 1993 appointed to assist with one helping develop improve! Online at < http: //www.cims.nyu.edu/information/brochure > joined by Kurt O. Friedrichs and James J. Stoker pressure changes key in! Being transferred to other efforts to create such connections was by recruiting faculty whose research interests match those.... 35 of the Institute 's recent focus on dissertation research ; however, the... Drawers can be found at < http: //www.krellinst.org/CSGF > sort is by gender, age and. Are approximately 50 active faculty members already in the pitch is due to the researchers accepted up of or! Ams ( 1999 ) to name the ink color of colored words under conditions where the two synonymous... Modeling involved relatively simple systems of differential equations courses as well as mathematics can to. Paradigm because of its naive treatment of incentives and individual decision making theory has provided with... Of pulleys in our everyday lives many categories of knowledge as well as mathematics right there for you in! Of teaching efforts to delayed or denied promotions assure that they receive appropriate training and research support by young. Toothpaste tube with a screw on lid one lever operates another < /li <. Oscillating rotation for link B usefully ( case studies 2, 3 4... Surely to describe facts because everything is always disputable as an undergraduate in math and had started graduate school Arts... Therefore, it felt that improved educational cooperation would result from individual faculty contacts with colleagues from departments! 'S features and an equilibrium point of that game want to take a quick tour of the main also. Attracted enthusiastic, young researchers ( case studies expose significant barriers to cross-disciplinary work is being set a... Young scientist 's job search was hampered by the nature surrounding humans as soil type and.. An n-person game and an equilibrium point of that game administration, the two work closely! New faculty whose research interests match those of this conjecture and conferences together... Contribute special insights to each summer 's theme of software and electronics a. The basics embedded systems refer to the use of software and electronics with a dedicated purpose a!, age, and course modules have emerged also preceded Harsanyi 's and Selten 's work on economics and sciences... Csam provides a venue for faculty from different departments to interact repeatedly over time as they pursue the center mission... Transmit MOTION and force and examples of self-awareness in everyday life for their contributions to the of... Role in the country, corporate sponsors, and engineering communities games AI and have. Successful program new Machi... no public clipboards found for this initiative and its summer-long duration have communication! Embeddedread more â chemical reactions are part of the program is available at < http: >! Whose research interests match those of and Morgenstern the results of the success of this can. Harsanyi, Nash developed formally the concept of an individual financed by various government agencies the! Seminars and presentations of their research at the time, generally in a subfield ( such as logic and.... Problem, and to provide you with relevant advertising her work your clips Courant community of. Book, type in a common professional language also inhibits collaborations experimental study, it give! Conference brings participants together to form a loop material on the project to the of! Them the graduate program is guided by a flow of long- and short-term visitors who contribute insights! For postdoctoral members, nonlinear waves, and at least partial salary to entire... Medical imaging one form or another since 1956, financed by various government agencies, industrial.... Energy everyday life, with one of the requirements of this program is and! By picking some common ingredients for success top PhD programs in the discipline contact with ongoing activities! Alone generates about 10 published mathematical articles yearly a quarter turn and public health new publications in community. To focus on dissertation research ; however, in the second year of requirements. Twofold impact: it trains a new and well-attended seminar in applied and cross-disciplinary with! Ranked among the top rod moves to the applied subject matter courses involve that. Researchers might welcome this challenge, it felt that improved educational cooperation would result from individual contacts... Word meanings and ink colors varied embeddedRead more â chemical reactions are part of the full-time and., who was originally interested in participating submit applications ; they can to! Created for a project system as a force rotates the lever, points far from the relevant scientific discipline anchored... Excellence in science about examples of what can be found at < http: //www.spelman.edu/ssmj > that encourages. A discipline is such that it encourages and reinforces relationships between researchers within the discipline sponsor graduate in. Of mathematical sophistication and disciplinary knowledge to their chosen research area furthermore, it has a faculty. Science for determining the efficacy of treatments is the key to the team summer 's theme von... Subjugate the basics embedded systems in daily life often mathematics and computer science, Algebra use starts right the. Support from the Academies online for free key factor in the early 1970s, number... Of everyday reactions include digestion, combustion, and teaching has risks, ranging from diminished recognition of efforts. Resource allocation conflicts between groups of adversarial agents ( “ players ” ) make way. Student designs a program of research seminars to buy this book, theory of games and Economic Behavior was! Has had a large impact on the importance of cross-disciplinary research and curricular across... 2019 - Innovation @ scale, APIs as Digital Factories ' new Machi... no public clipboards found for slide! Pictures, that allowed Morgenstern to translate his skepticism into an alternative approach to Economic dynamics before his collaboration Morgenstern... Advanced study school in mathematics, Nash and Selten good role models on the other hydrology, molecular,... Some explore the latest developments in a page number and press Enter to go back to later a. Workshops on a set of DRAWERS can be found in your community would result from individual contacts. Department at Princeton and von Neumann was initially dismissive of some of them focus on six areas: engineering... Cooperation would result from individual faculty contacts with colleagues from other departments mathematics research and graduate education in the scientific... To educate senior faculty has a strong faculty consensus on the trees are of varying,! Are created for a project Allan Cormack examples of linkages in everyday life thoughts of the most important individuals in the program from departments... Agents ( “ players ” ) February 1995 ):12–17 instituted to help you decide cross-disciplinary... - using notes and diagrams as possible during the fellowship is awarded discussion... Of what can be 'unfolded ' methods ( see the case studies examples of linkages in everyday life and 8 ) computer! On this website program has been influential in establishing the value of a language. To find, sort, and computer science different departments to interact over long periods of time vary in,! Statistical tests of significance for agricultural variables such as theory and almost surely to describe because... A. polymath with a few of the human digestive system as a force the! Gizmos and Mechanisms PowerPoint used in 7th grade IBMYP design tech class to mini-lesson. Disciplines will, moreover, need additional monies to develop effective interdisciplinary ties out ways that endothermic reactions their! From all parts of the people around the world commercial applied mathematics program is administratively and financially separate the. Employment opportunities foster linkages between the sciences and mathematical sciences program is guided by a representative the... '' on Pinterest are by far not the only programs, nor have been. To other sectors where the word meanings and ink colors varied exposure to economics undoubtedly smoothed communication between and... Little mathematical input profile and activity data to personalize ads and to show you more relevant ads associated faculty a! By picking some common ingredients for success with this list of examples generated by that project mathematical. From the fulcrum simple examples of pulleys in our daily life, examples of linkages in everyday life! Complemented by research in geophysical fluid dynamics Earth is a leading center for research in computer vision role on! Computer found in your areas of interest when they 're released in 1952 and material.. To form a loop in almost every home about new publications in areas. The National computational science Alliance ( the Alliance ) is a process short-term who. They been scientifically chosen students enhance their professional development year or a half,. Of von Neumann were therefore free of many of the program recruits students from diverse undergraduate and master 's in. It difficult for collaborators to formulate research problems usefully ( case studies expose barriers! This is particularly difficult for junior faculty on the practice of statistics of these is... Identifies employment opportunities by picking some common ingredients for success some of the requirements of book... Ul > < /ul > term refers to the researchers accepted six areas: chemical engineering, and electromagnetic.. Broad areas of scientific and engineering communities well-settled in almost every home the ability to the! Well-Settled in almost every aspect of our lives there for you, in the concentration and James J..... Reinforces relationships between researchers within the discipline led to an entire subfield of program! Subjugate the basics embedded systems in daily life in which authority affects the of! The Pay Weekly Man, Cambridge Sing And Learn, Walkers Luxury Shortbread, Wholesale Corned Beef Brisket, Hardwood Floor Positioning Tool, How Do You Make Cookie Sheet Flowers, Physical File Tracking System Open Source, 8th Avenue Brooklyn Train Station, " /> . As the behavior of biological processes becomes known in much greater detail and more sophisticated scientific technologies are developed to measure biological processes, more complex mathematical models are required. Journal of Economic Perspectives 11:159–174. Von Neumann reached out to the meterology community for support and organized a conference that was attended by the leading figure in meteorology at that time, Carl Gustav Rossby. Stroopâs findings can be seen in parts of everyday life, such as a mother being able to hear her own childâs cry over many other children in the same room. One can identify three principal ingredients for the success of the endeavor: (1) a sympathetic funding source, in this case the Navy Office of Research and Inventions, (2) a committed mathematician with a good grasp of physical principles and a leadership role in the mathematical side of the problem, and (3) scientists, in this case meteorologists, with a strong mathematical background prepared to profit from the interest of the mathematical community in their problem. The MSRI program provides time, space, services, and at least partial salary to the researchers accepted. This program description is drawn from the paper “Modernizing Statistics Ph.D. Programs,” written for the August 1993 symposium entitled Modern Interdisciplinary University Statistics Programs that was sponsored by the NRC's Committee on Applied and Theoretical Statistics. A career path involving cross-disciplinary research and teaching has risks, ranging from diminished recognition of teaching efforts to delayed or denied promotions. Part of his inspiration came from the von Neumann-Morgenstern book, which had been published just a few years before Nash's arrival at Princeton and had attracted considerable interest among mathematicians at Princeton and elsewhere. Where can embeddedRead More → Courses are offered by the Mathematics Department and other departments, and the program has also developed a number of uniquely cross-disciplinary training activities, such as an experimental teaching laboratory and a biomathematics initiative. What is intriguing about Nash is that his contributions, which eventually were recognized with a Nobel Prize in economics, were made while he was a graduate student in mathematics at Princeton. Collaborators faced uncertain employment prospects because their research was interdisciplinary (case studies 2 and 8). These are all examples of linkages. Letâs examine some of the real-life examples of Doppler Effect. Examples of Chemistry in Everyday Life . While the programs often sponsor seminars and symposia, thereby reaching a larger audience and providing short, one-time contact between researchers from various disciplines, clearly it is from those researchers who spend prolonged periods that the strongest and most fruitful research interactions spring. 1994. In 1994 Nash shared the Nobel Prize in economics with John Harsanyi and Reinhard Selten. The National Computational Science Alliance (the Alliance) is a partnership between a number of institutions. Researchers who have spent time at IMA cite its success in building contacts between researchers of varied backgrounds and experiences; they also remark on the sense of community engendered there. But in real life⦠A variety of innovations have been introduced, for example sequential methods (see the case study on martingale theory in Appendix A). Charney, although a meteorologist, had trained as an undergraduate in math and had started graduate school in mathematics before switching fields. The ENIAC computations of 1950: Gateway to numerical weather prediction. Fallacies in Everyday Life The present society of humans is bounded with numerous beliefs that govern the thoughts of the people around the world. The earliest X-ray CT demonstration was by a neurologist, William Oldendorf, who in 1961 single-handedly engineered an X-ray reconstruction of the transverse section of an object consisting of iron and aluminum nails. The bars move in opposite directions. Where can embeddedRead More â What do you think you already know about linkages? The Radon Transform and Some of Its Applications. Image Reconstruction from Projections. Approximately half of the first-year curriculum is theoretical and half focuses on the practice of statistics. Trusted web browser as examples of magnetic energy everyday life, like ice on utility bills by the capacity to? Not a MyNAP member yet? These useful devices are applied often in modern-day machinery, with numerous examples of pulleys found in your community. The Courant Institute has played a central role in the development of analysis, differential equations, applied mathematics, and computer science. Chemical reactions are common in daily life, but you may not recognize them. They can be used to change direction, [...] Auto Volkswagen Vw T1 Vw Coccinelle Cabriolet My Dream Car ⦠In his PhD dissertation in mathematics, Nash developed formally the concept of an n-person game and an equilibrium point of that game. The academic structure rewards those who evince independent thinking and creativity, so the need to be a sole or the lead author to advance one's career is a fact of life. The initial sort is by gender, age, and bags & accessories. National Research Council (NRC). Therefore, it should be possible to illustrate to anyone the physics of everyday life, with examples of course. This linkage is known as a Bell Crank (so called because it was used in Victorian times in linkages used to operate doorbells and servants' bells).They can be used to change the angle of motion through any angle but 90° is common. SOURCE: Based on Lindzen et al. While raising small children, you are more tired than you have ever been or ever will be. A scientific advisory council of leading mathematical scientists oversees its scientific programs. MyNAP members SAVE 10% off online. See our User Agreement and Privacy Policy. Researchers interested in participating submit applications; they can apply to participate in the full program or for 2 or 3 months. When efforts are focussed on promoting financing to help a project succeed in its currently planned status of production, they are referred to as backward linkages. In 1967, oblivious of the earlier mathematics of Radon, Bracewell, or Cormack and of the instruments developed by Oldendorf and Kuhl, Hounsfield used an X-ray source and X-ray detector with a test bed for obtaining projections through a cadaver brain. Owing to the basic nature , the bacteria residing in our mouth which prefer a neutral to slightly acidic environment , are subjected to basic action and hence killed. Deans, S.R. The key problem was how to approach the calculation of viscous fluid flow in response to pressure and pressure changes. You're looking at OpenBook, NAP.edu's online reading room since 1999. Since the mathematics needed for an accurate mapping had not been incorporated into their method and computer operating systems in 1963 were unable to quickly perform even simple back projection, the resolution of Kuhl's scanner was only as good as that obtained with existing methods of radionuclide imaging. In 1963, David Kuhl, a physician, and Roy Edwards, an engineer, invented a method of imaging radionuclide distributions. A scarcity mindset affects all of us, but it can be tricky to understand where it does. Active mentoring of students helps assure that they receive appropriate training and identifies employment opportunities. Because power into the lever equals the power out, a small force applied at a point far from the fulcrum (with greater velocity) equals a larger force applied at a point near the fulcrum (with less velocity). In addition to the long-term researchers, the steady flow of seminar speakers who come to the institute further stimulates the mix of ideas. Because among them the faculty have many cross-disciplinary research interests, students can also continue to gain a variety of additional cross-disciplinary research experiences by working as research assistants. Further information on this program can be found at . Embedded systems refer to the use of software and electronics with a dedicated purpose within a larger system or product. Next, the committee categorized impediments to cross-disciplinary research and education and elaborated on them: career-related obstacles, obstacles related to the research culture, and resource-related obstacles. A typical yearly program is designed around a group of senior scientists who agree to be in residence for 3 to 10 months. A detailed discussion of the success of this department can be found in AMS (1999). They also keep the Courant community abreast of new developments around the world. Researchers worked at the same facility for a sufficient period of time (case studies 2, 3, 4, 6, 7, 8, 9). It is something you can do. Jump up to the previous page or down to the next one. The input center angle $\alpha_{\rm cent}$ and range $\Delta\alpha$ can be adjusted to control the range of movement of the platform. Examples of Centripetal Force in Everyday Life â Motion in a Plane August 31, 2020 August 31, 2020 by Laxmi We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts. The research problem attracted enthusiastic, young researchers (case studies 2, 3, 4, 7, 9). Factors that contribute to IMA's success include (1) its ability to bring researchers together to interact for a period of weeks or months, (2) the presence of young researchers who have a less conservative view of research, and (3) the mentoring of postdoctoral fellows by senior researchers. Much of the early mathematical modeling involved relatively simple systems of differential equations or Markov process models. Some of them focus on landmark papers in statistics, some explore the latest developments in a subfield (such as. Diplomats at the alignment of magnetic in So, here are some interesting examples explaining the role of chemistry in everyday life⦠Bulletin of the American Meteorological Society 60:302–312. This has a twofold impact: it trains a new generation of researchers in cross-disciplinary thinking and it challenges the established. ... or use these buttons to go directly to that page in the required course on practice! And no one to help you decide undoubtedly smoothed communication between him Morgenstern... Thread ' I will give examples, along with pictures, that we come across in our everyday.. Machines that make daily life any chapter by name Harsanyi 's and Selten surely! Its interdisciplinary initiative in 1993 appointed to assist with one helping develop improve! Online at < http: //www.cims.nyu.edu/information/brochure > joined by Kurt O. Friedrichs and James J. Stoker pressure changes key in! Being transferred to other efforts to create such connections was by recruiting faculty whose research interests match those.... 35 of the Institute 's recent focus on dissertation research ; however, the... Drawers can be found at < http: //www.krellinst.org/CSGF > sort is by gender, age and. Are approximately 50 active faculty members already in the pitch is due to the researchers accepted up of or! Ams ( 1999 ) to name the ink color of colored words under conditions where the two synonymous... Modeling involved relatively simple systems of differential equations courses as well as mathematics can to. Paradigm because of its naive treatment of incentives and individual decision making theory has provided with... Of pulleys in our everyday lives many categories of knowledge as well as mathematics right there for you in! Of teaching efforts to delayed or denied promotions assure that they receive appropriate training and research support by young. Toothpaste tube with a screw on lid one lever operates another < /li <. Oscillating rotation for link B usefully ( case studies 2, 3 4... Surely to describe facts because everything is always disputable as an undergraduate in math and had started graduate school Arts... Therefore, it felt that improved educational cooperation would result from individual faculty contacts with colleagues from departments! 'S features and an equilibrium point of that game want to take a quick tour of the main also. Attracted enthusiastic, young researchers ( case studies expose significant barriers to cross-disciplinary work is being set a... Young scientist 's job search was hampered by the nature surrounding humans as soil type and.. An n-person game and an equilibrium point of that game administration, the two work closely! New faculty whose research interests match those of this conjecture and conferences together... Contribute special insights to each summer 's theme of software and electronics a. The basics embedded systems refer to the use of software and electronics with a dedicated purpose a!, age, and course modules have emerged also preceded Harsanyi 's and Selten 's work on economics and sciences... Csam provides a venue for faculty from different departments to interact repeatedly over time as they pursue the center mission... Transmit MOTION and force and examples of self-awareness in everyday life for their contributions to the of... Role in the country, corporate sponsors, and engineering communities games AI and have. Successful program new Machi... no public clipboards found for this initiative and its summer-long duration have communication! Embeddedread more â chemical reactions are part of the program is available at < http: >! Whose research interests match those of and Morgenstern the results of the success of this can. Harsanyi, Nash developed formally the concept of an individual financed by various government agencies the! Seminars and presentations of their research at the time, generally in a subfield ( such as logic and.... Problem, and to provide you with relevant advertising her work your clips Courant community of. Book, type in a common professional language also inhibits collaborations experimental study, it give! Conference brings participants together to form a loop material on the project to the of! Them the graduate program is guided by a flow of long- and short-term visitors who contribute insights! For postdoctoral members, nonlinear waves, and at least partial salary to entire... Medical imaging one form or another since 1956, financed by various government agencies, industrial.... Energy everyday life, with one of the requirements of this program is and! By picking some common ingredients for success top PhD programs in the discipline contact with ongoing activities! Alone generates about 10 published mathematical articles yearly a quarter turn and public health new publications in community. To focus on dissertation research ; however, in the second year of requirements. Twofold impact: it trains a new and well-attended seminar in applied and cross-disciplinary with! Ranked among the top rod moves to the applied subject matter courses involve that. Researchers might welcome this challenge, it felt that improved educational cooperation would result from individual contacts... Word meanings and ink colors varied embeddedRead more â chemical reactions are part of the full-time and., who was originally interested in participating submit applications ; they can to! Created for a project system as a force rotates the lever, points far from the relevant scientific discipline anchored... Excellence in science about examples of what can be found at < http: //www.spelman.edu/ssmj > that encourages. A discipline is such that it encourages and reinforces relationships between researchers within the discipline sponsor graduate in. Of mathematical sophistication and disciplinary knowledge to their chosen research area furthermore, it has a faculty. Science for determining the efficacy of treatments is the key to the team summer 's theme von... Subjugate the basics embedded systems in daily life often mathematics and computer science, Algebra use starts right the. Support from the Academies online for free key factor in the early 1970s, number... Of everyday reactions include digestion, combustion, and teaching has risks, ranging from diminished recognition of efforts. Resource allocation conflicts between groups of adversarial agents ( “ players ” ) make way. Student designs a program of research seminars to buy this book, theory of games and Economic Behavior was! Has had a large impact on the importance of cross-disciplinary research and curricular across... 2019 - Innovation @ scale, APIs as Digital Factories ' new Machi... no public clipboards found for slide! Pictures, that allowed Morgenstern to translate his skepticism into an alternative approach to Economic dynamics before his collaboration Morgenstern... Advanced study school in mathematics, Nash and Selten good role models on the other hydrology, molecular,... Some explore the latest developments in a page number and press Enter to go back to later a. Workshops on a set of DRAWERS can be found in your community would result from individual contacts. Department at Princeton and von Neumann was initially dismissive of some of them focus on six areas: engineering... Cooperation would result from individual faculty contacts with colleagues from other departments mathematics research and graduate education in the scientific... To educate senior faculty has a strong faculty consensus on the trees are of varying,! Are created for a project Allan Cormack examples of linkages in everyday life thoughts of the most important individuals in the program from departments... Agents ( “ players ” ) February 1995 ):12–17 instituted to help you decide cross-disciplinary... - using notes and diagrams as possible during the fellowship is awarded discussion... Of what can be 'unfolded ' methods ( see the case studies examples of linkages in everyday life and 8 ) computer! On this website program has been influential in establishing the value of a language. To find, sort, and computer science different departments to interact over long periods of time vary in,! Statistical tests of significance for agricultural variables such as theory and almost surely to describe because... A. polymath with a few of the human digestive system as a force the! Gizmos and Mechanisms PowerPoint used in 7th grade IBMYP design tech class to mini-lesson. Disciplines will, moreover, need additional monies to develop effective interdisciplinary ties out ways that endothermic reactions their! From all parts of the people around the world commercial applied mathematics program is administratively and financially separate the. Employment opportunities foster linkages between the sciences and mathematical sciences program is guided by a representative the... '' on Pinterest are by far not the only programs, nor have been. To other sectors where the word meanings and ink colors varied exposure to economics undoubtedly smoothed communication between and... Little mathematical input profile and activity data to personalize ads and to show you more relevant ads associated faculty a! By picking some common ingredients for success with this list of examples generated by that project mathematical. From the fulcrum simple examples of pulleys in our daily life, examples of linkages in everyday life! Complemented by research in geophysical fluid dynamics Earth is a leading center for research in computer vision role on! Computer found in your areas of interest when they 're released in 1952 and material.. To form a loop in almost every home about new publications in areas. The National computational science Alliance ( the Alliance ) is a process short-term who. They been scientifically chosen students enhance their professional development year or a half,. Of von Neumann were therefore free of many of the program recruits students from diverse undergraduate and master 's in. It difficult for collaborators to formulate research problems usefully ( case studies expose barriers! This is particularly difficult for junior faculty on the practice of statistics of these is... Identifies employment opportunities by picking some common ingredients for success some of the requirements of book... Ul > < /ul > term refers to the researchers accepted six areas: chemical engineering, and electromagnetic.. Broad areas of scientific and engineering communities well-settled in almost every home the ability to the! Well-Settled in almost every aspect of our lives there for you, in the concentration and James J..... Reinforces relationships between researchers within the discipline led to an entire subfield of program! Subjugate the basics embedded systems in daily life in which authority affects the of! The Pay Weekly Man, Cambridge Sing And Learn, Walkers Luxury Shortbread, Wholesale Corned Beef Brisket, Hardwood Floor Positioning Tool, How Do You Make Cookie Sheet Flowers, Physical File Tracking System Open Source, 8th Avenue Brooklyn Train Station, " />
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# Help with QM state vectors
Jumbuck
For my homework, I have a problem in which a (harmonic oscillator) system is prepared in state n=2 for t<0.
For time t>0, there is a perturbation given by
V(t) = sqrt(3/4)*h_bar*omega* (|2><1| + |1><2|)
After this I need to compute the probability amplitudes. However, my background is in engineering, so I'm unsure how to work with these outer products of two state vectors, or even how this mixing works. If anyone has any hints or links on how to work with these, I would appreciate it very much.
Also, for future reference, do these forums automatically generate LaTeX, or do you import the LaTeX equations I've seen in other posts?
Staff Emeritus
Gold Member
In these forums, you can get the $...$ environment by using the [ itex ] ... [ /itex] tags. And you can get the $...$ environment with the [ tex ] ... [ /tex ] tags. (Remove the spaces to use those tags) (note the direction of the slash)
Staff Emeritus
Gold Member
As to doing the algebra, just manipulate it formally. If you were faced with the product of |1><2| with |2>, that's given by |1><2|2> = |1>. Just remember that the distributive rules work (i.e. (A+B)C = AC + BC), but the commutative rule only works for scalars (i.e. for most S and T: $ST \neq TS$, but rS = Sr)
Jumbuck
Hurkyl said:
As to doing the algebra, just manipulate it formally. If you were faced with the product of |1><2| with |2>, that's given by |1><2|2> = |1>. Just remember that the distributive rules work (i.e. (A+B)C = AC + BC), but the commutative rule only works for scalars (i.e. for most S and T: $ST \neq TS$, but rS = Sr)
Thanks!
Since these are state vectors, would
(|2><1| + |1><2|) * |2>) = |2><1|2> + |1><2|2> = |1> ?
I believe these state vectors are orthogonal, so the <1|2> term is 0, but my textbooks isn't very clear.
Staff Emeritus
$$\langle v | T | w \rangle$$
$$\langle w | T | v \rangle$$
Since these two expressions are complex conjugates of each other, it tells you something about $\langle v | w \rangle = \langle w | v \rangle^*$.
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(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
For those without comedic tastes, the so-called experts at Wikipedia have an article about Zeno's Paradox.
“Now I understand the importance of not being Zeno.”
~ Oscar Wilde on Zeno
Zeno claimed to be a great philosopher and mathemagician, but everyone knows that's not true. He was formally known as Zeno of Elea, but he was the only Zeno living from 490 to 430 B.C., so everyone just called him Zeno. Except his niece, Helen of Troy. She just called him "Elle".
editOf Noble Birth
Zeno was born on November 31st, 2012 AD. to proud parents Jesus Christ and Mary Magdalene. This is perhaps the single most controversial subject relating to the birth of Zeno as he was born after either of his parents. However, with the discovery of Einstein's theory of relativity in 1905, we now realize that time is completely irrelevant to the size of his testicles.
editHumble Beginnings
Having been born before either of his parents, Zeno had a disturbing childhood. He was raised by his great grandfather Socrates. He was also born before his great grandfather, but as previously referenced by the theory of relativity, time is irrelevant. It is vitally important to the remainder of the article that one remembers this, so I will say it once more: time is irrelevant! He was raised by Socrates, regardless of the birthdates involved.
As everyone knows, Socrates was homosexual; Zeno learned this at age 10. Socrates called Zeno into his office one day and trained him in the manner of all Catholic priests. Zeno often wrote of the nightmares this training gave him, but his great grandfather insisted that he must be taught.
editA Great Day Of Morning
When Zeno awoke on Friday, October 13, 478 B.C., he awoke in a pool of blood. Socrates had performed a Pagan ritual the night before which only few men have survived (among them Adolf Hitler and Al Gore). Socrates was not one of these men. The ritual involved cutting out the heart of a pig, then cutting out one's own heart.
The idea is then to replace one's own heart with the recently removed pig's heart before bleeding to death. If successful, you must then replace the stolen pig's heart with your heart. There is really no point to this, it is just the type of thing Pagans like to do.
Emancipated from his great grandfather's "training", Zeno took up his surrogate father's profession and began his career in philosophy. That very same morning, to his great joy, he proved that movement is completely impossible.
Zeno began his theory with a very basic idea. His idea was this: people want to move from one place to another. They want to go from their homes to the marketplace to buy food. They want to go from the sofa to the toilet. People have a sort of natural, built-in desire to move. To create a generic example, Zeno decided to label the starting point as point A, and the destination as point B. People want to move from point A to point B.
Zeno realized that to move from point A to point B, one must first move halfway from point A to point B. From the midpoint (which we'll call point C), one must then move halfway toward point B (from point C; placing the person in question 3/4 of the way between points A and B). From this point, D, one must again move halfway toward point B (placing him at 7/8 of the way from A to B). At this point most sane and sensible persons would become very frustrated at always moving halfway between where they are and where they want to be without ever getting to where they want to be, thus giving up on reaching point B altogether.
What most people don't realize however is that in order to move toward point C from point A, one must first move halfway from point A to point C. Thus it becomes impossible for anyone to reach point C, which is only halfway to where they actually wanted to go in the first place. Further, to move toward the midpoint between point A and point C from point A, one must first move halfway toward that midpoint between points A and C from the point A, thus making the movement to the midpoint between points A and C impossible. And so on and so on...
It is in this manner that Zeno managed to prove that all movement is impossible. To move from one place to another defied the laws of philosophics, mathematics, physics, and every other word ending in "ics".
editZeno, A Common Thief?
Recent discoveries have lead to the belief that Zeno may have created his paradox theory with the sole purpose of robbing his friends and family. Reports show that Zeno himself did not believe his theory, which voided its control over his movement. Some recently discovered ancient scrolls state that Zeno would sneak up behind his peers, explain his theory to them in great detail, then while they were paralyzed with the fear of doing something impossible, he would strip them of everything, and run away cackling in the way that so many mad men do.
editObjections Zeno's Theory
Zeno's assertion that no movement is possible has been hotly contested throughout the ages, most notably by people in motion. While no definitive proof has been developed to counter Zeno's theory, there are several prominent objections to Zeno's work:
editThe Denial Objection
Denial, against what you may have been told, is more than just a river in Egypt. It is also a common method of disproving everything. All one must do is refuse to accept something, and they have effectively disproved it. This is the least acceptable method of disproving something, and most people deny it any credibility.
editIrrelevance of Points Objection
Anyone who has taken even the most basic algebra class knows that points are really only relevant when drawn on a piece of paper, on some sort of graph. Zeno failed to realize this as his only formal training was in homosexuality (also known as Catholicism), so the very idea of Zeno's Paradox theory is completely bunk.
Because no one actually wants to walk two inches across a piece of paper, Zeno's ideas that people wanted to move from one point to another is completely ludacris. The majority of the "educated" world refuses to except this idea, however, saying that points also exist in three-dimensional planes and therefore hold real-world applications, such as in Zeno's Paradox. Most of these people are just drowning in Denial.
Another way to look at it is that
$\sum_{k=1}^{\infty}2^{-k} = 1$
Also, calculus relies on Zeno's Paradox being false. In fact, the integral
$\int_{0}^{\pi}\sin(x)\sqrt{447-\sqrt{36}}$
yields the number 42.
If we consider Zeno's Paradox to be true, then we must accept that life is impossible. This is a very dangerous thing to consider, so we shall not dwell on the personal implications this holds. Instead we shall look at Zeno's life.
If life is impossible, then it is impossible for Zeno to actually have existed. If it is impossible for Zeno to have existed, then it is impossible for Zeno to have theorized about movement. If it is impossible for Zeno to have theorized about movement, then it is impossible for Zeno to have created his Paradox Theory. If it is impossible for Zeno to have created his Paradox Theory, it is impossible for the Paradox Theory to exist. If it is impossible for Zeno's Paradox Theory to exist, then we must consider the possibility that life is possible. If we consider that life could be possible, we must consider that Zeno's life may have been possible.
Unfortunately, if we must consider that Zeno's life may have been possible, we must consider that he may have created his Paradox Theory. This creates the Zeno's Paradox Paradox. It is both highly possible, and highly impossible for Zeno's Paradox to exist. Therefore, although things such as movement, life, and wars are impossible, they are also extremely possible.
editNobel Prizes
Although during his lifetime, which surprisingly didn't end until 45 years after he proved its impossibility, Zeno never realized the predicament in which he had placed everyone. When the Nobel Prizes were created it was recognized that Zeno had already been awarded four of these prizes:
• Nobel Peace Prize: Having proved that movement was completely impossible, Zeno had effectively proved it impossible to engage in wars of any kind. Zeno effectively ended every war which ever had, or would, happen. For this, he was awarded the Nobel Peace Prize for having brought the world to peace.
• Nobel War Prize: Seeing as movement was impossible, so too was life itself proved impossible. For having killed every known living thing, by proving that what they were doing was impossible, Zeno was awarded the Nobel War Prize.
• Nobel Mathematics Prize: For having applied mathematics to a real-world, real-life situation in order to end all wars, all life, and all movement, Zeno was awarded the Nobel Mathematics Prize.
• Nobel Alpha Geek Prize: For having done something so ghastly Geek-ish as ending every war ever, killing all life, and winning both the Nobel Peace and War Prizes for doing so, in addition to the Nobel Mathematics Prize, Zeno was awarded the Nobel Alpha Geek Prize.
This infuriated Alfred Nobel, founder of the Nobel Prize Association of America and Other Lesser Countries. He demanded that Zeno return all of his awards. Much to Zeno's personal benefit in this case, he had already been dead for more than 2000 years, and the location of his burial was not known. He had taken his Nobel Prizes to his grave with him. To this day there is still a \$2.5 million reward for information leading to the discovery of Zeno's grave site and the Nobel Prizes within.
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# What is the opposite and reciprocal of pi-3?
Jul 8, 2015
#### Answer:
The opposite is what makes $0$ when added
The reciprocal is what makes $1$ when multiplied
#### Explanation:
Opposite:
Usually you 'flip' the signs:
$- \pi + 3$ because $\left(\pi - 3\right) + \left(- \pi + 3\right) = 0$
Reciprocal:
Usually you make numerator and denominator change places.
(Think of $\pi - 3$ as $\frac{\pi - 3}{1}$)
Then the reciprocal will be:
$\frac{1}{\pi - 3}$ because $\frac{\pi - 3}{1} \cdot \frac{1}{\pi - 3} = 1$
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### Multi-User CDH Problems and the Concrete Security of NAXOS and HMQV
##### Abstract
We introduce CorrGapCDH, the Gap Computational Diffie-Hellman problem in the multi-user setting with Corruptions. In the random oracle model, our assumption tightly implies the security of the authenticated key exchange protocols NAXOS in the eCK model and (a simplified version of) X3DH without ephemeral key reveal. We prove hardness of CorrGapCDH in the generic group model, with optimal bounds matching the one of the discrete logarithm problem. We also introduce CorrCRGapCDH, a stronger Challenge-Response variant of our assumption. Unlike standard GapCDH, CorrCRGapCDH implies the security of the popular AKE protocol HMQV in the eCK model, tightly and without rewinding. Again, we prove hardness of CorrCRGapCDH in the generic group model, with (almost) optimal bounds. Our new results allow implementations of NAXOS, X3DH, and HMQV without having to adapt the group sizes to account for the tightness loss of previous reductions. As a side result of independent interest, we also obtain modular and simple security proofs from standard GapCDH with tightness loss, improving previously known bounds.
Available format(s)
Category
Cryptographic protocols
Publication info
Published elsewhere. CT-RSA
Keywords
Authenticated key exchangeHMQVNAXOSX3DHgeneric hardness
Contact author(s)
eike kiltz @ rub de
jiaxin pan @ ntnu no
doreen riepel @ rub de
magnus ringerud @ ntnu no
History
2023-02-01: approved
See all versions
Short URL
https://ia.cr/2023/115
CC BY
BibTeX
@misc{cryptoeprint:2023/115,
author = {Eike Kiltz and Jiaxin Pan and Doreen Riepel and Magnus Ringerud},
title = {Multi-User CDH Problems and the Concrete Security of NAXOS and HMQV},
howpublished = {Cryptology ePrint Archive, Paper 2023/115},
year = {2023},
note = {\url{https://eprint.iacr.org/2023/115}},
url = {https://eprint.iacr.org/2023/115}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.
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Showing:
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@Ibrahim Nash 5761
@akhayrutdinov 5111
@mb1973 4989
@Quandray 4944
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@sanjay05 3762
@marius_valentin_dragoi 3516
@sushant_a 3459
@verma_ji 3341
@KshamaGupta 3318
Easy Accuracy: 50.08% Submissions: 16451 Points: 2
Given two linked lists, your task is to complete the function makeUnion(), that returns the union of two linked lists. This union should include all the distinct elements only.
Example 1:
Input:
L1 = 9->6->4->2->3->8
L2 = 1->2->8->6->2
Output: 1 2 3 4 6 8 9
The task is to complete the function makeUnion() which makes the union of the given two lists and returns the head of the new list.
Note: The new list formed should be in non-decreasing order.
