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https://codereview.stackexchange.com/questions/148793/topologial-sort-in-python | Topologial Sort in Python
I am trying to self study algorithms and this is my attempt at topological sort using Tarjan's version of DFS. It runs correctly for the graph I included. Can someone tell me if this is correct and if/what optimizations I can make?
graph = {
'A':['C'],
'B': ['C', 'D'],
'C': ['E'],
'D': ['F'],
'E': ['F'],
'F': ['G'],
'G': [],
}
def topological_sort(graph):
visited, stack = set(), []
for vertex in graph:
handle_vertex(graph, vertex, visited, stack)
print(stack)
def handle_vertex(graph, vertex, visited, stack):
if vertex not in visited:
for neighbors in graph[vertex]:
if neighbors not in visited:
handle_vertex(graph, neighbors, visited, stack)
if vertex not in stack:
stack.insert(0, vertex)
return stack
print(topological_sort(graph))
• Do you only have one test case? Does the code work correctly for other test cases? – Phrancis Dec 2 '16 at 23:27
• @Phrancis it also works for the test case: graph2={ 0 : [], 1: [], 2: [3], 3: [1], 4: [0, 1], 5: [2, 0] } – driftdrift Dec 2 '16 at 23:36
• Excuse my ignorance, but can you please explain what's the output you expect for the graph in the code ? More, could you please add more context to what your code does ? Having some recursion going on there doesn't help :). PS: for example, when I'm testing you're first graph, I receive different outputs at every run, so I suspect your code as being broken. – Grajdeanu Alex Dec 3 '16 at 6:11
• @Dex'ter A topological sort returns things in order, where given an edge U->V in a graph, U always comes before V. – driftdrift Dec 3 '16 at 15:48
1. I don't get the point of returning the stack from the handle_vertex function. The return value is never used (and the stack which is passed as an argument is modified in place).
2. The topological_sort functions prints the stack, doesn't return anything and then it's returned value is printed: print(topological_sort(graph)). There're two reasonable options here:
• not printing the returned value of the topological_sort and just calling it instead
• Returning the stack from it instead of printing it inside the function.
3. The neighbors name here: for neighbors in graph[vertex] is misleading. This variable represent just one neighbor at a time. I don't see why would you make it plural. | 2021-01-23 20:29:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19622094929218292, "perplexity": 3112.3046811266922}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703538431.77/warc/CC-MAIN-20210123191721-20210123221721-00017.warc.gz"} |
https://www.math.purdue.edu/~otsymbal/ | ### Russian War and Invasion of Ukraine
On 24.2.2022, at 5am local time Ukraine has been attacked from the sky by barbarian russia, whose final goal is to diminish the independence of Ukraine, reunite it back with Russia, thus reviving the hell of Soviet Union. Since the Ukrainian army did not let them take any single major city during the first three days, they started shooting hospitals, oncological centers, birthing centers, child cares, schools, ambulance teams, and homes of civilians using unguided missiles and prohibitted cluster/vacuum bombs.
The truth shall win over the evil of russia's genocide of the peaceful Ukraine, but the cost of human life that civilian Ukrainians will pay for this is going to be incredibly high. You may support UKRAINIAN ARMY at savelife.in.ua
### Research Interests
My research interests are in Representation Theory (particularly, in Cherednik algebras, quantum affine and toroidal algebras, shuffle algebras, shifted quantum affine algebras and Yangians) and its connection to Algebraic Geometry (via Laumon spaces, Nakajima quiver varieties, Coulomb branches) and Integrable Systems (the Toda-like integrable systems and the quantum inverse scattering method)
My research has been supported by NSF Grants No. DMS-1502497 (changed to DMS-1821185) and DMS-2001247 (changed to DMS-2037602)
### Employment
Assistant Professor, Purdue University, 2020‒present
Gibbs Assistant Professor, Yale University, 2017‒2020
Research Assistant Professor, Simons Center for Geometry and Physics, 2014‒2017
### Education
PhD, Mathematics, Massachusetts Institute of Technology, 2014
MS, Mathematics, Moscow State University, 2009
MS, Mathematics, Independent University of Moscow, 2009
Kharkiv Physics and Mathematics Lyceum #27, Ukraine, 2004
### Publications (arXiv)
• Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems
Submitted; Preprint arXiv:2207.02804 (32pp, last update on 07/06/2022) (arXiv)
• Shuffle approach towards quantum affine and toroidal algebras
These are the lecture notes for the crash-course on shuffle algebras delivered in March 2019 in Tokyo
Commissioned by SpringerBriefs in Mathematical Physics
Preprint arXiv:2209.04294 (126pp, last update on 05/23/2022) (preliminary version) (arXiv)
• Transfer matrices of rational spin chains via novel BGG-type resolutions (with R. Frassek and I. Karpov)
Submitted; Preprint arXiv:2112.12065 (61pp, last update on 12/22/2021) (arXiv) (better aligned version)
• Rational Lax matrices from antidominantly shifted extended Yangians: BCD types (with R. Frassek)
Communications in Mathematical Physics 392 (2022), 545‒619 (journal) (arXiv)
• Surface defects in gauge theory and KZ equation (with N. Nekrasov)
Letters in Mathematical Physics 112 (2022), Paper No. 28, 53pp (journal) (arXiv)
• Quantum loop groups and shuffle algebras via Lyndon words (with A. Negut)
Submitted; Preprint arXiv:2102.11269 (75pp, last update on 10/18/2021) (arXiv)
• Lax matrices from antidominantly shifted Yangians and quantum affine algebras: A-type (with R. Frassek and V. Pestun)
Advances in Mathematics 401 (2022), Paper No. 108283, 73pp (journal) (arXiv)
• Shuffle algebra realizations of type A super Yangians and quantum affine superalgebras for all Cartan data
Letters in Mathematical Physics 110 (2020), no. 8, 2083‒2111 (journal) (arXiv)
• Duality of Lusztig and RTT integral forms of $U_v(L\mathfrak{sl}_n)$
Journal of Pure and Applied Algebra 225 (2021), no. 1, Paper No. 106469, 14pp (journal) (arXiv)
• Shifted quantum affine algebras: integral forms in type A (with M. Finkelberg; appendices joint with A. Weekes)
Arnold Mathematical Journal 5 (2019), no. 2, 197‒283 (journal) (arXiv)
• PBWD bases and shuffle algebra realizations for $U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ and their integral forms
Selecta Mathematica (New Series) 27 (2021), no. 3, Paper No. 35, 48pp (journal) (arXiv)
• On Sevostyanov's construction of quantum difference Toda lattices (with R. Gonin)
International Mathematics Research Notices (2021), no. 12, 8885‒8945 (journal) (arXiv)
• Multiplicative slices, relativistic Toda and shifted quantum affine algebras (with M. Finkelberg)
Representations and Nilpotent Orbits of Lie Algebraic Systems (special volume in honour of the 75th birthday of Anthony Joseph), Progress in Mathematics 330 (2019), 133‒304 (journal) (arXiv)
• Classical limits of quantum toroidal and affine Yangian algebras
Journal of Pure and Applied Algebra 221 (2017), no. 10, 2633‒2646 (journal) (arXiv)
• Several realizations of Fock modules for toroidal $\ddot{U}_{q,d}(\mathfrak{sl}_n)$
Algebras and Representation Theory 22 (2019), no. 1, 177‒209 (journal) (arXiv)
• Homomorphisms between different quantum toroidal and affine Yangian algebras (with M. Bershtein)
Journal of Pure and Applied Algebra 223 (2019), no. 2, 867‒899 (journal) (arXiv)
• Bethe subalgebras of $U_q(\widehat{\mathfrak{gl}}_n)$ via shuffle algebras (with B. Feigin)
Selecta Mathematica (New Series) 22 (2016), no. 2, 979‒1011 (journal) (arXiv)
• The affine Yangian of $\mathfrak{gl}_1$ revisited
Advances in Mathematics 304 (2017), 583‒645 (journal) (arXiv)
• Infinitesimal Hecke algebras of $\mathfrak{so}_N$
Journal of Pure and Applied Algebra 219 (2015), no. 6, 2046‒2061 (journal) (arXiv)
• Infinitesimal Cherednik algebras as W-algebras (with I. Losev)
Transformation Groups 19 (2014), no. 2, 495‒526 (journal) (arXiv)
• Representations of infinitesimal Cherednik algebras (with F. Ding)
Representation Theory (electronic) 17 (2013), 557‒583 (journal) (arXiv)
• Equivariant K-theory of Hilbert schemes via shuffle algebra (with B. Feigin)
Kyoto Journal of Mathematics 51 (2011), no. 4, 831‒854 (journal) (arXiv) (updated)
• Quantum affine Gelfand-Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces
Selecta Mathematica (New Series) 16 (2010), no. 2, 173‒200 (journal) (errata) (arXiv) (updated)
### Teaching Experience
• Fall 2022: Lecturer for MA 26500 (Linear Algebra) at Purdue web-page
• Spring 2022: Lecturer for MA 26500 (Linear Algebra) at Purdue web-page
• Spring 2021: Lecturer for MA 59800CIDLA (Infinite-dimensional Lie algebras and applications) at Purdue web-page
• Fall 2020: Lecturer for MA 26500 (Linear Algebra) at Purdue web-page
• Fall 2019: Lecturer for MATH 120 (Calculus of Functions of Several Variables) at Yale web-page
• Spring 2019: Lecturer for MATH 754 (Infinite-dimensional Lie algebras and applications) at Yale web-page
• Fall 2018: Lecturer for MATH 353 (Introduction to Representation Theory) at Yale web-page
• Fall 2018: Lecturer for MATH 120 (Calculus of Functions of Several Variables) at Yale web-page
• Spring 2018: Lecturer for MATH 667 (Topics in Quantum Groups and Representation Theory) at Yale web-page
• Fall 2017: Lecturer for MATH 120 (Calculus of Functions of Several Variables) at Yale web-page
• Fall 2016: Head instructor for MAT 118 (Mathematical Thinking) at SBU web-page
• Fall 2015: Lecturer for MAT 126 (Calculus B) at SBU
• Fall 2014: Section leader for MAT 303 (Calculus IV with Applications) at SBU
• Spring 2014: Teaching assistant for Math 18.100B (Real Analysis) at MIT web-page
• Winter 2014: Mentor in the MIT Directed Reading Program (Representation Theory) web-page
• Fall 2012: Section leader for Math 18.02 (Multivariable Calculus) at MIT web-page
• 2011‒2013: Grader for MIT courses 18.100B (Real Analysis), 18.125 (Real and Functional Analysis), 18.01 (Calculus), 18.782 (Introduction to Arithmetic Geometry), 18.705 (Commutative Algebra), and 18.737 (Algebraic Groups)
### Mentoring at MIT PRIMES and Yulia's Dream programs
I had mentored Fengning Ding during 2011‒2013 in the first two years of MIT PRIMES program. With our project "Representations of infinitesimal Cherednik algebras", Fengning won the 4th Prize at 2012 Intel STS US national competition (\$40,000 award) and became 2012 Davidson Fellow Laureate (\$50,000 award).
MIT PRIMES is a free, year-long after-school research program for high school students from the Boston area. Program participants work with MIT researchers on exciting unsolved problems in mathematics, computer science, and computational biology.
Since May 2022, I have been mentoring in the Yulia’s Dream program, an initiative under MIT PRIMES.
Yulia's Dream is a research program for exceptional high school students from Ukraine. It is dedicated to the memory of Yulia Zdanovska, a 21-year-old graduate of the National University of Kyiv, a silver medalist at the 2017 European Girls' Mathematical Olympiad, and a teacher for the “Teach for Ukraine” program who was killed by a Russian-fired missile in her home city of Kharkiv.