Expected Time Complexity: O(N * Log(N))
Expected Auxiliary Space: O(N)
Constraints:
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You will be given two arrays of floating-point numbers. Your task is to pair the corresponding elements of the two arrays, and get the maximum of each pair. However, if the two corresponding elements are equal, you must take their sum instead.
For example, given the lists [1, 3, 3.2, 2.3] and [3, 1, 3.2, 2.6], you must do the following:
• Pair the elements (or zip): [[1, 3], [3, 1], [3.2, 3.2], [2.3, 2.6]].
• Go through each pair and apply the process above: [3, 3, 6.4, 2.6].
Specs
• The arrays / lists will always have equal length. They may however be empty.
• The numbers they contain will always fit your language's capabilities, as long as you do not abuse that. They may be positive, zero or negative, you must handle all types.
• If it helps you reduce your byte count, you may also take the length of the lists as input.
Test Cases
Array_1, Array_2 -> Output
[], [] -> []
[1, 2, 3], [1, 3, 2] -> [2, 3, 3]
[1, 3, 3.2, 2.3], [3, 1, 3.2, 2.6] -> [3, 3, 6.4, 2.6]
[1,2,3,4,5,5,7,8,9,10], [10,9,8,7,6,5,4,3,2,1] -> [10, 9, 8, 7, 6, 10, 7, 8, 9, 10]
[-3.2, -3.2, -2.4, 7, -10.1], [100, -3.2, 2.4, -7, -10.1] -> [100, -6.4, 2.4, 7, -20.2]
• You say that the numbers will always fit "within your language's" capabilities". As long as you do not "abuse" that. Would only supporting integers in a language that does not have floats be considered an abuse? The question does say floating point but I don't really see a reason why it has to be floats. The same process can be done on integers. I would like to solve this in Brain-Flak but Brain-flak only supports ints. – Wheat Wizard Aug 28 '17 at 17:03
• @WheatWizard I can make an exception for that. Go ahead and post your answer and mention I allowed it to avoid confusion. – user70974 Aug 28 '17 at 18:06
Jelly, 4 bytes
=‘×»
Try it online!
This uses the exact same approach as my APL answer, except Jelly has a builtin for adding one to a number!
• Hate to be a spoilsport, but aren't some of those characters more than one byte each in any sensible encoding? – Cedric Knight Aug 30 '17 at 1:26
• This uses the jelly codepage. – Zacharý Aug 30 '17 at 1:37
• I finally won against competition! – Zacharý Sep 7 '17 at 0:39
• @Zacharý ONE MAN, 4 btytes... THIS SUMMER... You... WILL... BE... JELLY OF HIM... rated J for Jelly. – Magic Octopus Urn Oct 27 '17 at 18:58
Kotlin, 7875716665 59 bytes
It's my first attempt, be cool :D
a.zip(b).map{(a,b)->when{b>a->b;a>b->a;else->a*2}}.toList()
TIO doesn't work with this solution (and i don't know why), source code for testing below
fun main(args: Array<String>) {
bestOfTwo(floatArrayOf(), floatArrayOf()).print()
bestOfTwo(floatArrayOf(0F), floatArrayOf(0F)).print()
bestOfTwo(floatArrayOf(1F,2F,3F), floatArrayOf(1F,3F,2F)).print()
bestOfTwo(floatArrayOf(1F,3F,3.2F,2.3F), floatArrayOf(3F,1F,3.2F,2.6F)).print()
bestOfTwo(floatArrayOf(1F,2F,3F,4F,5F,5F,7F,8F,9F,10F), floatArrayOf(10F,9F,8F,7F,6F,5F,4F,3F,2F,1F)).print()
bestOfTwo(floatArrayOf(-3.2F,-3.2F,-2.4F,7F,-10.1F), floatArrayOf(100F,-3.2F,2.4F,-7F,-10.1F)).print()
}
fun bestOfTwo(a :FloatArray, b :FloatArray): List<Float> =
a.zip(b).map{(a,b)->when{b>a->b;a>b->a;else->a*2}}.toList()
fun List<Float>.print() {
this.forEach { print("$it, ") }; println("") } EDIT: -3 by replace "a+b[i]" by "a*2" -4 by replace "mapIndexed" method by "zip" (Thanks to @AnonymousReality Swift solution) -5 by replace "Math.max" method by when condition -1 by change when condition order -6 by change toFloatArray() by toList() • Welcome to PPCG! Please don't be discouraged by the downvote (it's the result of a slight quirk of the system that happens when a new user's first post is auto-flagged for quality and then they improve said post!!) – Jonathan Allan Aug 28 '17 at 13:23 • The worst "feature" ever...btw don't feel bad about it. – Erik the Outgolfer Aug 28 '17 at 13:26 Python 2, 45 bytes A mix of my initial solution and @ovs'. lambda*a:map(lambda x,y:max(x,y)*-~(x==y),*a) Try it online! Python 2, 49 bytes lambda x,y:[max(a,b)*-~(a==b)for a,b in zip(x,y)] Try it online! Python 2, 46 bytes @ovs suggested this method to save 3 bytes. lambda*x:[max(a,b)*-~(a==b)for a,b in zip(*x)] Try it online! How? First off, we pair the corresponding elements, by using either * or zip(). That allows us to do our further golfing by working either with a map or a list comprehension. The cool trick in this answer is this part: max(x,y)*-~(x==y). How does that work? - Well, as most of you already know, Python auto-converts bool values to integers when they are used in arithmetic operations. Hence, (x==y) gets evaluated as 1, if the condition is met. However, if the two values are not equal, it returns 0 instead. Then, the bitwise operation -~ increments the value returned from the bool by 1, giving us either 2 or 1. max(a,b) gives the maximum value of the pair and * multiplies it by the value returned above (so it gets multiplied by 2 only if they are equal, in which case max() returns the value of both). This is based on the fact that the sum of two equal numbers is in fact either of them doubled, and kind of "abuses" Python's bool class being a subclass of int. • Wow, that was really fast! – user70974 Aug 28 '17 at 12:05 • more straightforward, same number of bytes: lambda*a:map(lambda x,y:(x<=y)*y+(x>=y)*x,*a) – jferard Aug 28 '17 at 20:09 • @jferard I fact, that's already Luis' solution. – Mr. Xcoder Aug 28 '17 at 20:12 • @Mr.Xcoder Oops! I didn't read the whole page... – jferard Aug 28 '17 at 20:18 • Never say "above," as the ordering can change (I don't see your solution above) – Zacharý Aug 28 '17 at 21:53 JavaScript (ES6), 534945 43 bytes a=>b=>a.map((x,y)=>(y=b[y])>x?y:y<x?x:x+y) • 4 bytes saved by borrowing a trick from Mr. Xcoder. • 2 bytes saved thanks to Arnauld. Try it o.innerText=(f= a=>b=>a.map((x,y)=>(y=b[y])>x?y:y<x?x:x+y) )(i.value=[1,3,3.2,2.3])(j.value=[3,1,3.2,2.6]);oninput=_=>o.innerText=f(i.value.split,.map(eval))(j.value.split,.map(eval)) <input id=i><input id=j><pre id=o> Explanation a=>b=> Anonymous function taking the 2 arrays as arguments via parameters a and b, in currying syntax (i.e., call with f(a)(b) a.map((x,y)=> ) Map over the first array, passing each element through a function where x is the current element and y is the current index. (y=b[y]) Get the element at index y in the second array and assign that as the new value of y. >x?y Check if y is greater than x and, if so, return y. :y<x?x Else, check if y is less than x and, if so, return x :x+y Else, return the sum of x and y. (Multiplying x or y by 2 would also work here, for the same number of bytes.) • j.value.split,.map(eval) or eval('['+j.value+']')? Also x+y would look neater IMHO. – Neil Aug 28 '17 at 14:56 • @Neil: 1) I find the former easier to type. Also, I have a couple of Snippet templates on one of my machines; it's easier just to tack .map(eval) onto them. 2) Agreed, will edit in momentarily. – Shaggy Aug 28 '17 at 16:13 Haskell, 34 bytes x!y|x>y=x|x<y=y|1<2=x+y zipWith(!) Try it online. • Same byte count: x!y=max x y+sum[x|x==y]. – nimi Aug 28 '17 at 19:46 R, 31 29 bytes function(a,b)pmax(a,b)+a*!a-b pmax takes the parallel maximum of the two (or more) arrays (recycling the shorter as needed). I was looking at Luis Mendo's comment and obviously I realized the approach could work for R as well. That got me to 30 bytes, but then I started playing around with different ways of getting indices instead to improve my original answer, and stumbled upon !a-b as TRUE where a==b and FALSE otherwise, equivalent to a==b. However, for whatever reason, R doesn't require parentheses around !a-b as it does for a==b, which saved me two bytes. As mentioned by JDL in the comments, this works because ! (negation) has lower precedence than the binary operator - in R, which is strange. Try it online! (new version) Try it online! (original) • It turns out that unary "!" has lower precedence in R than binary "-", which I think is quite unusual (and I hadn't realised until reading this answer!) – JDL Aug 29 '17 at 15:29 • @JDL yeah I almost always have to open up the R Syntax page while golfing in case of weird quirks like this...and also because I can never remember the precedence of : when interacting with arithmetic. – Giuseppe Aug 29 '17 at 15:55 Python 3, 4846 44 bytes -2 bytes thanks to @nwellnhof lambda*a:map(lambda*x:max(x)*2/len({*x}),*a) Try it online! Dyalog APL, 5 bytes ⌈×1+= Try it online! How? • ⌈, element-wise maximum of the arguments • ×, element-wise multiply • 1+=, 1 added to the element-wise equality of the arguments This works because if the numbers are unequal, 1+= will be 1, which when multiplied by the maximum, is the maximum. When the numbers are equal, 1+= will return 2, when that is multiplied by the maximum, we get twice the maximum, or the maximum added to itself. Jelly, 6 bytes żSṀE?€ A dyadic link taking a list of numbers on each side and returning the resulting list. Try it online! or see a test-suite*. How? żSṀE?€ - Link: list of numbers L, list of numbers R e.g. [1,3,3.2,2.3], [3,1,3.2,2.6] ż - zip - interleave L & R [[1,3],[3,1],[3.2,3.2],[2.3,2.6]] € - for each pair: ? - { if: E - ...condition: equal 0 0 1 0 S - ...then: sum 6.4 Ṁ - ...else: maximum 3 3 2.6 - } ... -> [3 ,3 ,6.4 ,2.6] An alternative is this monadic link taking a list of the two lists, also 6 bytes: +»⁼?"/ * I don't think I've ever created a test-suite footer almost three times the byte count of the code before! • Outgolfed!. +1 for the practically verbatim interpretation of the question. – Zacharý Aug 28 '17 at 20:47 • ...and I've been caught out be forgetting that » vectorises before! – Jonathan Allan Aug 28 '17 at 20:55 • What else would it do, take the maximum array in some convoluted way? – Zacharý Aug 28 '17 at 21:13 • No need for any convoluted definitions, Python manages - for example max([1,1,0],[1,0,3]) -> [1,1,0] (not [1,1,3]). – Jonathan Allan Aug 28 '17 at 21:20 • So, basically infinite-base? – Zacharý Aug 28 '17 at 21:21 05AB1E, 5 bytes øεMÃO Try it online! -1 thanks to Emigna. • Nice idea using γ! – Emigna Aug 28 '17 at 13:20 • @Emigna I really wanted "maximal elements", and {γθ is probably the shortest I can get to for that. – Erik the Outgolfer Aug 28 '17 at 13:21 • How about øεMÃO? – Emigna Aug 28 '17 at 13:23 • @Emigna Cool, thanks! (duh why didn't I think of MÃ) yay got the lead now :p btw øεZÃO would work too – Erik the Outgolfer Aug 28 '17 at 13:23 MATL, 7 bytes X>tG=s* Input is a two-row matrix, where each row is one of the arrays. Try it online! Explanation X> % Implicit input. Maximum of each column t % Duplicate G % Push input = % Is equal? Element-wise with broadcast. Gives a two-row matrix s % Sum of each column. Gives a row vector containing 1 and 2 * % Multiply, element-wise. Implicit display Java 8, 806967666564 63 bytes (a,b,l)->{for(;l-->0;)if(a[l]>=b[l])b[l]=a[l]*(a[l]>b[l]?1:2);} Modifies the second input-array instead or returning a new float-array to save bytes. -11 bytes by taking the length as additional integer-input, which is allowed according to the challenge rules. -5 bytes thanks to @OliverGrégoire (one byte at a time.. xD) -1 byte indirectly thanks to @Shaggy's JS answer, by using a[l]*2 instead of a[l]+b[l]. Explanation: Try it here. (a,b,l)->{ // Method with 2 float-array and integer parameters and no return-type for(;l-->0;) // Loop over the array if(a[l]>=b[l]) // If the current value in a is larger or equal to b: b[l]= // Modify the second input-array: a[l]* // Use a multiplied by: (a[l]>b[l]? // If the current value in a is larger than b: 1 // Multiply by 1 : // Else (a is smaller of equal to b): 2) // Multiply by 2 // End of loop (implicit / single-line body) } // End of method • "If it helps you reduce your byte count, you may also take the length of the lists as input." It will definitely reduce your byte-count ;) – Olivier Grégoire Aug 28 '17 at 12:18 • Also, 2 bytes shorter: a->b->l->{float A,B;for(;l-->0;b[l]=(A=a[l])<B?B:A>B?A:A+B)B=b[l];} – Olivier Grégoire Aug 28 '17 at 12:25 • And you can save one more byte by putting float A, B in the for initialization. – Olivier Grégoire Aug 28 '17 at 12:28 • Or this: (a,b,l)->{for(;l-->0;)if(a[l]>=b[l])b[l]=a[l]*(a[l]>b[l]?1:2);} (63 bytes) – Olivier Grégoire Aug 28 '17 at 12:44 • @OlivierGrégoire Lol.. with golfing every byte helps, but that doesn't mean you need to golf it one byte at a time. ;p – Kevin Cruijssen Aug 28 '17 at 12:53 Pyth, 11 bytes m*eSdhnd{dC Try it here! Pyth, 12 bytes m*eSdhqhdedC Try it here! or m*eSdh!tl{dC Try it here! • @Jakube That's Erik's solution already, sadly. I wanted to use that too, but I can't now – Mr. Xcoder Aug 29 '17 at 13:56 • Oh, didn't see that one. – Jakube Aug 29 '17 at 13:57 05AB1E, 98 7 bytes Saved a byte as Erik the Outgolfer pointed out that a list of lists is valid input. øεMsËi· Try it online! Explanation ø # zip the lists ε # apply to each pair M # get max s # swap the top 2 elements on the stack Ëi # if all elements are equal · # double the max • Wow, that was really fast! – user70974 Aug 28 '17 at 12:05 • You can save a byte by removing the ‚ and inputting as a pair of a list and a list. – Erik the Outgolfer Aug 28 '17 at 13:01 • @EriktheOutgolfer: True. I assumed we weren't allowed to, but I see that the challenge does specify standard I/O rules. Thanks for notifying :) – Emigna Aug 28 '17 at 13:16 • @Emigna Tip: don't make rules out of your mind ;) – Erik the Outgolfer Aug 28 '17 at 13:19 • @EriktheOutgolfer: Yeah I really need to stop doing that. Especially rules which make my programs longer ;) – Emigna Aug 28 '17 at 13:19 Mathematica, 31 bytes MapThread[If[#==#2,2#,Max@##]&] J, 7 bytes >.+@.= Try it online! Takes one list as the left argument and the other as the right. Luckily, equality is a rank zero operation. Explanation >.+@.= = Compare equality pairwise @. If equal + Sum >. (Else) take greater value @. isn't really an if statement, but in this case it functions as one (it indexes into the gerund >.+ based on the result of its right argument and applies that to the input). • Nice job, I know I couldn't do this, even though you have beenoutgolfed by my translation of my APL. >_< – Zacharý Aug 29 '17 at 0:15 • J really shines here – Jonah Aug 29 '17 at 8:50 • @Zacharý rats, well-golfed nonetheless. – cole Aug 29 '17 at 13:49 Ruby, 42 bytes ->a,b{a.zip(b).map{|x,y|[x+y,x,y][x<=>y]}} Try it online! The spaceship operator is great. TI-Basic, 23 21 bytes Prompt A,B (ʟA=ʟB)ʟA+max(ʟA,ʟB Too bad lists take up two bytes each... • You can save two bytes by prompting for X and Y, then using ʟX and ʟY to access them, i.e. "Prompt X,Y:ʟX(ʟX=ʟY)+max(ʟ1,ʟ2". – Scott Milner Aug 29 '17 at 21:53 • Also, this is currently invalid, since L1(L1=L2) attempts to get the element of L1 at a list, which throws an error. To fix that, swap the order, i.e. (L1=L2)L1. – Scott Milner Aug 29 '17 at 21:58 • @ScottMilner Thanks for pointing both of those out. – Timtech Aug 29 '17 at 23:04 Octave, 36 Byte @(a,b)a.*(a>b)+b.*(b>a)+2.*a.*(a==b) Pyth, 7 bytes ms.MZdC Try it here. Python 3, 4946 45 bytes 3 bytes removed thanks to @Mr.Xcoder (splat instead of two arguments), and 1 byte thanks to @ovs (map instead of list comprehension) lambda*x:map(lambda a,b:a*(a>=b)+b*(b>=a),*x) Try it online! Common Lisp, 60 59 bytes (mapcar(lambda(x y)(*(max x y)(if(= x y)2 1)))(read)(read)) Try it online! -1 byte thanks to @Zacharý! • 59 bytes: (mapcar(lambda(x y)(*(max x y)(if(= x y)2 1)))(read)(read)). – Zacharý Aug 29 '17 at 0:42 • You're welcome, I don't know lisp that well, I just translated my other answers into Lisp which ended up saving a byte. – Zacharý Aug 29 '17 at 8:11 Python with numpy, 28 bytes lambda a,b:a*(a>=b)+b*(b>=a) Assumes input is given as two numpy arrays. • If we are using numpy then here is my worse solution: lambda a,b:n.fmax(a,b)*((a==b)+1) – Erich Aug 29 '17 at 18:09 • @Erich I like the idea, but to do that I would need to import numpy as n. I get away without it here because it's implicit in the input. – user48543 Aug 29 '17 at 18:22 • I guess i'm a bit shaky on the explicit byte counting, often python answers are simply lambdas, when an actual implementation of an answer would require assigning it to something. for this reason I wonder if it is allowable to get away with an implicit import statement as well? – Erich Aug 29 '17 at 18:32 • @Erich In general, you can only refer to a variable n if you've defined n in your code, so imports must be explicit. By default, we allow functions or full programs as answers, which includes anonymous functions. – user48543 Aug 29 '17 at 18:39 • Well, this only needs input as numpy arrays, rather than importing numpy. But does this even work without using return? – Zacharý Aug 29 '17 at 20:29 C# (.NET Core), using Linq 47+18=65 bytes x=>y=>x.Zip(y,(a,b)=>a>b?a:b>a?b:b+a).ToArray() Try it online! C# (.NET Core), 82 bytes x=>y=>l=>{for(int i=0;i<l;i++)x[i]=x[i]>y[i]?x[i]:y[i]>x[i]?y[i]:y[i]*2;return x;} Try it online! • You can drop the LINQ answer by a few bytes by changing namespace System.LINQ to using System.LINQ – jkelm Aug 28 '17 at 18:27 • @jkelm yeah, I've been wondering if the 'using System; is to be included or not like that, I guess not. I'll clean it up – Dennis.Verweij Aug 28 '17 at 19:23 • System.Linq is included in the "Visual C# Interactive Compiler". I am not totally sure about returning Array vs IList vs IEnumerable, but if all are eligible then you can get the byte count to 37 - tio.run/##Sy7WTS7O/… – dana Dec 17 '18 at 4:22 Perl 6, 34 28 bytes {map {.max*2/set$_},[Z] $_} Try it online! Swift 3, 81 79 Bytes func n(a:[Double],b:[Double]){for(x,y)in zip(a,b){print((x==y) ?x+y:max(x,y))}} Swift has an interesting property in that an Int isn't directly castable to a Double, so you have to specify any arrays as being arrays of Doubles before passing them to the function. (e.g.) var k:[Double] = [1,2,3,4,5,5,7,8,9,10] Edit: -2 bytes thanks to @EriktheOutgolfer • Do you need spaces around (x,y) and before ?? – Erik the Outgolfer Aug 28 '17 at 13:28 • @EriktheOutgolfer The one before ? is needed because Swift would treat them as optional types instead of ternaries (which they aren't). The others aren't. Apart from that, this can be drastically golfed. – user70974 Aug 28 '17 at 13:31 • @EriktheOutgolfer - TheIOSCoder has already answered you partly, but you're right, you don't need the ones in the for loop, interesting! – AnonymousReality Aug 28 '17 at 13:34 • 73 bytes: func n(a:[Float],b:[Float]){print(zip(a,b).map{$0==$1 ?2*$0:max($0,$1)})} (the float inaccuracies need not to be handled by default) – Mr. Xcoder Aug 28 '17 at 13:38
• Or 74 bytes: func n(a:[Float],b:[Float]){print(zip(a,b).map{($0==$1 ?2:1)*max($0,$1)})} – Mr. Xcoder Aug 28 '17 at 13:40
C, 76 75 bytes
Thanks to @Kevin Cruijssen for saving a byte!
f(a,b,n)float*a,*b;{for(;n--;++a,++b)printf("%f ",*a>*b?*a:*b>*a?*b:*a*2);}
Try it online!
Japt, 13 bytes
íV,È¥Y Ä *XwY
Try it online! with the -Q flag to format the output array.
• Nicely done. I made 2 attempts at this earlier with both coming out at 17 bytes. I'd forgotten í could take a function as a second argument. – Shaggy Aug 28 '17 at 16:39
Rust, 107 97 bytes
|a:V,b:V|a.iter().zip(b).map(|(&x,y)|if x==y{x+y}else{x.max(y)}).collect::<V>();
type V=Vec<f32>;
Try it online!
Saved 8 bytes thanks to @mgc
• I guess you can save 8 bytes by using type inference on the collected Vec and by using the max method of f32s: |a:Vec<f32>,b:Vec<f32>|a.iter().zip(b).map(|(&x,y)|if x==y{x+y}else{x.max(y)}).collect::<Vec<_>>(); – mgc Aug 28 '17 at 22:26
• @mgc Thanks! Type inference was a good idea, but in this case type alias is even shorter. – jferard Aug 29 '17 at 4:41
Swift 4, 41 bytes
{zip($0,$1).map{$0==$1 ?2*$0:max($0,$1)}} Test cases: let f: ([Float], [Float]) -> [Float] = {zip($0,$1).map{$0==$1 ?2*$0:max($0,$1)}}
let testcases: [(inputA: [Float], inputB: [Float], expected: [Float])] = [
(
inputA: [],
inputB: [],
expected: []
),
(
inputA: [1, 2, 3],
inputB: [1, 3, 2],
expected: [2, 3, 3]
),
(
inputA: [1, 3, 3.2, 2.3],
inputB: [3, 1, 3.2, 2.6],
expected: [3, 3, 6.4, 2.6]
),
(
inputA: [1,2,3,4,5,5,7,8,9,10],
inputB: [10,9,8,7,6,5,4,3,2,1],
expected: [10, 9, 8, 7, 6, 10, 7, 8, 9, 10]
),
(
inputA: [-3.2, -3.2, -2.4, 7, -10.1],
inputB: [100, -3.2, 2.4, -7, -10.1],
expected: [100, -6.4, 2.4, 7, -20.2]
),
]
for (caseNumber, testcase) in testcases.enumerated() {
let actual = f(testcase.inputA, testcase.inputB)
assert(actual == testcase.expected,
"Testcase #\(caseNumber) \((testcase.inputA, testcase.inputB)) failed. Got \(actual), but expected \(testcase.expected)!")
print("Testcase #\(caseNumber) passed!")
}
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Algebraically closed fields of positive characteristic - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:27:14Z http://mathoverflow.net/feeds/question/3551 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3551/algebraically-closed-fields-of-positive-characteristic Algebraically closed fields of positive characteristic Harrison Brown 2009-10-31T15:01:36Z 2010-03-31T02:05:12Z <p>I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far as intuition goes, I might as well add "...of characteristic 0" at the end of that.</p> <p>I know the complex numbers from kindergarten algebra, so I have a fairly good idea of how at least one algebraically closed field of characteristic 0 looks and feels. And while I don't have nearly the same handle on the field of algebraic numbers, I can pretty much do arithmetic in it, so that's two examples.</p> <p>But I've never (<em>really</em>) seen an algebraically closed field of characteristic p > 0! I can build one just fine, and if you put a gun to my head I could probably even do some arithmetic in it, but there's no intuition of the sort that you get with the complex numbers. So: does anyone know of an intuitive description of such a field, that it's possible to get a real sense of in the same way as C?</p> http://mathoverflow.net/questions/3551/algebraically-closed-fields-of-positive-characteristic/3555#3555 Answer by John Goodrick for Algebraically closed fields of positive characteristic John Goodrick 2009-10-31T15:44:59Z 2009-10-31T15:44:59Z <p>I doubt I can give an "intuitive" description of such a field, but I hope this is useful anyhow:</p> <p><em>Any two algebraically closed fields of the same characteristic and same transcedence degree (over their prime subfields) are isomorphic.</em></p> <p>So, in some sense, there really aren't too many examples of the kind of fields you're looking for.</p> <p>(This is one of those facts that model-theorists seem to like more than geometers, even though it has a very nice proof, IMHO. It can be proved in essentially the same way that you prove any two vector spaces of the same dimension over the same field are isomorphic, replacing spans by algebraic closures and "linearly independent" by "algebraically independent." The key point is that after making this translation, the Steinitz Exchange Lemma works just fine inside of any algebracially closed field, of any characteristic.)</p> http://mathoverflow.net/questions/3551/algebraically-closed-fields-of-positive-characteristic/3564#3564 Answer by Ben Webster for Algebraically closed fields of positive characteristic Ben Webster 2009-10-31T16:44:48Z 2009-10-31T16:44:48Z <p>So first of all, there's only one algebraically closed field of characteristic p that you should think about for the sake of intuition, the algebraic closure K of F_p. And that's very easy, it's just the limit of the finite fields of order p^n as n goes to infinity.</p> <p>In fact, almost any construction involving K can be done in a sufficiently large finite field. For example, a statement like "every variety over <code>F_p</code> has a point over K" is equivalent to "every variety over <code>F_p</code> has a point over a sufficiently large finite field."</p> <p>Also, the multiplicative group of K is easy to understand: it's just the direct limit over all numbers n coprime to p of Z/nZ, where Z/nZ -> Z/mZ if n|m by 1 -> m/n. Which is another way of saying that K is gotten from F_p by just adding all roots unity of order coprime to p.</p> http://mathoverflow.net/questions/3551/algebraically-closed-fields-of-positive-characteristic/3568#3568 Answer by Sam Lichtenstein for Algebraically closed fields of positive characteristic Sam Lichtenstein 2009-10-31T17:01:28Z 2009-10-31T17:01:28Z <p>It's certainly not too hard to understand everything there is to understand about the algebraic closure of Fp. Perhaps the reason this is unsatisfying as an example for founding intuition is because it doesn't really have a nice topological structure; it lacks anything like a natural metric. So here's an attempt to explain why what is in some sense the next simplest example puts you in a better situation, intuition-wise.</p> <p>If you have some intuition about the p-adic numbers look and feel (for example, topologically), then you secretly have intuition for the t-adic topology on the complete local field K=Fp((1/t)). Now, as far as characteristic p fields go, this sort of puts you in the position of (in your parlance) a "preschooler" who knows about R but hasn't yet gotten to kindergarten to learn about C. Why is K like R? First, it is locally compact. Second, it is at least analogous to completing Fp(t), which is very much like Q with Fp[t] as the analogue of Z, at an "infinite" valuation, namely the degree or (1/t)-adic valuation, rather than a "finite" place like a prime polynomial in Fp[t]. (The (1/t)-adic valuation corresponds to the point at infinity on the projective line over F_p. Likewise, number theorists love to say, perhaps partly to annoy John Conway, that the real and complex absolute values on Q correspond to "archimedean primes" or "primes dividing infinity". This is actually a pretty lame analogy, though, since K=Fp((1/t)) looks a lot more like Fp((t)), say, than R or C looks like Q2.)</p> <p>Unfortunately there are two extra difficulties in the characteristic p case. First, upon passing to the algebraic closure L of K we lose completeness. Second, we make an infinite field extension, unlike the degree 2 extension C/R. Thus, while L is an algebraically closed field of characteristic p, it bears little resemblance to R. In fact, it's a lot more like an algebraically closed field of characteristic 0 that is a bit scarier (at least to me) than C, namely Cp, or what you get when you complete the algebraic closure of Qp with respect to the topology coming from the unique extension of the p-adic valuation. While this may seem bad, I think it's actually good, because one can really get a handle on some of the properties of Cp. [Note that as another answerer pointed out, Cp = C as a field, but not as a topological or valued field, which is really a more interesting structure to consider from the viewpoint of intuition anyway.]</p> <p>For example of some similarities, miraculously Cp turns out to still be algebraically closed, and I believe the same proof goes through for L above. Another property L and Cp share is that in addition to "geometric" field extensions K'/K obtained by considering function fields of plane curves over Fp, there are also "stupider" extensions coming from extending the coefficient field. This is like passing to unramified extensions of p-adic fields, where one ramps up the residue field. (In fact, it's exactly the same thing.) Both L and Cp are complete valued fields with residue field the algebraic closure of Fp. (But the valuation is NOT discrete; it takes values in Q.) There are some dangerous bends to watch out for topologically, however. Some cursory googling tells me that Cp is not locally compact, although it is topologically separable. </p> <p>In addition, positive characteristic inevitably brings along the problem of inseparable field extensions sitting side L. This is, of course, an aspect where L/K is unlike Cp/Qp. Notwithstanding such annoyances, I would argue that the picture sketched above actually does give an example of an algebraically closed field of characteristic p for which it is possible to have some real intuition. </p> http://mathoverflow.net/questions/3551/algebraically-closed-fields-of-positive-characteristic/3584#3584 Answer by Qiaochu Yuan for Algebraically closed fields of positive characteristic Qiaochu Yuan 2009-10-31T19:46:14Z 2009-10-31T20:03:00Z <p>The algebraic closure of F<sub>p</sub> has one really important property, which is that its Galois group over F<sub>p</sub> is <strike>isomorphic to</strike> the profinite completion of Z and is topologically generated by the Frobenius map x -> x^p. The fixed field of the subgroup nZ is precisely F<sub>p^n</sub>; in other words, there is exactly one copy of each finite field of characteristic p and they are nested according to divisibility. Any statement you can make about a finite number of elements in this field takes place, as others have mentioned, in a sufficiently large finite subfield.</p> <p>One interesting way to get a grip on the elements of the algebraic closure is to think of them as <strong>aperiodic necklaces.</strong> As explained in <a href="http://mathoverflow.net/questions/769/exhibit-an-explicit-bijection-between-irreducible-polynomials-over-finite-fields" rel="nofollow">this question I asked</a>, there is a (non-canonical) bijection between irreducible polynomials of degree n over F<sub>p</sub> and Lyndon words of length n on an alphabet with p letters. One should think of conjugating as analogous to cyclic rotation, and the choice of a particular representative of a necklace up to rotation as the choice of a particular representative of a point up to conjugation. This is the categorification of the fact that as dynamical systems, words on an alphabet of length p and the algebraic closure of F<sub>p</sub> have the same zeta function.</p> <p>(The upshot of all of this is that some people think there should be a proof of the rationality of the zeta function of a variety over F<sub>p</sub> using formal language theory, by finding a regular language that describes those points. Unfortunately, all known translations give non-regular languages.)</p> http://mathoverflow.net/questions/3551/algebraically-closed-fields-of-positive-characteristic/8325#8325 Answer by S. Carnahan for Algebraically closed fields of positive characteristic S. Carnahan 2009-12-09T07:18:07Z 2009-12-09T07:18:07Z <p>The spherical completion (aka maximal completion) of $\overline{\mathbb{F}_p((t))}$ is an example of an algebraically closed field of characteristic p that hasn't been mentioned here yet. The "spherically complete" property means that any sequence of nested nonempty balls has nonempty intersection (if the radii converge to zero, this is just completeness). One nice property of this field is that you can describe the elements "explicitly": take any power series with rational powers of t and coefficients in $\overline{\mathbb{F}_p}$, such that the set of exponents with nonzero coefficients is a well-ordered subset of Q. (It is rather tricky to verify that this set is closed under multiplication)</p> <p>If you like nonarchimedean analysis, you might have a reason to use this field instead of Cp or an algebraic closure of Fp((t)).</p> http://mathoverflow.net/questions/3551/algebraically-closed-fields-of-positive-characteristic/16550#16550 Answer by L Spice for Algebraically closed fields of positive characteristic L Spice 2010-02-26T19:46:34Z 2010-03-31T02:05:12Z <p>It is a little unfair to pass this off as an answer to your question, but it also seems too interesting to be buried as a comment. (I can say that because I am citing other people's work!)</p> <p>The comments following <a href="http://mathoverflow.net/questions/3551/algebraically-closed-fields-of-positive-characteristic/3564" rel="nofollow">Ben's answer</a> point out some nice senses in which ‘any’ calculation over the algebraic closure of $\mathbb F_p$ is really a calculation over a suitably large finite extension of $\mathbb F_p$, but I didn't see a precise statement in the comments. I think it is very much worthwhile to note that, not just algebraically-closed-positive-characteristic computations, but even algebraically-closed-characteristic-$0$ calculations can be viewed this way. For example, this is one way to prove that an injective, polynomial self-map of $\mathbb C^n$ is bijective.</p> <p>See <a href="http://arxiv.org/abs/arxiv:0903.0517" rel="nofollow">Serre's lovely article</a> and <a href="http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem" rel="nofollow">Tao's lovely summary of it</a> for more details on this point of view.</p>
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Magyar Information Contest Journal Articles
# Problem P. 3703. (April 2004)
P. 3703. Subscribers can reach the text of the problem after signing in. The text will be public from April 15, 2004.]
(4 pont)
Deadline expired on May 11, 2004.
### Statistics:
139 students sent a solution. 4 points: 64 students. 3 points: 36 students. 2 points: 18 students. 1 point: 11 students. 0 point: 3 students. Unfair, not evaluated: 7 solutions.
Our web pages are supported by: Morgan Stanley
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# Tuple
A tuple is an ordered multiset that is a finite sequence with $n \ge 1$ tuple members.
## References
### 2015
• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/tuple Retrieved:2015-2-23.
• A tuple is an ordered list of elements. In mathematics, an n-tuple is a sequence (or ordered list) of $n$ elements, where $n$ is a non-negative integer. There is only one 0-tuple, an empty sequence. An $n$ -tuple is defined inductively using the construction of an ordered pair. Tuples are usually written by listing the elements within parentheses " $(\text{ })$ " and separated by commas; for example, $(2, 7, 4, 1, 7)$ denotes a 5-tuple. Sometimes other delimiters are used, such as square brackets " $[\text{ }]$ " or angle brackets " $\langle\text{ }\rangle$ ". Braces " $\{\}$ " are almost never used for tuples, as they are the standard notation for sets. Tuples are often used to describe other mathematical objects, such as vectors. In computer science, tuples are directly implemented as product types in most functional programming languages. More commonly, they are implemented as record types, where the components are labeled instead of being identified by position alone. This approach is also used in relational algebra. Tuples are also used in relation to programming the semantic web with Resource Description Framework or RDF. Tuples are also used in linguistics [1] and philosophy. [2]
• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/tuple#Properties Retrieved:2015-2-23.
• The general rule for the identity of two $n$ -tuples is : $(a_1, a_2, \ldots, a_n) = (b_1, b_2, \ldots, b_n)$ if and only if $a_1=b_1,\text{ }a_2=b_2,\text{ }\ldots,\text{ }a_n=b_n.$ Thus a tuple has properties that distinguish it from a set.
1. A tuple may contain multiple instances of the same element, so tuple $(1,2,2,3) \neq (1,2,3)$ ; but set $\{1,2,2,3\} = \{1,2,3\}$ .
2. Tuple elements are ordered: tuple $(1,2,3) \neq (3,2,1)$ , but set $\{1,2,3\} = \{3,2,1\}$ .
3. A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.
• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/tuple#Tuples_as_functions Retrieved:2015-2-23.
• If we are dealing with sets, an $n$ -tuple can be regarded as a function, F, whose domain is the tuple's implicit set of element indices, X, and whose codomain, Y, is the tuple's set of elements. Formally: : $(a_1, a_2, \dots, a_n) \equiv (X,Y,F)$ where: : \begin{align} X & = \{1, 2, \dots, n\} \\ Y & = \{a_1, a_2, \ldots, a_n\} \\ F & = \{(1, a_1), (2, a_2), \ldots, (n, a_n)\}. \\ \end{align} In slightly less formal notation this says: : $(a_1, a_2, \dots, a_n) := (F(1), F(2), \dots, F(n)).$
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11.1 Cost-Sensitive Classification
Imagine you are an analyst for a big credit institution. Let’s also assume that a correct decision of the bank would result in 35% of the profit at the end of a specific period. A correct decision means that the bank predicts that a customer will pay their bills (hence would obtain a loan), and the customer indeed has good credit. On the other hand, a wrong decision means that the bank predicts that the customer’s credit is in good standing, but the opposite is true. This would result in a loss of 100% of the given loan.
Good Customer (truth) Bad Customer (truth)
Good Customer (predicted) + 0.35 - 1.0
Expressed as costs (instead of profit), we can write down the cost-matrix as follows:
costs = matrix(c(-0.35, 0, 1, 0), nrow = 2)
print(costs)
## truth
## good -0.35 1
## bad 0.00 0
An exemplary data set for such a problem is the German Credit task:
library(mlr3)
task = mlr_tasks$get("german_credit") table(task$truth())
##
## 700 300
The data has 70% customers who are able to pay back their credit, and 30% bad customers who default on the debt. A manager, who doesn’t have any model, could decide to give either everybody a credit or to give nobody a credit. The resulting costs for the German credit data are:
# nobody:
(700 * costs[2, 1] + 300 * costs[2, 2]) / 1000
## [1] 0
# everybody
(700 * costs[1, 1] + 300 * costs[1, 2]) / 1000
## [1] 0.055
If the average loan is $20,000, the credit institute would lose more than one million dollar if it would grant everybody a credit: # average profit * average loan * number of customers 0.055 * 20000 * 1000 ## [1] 1100000 Our goal is to find a model which minimizes the costs (and thereby maximizes the expected profit). 11.1.1 A First Model For our first model, we choose an ordinary logistic regression (implemented in the add-on package mlr3learners). We first create a classification task, then resample the model using a 10-fold cross validation and extract the resulting confusion matrix: library(mlr3learners) learner = mlr_learners$get("classif.log_reg")
confusion = rr$prediction$confusion
print(confusion)
## truth
## good 607 156
## bad 93 144
To calculate the average costs like above, we can simply multiply the elements of the confusion matrix with the elements of the previously introduced cost matrix, and sum the values of the resulting matrix:
avg_costs = sum(confusion * costs) / 1000
print(avg_costs)
## [1] -0.05645
With an average loan of $20,000, the logistic regression yields the following costs: avg_costs * 20000 * 1000 ## [1] -1129000 Instead of losing over$1,000,000, the credit institute now can expect a profit of more than $1,000,000. 11.1.2 Cost-sensitive Measure Our natural next step would be to further improve the modeling step in order to maximize the profit. For this purpose we first create a cost-sensitive classification measure which calculates the costs based on our cost matrix. This allows us to conveniently quantify and compare modeling decisions. Fortunately, there already is a predefined measure Measure for this purpose: MeasureClassifCosts: cost_measure = MeasureClassifCosts$new("credit_costs", costs)
print(cost_measure)
## <MeasureClassifCosts:credit_costs>
## Packages: -
## Range: [-Inf, Inf]
## Minimize: TRUE
## Predict type: response
We set this measure as new default measure for our cost-sensitive task.
task$measures = list(cost_measure) If we now call resample() or benchmark(), the cost-sensitive measures will be evaluated. We compare the logistic regression to a simple featureless learner and to a random forest from package ranger : learners = mlr_learners$mget(c("classif.log_reg", "classif.featureless", "classif.ranger"))
print(bmr)
## <BenchmarkResult> of 30 experiments in 3 resamplings:
## german_credit classif.log_reg cv -0.05665
## german_credit classif.ranger cv -0.04865
## german_credit classif.featureless cv 0.05500
As expected, the featureless learner is performing comparably bad. The logistic regression and the random forest work equally well.
11.1.3 Thresholding
Although we now correctly evaluate the models in a cost-sensitive fashion, the models themselves are unaware of the classification costs. They assume the same costs for both wrong classification decisions (false positives and false negatives). Some learners natively support cost-sensitive classification (e.g., XXX). However, we will concentrate on a more generic approach which works for all models which can predict probabilities for class labels: thresholding.
Most learners can calculate the probability $$p$$ for the positive class. If $$p$$ exceeds the threshold $$0.5$$, they predict the positive class, and the negative class otherwise.
For our binary classification case of the credit data, the we primarily want to minimize the errors where the model predicts “good”, but truth is “bad” (i.e., the number of false positives) as this is the more expensive error. If we now increase the threshold to values $$> 0.5$$, we reduce the number of false negatives. Note that we increase the number of false positives simultaneously, or, in other words, we are trading false positives for false negatives.
# fit models with probability prediction
learner = mlr_learners$get("classif.log_reg", predict_type = "prob") rr = resample(task, learner, "cv") p = rr$prediction
print(p)
## <PredictionClassif> for 1000 observations:
## row_id truth response prob.good prob.bad
## 1: 7 good good 0.9364 0.06356
## 3: 43 good good 0.6406 0.35944
## ---
## 998: 963 good good 0.8204 0.17963
## 999: 976 good good 0.9167 0.08334
## 1000: 993 good good 0.8080 0.19201
# helper function to try different threshold values interactively
with_threshold = function(p, th) {
p$response = p$set_threshold(th)$response list(confusion = p$confusion, costs = cost_measure$calculate(prediction = p)) } with_threshold(p, 0.5) ##$confusion
## truth
## good 603 159
##
## $costs ## [1] -0.05205 with_threshold(p, 0.75) ##$confusion
## truth
## good 471 72
##
## $costs ## [1] -0.09285 with_threshold(p, 1.0) ##$confusion
## truth
## good 1 1
##
## $costs ## [1] 0.00065 # TODO: include plot of threshold vs performance Instead of manually trying different threshold values, we here use optimize() to find a good threshold value w.r.t. our performance measure: # simple wrapper function which takes a threshold and returns the resulting model performance # this wrapper is passed to optimize() to find its minimum for thresholds in [0.5, 1] f = function(th) { with_threshold(p, th)$costs
}
best = optimize(f, c(0.5, 1))
print(best)
## $minimum ## [1] 0.7661 ## ##$objective
## [1] -0.0956
# optimized confusion matrix:
with_threshold(p, best$minimum)$confusion
## truth
## bad 244 236
The function optimize() is intended for unimodal functions and therefore may converge to a local optimum here. See below for better alternatives to find good threshold values.
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# 134.2 kHz RFID reader for Arduino
I am looking for an Arduino-compatible RFID reader supporting ISO 11784 & 11785, in order to read tags used for animals (mostly cats and dogs) identification.
I found both RMD660 and and Sniffer Nano v2.0 from ITead Studio, but they are out of stock. Is there any alternative?
The final goal is to build an intelligent cat door, that does not require adding a new RFID tag to my cat (he already had one implanted, and doesn't wear a collar).
-
If your goal is not to require an RFID tag why do you need an RFID reader? – Curd Jan 3 '12 at 12:54
He meant "does not require adding ANOTHER RFID tag to my cat". See this Wikipedia page: Microchip implant (animal) – Axeman Jan 3 '12 at 13:32
Thanks Axeman, that's exactly what I meant. I edited my question to reflect this. – Wookai Jan 3 '12 at 16:39
BTW I'm not an expert, but I don't think those tags can be easily (and reliably) read beyond the usual 10-15cm range... – Axeman Jan 3 '12 at 18:31
That's enough, the chip is located on the left shoulder, and I plan on putting the reader at the top of the cat door. If this does not work, I may look into another alternative though. – Wookai Jan 3 '12 at 22:03
I got a reader dirt cheap from Priority1 Design. There based in Australia but with shipping it only cost $61 ($43 for the reader + shipping). The electronics are exposed so I needed to make a custom case for it but it works great. They have readers for both 125 KHz and 134 KHz but both seem to read the 134 KHz chips fine. I get my RFID tags from Oregon RFID. Links below.
http://www.priority1design.com.au
http://www.oregonrfid.biz
-
Thanks a lot, this is exactly what I was looking for! – Wookai Oct 4 '12 at 8:45
@Wookai Those Priority 1 Design boards' antennas are on the PCB, resulting in a short reading distance, as it can be seen in this technical sheet. Check out my tutorial if you want to build it from scratch for very cheap. – abdullah kahraman Mar 4 at 8:37
Thanks, your tutorial looks very interesting! I'll give it a try! – Wookai Mar 4 at 16:43
1) Are you sure it's a 134.2 kHz tag? Companion animal tags in the US are commonly 125 kHz. Some tags are non-ISO/proprietary and can only be read with the manufacturer's reader.
2) Many 125 kHz readers can be modified to read 134.2 kHz by replacing the crystal.
3) There are higher power readers that can read 12mm tags at much larger distances than what the Arduino and other simple readers can do. They cost a couple thousand dollars though. FYI, the world's most powerful ISO FDX reader is at Bonneville dam on the Columbia River and can read 12mm FDX-B tags at 7 feet. It has its own HVAC system. http://php.ptagis.org/wiki/index.php/Site_Page_BCC_-_Bonneville_PH2_Corner_Collector
-
Yes, I'm quite sur it's 134.2 kHz. The specs (anis.ch/en/microchip) say it uses ISO 11784 & 11785. – Wookai Jan 3 '12 at 22:04
I have been using an arduino UNO to decode FDX-B with some success in conjunction with the EM4095 chip tuned for 134.2kHz. I purchased a populated board from Elektor, see the article link: http://outlet.elektor.com/contents/en-us/50-00023%20manual.pdf.
BTY you can buy a cat flap from: http://www.sureflap.co.uk/ , a bit expensive, it uses a patented technique to extend the detection range.
As for my progress, I can currently detect the cats tags as well as a test tag and locate the header in the data stream, working on the CRC and data payload formatting.
-
These OEM boards will read ISO FDX-B at close range
• HID 0000-USM-01-0-01
• Priority 1 RFIDRW-E-232
-
Links would be nice. – stevenvh May 3 '12 at 12:10
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Register now and save up to $200 Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code ## gmat strategy help This topic has 2 expert replies and 2 member replies krishhxa Senior | Next Rank: 100 Posts Joined 01 Jan 2016 Posted: 50 messages Followed by: 2 members #### gmat strategy help Mon Feb 08, 2016 10:40 am Hi, I have been facing issues in certain type of quant questions which is being a major obstacle in acheiving my dream score. Kindly guide. 1. Certain type of DS questions which have inequality signs. such as is x>0 or any question with inequality in the answer choices as well as in the proving. I fail to understand what values should i choose in the limited 2 minutes to figure out whether the situation is true/false. 2.In PS Certain type of percentage questions which consist of either tax or some kind of increase or decrease and the answer choices are given in variable terms such as p+r/t+q I freak out in such questions because each sentence of the problem has something happening. It becomes hard for me to comprehend the problem and solve it in such variables. Kindly guide any easy way to go about such complex questions. 3. The mixing questions. can hardly solve them. 4. General inequality questions. Can't figure out the numbers i should choose because there mite be a case where the response comes out false. Many cases can be made. Kindly guide me fore the above mentioned problems. further just wanted to enquire out of the sc questions that come up in gmat. About how many can be solved correctly with plain logic. without getting into the technical aspects? Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums! ### GMAT/MBA Expert DavidG@VeritasPrep Legendary Member Joined 14 Jan 2015 Posted: 2301 messages Followed by: 115 members Thanked: 1072 times GMAT Score: 770 Mon Feb 08, 2016 11:12 am krishhxa wrote: Hi, I have been facing issues in certain type of quant questions which is being a major obstacle in acheiving my dream score. Kindly guide. 1. Certain type of DS questions which have inequality signs. such as is x>0 or any question with inequality in the answer choices as well as in the proving. I fail to understand what values should i choose in the limited 2 minutes to figure out whether the situation is true/false. 2.In PS Certain type of percentage questions which consist of either tax or some kind of increase or decrease and the answer choices are given in variable terms such as p+r/t+q I freak out in such questions because each sentence of the problem has something happening. It becomes hard for me to comprehend the problem and solve it in such variables. Kindly guide any easy way to go about such complex questions. 3. The mixing questions. can hardly solve them. 4. General inequality questions. Can't figure out the numbers i should choose because there mite be a case where the response comes out false. Many cases can be made. Kindly guide me fore the above mentioned problems. further just wanted to enquire out of the sc questions that come up in gmat. About how many can be solved correctly with plain logic. without getting into the technical aspects? There's a lot to chew on here. I think it might be a better idea to post individual examples of actual questions that you struggled with, and then we can talk those through step-by-step. (So start one thread with a particular inequality question, one thread with a particular mixture question, etc.) _________________ Veritas Prep | GMAT Instructor Veritas Prep Reviews Save$100 off any live Veritas Prep GMAT Course
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krishhxa Senior | Next Rank: 100 Posts
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Mon Feb 08, 2016 11:17 am
Thanx for the promt respnse.I will surely do that.But can you kindly guide me how to solve inequality questions that appear in ds.I get confused on how to solve the same.
### GMAT/MBA Expert
DavidG@VeritasPrep Legendary Member
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Mon Feb 08, 2016 12:13 pm
krishhxa wrote:
Thanx for the promt respnse.I will surely do that.But can you kindly guide me how to solve inequality questions that appear in ds.I get confused on how to solve the same.
It'll depend on the question. If the question were, say, "Is x > 3?" you may end up picking numbers.
If the question were "Is x + y > x - y?" you'd simplify to get "Is y > -y?" which is really just "Is y > 0 ?"
This is all to say that in some cases, you'll pick numbers. In other cases, you'll do algebra. It's possible that on another question you'll do a combination of algebra and number-picking. So it's hard to give a piece of prescriptive advice, such as "Always do x," when the best approach will depend on the context of a given question
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Marty Murray Legendary Member
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Mon Feb 08, 2016 6:21 pm
krishhxa wrote:
further just wanted to enquire out of the sc questions that come up in gmat. About how many can be solved correctly with plain logic. without getting into the technical aspects?
Good question.
I am not sure of the exact answer to that question, but in a way most of them can be solved correctly via the use of "plain logic". In fact, the last time I took the test, I was happy to see that all of the SC questions I saw were rather hackable.
I can also say that I have seen someone who does not have a perfect command of English score 47 on the verbal section of the GMAT.
Basically, GMAT sentence correction IS ABOUT logic, and increasingly the questions reflect that situation.
Maybe on a given test a test taker will see three or four SC questions such that in some way getting them right requires or almost requires some specific knowledge of grammatical or idiomatic construction. Then again, is a subject matching a verb "technical"? I guess maybe what you are asking is "How many SC questions are such that getting them right requires the use of advanced knowledge of grammar and other sentence construction rules and conventions?" The answer to that question is "Not very many."
Having said that, there are key things that one needs to notice in order to get SC questions right. So some preparation for SC definitely helps. For instance, parallelism is logical, but if you haven't thought much about parallelism then you might not notice that a construction is not parallel, and therefore not logical.
Also, this discussion is a little bit tautological in that, if one doesn't understand certain things related to sentence construction, one may have difficulty sorting out what is logical and what is not.
Taking a practice test and seeing how you do may be the best way to answer your question. I personally was able to get most SC questions right without doing much preparation and without having ever thought about many of the rules and conventions that people talk about. I used practice test results to see where there were some knowledge gaps that I could fill in order to get more SC questions right, and then I did some focused work to fill those gaps.
Even that method won't give you a perfect answer though as I think that the questions in GMAT Prep are less logic based than the more recently created ones that show up on the actual test.
Another thing that comes to mind is that from what I have seen, most of the time people miss SC questions more because they don't notice something than because they don't know about certain rules.
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# How do you find the asymptotes for 1/3(x-1)^3+2?
##### 1 Answer
Dec 11, 2016
No asymptote. This is an increasing function, making an x-intercept -0,8172 and y-intercept 5/3, nearly. See graph.
#### Explanation:
$y = \frac{1}{3} {\left(x - 1\right)}^{3} + 2 = 0$, when $x = - {6}^{\frac{1}{3}} + 1 = - 0.81712$, nearly
$y ' = {\left(x - 1\right)}^{2} \ge 0$. So, y is an increasing function, excepting at x = 1.
$y ' = 0 , w h e n x = 1$. Here, $y ' ' = 0 \mathmr{and} y ' ' ' = 2 > 0$
So, (1, 2) is a point of inflexion.
As $x \to \pm \infty , y \to \pm \infty$.
There is no asymptote.
graph{y-(x-1)^3/3-2=0 [-10, 10, -5, 5]}
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# Equivalence of norms on finite dimensional subset ($L^2$ and $H^1$ norms)
Let $b_i$ be a basis of $H^1(\Omega)$ and define $X=\text{span}(b_1, ..., b_n)$ for $n$ fixed.
Consider $X_1 = (X, \lVert \cdot \rVert_{H^1(\Omega)})$ and $X_1 = (X, \lVert \cdot \rVert_{L^2(\Omega)})$.
Both $X_1$ and $X_2$ are finite-dimensional sets equipped with different norms. So we know that by equivalence of norms, the $L^2$ norm and $H^1$ norm are equivalent (restricted to elements of $X$).
This seems a bit weird to me. I can put any norm on it and I will get a equivalence of norms result to any other norm. Is it correct to think like this?
First of all, there is no finite(!) basis of $H^1(\Omega)$. I assume that you take the $b_i$ to be just linearly independent.
In that case, what you say is correct, but the point here is that the constant of equivalence of the norms is dependent on the $b_i$ and in particular on the number of elements $n$.
If you let $n \rightarrow \infty$, your constants will deteriorate.
• Yeah I just meant $\{b_i\}_{i=1}^\infty$ is a basis. I am taking the first $n$ elements to make $X$. – studentX May 24 '14 at 16:48
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Under the auspices of the Computational Complexity Foundation (CCF)
REPORTS > DETAIL:
### Paper:
TR16-128 | 13th August 2016 18:23
#### Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover
TR16-128
Authors: Irit Dinur
Publication: 13th August 2016 18:23
Downloads: 798
Keywords:
Abstract:
We show that if gap-3SAT has no sub-exponential time algorithms then a weak form of the sliding scale conjecture holds. Namely, for every $\alpha>0$ any algorithm for $n^\alpha$-approximating the value of label cover must run in time at least $n^{\Omega(\exp(1/\alpha))}$, where $n$ is the size of the instance.
Put differently, if there is a polynomial time algorithm for approximating the value of label cover to within a factor of approximation of $n^{o(1)}$ then gap-3SAT (and thus, random-3SAT) must have faster-than-exponential-time algorithms.
Our proof is a twist on the well-studied parallel repetition reduction from 3SAT to label cover. Our key observation is that if we take unordered repetition, replacing tuples by sets, then we can afford to take the number of repetitions to be linear in $n$, and generate an instance of size $\exp(n)$ which results in the claimed parameters.
ISSN 1433-8092 | Imprint
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# Theta cos theta sin theta
our channel provide easy and simple solutions of maths and science problems. so guys join our channel subscribe it.Send your maths and science problems direc
if sin theta =cos theta ,find the value of theta and tan theta. Ask questions, doubts, problems and we will help you. 2015-06-12 As sin^2 theta = 1 - cos^2 theta = 1- (7/25)^2 =1-49/625 =(625-49)/625 ⇒ sin^2 theta = 576/625 ⇒ sintheta = sqrt(576/625) ⇒sin theta = 24/25 Now , tan theta + cot theta =sin theta / cos theta+ cos theta /sin theta =(sin^2 Y = cos theta + 2sin theta. Share with your friends. Share 15. I f y = 3 cos θ + 2 sin θ i 2 s q u a r i n g a n d a d d i n g a b o v e b o t h e q u a t i o n s, w e h a v e, x 2 + y 2 = 2 cos θ-3 sin θ 2 + 3 cos θ + 2 sin θ 2 ⇒ x 2 + y 2 = 4 cos 2 θ + 9 sin 2 θ-12 sin θ cos PROVE THAT (sin theta + sec theta)^2 + (cos theta + cosec theta)^2 = (1+ sec theta cosec theta)^2 - Math - Introduction to Trigonometry The functions sine, cosine and tangent of an angle are sometimes referred to as the primary or basic trigonometric functions. Their usual abbreviations are sin(θ), cos(θ) and tan(θ), respectively, where θ denotes the angle.
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## Provably Efficient Exploration in Policy Optimization
### Qi Cai · Zhuoran Yang · Chi Jin · Zhaoran Wang
Keywords: [ Non-convex Optimization ] [ Reinforcement Learning Theory ] [ Optimization - Non-convex ]
[ Abstract ]
Tue 14 Jul 7 a.m. PDT — 7:45 a.m. PDT
Tue 14 Jul 6 p.m. PDT — 6:45 p.m. PDT
Abstract: While policy-based reinforcement learning (RL) achieves tremendous successes in practice, it is significantly less understood in theory, especially compared with value-based RL. In particular, it remains elusive how to design a provably efficient policy optimization algorithm that incorporates exploration. To bridge such a gap, this paper proposes an \underline{O}ptimistic variant of the \underline{P}roximal \underline{P}olicy \underline{O}ptimization algorithm (OPPO), which follows an optimistic version'' of the policy gradient direction. This paper proves that, in the problem of episodic Markov decision process with unknown transition and full-information feedback of adversarial reward, OPPO achieves an $\tilde{O}(\sqrt{|\cS|^2|\cA|H^3 T})$ regret. Here $|\cS|$ is the size of the state space, $|\cA|$ is the size of the action space, $H$ is the episode horizon, and $T$ is the total number of steps. To the best of our knowledge, OPPO is the first provably efficient policy optimization algorithm that explores.
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# Kronecker delta expansion
## Main Question or Discussion Point
If given δ_ijδ_kk what would the expansion of that be? I thought it was nine but have been told that is incorrect. I know that i=j =1 else zero so I thought that the δ_kk would equal 3 times 3 from the expansion of δ_ij but that isn't the answer
tiny-tim
Homework Helper
Welcome to PF!
Hi cgstu! Welcome to PF!
(try using the X2 tag just above the Reply box )
In δijδkk, which indices are you summing over?
Hi cgstu! Welcome to PF!
(try using the X2 tag just above the Reply box )
In δijδkk, which indices are you summing over?
I guess thats what I am not sure of. I know that if i=j then the delta function =1 else delta =0 so my thinking was
δ11δ11 + δ12δ11 +
δ13δ11 + δ21δ11 +
δ22δk11 + δ23δ11 + ...... where only when the indices matched is the entire function = 1
δ11δ11, δ22δ11,
δ33δ11.... etc
This would give me a total of 9. However, this is incorrect and I do not understand why.
tiny-tim
Homework Helper
Hi cgstu!
Nooo …
the "Einstein summation convention" is that only repeated indices are summed over.
In this case, k is repeated (ie, there's two of them!), so you sum over k, but i and j are not repeated, so you don't sum over them, and they'll still be in the final result.
In other words, δijδkk is shorthand for ∑k δijδkk.
See http://en.wikipedia.org/wiki/Einstein_summation_convention" [Broken] for details.
Last edited by a moderator:
thanks tiny tim,
so if I understand correctly now the answer should be three?
tiny-tim
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# Difference between revisions of "2006 AMC 10B Problems/Problem 23"
## Problem
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7 as shown. What is the area of the shaded quadrilateral?
$[asy] unitsize(1.5cm); defaultpen(.8); pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A); pair F = intersectionpoint( A--D, B--Ep ); draw( A -- B -- C -- cycle ); draw( A -- D ); draw( B -- Ep ); filldraw( D -- F -- Ep -- C -- cycle, mediumgray, black ); label("7",(1.25,0.2)); label("7",(2.2,0.45)); label("3",(0.45,0.35)); [/asy]$
$\mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 17\qquad \mathrm{(C) \ } \frac{35}{2}\qquad \mathrm{(D) \ } 18\qquad \mathrm{(E) \ } \frac{55}{3}$
## Solution 1
Label the points in the figure as shown below, and draw the segment $CF$. This segment divides the quadrilateral into two triangles, let their areas be $x$ and $y$.
$[asy] unitsize(2cm); defaultpen(.8); pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A); pair F = intersectionpoint( A--D, B--Ep ); draw( A -- B -- C -- cycle ); draw( A -- D ); draw( B -- Ep ); filldraw( D -- F -- Ep -- C -- cycle, mediumgray, black ); label("7",(1.45,0.15)); label("7",(2.2,0.45)); label("3",(0.45,0.35)); draw( C -- F, dashed ); label("A",A,SW); label("B",B,SE); label("C",C,N); label("D",D,NE); label("E",Ep,NW); label("F",F,S); label("x",(1,1)); label("y",(1.6,1)); [/asy]$
Since triangles $AFB$ and $DFB$ share an altitude from $B$ and have equal area, their bases must be equal, hence $AF=DF$.
Since triangles $AFC$ and $DFC$ share an altitude from $C$ and their respective bases are equal, their areas must be equal, hence $x+3=y$.
Since triangles $EFA$ and $BFA$ share an altitude from $A$ and their respective areas are in the ratio $3:7$, their bases must be in the same ratio, hence $EF:FB = 3:7$.
Since triangles $EFC$ and $BFC$ share an altitude from $C$ and their respective bases are in the ratio $3:7$, their areas must be in the same ratio, hence $x:(y+7) = 3:7$, which gives us $7x = 3(y+7)$.
Substituting $y=x+3$ into the second equation we get $7x = 3(x+10)$, which solves to $x=\frac{15}{2}$. Then $y=x+3 = \frac{15}{2}+3 = \frac{21}{2}$, and the total area of the quadrilateral is $x+y = \frac{15}{2}+\frac{21}{2} = \boxed{\textbf{(D) }18}$.
## Solution 2
Connect points $E$ and $D$. Triangles $EFA$ and $FAB$ share an altitude and their areas are in the ration $3:7$. Their bases, $EF$ and $FB$, must be in the same $3:7$ ratio.
Triangles $EFD$ and $FBD$ share an altitude and their bases are in a $3:7$ ratio. Therefore, their areas are in a $3:7$ ratio and the area of triangle $EFD$ is $3$.
Triangle $CED$ and $DEA$ share an altitude. Therefore, the ratio of their areas is equal to the ratio of bases $CE$ and $EA$. The ratio is $A:(3+3) \Rightarrow A:6$ where $A$ is the area of triangle $CED$
Triangles $CEB$ and $EAB$ also share an altitude. The ratio of their areas is also equal to the ratio of bases $CE$ and $EA$. The ratio is $(A+3+7):(3+7) \Rightarrow (A+10):10$
Because the two ratios are equal, we get the equation $\frac{A}{6} = \frac {A+10}{10} \Rightarrow 10A = 6A+60 \Rightarrow A = 15$. We add the area of triangle $EDF$ to get that the total area of the quadrilateral is $\boxed{\textbf{(D) }18}$.
~Zeric Hang
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## RS Aggarwal Class 8 Solutions Chapter 1 Rational Numbers Ex 1D
These Solutions are part of RS Aggarwal Solutions Class 8. Here we have given RS Aggarwal Solutions Class 8 Chapter 1 Rational Numbers Ex 1D.
Other Exercises
Question 1.
Solution:
Question 2.
Solution:
Question 3.
Solution:
Question 4.
Solution:
Question 5.
Solution:
Question 6.
Solution:
Question 7.
Solution:
Question 8.
Solution:
Question 9.
Solution:
(i) The product of a rational number and its reciprocal is 1.
(ii) Zero has no reciprocal.
(iii) The numbers 1 and -1 are their own reciprocal.
(iv) Zero is not the reciprocal of any number.
(v) The reciprocal of a, where a≠0, is $$\\ \frac { 1 }{ a }$$
(vi) The reciprocal of $$\\ \frac { 1 }{ a }$$ where a≠0 is a
(vii) The reciprocal of a positive rational number is positive.
(viii) The reciprocal of a negative rational number is negative.
Hope given RS Aggarwal Solutions Class 8 Chapter 1 Rational Numbers Ex 1D are helpful to complete your math homework.
If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.
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# How does randomization avoid entering infinite loops in the vacuum cleaner problem?
Suppose we have a vacuum cleaner operating in a $$1 \times 2$$ rectangle consisting of locations $$A$$ and $$B$$. The cleaner's actions are Suck, Left, and Right and it can't go out of the rectangle and the squares are either empty or dirty. I know this is an amateur question but how does randomization (for instance flipping a fair coin) avoid entering the infinite loop? Aren't we entering such a loop If the result of the toss is heads in odd tosses and tails in even tosses?
This is the text from the book "Artificial Intelligence: A Modern Approach" by Russell and Norvig
We can see a similar problem arising in the vacuum world. Suppose that a simple reflex vacuum agent is deprived of its location sensor and has only a dirt sensor. Such an agent has just two possible percepts: [Dirty] and [Clean]. It can Suck in response to [Dirty]; what should it do in response to [Clean]? Moving Left fails (forever) if it happens to start in square A, and moving Right fails (forever) if it happens to start in square B. Infinite loops are often unavoidable for simple reflex agents operating in partially observable environments. Escape from infinite loops is possible if the agent can randomize its actions. For example, if the vacuum agent perceives [Clean], it might flip a coin to choose between Right and Left. It is easy to show that the agent will reach the other square in an average of two steps. Then, if that square is dirty, the agent will clean it and the task will be complete. Hence, a randomized simple reflex agent might outperform a deterministic simple reflex agent.
And this is the agent program from the same source:
function REFLEX-VACUUM-AGENT([location,status]) returns an action
if status = Dirty then return Suck
else if location = A then return Right
else if location = B then return Left
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Acta Phys. -Chim. Sin. ›› 2018, Vol. 34 ›› Issue (12): 1321-1333.
Special Issue: 表面物理化学
• REVIEW •
### Interactions between Bases and Metals on Au(111) under Ultrahigh Vacuum Conditions
Xinyi WANG,Lei XIE,Yuanqi DING,Xinyi YAO,Chi ZHANG,Huihui KONG,Likun WANG,Wei XU*()
• Received:2018-01-09 Online:2018-12-15 Published:2018-04-27
• Contact: Wei XU E-mail:xuwei@tongji.edu.cn
• Supported by:
the National Natural Science Foundation of China(21473123);the National Natural Science Foundation of China(21622307)
Abstract:
Nucleobases (guanine (G), adenine (A), thymine (T), cytosine (C), and uracil (U)) are important constituents of nucleic acids, which carry genetic information in all living organisms, and play vital roles in structure formation, functionalization, and biological catalytic processes. The principle of complementary base pairing is significant in the high-fidelity replication of DNA and RNA. In addition to their specific recognition, the interaction between bases and other reactants, such as metals, salts, and certain small molecules, may cause distinct effects. Specifically, the interactions between bases and certain metal atoms or ions could damage nucleic acids, inducing gene mutation and even carcinogenesis. In the meantime, nanoscale devices based on metal-DNA interactions have become the focus of research in nanotechnology. Therefore, extensive researches on the interactions between metals and bases and the corresponding mechanism are of great importance and may make improvements in the fields of both biochemistry and nanotechnology. Scanning tunneling microscopy (STM) is a powerful tool for effectively resolving nanostructures in real space and on the atomic scale under ultrahigh vacuum (UHV) conditions. Moreover, density functional theory (DFT) calculations could help elucidate the reaction pathways and their mechanisms. In this review, we summarize the distinct interactions between bases (including their derivatives) and various metal species (comprising alkali, alkaline earth, and transition metals) derived from metal sources and the corresponding salts on the Au(111) substrate reported recently based on the results obtained by a combination of above two methods. In general, bases afford N and/or O binding sites to interact with metal atoms, resulting in various motifs via coordination or electrostatic interactions, and form intermolecular hydrogen bonds to stabilize the whole system. On the basis of high-resolution STM images and DFT calculations, structural models and the possible reaction pathways are proposed, and their underlying mechanisms, which reveal the nature of the interactions, are thus obtained. Among them, we summarize the construction of G-quartet structures with different kinds of central metals like Na, K, and Ca, which are directly introduced by salts, and their relative stabilities are compared. In addition, salts can provide not only metal cations but also halogen anions in modulating the structure formation with bases. The halogen species enable the regulation of metal-organic motifs and induce phase transition by locating at specific hydrogen-rich sites. Moreover, reversible structural transformations of metal-organic nanostructures are realized owing to the intrinsic dynamic characteristic of coordination bonds, together with the coordination priority and diversity. Furthermore, the controllable scission and seamless stitching of metal-organic clusters, which contain two types of hierarchical interactions, have been successfully achieved through STM manipulations. Finally, this review offers a thorough comprehension on the interaction between bases and metals on Au(111) and provide fundamental insights into controllable fabrication of nanostructures of DNA bases. We also admit the limitation involved in detecting biological processes by on-surface model system, and speculate on future studies that would use more complicated biomolecules together with other technologies.
MSC2000:
• O647
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# C#LeetCode刷题之#231-2的幂（Power of Two）
Given an integer, write a function to determine if it is a power of two.
Credits:
Special thanks to @jianchao.li.fighter for adding this problem and creating all test cases.
```public class Program {
public static void Main(string[] args) {
var n = 8;
var res = IsPowerOfTwo(n);
Console.WriteLine(res);
n = 513;
res = IsPowerOfTwo2(n);
Console.WriteLine(res);
}
private static bool IsPowerOfTwo(int n) {
//先看原值是否能被2整除
//若不能整除，不是2的幂；
//若能整除，继续往下，直接<=1时为止
//最后判断值是否为1即可
while(n % 2 == 0 && (n /= 2) > 1) { }
return n == 1;
}
private static bool IsPowerOfTwo2(int n) {
//2为10,4为100
//2-1为01,4-1为011
//对它们进行“与”运算
//10 & 01 = 0
//100 & 011 = 0
//得出结论，如果一个数n为2的幂，则n & (n - 1) = 0
return ((n > 0) && (n & (n - 1)) == 0);
}
}```
```True
False```
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Volume 9 Issue 2 pp. 164-176 • doi: 10.15627/jd.2022.13
# Daylight Performance of the Modified Double Light Pipe (MDLP) Through Experimental Analysis on a Reduced Scale Model
Paolo Zazzini,a,* Alessandro Di Crescenzo,a Roberto Giammicheleb
Author affiliations
a Department INGEO, University “G. D’Annunzio”, Viale Pindaro 42, 65127 Pescara, Italy
b Engineer, Via A.Gramsci n°10/A, Vasto (CH), Italy
*Corresponding author.
zazzini@unich.it (P. Zazzini)
dicrescenzo.a@live.it (A. Di Crescenzo)
robertogiamm.rg@gmail.com (R. Giammichele)
History: Received 21 June 2022 | Revised 30 July 2022 | Accepted 8 August 2022 | Published online 6 September 2022
Copyright: © 2022 The Author(s). Published by solarlits.com. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Citation: Paolo Zazzini, Alessandro Di Crescenzo, Roberto Giammichele, Daylight Performance of the Modified Double Light Pipe (MDLP) Through Experimental Analysis on a Reduced Scale Model, Journal of Daylighting 9 (2022) 164-176. https://dx.doi.org/10.15627/jd.2022.13
Figures and tables
## Abstract
This paper is focused on the Modified Double Light Pipe (MDLP), an innovative daylighting system set up by the authors in the Laboratory of Technical Physics of the University “G. D’Annunzio” of Pescara (Italy). It is an evolution of the Double Light Pipe (DLP), designed by the authors to distribute natural light in two underground levels of a building and tested through an experimental activity on a reduced scale model. The MDLP has been designed by modifying the same scale model of the DLP with the goal to solve some problems related to its size and the risk of glare, thanks to its smaller encumbrance, and a light shelf applied around the external tube to prevent the occupants of the room from seeing the upper brightest portion of the device. Furthermore, the light shelf reflects light towards the ceiling spreading it more uniformly on the horizontal work-plan. The technological components of the MDLP are shown demonstrating the possibility of installing it in both new and refurbished buildings. Then, the first results of an experimental activity carried out on the model of the MDLP in winter climatic conditions are shown. Although the unfavorable values of external illuminance, the MDLP contributes to distributing natural light in the passage room both with overcast and clear sky conditions. On sunny days, direct solar radiation hits the measure positions on the corners of the room producing peak illuminance values. The illuminance uniformity is calculated according to the EN 12464-1. The results can be considered satisfactory both in terms of internal illuminance and uniformity of light distribution.