### Notes for Old Selected Talks
• Temple University, Algebra Seminar, November 2015
Relation between quantum toroidal algebras of $\mathfrak{sl}_n$ and affine Yangians of $\mathfrak{sl}_{nm}$ (handwritten notes)
• Yale University, Geometry, Symmetry and Physics Seminar, April 2015
Shuffle realization of $\ddot{U}_{q,d}(\mathfrak{sl}_n)$ and Bethe subalgebras of $U_q(\widehat{\mathfrak{gl}}_n)$ (handwritten notes)
• Northeastern University, Graduate student seminar, April 2014
The affine Yangian and the quantum toroidal of $\mathfrak{gl}_1$ (handwritten notes)
• MIT-NEU, Graduate seminar on Quantum cohomology and Representation theory, February 2014
Geometric representation theory of the Hilbert schemes (pdf notes I) (pdf notes II) (pdf notes III)
• Northeastern University, Graduate student seminar, April 2013
Infinitesimal Cherednik algebras (handwritten notes)
• Harvard-MIT, Graduate student seminar in Geometric Representation theory, September 2011
Category $\mathcal{O}$ at the negative level (pdf notes)
• MIT, Infinite Dimensional Algebra Seminar, March 2010
Ding-Iohara algebras and their action on the K-theory of the Hilbert scheme (handwritten notes)
• Clay Mathematics Institute, Workshop "Macdonald Polynomials and Geometry", March 2010
Gelfand-Tsetlin bases via Laumon spaces (handwritten notes) | 2022-09-30 19:04:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31529703736305237, "perplexity": 10560.688265704888}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335504.22/warc/CC-MAIN-20220930181143-20220930211143-00391.warc.gz"} |
https://brilliant.org/discussions/thread/de-moivres-theorem-H/ | ×
# de Moivre's theorem
$\ For\ example\ (cos ( \frac{ \pi }{4}\ ) - \ i\sin ( \frac { \pi }{4}\ ))^6$ $\ Based\ on\ de\ Moivre's\ theorem,\ we\ must\ rewrite\ (cos θ - \ i\ sin θ)^{n}\ as\ (cos (-θ) + \ i\ sin (-θ) )^{n}$ $\ Since\ we\ know\ that\ cos (- \pi ) = \cos \pi \ and\ sin (- \pi ) = - \sin \pi$ $\ Why\ we\ can't\ just\ write\ the\ solution\ as$ $\ (cos\ 6( \frac{ \pi }{4}\ ) - \ i\sin\ 6( \frac{ \pi }{4}\ ))\ instead\ of \ ( cos\ 6( -\frac{ \pi }{4}\ ) + \ i\sin\ 6( -\frac{ \pi }{4}\ ))$ $\ After\ all\ at\ the\ end,\ the\ answer\ still\ the\ same.$
Note by Michael Loh
3 years ago | 2016-10-23 18:11:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4932617247104645, "perplexity": 4664.293952378286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719397.0/warc/CC-MAIN-20161020183839-00120-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://blog.devnode.pl/blog/2012/07/16/swank-js/ | Swank-js
Now, I’m extremally happy to have Chrome opened on one monitor, Emacs on the other and immediately visualise changes on my web page just with one C-c C-c. | 2018-12-12 13:49:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2887866497039795, "perplexity": 10354.654888910962}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823895.25/warc/CC-MAIN-20181212134123-20181212155623-00269.warc.gz"} |
http://pwnagetool-windows-live-messenger.com/montrer-matrice-nilpotente-non-invertible-matrix.php | ## Montrer matrice nilpotente non invertible matrix
Unfortunately if the column vectors $(a c) = \alpha (b d)$ then the denominator of the scalar for the inverse formula is 0, making the inverse non-finite. For computational purposes a matrix can also be 'computationally singular' where the precision of the discrete representation on the computer isn't sufficient to calculate the inverse. In practice however, one may encounter non-invertible matrices. And in numerical calculations, matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be ill-conditioned. Examples. Consider the following 2-by-2 matrix. Dec 18, · Matrix A is invertible if we can find another matrix B of same order such that AB = I where I is the identity matrix of same order. A matrix is invertible only if it is a square matrix and its.
# Montrer matrice nilpotente non invertible matrix
This form is a special case of the Jordan canonical form for matrices. For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix []. That is, if is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b 1, b 2 such that Nb 1 = 0 and Nb 2 = b 1. Aug 31, · If you have a matrix A and you are able to find a matrix B(which you can find by some methods) such that AB=I identity matrix then B is called the inverse of A or vice versa. Then you call A as invertible. Note:A and B are square matrices of same order. There are matrices for which you cannot find pwnagetool-windows-live-messenger.com they are called non invertible 1 2 2 4 is one such matrix. abelian group augmented matrix basis basis for a vector space characteristic polynomial commutative ring determinant determinant of a matrix diagonalization diagonal matrix eigenvalue eigenvector elementary row operations exam field theory finite group group group homomorphism group theory homomorphism ideal inverse matrix invertible matrix.le remercie pour l'intérêt qu'il a su montrer à cette thèse. Mes plus . Non- homogeneous linear differential systems with constant coefficients 44 .. are not necessarily supposed to be invertible, i.e, arbitrary matrices. where κ is a partial multiplicity of L(λ) associated with eigenvalue 0, J is a nilpotent Jordan . Représentations modulaires, matrices de décomposition. La théorie Le morphisme e s'obtient en composant avec l'inverse . des µ tels que ce quotient soit non nul (celui-ci sera alors noté Dµ) est l'ensemble P. ℓ-reg . variété nilpotente [KP89], nous permet de montrer la nullité de certains nombres de. 6 Springer correspondence and decomposition matrices The geometric .. Le morphisme es'obtient en composant avec l'inverse .. des µtels que ce quotient soit non nul (celui-ci sera alors not´e Dµ) est l'ensemble Pℓ-reg. n .. vari´et´e nilpotente [KP89], nous permet de montrer la nullit´e de certains nombres de.
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Matrices 19 (Nilpotent matirces), time: 3:17
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[pwnagetool-windows-live-messenger.com] - Matrices 3x3 : matrices semblables, time: 2:08
Tags: Montrer matrice nilpotente non invertible matrix,Montrer matrice nilpotente non invertible matrix,Montrer matrice nilpotente non invertible matrix. | 2021-09-22 05:53:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9480568766593933, "perplexity": 3436.2859606328557}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057329.74/warc/CC-MAIN-20210922041825-20210922071825-00047.warc.gz"} |
https://chemistry.stackexchange.com/questions/126478/how-is-the-bonding-in-the-au6cpph362-cluster-explained?noredirect=1 | # How is the bonding in the [Au6C(PPh3)6]2+ cluster explained?
How can carbon form six bonds in such a compound as $$\ce{[Au6C(PPh3)6]^2+}?$$ I understand how carbon with five bonds can be formed $$(\ce{CH5+},$$ for instance), but it shouldn't have enough electrons to form six bonds.
• Carbon can well be hexacoordinate, or even hepta. This case isn't all that different from methanium you mention. Jan 12, 2020 at 19:47
• Jan 12, 2020 at 19:50
• Mildly related: I explain a (relatively simplistic) model of bonding for $\ce{SF6}$ in Why do compounds like SF6 and SF4 exist but SH6 and SH4 don't?. The discussion following the MO diagram briefly touches on species such as $\ce{CH6^2+}$. Now, if you treat $\ce{Au(PPh3)+}$ as a replacement for a single proton (the proper term is isolobal), then that becomes $\ce{C[Au(PPh3)]6^2+}$. Note the qualitative similarity in the MO diagram there to the one andselisk has drawn. Jan 13, 2020 at 1:26
This is by no means a trivial case. You are dealing with a so-called template or interstitial atom placement within a cluster shell. There is a thorough review available [1, pp. 18–20] (reference numbers updated):
An alternative way of satisfying the closed shell electronic requirements and retaining a positive charge on the cluster is possible if an interstitial atom is introduced. For an octahedral gold cluster an additional gold atom is too large to satisfy the steric requirements of the interstitial cavity, but a small main group atom, e.g. C, N or O, is able to satisfy these steric requirements. Figure 11 gives the relevant interaction diagram for octahedral $$\ce{[Au6C(PPh3)6]^2+}$$ and an interstitial carbon atom, which contributes just the right number of electrons to lead to a $$\mathrm{[S^σ]^2[P^σ]^6}$$ pseudo-spherical ground state configuration. It is noteworthy that this simple theoretical analysis was first published in 1976 [2], and the existence of $$\ce{[Au6C(PPh3)6]^2+}$$ was confirmed by Schmidbaur and his coworkers in 1989 [3–5]. This compound had been isolated and structurally characterised previously, but the interstitial carbon had not been identified. The introduction of the interstitial atom strengthens the radial interactions significantly as a result of effective overlaps between the carbon $$\mathrm{2s}$$ and $$\mathrm{2p}$$ orbitals and the matching $$\mathrm{S^σ}$$ and $$\mathrm{P^σ}$$ cluster molecular orbitals. In broad brush terms the stabilisation of the valence orbitals of the central atom are stabilised by $$\mathrm{nβ^σ}$$/degeneracy of the molecular orbitals. If $$\mathrm{β^σ(s)} = \mathrm{β^σ(p)} = \mathrm{β^σ(d)},$$ the relative stabilisations are $$\mathrm{6β^σ(s)},$$ $$\mathrm{2β^σ(p)},$$ and $$\mathrm{3β^σ(d)}.$$ Therefore, the greatest stabilisation involves the $$\mathrm{s}$$ orbitals of the central atom and increases as the number of metal atoms, $$n,$$ increases. For filled shells the stabilisation energies are independent of geometry as long as the ligand polyhedron approximates to a sphere. It follows that gold clusters with main group interstitial atoms are characterised by a pec of $$12n + 8$$ (sec 8) valence electrons, since each $$\ce{AuPPh3}$$ fragment is associated with a filled $$\mathrm{d}$$ shell containing $$10$$ electrons and a bonding $$\ce{Au–P}$$ bonding molecular orbital.
Fig. 11 Molecular orbital interaction diagram for $$\ce{[Au6C(PPh3)6]^2+}.$$ Similar analyses may be constructed for trigonal bipyramidal $$\ce{[Au5N(PPh3)5]^2+}$$ and tetrahedral $$\ce{[Au4O(PPh3)4]^2+}$$
### References
1. Gold Clusters, Colloids and Nanoparticles II; Mingos, D. M. P., Ed.; Structure and Bonding; Springer International Publishing: Cham, 2014; Vol. 162. DOI: 10.1007/978-3-319-07845-8.
2. Mingos, D. M. P. Molecular-Orbital Calculations on Cluster Compounds of Gold. J. Chem. Soc., Dalton Trans. 1976, No. 13, 1163–1169. DOI: 10.1039/DT9760001163.
3. Schmidbaur, H., Grohmann, A., Olmos, M. E.; Organogold chemistry. In: Schmidbaur, H. (ed) Gold progress in chemistry, biochemistry and technology. Wiley, Chichester, 1999. pp 648–731.
4. Steigelmann, O.; Bissinger, P.; Schmidbaur, H. Assembly of the $$\ce{[CAu6]^{2⊕}}$$ Cluster with a Tailor-Made Diphosphane Spanning the Octahedral Edges. Angewandte Chemie International Edition in English 1990, 29 (12), 1399–1400. DOI: 10.1002/anie.199013991.
5. Schmidbaur, H.; Brachthäuser, B.; Steigelmann, O. Direct Observation of the Central Atom in $$\ce{[C\{Au[(C6H5)2(PC6H4NMe2)]\}6](BF4)2}$$ by $$\ce{^{13}C}$$ NMR Spectroscopy. Angewandte Chemie International Edition in English 1991, 30 (11), 1488–1490. DOI: 10.1002/anie.199114881.