## Keywords
Daytime, Alertness, Non-visual photoreception, Lighting
## 1. Introduction
Natural light can give a contribution to guaranteeing visual comfort conditions in buildings [1], even if it involves the risk of some discomfort conditions due to visual glare or veil reflections. Tabadkani et al. [2] have recently underlined the role of facades equipped with daylight sources (windows) and daylighting strategies able to block or redirect light to get visual comfort condition of the occupants and analyzed a great number of parameters to underline the influence of daylight on visual comfort of occupants.
Visual comfort in internal areas of buildings depends on various physical aspects such as light quantity, absence of glare phenomena, uniformity in illuminance and luminance distribution, and adequate view outdoor, as well as psychological factors that influence the occupants’ perception of daylight [3,4].
The presence of daylight is particularly important in office buildings, as underlined by Galasiu and Veitch [5], so much that it can have a positive influence on productivity, as shown by De Carli et al. [6].
Recent studies published by IEA SHC Task 61 show that combing daylight with LED ceiling panels able to tune light colour during the day increases the office workers’ alertness [7], as well as improves the satisfaction of customers of shopping centers [8].
Daylight in buildings is very significant also for sanitary purposes. The lack of daylight can severely penalize human health, producing modification of the circadian rhythms, weakening of the immune system, alteration of mood, and depression [9,10].
Natural light also allows effective energy savings in buildings, thanks to less use of artificial light, taking into account that about 14% of electrical consumption in EU and 19 % in the world is due to the use of artificial light. [11-18]
In many cases, particularly in public buildings, this is partially due to the occupants’ bad habit of using artificial light also in presence of natural light taking curtains or blinds shut to avoid glare from the windows hit by direct solar radiation [19].
The electric energy consumption in Europe must be significantly reduced as requested by the European Directives that fixed the Nearly Zero Energy target for buildings and daylight can effectively contribute to achieving this.
For these reasons, the use of natural light in buildings is growing in importance. In this perspective, numerous visual comfort metrics have been proposed to quantify the daylight availability in the design process of buildings and thus guide the design choices [20].
When traditional sources of daylight are absent, such as in underground areas of buildings, or unable to provide an adequate light level, such as in large plan area environments (i.e. industrial or commercial buildings), daylight can be introduced and transported by technological light transport systems. Among these, light pipes or similar technological devices are very widespread [21-23]. Obradovic and Matusiak [17] propose “a literature study of daylight transport systems aiming at selecting the most appropriate ones for application at high latitudes”.
Although vertical light pipes and similar daylighting strategies are most suitable at high latitudes, being particularly apt to catch zenithal light, and less effective in the Mediterranean latitudes as suggested by Obradovic and Matusiak [17,18], these systems can still contribute to energy saving by allowing underground or basement environments to be illuminated with daylight.
In environments equipped with windows, an uneven spatial distribution of natural light usually occurs, with very high values near the window rapidly decreasing away from it. In these cases, light shelves can be effectively used to improve the spatial distribution of light [24-30].
Many authors have investigated the performance of daylighting technological devices through numerical or experimental methods [31-39].
Experimental data can be collected under real or artificial sky, using the scale model approach. Boccia and Zazzini [40] propose a critical analysis of the use of the scale model approach, underlying its simplicity and effectiveness but its reduced accuracy due to some accidental factors, such as the presence of direct solar radiation.
In previous years, the authors of this paper carried out research activities on daylight transport systems and they developed an innovative device called Double Light Pipe (DLP). The DLP is an evolution of the traditional light pipe, particularly suitable for large showrooms or museums, able to illuminate contemporarily two levels of underground buildings or buildings not equipped with traditional daylight sources [1,41].
Recently, they developed the idea of combining the technology of the DLP with that of light shelves and they set up a new system named Modified Double Light Pipe (MDLP), a technological device designed to reduce the encumbrance and the brightness of the Double Light Pipe [42].
In this paper, the authors present the technological components of the MDLP and the first results of an experimental analysis carried out on a 1:2 scale model of the system.
## 2. The Modified Double Light Pipe (MDLP)
The MDLP has been designed to remove some defects of the DLP, a device developed by the authors to distribute natural light in underground buildings. The authors carried out a numerical and experimental analysis on a 1:2 scale model of the DLP [1]. It can light two hypogeal levels by two coaxial tubes. The internal one brings natural light into the second underground level as a traditional light pipe, while the external transparent one allows the light to enter the intermediate room. The DLP presents some troubles: it has a considerable encumbrance and involves the risk of glare from the upper portion of the system. Moreover, it produces an uneven distribution of light, more concentrated near the tube. The illuminance values on a horizontal work-plan in the passage room decrease from the center to the corners and the upper portion of the system is very bright. This leads to a high risk of visual glare, as shown in Baroncini et al. 2010.
Starting from these considerations, the authors decided to modify the 1:2 scale model of the DLP with the intent to solve these problems and they named this new device MDLP.
The DLP has been modified by fixing to the ceiling a reflecting panel and equipping it with a circular light shelf, 600 mm distant from the ceiling, able to reflect light toward the ceiling and improve the uniformity of light distribution in the environment. The light shelf also prevents the occupants from seeing the upper portion of the device with the highest luminance, avoiding the risk of glare. Furthermore, the encumbrance of the external pipe is significantly reduced than in the DLP, because the lower portion of the tube is cut.
Figure 1
Fig. 1. Comparison between the features of (a) the DLP and (b) the MDLP.
Figure 1 shows a comparison between the DLP (Fig. 1(a)) and the MDLP (Fig. 1(b)), underlying the different visual perceptions of the two devices, and the reduced encumbrance of the MDLP if compared to the DLP.
A similar idea has been proposed in Garcia-Hansen, Edmonds [43], where the results of a study on a vertical light pipe that illuminates five levels of a building are shown. This interesting system is more complex than that illustrated in this paper. It consists of an alternation of transparent and opaque parts and, at each level, it is equipped with a partially transparent and partially reflective cone, as well as a light shelf that reflects towards the ceiling solar radiation coming from the sun with low elevation angle.
The MDLP is a simpler system, having been designed to illuminate only two underground levels. The central tube is completely opaque and coated both internally and externally with highly reflective film, while the external one is completely transparent. Moreover, the light shelf is smaller in size as it has been designed mainly to reduce the risk of glare, even if experimental tests carried out in spring and summer, whose results are not shown in this paper, demonstrated that it helps to reflect light coming from the sun with high elevation angle. Figure 2 shows all the steps that led to the construction of the system and the definition of its components as well as the method used to determine its performance.
Figure 2
Fig. 2. Sequence of the steps that led to the creation of the MDLP and the determination of its performance.
## 3. Technological components and installation procedure of the MDLP
The MDLP consists of the following components:
• 500 mm diameter light collector positioned on the flat roof of the building;
• 500 mm diameter outer tube of transparent polycarbonate with a length equal to the depth of the attic plus 600 mm;
• 250 mm diameter inner tube internally and partly externally coated with a highly reflective film;
• 140 mm diameter circular reflector panel applied on the ceiling;
• 100 mm diameter light shelf, 60 mm distant from the ceiling.
The MDLP can be installed in both new and existing buildings. In this paragraph, the authors describe the device applied to a generic brick concrete roof slab as it is the most used in common constructions.
The first step of the installation process consists in making a hole over the roof slab. It must have a 505 mm diameter to allow easy fixing of the external tube. In the case of an existing building, the hole should be positioned so that only one joist is removed and an appropriate stiffening should be created to restore the structural continuity offered by the joist.
At this point, the fixing surface should be prepared by removing part of the surface layer to “hook” the flashing to the covering screed (Fig. 3). The flashing should be fixed on the screed by mechanical anchors, which allow a firm grip and high resistance thanks to friction and shape. The fixing of the flashing should be carried out using sealants and sheaths to avoid long-term corrosion phenomena.
Figure 3
Fig. 3. Axonometric view of the flashing that allows anchor the system to the roof slab.
The upper part of the system consists of the elements shown in Fig. 4, which also displays how it can be fixed to the roof slab, inserting it into the flashing previously prepared. In Fig. 5(a) the connection of the system to the intermediate floor is shown, while Fig. 5(b) shows the telescopic shelf that is a fundamental element of the system. In addition, it allows setting the shelf at the correct height to facilitate the installation of the apparatus making the system extremely versatile and applicable at different heights.
Figure 4
Fig. 4. Connection system to the roof slab.
Figure 5
Fig. 5. (a) Fixing mode of the system to the inter-floor slab and (b) lower telescopic shelf.
Considering that the internal pipe is in contact with the users of the intermediate room, it is necessary to cover it with a steel coating, that protects it from any crashes with people or things. In addition, it supports the overlying telescopic element that can be used for changing the distance of the light shelf from the ceiling. Overall, the system looks like a long classic tube to which a steel circular protection has been applied to prevent damage. Finally, by assembling the components, the configuration shown in Fig. 6 is obtained.
Table 1 shows the luminous reflectance of the components of the MDLP.
Figure 6
Fig. 6. Complete configuration of the MDLP, overall and detailed renderings.
Table 1
Table 1. Luminous reflectance of the components of the MDLP.
## 4. Description of the experimental apparatus
A wood 1:2 scale model of a 3.8×3.8 m plan area room, 3.0 m high was built by the authors. The vertical walls are made of unpainted multilayer wood, with a luminous reflectance equal to 50%. Sheets of grey drawing paper are applied to the floor of the room (luminous reflectance = 49.1%) and a circular grey panel (100 mm diameter) with the same luminous reflectance is applied over the ceiling around the external tube. The internal tube of the MDLP is made of PVC. The upper portion of it is externally covered by a reflective film (3 M Radiant Mirror Film LRF) with luminous reflectance r = 99.5%. The external tube is made of transparent polycarbonate. Figure 7 shows some photos of the device and the test room.
Figure 7
Fig. 7. Some photos of the MDLP and the test room.
The model simulates the passage room of a two-levels hypogeal construction illuminated by the MDLP. The room is not equipped with any windows or skylights, so the MDLP is the only source of natural light.
Twelve CIE Lux-meters sensors type LSI-BSR001, range 0–25 klx, accuracy 3% of the reading value for illuminance, have been positioned in the room on a horizontal work-plan (see Fig. 8). This last is 0.4 m high on the floor and simulates a 0.8 m high work-plan in the real scale room. A CIE sensor type LSI-DPA 503, range 0–100 klx, tolerance 1.5%, has been used to measure the external horizontal illuminance Eout. Figure 9 shows the external CIE sensor placed on the roof of the building during the experimental activity. Note that no obstructions take place from nearby buildings.
A data-logger type LSI Lastem ELO 310 has been used to register data.
Figure 8
Fig. 8. Real scale dimensions of the test room and positions of the luxmeters on the horizontal work plane.
Figure 9
Fig. 9. The external luxmeter on the roof of the building.
## 5. Experimental results
The experimental activity was carried out from December the 22nd to January the 19th for 24 hours a day, collecting data of illuminance every one minute and elaborating them every ten minutes.
Figures 10 and 11 show the results of typical situations, respectively: a cloudy day (22 December), and a sunny day (5 January).
Figure 10
Fig. 10. Illuminance data of a typical overcast day (22 December). Data in positions other than 8 (minimum) and 2 (maximum) are overlapping between them - External illuminance referred to the right axis.
Figure 11
Fig. 11. Illuminance data of a typical overcast day (22 December). Data in positions other than 8 (minimum) and 2 (maximum) are overlapping between them - External illuminance referred to the right axis.
Under overcast sky (i.e., 22 December) the internal illuminance trend is very similar to the external one. The maximum external illuminance is about 21 klx and takes place at 12.30, while the internal illuminance is generally ranging between about 30 and 60 lx, except for the period from 12.30 to 13.30 during which it significantly decreases due to very low external illuminance values.
On the contrary, under clear sky with sun (i.e., 5 January) although the internal illuminance trend generally follows the external one, illuminance in positions 2 and 12 (in the right corners of the room) is significantly higher than in the other points. In particular, a peak value of about 350 lx takes place in position 12 at 12.30. Due to the low elevation of sun in winter conditions, in the central hours of the day, points 2 and 12 are hit by direct solar radiation without any interceptions by the light shelf. On January the 5th, the sun maximum elevation is ranging from 23.07° to 24.93° between 12.00 and 13.00, with azimuth equal to 162.82° - 177.83 °.
This does not occur with overcast sky, such as on December the 22nd, due to quite a complete absence of direct solar radiation.
Note that in Fig. 9 data in positions other than 8 (minimum) and 2 (maximum) have not been shown because they are overlapping and included between them.
Moreover, note that while on December the 22nd the external and internal maximum values of illuminance are perfectly simultaneous, on January the 5th there is a time shift of 30 minutes between them.
Moving from the previous considerations, the authors analyzed data of all the test days to determine the correlation between internal and external illuminance.
Table 2 shows the internal and external average and maximum illuminance values for each day between sunrise and sunset, the measure positions in which the maximum value happens, and the time shift between internal and external maximum values.
Table 2
Table 2. Average and maximum illuminance values, measure positions of maximum values and time shift between internal and external maximum illuminances.
Note that the table is lacking external illuminance data for five days (28 December, 1-3 January, 12 January) due to an accidental breakdown of the external luxmeter.
Maximum values take always place in positions 2 (13 times) and 12 (16 times), on the right side of the room.
For twelve days, the internal and external values are simultaneous, while for eleven days, there is a time shift between them, six times late and five times early. The maximum delay occurs on January the 15th (70 min), while the maximum advance occurs on January the 18th (60 min). The contemporaneity generally happens with low external illuminance on partially or fully cloudy days, while the time shift generally takes place on sunny days under clear sky with sun.
The reason for these time lags is not easily determined. They are probably due to many causes. First, both the internal and external illuminances are not instantaneous values but average values every ten minutes. Secondly, in addition to direct radiation, diffuse radiation and multiple reflections from some parts of the system and the walls contribute to determining the illuminance in an internal point. Finally, the external illuminance sometimes is maximum when the position of the sun on the sky is not such as to directly interface the most illuminated points (i.e., 2 and 12). In these cases, direct solar radiation does not hit any sensor and it is not registered.
### 5.1. Illuminance uniformity
The authors considered the parameter “Illuminance Uniformity”. According to the EN12464-1, it can be defined in two different ways, as shown respectively in Eqs. (1) and (2):
$U_0=\frac{E_{min}}{E_{avg}}$
$U_0^\prime=\frac{E_{min}}{E_{max}}$
Table 3 and Figs. 12 and 13 show the calculated values of U0 and U0' for all the test days at 10.00, 12.30 and 15.30.
Table 3
Table 3. Calculated values of Illuminance uniformity.
Figure 12
Fig. 12. Illuminance uniformity, U0, on the work plane for all the test days at 10.00, 12.30, and 15.30.
Figure 13
Fig. 13. Illuminance uniformity, Uo', prime on the work plane for all the test days at 10.00, 12.30, and 15.30.
From data reported in Table 3 and Figs. 12 and 13, we can deduce that the illuminance uniformity is not ideal because U0' is higher than 0.5 only in 38 % of cases, it is never higher than 0.8 and it is lower than 0.4 in 39 % of cases. Furthermore, U0 is higher than 0.8 only in 18 % of cases and lower than 0.5 in 16 % of cases, while in 66 % of cases it ranges between 0.5 and 0.8 (31 % between 0.7 and 0.8 - 29 % between 0.6 and 0.7 - 5 % between 0.5 and 0.6).
On the other hand, some authors underline that these criteria seem to be too restrictive for environments illuminated with natural light where a lower degree of uniformity is tolerated by users if compared to similar situations but with the use of artificial light [44].
Table 3 and Figs. 12 and 13 show that U0 and U0' have similar values for some days (i.e., 13-19 Jan). This trend is typical of sunny days with high external illuminance. In these cases, the illuminance uniformity is very similar at 10.00 and 15.30 and it is significantly lower at 12.30, probably due to the high value of solar elevation that causes less spatial penetrating reflections of solar radiation.
Figure 14 shows the results on January the 16th as an example of a sunny day. Sensors in positions 2 and 12 have the maximum values all the time with high illuminance between 12.00 and 13.00, respectively 287 lx in position 12 at 12.20, and 242 lx in position 2 at 13.00. In the other measure positions, illuminance ranges between 100 and 160 lx. It is noteworthy that the maximum values of illuminance take place further away from the system, so improving the light uniformity on the work-plan.
Figure 14
Fig. 14. Illuminance data on January the 16th - External illuminance referred to the right axis.
When direct solar radiation is quite absent and external illuminance is low, such as during cloudy days, the internal illuminance trend is very similar to the external one, without peak values. Figures 15 and 16 show the results on December 25 and 26 as an example of this situation.
Figure 15
Fig. 15. Illuminance data on December the 25th - External illuminance referred to the right axis.
Figure 16
Fig. 16. Illuminance data on December the 26th - External illuminance referred to the right axis.
Note that also in these cases, the maximum values of internal illuminance take place in the corners of the room (position 1, 2, 11 or 12) as evidence of the fact that even in the absence of intense direct solar radiation, the light shelf effectively contributes to the diffusion of light in the environment.
## 6. Conclusions
This paper describes the results of a preliminary experimental activity carried out on a reduced scale model of the MDLP, an innovative daylighting system, set up by modifying the DLP previously developed by the authors.
The experimental activity has been carried out in winter climatic conditions, between December 2021 and January 2022. The first results seem to be appreciable and encourage further investigation to better define the performance of the device.
Some problems of the DLP have been attenuated: the cut of the lower part of the external tube decreases the overall dimensions of the system and the application of a light shelf on its upper portion reduces the risk of glare and improves the illuminance uniformity on the horizontal work-plan.
Although the tests were carried out in winter conditions, with low external illuminance, the system contributes to light distribution in the passage room of underground buildings. On sunny days, with maximum external illuminance of about 45 klx, internal illuminance is around 100 - 150 lx most of the time between sunrise and sunset. Peak values of illuminance have been registered in positions close to the right side of the room, caused by intense direct solar radiation not intercepted by the light shelf, due to the low elevation angle of the sun. Taking into account that the tests have been carried out in winter climatic conditions, illuminance data on the work-plan can be considered satisfactory for underground environments.
The illuminance uniformity is not completely satisfactory, but it can be judged acceptable considering that a lower degree of uniformity is tolerated by users of environments illuminated with natural light instead of artificial light.
Finally, the technological components of the system have been described in detail, as well as the installation procedure for a generic brick concrete roof slab.
The authors intend to continue the experimental activity on the scale model of the MDLP to collect a sufficient amount of data available to define the principal daylight dynamic parameters, as the Spatial Daylight Autonomy or the Useful Daylight Illuminance. In addition, they are going to carry out a parametric analysis of the performance of the MDLP through a numerical activity.
## Contributions
All the authors contributed equally.
## Declaration of competing interest
The authors declare no conflict of interest.
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8. R. Campama Pizarro, N. Gentile, A case study addressing the benefits of integrated solutions for daylighting and electric lighting in the retail sector, in: ISES Solar World Congress, 2019, pp. 1825-1836, Santiago, Chile. https://doi.org/10.18086/swc.2019.38.02
9. M. Boubekri, R. Hull, L. Boyer, Impact of window size and sunlight penetration on office workers' mood and satisfaction: A novel way of assessing sunlight, Environment and Behaviour 23 4 (1991) 474-493. https://doi.org/10.1177/0013916591234004
10. P. Leather, M. Pyrgas, D. C. Beale. C. Lawrence, Windows in the workplace: sunlight, view, and occupational stress, Environment and Behaviour 30 (1998) 739-762. https://doi.org/10.1177/001391659803000601
11. L. Bellia, F. Fragliasso, A. Pedace, Evaluation of daylight availability for energy savings, Journal of Daylighting 2 (2015) 12-20. https://doi.org/10.15627/jd.2015.2
12. M. Bodart, A. De Herde, G. Lobal, Energy savings in offices buildings by the use of daylighting, Energy and Buildings 34 (2002) 421-429. https://doi.org/10.1016/S0378-7788(01)00117-7
13. E. J. Gago, T. Muneer, M. Knez, H. Köster, Natural light controls and guides in buildings. Energy saving for electrical lighting, reduction of cooling load, Renewable and Sustainable Energy Reviews 41 (2015) 1-13. https://doi.org/10.1016/j.rser.2014.08.002
14. J. Mardaljevic, L. L. Heschong, E. S. Lee, Daylight metrics and energy savings, Lighting Research and Technology 41 (2009) 261-283. https://doi.org/10.1177/1477153509339703
15. A. M. Momani, B. Yatim, M. Alauddin, M. Ali, The impact of the daylight saving time on electricity consumption a case study from Jordan, Energy Policy 37 (2009) 2042-2051. https://doi.org/10.1016/j.enpol.2009.02.009
16. I. Pyonchan, N. Abderrezek, K. Moncef, Estimation of lighting energy savings from daylighting, Building and Environment 44 (2009) 509-514. https://doi.org/10.1016/j.buildenv.2008.04.016
17. B. Obradovic, B. S. Matusiak, Daylight Transport Systems for Buildings at High Latitudes, Journal of Daylighting 6 2 (2019) 60-79. https://doi.org/10.15627/jd.2019.8
18. B. Obradovic, B. S. Matusiak, Daylight autonomy improvement in buildings at high latitudes using horizontal light pipes and light-deflecting panels, Solar Energy 208 (2020) 493-514. https://doi.org/10.1016/j.solener.2020.07.074
19. O. T. Masoso, L. J. Grobler, The dark side of occupants' behaviour on building energy use, Energy and Buildings 42 (2010) 173-177. https://doi.org/10.1016/j.enbuild.2009.08.009
20. S. Carlucci, F. Causone, F. De Rosa, L. Pagliano, A review of indices for assessing visual comfort with a view to their use in optimization processes to support building integrated design, Renewable and Sustainable Energy Reviews 47 (2015) 1016-1033. https://doi.org/10.1016/j.rser.2015.03.062
21. R. Canziani, F. Peron, G. Rossi, Daylight and energy performances of a new type of light pipe, Energy and Buildings 36 11 (2004) 1163-1176. https://doi.org/10.1016/j.enbuild.2004.05.001
22. D. Jenkins, T. Muneer, J. Kubie, A design tool for predicting the performances of light pipes, Energy and Buildings 37 5 (2005) 485-492. https://doi.org/10.1016/j.enbuild.2004.09.014
23. H. von Wachenfelt, V. Vakouli, A. P Diéguez, N. Gentile, M. C. Dubois, K. H. Jeppsson, Lighting Energy Saving with Light Pipe in Farm Animal Production, Journal of Daylighting 2 2 (2015) 21-31. https://doi.org/10.15627/jd.2015.5
24. A. A. Freewan, L. Shao, S. Riffat, Optimizing performance of the light shelf by modifying ceiling geometry in highly luminous climates, Solar Energy 82 (2008) 343-353. https://doi.org/10.1016/j.solener.2007.08.003
25. A. Ganga, B. R. Warrier, Performance evaluation of light shelves, Energy and Buildings 140 (2017) 19-27. https://doi.org/10.1016/j.enbuild.2017.01.068
26. A. Kontadakis, A. Tsangrassoulis, L. Doulos, S. Zerefos, A Review of Light Shelf Designs for Daylit Environments, Sustainability 10 1 (2018) 71-94. https://doi.org/10.3390/su10010071
27. A. Meresi, Evaluating daylight performance of light shelves combined with external blinds in south-facing classrooms in Athens, Greece, Energy and Buildings 116 (2016) 190-205. https://doi.org/10.1016/j.enbuild.2016.01.009
28. G. A. Warrier, B. Raphael, Performance evaluation of light shelves, Energy and Buildings 140 (2017) 19-27. https://doi.org/10.1016/j.enbuild.2017.01.068
29. Y. W. Lim, M. H. Ahmad, The effects of direct sunlight on light shelf performance under tropical sky, Indoor and Built Environment 24 6 (2015) 788-8022. https://doi.org/10.1177/1420326X14536066
30. P. Zazzini, A. Romano, A. Di Lorenzo, V. Portaluri, A. Di Crescenzo, Experimental Analysis of the Performance of Light Shelves in Different Geometrical Configurations through the Scale Model Approach, Journal of Daylighting 7 (2020) 37-56. https://doi.org/10.15627/jd.2020.4
31. S. Ahmed, A. Zain-Ahmed, S. A. Rahman, M. H. Sharif, Predictive tools for evaluating daylighting performance of light pipes, International Journal of Low-Carbon Technologies 1 (2006) 315-328. https://doi.org/10.1093/ijlct/1.4.315
32. D. J. Carter, The measured and predicted performance of passive solar light pipe systems, Lighting Research and Technology 34 1 (2002) 39-51. https://doi.org/10.1191/1365782802li029oa
33. S. Dutton. L. Shao, Raytracing simulation for predicting light pipe transmittance, International Journal of Low-Carbon Technologies 2 4 (2007) 339-358. https://doi.org/10.1093/ijlct/2.4.339
34. D. H.W. Li, E. K.W. Tsang, C. C. O. Tam, An analysis of light-pipe system via full-scale measurements, Applied Energy 87 3 (2010) 799-805. https://doi.org/10.1016/j.apenergy.2009.09.008
35. G. Oakley, S. B. Riffat, L. Shao, Daylight performance of light pipes, Solar Energy (2000) 69 2 89-98. https://doi.org/10.1016/S0038-092X(00)00049-9
36. M. Paroncini, B. Calcagni, F. Corvaro, Monitoring of a light-pipe system. Solar Energy 81 9 (2007) 1180-1186. https://doi.org/10.1016/j.solener.2007.02.003
37. A. Rosemann, H. Kaase, Light pipe applications for daylighting systems Solar Energy 78 (2005) 772-780. https://doi.org/10.1016/j.solener.2004.09.002
38. Y. Su, N. Khan, S. B. Riffat, O. Gareth, Comparative monitoring and data regression of various sized commercial light pipes, Energy and Buildings 50 (2012) 308-314. https://doi.org/10.1016/j.enbuild.2012.03.053
39. K. Vasilakopouloua, D. Kolokotsa, M. Santamouris, I. Kousis, H. Asproulias, I. Giannaraki, Analysis of the experimental performance of light pipes, Energy and Buildings 151 (2017) 242-249. https://doi.org/10.1016/j.enbuild.2017.06.061
40. O. Boccia, P. Zazzini, Daylight in buildings equipped with traditional or innovative sources: a critical analysis on the use of the scale model approach, Energy and Buildings 86 (2015) 376-393. https://doi.org/10.1016/j.enbuild.2014.10.003
41. C. Baroncini, F. Chella, P. Zazzini, Numerical and experimental analysis of the double light pipe: a new system for daylight distribution in interior spaces, International Journal of Low Carbon Technologies 3 2 (2008) 110-125. https://doi.org/10.1093/ijlct/3.2.110
42. P. Zazzini, A. Di Crescenzo, R. Giammichele, Numerical analysis of the performance of an innovative daylighting system named Modified Double Light Pipe, in: 6th AIGE IIETA International Conference, 2021, Ancona (Italy). https://doi.org/10.18280/ti-ijes.652-432
43. V. Garcia-Hansen, I. Edmonds, Methods for the illumination of multilevel buildings with vertical light pipes, Solar Energy 117 (2015) 74-88. https://doi.org/10.1016/j.solener.2015.04.017
44. M. C. Dubois, Impact of shading devices on daylight quality in offices - simulations with radiance, Research report No. TABK--01/3061Energy Build Des. Lund (SE) (2001), Lund University Sweden.
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# The Erasmus chessboard theorem
Professor Erasmus informed me today that he has proved another fascinating theorem about chessboards.
Consider a standard $8\times8$ chessboard with 32 black and 32 white squares. Pick an arbitrary (black or white) square $x$. Compute the sum $B$ of the 32 squared distances between the center of $x$ and the centers of the 32 black squares. Compute the sum $W$ of the 32 squared distances between the center of $x$ and the centers of the 32 white squares. Then independently of the choice of $x$, you will always find that $B=W$.
Has the professor once again made one of his well-known mathematical blunders, or does his claimed theorem indeed hold true?
Place a white 1kg weight at the center of each white square and a black 1kg weight at the center of each black square. Then $B$ is the rotational inertia of the black weights around $x$, and $W$ is the rotational inertia of the white weights around $x$.
By the Parallel Axis Theorem, the rotational inertia of an object depends only on (1) its total mass, (2) its moment about the center of mass, and (3) the distance from its center of mass to the axis of rotation. These quantities are evidently the same for the set of black weights and the set of white weights, by symmetry. So their rotational inertia around any point $x$ is equal for both sets. It follows that $B=W$.
• The rotational inertia also depends on the moment about the center of mass. – Julian Rosen Oct 4 '15 at 17:03
• @JulianRosen - correct, this should be added (together with the remark that both moments of inertia are equal). Also, it is worthwhile to note that Lopsy's reasoning generalizes the Erasmus checkerboard theorem to locations non-central to square X. – Johannes Oct 4 '15 at 17:16
• Good catch Julian, edited. – Lopsy Oct 5 '15 at 2:12
• The first sentence is a bit ambiguous: do you place thirty-two weights of each color or just one? This ambiguity is immediately cleared up, but it might still be worth rewording. – Lynn Oct 14 '15 at 6:33
The professor
is not as dumb as some people may be tempted to think.
Let the intersection cell of row $i$ and column $j$ be $(i,j)$. Let the chosen cell be $(x,y)$.
Therefore,
\begin{align} B &=\sum_{(i,j) \mathrm{ is\ black}} (x-i)^2 + (y-j)^2\\&= \sum_{(i,j) \mathrm{\ is\ black}}(x-i)^2 + \sum_{(i,j) \mathrm{\ is\ black}}(y-j)^2\\&=x_b+y_b \end{align}
Similarly,
\begin{align} W&=\sum_{(i,j) \mathrm{\ is\ white}} (x-i)^2 + \sum_{(i,j) \mathrm{\ is\ white}}(y-j)^2\\&=x_w+y_w \end{align}
Since, there are exactly 4 black and 4 white cells each row,
$$\sum_{(i,j) \mathrm{\ is\ black}} (x-i)^2=4\sum_{i=1}^8(x-i)^2=\sum_{(i,j) \mathrm{\ is\ white}} (x-i)^2$$
So, $x_w=x_b$ Similarly, $y_w=y_b$
Therefore, $B=W$
The first thing to understand is the sum of squares of distances from a cell to all black cells is the sum of squares of distances of the x coordinates $(x_b)$ plus the sum of squares of distances of the y coordinates $(y_b)$. This follows from Pythagoras/distance formula.
But since there are an equal number of black and white cells each row, The set of distances from black cells is actually the same set as the set of distances from white cells. Therefore $x_b$ and $x_w$ are summing the squares of the same set and therefore are equal. Similarly, $y_b$ and $y_w$ are equal.
Therefore B and W are equal.
Let us take a cell $(x,y)$ and four cells in a block $(a,b),(a+1,b),(a,b+1),(a+1,b+1)$. We know that two of them are black and two are white. We can prove that the sum of squared distances of $(x,y)$ from the black and white cells is equal.
Using Pythagoras theorem, $$B=[d\ to\ (a,b)]+[d\ to\ (a+1,b+1)]$$ $$=[(x-a)^2+(y-b)^2]+[(x-a-1)^2+(y-b-1)^2]$$
and $$W=[d\ to\ (a,b+1)]+[d\ to\ (a+1,b)]$$ $$=[(x-a)^2+(y-b-1)^2]+[(x-a-1)^2+(y-b)^2]$$
On comparinging, we find these two equal. Hence we conclude that for any cell and a block of $4$ other cells, this theorem is true.
We divide the $8\times 8$ checkerboard into $16\ (2\times 2)$ blocks. Since the theorem holds for distances to each of the blocks, it also holds for distances to all of the blocks.
The only block not taken into account is the block where the cell itself resides. It is easy to show that this theorem is true in the cell's own block. ($1^2+1^2=(\sqrt2)^2$)
Hence proved.
P.S.
I did not cheat from @Anachor 's answer. I couldn't understand it because I am yet to learn how to use the $\sum$ symbol properly. I knew Pythagoras theorem had to be used right from the beginning and made up the proof on my own.
• You don't have to take the cell's own block as a separate case. In that case "d to (.., ..)" equals 0, and that's fine. And by the way, the sigma symbol is very simple, if you just read it as "sum" you'll usually be fine. The subscript and superscript just tell us where the variables come from, like for example, $\sum_{v\in \mathbb{Z}, 1\le v^2\le 20}\, 1/v$ can be read as "sum over all integers v, whose squares are between 1 and 20, of 1/v". Sometimes some constraints on the variable are left out because they can be inferred from context. – Ben Frankel Oct 5 '15 at 2:25
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## Geometrization of 3-orbifolds of cyclic type. With an appendix: Limit of hyperbolicity for spherical 3-orbifolds by Michael Heusener and Joan Porti.(English)Zbl 0971.57004
Astérisque. 272. Paris: Société Mathématique de France. vi, 208 p. (2001).
Thurston’s geometrization conjecture in 3-dimensional topology states that every compact orientable irreducible 3-orbifold (including 3-manifolds) has a canonical decomposition along Euclidean 2-suborbifolds into geometric 3-orbifolds (mainly hyperbolic and Seifert fibered 3-orbifolds). For Haken 3-manifolds this follows from Thurston’s manifold hyperbolization theorem and the Jaco-Shalen-Johannson decomposition. The main results of the present monograph show that the conjecture holds for (very good) 3-orbifolds of cyclic type, i.e. with nonempty singular set consisting of circles and intervals (in general, the singular set of a compact orientable 3-orbifold is a 3-valent graph). In more technical terms, the first main result of the monograph is the following
Theorem 1: Let $$\mathcal O$$ be a compact connected orientable irreducible 3-orbifold of cyclic type. If $$\mathcal O$$ is very good, topologically atoroidal and acylindrical, then $$\mathcal O$$ is geometric (hyperbolic, Euclidean or Seifert fibered).
Here a 3-orbifold is called very good if it admits a finite covering by a 3-manifold. This hypothesis allows to apply the variety of known results about 3-manifolds (or their equivariant versions) and is presently mainly due to the technical structure of the proof. Theorem 1 applies, for example, to any 3-orbifold which is the 3-sphere with a hyperbolic knot as singular set, of singularity index $$p>2$$; in particular, it follows that the $$p$$-fold cyclic branched covering of a hyperbolic knot is a hyperbolic 3-manifold, with the only exception of the 3-fold cyclic branched covering of the figure-8 knot which is Euclidean. However, in general it is not easy to prove that an orbifold is very good so one would like to get rid of this hypothesis. In the present monograph, this is achieved by distinguishing two complementary classes of atoroidal 3-orbifolds: small orbifolds and Haken 3-orbifolds. For small orbifolds which, by definition, do not contain essential orientable 2-suborbifolds and whose boundary consists of “turnovers”, the methods of the proof of Theorem 1 apply to show that small 3-orbifolds of cyclic type are geometric, avoiding the hypothesis of being very good. For atoroidal Haken 3-orbifolds, a generalization of Thurston’s hyperbolization theorem for Haken 3-manifold is invoked, and the main steps of a proof are sketched. As any compact atoroidal 3-orbifold can be decomposed into small 3-orbifolds and Haken 3-orbifolds, these results combine to the orbifold geometrization theorem as announced by Thurston, for 3-orbifolds of cyclic type:
Let $$\mathcal O$$ be a compact connected orientable irreducible 3-orbifold of cyclic type; if $$\mathcal O$$ is topologically atoroidal, then $$\mathcal O$$ is geometric. (Also, any compact irreducible 3-orbifold possesses a canonical decomposition into geometric 3-orbifolds; then, a posteriori, such orbifolds are also very good).