• I wonder, if the LaTeX advanced editors use any handy reference generator, or if all is written down.. I bet for the former. When I was writing my diploma work, top tech were DOS based editors a latex was rubber. Jan 12, 2020 at 12:44
• @Poutnik I use Zotero (desktop)/Zbib (online) for references as there is no Bib(La)TeX support on any of the SE sites. Jan 12, 2020 at 13:33
• @Poutnik, the references in andselisk's answer are not in LaTeX (apart from the chemical formula in ref 5, which I assume is manually typed in). They are just plain Markdown. See chemistry.meta.stackexchange.com/a/4299/16683 for a tool which generates citations from a DOI. Jan 12, 2020 at 15:38
• @orthocresol I see. I had rather in mind perhaps just generated by LaTex aware word processing software, as I am not familiar with these systems. I will check the link, thanks. Jan 12, 2020 at 16:02
• You beat me to it, +1. Compare with SF6 where the e_g orbitals are occupied. As explained here this orbital pair becomes antibonding if the central spacer atom is too small. But if that level is empty, as in the gold cluster, you want the small central atom to promote ligand-ligand bonding in the still occupied lower orbitals. Jan 12, 2020 at 16:36 | 2022-05-22 04:02:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 27, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6037625074386597, "perplexity": 2888.913137616998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662543797.61/warc/CC-MAIN-20220522032543-20220522062543-00252.warc.gz"} |
http://physics.stackexchange.com/tags/terminology/hot?filter=week | # Tag Info
9
Your question is not specific to inflation, and really applies to any case where a bosonic quantum field behaves semiclassically due to macroscopically large occupation numbers. One very simple example of this is the Stark effect in quantum mechanics, where a Hyrodgen atom is placed in a uniform electric field. The atom is treated as a quantum mechanical ...
1
The photons released are individually massless, but all of them together have an effective mass equal to the original masses of the particle and antiparticle; see my answer here. This isn't some mathematical abstraction either -- you can put the photons in a reflective box and weigh it, and it'll have extra weight. It's safe to say that the phrase ...
1
I would look at this in a slightly different way. Rearranging it: $$m \ddot{x} = -(a|\dot{x}|+k) x = -k_{eff} x$$ If you look at it that way, it is really a variable, non-linear stiffness $k_{eff}$ that depends on the velocity, rather than a damping that depends on the position. In this respect (assuming $a > 0$), the stiffness coefficient has a lower ...
1
This is an every day term, with no meaning for physics. In the Oxford dictionary for "energy" one gets: Physics The property of matter and radiation which is manifest as a capacity to perform work (such as causing motion or the interaction of molecules) In science fiction one might separate zero mass particles, like the photon and the graviton ( ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2016-05-26 18:35:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7476395964622498, "perplexity": 316.8211268060058}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276131.97/warc/CC-MAIN-20160524002116-00161-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://brilliant.org/problems/an-electricity-and-magnetism-problem-by-manmeswar/?group=YE4AJIL1HxN6&ref_id=1120576 | # An electricity and magnetism problem by Manmeswar Patnaik
The current through 60 ohm resistor in the following network is
× | 2021-05-18 20:45:20 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8408154249191284, "perplexity": 3617.985826074324}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991514.63/warc/CC-MAIN-20210518191530-20210518221530-00446.warc.gz"} |
https://itectec.com/database/sql-server-stored-procedure-processing-and-error-log/ | # Sql-server – Stored Procedure Processing and Error Log
Architectureauditloggingsql server 2014stored-procedures
I used to work for a company where they had this 3rd party Data warehouse Solution. Obviously all the objects and tables were hidden in the Supporting database so I don't have clear idea on what exactly happened inside some of the stored procedure. I saw this interesting stored procedure there and want to replicate that in my own solution but I just can't get my head around on how it works. I am describing the stored procedure below and it will be really helpful if someone can give me some ideas on how to achieve this. Even better if you can suggest me how I can make this even better.
Stored procedure was called process log. It had Parameters like DBID, ObjectId, Step, Status, Remarks, Reads, Inserts, Updates, Deletes
What we had to do was, inside every stored procedure, we have to execute this stored procedure with Status as 2 (In Progress) During the process where can execute this stored procedure multiple time at the end of each step or section after increasing the value of variable step. based on the row counts of Insert update select and delete statements we were supposed to record values in respective stored procedure parameter variables.
In the end you can execute same stored procedure with Status of 3 (Completed) or if the procedure ended up in catch block, status will be 4 (Failed) in the remarks section we can copy SQL's error messages.
To see all this information, we were given access to a Report, For which obviously I didn't have the source code but the report showed what time stored procedure started when it finished, what was the status how many inserts updates deletes and reads it did. if it failed, what was the error message?
I already have few ideas for improvements to store, who started it?, what were the values of parameters? For who started the stored procedure part, I have a confusion. Most of these stored procedure run as part of different Jobs. All our jobs run as a Service Account user but jobs are started manually by various users. I need to find out which user started it, as inside stored procedure, as current user, it will always show service account. Also for Values of Paramters, is there a better dynamic way of finding this out? rather then setting a value for a variable manually. I was hopping to use output from INPUTBUFFER but it only shows the name of the parameter not values.
If someone can guide me regarding the back end table structure and script for this auditing SP, it will be really helpful. Also any more ideas for improvement are welcome as well.
My main confusion:
I believe they had some table where these stored procedure values were stored and if the SP is running already, they did update in the record then doing an Insert but how would they identify to do an insert instead of update in the scenario where stored procedure failed critically and catch block wasn't executed.
Here is a structure that is at least very close.
There is no programmatic way to get the parameters (unfortunately). You need to format them into XML to pass in.
The Login that initiates a SQL Agent job seems to only be logged in the message column of msdb.dbo.sysjobhistory, for step_id = 0. This value can be extracted, but not during the execution of the job.
You get ObjectID to pass in from @@PROCID.
Below is the schema (2 tables) and stored procedures (3 procs). The concept is to separate "init", "in process", and "completed (success or error)" logging. This allows for setting certain columns only at the appropriate time (e.g. only need to set DatabaseID, StartedAt, etc at the very beginning). Separating the type of event also makes it easier to have event-specific logic (yes, can have it even in a single proc, but then you still have all the input parameters when you only need a subset per each event-type).
A "process" record get updated via its IDENTITY (and clustered PK) value. This is another benefit of having the "event type" separation: it makes it much easier to handle capturing the SCOPE_IDENTITY() and passing it back to be used for the two other logging stored procedures. If a stored procedure fails and doesn't go to the CATCH block, then there is no need to worry about accidentally updating that process record as the next time any stored procedure (that is being logged) starts, it will get a new / unique ID to update.
Cleanup (optional) and Schema
/* -- optional cleanup
DROP PROCEDURE [dbo].[ProcessLogDemo];
DROP PROCEDURE [Logging].[ProcessLog_Log];
DROP PROCEDURE [Logging].[ProcessLog_Start];
DROP PROCEDURE [Logging].[ProcessLog_Stop];
DROP TABLE [Logging].[ProcessLog];
DROP TABLE Logging.[Status];
DROP SCHEMA [Logging];
*/
CREATE SCHEMA [Logging];
GO
Tables and Indexes
CREATE TABLE Logging.[Status]
(
[StatusID] TINYINT NOT NULL
CONSTRAINT [PK_Status] PRIMARY KEY CLUSTERED,
[StatusName] VARCHAR(50) NOT NULL
);
CREATE TABLE [Logging].[ProcessLog]
(
ProcessLogID INT NOT NULL IDENTITY(-2147483648, 1) -- start at INT min value
CONSTRAINT [PK_ProcessLog] PRIMARY KEY CLUSTERED,
DatabaseID INT NOT NULL,
ObjectID INT NULL, -- NULL = ad hoc query
SessionID SMALLINT NOT NULL
CONSTRAINT [DF_ProcessLog_SessionID] DEFAULT (@@SPID),
Step TINYINT NOT NULL, -- if you have more than 255 steps, consult psychiatrist
StatusID TINYINT NOT NULL
CONSTRAINT [FK_ProcessLog_Status]
FOREIGN KEY REFERENCES [Logging].[Status]([StatusID]),
Remarks NVARCHAR(MAX) NULL, -- or maybe VARCHAR(MAX)?
Params XML NULL,
RowsSelected INT NULL,
RowsInserted INT NULL,
RowsUpdated INT NULL,
RowsDeleted INT NULL,
StartedBy [sysname] NULL,
StartedAt DATETIME2 NOT NULL
CONSTRAINT [DF_ProcessLog_StartedAt] DEFAULT (SYSDATETIME()),
UpdatedAt DATETIME2 NULL, -- use to show progress / "heartbeat"
StoppedAt DATETIME2 NULL
);
GO
Stored Procedure to call at the very beginning of "logged" stored procedures
CREATE PROCEDURE [Logging].[ProcessLog_Start]
(
@DatabaseID INT,
@ObjectID INT,
@Params XML,
@ProcessLogID INT = NULL OUTPUT
)
AS
SET NOCOUNT ON;
-- First, capture the MAX "instance_id" from sysjobhistory if this process is a SQL
-- Server Agent job (use later to get the "invoked by" Login), else grab the Login.
DECLARE @StartedBy [sysname];
IF (EXISTS(
SELECT *
FROM sys.dm_exec_sessions sdes
WHERE sdes.[session_id] = @@SPID
AND sdes.[program_name] LIKE N'SQLAgent - TSQL JobStep (%'))
BEGIN
DECLARE @JobID UNIQUEIDENTIFIER;
SELECT @JobID = CONVERT(UNIQUEIDENTIFIER,
CONVERT(BINARY(16),
SUBSTRING(sdes.[program_name],
CHARINDEX(N'(Job 0x', sdes.[program_name]) + 5,
34), 1
)
)
FROM sys.dm_exec_sessions sdes
WHERE sdes.[session_id] = @@SPID;
--SELECT @JobID;
SELECT @StartedBy = N'sysjobhistory.instance_id: '
+ CONVERT(NVARCHAR(20), MAX(sjh.[instance_id]))
FROM msdb.dbo.sysjobhistory sjh
WHERE sjh.[job_id] = @JobID;
END;
ELSE
BEGIN
SET @StartedBy = ORIGINAL_LOGIN();
END;
-- Now it should be safe to create a new entry
INSERT INTO [Logging].[ProcessLog] ([DatabaseID], [ObjectID], [Step], [StatusID],
[Params], [StartedBy])
VALUES (@DatabaseID, @ObjectID, 0, 1, @Params, @StartedBy);
SET @ProcessLogID = SCOPE_IDENTITY();
GO
Stored Procedure to call after all but the final step
CREATE PROCEDURE [Logging].[ProcessLog_Log]
(
@ProcessLogID INT,
@Step TINYINT,
@RowsSelected INT = NULL,
@RowsInserted INT = NULL,
@RowsUpdated INT = NULL,
@RowsDeleted INT = NULL
)
AS
SET NOCOUNT ON;
UPDATE pl
SET pl.[StatusID] = 2, -- In process
pl.[Step] = @Step,
pl.[UpdatedAt] = SYSDATETIME(),
pl.[RowsSelected] = ISNULL(@RowsSelected, pl.[RowsSelected]),
pl.[RowsInserted] = ISNULL(@RowsInserted, pl.[RowsInserted]),
pl.[RowsUpdated] = ISNULL(@RowsUpdated, pl.[RowsUpdated]),
pl.[RowsDeleted] = ISNULL(@RowsDeleted, pl.[RowsDeleted])
FROM [Logging].[ProcessLog] pl
WHERE pl.[ProcessLogID] = @ProcessLogID;
IF (@@ROWCOUNT = 0)
BEGIN
RAISERROR('No initial or in-process record for ProcessLogID = %d !', 16, 1,
@ProcessLogID);
RETURN;
END;
GO
Stored Procedure to call after the final step and/or in a CATCH block
CREATE PROCEDURE [Logging].[ProcessLog_Stop]
(
@ProcessLogID INT,
@Step TINYINT,
@StatusID TINYINT,
@Remarks NVARCHAR(MAX) = NULL,
@RowsSelected INT = NULL,
@RowsInserted INT = NULL,
@RowsUpdated INT = NULL,
@RowsDeleted INT = NULL
)
AS
SET NOCOUNT ON;
UPDATE pl
SET pl.[StatusID] = @StatusID, -- 3 = Success, 4 = Fail
pl.[Step] = @Step,
pl.[Remarks] = @Remarks,
pl.[StoppedAt] = SYSDATETIME(),
pl.[RowsSelected] = ISNULL(@RowsSelected, pl.[RowsSelected]),
pl.[RowsInserted] = ISNULL(@RowsSelected, pl.[RowsInserted]),
pl.[RowsUpdated] = ISNULL(@RowsSelected, pl.[RowsUpdated]),
pl.[RowsDeleted] = ISNULL(@RowsSelected, pl.[RowsDeleted])
FROM [Logging].[ProcessLog] pl
WHERE pl.[ProcessLogID] = @ProcessLogID;
IF (@@ROWCOUNT = 0)
BEGIN
RAISERROR('No initial or in-process record for ProcessLogID = %d !', 16, 1,
@ProcessLogID);
RETURN;
END;
GO
Demo stored procedure (input parameters are formatted as XML)
The reason for putting "StepNumber" in a variable is so that the value can get passed to the CATCH block. The @StepNumber variable is incremented prior to each operation. If the operation succeeds, that value is used to call the "Log" stored procedure that captures the number of rows affected for that step and the time it was called. If the operation fails, that same @StepNumber value is used to call the "Stop" stored procedure that marks the process as "failed" and passes in the error message. This makes the data less confusing since the Step column on failed records will be the step it was actually working on when the error occurred.
CREATE PROCEDURE [dbo].[ProcessLogDemo]
(
@Param1 INT,
@Param2 DATETIME,
@Param3 NVARCHAR(50) = NULL
)
AS
SET NOCOUNT ON;
DECLARE @ProcessID INT,
@DB_ID INT = DB_ID(),
@Params XML,
@StepNumber TINYINT;
SET @Params = (
SELECT @Param1 AS [Param1],
@Param2 AS [Param2],
@Param3 AS [Param3]
FOR XML PATH(N'Params')
); -- missing elements mean the value == NULL
--SELECT @Params;
BEGIN TRY
EXEC [Logging].[ProcessLog_Start]
@DatabaseID = @DB_ID,
@ObjectID = @@PROCID,
@Params = @Params,
@ProcessLogID = @ProcessID OUTPUT;
SET @StepNumber = 1;
-- do something
EXEC [Logging].[ProcessLog_Log]
@ProcessLogID = @ProcessID,
@Step = @StepNumber,
@RowsSelected = @@ROWCOUNT;
SET @StepNumber = 2;
-- do something else
EXEC [Logging].[ProcessLog_Log]
@ProcessLogID = @ProcessID,
@Step = @StepNumber,
@RowsUpdated = @@ROWCOUNT;
SET @StepNumber = 3;
-- do final thingy
EXEC [Logging].[ProcessLog_Stop]
@ProcessLogID = @ProcessID,
@Step = @StepNumber,
@StatusID = 3, -- success
@RowsInserted = @@ROWCOUNT;
END TRY
BEGIN CATCH
DECLARE @ErrorMessage NVARCHAR(MAX) = ERROR_MESSAGE();
EXEC [Logging].[ProcessLog_Stop]
@ProcessLogID = @ProcessID,
@Step = @StepNumber,
@StatusID = 4, -- fail
@Remarks = @ErrorMessage;
END CATCH;
GO
NOTES:
• With regards to getting the "invoked by" Login for SQL Server Agent jobs: the step_id = 0 record (which is the only place that this info exists) does not exist until the job completes (success or failure). Hence, it isn't available while the stored procedure is running, let alone at the beginning. For now we capture the MAX(sjh.[instance_id]) FROM msdb.dbo.sysjobhistory sjh for the currently executing job for the current session. Later (i.e. after the job completes), that can get replaced with the job invoker Login.
• I would generally recommend against adding this type of logging to Stored Procedures that are executed very frequently as the additional read and write operations will have a negative impact on performance.
Here is an Inline Table-Valued Function (ITVF) to get the job outcome info (including the "invoked by" user or schedule or whatever) based on the instance_id value that was captured into the ProcessLog.StartedBy column. The instance_id value returned in the result set is the row for step_id = 0.
CREATE FUNCTION dbo.GetSqlServerAgentJobOutcome
(
@InstanceID INT
)
RETURNS TABLE
AS RETURN
WITH cte AS
(
SELECT TOP (1)
sjh.[instance_id],
sjh.job_id,
sjh.[message],
sjh.[run_date],
sjh.[run_time],
sjh.[run_duration],
sjh.[run_status],
sjh.[sql_message_id],
sjh.[sql_severity],
(CHARINDEX(N' was invoked by ', sjh.[message]) + 16) AS [invoker_begin],
CHARINDEX(N'. The last step to run', sjh.[message]) AS [invoker_end]
FROM msdb.dbo.sysjobhistory sjh
WHERE sjh.[job_id] = (SELECT sjh2.[job_id]
FROM msdb.dbo.sysjobhistory sjh2
WHERE sjh2.[instance_id] = @InstanceID)
AND sjh.[step_id] = 0
AND sjh.[instance_id] >= @InstanceID
ORDER BY instance_id ASC
)
SELECT [instance_id], [job_id],
--[message],
[run_date], [run_time],
[run_duration], [run_status],
[sql_message_id], [sql_severity],
SUBSTRING([message], invoker_begin, ([invoker_end] - [invoker_begin]))
AS [InvokedBy]
FROM cte;
GO | 2022-01-23 02:26:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29791542887687683, "perplexity": 8967.543810201803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303956.14/warc/CC-MAIN-20220123015212-20220123045212-00358.warc.gz"} |
https://openstax.org/books/introductory-statistics/pages/8-1-a-single-population-mean-using-the-normal-distribution | Introductory Statistics
# 8.1A Single Population Mean using the Normal Distribution
Introductory Statistics8.1 A Single Population Mean using the Normal Distribution
A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of and we have constructed the 90% confidence interval (5, 15) where EBM = 5.
### Calculating the Confidence Interval
To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need $x ¯ x ¯$ as an estimate for μ and we need the margin of error. Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM). The sample mean $x ¯ x ¯$ is the point estimate of the unknown population mean μ.
The confidence interval estimate will have the form:
(point estimate - error bound, point estimate + error bound) or, in symbols,($x ¯ –EBM, x ¯ +EBM x ¯ –EBM, x ¯ +EBM$)
The margin of error (EBM) depends on the confidence level (abbreviated CL). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because that person wants to be reasonably certain of his or her conclusions.
There is another probability called alpha (α). α is related to the confidence level, CL. α is the probability that the interval does not contain the unknown population parameter.
Mathematically, α + CL = 1.
### Example 8.1
• Suppose we have collected data from a sample. We know the sample mean but we do not know the mean for the entire population.
• The sample mean is seven, and the error bound for the mean is 2.5.
$x ¯ x ¯$ = 7 and EBM = 2.5
The confidence interval is (7 – 2.5, 7 + 2.5), and calculating the values gives (4.5, 9.5).
If the confidence level (CL) is 95%, then we say that, "We estimate with 95% confidence that the true value of the population mean is between 4.5 and 9.5."
Try It 8.1
Suppose we have data from a sample. The sample mean is 15, and the error bound for the mean is 3.2.
What is the confidence interval estimate for the population mean?
A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of $x ¯ x ¯$ = 10, and we have constructed the 90% confidence interval (5, 15) where EBM = 5.
To get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. If we include the central 90%, we leave out a total of α = 10% in both tails, or 5% in each tail, of the normal distribution.
Figure 8.2
To capture the central 90%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean. The value 1.645 is the z-score from a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail.
It is important that the "standard deviation" used must be appropriate for the parameter we are estimating, so in this section we need to use the standard deviation that applies to sample means, which is $σ n σ n$. The fraction $σ n σ n$, is commonly called the "standard error of the mean" in order to distinguish clearly the standard deviation for a mean from the population standard deviation σ.
In summary, as a result of the central limit theorem:
• $X ¯ X ¯$ is normally distributed, that is, $X ¯ X ¯$ ~ N$( μ X , σ n ) ( μ X , σ n )$.
• When the population standard deviation σ is known, we use a normal distribution to calculate the error bound.
#### Calculating the Confidence Interval
To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. The steps to construct and interpret the confidence interval are:
• Calculate the sample mean $x ¯ x ¯$ from the sample data. Remember, in this section we already know the population standard deviation σ.
• Find the z-score that corresponds to the confidence level.
• Calculate the error bound EBM.
• Construct the confidence interval.
• Write a sentence that interprets the estimate in the context of the situation in the problem. (Explain what the confidence interval means, in the words of the problem.)
We will first examine each step in more detail, and then illustrate the process with some examples.
#### Finding the z-score for the Stated Confidence Level
When we know the population standard deviation σ, we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. We need to find the value of z that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution Z ~ N(0, 1).
The confidence level, CL, is the area in the middle of the standard normal distribution. CL = 1 – α, so α is the area that is split equally between the two tails. Each of the tails contains an area equal to $α 2 α 2$.
The z-score that has an area to the right of $α 2 α 2$ is denoted by $z α 2 z α 2$.
For example, when CL = 0.95, α = 0.05 and $α 2 α 2$ = 0.025; we write $z α 2 z α 2$ = z0.025.
The area to the right of z0.025 is 0.025 and the area to the left of z0.025 is 1 – 0.025 = 0.975.
, using a calculator, computer or a standard normal probability table.
### Using the TI-83, 83+, 84, 84+ Calculator
invNorm(0.975, 0, 1) = 1.96
### Note
Remember to use the area to the LEFT of $z α 2 z α 2$; in this chapter the last two inputs in the invNorm command are 0, 1, because you are using a standard normal distribution Z ~ N(0, 1).
#### Calculating the Error Bound (EBM)
The error bound formula for an unknown population mean μ when the population standard deviation σ is known is
• EBM = $( z α 2 )( σ n ) ( z α 2 )( σ n )$
#### Constructing the Confidence Interval
• The confidence interval estimate has the format $( x ¯ –EBM, x ¯ +EBM) ( x ¯ –EBM, x ¯ +EBM)$.
The graph gives a picture of the entire situation.
CL + $α 2 α 2$ + $α 2 α 2$ = CL + α = 1.
Figure 8.3
#### Writing the Interpretation
The interpretation should clearly state the confidence level (CL), explain what population parameter is being estimated (here, a population mean), and state the confidence interval (both endpoints). "We estimate with ___% confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units)."
### Example 8.2
Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. Find a confidence interval estimate for the population mean exam score (the mean score on all exams).
Find a 90% confidence interval for the true (population) mean of statistics exam scores.
### Try It 8.2
Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes.
Find a 90% confidence interval estimate for the population mean delivery time.
### Example 8.3
The Specific Absorption Rate (SAR) for a cell phone measures the amount of radio frequency (RF) energy absorbed by the user’s body when using the handset. Every cell phone emits RF energy. Different phone models have different SAR measures. To receive certification from the Federal Communications Commission (FCC) for sale in the United States, the SAR level for a cell phone must be no more than 1.6 watts per kilogram. Table 8.1 shows the highest SAR level for a random selection of cell phone models as measured by the FCC.