In two recent preprints the two authors, together with B. Leeb, proved the orbifold geometrization theorem for compact orientable 3-orbifolds with arbitrary singular sets, i.e. not necessarily of cyclic type. An alternative approach to the orbifold geometrization has been worked out by D. Cooper, C. D. Hodgson and St. P. Kerckhoff [Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5 (2000; Zbl 0955.57014)].
The main tools of the proofs in the present monograph are Thurston’s hyperbolization theorem for Haken 3-manifolds, the hyperbolic Dehn surgery (or Dehn filling) theorem and results about the geometric convergence of sequences of hyperbolic cone-manifolds. Following Thurston’s original approach, the structure of the proof of Theorem 1 is as follows. One reduces to the case that the complement of the singular set of the orbifold has a complete hyperbolic structure. The hyperbolic Dehn surgery theorem implies that, for small angles around the singular set, the corresponding cone manifolds have hyperbolic structures (in contrast to geometric 3-orbifolds whose angles around the singular set are submultiples of $$2\pi$$, for a geometric cone manifold arbitrary angles are allowed). Then one tries to increase these cone angles in order to arrive at the cone angles of the original orbifold, studying the geometric limits and possible types of degeneration of the corresponding sequences of hyperbolic cone manifolds. This uses a cone manifold version of Gromov’s compactness theorem for Riemannian manifolds with pinched sectional curvature. In the nicest (non-collapsing) case the hyperbolic cone manifold structures converge to a hyperbolic structure on the original orbifold, in the collapsing case one has to study carefully the possible limits (possibly after rescaling the hyperbolic cone metrics) which may be Euclidean 3-orbifolds, Seifert orbifolds or may contain Euclidean 2-suborbifolds. These geometric convergence theorems are among the main results of the monograph, their proofs occupy chapters 3-6 and are the heart of the monograph. “The authors’ main contribution takes place in the analysis of the so-called collapsing case. There they use the notion of simplicial volume due to Gromov and a cone manifold version of his isolation theorem. This gives a simpler combinatorial approach to collapses than Thurston’s original one. In particular, it spares us the difficult task of establishing a suitable Cheeger-Gromov theory for collapses of cone manifolds.” In the last chapter (9), explicit examples of collapses of hyperbolic cone structures to other geometric structures are discussed. In two appendices, limits of hyperbolicity for spherical 3-orbifolds are considered (when deforming the hyperbolic structures of the complement of the singular set, these cannot converge directly to the spherical structure but one arrives at a Euclidean structure first), and a proof of Thurston’s hyperbolic Dehn surgery theorem is given.
### MSC:
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 57M50 General geometric structures on low-dimensional manifolds 53C20 Global Riemannian geometry, including pinching 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
### Keywords:
3-orbifold; geometric structure; orbifold geometrization
Zbl 0955.57014
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# 【matplotlib, scipy】How to plot normal distribution by python
The code to plot above graph.
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats
mean = 50
std = 10
x = np.linspace(0,100,100)
y = scipy.stats.norm.pdf(x,mean,std)
plt.plot(x,y)
plt.show()
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BBO Discussion Forums: WHICH BROWESRS BEST - BBO Discussion Forums
Page 1 of 1
WHICH BROWESRS BEST
#1RapidRob56
• Group: Members
• Posts: 1
• Joined: 2021-January-22
Posted 2021-January-22, 23:37
I am trying to switch browsers for BBO but am having trouble. The Settings under the Accounts Tab does NOT show the check boxes for choosing an option under Firefox and Edge. What can I do to show the check boxes ? Right now I have to scroll over the area and hope I am
selecting the correct option.
0
#2otangu
• Group: Members
• Posts: 27
• Joined: 2018-October-08
Posted 2021-January-24, 04:47
RapidRob56, on 2021-January-22, 23:37, said:
I am trying to switch browsers for BBO but am having trouble. The Settings under the Accounts Tab does NOT show the check boxes for choosing an option under Firefox and Edge. What can I do to show the check boxes ? Right now I have to scroll over the area and hope I am
selecting the correct option.
Try Chrome, it hasn't given me any trouble
0
#3stanspjr
• Group: Members
• Posts: 1
• Joined: 2021-January-24
Posted 2021-January-24, 11:05
I use EDGE which is Chrome based. Seems to work well for me.
0
#4undoubling
• Group: Members
• Posts: 10
• Joined: 2020-July-27
Posted 2021-January-24, 11:53
Edge, which is standard for Windows systems, works just fine. I'm not sure what works with Apple systems
0
#5Gazumper
• Group: Members
• Posts: 46
• Joined: 2003-December-28
Posted 2021-January-24, 16:23
Mozilla Firefox works for me.
2
#6sfi
• Posts: 1,932
• Joined: 2009-May-18
• Location:Oz
Posted 2021-January-24, 16:39
Firefox has worked well for me on both Mac and Windows.
1
#7RayFink
• Group: Members
• Posts: 4
• Joined: 2016-November-15
Posted 2021-January-24, 21:48
Slide the gray vertical divider bar -- between Settings and the "table area" -- further to the left to enlarge the Settings area. Click left and hold, drag left.
0
#8fromageGB
• Posts: 2,618
• Joined: 2008-April-06
Posted 2021-January-28, 11:32
Sliding the divider does nothing, the selector blobs are always there on the right, and the words slide off if the pane is narrowed. But they are not check-boxes in the traditional sense, but slider blobs that slide left or right when dragged. (Could this be dependent on your choice of desktop environment? ie Windows looks one way and KDE another?) I am using firefox very happily, before that I had no problems with chromium. Friends have problems with communications dropping in safari.
0
#9barmar
• Posts: 20,501
• Joined: 2004-August-21
• Gender:Male
Posted 2021-January-28, 12:45
We adopted a consistent style from the mobile app. So instead of checkboxes all the options use those slider toggles. It's the same in all browsers.
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# Product of Odd Factorials: Do They Exist?
Case 1 ($n=1$): $$x_1!=1!\cdot3!=6=3!$$ Case 2 ($n=2$): $$x_2!=1!\cdot3!\cdot 5!=(1)\cdot(1\cdot 2\cdot 3)\cdot (1\cdot 2\cdot 3\cdot 4\cdot 5)=1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6=6!$$ Case 3 ($n=3$): $$x_3!=1!\cdot3!\cdot 5!\cdot 7!=(1)\cdot(1\cdot 2\cdot 3)\cdot (1\cdot 2\cdot 3\cdot 4\cdot 5) \cdot (1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7)$$ $$=1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9 \cdot 10=10!$$ Case 4 ($n=4$): $$x_4!=1!\cdot3!\cdot 5!\cdot 7!\cdot 9!$$ It is sufficient to say that: $$x_4!>11!\implies 11\mid x_4!$$ However: $$11 \mid 1!\cdot3!\cdot 5!\cdot 7!\cdot 9!$$ Case 5 ($n=5$): $$x_5!=1!\cdot3!\cdot 5!\cdot 7!\cdot 9!\cdot 11!$$ It is sufficient to say that: $$x_5!>13!\implies 13\mid x_5!$$ However: $$13 \mid 1!\cdot 3!\cdot 5!\cdot 7!\cdot 9!\cdot 11!$$ Case $n$ ($n \ge 6$): $$1!\cdot3!\cdot 5! \cdots (2n+1)!=x_{\ge 6}!$$ It is sufficient to say that: $$x_{\ge 6}!>(4n+2)!$$
Bertrand’s Postulate (should be a theorem) states that $\forall n>1, n< p< 2n$ for some prime, $p$. Generalizing this to our argument, we have the following: $$2n+1< p < 4n+2$$ $$\implies (2n+1)!< p!<(4n+2)!$$ $$\implies (2n+1)!< p!<(4n+2)!< x_{\ge 6}!$$ $$\implies p \mid x!$$ However, since $p!>(2n+1)!$, there exists no $p$ such that $p \mid x!$. Therefore, $p \mid x!$.
Proposition does not hold past $n=1,2,3$. So, no they do not exist.
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## Thursday, November 24, 2016
### Author: Dhananjay Ghei
Dhananjay Ghei is a researcher at the National Institute for Public Finance and Policy. On this blog:
Please note: LaTeX mathematics works. This means that if you want to say $10 you have to say \$10.
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# Image segmentation
(Redirected from Segmentation (image processing))
Model of a segmented femur. It shows the outer surface (red), the surface between compact bone and spongy bone (green) and the surface of the bone marrow (blue).
In computer vision, image segmentation is the process of partitioning a digital image into multiple segments (sets of pixels, also known as superpixels). The goal of segmentation is to simplify and/or change the representation of an image into something that is more meaningful and easier to analyze.[1][2] Image segmentation is typically used to locate objects and boundaries (lines, curves, etc.) in images. More precisely, image segmentation is the process of assigning a label to every pixel in an image such that pixels with the same label share certain characteristics.
The result of image segmentation is a set of segments that collectively cover the entire image, or a set of contours extracted from the image (see edge detection). Each of the pixels in a region are similar with respect to some characteristic or computed property, such as color, intensity, or texture. Adjacent regions are significantly different with respect to the same characteristic(s).[1] When applied to a stack of images, typical in medical imaging, the resulting contours after image segmentation can be used to create 3D reconstructions with the help of interpolation algorithms like Marching cubes.
## Applications
Some of the practical applications of image segmentation are:
Several general-purpose algorithms and techniques have been developed for image segmentation. To be useful, these techniques must typically be combined with a domain's specific knowledge in order to effectively solve the domain's segmentation problems.
## Thresholding
The simplest method of image segmentation is called the thresholding method. This method is based on a clip-level (or a threshold value) to turn a gray-scale image into a binary image. There is also a balanced histogram thresholding.
The key of this method is to select the threshold value (or values when multiple-levels are selected). Several popular methods are used in industry including the maximum entropy method, Otsu's method (maximum variance), and k-means clustering.
Recently, methods have been developed for thresholding computed tomography (CT) images. The key idea is that, unlike Otsu's method, the thresholds are derived from the radiographs instead of the (reconstructed) image [4] .[5]
## Clustering methods
Main article: Data clustering
Source image.
Image after running k-means with k = 16. Note that a common technique to improve performance for large images is to downsample the image, compute the clusters, and then reassign the values to the larger image if necessary.
The K-means algorithm is an iterative technique that is used to partition an image into K clusters.[6] The basic algorithm is
1. Pick K cluster centers, either randomly or based on some heuristic
2. Assign each pixel in the image to the cluster that minimizes the distance between the pixel and the cluster center
3. Re-compute the cluster centers by averaging all of the pixels in the cluster
4. Repeat steps 2 and 3 until convergence is attained (i.e. no pixels change clusters)
In this case, distance is the squared or absolute difference between a pixel and a cluster center. The difference is typically based on pixel color, intensity, texture, and location, or a weighted combination of these factors. K can be selected manually, randomly, or by a heuristic. This algorithm is guaranteed to converge, but it may not return the optimal solution. The quality of the solution depends on the initial set of clusters and the value of K.
## Compression-based methods
Compression based methods postulate that the optimal segmentation is the one that minimizes, over all possible segmentations, the coding length of the data.[7][8] The connection between these two concepts is that segmentation tries to find patterns in an image and any regularity in the image can be used to compress it. The method describes each segment by its texture and boundary shape. Each of these components is modeled by a probability distribution function and its coding length is computed as follows:
1. The boundary encoding leverages the fact that regions in natural images tend to have a smooth contour. This prior is used by Huffman coding to encode the difference chain code of the contours in an image. Thus, the smoother a boundary is, the shorter coding length it attains.
2. Texture is encoded by lossy compression in a way similar to minimum description length (MDL) principle, but here the length of the data given the model is approximated by the number of samples times the entropy of the model. The texture in each region is modeled by a multivariate normal distribution whose entropy has closed form expression. An interesting property of this model is that the estimated entropy bounds the true entropy of the data from above. This is because among all distributions with a given mean and covariance, normal distribution has the largest entropy. Thus, the true coding length cannot be more than what the algorithm tries to minimize.
For any given segmentation of an image, this scheme yields the number of bits required to encode that image based on the given segmentation. Thus, among all possible segmentations of an image, the goal is to find the segmentation which produces the shortest coding length. This can be achieved by a simple agglomerative clustering method. The distortion in the lossy compression determines the coarseness of the segmentation and its optimal value may differ for each image. This parameter can be estimated heuristically from the contrast of textures in an image. For example, when the textures in an image are similar, such as in camouflage images, stronger sensitivity and thus lower quantization is required.
## Histogram-based methods
Histogram-based methods are very efficient when compared to other image segmentation methods because they typically require only one pass through the pixels. In this technique, a histogram is computed from all of the pixels in the image, and the peaks and valleys in the histogram are used to locate the clusters in the image.[1] Color or intensity can be used as the measure.
A refinement of this technique is to recursively apply the histogram-seeking method to clusters in the image in order to divide them into smaller clusters. This is repeated with smaller and smaller clusters until no more clusters are formed.[1][9]
One disadvantage of the histogram-seeking method is that it may be difficult to identify significant peaks and valleys in the image.
Histogram-based approaches can also be quickly adapted to occur over multiple frames, while maintaining their single pass efficiency. The histogram can be done in multiple fashions when multiple frames are considered. The same approach that is taken with one frame can be applied to multiple, and after the results are merged, peaks and valleys that were previously difficult to identify are more likely to be distinguishable. The histogram can also be applied on a per pixel basis where the information result are used to determine the most frequent color for the pixel location. This approach segments based on active objects and a static environment, resulting in a different type of segmentation useful in Video tracking.
## Edge detection
Edge detection is a well-developed field on its own within image processing. Region boundaries and edges are closely related, since there is often a sharp adjustment in intensity at the region boundaries. Edge detection techniques have therefore been used as the base of another segmentation technique.
The edges identified by edge detection are often disconnected. To segment an object from an image however, one needs closed region boundaries. The desired edges are the boundaries between such objects or spatial-taxons. [10] [11]
Spatial-taxons [12] are information granules.,[13] consisting of a crisp pixel region, stationed at abstraction levels within a hierarchical nested scene architecture. They are similar to the Gestalt psychological designation of figure-ground, but are extended to include foreground, object groups, objects and salient object parts. Edge detection methods can be applied to the spatial-taxon region, in the same manner they would be applied to a silhouette. This method is particularly useful when the disconnected edge is part of an illusory contour [14][15]
Segmentation methods can also be applied to edges obtained from edge detectors. Lindeberg and Li [16] developed an integrated method that segments edges into straight and curved edge segments for parts-based object recognition, based on a minimum description length (MDL) criterion that was optimized by a split-and-merge-like method with candidate breakpoints obtained from complementary junction cues to obtain more likely points at which to consider partitions into different segments.
## Region-growing methods
Region-growing methods mainly rely on the assumption that the neighboring pixels within one region have similar values. The common procedure is to compare one pixel with its neighbors. If a similarity criterion is satisfied, the pixel can be set belong to the cluster as one or more of its neighbors. The selection of the similarity criterion is significant and the results are influenced by noise in all instances.
The method of Statistical Region Merging[17] (SRM) starts by building the graph of pixels using the 4-connectedness with edges weighted by the absolute value of the intensity difference. Initially each pixel forms a single pixel region. SRM then sorts those edges in a priority queue and decide whether to merge or not the current regions belonging to the edge pixels using a statistical predicate.
One region-growing method was the seeded region growing method. This method takes a set of seeds as input along with the image. The seeds mark each of the objects to be segmented. The regions are iteratively grown by comparing all unallocated neighboring pixels to the regions. The difference between a pixel's intensity value and the region's mean, $\delta$, is used as a measure of similarity. The pixel with the smallest difference measured this way is allocated to the respective region. This process continues until all pixels are allocated to a region. Because Seeded region growing requires seeds as additional input, the segmentation results are dependent on the choice of seeds, and noise in the image can cause the seeds to be poorly placed.
Another region-growing method was the unseeded region growing method. It is a modified algorithm that doesn't require explicit seeds. It starts off with a single region $A_1$ – the pixel chosen here does not significantly influence final segmentation. At each iteration it considers the neighboring pixels in the same way as seeded region growing. It differs from seeded region growing in that if the minimum $\delta$ is less than a predefined threshold $T$ then it is added to the respective region $A_j$. If not, then the pixel is considered significantly different from all current regions $A_i$ and a new region $A_{n+1}$ is created with this pixel.
One variant of this technique, proposed by Haralick and Shapiro (1985),[1] is based on pixel intensities. The mean and scatter of the region and the intensity of the candidate pixel is used to compute a test statistic. If the test statistic is sufficiently small, the pixel is added to the region, and the region’s mean and scatter are recomputed. Otherwise, the pixel is rejected, and is used to form a new region.
A special region-growing method is called $\lambda$-connected segmentation (see also lambda-connectedness). It is based on pixel intensities and neighborhood-linking paths. A degree of connectivity (connectedness) will be calculated based on a path that is formed by pixels. For a certain value of $\lambda$, two pixels are called $\lambda$-connected if there is a path linking those two pixels and the connectedness of this path is at least $\lambda$. $\lambda$-connectedness is an equivalence relation.[18]
Split-and-merge segmentation is based on a quadtree partition of an image. It is sometimes called quadtree segmentation.
This method starts at the root of the tree that represents the whole image. If it is found non-uniform (not homogeneous), then it is split into four son-squares (the splitting process), and so on so forth. Conversely, if four son-squares are homogeneous, they can be merged as several connected components (the merging process). The node in the tree is a segmented node. This process continues recursively until no further splits or merges are possible.[19][20] When a special data structure is involved in the implementation of the algorithm of the method, its time complexity can reach $O(n\log n)$, an optimal algorithm of the method.[21]
## Partial differential equation-based methods
Using a partial differential equation (PDE)-based method and solving the PDE equation by a numerical scheme, one can segment the image.[22] Curve propagation is a popular technique in this category, with numerous applications to object extraction, object tracking, stereo reconstruction, etc. The central idea is to evolve an initial curve towards the lowest potential of a cost function, where its definition reflects the task to be addressed. As for most inverse problems, the minimization of the cost functional is non-trivial and imposes certain smoothness constraints on the solution, which in the present case can be expressed as geometrical constraints on the evolving curve.
### Parametric methods
Lagrangian techniques are based on parameterizing the contour according to some sampling strategy and then evolve each element according to image and internal terms. Such techniques are fast and efficient, however the original "purely parametric" formulation (due to Kass and Terzopoulos in 1987 and known as "snakes"), is generally criticized for its limitations regarding the choice of sampling strategy, the internal geometric properties of the curve, topology changes (curve splitting and merging), addressing problems in higher dimensions, etc.. Nowadays, efficient "discretized" formulations have been developed to address these limitations while maintaining high efficiency. In both cases, energy minimization is generally conducted using a steepest-gradient descent, whereby derivatives are computed using, e.g., finite differences.
### Level set methods
The level set method was initially proposed to track moving interfaces by Osher and Sethian in 1988 and has spread across various imaging domains in the late nineties. It can be used to efficiently address the problem of curve/surface/etc. propagation in an implicit manner. The central idea is to represent the evolving contour using a signed function, where its zero level corresponds to the actual contour. Then, according to the motion equation of the contour, one can easily derive a similar flow for the implicit surface that when applied to the zero-level will reflect the propagation of the contour. The level set method encodes numerous advantages: it is implicit, parameter free, provides a direct way to estimate the geometric properties of the evolving structure, can change the topology and is intrinsic. Furthermore, they can be used to define an optimization framework as proposed by Zhao, Merriman and Osher in 1996. Therefore, one can conclude that it is a very convenient framework to address numerous applications of computer vision and medical image analysis.[23] Furthermore, research into various level set data structures has led to very efficient implementations of this method.
### Fast marching methods
The fast marching method has been used in image segmentation,[24] and this model has been improved (permitting a both positive and negative speed propagation speed) in an approach called the generalized fast marching method.[25]
## Graph partitioning methods
Graph partitioning methods can effectively be used for image segmentation. In these methods, the image is modeled as a weighted, undirected graph. Usually a pixel or a group of pixels are associated with nodes and edge weights define the (dis)similarity between the neighborhood pixels. The graph (image) is then partitioned according to a criterion designed to model "good" clusters. Each partition of the nodes (pixels) output from these algorithms are considered an object segment in the image. Some popular algorithms of this category are normalized cuts,[26] random walker,[27] minimum cut,[28] isoperimetric partitioning,[29] minimum spanning tree-based segmentation,[30] and segmentation-based object categorization.
## Watershed transformation
The watershed transformation considers the gradient magnitude of an image as a topographic surface. Pixels having the highest gradient magnitude intensities (GMIs) correspond to watershed lines, which represent the region boundaries. Water placed on any pixel enclosed by a common watershed line flows downhill to a common local intensity minimum (LIM). Pixels draining to a common minimum form a catch basin, which represents a segment.
## Model based segmentation
The central assumption of such an approach is that structures of interest/organs have a repetitive form of geometry. Therefore, one can seek for a probabilistic model towards explaining the variation of the shape of the organ and then when segmenting an image impose constraints using this model as prior. Such a task involves (i) registration of the training examples to a common pose, (ii) probabilistic representation of the variation of the registered samples, and (iii) statistical inference between the model and the image. State of the art methods in the literature for knowledge-based segmentation involve active shape and appearance models, active contours and deformable templates and level-set based methods.[citation needed]
## Multi-scale segmentation
Image segmentations are computed at multiple scales in scale space and sometimes propagated from coarse to fine scales; see scale-space segmentation.
Segmentation criteria can be arbitrarily complex and may take into account global as well as local criteria. A common requirement is that each region must be connected in some sense.
### One-dimensional hierarchical signal segmentation
Witkin's seminal work[31][32] in scale space included the notion that a one-dimensional signal could be unambiguously segmented into regions, with one scale parameter controlling the scale of segmentation.
A key observation is that the zero-crossings of the second derivatives (minima and maxima of the first derivative or slope) of multi-scale-smoothed versions of a signal form a nesting tree, which defines hierarchical relations between segments at different scales. Specifically, slope extrema at coarse scales can be traced back to corresponding features at fine scales. When a slope maximum and slope minimum annihilate each other at a larger scale, the three segments that they separated merge into one segment, thus defining the hierarchy of segments.
### Image segmentation and primal sketch
There have been numerous research works in this area, out of which a few have now reached a state where they can be applied either with interactive manual intervention (usually with application to medical imaging) or fully automatically. The following is a brief overview of some of the main research ideas that current approaches are based upon.
The nesting structure that Witkin described is, however, specific for one-dimensional signals and does not trivially transfer to higher-dimensional images. Nevertheless, this general idea has inspired several other authors to investigate coarse-to-fine schemes for image segmentation. Koenderink[33] proposed to study how iso-intensity contours evolve over scales and this approach was investigated in more detail by Lifshitz and Pizer.[34] Unfortunately, however, the intensity of image features changes over scales, which implies that it is hard to trace coarse-scale image features to finer scales using iso-intensity information.
Lindeberg[35][36] studied the problem of linking local extrema and saddle points over scales, and proposed an image representation called the scale-space primal sketch which makes explicit the relations between structures at different scales, and also makes explicit which image features are stable over large ranges of scale including locally appropriate scales for those. Bergholm proposed to detect edges at coarse scales in scale-space and then trace them back to finer scales with manual choice of both the coarse detection scale and the fine localization scale.
Gauch and Pizer[37] studied the complementary problem of ridges and valleys at multiple scales and developed a tool for interactive image segmentation based on multi-scale watersheds. The use of multi-scale watershed with application to the gradient map has also been investigated by Olsen and Nielsen[38] and been carried over to clinical use by Dam[39] Vincken et al.[40] proposed a hyperstack for defining probabilistic relations between image structures at different scales. The use of stable image structures over scales has been furthered by Ahuja[41][42] and his co-workers into a fully automated system. A fully automatic brain segmentation algorithm based on closely related ideas of multi-scale watersheds has been presented by Undeman and Lindeberg [43] and been extensively tested in brain databases.
These ideas for multi-scale image segmentation by linking image structures over scales have also been picked up by Florack and Kuijper.[44] Bijaoui and Rué[45] associate structures detected in scale-space above a minimum noise threshold into an object tree which spans multiple scales and corresponds to a kind of feature in the original signal. Extracted features are accurately reconstructed using an iterative conjugate gradient matrix method.
## Semi-automatic segmentation
In one kind of segmentation, the user outlines the region of interest with the mouse clicks and algorithms are applied so that the path that best fits the edge of the image is shown.
Techniques like SIOX, Livewire, Intelligent Scissors or IT-SNAPS are used in this kind of segmentation. In an alternative kind of semi-automatic segmentation, the algorithms return a spatial-taxon (ie. foreground, object-group, object or object-part) selected by the user or designated via prior probabilities.[46][47]
## Trainable segmentation
Most segmentation methods are based only on color information of pixels in the image. Humans use much more knowledge than this when doing image segmentation, but implementing this knowledge would cost considerable computation time and would require a huge domain-knowledge database, which is currently not available. In addition to traditional segmentation methods, there are trainable segmentation methods which can model some of this knowledge.
Neural Network segmentation relies on processing small areas of an image using an artificial neural network[48] or a set of neural networks. After such processing the decision-making mechanism marks the areas of an image accordingly to the category recognized by the neural network. A type of network designed especially for this is the Kohonen map.
Pulse-coupled neural networks (PCNNs) are neural models proposed by modeling a cat’s visual cortex and developed for high-performance biomimetic image processing. In 1989, Eckhorn introduced a neural model to emulate the mechanism of a cat’s visual cortex. The Eckhorn model provided a simple and effective tool for studying the visual cortex of small mammals, and was soon recognized as having significant application potential in image processing. In 1994, the Eckhorn model was adapted to be an image processing algorithm by Johnson, who termed this algorithm Pulse-Coupled Neural Network. Over the past decade, PCNNs have been utilized for a variety of image processing applications, including: image segmentation, feature generation, face extraction, motion detection, region growing, noise reduction, and so on. A PCNN is a two-dimensional neural network. Each neuron in the network corresponds to one pixel in an input image, receiving its corresponding pixel’s color information (e.g. intensity) as an external stimulus. Each neuron also connects with its neighboring neurons, receiving local stimuli from them. The external and local stimuli are combined in an internal activation system, which accumulates the stimuli until it exceeds a dynamic threshold, resulting in a pulse output. Through iterative computation, PCNN neurons produce temporal series of pulse outputs. The temporal series of pulse outputs contain information of input images and can be utilized for various image processing applications, such as image segmentation and feature generation. Compared with conventional image processing means, PCNNs have several significant merits, including robustness against noise, independence of geometric variations in input patterns, capability of bridging minor intensity variations in input patterns, etc.
Open-source implementations of trainable segmentation:
## Segmentation benchmarking
Several segmentation benchmarks are available for comparing the performance of segmentation methods with the state-of-the-art segmentation methods on standardized sets
## Notes
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21. ^ L. Chen, The lambda-connected segmentation and the optimal algorithm for split-and-merge segmentation, Chinese J. Computers, 14(1991), pp 321-331
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23. ^ S. Osher and N. Paragios. Geometric Level Set Methods in Imaging Vision and Graphics, Springer Verlag, ISBN 0-387-95488-0, 2003.
24. ^ James A. Sethian. "Segmentation in Medical Imaging". Retrieved 15 January 2012.
25. ^ Forcade, Nicolas; Le Guyader, Carole; Gout, Christian (July 2008), "Generalized fast marching method: applications to image segmentation", Numerical Algorithms 48 (1-3): 189–211, doi:10.1007/s11075-008-9183-x
26. ^ Jianbo Shi and Jitendra Malik (2000): "Normalized Cuts and Image Segmentation", IEEE Transactions on pattern analysis and machine intelligence, pp 888-905, Vol. 22, No. 8
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28. ^ Z. Wu and R. Leahy (1993): "An optimal graph theoretic approach to data clustering: Theory and its application to image segmentation", IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 1101–1113, Vol. 15, No. 11
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30. ^ C. T. Zahn (1971): "Graph-theoretical methods for detecting and describing gestalt clusters", IEEE Transactions on Computers, pp. 68–86, Vol. 20, No. 1
31. ^ Witkin, A. P. "Scale-space filtering", Proc. 8th Int. Joint Conf. Art. Intell., Karlsruhe, Germany,1019–1022, 1983.
32. ^ A. Witkin, "Scale-space filtering: A new approach to multi-scale description," in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), vol. 9, San Diego, CA, Mar. 1984, pp. 150–153.
33. ^ Koenderink, Jan "The structure of images", Biological Cybernetics, 50:363–370, 1984
34. ^ Lifshitz, L. and Pizer, S.: A multiresolution hierarchical approach to image segmentation based on intensity extrema, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12:6, 529–540, 1990.
35. ^ Lindeberg, T.: Detecting salient blob-like image structures and their scales with a scale-space primal sketch: A method for focus-of-attention, International Journal of Computer Vision, 11(3), 283–318, 1993.
36. ^
37. ^ Gauch, J. and Pizer, S.: Multiresolution analysis of ridges and valleys in grey-scale images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 15:6 (June 1993), pages: 635–646, 1993.
38. ^ Olsen, O. and Nielsen, M.: Multi-scale gradient magnitude watershed segmentation, Proc. of ICIAP 97, Florence, Italy, Lecture Notes in Computer Science, pages 6–13. Springer Verlag, September 1997.
39. ^ Dam, E., Johansen, P., Olsen, O. Thomsen,, A. Darvann, T. , Dobrzenieck, A., Hermann, N., Kitai, N., Kreiborg, S., Larsen, P., Nielsen, M.: "Interactive multi-scale segmentation in clinical use" in European Congress of Radiology 2000.
40. ^ Vincken, K., Koster, A. and Viergever, M.: Probabilistic multiscale image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 19:2, pp. 109–120, 1997.]
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42. ^ E. Akbas and N. Ahuja, "From ramp discontinuities to segmentation tree"
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47. ^ Barghout, Lauren (2014). Vision: How Global Perceptual Context Changes Local Contrast Processing (Ph.D. Dissertation 2003). Updated to include Computer Vision Techniques. Scholars Press. ISBN 978-3-639-70962-9.
48. ^ Mahinda Pathegama & Ö Göl (2004): "Edge-end pixel extraction for edge-based image segmentation", Transactions on Engineering, Computing and Technology, vol. 2, pp 213–216, ISSN 1305-5313
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50. ^ D.Martin; C. Fowlkes and D. Tal and J. Malik (July 2001). "A Database of Human Segmented Natural Images and its Application to Evaluating Segmentation Algorithms and Measuring Ecological Statistics". Proc. 8th Int'l Conf. Computer Vision 2. pp. 416–423.
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### Session GO6: Z Pinches
9:30 AM–12:30 PM, Tuesday, November 15, 2011
Room: Ballroom G
Chair: Stephanie Hansen, Sandia National Laboratories
Abstract ID: BAPS.2011.DPP.GO6.4
### Abstract: GO6.00004 : Experimental Study of an Inverse Wire Array Z-Pinch Operating as a Current Switch
10:06 AM–10:18 AM
Preview Abstract MathJax On | Off Abstract
#### Authors:
Sergey Lebedev
(Imperial College)
A. Harvey-Thomson
G.N. Hall
E.M. Waisman
E. Khoory
G. Burdiak
J.P. Chittenden
P. de Grouchy
F.A. Suzuki-Vidal
We will present experiments on the MAGPIE facility (1.5MA, 250ns) in which an inverse wire array [1] (with the wires acting as a return current cage placed around a central current conductor) operated as a fast current switch. This allowed to significantly reduce the rise-time of the current pulse ($<$100ns) delivered to a separate, standard imploding wire array z-pinch load. Experimental studies of the operation of this arrangement as a current switch will be discussed and new measurements of current switching into the load array will be presented. We will also discuss how pre-conditioning of the load array wires by the current pre-pulse [2] depends on wire materials (Al, Cu, W) used in the load and the exploding wire arrays. \\[4pt] [1] A. Harvey-Thompson, S.V. Lebedev, S.N. Bland et al., PoP 16, 022701 (2009).\\[0pt] [2] A. Harvey-Thompson, S.V. Lebedev, G. Burdiak, et al., PRL 106, 205002 (2011)
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## wcaprar 3 years ago I just don't seem to be able to understand how to get the second derivative of a parametric equation. Can someone walk me through it?
1. wcaprar
Okay, my parametric equation is x=3+3t^2, y+3t^2+t^3. to find dy/dx you divide (dy/dt)/(dx/dt). Which I can do no problem.
2. wcaprar
For the first derivative I got for 1-((3t^2)/(4t)). Which the online homework problem says is correct.
3. wcaprar
To find the second derivative I have to find d/dt (dy/dx). I separated the two parts and got 0 for the one, of course, and then used the quotient rule on the -3t^2/4t part. Is this correct?
4. amistre64
a paramentric eqution is also a vector equation: r(t) = <x(t),y(t)> right?
5. wcaprar
Yes
6. amistre64
x=3+3t^2 , y=3t^2+t^3 x'=6t , y=6t+3t r'(t) = < 6t , 6t+3t^2 > which is what youve got for the tangent
7. wcaprar
Lol, I mixed two problems in what I explained, but yes, go on
8. amistre64
r''(t) is just the next derivative. r''(t) = <6, 6+6t> if im reading it right
9. amistre64
10. amistre64
bout 3/4 of the way down it goes into 2nd derivative
11. wcaprar
Right, but the question is asking for it in a certain form. Here is the question:
12. amistre64
a tiff? really?
13. amistre64
you got that in a jpeg?
14. wcaprar
Lol, here it is in jpg
15. wcaprar
I just sliced a portion of the window to get the picture
16. amistre64
$\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})=\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$
17. amistre64
soo dy/dx = 1+ t/2 stick this in the derivative thing for: 1/2 right?
18. amistre64
$\frac{d}{dt}(\frac{dy}{dx})\to \frac{d}{dt}(1+t/2)$
19. wcaprar
Yeah, you're right. But what I have is a dy/dx = 1-3t^2/4t
20. amistre64
or the right dy/dx .. lol
21. wcaprar
Sorry, I starting typing the first, which is pretty easy when I was actually going for a second problem. But this is the spot I always get confused. With that dy/dx I would lose the one and use quotient rule to solve for -3t^2/4t, wouldn't I? But when I do that it says I got the problem wrong
22. amistre64
reduce it by t and you got -3t/4 which is simpler to derive
23. amistre64
not reduce ..... simplify
24. wcaprar
Oh my Gosh!! I can't believe I wasn't seeing that! Thank you!