Phone Model SAR Phone Model SAR Phone Model SAR
Apple iPhone 4S 1.11 LG Ally 1.36 Pantech Laser 0.74
BlackBerry Pearl 8120 1.48 LG AX275 1.34 Samsung Character 0.5
BlackBerry Tour 9630 1.43 LG Cosmos 1.18 Samsung Epic 4G Touch 0.4
Cricket TXTM8 1.3 LG CU515 1.3 Samsung M240 0.867
HP/Palm Centro 1.09 LG Trax CU575 1.26 Samsung Messager III SCH-R750 0.68
HTC One V 0.455 Motorola Q9h 1.29 Samsung Nexus S 0.51
HTC Touch Pro 2 1.41 Motorola Razr2 V8 0.36 Samsung SGH-A227 1.13
Huawei M835 Ideos 0.82 Motorola Razr2 V9 0.52 SGH-a107 GoPhone 0.3
Kyocera DuraPlus 0.78 Motorola V195s 1.6 Sony W350a 1.48
Kyocera K127 Marbl 1.25 Nokia 1680 1.39 T-Mobile Concord 1.38
Table 8.1
Find a 98% confidence interval for the true (population) mean of the Specific Absorption Rates (SARs) for cell phones. Assume that the population standard deviation is σ = 0.337.
### Try It 8.3
Table 8.2 shows a different random sampling of 20 cell phone models. Use this data to calculate a 93% confidence interval for the true mean SAR for cell phones certified for use in the United States. As previously, assume that the population standard deviation is σ = 0.337.
Phone Model SAR Phone Model SAR
Blackberry Pearl 8120 1.48 Nokia E71x 1.53
HTC Evo Design 4G 0.8 Nokia N75 0.68
HTC Freestyle 1.15 Nokia N79 1.4
LG Ally 1.36 Sagem Puma 1.24
LG Fathom 0.77 Samsung Fascinate 0.57
LG Optimus Vu 0.462 Samsung Infuse 4G 0.2
Motorola Cliq XT 1.36 Samsung Nexus S 0.51
Motorola Droid Pro 1.39 Samsung Replenish 0.3
Motorola Droid Razr M 1.3 Sony W518a Walkman 0.73
Nokia 7705 Twist 0.7 ZTE C79 0.869
Table 8.2
Notice the difference in the confidence intervals calculated in Example 8.3 and the following Try It exercise. These intervals are different for several reasons: they were calculated from different samples, the samples were different sizes, and the intervals were calculated for different levels of confidence. Even though the intervals are different, they do not yield conflicting information. The effects of these kinds of changes are the subject of the next section in this chapter.
### Example 8.4
Suppose we change the original problem in Example 8.2 by using a 95% confidence level. Find a 95% confidence interval for the true (population) mean statistics exam score.
Try It 8.4
Refer back to the pizza-delivery Try It exercise. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes. Use a sample size of 20. Find a 95% confidence interval estimate for the true mean pizza delivery time.
### Example 8.5
Suppose we change the original problem in Example 8.2 to see what happens to the error bound if the sample size is changed.
Leave everything the same except the sample size. Use the original 90% confidence level. What happens to the error bound and the confidence interval if we increase the sample size and use n = 100 instead of n = 36? What happens if we decrease the sample size to n = 25 instead of n = 36?
• $x ¯ x ¯$ = 68
• EBM = $( z α 2 )( σ n ) ( z α 2 )( σ n )$
• σ = 3; The confidence level is 90% (CL=0.90); $z α 2 z α 2$ = z0.05 = 1.645.
Summary: Effect of Changing the Sample Size
• Increasing the sample size causes the error bound to decrease, making the confidence interval narrower.
• Decreasing the sample size causes the error bound to increase, making the confidence interval wider.
Try It 8.5
Refer back to the pizza-delivery Try It exercise. The mean delivery time is 36 minutes and the population standard deviation is six minutes. Assume the sample size is changed to 50 restaurants with the same sample mean. Find a 90% confidence interval estimate for the population mean delivery time.
### Working Backwards to Find the Error Bound or Sample Mean
When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. However, sometimes when we read statistical studies, the study may state the confidence interval only. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean.
Finding the Error Bound
• From the upper value for the interval, subtract the sample mean,
• OR, from the upper value for the interval, subtract the lower value. Then divide the difference by two.
Finding the Sample Mean
• Subtract the error bound from the upper value of the confidence interval,
• OR, average the upper and lower endpoints of the confidence interval.
Notice that there are two methods to perform each calculation. You can choose the method that is easier to use with the information you know.
### Example 8.6
Suppose we know that a confidence interval is (67.18, 68.82) and we want to find the error bound. We may know that the sample mean is 68, or perhaps our source only gave the confidence interval and did not tell us the value of the sample mean.
Calculate the Error Bound:
• If we know that the sample mean is 68: EBM = 68.82 – 68 = 0.82.
• If we don't know the sample mean: EBM = $(68.82−67.18) 2 (68.82−67.18) 2$ = 0.82.
Calculate the Sample Mean:
• If we know the error bound: $x ¯ x ¯$ = 68.82 – 0.82 = 68
• If we don't know the error bound: $x ¯ x ¯$ = $(67.18+68.82) 2 (67.18+68.82) 2$ = 68.
Try It 8.6
Suppose we know that a confidence interval is (42.12, 47.88). Find the error bound and the sample mean.
### Calculating the Sample Size n
If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.
The error bound formula for a population mean when the population standard deviation is known is
EBM = $( z α 2 )( σ n ) ( z α 2 )( σ n )$.
The formula for sample size is n = $z 2 σ 2 EB M 2 z 2 σ 2 EB M 2$, found by solving the error bound formula for n.
In this formula, z is $z α 2 z α 2$, corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.
### Example 8.7
The population standard deviation for the age of Foothill College students is 15 years. If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed?
• From the problem, we know that σ = 15 and EBM = 2.
• z = z0.025 = 1.96, because the confidence level is 95%.
• n = $z 2 σ 2 EB M 2 z 2 σ 2 EB M 2$ = $( 1.96 ) 2 ( 15 ) 2 2 2 ( 1.96 ) 2 ( 15 ) 2 2 2$ = 216.09 using the sample size equation.
• Use n = 217: Always round the answer UP to the next higher integer to ensure that the sample size is large enough.
Therefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students.
Try It 8.7
The population standard deviation for the height of high school basketball players is three inches. If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed? | 2020-08-04 06:18:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 60, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8151518106460571, "perplexity": 697.5784790917193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735860.28/warc/CC-MAIN-20200804043709-20200804073709-00491.warc.gz"} |
https://history.jes.su/missionerskaya-deyatelnost-nikolaya-ivanovicha-ilminskogo-v-ufimskoy-gubernii/?sl=en | Scale, Meaning, Pattern, Form: Conceptual Challenges for Quantitative Literary Studies
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Scale, Meaning, Pattern, Form: Conceptual Challenges for Quantitative Literary Studies
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S207987840001646-9-1
DOI
10.18254/S0001646-9-1
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Abstract
Digital format introduced in research of literature the new scale of the archive: previously several hundred novels of the nineteenth century could be studied, now it’s possible to analyze thousands, tens of thousands, and soon hundreds of thousands of texts. The article observes the problem to identify the meaning in statistically large sets, establishing patterns and their relationship with forms. If we turn the daily experience of reading literature in the abstract distributions, after we find the patterns and isolate patterns of formal relations that underpin them, it would be the last, and for many of us the main step: use all these innovations to return to the sociological understanding of literature in a completely new basis.
Keywords
quantitative literary studies, distant reading, scale, meaning, pattern, form
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11.11.2016
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01.12.2016
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References
Additional sources and materials
André Leroi-Gourham. Gesture and Speech. MIT Press, 1993. P. 148.
Chris Anderson. The End of The Data Deluge Makes the Scientific Method Obsolete [Ehlektronnyj resurs]. URL: http://archive.wired.com/ science/discoveries/magazine/16–07/pb_theory/
d’Arcy W. Thompson. On Growth and Form. Cambridge, 1917. P. 11.
Elena Tognini Bonelli. The overview of the evolution of corpus linguistics // Anne O’Keeff and Michael McCarthy / eds. The Routledge Handbook of Corpus Linguistics. Routledge; New York, 2010. P. 19.
Fernand Braudel. History and the social sciences. The longue durée // On History, Chicago University Press. 1980. P. 29.
Krzysztof Pomian. L’Ordre du temps. Paris, 1984. P. 31.
Leo Spitzer. Linguistics and Literary History, 1948 // Representative essays. Stanford University Press, 1988. P. 32.
Max Weber. “Objectivity” in social science and social policy // The Methodology of Social Sciences. Glencoe: The Free Press. P. 90, 97. | 2020-01-27 21:59:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24431955814361572, "perplexity": 9660.977482823526}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251728207.68/warc/CC-MAIN-20200127205148-20200127235148-00117.warc.gz"} |
https://en.wikibooks.org/wiki/Sound_Synthesis_Theory/Sound_in_the_Digital_Domain | # Sound in the Digital Domain
## Introduction
Digital systems (e.g. computers) and formats (e.g. CD) are clearly the most popular and commonplace methods of storing and manipulating audio. Since the introduction of the compact disc in the early 1980s, the digital format has provided increasingly greater storage capacity and the ability to store audio information at an acceptable quality. Although analogue formats still exist (vinyl, tape), they typically serve a niche audience. Digital systems are ubiquitous in modern music technology. It must be stressed that there is no argument as to whether one domain, be it analogue or digital is superior, but the following provides some desirable features of working with audio in the digital domain.
• Storage. The amount of digital audio data capable of being stored on a modern hard drive is far greater than a tape system. Furthermore, we can choose the quality of the captured audio data, which relates directly to file size and other factors.
• Control. By storing audio information in digital, we can perform powerful and complex operations on the data that would be extremely difficult to realise otherwise.
• Durability. Digital audio can be copied across devices without any loss of information. Furthermore, many systems employ error correction codes to compensate for wear and tear on a physical digital format such as a compact disc.
## Digital <-> Analogue Conversion
Acoustic information (sound waves) are treated as signals. As demonstrated in the previous chapter, we traditionally view these signals as varying amplitude over time. In analogue systems, this generally means that the amplitude is represented by a continuous voltage; but inside a digital system, the signal must be stored as a stream of discrete values.
Figure 2.1. An overview of the digital <-> analogue conversion process.
Digital data stored in this way has no real physical meaning; one could describe a song on a computer as just an array of numbers; these numbers are meaningless unless there exists within the system a process that can interpret each number in sequence appropriately. Fig. 2.1 shows an overview of the process of capturing analogue sound and converting it into a digital stream of numbers for storage and manipulation in such a system. The steps are as follows:
1. An input such as a microphone converts acoustic air pressure variations (sound waves) into variations in voltage.
2. An analogue to digital converter (ADC) converts the varying voltage into a stream of digital values by taking a 'snapshot' of the voltage at a point in time and assigning it a value depending on its amplitude. It typically takes these 'snapshots' thousands of times a second, the rate at which is known as the sample rate.
3. The numerical data is stored on the digital system and then subsequently manipulated or analysed by the user.
4. The numerical data is re-read and streamed out of the digital system.
5. A digital to analogue converter (DAC) converts the stream of digital values back to a varying voltage.
6. A loudspeaker converts the voltage to variations in air pressure (sound).
Although the signal at each stage comes in a different form (sound energy, digital values etc.), the information is analogous. However, due to the nature of the conversion process, this data may become manipulated and distorted. For instance, low values for sample rates or other factors at the ADC might mean that the continuous analogue signal is not represented with enough detail and subsequently the information will be distorted. There are also imperfections in physical devices such as microphones which further "colour" the signal in some way. It is for this reason that musicians and engineers aim to use the most high-quality equipment and processes in order to preserve the integrity of the original sound throughout the process. Musicians and engineers must consider what other processes their music will go through before consumption, too (radio transmission etc.).
## Sampling
Sound waves in their natural acoustic form can be considered continuous; that is, their time-domain graphs are smooth lines on all zoom factors without any breaks or jumps. We cannot have these breaks, or discontinuities because sound cannot switch instantaneously between two values. An example of this may be an idealised waveform like a square wave - on paper, it switches between 1 and -1 amplitude at a point instantaneously; however a loudspeaker cannot, by the laws of physics, jump between two points in no time at all, the cone has to travel through a continuous path from one point to the next.
Figure 2.2. Discrete samples (red) of a continuous waveform (grey).
Sampling is the process of taking a continuous, acoustic waveform and converting it into a digital stream of discrete numbers. An ADC measures the amplitude of the input at a regular rate creating a stream of values which represent the waveform in digital. The output is then created by passing these values to the DAC, which drives a loudspeaker appropriately. By measuring the amplitude many thousands of times a second, we create a "picture" of the sound which is of sufficient quality to human ears. The more and more we increase this sample rate, the more accurately a waveform is represented and reproduced.
### Nyquist-Shannon sampling theorem
The frequency of a signal has implications for its representation, especially at very high frequencies. As discussed in the previous chapter, the frequency of a sine wave is the number of cycles per second. If we have a sample rate of 20000 samples per second (20 kHz), it is clear that a high frequency sinusoid such as 9000 Hz is going to have fewer "snapshots" than a sinusoid at 150 Hz. Eventually there reaches a point where there are not enough sample points to be able to record the cycle of a waveform, which leads us to the following important requirement:
The sample rate must be greater than twice the maximum frequency represented.
Why is this? The minimum number of sample points required to represent a sine wave is two, but we need at least slightly more than this so that we're not dependent phase (samples at exactly twice the sine wave frequency, the samples may fall on the peaks of the sine wave, or on the zero crossings). It may seem apparent at this time that using just two points to represent a continuous curve such as a sinusoid would result in a crude approximation - a square wave. And, inside the digital system, this is true. However, both ADCs and DACs have low-pass filters set at half the sample rate (the highest representable frequency). What this means for input and output is that any frequency above the cut-off point is removed and it follows from this that the crude sine representation - a square wave in theory - becomes filtered down to a single frequency (i.e. a sine wave). From this, we have two mathematical results:
${\displaystyle F_{s}>2f_{max}}$ and ${\displaystyle F_{N}={\frac {F_{s}}{2}}}$
Where ${\displaystyle F_{s}}$ is the sample rate, ${\displaystyle f_{max}}$ is the highest frequency in the signal. ${\displaystyle F_{N}}$ is the Nyquist frequency. Frequencies over the Nyquist frequency are normally blocked by filters before conversion to the digital domain when recording; without such processes there would be frequency component foldover, otherwise known as aliasing.
### Sampling accuracy and bit depth
It has been established that the higher the sample rate, the more accurate the representation of a waveform in a digital system. However, although there are many reasons and arguments for higher sample rates, there are two general standards: 44100 samples per second and 48000 samples per second, with the former being most commonplace. The main consideration for this is the fact that the human hearing range extends, at maximum, to an approximate limit (that varies from person to person) of 20000 Hz. Frequencies above this are inaudible. Considering the example of 44.1 kHz, we find that the Nyquist frequency evaluates to 22050 Hz, which is more than the human hearing system is capable of perceiving. There are other reasons for this particular sample rate, but that is beyond the scope of this book.
Figure 2.3. Effects of increased sample rate and bit depth on representing a continuous analogue signal.
There is one more important factor to consider when considering the sampling process: bit depth. Bit depth represents the precision with which the amplitude is measured. In the same way that there are a limited amount of samples per second in a conversion process, there are also a limited amount of amplitude values for a sample point, and the greater the number, the greater the accuracy. A common bit resolution found in most standard digital audio systems (Hi-Fi, Compact Disc) is 16 binary bits which allows for a range of 65536 (${\displaystyle 2^{16}}$) individual amplitude values at a point in time. Lower bit values result in a greater distortion of the sound - a two bit system (${\displaystyle 2^{2}}$) only allows for four different amplitudes, which results in a massively inaccurate approximation of the input signal. | 2017-10-17 13:28:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5379524827003479, "perplexity": 604.8778653449676}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187821189.10/warc/CC-MAIN-20171017125144-20171017145144-00499.warc.gz"} |
https://www.studyadda.com/ncert-solution/11th-chemistry-classification-of-elements-and-periodicity-in-properties_q77/492/38235 | • # question_answer 77) Ionisation enthalpies of elements of second period are given below : Ionisation enthalpy (kcal$mo{{l}^{-1}}$): 520, 899, 801, 1086, 1402, 1314, 1681, 2080 Match the correct enthalpy with the elements and complete the graph given in the following figure. Also symbols of elements with their atomic numbers.
Symbol Li Be B C N O F Ne At. Number 3 4 5 6 7 8 9 10 $\Delta {{H}_{1}}(kJ\,mo{{l}^{-1}})$ 520 899 801 1086 1402 1314 1681 2080
In a period, the value of ionisation enthalpy increases from left to right with breaks where the atoms have somewhat stable configurations. Be and N have higher values than expected. Be has fully filled orbital while N has half filled orbitals. Be: $1{{s}^{2}},2{{s}^{2}}$ N : $1{{s}^{2}},2{{s}^{2}},2p_{x}^{1},2p_{y}^{1}2p_{z}^{1}$ Complete the graph yourself and compare with the graph given in problem 38. | 2020-07-06 10:12:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6303042769432068, "perplexity": 837.2629713943791}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655890157.10/warc/CC-MAIN-20200706073443-20200706103443-00533.warc.gz"} |
https://web2.0calc.com/questions/help_14302 | +0
help
0
171
2
+626
How many three-digit positive integers exist, all of whose digits are 2's and/or 5's?
Oct 9, 2018
#1
+99618
+1
222
225
252
255
522
525
552
555
Oct 9, 2018
#2
+2340
0
Listing all the possible arrangements is one way to approach this problem; however, if I told you that this number has 10 or 100 digits, would you be as willing to write out all of them? I doubt it.
The first digit has two possibilities; it can either be a 2 or a 5.
The second digit has two possibilities; it can either be a 2 or a 5.
The third digit has two possibilities; it can either be a 2 or a 5.
$$2*2*2=8 \text{ integers}$$
.
Oct 9, 2018 | 2019-04-24 17:08:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6067850589752197, "perplexity": 460.86455002814523}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578650225.76/warc/CC-MAIN-20190424154437-20190424180437-00141.warc.gz"} |
http://math.stackexchange.com/questions/195978/the-discriminant-of-an-integral-binary-quadratic-form-and-the-discriminant-of-a | # The discriminant of an integral binary quadratic form and the discriminant of a quadratic number field
Let $ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. Let $D = b^2 - 4ac$ be its discriminant. It is easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$).
Conversely suppose $D$ is a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Then there exists an integral binary quadratic form of discriminant $D$(see this question).
Is the following proposition true? If yes, how do we prove it?
Proposition Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Then $D$ can be written uniquely as $D = f^2 d$, where $f$ is a positive integer and $d$ is the discriminant of a quadratic number field.
- | 2013-05-21 19:09:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.994805634021759, "perplexity": 28.426787644315574}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368700438490/warc/CC-MAIN-20130516103358-00004-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-evaluate-4-50-3-8 | # How do you evaluate (4+√50) - (3-√8)?
Mar 10, 2018
See a solution process below:
#### Explanation:
First, we can rewrite the expression as:
$4 + \sqrt{50} - 3 + \sqrt{8} \implies$
$4 - 3 + \sqrt{50} + \sqrt{8} \implies$
$1 + \sqrt{50} + \sqrt{8}$
Next, we can rewrite the radicals as:
$1 + \sqrt{25 \cdot 2} + \sqrt{4 \cdot 2}$
Then, we can use this rule for radicals to simplify the radicals:
$\sqrt{\textcolor{red}{a} \cdot \textcolor{b l u e}{b}} = \sqrt{\textcolor{red}{a}} \cdot \sqrt{\textcolor{b l u e}{b}}$
$1 + \sqrt{\textcolor{red}{25} \cdot \textcolor{b l u e}{2}} + \sqrt{\textcolor{red}{4} \cdot \textcolor{b l u e}{2}} \implies$
$1 + \sqrt{\textcolor{red}{25}} \sqrt{\textcolor{b l u e}{2}} + \sqrt{\textcolor{red}{4}} \sqrt{\textcolor{b l u e}{2}} \implies$
$1 + 5 \sqrt{\textcolor{b l u e}{2}} + 2 \sqrt{\textcolor{b l u e}{2}}$
Now, we can factor out the common term to complete the evaluation:
$1 + \left(5 + 2\right) \sqrt{\textcolor{b l u e}{2}}$
$1 + 7 \sqrt{\textcolor{b l u e}{2}}$ | 2021-09-19 11:03:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 10, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9740211963653564, "perplexity": 1442.5445691146517}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056856.4/warc/CC-MAIN-20210919095911-20210919125911-00473.warc.gz"} |
http://lesswrong.com/lw/1om/bizarre_illusions/ | # Bizarre Illusions
11 27 January 2010 06:25PM
Illusions are cool. They make me think something is happening when it isn't. When offered the classic illusion pictured to the right, I wonder at the color of A and B. How weird, bizarre, and incredible.
Today I looked at the above illusion and thought, "Why do I keep thinking A and B are different colors? Obviously, something is wrong with how I am thinking about colors." I am being stupid when my I look at this illusion and I interpret the data in such a way to determine distinct colors. My expectations of reality and the information being transmitted and received are not lining up. If they were, the illusion wouldn't be an illusion.
The number 2 is prime; the number 6 is not. What about the number 1? Prime is defined as a natural number with exactly two divisors. 1 is an illusionary prime if you use a poor definition such as, "Prime is a number that is only divisible by itself and 1." Building on these bad assumptions could result in all sorts of weird results much like dividing by 0 can make it look like 2 = 1. What a tricky illusion!
An optical illusion is only bizarre if you are making a bad assumption about how your visual system is supposed to be working. It is a flaw in the Map, not the Territory. I should stop thinking that the visual system is reporting RGB style colors. It isn't. And, now that I know this, I am suddenly curious about what it is reporting. I have dropped a bad belief and am looking for a replacement. In this case, my visual system is distinguishing between something else entirely. Now that I have the right answer, this optical illusion should become as uninteresting as questioning whether 1 is prime. It should stop being weird, bizarre, and incredible. It merely highlights an obvious reality.
Addendum: This post was edited to fix a few problems and errors. If you are at all interested in more details behind the illusion presented here, there are a handful of excellent comments below.
Sort By: Best
Comment author: 28 January 2010 01:26:42AM 18 points [-]
Why do I keep thinking A and B are different colors?
Meanwhile I am thinking 'Wow! My brain can automatically reconstruct a 3D image from limited 2D input and even compensate for shadows and lighting. That is orders of magnitude more complex than the reverse, generating such images from a model such as those we add 3D cards to computers for'.
I don't particularly consider this an 'illusion', especially when it is not simultaneously acknowledged that it is an 'illusion' that A and B are squares on a 3D 'square X a bit' thing that also has a cylinder on top of it.
Comment author: 28 January 2010 04:11:25PM 8 points [-]
Wow, good point! I never thought about it like that. It raises the question: Why are people amazed when you say, "Tiles A and B are actually the same color -- check for yourself!" but they roll their eyes when you say, "There are no squares in this image -- check for yourself!"? In both cases, you can respond with, "Well, yeah -- if you don't interpret it like the scene it's trying to represent!"
I'm not a very good artist, so learning about how to create these illusions sounds like a good reason to take an art class, and help me appreciate what artists are doing. (Why didn't the first major breakthrough in cognitive science come from painters and sketchers?)
Of course, it probably wouldn't do much to help me understand why they can count random smears on a canvas as "art"...
Comment author: 28 January 2010 04:41:10PM 0 points [-]
That last line is coming from a decidedly unrational state of mind!
Comment author: 28 January 2010 07:22:56PM * 2 points [-]
How so? I wasn't spouting the usual greedy/fake reductionist cliches; I was talking about the paintings that look like a 3-year-old made a mess, yet get classified as art, and noting that an art class probably wouldn't convince me this is appropriate.
What specific criticism of that claim do you have?