25. amistre64
youre welcome
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Reasonable "Random" matrices to test numerical algorithms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:12:31Z http://mathoverflow.net/feeds/question/50822 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/50822/reasonable-random-matrices-to-test-numerical-algorithms Reasonable "Random" matrices to test numerical algorithms Martin 2010-12-31T18:37:48Z 2011-02-10T15:51:20Z <p>Hello,</p> <p>in numerical analysis, it is common to compare the behavior of different algorithms, and of different implementation of algorithms. This occurs not only on the theoretical level, but also on the concrete level of implementation - and not to forget, it serves the purpose of demonstration.</p> <p>A prominent problem is the solution of linear systems, both general as well as various subcases.</p> <p>To test, benchmark and profile numerical implementations, you run your work on several instances of the problem. However, it is difficult question to obtain a good set of these instances. You want to inspect pathological cases (diff. degrees of ill-conditionedness) as well as "real-life" examples (whatever this may mean). Ideally, you have an algorithm which puts out matrices with certain properties in a "reasonable" probability measure. A good notion of "reasonable" might be accessible, as most such LSE problems from physics or simulations have much more structure as is actually demanded by the algorithms in theory.</p> <p>In so far, I wonder whether there are works in numerical analysis how to, given $n \in \mathbb N$ randomly produce</p> <ul> <li>a sequence of ${n \times n}$-matrices</li> <li>optionally constraint to be symmetric, positive definite, well-conditioned</li> <li>which is reasonable in whatever sense</li> </ul> <p>This is probably an interesting topic within the theory of numerical algorithms.</p> <p>Thanks, Martin</p> http://mathoverflow.net/questions/50822/reasonable-random-matrices-to-test-numerical-algorithms/50830#50830 Answer by Igor Rivin for Reasonable "Random" matrices to test numerical algorithms Igor Rivin 2010-12-31T20:12:49Z 2010-12-31T20:12:49Z <p>This is an impossible question to answer (so maybe this should be a comment...), since (for example) the sparsity patterns of the matrices encountered are completely different in statistical applications and in finite elements -- each problem class leads to a completely different distribution on the space of matrices. As a result, all the papers I have seen in numerical analysis are of the sort: we tried heuristic X and it worked well for problem Y. This gives us reasonable confidence that it will work for problem Z, such that $|Z-Y| < \epsilon.$ People are very careful not to make general statements.</p> <p>DISCLAIMER: I am not a professional numerical analyst, nor do I play one on TV.</p> http://mathoverflow.net/questions/50822/reasonable-random-matrices-to-test-numerical-algorithms/50871#50871 Answer by Federico Poloni for Reasonable "Random" matrices to test numerical algorithms Federico Poloni 2011-01-01T18:02:44Z 2011-01-01T18:02:44Z <p>A good set of benchmark matrices often depends on the problem being solved (sparse solvers, eigenvalue problems, special structures, et cetera). Good sets of benchmarks are hand-picked, and are publication-worthy in themselves (they get lots of citations).</p> <p>Among many of them, I shall mention here the all-purpose Matlab's <code>gallery</code>, based on Higham's books (and in particular tweaking the parameters in <code>randsvd</code> could suit your needs well), the University of Florida's sparse matrix collection , <code>carex</code> and <code>darex</code> by Benner for control problems, and several image sets for image reconstruction and compression algorithms, such as the baboon or the long-debated <a href="http://en.wikipedia.org/wiki/Lenna" rel="nofollow">Lenna</a>.</p> http://mathoverflow.net/questions/50822/reasonable-random-matrices-to-test-numerical-algorithms/54906#54906 Answer by Diego de Estrada for Reasonable "Random" matrices to test numerical algorithms Diego de Estrada 2011-02-09T17:26:10Z 2011-02-10T15:51:20Z <p>One way to generate random matrices while constraining it, is to generate its LU decomposition first. That way you can restrict it to be symmetric ($L=U^T$) and gives you the control over its spectrum.</p> <p>In [S.M. Rump. A Class of Arbitrarily Ill-conditioned Floating-Point Matrices. SIAM J. Matrix Anal. Appl. (SIMAX), 12(4):645-653, 1991] there is a related method for generating very ill-conditioned matrices.</p> http://mathoverflow.net/questions/50822/reasonable-random-matrices-to-test-numerical-algorithms/54950#54950 Answer by Andrew D. King for Reasonable "Random" matrices to test numerical algorithms Andrew D. King 2011-02-09T23:08:52Z 2011-02-09T23:08:52Z <p>One way to get useful information on a numerical algorithm is to choose by hand a well-conditioned or even more an ill-conditioned example and perturb it randomly. In fact it is useful to perturb it randomly such that the perturbations form a sequence approaching zero; you can then look at the sequence of solutions that you get and see what you can say about error propagation and the like.</p>
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Accelerating Innovation Through Analogy Mining
# Accelerating Innovation Through Analogy Mining
Tom Hope The Hebrew University of Jerusalem Joel Chan Carnegie Mellon University Aniket Kittur Carnegie Mellon University and Dafna Shahaf The Hebrew University of Jerusalem
###### Abstract.
The availability of large idea repositories (e.g., the U.S. patent database) could significantly accelerate innovation and discovery by providing people with inspiration from solutions to analogous problems. However, finding useful analogies in these large, messy, real-world repositories remains a persistent challenge for either human or automated methods. Previous approaches include costly hand-created databases that have high relational structure (e.g., predicate calculus representations) but are very sparse. Simpler machine-learning/information-retrieval similarity metrics can scale to large, natural-language datasets, but struggle to account for structural similarity, which is central to analogy. In this paper we explore the viability and value of learning simpler structural representations, specifically, “problem schemas”, which specify the purpose of a product and the mechanisms by which it achieves that purpose. Our approach combines crowdsourcing and recurrent neural networks to extract purpose and mechanism vector representations from product descriptions. We demonstrate that these learned vectors allow us to find analogies with higher precision and recall than traditional information-retrieval methods. In an ideation experiment, analogies retrieved by our models significantly increased people’s likelihood of generating creative ideas compared to analogies retrieved by traditional methods. Our results suggest a promising approach to enabling computational analogy at scale is to learn and leverage weaker structural representations.
Computational analogy; innovation; creativity; product dimensions; text mining; text embedding
## 1. Introduction
The ability to find useful analogies is critical to driving innovation in a variety of domains. Many important discoveries in science were driven by analogies: for example, an analogy between bacteria and slot machines helped Salvador Luria advance the theory of bacterial mutation. Analogical reasoning forms the foundation of law, with the effectiveness of an argument often dependent on its legal precedents (Spellman and Schauer, 2012). Innovation is often spurred by analogy as well: an analogy to a bicycle allowed the Wright brothers to design a steerable aircraft. Whether architecture, design, technology, art, or mathematics, the ability to find and apply patterns from other domains is fundamental to human achievement (Hesse, 1966; Pask, 2003; Markman and Loewenstein, 2010; Dahl and Moreau, 2002).
The explosion of available online data represents an unprecedented opportunity to find new analogies and accelerate human progress across domains. For example, the US Patent database has full text for more than 9 million patents issued from 1976 to the present. InnoCentive111innocentive.com contains more than 40,000 business, social, policy, scientific, and technical problems and solutions. Quirky222quirky.com, a company that assists inventors in the development process, has had over 2 million product idea submissions. OpenIDEO333OpenIDEO.com receives hundreds of solutions for a variety of social problems. Millions of scientific papers and legal cases are searchable on Google Scholar.
We believe these data form a treasure trove of analogies that can accelerate problem solving, innovation and discovery. In a striking recent example, a car mechanic invented a simple device to ease difficult childbirths by drawing an analogy to extracting a cork from a wine bottle, which he discovered in a YouTube video. This award-winning device could save millions of lives, particularly in developing countries. We imagine a future in which people could search through data based on deep analogical similarity rather than simple keywords; lawyers or legal scholars could find legal precedents sharing similar systems of relations to a contemporary case; and product or service designers could mine myriad potential solutions to their problem.
However, sifting through these massive data sources to find relevant and useful analogies poses a serious challenge for both humans and machines. In humans, memory retrieval is highly sensitive to surface similarity, favoring near, within-domain analogs that share object attributes over far, structurally similar analogs that share object relations (Gentner et al., 1985; Holyoak and Thagard, 1996; Gentner et al., 1993; Gick and Holyoak, 1983). Analogical processing also incurs a heavy cognitive load, taxing working memory when even a few relations are required to be processed at once (Halford et al., 2005). Thus searching through datasets with thousands or millions of items for structurally similar ones may be a daunting prospect.
Finding analogies is challenging for machines as well, as it is based on having an understanding of the deep relational similarity between two entities that may be very different in terms of surface attributes (Gentner, 1983). For example, Chrysippus’ analogy between sound waves and water waves required ignoring many different surface features between the two (Holyoak and Thagard, 1996). Recent advances in data mining and information retrieval include a variety of natural language techniques that use words, parts of speech or other language feature-based vector representations in order to calculate similarity measures (see (Shutova, 2010)). Examples include word embedding models like Word2Vec (Mikolov et al., 2013), vector-space models like Latent Semantic Indexing (Deerwester et al., 1990), and probabilistic topic modeling approaches like Latent Dirichlet Allocation (Blei et al., 2003). These approaches excel at detecting surface similarity, but are often unable to detect similarity between documents whose word distributions are disparate. The problem is especially acute when the source and target domains are different (for example, bacterial mutation and slot machines).
Another approach to finding analogies has been to use the structural similarity of sentences or texts, such as using coupled clustering for detecting structural correspondence of text (Bollegala et al., 2009; Turney, 2006). However, these approaches typically require rich data sets with clear substructures, whereas most descriptions of problems or ideas in existing online databases are short, sparse, or lack consistent structure. Other current methods focus on very narrow analogy tasks, such as four-term analogy problems (teacher:student = doctor:?), in particular with short strings (ABC:ABD = KJI:?) (Hofstadter et al., 1994). In contrast, we wish to find analogies in real world data, which involve complex representations and a diverse set of analogical relations.
In this paper, we are interested in automatically discovering analogies in large, unstructured data sets. In particular, we focus on a corpus of product innovations. There are two insights behind our approach that we believe may make this problem tractable despite its longstanding status as a “holy grail” in both cognitive science and AI. First, rather than trying to solve the problem of fully structured analogical reasoning, we instead explore the idea that for retrieving practically useful analogies, we can use weaker structural representations that can be learned and reasoned with at scale (in other words, there is a tradeoff between the ease of extraction of a structure and its expressivity). Specifically, we investigate the weaker structural representation of an idea’s purpose and mechanism as a way to find useful analogies. The second insight is that advances in crowdsourcing have made it possible to harvest rich signals of analogical structure that can help machine learning models learn in ways that would not be possible with existing datasets alone.
This paper combines these two ideas to contribute a technique for computationally finding analogies from unstructured text datasets that go beyond surface features. At a high level, our approach uses the behavioral traces of crowd workers searching for analogies and identifying the purpose and mechanisms of ideas, then developing machine learning models that develop similarity metrics suited for analogy mining. We demonstrate that learning purpose and mechanism representations allows us to find analogies with higher precision and recall than traditional information-retrieval methods based on TF-IDF, LSA, LDA and GloVe, in challenging noisy settings. Furthermore, we use our similarity metrics to automatically find far analogies – products with high purpose similarity, and low mechanism similarity. In a user study, we show that we are able to “inspire” participants to generate more innovative ideas than alternative baselines, increasing the relative proportion of positively-rated ideas by at least .
## 2. Learning a Representation for Analogies
### 2.1. Motivation
Much work in computation analogy has focused on fully structured data, often with logic-based representations. For example (Falkenhainer et al., 1989),
CAUSE(GREATER-THAN[TEMPERATURE(coffee),
TEMPERATURE (ice-cube)],
FLOW(coffee, ice-cube, heat, bar))
These representations, while very expressive, are notoriously difficult to obtain. In this section, we investigate a weaker structural representation. Our goal is to come up with a representation that can be learned, while still being expressive enough to allow analogical mining.
Analogies between product ideas are intricately related to their purpose and mechanism. Informally, we think of a product’s purpose as “what it does, what it is used for”, and a product’s mechanism is “how it does it, how it works”. The importance of a product’s purpose and mechanism as core components of analogy are theoretically rooted in early cognitive psychology work on schema induction which define the core components of a schema as a goal and proposed solution to it (e.g., (Gick and Holyoak, 1983)). More recently, the practical value of defining a problem schema as a purpose and mechanism has been demonstrated to have empirical benefits in finding and using analogies to augment idea generation (e.g., (Yu et al., 2016, 2016, 2014, 2014)).
Separating an idea into purpose and mechanisms enables core analogical innovation processes such as re-purposing: For a given product (such as a kitchen-sink cleaner) and its purpose, finding another way to put it to use (cleaning windows). To that end, assume (for the moment) that we have for each product two vectors, and , representing the product’s purpose and mechanism, respectively. Using this representation, we are able to apply rich queries to our corpus of products, such as:
• Same purpose, different mechanism. Given the corpus of all products , a product with (normalized) purpose and mechanism vectors , and distance metrics between purpose and mechanism vectors (respectively), solve:
(1) argmin ~i∈P dp(pi,p~i) s.t. dm(mi,m~i)≥threshold,
• Same Mechanism, different purpose. Solve:
(2) argmin ~i∈P dm(mi,m~i) s.t. dp(pi,p~i)≥threshold,
The decomposition of products into purpose and mechanism also draws inspiration from engineering functional models and ontologies for describing products (Hirtz et al., 2002). Although there is no set common definition of functions (Ookubo et al., 2007), much research on functionality has been conducted in areas such as functional representation, engineering design and value engineering. The scope of these ontologies, however, is highly “mechanistic” or engineering-oriented, while in many cases we observe in product data the purpose of a product is more naturally understood in others term – such as whether it is for entertainment, leisure, or more serious purposes, who is the target user (adults, children), and so forth.
Importantly, our dataset of product descriptions contains noisy texts, often written informally by non-professional people. In these texts product descriptions are often lacking detail or are ill-defined. To automatically describe a product in terms of a formal functional model would require an inordinate amount of meticulous data annotation and collection by professional engineers over a large number of product descriptions. We thus resort to a softer approach, hoping that a compromise on the level of detail will enable data-driven methods to automatically extract useful representations of product purpose and mechanism.
Finally, we also make note of the potentially wider applicability of automatically extracting these representations from real-word product descriptions. Identifying the key components and functions of products could conceivably improve (or augment) search capabilities in internal or external product databases, and perhaps enhance recommender systems by better understanding what a user is looking for in a product and what a product offers. This last idea is connected to a line of work on “product dimensions” (McAuley and Leskovec, 2013), in which it is shown that implicitly identifying the properties of products (such as that Harry Potter is a book about wizards), helps in improving recommendations. The authors propose a method that combines ratings data with textual product reviews, hoping to implicitly recover topics in the text that inform recommendations. We too look at product dimensions, but target only two that are more abstract and broad, and directly learn them in a supervised fashion from annotated data.
### 2.2. Data
Innovation Corpus. We test our approach with a corpus of product descriptions from Quirky.com, an online crowdsourced product innovation website. Quirky is representative of the kinds of datasets we are interested in, because it is large (at the time of writing, it hosts upwards of 10,000 product ideas, of which our corpus included 8500), unstructured (ideas are described in natural language), and covers a variety of domains (invention categories) which makes cross-domain analogies possible. The following example illustrates the typical length and “messiness” of product ideas in this dataset:
Thirsty too
Pet water bowl/dispenser for your vehicle cup holder.
Over spill lip to catch water
Has optional sleeve for larger cup holders
Optional floor base
One way valve so water cant over flow from bottle
Small reservoir
Reservoir acts as backsplash
Water bottle attachment
Holds water in your vehicle cupholder for pet
Foldable handle to get unit out of holder
Dishwasher safe
Optional sleeve for larger cup holders
Collecting Purpose and Mechanism Data. In addition to the Quirky innovation corpus, we needed to collect analogy-specific data to train our model. Previous approaches to creating structured representations of items for analogical computation, for example predicate logic, are extremely heavyweight and can take tens of person-hours for complex items (Vattam et al., 2011). Instead, we aim to develop a lightweight task that avoids complex structure but instead relies on the cognitive expertise and intuitions of people to be able to separate the purpose of a product from its mechanism. By doing so we can scale up the collection of purpose and mechanism labels through the use of microtasks and crowdsourcing markets (Kittur et al., 2008). Specifically, we show Amazon Mechanical Turk (AMT) crowd workers a product description, asking them to annotate the parts of the text they consider to be about the purposes of the product, and the parts related to mechanisms. We frame the problem in simple terms, guiding workers to look for words/phrases/chunks of text talking about “what the product does, what it is good for” (purposes), and “how it works, what are its components” (mechanisms). As seen in Figure 1, we juxtapose two copies of the product text side-by-side, to ease cognitive load and encourage workers not to give purpose and mechanism tags that are too similar or overlapping, thus capturing a potentially richer and more distinct signal. Our corpus consisted of products. Each product was annotated by four workers.
Collecting Analogies. In previous, preliminary work (Chan et al., 2016), we explored the use of crowdsourcing to label analogies, collecting labeled examples of analogies (product pairs) that were fed into a metric-learning deep learning model. While showing promising results, the process of collecting labeled analogies proved expensive, requiring considerable cognitive effort from workers and thus more time, limiting the number of labels that can realistically be collected. In addition, in that work, the deep learning model was blind to the rich structures of purpose and mechanism, and had no hope of recovering them automatically due to relative data scarcity. In this paper we take a different approach, focusing our resources on collecting purpose and mechanism annotations from the crowd, while collecting only a small number of curated labeled analogies strictly for the purpose of evaluation (see Section 3.1 for details).
### 2.3. Method
#### 2.3.1. Extracting Purpose and Mechanism vectors
In this section, we describe our approach to learning to extract purpose and mechanism product representations. We begin with a set of training product texts , where each is a variable-length sequence of tokens . For each document , we collect a set of purpose annotations and mechanism annotations, where is the number of workers who annotate each document. We define the purpose annotation to be a binary vector
~pik=(~p1ik,~p2ik,…,~pTik)
of the same length as , with if token is annotated as purpose by annotator , if not. In the same way, we denote the mechanism annotation with
~mik=(~m1ik,~m2ik,…,~mTik)
While on the surface this setting appears to lend itself naturally to sequence-to-sequence learning (Sutskever et al., 2014), there are a few important differences. A key difference is that in our setting the problem of interest is not to learn to recover the latent () exactly for unseen products, but rather to extract some form of representation that captures the overall purpose and mechanism. That is, what we do care about is the semantic meaning or context our representation captures with respect to product purposes or mechanisms, rather than predicting any individual words. Additionally, sequence-to-sequence models typically involve heavier machinery and work well on large data sets, not suitable for our scenario were we have at most a few thousands of tagged examples.
On a more technical note, instead of one sequence in the output, we now have . A simple solution is to aggregate the annotations, for example by taking the union or intersection of annotations, or considering a token to be positively annotated if it has at least positive labels. Richer aggregations may also be used.
Considering that our focus here is to capture an overall representation (not predict a precise sequence), however, we resort to a simple and soft aggregation of the annotations. In simple terms, we look at all words that were annotated, and take a TF-IDF-weighted average of their word vectors.
In more formal terms, let be the sequence of GloVe (Pennington et al., 2014) word vectors (pre-trained on Common Crawl web data), representing . We select all word vectors for which () for some , and concatenate them into one sequence. We then compute the TF-IDF scores for tokens in this sequence, find the tokens with top TF-IDF scores ( in our experiments), and take the TF-IDF-weighted average of their corresponding GloVe vectors. We denote the resulting weighted-average vectors as and for purpose and mechanism annotations, respectively. We consider to be target vectors we aim to predict for unseen texts.
Embedding (short) texts with a weighted-average of word vectors (such as with TF-IDF) can lead to surprisingly good results across many tasks (Arora et al., 2016b). Furthermore, in our case this simple weighted-average has several advantages. As we will next see, it lends itself to a straightforward machine learning setting, suitable for our modestly-sized data set, and for the objective of finding an overall vector representation that can be used in multiple ways, chiefly the computation of purpose-wise and mechanism-wise distances between products. Additionally, by concatenating all annotations and weighting by TF-IDF, we naturally give more weight in the average vector to words that are more frequently annotated – thus giving higher impact to words considered important by all annotators with respect to purpose/mechanism.
#### 2.3.2. Learning purpose and mechanism
We now have training product texts , and corresponding target tuples . We represent each with its pre-trained GloVe vectors, . Our goal is to learn a function that predicts . To this end, we model with a Recurrent Neural Network as follows. The network takes as input the variable-length sequence . This sequence is processed with a bidirectional RNN (BiRNN) (Bahdanau et al., 2014) with one GRU layer. The BiRNN consists of the forward GRU which reads the sequence from to , and a backward GRU which reads from to , thus in practice capturing the neighborhood of from “both directions”:
→hji=−−−→GRU(wji), ←hji=←−−−GRU(wji), hji=[→hji,←hji],
where we concatenate forward and backward GRU hidden states to obtain , our representation for word in product . In our case, we are interested in which captures the entire product text.
Next, let and be purpose and mechanism weight matrices, respectively. is a shared representation of the document, which we now transform into two new vectors, , forming our purpose and mechanism predictions for product :
(3) ^pi=WphTi, ^mi=WmhTi.
Parameters in this network are then tuned to minimize the loss averaged over . In some scenarios, we may care more about predicting either purpose or mechanism, and in that case could incorporate a weight term in the loss function, giving more weight to either or .
#### 2.3.3. Purpose and Mechanism vector interpretations
Here, we give intuition about the kinds of representations extracted and the ability to interpret them with very simple tools. We first compute for held-out product texts. Then, in the first approach to interpreting purpose and mechanism predictions, we find the top GloVe word vectors most similar to each of , among all vectors that appear in our vocabulary.
In the second approach, we aim to recover a set of word vectors such that their sparse linear combination approximately gives or . More formally, in the spirit of the sparse coding approach in (Arora et al., 2016a), consider the collection of all word vectors in our vocabulary , . We stack them into a matrix . We aim to solve the following optimization problem:
(4) argmina ||^pi−Wa||22 s.t. ||a||0≤10,
where is a weight vector. Optimization can be done with the Orthogonal Matching Pursuit (OMP) (Cai and Wang, 2011) greedy algorithm.
In Table 1, we display some examples of applying these two simple methods, to product texts in test data (not seen during training). The first product is a yogurt maker machine, used for concentrating yogurt under heat, and to reduce time and energy. We observe that words selected as most related to our purpose vector representation include food, produce, concentrate, making, energy, reduce and also words that are typical in the language used in describing advantages of products in our data, such as especially, whole, enough, much. Mechanism words, are indeed overall of a much more “mechanical” nature, including liquid, heat, cooling, pump, steel, machine. In the other examples too, we observe the same pattern: Words selected as most closely-related to purpose or mechanism representations, using simple techniques, empirically appear to reflect corresponding properties in the product text, both in language and in deeper meaning.
## 3. Evaluation: Analogies
As typically done in the context of learning document representations, the key approach to quantitative evaluation is a down-stream task such as document classification. We now evaluate the predicted in the context of their ability to capture distances that reflect analogies, which is the primary focus of this paper. To do so, we first create a dataset of analogies and non-analogies.
### 3.1. Collecting analogies via crowdsourcing
We crowdsourced analogy finding within a set of about Quirky products. AMT crowd workers used our search interface to collect analogies – pairs of products – for about seed documents. The search task is powered by a simple word-matching approach. To deal with word variants, we added lemmas for each word to the bag-of-words associated with each product. Each search query was also expanded with lemmas associated with each query term. Search results were ranked in descending order of number of matching terms. Median completion time for each seed was 7 minutes (workers could complete as many seeds as they wanted). Further, to deal with potential data quality issues, we recruited 3 workers for each seed (to allow for majority-vote aggregation).
Pairs that were tagged as matches became positive examples in our analogy dataset. However, coming up with negative examples was more difficult. Borrowing from information retrieval, we assume that people read the search results sequentially, and treat the implicitly rejected documents (i.e., documents that were not matches, despite appearing before matches) as negatives. It is important to remember that these documents are not necessarily real negatives. To further increase the chance that the document has actually been read, we restrict ourselves to the top-5 results.
Challenges. Getting workers to understand the concept of analogies and avoiding tagging products that are superficially similar (e.g., “both smartphone-based”) as analogies proved a challenge. To address this, we scaffolded the search task by first requiring workers to generate a schema (or “pattern”) to describe the core purpose and mechanism of the product, first in concrete terms, and then in more abstract terms (see an example pattern Figure 2). Workers were then instructed to find other products that matched the abstract schema they created. We found that this scaffolded workflow reduced the number of superficial matches; yet, a non-negligible portion of the pairs labeled as positive were either superficial matches or near analogies (i.e., analogies with many shared surface features), likely due to the strong tendency towards surface features in analogical retrieval (Gentner et al., 1993). Further, because products were multifaceted, search results may have been implicitly rejected even if they were analogous to the seed if the matching schema was different from the one initially identified by the worker.
### 3.2. Quantitative results
In Table 2, we present precision and recall @ K results. We rank all pairs in the test data (, with training done on about products) based on their distances, according to various metrics, including our own. In summary, across all levels our approach outperformed the baselines, despite a challenging noisy setting. A considerable portion of test product pairs were tagged by workers as analogies despite having only surface similarity, creating mislabeled positive examples that favor the surface-based baselines. In addition to ranking by purpose-only and mechanism-only, we also concatenate both representations in a vector for product , and observe an overall improvement in results, although the “one-dimensional” use of either purpose or mechanism alone still beats the baselines. Using only led to considerably better results when looking at precision @ top , perhaps indicating a tendency by workers to find more mechanism-based analogies.
## 4. Evaluation: Ideation by analogy
Since a major application of the enhanced search and retrieval capabilities of analogy is enhanced creativity, we now evaluate the usefulness of our algorithms. We examine the degree to which our model’s retrieved output improves people’s ability to generate creative ideas, compared to other methods. To do so we use a standard ideation task in which participants redesign an existing product (Ullman, 2002), and are given inspirations to help them – either from our approach, a TF-IDF baseline, or a random baseline.
See Figure 3 for an example task given to crowdworkers. Here, the task was to redesign a cell phone case that can charge the phone. The middle part shows the top 3 inspirations per condition.
Our assumption is that our approach will provide participants with useful examples that are similar in purpose but provide diverse mechanisms that will help them explore more diverse parts of the design space in generating their ideas. We hypothesize that this approach will lead to better results than the TF-IDF baseline (highly relevant but non-diverse inspirations, focusing on surface features) and the random baseline (highly diverse but low relevance).
### 4.1. Generating near-purpose far-mechanism analogies
To generate inspirations for the redesign task, we start by using the learned purpose and mechanism representations for each document (in the test set) to apply rich queries to our corpus of products. In particular, assuming all vectors are normalized to unit euclidean norm, we can find pairs of products such that is high (near purpose), while is low (far mechanism). This type of reasoning, as discussed above, is a core element of analogical reasoning.
We take this idea one step forward by clustering by purpose and diversifying by mechanism. In more detail, we take a set of products not seen during training, and follow a simple and intuitive procedure as follows. Let denote our corpus of test-set products. Let denote the number of seed products we wish to use in our experiment. Let denote the number of inspirations we wish to produce for each seed .
Clustering by purpose. First, we find groups of products with similar purpose by clustering by our purpose representation.
• Run K-means (), based on vectors . (Note that when all vectors are normalized, the euclidean norm on which K-means is based is equivalent to the cosine distance).
• For each cluster , compute an intra-distance measure (purpose homogeneity) . We use the . Prune clusters with less than instances. Rank clusters by in descending order, pick top . Call this set of clusters , with corresponding cluster centers .
• For each cluster in , select the product whose vector is nearest to the cluster center . This is our seed product, denoted by .
Result diversification by mechanism. We now have a set of seed products, each with a corresponding cluster of products with similar purposes. Next, we need to pick inspirations per seed.
For each seed , we have a set of candidate matches , all from the same purpose cluster. We empirically observe that in the purpose-clusters we generate, some vectors are highly similar to the seed with respect to mechanism, and some less so. In order to generate far-mechanism results for each seed from candidate set , we now turn to diversification of results.
The problem of extracting a well-diversified subset of results from a larger set of candidates has seen a lot of work, prominently in the context of information retrieval (which is closely related to our setting). In our case, we assume to have found a set of relevant results according to purpose metric , and diversify by mechanism metric .
There are many ways to diversify results, mainly differing by objective function and constraints. Two canonical measures are the MAX-MIN and MAX-AVG dispersion problems (Ravi et al., 1994). In the former, we aim to find a subset such that , and
minmi1,mi2∈Mdm(mi1,mi2)
is maximized. In the latter, we aim to find a subset such that , and
2M(M−1)∑mi1,mi2∈Mdm(mi1,mi2)
is maximized. In other words, in the MAX-MIN problem we find a subset of products such that the distance between the two nearest products is maximized. In the MAX-AVG problem, we find a subset such the average distance between pairs is maximized. Both problems admit simple greedy algorithms with constant-factor approximations (Ravi et al., 1994). We choose the MAX-MIN problem, since we want to avoid displaying too-similar results even once to a user (who may become frustrated and not proceed to read more inspirations).
We solve the problem using the GMM algorithm mentioned in (Ravi et al., 1994). Each iteration of GMM selects a candidate such that the minimum distance from to an already-selected product in is the largest among remaining candidates in , where we measure distance according to our mechanism metric .
In our experiments, we set , for seeds and matches each, respectively.
### 4.2. Experiment design
We recruited 38 AMT workers to redesign an existing product, a common creative task in design firms (Ullman, 2002). To ensure robustness of effects, the experiment included 12 different “seed” products. Participants were paid \$1.5 for their participation.
To maximize statistical power, we utilized a within-subjects design with a single manipulated factor, inspiration_type:
• ANALOGY: participants receive product inspirations retrieved by our method detailed above, using near-purpose far-mechanism clustering and diversification.
• BASELINE: SURFACE: participants receive product inspirations retrieved using TF-IDF, by finding the top products similar to the seed. This baseline is meant to simulate current search engines.
• BASELINE: RANDOM: participants receive product inspirations sampled at random from our product corpus.
Since we used a within-subjects design, participants completed the redesign task under each of the 3 inspiration_type conditions. The order of conditions was counterbalanced to prevent order effects. To ensure unbiased permutations, we used the Fisher-Yates shuffle to assign seeds to conditions, so that every seed would be seen in all conditions (by different users).
Since prior work has shown that people benefit more from analogies if they receive them after ideation has begun (Tseng et al., 2008), the ideation task proceeded in two phases: 1) generating ideas unassisted for one minute, then 2) receiving 12 inspirations and generating more ideas for 6 minutes. The inspirations were laid out in four pages, 3 inspirations per page, and the users could freely browse them.
Figure 3 provides an overview of the experiment and an excerpt from the data. The original task was to redesign an existing product, in this case a cell phone charger case. The SURFACE baseline retrieves products that are very phone-related (or case related). In contrast, our algorithm retrieves diverse results such as a human pulley-powered electricity generator suit. The bottom of the figure shows ideas generated by users in each condition. Interestingly, the user exposed to our approach suggested a case that generates power using movement, potentially inspired by the suit.
### 4.3. Results
Measures. We are interested in the ability of our approach to enhance people’s ability to generate creative ideas. Following (Reinig et al., 2007), we measured creative output as the rate at which a participant generates good ideas. We recruited five graduate students to judge each idea generated by our participants as good or not. Our definition of “good” follows the standard definition of creativity in the literature as a combination of novelty, quality, and feasibility (Runco and Jaeger, 2012). Each judge was instructed to judge an idea as good if it satisfied all of the following criteria: 1) it uses a different mechanism/technology than the original product (novelty), 2) it proposes a mechanism/technology that would achieve the same purpose as the original product (quality), and 3) could be implemented using existing technology and does not defy physics (feasibility).
Agreement between the judges was substantial, with a Fleiss kappa of 0.51, lending our measure of creativity acceptable inter-rater reliability. The final measure of whether an idea was good or not was computed by thresholding the number of votes, so that good = 1 if at least judges rated it as good. We report results for both liberal and strict settings .
Evaluation. For , out of total ideas collected, ideas were judged as good by this measure. As mentioned above, we use the Fisher-Yates shuffle to assign seeds to conditions. To take a conservative approach, as a first step we look only at seeds that appeared across all three conditions ( such seeds), to put the conditions on par with one another. By this slicing of the data, there were good ideas. The proportion of good ideas in our condition was (). Next was the random baseline with (), and finally the TF-IDF baseline achieved (N=). These results are significant by a proportion test (). We thus observe that both in terms of the absolute number of positively-rated ideas and in terms of proportions, our approach was able to generate a considerably large relative positive effect, leading to better ideas.
For (the majority vote), out of total ideas collected, ideas were judged as good. Again, we start by looking only at seeds that appeared across all three conditions ( such seeds). This leaves good ideas. The proportion of good ideas in our condition was (). Next-up was the random baseline with (), and finally the TF-IDF baseline achieved (N=), with . By looking at the more conservative majority-vote threshold, the observed effect of our method only increases.
Looking only at seeds that appeared across all conditions was a basic way to make sure we cancel out possible confounding factors. A more refined way is attempting to model these effects and condition on them, as follows.
We are interested in the likelihood that a given idea is good, or pr(good), as a function of inspiration condition. However, ideas are not independent: each participant generated multiple ideas, and ideas were proposed for different seeds. Failing to account for these dependencies would lead to inaccurate estimates of the effects of the inspirations: some participants may be better at generating ideas than others, while some seeds might be more easy/difficult than others. Therefore, we used a generalized linear mixed model, with a fixed effect of inspiration condition, and random effects of participant and seed (to model within-participant and within-seed dependencies between ideas).
For , our resulting model (with fixed effect of inspiration condition) yields a significant reduction in variance compared to a null model with no fixed effects, Likelihood ratio , . The model also yields a reduction in Akaike Information Criterion (AIC), from in the null model to , indicating that the improved fit to the data is not due to overfitting.
For , the model also yields a significant drop in variance compared to a null model, Likelihood ratio , , with AIC dropping from in the null model to .
As Figure 4 shows, our method led to a significantly higher probability for good ideas. For , pr(Good) = , confidence interval = [, ] in our condition. TF-IDF had pr(Good) = [, ], and random had pr(Good) = [, ]. The advantages of the analogy condition over each baseline are both substantial and statistically significant, B = , vs. TF-IDF, and B = , vs. random. For , we had pr(Good) = , [, ]. TF-IDF had pr(Good) = [, ], and random had pr(Good) = [, ], B = , vs. TF-IDF, and B = , vs. random.
Note that confidence intervals for the probability estimates are relatively wide (more so, unsurprisingly, for ). Replications of this experiment, possibly with more data, could yield results somewhere in between, with more precise estimates on the true size of the effect. The main take-away of this study is that our approach yields a reliable increase in participants’ creative ability.
## 5. Discussion and Conclusion
In this paper, we sought to develop a scalable approach to finding analogies in large, messy, real-world datasets. We explored the potential of learning and leveraging a weak structural representation (i.e., purpose and mechanism vectors) for product descriptions. We leverage crowdsourcing techniques to construct a training dataset with purpose/mechanism annotations, and use an RNN to learn purpose and mechanism vectors for each product. We demonstrate that these learned vectors allow us to find analogies with higher precision than traditional information-retrieval similarity metrics like TF-IDF, LSA, GloVe and LDA.
Our ideation evaluation experiment further illustrates the effectiveness of our approach: participants had a higher likelihood of generating good ideas for the redesign ideation task when they received inspirations sampled by our analogy approach (tuned to be similar in purpose, but different in mechanism), compared to a traditional (TF-IDF) baseline or random sampling approach. From a psychological perspective, the benefits of our inspirations are likely due to our approach’s superior ability to sample diverse yet still structurally similar inspirations, since diversity of examples is a known robust booster for creative ability (Chan and Schunn, 2015). The TF-IDF approach yielded inspirations likely to be relevant, but also likely to be redundant and homogeneous, while the random sampling approach yields diversity but not relevance.
While moving to a “weak” structural representation based on purpose and mechanism significantly increased the feasibility of analogy-finding, extensions may be necessary to generalize to other domains besides product descriptions. For example, our purpose and mechanism vectors did not distinguish between higher and lower level purposes/mechanisms, or core/peripheral purposes/ mechanisms, and also did not encode dependencies between particular purposes/mechanisms. These are potentially fruitful areas for future work and may be especially important when moving from relatively simple product descriptions to more complex data such as scientific papers, in which purposes and mechanisms can exist at multiple hierarchical levels (e.g., “accelerate innovation” vs. “learn a vector representation of the purpose of an item”). More generally, we believe exploring the tradeoffs between degree of structure, learnability (including costs of generating training data, accuracy, and generalizability) and utility for augmenting innovation could lead to interesting points in the design space that could have both theoretical and practical value.
Acknowledgments. The authors thank the anonymous reviewers for their helpful comments. This work was supported by NSF grants CHS-1526665, IIS-1149797, IIS-1217559, Carnegie Mellon’s Web2020 initiative, Bosch, Google, ISF grant 1764/15 and Alon grant. Dafna Shahaf is a Harry&Abe Sherman assistant professor.
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# Need help converting units, momentum of quark question
1. Jan 2, 2013
### DunWorry
1. The problem statement, all variables and given/known data
A proton has a diameter of about 1fm. Estimate the minimum momentum in units of MeV/C of a quark confined in a proton.
2. Relevant equations
3. The attempt at a solution
I used the uncertainty relation $\Delta$P$\Delta$X ~ $\frac{\hbar}{2}$
$\Delta$P ~ $\frac{\hbar}{2 x 1x10^{-15}}$ = 5.276x10$^{-20}$ kgm/s
I'm unsure how to put this into MeV/C
thanks
2. Jan 2, 2013
### Staff: Mentor
You can convert kg in J/c^2 and that to MeV/c^2. Isolate a factor of c (based on m/s) and use this to cancel one c in the denominator.
Alternatively, use ℏ given in MeV/c * length.
3. Jan 2, 2013
### DunWorry
Hmmmm I tried 5.276x10$^{-20}$ x (3x10$^{8}$)$^{2}$ = 4.7484x10$^{-3}$ Joules / c$^{2}$ Then to convert to eV I divided by 1.6x10$^{-19}$
And I ended up getting 2.96775x10$^{16}$eV / C. the answer should be around 200 MeV/C
4. Jan 2, 2013
### Staff: Mentor
Did you consider m/s -> c?
If I divide your value by 3*108, I get 108eV/c or ~100MeV/c.
5. Jan 3, 2013
### DunWorry
Hmm that seems more correct, did I do the sum incorrectly?
6. Jan 3, 2013
### Staff: Mentor
I don't see a sum, but a missing factor of 3*10^8 would explain the difference.
7. Jan 3, 2013
### HallsofIvy
Staff Emeritus
I suspect that Dunworry is British (or learned "British" English) and they, for some strange reason, use the word "sum" to refer to any arithmetic calcluation!
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# Diff of /trunk/doc/cookbook/example01.tex
revision 3369 by jfenwick, Tue Oct 26 03:24:54 2010 UTC revision 3370 by ahallam, Sun Nov 21 23:22:25 2010 UTC
# Line 11 Line 11
11 % %
12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13
\begin{figure}[h!]
\centerline{\includegraphics[width=4.in]{figures/onedheatdiff001}}
\caption{Example 1: Temperature differential along a single interface between
two granite blocks.}
\label{fig:onedgbmodel}
\end{figure}
14 \section{Example 1: One Dimensional Heat Diffusion in Granite} \section{Example 1: One Dimensional Heat Diffusion in Granite}
15 \label{Sec:1DHDv00} \label{Sec:1DHDv00}
16
17 The first model consists of two blocks of isotropic material, for instance The first model consists of two blocks of isotropic material, for instance
18 granite, sitting next to each other. granite, sitting next to each other (\autoref{fig:onedgbmodel}).
19 Initial temperature in \textit{Block 1} is \verb|T1| and in \textit{Block 2} is Initial temperature in \textit{Block 1} is \verb|T1| and in \textit{Block 2} is
20 \verb|T2|. \verb|T2|.
21 We assume that the system is insulated. We assume that the system is insulated.
# Line 31 Intuition tells us that heat will be tra Line 24 Intuition tells us that heat will be tra
24 cooler one until both cooler one until both
25 blocks have the same temperature. blocks have the same temperature.
26
27 \begin{figure}[ht]
28 \centerline{\includegraphics[width=4.in]{figures/onedheatdiff001}}
29 \caption{Example 1: Temperature differential along a single interface between
30 two granite blocks.}
31 \label{fig:onedgbmodel}
32 \end{figure}
33
34 \subsection{1D Heat Diffusion Equation} \subsection{1D Heat Diffusion Equation}
35 We can model the heat distribution of this problem over time using the one We can model the heat distribution of this problem over time using the one
36 dimensional heat diffusion equation\footnote{A detailed discussion on how the dimensional heat diffusion equation\footnote{A detailed discussion on how the
# Line 668 The last statement generates the plot an Line 668 The last statement generates the plot an
668 As expected the total energy is constant over time, see As expected the total energy is constant over time, see
669 \reffig{fig:onedheatout1}. \reffig{fig:onedheatout1}.
670
671 \begin{figure} \begin{figure}[ht]
672 \begin{center} \begin{center}
673 \includegraphics[width=4in]{figures/ttblockspyplot150} \includegraphics[width=4in]{figures/ttblockspyplot150}
674 \caption{Example 1b: Total Energy in the Blocks over Time (in seconds)} \caption{Example 1b: Total Energy in the Blocks over Time (in seconds)}
# Line 723 want to substitute a value into the name Line 723 want to substitute a value into the name
723 and and
724 \item \verb d means that the value to substitute will be a decimal integer. \item \verb d means that the value to substitute will be a decimal integer.
725 \end{itemize} \end{itemize}
726
727 To actually substitute the value of \verb|i| into the name write \verb|%i| after To actually substitute the value of \verb|i| into the name write \verb|%i| after
728 the string. the string.
729 When done correctly, the output files from this command will be placed in the When done correctly, the output files from this command will be placed in the
Legend:
Removed from v.3369 changed lines Added in v.3370
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CS 440/ECE 448
Margaret Fleck
## Reinforcement Learning 1
Gridworld pictures are from the U.C. Berkeley CS 188 materials (Pieter Abbeel and Dan Klein) except where noted.
## Reinforcement Learning
Reinforcement learning is a technique for deciding which action to take in an environment where your feedback is delayed and your knowledge of the environment is uncertain. Consider, for example, a driving exam. You excute a long series of maneuvers, with a stony-faced examiner sitting next to you. At the end (hopefully) he tells you that you've passed. Reinforcement Learning can be used on a variety of AI tasks, notably board games. But it is frequently associated with learning how to control mechanical systems. It is often one component in an AI system that also uses other techniques that we've seen.
For example, we can use reinforcement learning (plus neural networks) to learn how to park a car. The car starts off behaving randomly, but is rewarded whenever it parks the car, or comes close to doing so. Eventually it learns what's required to get this reward.
## Markov Decision Processes
The environment for reinforcement learning is typically modelled as a Markov Decision Process (MDP). The basic set-up is an agent (e.g. imagine a robot) moving around a world like the one shown below. Positions add or deduct points from your score. The Agent's goal is to accumulate the most points.
It's best to imagine this as a game that goes on forever. Many real-world games are finite (e.g. a board game) or have actions that stop the game (e.g. robot falls down the stairs). We'll assume that reaching an end state will automatically reset you back to the start of a new game, so the process continues forever.
Actions are executed with errors. So if our agent commands a forward motion, there's some chance that they will instead move sideways. In real life, actions might not happen as commanded due to a combination of errors in the motion control and errors in our geometrical model of the environment.
## Mathematical setup
Eliminating the pretty details gives us a mathematician-friendly diagram:
Our MDP's specification contains
• set of states $$s \in S$$
• set of actions $$a \in A$$
• reward function R(s)
• transition function P(s' | s,a)
The transition function tells us the probability that a commanded action a in a state s will cause a transition to state s'.
Our solution will be a policy $$\pi(s)$$ which specifies which action to command when we are in each state. For our small example, the arrows in the diagram below show the optimal policy.
## The reward function
The reward function could have any pattern of negative and positive values. However, the intended pattern is
• A few states have big rewards or negative consequences.
• The rest of the states have some small background reward, often constant across all of these background states.
For our example above, the unmarked states have reward -0.04.
The background reward for the unmarked states changes the personality of the MDP. If the background reward is high (lower right below), the agent has no strong incentive to look for the high-reward states. If the background is strongly negative (upper left), the agent will head aggressively for the high-reward states, even at the risk of landing in the -1 location.
## What makes a policy good?
We'd like our policy to maximize reward over time. So something like
$$\sum_\text{sequences of states}$$ P(sequence)R(sequence)
R(sequence) is the total reward for the sequence of states. P(sequence) is how often this sequence of states happens.
However, sequences of states might be extremely long or (for the mathematicians in the audience) infinitely long. Our agent needs to be able to learn and react in a reasonable amount of time. Worse, infinite sequences of values might not necessarily converge to finite values, even with the sequence probabilities taken into account.
So we make the assumption that rewards are better if they occur sooner. The equations in the next section will define a "utility" for each state that takes this into account. The utility of a state is based on its own reward and, also, on rewards that can be obtained from nearby states. That is, being near a big reward is almost as good as being at the big reward.
So what we're actually trying to maximize is
$$\sum_\text{sequences of states}$$ P(sequence)U(sequence)
U(sequence) is the sum of the utilities of the states in the sequence and P(sequence) is how often this sequence of states happens.
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# BMI Calculator
Not Reviewed
BMI =
Tags:
Rating
ID
Zoozoodude.BMI Calculator
UUID
ec2dac73-7138-11e5-a3bb-bc764e2038f2
Body mass index (BMI) is a way to calculate the amount of fat in a person's body.
Below is a table of the current health of the individual with the given values:
BMIWeight Health
Below 18.5Underweight
18.5 – 24.9Normal or Healthy Weight
25.0 – 29.9Overweight
30.0 and AboveObese
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NEW
New Website Launch
Experience the best way to solve previous year questions with mock tests (very detailed analysis), bookmark your favourite questions, practice etc...
1
### IIT-JEE 1998
If $$f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},$$ for every real number $$x$$, then the minimum value of $$f$$
A
does not exist because $$f$$ is unbounded
B
is not attained even though $$f$$ is bounded
C
is equal to 1
D
is equal to -1
2
### IIT-JEE 1998
The number of values of $$x$$ where the function
$$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$$ attains its maximum is
A
$$0$$
B
$$1$$
C
$$2$$
D
infinite
3
### IIT-JEE 1997
If $$f\left( x \right) = {x \over {\sin x}}$$ and $$g\left( x \right) = {x \over {\tan x}}$$, where $$0 < x \le 1$$, then in this interval
A
both $$f(x)$$ and $$g(x)$$ are increasing functions
B
both $$f(x)$$ and $$g(x)$$ are decreasing functions
C
$$f(x)$$ is an increasing functions
D
$$g(x)$$ is an increasing functions
4
### IIT-JEE 1995 Screening
The slope of the tangent to a curve $$y = f\left( x \right)$$ at $$\left[ {x,\,f\left( x \right)} \right]$$ is $$2x+1$$. If the curve passes through the point $$\left( {1,2} \right)$$, then the area bounded by the curve, the $$x$$-axis and the line $$x=1$$ is
A
$${5 \over 6}$$
B
$${6 \over 5}$$
C
$${1 \over 6}$$
D
$$6$$
### Joint Entrance Examination
JEE Main JEE Advanced WB JEE
### Graduate Aptitude Test in Engineering
GATE CSE GATE ECE GATE EE GATE ME GATE CE GATE PI GATE IN
NEET
Class 12
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## Introduction
It is observed that the reported MBI systems used different types of antennas for identifying the target tumor object. The reconstructed images from the systems are low resolution based, noisy, and blurry. As a result, tumor identification by non-expert physicians and radiologists can be difficult. These problems have occurred due to deficiency of antenna’s radiation directivity, low gain, incapability of signal penetration, and high SAR. Also, the systems are examined to detect only a single tumor. In most cases, both benign and malignant tumors can be formed in the brain. In those cases, the existing systems can be failed to detect both tumors. Therefore, there is a demand to implement an experimental MBI system by utilizing a significant wideband antenna that can detect benign and malignant tumor in the brain in the early stage. So, an efficient wideband antenna is required in an MBI system that has high gain and efficiency, directional radiation capability for signal penetration, high bandwidth, and low SAR. In this research, we developed an MBI system by utilizing a 3D stacked antenna array that can detect both tumors. The novelties of this research are as follows:
1. 1.
According to authors best knowledge, this is the first research work, where a new metamaterial-loaded three-dimensional (3D) stacked wideband antenna is proposed for detecting the two tumors (i.e., benign and malignant tumor) in microwave brain imaging system.
2. 2.
The metamaterial (MTM) array is utilized in staked layers of the antenna to enhance the bandwidth, realized gain, efficiency, and capability of radiation directivity for adequate signal penetration to the brain. In addition, the new designed MTM unit cell has reduced the SAR and improved the fidelity factor that ensures less distortion for receiving the backscattered signals.
3. 3.
Implemented a nine-antenna array-based portable microwave brain imaging system that can reconstruct noiseless and high-resolution brain images for detecting the benign and malignant tumors.
4. 4.
Fabricated a six-layered tissue-mimicking head phantom model, including tumors, to verify the system performance.
5. 5.
Compared the outcomes of the implemented system with the state-of-the-art MBI system to ensure better performance of the proposed antenna.
The remaining part of the research is organized as follows: The proposed antenna design and analysis are explained in “Metamaterial loaded 3D stacked antenna design” section. Performance analysis of the proposed antenna is discussed in “Antenna performance analysis” section. “Sensitivity measurements of the head model” section discussed the sensitivity analysis of the head model by the antenna. A detailed experimental investigation of the microwave brain imaging system, phantom fabrication and measurement, and image reconstruction results are explained in “Microwave brain imaging results and discussion” section and concluded in “Conclusion” section.
## Metamaterial loaded 3D stacked antenna design
### Antenna structure and MTM unit cell analysis
The schematic diagram of the metamaterial (MTM) loaded 3D stacked antenna’s structure is depicted in Fig. 1. The antenna consists of three layers: Top Layer (TL), Middle Layer (ML), and Bottom Layer (BL). The MTM array unit cells are used in three layers. The 2 mm thickness of the air gap is considered between TL and ML, ML and BL. The Rogers RT5880 substrate (Relative permittivity εr = 2.2, loss tangent δ = 0.0009, thickness h1 = h2 = 1.575 mm) material is used in TL and ML, whereas the Rogers RO4350B substrate (Relative permittivity εr = 3.66, loss tangent δ = 0.0037, thickness h3 = 1.524 mm) material is used in BL. The ML is attached with the TL, and the BL is attached with the ML by using a thickness of 2 mm double-sided foam tape. The main radiating patch and feed line are designed on the TL. A 50Ω SMA port is connected to the feeding line that feds the signal to the antenna. The antenna’s feed line width is responsible for 50Ω impedance matching. The spider net-shaped MTM array is employed on the TL, ML, and BL to enhance the antenna’s performance. A 1 × 4 MTM array (m1 to m4) is used on the top layer and backside of the middle layer, and a 3 × 2 MTM array (m1 to m6) is used on the backside the bottom layer to enhance the radiation directionality, gain, and bandwidth of the antenna. The proposed antenna’s overall 3D layer-based stacked structure layout is illustrated in Fig. 1a, and its model is illustrated in Fig. 1b. The three layers of the antenna, including two air gaps (a1 and a2) made it a three-dimensional (3D) stacked antenna structure. The stacked structure is responsible for the high performance of the antenna. The spider net-shaped MTM unit cell’s geometric structure and simulated effective parameters are depicted in Fig. 2. The MTM unit cell is depicted in Fig. 2a. The simulation is performed by the FDS (Frequency-domain solver) based simulator CST (Computer Simulator Tool) software to design, investigate, and analyze the cell's properties. For achieving the optimal electromagnetic field distribution, the MTM unit cell is positioned between the + Z axis and -Z axis waveguide ports and powered towards the Z-axis by the wave.
The PEC (perfect electric conducting) and PMC (perfect magnetic conducting) walls are imposed on ± x and ± y directions of the unit cell to emulate the periodic nature of the cell35. The unit cell is simulated by using both RT5880 and RO4350B substrate materials and analysis of the effective parameters. The Nicolson-Ross-weir (NRW) method36 is utilized to extract the effective parameters (real and imaginary) from the scattering parameters as follows:
$$Permittivity\,(\varepsilon_{r} ) = \frac{2}{{jk_{0} d}} \times \left( {\frac{{1 - S_{21} + S_{11} }}{{1 + S_{21} - S_{11} }}} \right)$$
(1)
$$Permeability\,(\mu_{r} ) = \frac{{j2S_{11} }}{{jk_{0} d}} + \mu {}_{0}$$
(2)
$$Refractive\,Index\,(\eta_{r} ) = \frac{2}{{jk_{0} d}}\sqrt {\frac{{(S_{21} - 1)^{2} - S_{11}^{2} }}{{(S_{21} + 1)^{2} - S_{11}^{2} }}}$$
(3)
$$k_{0} = \omega \sqrt {\mu_{0} \varepsilon_{0} }$$
(4)
where S11 denotes the reflection coefficient, S21 denotes the transmission coefficient, k0 denotes the wave number, and d denotes the substrate thickness. The frequency range for the MTM in the proposed antenna is 1.30–3.20 GHz. The real and imaginary permittivity of the unit cell is shown in Fig. 2b. In Fig. 2b, the −ve real permittivity frequency ranges are 2.65–2.86 GHz for RT5880 and 2.57–2.74 GHz for RO4350B, whereas the −ve imaginary permittivity frequency ranges are 1.30–2.65, 2.97–3.12 GHz for RT5880 and 1.30–2.50 GHz, 2.91–3.12 GHz for RO4350B. The real and imaginary permeability and refractive index of the unit cell for both RT5880 and RO4350B are shown in Fig. 2c.
The real permeability of the unit cell shows positive permeability for both substrates, whereas the imaginary permeability of the unit cell displays negative permeability in the range of 2.69–2.95 GHz for RT5880 and 2.57–2.84 GHz for RO4350B. In contrast, the −ve real and imaginary frequency ranges are 2.69–2.84 GHz for RT5880, while 2.61–2.72 GHz for RO4350B substrate. However, the negative properties of the proposed MTM unit cell are the special features that might improve antenna performances. When the MTM unit cell is employed in the antenna, it enhances the bandwidth, radiation directionality, realized gain, and efficiency.
### Proposed antenna geometry and design evolution analysis
The wideband antenna is required in microwave brain imaging (MBI) systems with special features such as higher gain, efficiency, and directional radiation characteristics15,16,17,18,26,37. A wideband antenna for a microwave brain imaging system must be capable of producing a wideband operating frequency between 1 and 4 GHz with higher gain, higher efficiency, and high fidelity. If the antenna operates at wideband frequencies with the mentioned features, the signals are able to adequately penetrate to the brain tissues and the backscattered signals from the brain tissues are easily received by the receiving antennas of the system. As a result, desired high resolution-based microwave brain image generation is possible. However, frequencies below 1 GHz cause the image resolution to decrease, resulting in a noisy and indistinct image. Thus, wideband frequency is required in MBI system for detecting abnormalities (e.g., tumors) in the brain. In this article, a new MTM-loaded three-dimensional (3D) stacked wideband antenna is designed, which achieves microwave brain imaging features. The geometry and evolution of the proposed antenna are depicted in Fig. 3. The antenna consists of mainly three substrate layers: Top layer (TL), Middle layer (ML), and Bottom layer (BL), including two air gaps. The air gaps are considered between the TL and ML, ML and BL. The optimized parameters with their values are presented in Table 1. The antenna is printed on cost-effective Rogers substrate materials. The TL and ML are printed on RT5880 substrate, and the BL is printed on RO4350B substrate. In the initial step, a main radiating patch and feed line are designed on TL. Two triangle-shaped patches are attached in opposite directions to make the main radiating patch. The dimension of TL is 50 × 40 mm2. The partial ground (l3 × l4 mm2) and a rectangle-shaped parasitic element (l1 × l2 mm2) are attached to the backside of the TL. The lowest operating frequency (fl), patch width (l), and feed line width (fw) are calculated by the following equations26,38,39:
$$l = \frac{2c}{{3f_{r} \sqrt {\varepsilon_{r} } }}$$
(5)
$$f_{w} = \frac{7.48 \times h}{{e^{{\left( {z_{0} \frac{{\sqrt {\varepsilon_{r} + 1.41} }}{87}} \right)}} }} - 1.25 \times t$$
(6)
$$\varepsilon_{eff} = \frac{{(\varepsilon_{r} + 1)}}{2} + \frac{{(\varepsilon_{r} - 1)}}{2}\left[ {1 + \frac{10 \times h}{l}} \right]^{ - 0.5}$$
(7)
$$f_{l} = \frac{c}{{2 \times l\sqrt {\varepsilon_{eff} } }}$$
(8)
where fr denotes the resonance frequency, h denotes the thickness of the substrate, t denotes the copper thickness, c denotes the speed of light in free space, εeff denotes the substrate’s effective dielectric constant. The CST simulator software is used to optimize the geometric parameters for achieving the required band and special features of the antenna. The feed line width is responsible for 50Ω impedance matching.
Figure 3 depicts the evolution of the proposed antenna. The simulated reflection coefficient |S11|, and efficiency vs. realized gain curves of the stacked antenna's evolution are illustrated in Fig. 4a, b, respectively. Figure 3a shows the top view of antenna-1 (TL without MTM). It is observed from Fig. 4 that the frequency band of antenna-1 is 1.94–2.94 GHz with a resonance frequency at 2.29 GHz and the maximum realized gain is 3.36dBi at 2.96 GHz with 79.13% efficiency. Thereafter, when a 1 × 4 MTM array is applied at top of the patch in antenna-2 (TL with MTM) shown in Fig. 3b, the frequency band increases towards both lower and upper frequencies due to MTM characteristics and achieves a frequency band of 1.89–3.04 GHz with a resonance at 2.24 GHz. The attained maximum realized gain of antenna-2 is 3.69 dBi at 2.96 GHz with 81.32% efficiency. In the second step, the ML (antenna-3) without MTM array is placed at 2 mm away and attached with antenna-2 by using double-sided foam tape shown in Fig. 3c. The air gap (a1) between antenna-2 and antenna-3 is 2 mm. The dimension of ML is 43 × 40 mm2 (L1 × W1). The observed operating band, gain, and efficiency of the antenna-3 are 1.69–3.10 GHz, 5.38 dBi at 2.96 GHz, and 87.84%, respectively. When another 1 × 4 MTM array (m1–m4) and a rectangular-shaped (l5 × l6) parasitic element are attached at the backside of the ML (antenna-4), as shown in Fig. 3d, the gain is increased. In addition, the radiation directivity and bandwidth are also increased. The operating frequency band is 1.61–3.10 GHz, with two resonances at 2.21 GHz and 2.92 GHz. The recorded simulated realized gain is 5.68 dBi at 2.96 GHz and 88.59% efficiency. The right view, the perspective view of the two-layered stacked antenna (antenna-4) is shown in Fig. 3d. In the third step, the BL (antenna-5) without an MTM array is positioned at 2 mm away from the ML, which is shown in Fig. 3e. The air gap (a2) between antenna-4 and antenna-5 is 2 mm. The dimension of BL is 43 × 40 mm2 (L2 × W2). The achieved frequency band, realized gain, and efficiency of antenna-5 are 1.55–3.06 GHz, 5.54 dBi, and 93.89% at 2.96 GHz, respectively. Finally, when a 3 × 2 MTM array (m1–m6) is applied at the backside of the BL (proposed antenna), the proposed antenna shows better outcomes than other antennas in terms of bandwidth, radiation directionality, realized gain, and efficiency.
The resultant simulated operating band of the proposed antenna is 1.43–3.12 GHz with a fractional bandwidth (FBW) of 74.45% for the center frequency of 2.27 GHz. It also produced three resonances at 2.24 GHz, 2.66 GHz, and 2.84 GHz. The back, right, and perspective view of the three layered-based proposed antenna is illustrated in Fig. 3f. The recorded maximum realized gain and efficiency are 6.74 dBi and 95.06% at 2.96 GHz, respectively, shown in Fig. 4b. The overall performance summary of the antenna evolutions is presented in Table 2. In the antenna design, it is noticeable that the overall performance of the antenna is enhanced due to the MTM array element effectiveness in all layers and stacked architecture. Because the negative permittivity characteristics of MTM array elements enhance the overall bandwidth. Also, it produces extra resonances and improves radiation efficiency by resulting in an additional electromagnetic coupling between radiating patches and layers. In addition, MTM unit cells increase the electrical length and enhance the current flow on the surface of the layer. As a result, it produces strong electrical coupling between layers of the antenna, which leads to improved gain, radiation directivity, and efficiency.
### Parametric analysis
In Table 1, 22 parameters are presented, which are used to demonstrate the overall antenna structure. The antenna consists of three layers, and a feeding line is attached to the top layer. The ground plane is attached to the backside of the top layer. Especially, two air gaps (a1 and a2), the length of the ground plane (l4), and used MTM array elements in the layers are responsible for getting desired frequency band. The air gaps (a1 and a2) between layers are significant parameters to make the antenna a 3D stacked structure for better performance. The reflection coefficient is changed due to variations of air gaps between the layers. So, the air gaps must optimize to check the performance of the antenna. In simulation, we optimize the air gaps by using scientific formula. For air gaps optimization analysis, the side view simulation structure of the antenna is shown in Fig. 5a and corresponding side layout is illustrated in Fig. 5b. It is notable that the dielectric properties of the three layers (εr1, εr2, and εr3) and air (εair) are known parameters.
Also, thickness of height (h1, h2, and h3) of the used substrates are also known parameters, which are shown in Fig. 5b, but the conductivity of the substrates is different due to the different substrate layers properties and conductance. Thus, it effects the air gap analysis. In this case, we have to calculate the optimization value of two air gaps: a1 and a2, which are unknown parameters. According to Fig. 5b, we can calculate optimize equation for air gaps a1 by the following formula:
$$H_{1} = h_{1} + h_{2} + a_{1}$$
(9)
here H1 is the total distance or thickness of TL, ML, and air gap a1. Then, we find out the value of a1 from the Eq. (9) as follows:
\begin{aligned} a_{1} & = H_{1} - h_{1} - h_{2} \\ & = \left( {1.575 + 1.575 - \frac{{4\pi \sigma_{1} }}{{\omega_{1} }} - \frac{{4\pi \sigma_{2} }}{{\omega_{2} }}} \right) \\ \end{aligned}
(10)
where σ1 and σ2 are the conductivity of TL and ML respectively. The TL and ML also have conductive parts on the top and bottom of the substrate layers. These conductivities are same because both are same substrate. In addition, the ω1 and ω2 are the operating frequency within the frequency band 1.43–3.12 GHz and are also same because both are same substrate. However, we can choose arbitrarily any resonance frequency from this operating band. Now, arbitrarily we can calculate the optimization air gap a1 formula from Eq. (10) as follows:
$$a_{1} = 3.15 - 2 \times \frac{{4\pi \sigma_{1} }}{{\omega_{1} }};\quad \because \;\sigma_{1} = \sigma_{2} \;and\;\omega_{1} = \omega_{2} {\mkern 1mu}$$
for any particular frequency band. Hence, the optimization formula for the air gap a1 as follows:
$$a_{1} = 3.15 - \frac{{8\pi \sigma_{1} }}{{\omega_{1} }}.$$
(11)
Similarly, we can calculate optimization equation for air gap a2 by the following formula:
$$H_{2} = h_{2} + h_{3} + a_{2}$$
(12)
Here H2 is the total distance or thickness of ML, BL, and air gap a2. Then, we find out the value of a2 from the Eq. (12) as follows:
\begin{aligned} a_{2} & = H_{2} - h_{2} - h_{3} \\ & = \left( {1.575 + 1.524} \right) - \frac{{4\pi \sigma_{2} }}{{\omega_{2} }} - \frac{{4\pi \sigma_{3} }}{{\omega_{3} }} \\ \end{aligned}
(13)
where σ2 and σ3 are the conductivity of ML and BL respectively. The ML and BL also have conductive parts on the top and bottom of the substrate layers. These conductivities are different because both are different substrate. In addition, the ω2 and ω3 are the operating frequency within the frequency band 1.43–3.12 GHz and are also different because both are different substrate. However, we can choose arbitrarily any resonance frequency from this operating band. Now, arbitrarily we can calculate the optimization gap a2 formula from Eq. (13) as follows:
$$a_{2} = 3.1 - 4\pi \left( {\frac{{\sigma_{2} }}{{\omega_{2} }} - \frac{{\sigma_{3} }}{{\omega_{3} }}} \right).$$
(14)
So, the Eqs. (11) and (14) are the optimization formula for air gaps a1 and a2 respectively. It is also seen that these formulas are analyzed and used during simulation. The simulated results give the optimal value as practically 2 mm for both air gaps. The simulated analysis results are illustrated in Fig. 6a. When no air gaps (a1 = 0 and a2 = 0) are considered, then the antenna shows a narrow band with a low reflection coefficient. If the middle air gap is gradually increased, the operating band is increased due to the effect of the substrate and MTM unit cell, but it is saturated at a certain value. When a1 = a2 = 1 mm, then the operating frequency band of the antenna is 1.64–3.13 GHz, with two resonances at 2.20 GHz and 2.90 GHz providing a low reflection coefficient. When a1 = a2 = 2 mm, the antenna achieved the highest operating band from the 3 mm air gap due to the air gap optimization.
The parametric analysis by considering the variations in the length of the air gaps, ground plane, and MTM unit cell are presented in Fig. 6a–c respectively. When the length of the ground plane is varied, and the remaining parameters remain constant, the reflection coefficient is demonstrated in Fig. 6b. When l4 = 4 mm, the achieved operating band is 1.77–2.95 GHz with a resonance at 2.05 GHz, whereas when l4 = 5 mm, the achieved operating band is 1.80–3.02 GHz with a resonance at 2.11 GHz. When l4 = 6 mm, the attained operating band is 1.71–3.08 GHz with a resonance at 2.18 GHz, whereas when l4 = 8 mm, the accomplished operating band is 1.87–3.09 GHz with a resonance at 2.33 GHz. If l4 = 9 mm is set, then the achieved operating band is 2.06–3.03 GHz with very low reflection coefficient without resonance frequency. But if the length is set l4 = 7 mm, then the antenna shows better outcomes than others, and the desired band is 1.43–3.12 GHz with three resonances at 2.24 GHz, 2.66 GHz, and 2.84 GHz, respectively. So, it is observed that if the ground plane length is increased the reflection coefficient is decreased.
The MTM unit cell is an important element for the antenna because the MTM array elements can change and enhance the antenna performance and increase the bandwidth, radiation directivity, and gain. In this analysis, we only showed the MTM cell effects when used in the bottom layer of the antenna. The reflection coefficient of the antenna by using the MTM array in BL layers is depicted in Fig. 6c. Initially, when no MTM unit cell is used in layers of the antenna, the antenna shows a resonance at 2.35 GHz, and the functioning band is 1.96–3.04 GHz. When 3 × 3 MTM array elements are applied to the backside of the bottom layer, the antenna's operating frequency is 1.73–3.09 GHz with two resonances at 2.27 GHz and 2.86 GHz, respectively, providing a high reflection coefficient. The operating frequency is shifted towards the lower frequencies due to MTM effects, but the resonances are distorted at the upper frequencies. The distorted resonances are also observed when 4 × 4 MTM array elements are applied and show a narrow operating band from others. But, if 3 × 2 MTM array elements are applied, then the antenna shows distorted less frequency band and it covers 1.43–3.12 GHz with three resonance at 2.24 GHz, 2.66 GHz and 2.84 GHz respectively.
## Antenna performance analysis
Initially, top, middle, and bottom layers of the antenna have been printed on proposed substrate materials. Figure 7 illustrates the fabricated top view and back view of three layers of the proposed antenna. The fabricated antenna consists of three layers: Top Layers (TL), Middle Layers (ML), and Botom Layers (BL). The complete fabrication process of the proposed antenna is depicted in Fig. 8a. The ML is attached at 2 mm away from the TL using double-sided foam tape, which shown in Fig. 8a. Then, the BL has also been attached at 2 mm away from the ML by using double-sided foam tape. The thickness of the double-sided foam tape is 2 mm. Finally, the fabrication process of the antenna is completed by attaching the BL with the ML. The back view, side view, and perspective view of the fabricated antenna is illustrated in Fig. 8b–d, respectively. The fabricated antenna prototype measurement is performed by using PNA (Power Network Analyzer, PNA-N5227A). At first, the PNA is turned on and the coaxial cable is connected with the port B of PNA. After that, the frequency is set to 1–4 GHz on the PNA. Later, the PNA is calibrated by the calibration kit within the frequency range of 1–4 GHz. After that, the measurement connector is connected with the coaxial cable and then fabricated antenna is connected tightly with the connector for measurement purposes. Then, the S-parameters or reflection coefficient is observed from the PNA and recorded for further uses. The antenna measurement process is portrayed in Fig. 8e.
The simulated and measured reflection coefficient is depicted in Fig. 9a. From Fig. 9a, it is observed that the simulated operating band is 1.43–3.12 GHz with 74.45% fractional bandwidth (FBW), whereas the measured operating band is 1.37–3.16 GHz with 79.20% FBW. It is seen in the measured result that the antenna generates three resonances at 2.39 GHz, 2.57 GHz, and 2.81 GHz under − 35 dB. The first resonance has shifted towards the upper frequency from 2.24 to 2.39 GHz (150 MHz). The second resonance has shifted towards a lower frequency from 2.66 to 2.57 GHz (90 MHz) third resonance is approximately the same as the simulated resonance. In Fig. 9a, it is seen that the there is a difference between simulated and measured results. This scenario may be happened due to the fabrication tolerance, calibration inaccuracy, and mutual coupling effect of layers. It is also seen that the measured reflection peak at first resonance is dipper compared to the simulated results. This minor discrepancy exists due to some factors: (i) changing loss tangent of substrate materials when measured in free environment, (ii) stacked structure of the antenna when the layers are attached using foam tape, (iii) mutual coupling effect between layers and MTM effectiveness. The loss tangent of substrate can be changed due to the change of temperature, and mutual coupling effect of MTM array always produce minor variations. As a result, it is shown difference between simulated and measured results and produce dipper reflection peak. However, the measured and simulated outcomes demonstrated good agreement between them. In addition, the Satimo near filed chamber has been used to perform the measurement of the radiation pattern, realized gain, and efficiency of the fabricated prototype. Figure 9b,c illustrate the antenna’s measured and simulated realized gain and efficiency, respectively. It is observed from Fig. 9b that the measured maximum realized gain is 6.67 GHz at 3.13 GHz, whereas the simulated maximum realized gain is 6.74 GHz at 2.96 GHz. Also, Fig. 9c investigated the maximum measured efficiency of 94%, whereas the simulated maximum efficiency is 95.06%. The simulated and measured outcomes exhibited good agreement between them in both gain and efficiency.
The simulated and measured far-field radiation pattern at 2.24 GHz is shown in Fig. 12a. Figure 12b illustrates the simulated and measured far-field radiation pattern at 2.66 GHz, and simulated and measured far-field radiation pattern at 2.84 GHz is shown in Fig. 12c. The measured and simulated radiation patterns showed good agreement between them in the entire operating band. Moreover, the antenna will be placed near the head model for imaging purposes; hence it is needed to verify the near field characteristics of the antenna. Thus, the antenna is measured at the near field lab. Figure 12d presents the near-field radiation measurement in the H-plane scenario. It is examined proposed prototype demonstrates a directional characteristic for 2.24 GHz, 2.66 GHz, and 2.84 GHz, respectively, and shows almost symmetrical for both far-field and near-field. Furthermore, the fidelity factor (FF) is calculated to investigate the antenna near field performances by applying two methods: (i) using E- field probs, and (ii) using transmitting and receiving antenna by considering the near-field distance between them in four scenarios. Typically, the FF is used to evaluate the level of distortion in the near-field region, which is defined by the correlation coefficient between the radiated received E-field pulses Erad in various directions (φ, θ) and the exciting Gaussian pulses Gt at the input of the antenna. In this work, the finite difference time domain (FDTD) method is utilized to investigate the level of distortion in both methods. Firstly, to observe the FF and time-domain amplitude responses around the antenna, E-field probes are situated around the antenna at distances of 50 mm from the center of the antenna and arranged in both the E-plane and H-plane with angular differences of 15° apart from each other. The FF at a specific direction is calculated as14,26:
$$FF(\varphi ,\theta ) = \max \frac{{\int\limits_{ - \infty }^{ + \infty } {E_{rad} (t,\varphi ,\theta ).G_{t} (t - \tau ,\varphi ,\theta )dt} }}{{\sqrt {\int\limits_{ - \infty }^{ + \infty } {\left| {E_{rad} (t,\varphi ,\theta )} \right|^{2} dt\int\limits_{ - \infty }^{ + \infty } {\left| {G_{t} (t,\varphi ,\theta )} \right|^{2} dt} } } }}.$$
(15)
Figure 13a,b depicts the radiated received E-field pulses in Near field regions at different angles. It shows the antenna radiates almost distortionless pulses. The calculated fidelity factors for E- and H-plane are illustrated in Fig. 14. It is seen from Fig. 14 that the FF is greater than 93% in both E and H-planes, and the maximum FF is 98.01% for H-plane and 96% for E-plane. Secondly, since the antenna will be utilized in the imaging system to receive scattered signals, where the distance of antenna to antenna is 100 mm, thus, it is needed to investigate the fidelity factor of the antenna at a different axis. In this work, four cases are considered: (i) Side by side at X-axis (SBS, X-axis), (ii) Side by side at Y-axis (SBS-Y-axis, (iii) Front to Front at Z-axis (FTF, Z-axis), and (iv) Back-to-back at Z-axis (BTB, Z-axis). The normalized transmitting and receiving signals for four cases are illustrated in Fig. 15a,b. It is examined that when an MTM unit cell is applied to the antenna in different layers, the antenna receives almost distortionless pulses than the without an MTM-based antenna. As a result, the proposed antenna (MTM-based) exhibits a higher fidelity factor for four cases. The calculated FFs are 84.42%, 86.24%, 90.93%, and 91.01% for SBS-X-axis, SBS-Y-axis, FTF-Z-axis, BTB-Z-axis without MTM respectively, whereas 93.41%, 94.52%, 95.20% and 98% for SBS-X-axis, SBS-Y-axis, FTF-Z-axis, BTB-Z-axis with MTM respectively. However, it is concluded that the proposed antenna is performed better in both far-field and near-field. Thus, it is a suitable candidate in the microwave brain imaging system, where the antenna has to be set up in a Back-to-Back (BTB, Z-axis) orientation to get better performance.