Comment author: 28 January 2010 07:48:56PM * 4 points [-]
I was talking about the paintings that look like a 3-year-old made a mess, yet get classified as art, and noting that an art class probably wouldn't convince me this is appropriate.
What specific criticism of that claim do you have?
Short version:
High art is about a lot of things, not least of which is impact on the viewer.
In the case of Pollock, for instance, a lot of the interesting thing that was going on there, was that he depicted a process - not by painting a representation of himself doing it, but by actually doing it. You can look at a Pollock and see how he constructed it without being distracted by exactly what he was trying to construct. And being able to see that aspect of art and be aware of it, will in turn give you a greater appreciation of medieval cathedrals and Greek sculpture.
Possibly related: no one knows what science doesn't know
Comment author: 28 January 2010 08:34:00PM 2 points [-]
To add to this with a similar example, consider that some people prefer listening to foreign language vocalists because it allows one to appreciate the sound of the vocal instrument without focusing on the words.
Comment author: 29 January 2010 11:42:46PM 0 points [-]
To me, most music sounds like a foreign language (though one that sounds exactly like English), unless I'm familiar with the lyrics beforehand, in which case I can "hear" them just fine.
Comment author: 29 January 2010 11:49:08PM 2 points [-]
Like this?
Comment author: 31 January 2010 05:41:34AM 0 points [-]
Probably.
Comment author: 29 January 2010 11:54:02PM -1 points [-]
Yes, most people don't care much about the actual lyrics. Which explains the phenomenon thomblake was trying to use for tenuous support of another hypothesis, yet remains modded to 3 for some reason (7 if you include his parent comment).
Comment author: 28 January 2010 09:54:01PM 0 points [-]
Right, because the words (i.e. the lyrical semantics, as differentiated from the qualities of the sounds the words make) are a small, perhaps negligible component of what people like about many of these songs.
If you were trying to draw some other inference from this fact, you're going to have to be more specific about why that inference follows.
Comment author: 28 January 2010 08:06:12PM * 1 point [-]
Definitely related: Truly Part of You
If I erased your knowledge (and everyone else's) of what the kewl kids had classified as "good art", would it grow back? Would you eventually re-recognize the same works as being good, with the same relative merit, for the same reasons?
If your answer is no, that's a big red flag that you're dealing in bullshit.
(The correct reaction to the parable of The Emporer's New Clothes is not, "Well of course a kid isn't going to see the clothes! What's your point?")
Comment author: 28 January 2010 09:48:14PM 2 points [-]
If I erased your knowledge (and everyone else's) of what the kewl kids had classified as "good art", would it grow back? Would you eventually re-recognize the same works as being good, with the same relative merit, for the same reasons?
I imagine that the answer to this is yes for a great deal of art. I don't know much about it myself, but when I think about art that I like I can find reasons aside from cultural significance or peer pressure.
This assumes the question is ignoring the lens created by my limited expose to art. I highly doubt that any of the artists I like would have been experienced by me if others hadn't considered them worthwhile.
Music is an easier analogy for me to make. I can more accurately describe what I like in music because I know a few more terms. But also, when I listen to a song, I find that my opinions are more distinct. I assume this is because my tastes are becoming refined; I am open to other interpretations.
Comment author: 28 January 2010 08:20:10PM 2 points [-]
That's not related.
If you took away everyone's knowledge of English, and someone laid King Lear at your feet, what would you do with that? The fact that art is rooted in culture and context, some of which is the result of stochastic processes, does not mean you're dealing in bullshit.
Comment author: 28 January 2010 08:37:27PM 3 points [-]
I would be very surprised to discover that a King Lear in an unfamiliar language had been produced by an ape. I am not surprised by hoaxes like this. I think that is indicative of a meaningful difference.
Comment author: 28 January 2010 09:51:46PM * 1 point [-]
After Peter had created a number of paintings, Axelsson chose what he considered to be the four best and arranged to have them exhibited in an art show at the Christina Gallery.
Emphasis added to indicate flaw in experimental protocol.
Edit: This point is much weaker than it appears at first glance. See responses.
Comment author: 28 January 2010 10:03:00PM 1 point [-]
I would still be surprised if the monkey King Lear was chosen as the very best of the monkey's literary oeuvre.
Comment author: 28 January 2010 09:57:37PM 1 point [-]
Yeah, I noticed that too. I felt that it was still a valid test of critics' ability to interpret art considering that most artists will do the same thing with their collection before entering an exhibition.
Comment author: 28 January 2010 08:45:51PM 0 points [-]
That's a far cry away from "eventually re-recogniz[ing] the same works as being good, with the same relative merit, for the same reasons."
Comment author: 28 January 2010 08:49:43PM 0 points [-]
Erasing everyone's knowledge of English is a far cry from erasing their knowledge of "what the kewl kids had classified as 'good art'".
Comment author: 28 January 2010 08:44:28PM 1 point [-]
But it does mean that the writing of King Lear is less of an epistemic achievement than, say, the laws of physics, which are not dependent on a particular species' form of communication.
If King Lear is (claimed to be) a good work, given a certain language (humanity? evolutionary history? political history?), does the recognition of its supposed greatness survive deletion of the knowledge about what the kewl kids think is great?
If people continued to speak English, but King Lear fell out of fashion and later was found, but disconnected from anyone's recommendation, would people still decide it was better than most other works? Would they decide it for the same reason?
Do children spontaneously flock to King Lear at a certain age, even when it's not recommended to them by a True Literary Authority?
Comment author: 28 January 2010 08:49:32PM 0 points [-]
If King Lear is (claimed to be) a good work, given a certain language (humanity? evolutionary history? political history?), does the recognition of its supposed greatness survive deletion of the knowledge about what the kewl kids think is great?
Of course not. It doesn't even come with 3D special effects!
Comment author: 28 January 2010 08:16:28PM 0 points [-]
If your answer is no, that's a big red flag that you're dealing in bullshit.
I seriously doubt that the correct answer is "no".
Obviously, there would be a little bit of wobble - I might not care who Pollock is, but I expect there would be something else I'd find that would illuminate the same aspects of the aesthetic experience. But I think being the first to do it that way counts for something.
Comment author: 28 January 2010 08:36:23PM * 2 points [-]
Sorry, all I got out of that was a name-drop and (what seemed like) a dodge.
Could you answer again, and this time maybe explain it a little differently? Specifically:
-Are you claiming that Pollock discovered a way of satisfying the aesthetic senses that allowed generalization of the method in other forms?
-Let's say I knew a wacko who believed that "By historical accident, Pollock became a focal point for people of high-status to identify each other, despite there being nothing special about his work." What evidence would you point me to that has a low Bayes factor against that hypothesis?
Comment author: 28 January 2010 09:07:47PM 1 point [-]
I'm claiming that Pollock's work demonstrated easily-neglected and valuable parts of the aesthetic experience.
As for evidence against that hypothesis, I think that depends largely upon how seriously you take some of the relevant premises in your wacko's model. According to some, there is virtually nothing to all of culture other than status games (though in this case the clause "despite there being nothing special about his work" would make little sense). According to others, there really is quite a bit to aesthetics, and perhaps it's worth listening to the folks who've spent their lives studying it.
There are a lot of different kinds of things in the world, and many of them are valuable in unexpected ways.
Comment author: 28 January 2010 09:28:01PM * 1 point [-]
So you really can't think of anything that is less likely to be observed if "it's all bullshit" than if it's not? There isn't any kind of aesthetic feeling you could feed to the wacko that he couldn't help but burst out in appreciation for?
As for evidence against that hypothesis, I think that depends largely upon how seriously you take some of the relevant premises in your wacko's model.
Not when I'm asking for evidence with a low Bayes factor, rather than a guaranteed low posterior.
Maybe an example would be in order. Let's say Bob is the wacko, but about quantum physics. Bob believe that the claims of quantum physics are just a big status game, and so are the results of the particle accelerators and everything. I could point to evidence like the atom bomb. If they were just arguing over meaningless crap the whole time, and assigning truth purely based on who has the most status, how did they ever get the understanding necessary to build an atom bomb, Bob?
There are a lot of different kinds of things in the world, and many of them are valuable in unexpected ways.
Right, like Halo. Except that millions of people like Halo even in the absence of a well-funded indoctrination campaign, and the fact that expressing appreciation for Halo won't endear them to the kewl kids of art.
It's not very impressive if people start to enjoy something after they've
spent their lives studying it.
You have to adjust for stuff like that.
Comment author: 28 January 2010 09:09:51PM 0 points [-]
Well said!
Comment author: 28 January 2010 09:25:04PM 1 point [-]
I just realized we seem to be arguing over wine again. I fold.
Comment author: 28 January 2010 09:34:15PM * 1 point [-]
Yes, we are.
If you spend ten years associating wine with a good time, and are expected to have a refined palette for wine to be part of the kewl kids club, then guess what -- you can make yourself like wine! The fact that you like wine in such a scenario does not, to me, count as a genuine liking, in the sense in which I judge beverages. Any substance, even bat urine, will find connosieurs under those conditions!
What I want to do is, find out what's good about something, that isn't simply an artifact of practices that can make anything look good.
That's why I'm not impressed by "enjoy this because people are telling you to enjoy it", which the support for much high art and alcohol amounts to.
Instead of refusing to engage the issue, maybe you should start to think about the recursivity of your criteria for quality?
Comment author: 28 January 2010 07:53:01PM 0 points [-]
Comment author: 28 January 2010 08:05:11PM 1 point [-]
Meh.
Never trust a site with that many banners.
I, for one, have never heard seriously disparaging things about 19th-century or Academic art in general, and the essay sounds a bit... rabid. It seems like the writer was operating under the common heuristic "I don't understand it, so it must be stupid."
Comment author: 28 January 2010 10:53:51PM 1 point [-]
And I never trust anyone that says, "No, no, this stuff is good, it really is! Just pay for 10 years worth of education in this specific area, and then you'll see the light!"
(Note the similarity to Scientology practices...)
Comment author: 28 January 2010 11:04:32PM 2 points [-]
But then what do you do when something really does take that long to explain? People say category theory is beautiful; is the nonmathematician supposed to call them liars?
Comment author: 28 January 2010 11:09:41PM 2 points [-]
Category theory doesn't take 10 years to explain. You should be able to explain to a willing, intelligent friend in two full days, and get them to a point where they see the beauty.
I've done similar things, like explaining the elegant beauty of aircraft component structural analysis -- got a decent appreciation across in 10 minutes. ("You know how a chain is as strong as its weakest link? A component is as strong as its weakest failure mode...")
The point is, you can explain it. That's a lot more than you can do for (much of) art, it seems.
Comment author: 28 January 2010 11:12:27PM 0 points [-]
At least where category theory is concerned, you don't have to pay.
Comment author: 28 January 2010 07:35:38PM * 0 points [-]
Disliking Pollock is irrational. As is disliking Cage. Or Joyce. Or PEZ.
Comment author: 28 January 2010 07:37:07PM 4 points [-]
I love 4'33". It helps me get to sleep.
Comment author: 28 January 2010 07:47:42PM 1 point [-]
Edit: OMG! Is my humor too subtle people? PEZ!
It was, yes - maybe we need to make emoticons more normal here, since this is a recurring problem. :P
(Downvote removed)
Comment author: 28 January 2010 08:45:57PM * 2 points [-]
maybe we need to make emoticons more normal here, since this is a recurring problem.
But then how will we signal our sophistication and mutually affirm our status as an intellectual subculture? ;)
Comment author: 29 January 2010 02:32:51AM 2 points [-]
Anime references. Duh.
Comment author: 28 January 2010 08:47:55PM 2 points [-]
:P~~~~~
;D
Comment author: 28 January 2010 08:43:40PM 0 points [-]
Neutral vote. I like the PEZ juxtaposition but 'arational' would fit better. A simply false assertion doesn't fit well with the irony.