## Sensitivity measurements of the head model
In this section, we discuss the sensitivity analysis, which refers to investigating the antenna's scattering parameters (S-parameters) with a benign and a malignant tumor as brain abnormalities in the Hugo head model, signal penetration, and specific absorption rate (SAR). The Hugo head model is imported from the CST 2019 software’s voxel model. Different views and cutting plane views of the Hugo model are depicted in Fig. 16. It is notable that the impedance matching between antenna and Hugo head model is necessary for better imaging outcomes. So, the impedance matching is also investigated during simulation analysis. The stacked structure of the antenna is a combination of coupling and decoupling of patches and MTM elements with air gap between layers as a dielectric medium. Consequently, at any moment, in order to calculate the impedance between the transmitting antenna and the Hugo head model, the transmitting antenna with excitation at any point of the surrounding imaging setup can be analytically optimized. The Iteratively Corrected Coherence Factor Delay-Multiply-and-Sum (IC-CF-DMAS) imaging algorithm17 is used for impedance optimization by utilizing the Green’s function40. For instance, the real Green’s function does not reflect the insignificant surface current and voltage difference at an arbitrary point of any microstrip line40. The rectangular shape is judged for the antenna demonstrated in Fig. 17 assuming instantaneous transmitting antenna component and Hugo model for matching the impedance.
When defining any random surface point (∆sp) in the microstrip coupled line (MCL), the antenna's 3D structure plane is expected to be the same as in simulations, along with the antenna's length (L), width (W), and height (H). The effective permittivity (εeff), substrate permittivity (εsp), air permittivity (εair), and loss tangent (tanδ) considered between Hugo model and the antenna. So, random surface points (∆sp1 to ∆sp5) are decided on the MCL as the assessment location of current and voltage estimation for impedance matching. Rewrite the function as follows to distinguish it from the original Green's function and include precise current and voltage at a random MCL surface location in the simulated imaging setup with antenna and Hugo model:
$$Gn_{obs(1 \ldots 5)} = \frac{1}{LW}\sum\limits_{m = 0}^{\infty } {\sum\limits_{n = 0}^{\infty } {(2 - \delta_{m} )(2 - \delta_{n} ) \times \frac{{\varepsilon_{eff} \varepsilon_{air} \varepsilon_{sp} Cos(P_{x} x)Cos(P_{y} y)}}{{P_{mn}^{2} - P^{2} }}} }$$
(16)
where $$P_{mn}^{2} = P_{x}^{2} + P_{y}^{2}$$, $$P_{x} = (m\pi /L)$$, $$P_{y} = (n\pi /W)$$, and m, n = 0,1,2 … ∞ is the propagation mode deemed on the rapid transmission. The permittivity (µ) and antenna height (H) associated to higher-order frequency terms have been removed from the real functions because their convergence mechanisms slow down the calculation as a whole. The imaging algorithm now incorporates Eq. (16) and translates or postulates it as a different solution that uses coupling and decoupling lines for impedance matching. However, this technique works well when there are multiple radiating antennae or boundaries surrounding the Hugo model. If the boundary is ignored, the contribution from the excitation port in the electromagnetic field that reaches the observation point can be assumed, and effective dielectric parameters speed up the calculation accuracy of the provided imaging system for impedance matching.
A six-layered tissue-mimicking head phantom model with different layers and tumor(s) is fabricated to investigate the antenna's performance (described in “Microwave brain imaging results and discussion” section). The tumors are placed at different positions in the model. Figure 18a represents the head model with a single benign tumor, and Fig. 18b represents the head model with a benign and a malignant tumor. The simulated and measured (details are discussed in “Microwave brain imaging results and discussion” section) S-parameters of the head model with and without tumor using a single antenna are presented in Fig. 18c. The reflection coefficients are slightly decreased with resonance frequencies and shifted towards higher frequency. It is observed from Fig. 18c, there is slight difference between simulated and measured outcomes. This scenario can be occurred due to the change of head tissues and tumor dielectric properties. In this research, we have simulated a realistic Hugo head model, including tumors, where the thickness and dielectric properties of the tissues are fixed. On the other hand, the fabricated tissues thickness and dielectric properties are slightly changed due to fabrication tolerance (i.e., thickness and combination of ingredients). Also, the free space measurement, environmental temperature, time variant factors are affected the measured result. As a result, there is a difference between simulated and measured results, but remain stable within the required operating band because of the effectiveness of the MTM array. However, the antenna showed a good agreement between simulated and measured outcomes. In this work, the microwave signal penetration to the head is also investigated to evaluate antenna effectiveness. Figure 19 illustrates the E-and H-filed penetration inside the head model. The E-field and H-field distribution in the xz-plane at 2.24 GHz and 2.84 GHz are shown in Fig. 19a–d, respectively. Besides, The E-field and H-field distribution in the yz-plane at 2.24 GHz and 2.84 GHz are shown in Fig. 19e–h, respectively. It is observed from Fig. 19; that the antenna continues to show the directionality to the model, and the signal is able to penetrate the head tissues covering a two-thirds portion of the head.
Figure 20a illustrates the simulated nine-antenna array set up, where antenna 1 acts as a transmitter and the remaining eight antennas act as receivers. The received S-parameters without tumor (i.e., Healthy brain), with a single benign tumor, and a single benign and a malignant tumor are depicted in Fig. 20b,d, respectively. It is examined from the S-parameters that there is significant distortion of the backscattered signals for the benign and malignant tumors. These distortion differences happened due to the presence of high dielectric properties of the tumors and peak resonance frequencies algorithm. However, the nine-antenna array setup covers the complete area of the head, which carries all the sufficient information for reconstructing the brain images. Also, the specific absorption rate (SAR) is analysed because microwave radiation is extremely harmful to the human brain when exposed to the brain. So, the SAR analysis is an essential consideration for microwave brain imaging modalities to ensure operational safety. The SAR is measured by the following formula41:
$$SAR = \frac{{\left| {E_{f} } \right|^{2} .\sigma }}{{M_{d} }}$$
(17)
where $$E_{f}$$ represents the electric fields, $$M_{d}$$ represents the mass density, and $$\sigma$$ represents the conductivity of the human brain tissue. According to the IEEE radiation exposure standard regulations, the highest SAR must not exceed 1.6 W/kg for 1 gm and 2 W/kg for 10 gm of tissue41,42. In this research, 1mW power is applied as an input to the antenna (positions: 1, 3,5,7, and 8) and observed the SAR at 2.24 GHz, 2.66 GHz, and 2.84 GHz, respectively, for 1 gm and 10 gm of tissue. The measured SAR values are presented in Table 3. The investigation shows that the observed maximum SAR value is 0.0020 W/Kg for 1gm and 0.0018 W/kg for 10 gm of tissue with the proposed MTM loaded 3D stacked antenna at 2.66 GHz, which satisfies the IEEE radiation exposure limit. Also, it is realized from Table 3, that when MTM array elements are applied to the antenna, the SAR value is decreased. The observed highest SAR value is 0.0047 W/kg and 0.0029 W/kg for 1 gm and 10 gm of tissue without MTM array components, respectively, whereas the value is 0.0020 W/kg and 0.0018 W/kg for 1 gm and 10 gm of tissue for the proposed stacked antenna at 2.66 GHz. Therefore, it is concluded that the proposed staked antenna is applicable in microwave brain imaging systems to reduce SAR. In this research, it is noticeable that the SAR value is observed from different side of the head model by placing the antenna at different positions. It is also observed from Table 3, higher SAR value is recorded for higher gain of the antenna whereas lower SAR value is recorded for lower gain of the antenna. This circumstance is occurred due to the absorbed microwave energy by the head tissues and effectiveness of dielectric properties (i.e., permittivity and conductivity) of brain tissues. The head model is a multilayer structure consists of six layers. The permittivity of brain tissues is decreased with respect to increasing the frequency, while conductivity is increased with respect to increasing frequency. If the permittivity and conductivity is varied with the variations of frequency and gain of the antenna, the backscattered signals are also reflected and scattered at different manner. Therefore, when the gain of the antenna is low at lower frequencies, then the radiation directivity is partial directional. So, the microwave signal cannot fully penetrate to the brain tissues and the absorbed energy by the tissues is also low. Hence, the calculated SAR value is low. On the other hand, when the gain gradually increased at higher frequencies, then the radiation directivity is fully directional which is controlled by MTM array. Consequently, the radiated microwave signal can sufficiently penetrate to the brain tissues and higher energy absorbed by the brain tissues. As a result, the calculated SAR value is high, but it is safe for human brain and not exceed IEEE standard safety limit.
## Microwave brain imaging results and discussion
### Phantom fabrication process and measurement
Initially, a six-layered (i.e., Dura, CSF, Grey Matter, White Matter, Fat, and Skin) tissue-mimicking phantom, benign, and malignant tumor tissues are fabricated. According to the presented recipe in43, the brain layers and tumors are fabricated. The dielectric properties of the tumors (i.e., benign and malignant) are considered as presented value in44,45. The benign tumor is fabricated as almost circular with a regular shape, whereas the malignant tumor is fabricated as an elliptical and triangular irregular shape45. The required ingredients for phantom fabrication are presented in Tables 4 and 5. The brain tissues of the human head are comprised with four layers: DURA, CSF, White Matter, and Gray Matter. The permittivity and conductivity of real human brain tissues26 are presented in Table 6. The permittivity and conductivity are changed with respect to the changing the frequency. The permittivity is gradually decreased with increasing the frequency. On the other hand, the conductivity is gradually increased with increasing the frequency. In this research, the permittivity and conductivity of brain tissues are considered as a reference of the human head model. The measured values of fabricated tissues can be 3 to 5% slightly increased or decreased due to ingredients tolerance. However, the permittivity and conductivity of brain tissues at 2 GHz are presented in Table 6. The fabricated phantoms are shown in Fig. 21a. Later, the tissue-mimicking phantoms and tumors are inserted layer by layer into the 3D head model shown in Fig. 21b.
The dielectric properties of the fabricated tissues are measured by the dielectric probe kit KEYSIGHT 85070E and a power network analyzer (PNA, Model: PNA-L N5232A; 300 kHz to 20 GHz). The fabricated tissue-imitating head phantom's dielectric characteristics are measured using the open-ended coaxial probe technique. This method can be used for both in-vivo and ex-vivo measurements over a wide frequency range and is straightforward, non-destructive, and effective. However, the complicated heterogeneous structures and uneven surfaces in the homogenous structures are seen as limits for reliable measurements. The main limiting elements for measuring contents are calibration methods and measurement tools like PNA. For an accurate measurement, additional environmental aspects like temperature, humidity, and pressure as well as system elements like the cleanliness of the probe tip should be considered. The transmission line where the microwaves travel through the coaxial wire is cut off for the dielectric probe kit. Reflected signals are produced by the impedance mismatch between the probe and the targeted tissue sample, and these signals are then transformed into complicated permittivity values. Figure 22a,b depicts the initial calibration phase utilizing the VNA and coaxial probe with 25 cm3 of sterile water. Next, to guarantee enough contact between the sample and the coaxial probe while completing the measurements, each sample of the head phantom is sliced separately. The visual inspection of the inner and outer sections of the sample is reviewed to verify the consistency after the outside part of the sample is polished flat to ensure there is no gap between the coaxial probe and sample component. For collecting data with greater accuracy, the coaxial probe is positioned on the sample surface multiple times at random. The final assessment is then made by calculating the mean value from the many data points that were collected. The sample measurement setup picture of the fabricated phantom components is depicted in Fig. 22c–e. The benign and malignant tumors are inserted at different positions in the model to perform the measurement by the brain imaging system. The geometrical orientation of the head model filled with tissues is depicted in Fig. 23a. The benign and malignant tumors are inserted at different locations in the head model, which is depicted in Fig. 23b–f. The measured dielectric properties of the fabricated tissues are illustrated in Fig. 24.
### Microwave brain imaging system implementation and imaging results
The proposed microwave brain imaging (MBI) system is implemented to verify the performance. Figure 25 shows the overall implemented MBI system. The proposed system consists of an MTM-loaded 3D stacked nine antenna arrays, a custom-made half-cut elliptical-shaped helmet, a stepper motor, a portable stand, RF switch, microcontroller, and a PNA E8358A transceivers. The stepper motor is attached to the portable stand, which rotates clockwise with a 7.2° angle at every step to cover the whole (360°) area. The helmet is connected with the motor by the motor shaft. The diameter of the helmet is 250 mm. The antenna is attached inside the helmet by double-sided foam tape. The angular distance from antenna to antenna is 40° to cover whole are of the system. The antenna position is set 100 mm up from the bottom point of the helmet to adjust the phantom head position. The PNA is connected with the computer through the GPIB port, port A is connected with the transmitting antenna, and Port B is connected to RF switch for receiving the backscattered signals. The fabricated six-layered 3D head phantom model is placed at the center position of the helmet to verify the system performance. For verifying the outcomes of the system, the simulated and measurement imaging setup and their corresponding reconstructed brain images comparison are presented in Table 7. The Hugo head model is considered for simulation setup and fabricated phantoms with human 3D head skull model is considered for measurement purposes. The benign and malignant tumor(s) with different shapes are inserted in both head models and investigated the imaging performances. For better understanding, the cartesian coordinate system is used to present the geometrical view of the images for finding the localization of the tumor in the images. From the cartesian coordinate (x, y), it is easy to find the location of the tumor in the reconstructed images. The x-axis and y-axis are assumed in mm. For instances, in simulation Setup 2, the location of the benign tumor is (− 12, 12) whereas, in measurement Setup 2, the location of benign tumor is (− 4, 4). However, the tumor locations (i.e., benign and malignant) as a cartesian coordinate in reconstructed images for both simulation and measurement setup are summarized in Table 8.
Furthermore, the backscattered signals (i.e., S11, S21, S31, … S91) are collected by the PNA in each 7.2 rotation from the system. Later, the collected signals are post-processed and utilized by the Iteratively Corrected Coherence Factor Delay-Multiply-and-Sum (IC-CF-DMAS) image reconstruction algorithm17 to reconstruct the brain tumor images. This is the improvement of IC-DMAS algorithm and novelties of the used IC-CF-DMAS algorithm compared to other image reconstruction algorithm are (i) It has ability to produce noiseless image with a clear tumor object localization in reconstructed images, (ii) It can reconstruct images with more than one tumor object, and (iii) It takes less computational time to reconstruct brain images. Figure 26a–f represent the non-tumor image (i.e., healthy brain), single benign tumor image, single malignant tumor image, two benign tumors image, two malignant tumors image, and a single benign tumor and a single malignant tumor image, respectively. It is observed that the reconstructed images exhibited very low noises within the regional head area. The proposed MBI system is able to generate images of the brain with target tumors and location. The circular red mark in the images presents the tumor(s) detection and position. The left and bottom side axis labels are used to detect the location of the tumor as a cartesian coordinate. Eight different tumor location(s) are considered in this work to examine and evaluate the performance. It is concluded that the proposed implemented MBI system can identify and trace the brain tumor(s) with a location inside the brain that verifies system capability. Finally, the overall comparison of the proposed antenna with the state-of-the-art in terms of the antenna structure, dimension, substrate name, substrate layer, MTM inclusion, operating band, FBW, realized gain (Re. Gain), fidelity factor (FF), Maximum SAR (Max. SAR), imaging system, detected no. of tumor by the system, and used phantom model for testing is presented in Table 9.
## Conclusion
A metamaterial (MTM) loaded three-dimensional (3D) stacked wideband antenna array is used in microwave brain imaging system to detect brain tumors inside the brain. A spider net-shaped metamaterial unit cell is designed and employed in the proposed antenna to enhance antenna performances. The antenna is fabricated on cost-effective Rogers substrate materials. The top layer and middle layer are fabricated on RT5880, and bottom layer is fabricated on RO4350B substrate material. The optimized dimension of the proposed antenna is 50 × 40 × 8.66 mm3 with an operating band of 1.37–3.16 GHz. The fabricated antenna achieved high radiation efficiency, gain with a high-fidelity factor. The fidelity factor is investigated for H-plane and E-plane as well as in four cases: SBS (X-axis and Y-axis), FTF, and BTB. The antenna showed a high-fidelity factor in the H-plane and in BTB cases. The specific absorption rate of the antenna is calculated with and without MTM and ensured satisfactory field penetration in the head tissue. The implemented system is investigated by utilizing a six-layered tissue-mimicking head phantom and reconstructed the brain images using IC-CF-DMAS algorithm to detect tumors. The benign and malignant tumors are effectively detected by the MBI system. Finally, it is decided that the proposed antenna can be a suitable candidate for the MBI system to successfully detect and locate benign and malignant tumors inside the brain.
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# Ahmed Arafat's research while affiliated with Mansoura University and other places
## Publications (4)
Article
Full-text available
We consider the class Ψd of continuous functions that define isotropic covariance functions in the d-dimensional sphere Sd. We provide a new recurrence formula for the solution of Problem 1 in Gneiting (2013b), solved by Fiedler (2013). In addition, we have improved the current bounds for the curvature at the origin of locally supported covariances...
Preprint
We consider the class $\Psi_d$ of continuous functions $\psi \colon [0,\pi] \to \mathbb{R}$, with $\psi(0)=1$ such that the associated isotropic kernel $C(\xi,\eta)= \psi(\theta(\xi,\eta))$ ---with $\xi,\eta \in \mathbb{S}^d$ and $\theta$ the geodesic distance--- is positive definite on the product of two $d$-dimensional spheres $\mathbb{S}^d$. We...
Article
Full-text available
The equivalence of Gaussian measures is a fundamental tool to establish the asymptotic properties of both prediction and estimation of Gaussian fields under fixed domain asymptotics. The paper solves Problem 18 in the list of open problems proposed by Gneiting (2013). Specifically, necessary and sufficient conditions are given for the equivalence o...
Article
We propose and define a family of marked point processes in noncompact semisimple Lie groups. We first generate Lévy processes via marked point processes by using jump-diffusion processes. Then we build a family of Markov processes in a maximal compact subgroup of a given semisimple Lie group.
## Citations
... The approach was then generalized to nonparametric spectral estimation (Castruccio and Genton, 2014), three-dimensional variables (Castruccio and Genton, 2016), different land/ocean behavior (Castruccio and Guinness, 2017) and also multivariate processes (Edwards et al., 2019). On the more theoretical side, substantial progress has been made in the determination of properties of high dimensional spheres for isotropic processes via basis decomposition see, e.g., Arafat et al. (2020);Porcu et al. (2020). We refer to Jeong et al. (2017); Porcu et al. (2018) for two recent reviews on the topic. ...
... The zonal kernel induced by any continuous function 1 1 possesses a Fourier-type expansion in spherical harmonics where 1 is a real orthonormal basis for the space of spherical harmonics of degree and the collection 1 0 forms a real orthonormal basis for 2 1 . Using Schoenberg's [30] pioneering work, it can be shown that if the expansion coefficients are positive for 0, then is a strictly positive definite kernel on 1 . ...
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# Worksheet on Operation of Multiplication
Practice the set of questions given in the worksheet on operation of multiplication. The questions are based on multiplying numbers to find the product of the given numbers, multiplication facts and word problems on multiplication.
1. Write the multiplication fact for each of the following addition facts:
(i) 4 + 4 + 4 + 4 + 4 + 4 = 24
(ii) 6 + 6 + 6 + 6 + 6 = 30
(iii) 7 + 7 + 7 + 7 + 7 + 7 + 7 = 49
(iv) 8 + 8 + 8 + 8 = 32
(v) 9 + 9 + 9 + 9 + 9 + 9 + 9 = 63
2. Fill in the blanks:
(i) 3 × 6 = (3 × 5) + __3__
(ii) 4 × 8 = (4 × 7) + ___
(iii) 5 × 5 = (5 × 4) + ___
(iv) 5 × 4 = (5 × ___) + 5
(v) 9 × 9 = (9 × ___) + 9
(vi) 3 × 7 = (3 × ___) + 3
3. Fill in the blanks:
(i) 7 × 12 = 12 × ___
(ii) 5 × 0 = ___
(iii) ___ × 8 = 8 × 18
(iv) 18 × 1 = _
(v) 14 × 7 = ___ × 14
(vi) ___ × 1 = 17
(vii) 9 × ___ = 21 × 9
(viii) 18 × ___ = 0
4. Multiply the following:
(i) 31 × 3 =
(ii) 22 × 4 =
(iii) 23 × 3 =
(iv) 31 × 5 =
(v) 82 × 1 =
(vi) 52 × 3 =
(vii) 24 × 2 =
(viii) 52 × 4 =
(ix) 81 × 4 =
(x) 75 × 0 =
5. (i) What is the product of 61 × 4?
(ii) How many fingers do 5 boys have altogether?
(iii) How many days are there in 5 weeks?
(iv) How many months are there in 6 years?
(v) A mosquito has 6 legs. How many legs do 9 mosquitoes have?
(vi) There are 8 tables in a row. If there are 16 such rows, how many tables are there altogether?
(vii) A book has 84 pages. How many pages are there in 5 such books?
Answers for the worksheet on operation of multiplication are given below to check the exact answers of the above set of questions.
1. (i) 4 × 6 = 24
(ii) 6 × 5 = 30
(iii) 7 × 7 = 49
(iv) 8 × 4 = 32
(v) 9 × 7 = 63
2. (i) 3
(ii) 4
(iii) 5
(iv) 3
(v) 8
(vi) 6
3. (i) 7
(ii) 0 × 5
(iii) 18
(iv) 1 × 18
(v) 7
(vi) 17
(vii) 21
(viii) 0
4. (i) 93
(ii) 88
(iii) 69
(iv) 155
(v) 82
(vi) 156
(vii) 48
(viii) 208
(ix) 324
(x) 0
5. (i) 244
(ii) 100 fingers
(iii) 35 days
(iv) 72 months
(v) 54 legs
(vi) 128 tables
(vii) 420 pages
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# Literature: Literate programming: presenting code in human order
## HackerNews Discussion
### Literate programming but with many different presentations or paths of reading the program based on experience/need/use-case
" Peter Norvig 6 July 2016 at 11:47 I think the problem with Literate Programming is that assumes there is a single best order of presentation of the explanation. I agree that the order imposed by the compiler is not always best, but different readers have different purposes. You don’t read documentation like a novel, cover to cover. You read the parts that you need for the task(s) you want to do now. What would be ideal is a tool to help construct such paths for each reader, just-in-time; not a tool that makes the author choose a single pth for all readers."
Has anyone attempted something like this?
I’ve heard you can do transclusion in org-mode which might be a starting point
Edit: Some initial ideas:
• Code can be deconstructed into blocks, and where the code is not self-documenting, prose can be added. Or even visualisations etc if a block is conceptually tricky to grok. You could even have MOOC-style validations to verify reader understanding for each block.
• Some kind of topology of the blocks should be generated based on how they interact and how they are conceptually related
• The author creates a few ‘starting points’ for different audiences, e.g. ‘if you’ve used X before, start at Y’
• From there next blocks to read are auto-suggested to the user. A map or network diagram of all blocks is also provided so the reader can chart their progress and see where the ‘big ideas’ lie.
Edit 2: This also really reminds me of Bret Victor’s ‘Humane Representation of Thought’ lecture (https://vimeo.com/115154289, 45:40) where he says there is a conflict between code being an engineering specification and an authored work meant to be read by humans. - source
I hope to expand on this one day in Literate Programming’s super power: Speaking to different audiences
### “Literate Programming gets in the way when you need to fix a bug”
literate prose and structure tends to get in the way when you are trying to, say, add a new feature or fix a bug.
How? Maybe I’m just thinking in an emacs-centric way here where I can always search via plain text? Perhaps their thought is of reading through a pdf without easy access to the source code to rebuild the pdf (program)?
I’d imagine instead of grepping for an error string I’d C-f for it instead inside of my literate org program. Then once I landed there be able to fix the issue, and pay some reasonable amount of attention to the prose that’s trying to teach me the context of this piece of code.
I still think it’s great tool to think clearly, especially when you have somethings like Emacs Org + Babel at your disposal, but LP it’s not a great fit for regular projects.
The conclusion doesn’t necessarily follow? Especially if those issues are relatively easy to overcome, which to me, they don’t seem too difficult.
### I think most concerns around Literate Programming boil down to “I don’t trust the person to communicate the right things to me”
I think that one of the main drawbacks of this style is the fact that the code is intermingled with the text and that makes the code more confusing. The code is never complete and you always have to jump back and forward to see the other parts of the algorithm and this just makes it harder to understand how the algorithm actually works.
### AND the presenter from Literature: Literate Programming in the Large - Timothy Daly :)
And he has a book recommendation that intrigues me:
If you want to see a real literate program done well, buy the book “Physically Based Rendering” (it won an Academy Award).
Ah it’s freely available online. I create and add Literature: Physically Based Rendering (literate program) to Best examples of Literate Programming
### “literate programming is for the poor souls who have to maintain your magic code piles” lol
Literate programs aren’t just for you. They aren’t just for now. They are there to capture the “WHY” for the poor souls who have to maintain your magic code piles.
### Discovery to a style of literate programming organization in Literature: A New Way to Organize Programs - akkartik.name
I use a layer-based approach described at http://akkartik.name/post/wart-layers in https://github.com/akkartik/mu. It’s 20kLoC of literate code, and I was super concerned about optimizing it for non-linear reading, because that’s how I need to browse the code for myself. It’s not a system to generate multiple views like you want; instead, it’s a single view that I find easy to skim and easy to reorganize at will: a) Each layer puts more important stuff up top. So you can start skimming from the first layer, but you don’t have to read all the way down each layer, just get a sense of what it provides and why.
b) There’s not much emphasis on lengthy comments (which would get linear). Instead just the order in which code is presented does a lot of heavy lifting.
c) The tangled code is intended to be readable (unlike Knuth’s original tools), so I often jump down into it when I feel the need. Error messages and debugger support do show lines in the original literate sources, however.
### More on Organizing programs in human order with literate programming
Has anyone attempted something like this? Yes, I write all my personal projects in human order. To me human order is when the flow control of human consumption and computer execution are most closely aligned. I attempt to achieve this with depth and order.
1. Have a giant library that defines your library or application.
2. Inside the giant function nest child functions for the primary tasks in the order with which they will execute. In the case of the following example the options evaluation is first, followed by the lexer/parser is the second child task, followed by various code presentation tasks, and finally by an analysis task. http://prettydiff.com/lib/jspretty.js
3. Break the child tasks down, as necessary, into reusable components.
4. I also believe a reference should always be declared before it used, which can dictate the order of reference declaration.
My opinion is that when the flow control of the application is unclear you are wasting time during maintenance. You may not know where a problem is occurring, but if the flow of the application is immediately clear from reading the code you immediately know where to start, where to step to next, and where to stop. There is minimal guessing and it doesn’t require breakpoints to figure it out.
### Cool project supposedly doing some of this literature programming stuff that’s taken down now
My hobby project, with dependency graph visualized: https://e8vm.io/e8vm
Source code:
https://github.com/e8vm/e8vm
Or maybe not? I don’t see any of the concepts in this wayback link anyway
### Someone saying basically saying “Literate Programming sounds nice but falls down in the real world”
His comment sums up exactly my thoughts wrt literate programming. It sounds like a nice idea, but in practice it falls far short. The worst program I’ve ever had to maintain was written in funnelweb. It was a large, complicated program, so the funnelweb-generated documentation was tens of thousands of pages long. Nobody was ever going to read that. Trying to find out what you needed to modify to fix a bug was impossible from that document. Most developers I know looked at the woven C code (after all, that’s where the line numbers in the stack trace pointed to), and worked from there.
Well if it’s just that reason, it doesn’t seem too difficult to get tracebacks to work with the tangled files. I suppose it’d be a naturaly but noiser extension of detangling.
### a very ANTI-Literate Programming commenter
#### quote
We have tools that let you examine code from various angles. There is no need to encode some “human order” in the code itself. Code should be organized in the way that harmonizes with the module structure and promotes maintainability, without regard for some “human order” nonsense. Who decides what is “human order”? Humans want things in different order for different purposes. For example, business software generates all sorts of reports of different kinds from the same data.
This is wrong:
Traditional source code, no matter how heavily commented, is presented in the order dictated by the compiler.
This is only true of one-pass, strict definition-before-use, single-module-only programming languages, like toy versions of Pascal.
In any decent language, we can take exactly the same program, and permute the order of its elements with a great deal of liberty, and present them in that order to the compiler. We can decide which functions go into which modules, and we can have those functions in different orders regardless of what calls what.
Ah, but some of the more crazy proponents of literate programming are not satisfied with that granularity. It bothers them that the individual statements of a program that are to be executed in sequence have to be presented to the compiler in that sequence: S1 ; S2.
Knuth’s outlandish version, in particular, stretches the meaning of “literate” by turning code into a dog’s breakfast in which functions are chopped into blocks. For compilation, these blocks are re-assembled by the “literate” processor into functions.
The result is difficult to understand. Yes, the nice explanations and presentation order may present something which makes sense. But here is the rub: I don’t just want to follow the presentation of a program, I want to understand it for myself and convince myself that it is correct. For that, I need to ignore all the text, which, for all I know, only expresses the author’s wishful belief about the code.
Perhaps some of these points could be explored or expanded on in The case against Literate Programming which for me now, would be a good opposing opinion to counteract my cognitive biases by playing devils advocate
#### “We have tools that let you examine code from various angles.”
Do we? What angles? The module hierarchy? Isn’t that just a less up to date hierarchy that’s harder to change than a literate program?
#### “There is no need to encode some “human order” in the code itself.”
LOL… that’s already going to get encoded… the question is will it be a primary or as is the case in the software industry I’ve experienced… very second-hand concern?
#### Who decides what is “human order”? Humans want things in different order for different purposes. For example, business software generates all sorts of reports of different kinds from the same data.
Humans of course ;)
The point about businesses and multiple views of data is a good one though, and makes me more excited about how good of a solution (if annoying for the one writing it) Literate programming but using many different presentations based on experience/use-case could be.
#### Ah, but some of the more crazy proponents of literate programming are not satisfied with that granularity.
It bothers them that the individual statements of a program that are to be executed in sequence have to be presented to the compiler in that sequence: S1 ; S2.
Yeah, perhaps in S1..S100, I’d like S25-70 to be a footnote because it’s not important to understanding the main idea. Perhaps I want to enable those who don’t know specifics of a domain the ability to criticize my architecture?
#### Yes, the nice explanations and presentation order may present something which makes sense.
But here is the rub: I don’t just want to follow the presentation of a program, I want to understand it for myself and convince myself that it is correct.
Not that I don’t get the mistrust, but that mindset keeps you stuck in what we have in the same way people were stuck in non-modular programming. Perhaps they could have said:
But here is the rub: I don’t just want to follow the presentation of a programmers module and “structure”, I want to all of the statements in one file to understand it for myself and convince myself that it is correct.
### There was some relation here to Literature: The trouble with ‘readability’ - akkartik.name as well, namely this quote
Here’s a paean to the software quality of Doom 3. It starts out with this utterly promising ideal:
“Local code should explain, or at least hint at, the overall system design.”
Which I believe is one of the most important things… typically you “start in main” and go to definition (“dumbly” with grep or with tags or editor support).
This is workable, but when you go-to definition, you have to rebuild the contextual link between those things. What if there are edge cases? Well there’s kind of an anti-commenting thing because “my code is self-documenting” thing that’s popular now… so what if it wasn’t deemed important enough to over come that?
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Browse Questions
+1 vote
# A spring has a natural length of 14 meters.If a force of 5N is required to keep the spring stretched 2 meters.How much work is done in stretching the spring from its natural length to a length of 18 meters.
$\begin{array}{1 1}20Joules\\25Joules\\40Joules\\34Joules\end{array}$
We must first find the spring constant $k$.
By Hooke's law,$F(x) =kx$
Thus,$F(2) =k(2)=5$ and so we can calculate k as follows
$k=\large\frac{F(2)}{2}$
$\;\;\;=\large\frac{5N}{2m}$
$\;\;\;=2.5N/m$
Therefore,$F(x)=2.5x$ and so the work done by the force is
Work done = $\int\limits_0^4 2.5xdx$
$\qquad\qquad\;=\big[\large\frac{5}{4}$$x^2\big]_0^4$
$\qquad\qquad\;=20$ Joules
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52
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Find all possible integer values of n such that the following system of equations has a solution for z: \begin{align*} z^n &= 1, \\ \left(z + \frac{1}{z}\right)^n &= 1. \end{align*}
I just have no idea what to do.
\(\begin{align*} z^n &= 1, \\ \left(z + \frac{1}{z}\right)^n &= 1. \end{align*}
Aug 21, 2020
edited by Melody Aug 21, 2020
#1
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You can check that the only possible values of n are 1, 2, and 3.
Aug 21, 2020
#2
+82
0
That is wrong.
I will make this equation a little easier to read now.
\begin{align*} z^n &= 1\end{align*} and \begin{align*}\left(z + \frac{1}{z}\right)^n &= 1. \end{align*}are the restrictions.
Look at what it means if z^n=1. n could be 0, hence z=All integers. z can be 1, hence n could be all integers. The next equation means that if n is 0, z IS all integers. If n is all integers, z=1, but 1+1/1=2!!! 2^integer=not 1, therefore n has to be ZERO (0)
An alternative is that if (z+1/z)^n=1, we do 1 to the nth root and get z+1/z=1, and z+1=z. This is impossible, therefore n has to be 0!
,
Pangolin14 Aug 21, 2020
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Home > CC1MN > Chapter cc21 > Lesson cc21.2.7 > Problem1-127
1-127.
Locate the coordinates of the three highlighted points on the graph of the triangle at right and write them as ordered pairs $(x,y)$. Homework Help ✎
The $x$ coordinate is the number of places going right or left to the point.
The $y$ coordinate is the number of places going up or down to the point.
One ordered pair is $(2,1)$.
Don't forget to put the $x$ coordinate before the $y$ coordinate in your ordered pair. $(x,y)$
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# Hydrolysis constant of $NH^+_4$ is $5.55 \times 10^{-10}$. The ionisation constant of $NH^+_4$ is
(a) $1.8 \times 10^9$
(b) $5.55 \times 10^{-10}$
(c) $5.55 \times 10^4$
(d) $1.8 \times 10^{-5}$
Answer: $5.55 \times 10^{-10}$
$NH^+_4 + H_2O \rightleftharpoons NH_4OH + H^+$
$K_h = \frac{[NH_4OH][H^+]}{[NH^+_4]}$
Since $[NH_4OH] = [H^+]$
$\therefore K_h = \frac{[H^+]^2}{[NH^+_4]}$
$NH^+_4$ ionises as follows:
$NH^+_4 \rightleftharpoons NH_3 + H^+$
$K_a = \frac{[NH_3][H^+]}{[NH^+_4]}$
Since, $[NH_3] = [H^+]$
$\therefore K_a = \frac{[H^+]^2}{[NH^+_4]}$
$\therefore K_a = K_h = 5.55 \times 10^{-10}$
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