Comment author: 29 January 2010 12:01:43AM 0 points [-]
As it was mocking bgrah's assertion, and bgrah used "unrational", and in my estimation his meaning was closer to "irrational" than "arational", I used the former. Perhaps using "unrational" would have been better, though.
Comment author: 28 January 2010 07:44:52PM 0 points [-]
Just consider it evidence of the level of culture you'll find hereabouts. Savages.
Comment author: 29 January 2010 01:53:09AM 1 point [-]
Why are people amazed when you say, "Tiles A and B are actually the same color -- check for yourself!" but they roll their eyes heir eyes when you say, "There are no squares in this image -- check for yourself!"?
It's about expectations. People expect to be able to take (physical) objects that appear to be different colors, examine them under a variety of contexts, and always perceive them as different. People incorrectly extrapolate that expectation to images, and thus find the fact that removing the context reveals these images to be the same colorRGB surprising. They also expect to be presented with representations, so pointing out the fact that they're looking at a representation seems silly to them.
Comment author: 28 January 2010 08:39:50PM 1 point [-]
Of course, it probably wouldn't do much to help me understand why they can count random smears on a canvas as "art"...
By way of reversing the ADBOC concept, I disagree denotationally but confirm your connotation. As you explain in the cousin several times removed post, many kinds of art are bullshit. Cultural preferences that would not be particularly likely to be rediscovered if all trace was removed. This differs from other forms of art which are more specifically directed at aesthetic preferences intrinsic to humans.
Of course, immersing yourself in a culture and experiencing the flow of status first hand is the perhaps the best way to get an intuitive anthropological understanding. I found, for example, that having done a research degree in a subfield of AI helps me understand how peer affiliation by persisting with researching silly ideas can be counted as 'science'.
http://lesswrong.com/lw/1om/bizarre_illusions/1iub
Comment author: 27 January 2010 10:41:38PM * 11 points [-]
I identified a useful and cogent point in your post and it was this: Whenever you receive data from any source (your brain, your eyes, a drug study, Less Wrong) you've got to be aware of how that data has already been packaged. Taking the data at face value -- for example imagining your brain is actually making a claim about the RGB values of the pixels -- can lead to problems, misconceptions, mistakes.
Comment author: 27 January 2010 07:12:28PM 8 points [-]
Your eye didn't evolve to report trivia like, "These two colors are actually the same." Your eye is reporting the most useful information - from which direction the light is coming, the shaded region under it, and the fact that the floor is tiled.
Which is more amazing - this picture, or a picture that somehow tricked the average person into noticing two colors were the same, but didn't notice the picture also had floor tiling, light directionality, and shading? I'd say this picture is pretty tame in comparison to the picture that could do that.
Comment author: 27 January 2010 07:55:46PM 13 points [-]
I am being stupid when my eye looks at this illusion and I interpret the data in such a way to determine distinct colors.
Not at all. In the context of the scene that this picture represents, A and B are absolutely different shades. On the contrary, I think your perceptual system would be poor indeed if it did not reconstruct context, and under-interpreted the picture as a meaningless 2D array of pixels.
(BTW, as with the necker cube, I find that I can consciously exert to experience the interpretation that I choose, without too much difficulty.)
Comment author: 27 January 2010 10:03:13PM 3 points [-]
Hmm. I can with the necker cube, but not at all with this one.
Comment author: 27 January 2010 10:13:39PM 6 points [-]
I was never able to do it with this one before, either. What I'm doing now is concentrating hard on the two tiles of interest, until the rest of the picture fades into the background. The two tiles then seem to be floating on a separate top layer, and appear to be the same shade.
Comment author: 28 January 2010 02:51:29AM 2 points [-]
That worked! Cool!
Comment author: 28 January 2010 08:12:03AM * 1 point [-]
If you go to an Art or Design school. Seeing and producing illusions like this is one of the assignments that they usually will give you in a 2D design class.
As it has been described above, if you can concentrate (if school, we learn how to look at them by squinting as we would when discerning simple shape or color - or, if you have ever learned how to look at one of those weird 3D images made out of what looks to be paint splatter) on the two squares, then you will be able to see that they are indeed the same shade (not color, color is used to describe something else)
Comment author: 28 January 2010 11:36:44AM * 2 points [-]
Ah, that worked for me. For people wondering how to do the technique to see "Magic Eye " images, you focus your eyes so that the image doubles and and overlaps the image. That causes a stereoscopic illusion when done on any things that overlap. You could practice it here. Focus your eyes so that the the first abc overlaps the second abc -- now you have three abc's in your vision, the 1st and 3rd abc are being seen out of one eye and the abc in the middle appears to be almost 3d.
a...b...c......................a...b...c
In this case, I could see that A and B are the same color by tilting my head and then focusing so I saw a double image of A overlapping B.
Comment author: 29 January 2010 08:02:25AM 1 point [-]
Exactly... We spent a total of 6 weeks in Art School design class learning how to do this specific trick with a variety of images. From color, to line length (you know those "which line is longer" tricks that make you think one line is longer when they are usually the same length), to line thickness, to shading and tinting aliasing.
We spent those weeks consuming a lot of aspirin and Tylenol.
Comment author: 28 January 2010 08:18:10AM 0 points [-]
Interesting, when I try this technique the shades seem even more distinct.
Comment author: 28 January 2010 11:21:12AM 1 point [-]
It takes some practice.
We were taught that if you put your nose right in the center of the image, and then let your focus go, and pull back from the image, that at a certain distance from the image (as your focus is still at ∞) various structures of the image will begin to resolve. So contrasts, similarities, and shades will all resolve at different focal lengths from the image.
It was rare that any one person would be able to pick up immediately upon all the effects perceptible in an image. I was able to pick up on certain shades of the color green that are used in contrast to red, but it took me a long time to get the shading of black-white (as in this optical illusion - and it is but one of many).
When we were tested on this, we would not be told what was similar, or where optical tricks were used, and we would have to pick them out of an image (and this was long before the internet, so we couldn't just go online to do a search for optical illusions to find images to study that had their illusions spelled out for us). So, it is a skill that can be learned. For me, eventually I had to learn how to focus upon each square with a different eye, while squinting, and letting the focus go back and forth between my right and left eye. eventually, I get the images resolved as a single shade as I go back and forth between my eyes.
Comment author: 27 January 2010 10:39:06PM * -1 points [-]
The point of the illusion is that they seem different in context. Ignoring context to make them appear similar isn't a proper resolution.
Comment author: 28 January 2010 03:56:49AM 0 points [-]
I found two ways to do it myself:
(A) Cover up areas of the image to see what causes what to change color in your perception. Slowly reveal the full image again and sometimes A and B look alike
(B) Let your focus drift until the lines of the image get fuzzy. Look at the two squares without actually looking at them. I find that the colors look alike here. If I "snap" focus back they still look alike but nothing is fuzzy anymore.
B works better.
Comment author: 27 January 2010 08:48:25PM 1 point [-]
I don't understand this. Are you saying that A and B are not the same color?
Comment author: 27 January 2010 09:22:07PM * 12 points [-]
AndyWood gave a good explanation, but let me elaborate. If you saw the scene depicted, but in real life -- rather than on a flat paper or 2D screen -- you would be correct to infer that the actual, invariant colors of the tiles are different. But, since they are just pixels on paper or a screen, their invariant colors are the same, and yet your eyes tell you otherwise.
So are the eyes "wrong" in any serious sense? Well, let me put it this way: do you want
a) a visual system that gives the right interpretation of scenes that you are actually going to encounter often, but is tripped up by carefully designed optical illusions?
or do you want:
b) a visual system that gives the right interpretation for carefully designed optical illusions, but fails to catch many attributes of common scenes?
(Yes, there is a tradeoff. Your visual system encounters an "inverse optics" problem: given the retina images, what is the scene you're looking at made of? This is ill-posed: many scenes can generate the same retinal images. E.g. a given square could be far away and big, or close and small. To constrain the solution set, you need assumptions, and any set of assumptions will get some scenes wrong.)
Yes, you are wrong to think that the tiles have different colors. You are not wrong to prefer a visual system that gets most scenes right at the cost of getting a few scenes (like this one) wrong.
(Incidentally, I really like this optical illusion, and have it by my desk at work. What's so great about it is that once you see it, you can actually strip away everything that you think is causing the illusion, and yet they still look different!)
Comment author: 27 January 2010 08:58:35PM 5 points [-]
Your understanding of the word 'colour' does not match what properties of the world your brain is trying to identify and categorize when it interprets 'colour'. The interesting constant property of objects in the world that makes 'colour' useful to your visual system for purposes of object identification and categorization is really the surface properties that interact with incident lighting. Your brain attempts to ignore effects due to lighting variation and assign a 'colour' label to objects that is more or less an invariant property of the surface under a variety of different lighting conditions. This is in general not a solvable problem since the same incident photons can be produced by a number of different lighting and material combinations. Optical illusions like this merely reveal the heuristics your visual system uses to identify the relevant constant aspects of the scene and ignore the irrelevant lighting variation. They generally work quite well.
Comment author: 27 January 2010 09:09:26PM 4 points [-]
When we covered this phenomenon in my psychology degree it was referred to as colour constancy. I now work as a 3D graphics programmer and so know a lot about the physics of light transport. The illusion does not surprise me any more, in fact it seems a little surprising that I ever could have thought that the RGB colour value of an onscreen pixel was directly related to the property of objects in the real world that we call 'colour'.
Comment author: 27 January 2010 09:47:50PM 0 points [-]
Well said (including your later comment about color constancy). Along the same lines, this is why cameras often show objects in shadows as blacked out -- because that's the actual image it's getting, and the image your own retinas get! It's just that your brain has cleverly subtracted out the impact of the shadow before presenting it to you, so you can still see significant contrast and colors in the shadowed objects.
Comment author: 27 January 2010 10:14:25PM 4 points [-]
Along the same lines, this is why cameras often show objects in shadows as blacked out -- because that's the actual image it's getting, and the image your own retinas get! It's just that your brain has cleverly subtracted out the impact of the shadow before presenting it to you
That doesn't explain why faithful reproductions of images with shadows don't prompt the same reinterpretation by your brain.
Comment author: 27 January 2010 11:32:35PM 4 points [-]
Blacked out shadows are generally an indication of a failure to generate a 'faithful' reproduction due to dynamic range limitations of the camera and/or display medium. There is a fair amount of research into how to work around these limitations through tone mapping. High Dynamic Range cameras and displays are also an area of active research. There's not really anything to explain here beyond the fact that we currently lack the capture or display capability to faithfully reproduce such scenes.
Comment author: 27 January 2010 11:00:48PM 0 points [-]
Sure it does -- Faithful reproductions give the shadowed portion the appropriate colors for matching how your brain would perceive a real-life shadowed portion of a scene.
Comment author: 27 January 2010 11:26:08PM * 0 points [-]
Umm, that's not what I meant by "faithful reproductions", and I have a hard time understanding how you could have misunderstood me. Say you took a photograph using the exact visual input over some 70 square degrees of your visual field, and then compared the photograph to that same view, trying to control for all the relevant variables*. You seem to be saying that the photograph would show the shadows as darker, but I don't see how that's possible. I am familiar with the phenomenon, but I'm not sure where I go wrong in my thought experiment.
* photo correctly lit, held so that it subtends 70 square degrees of your visual field, with your head in the same place as the camera was, etc.
Comment author: 27 January 2010 11:31:17PM 0 points [-]
I thought you meant "faithful" in the sense of "seeing this is like seeing the real thing", not "seeing this is learning what your retinas actually get". If you show a photograph that shows exactly what hit the film (no filters or processing), then dark portions stay dark.
When you see the scene in real life, you subtract off the average coloring that can be deceiving. When you see the photo, you see it as a photo, and you use your current real-life-background and lighting to determine the average color of your visual field. The darkness on the photo deviates significantly from this, while it does not so deviate when you're immersed in the actual scene, and have enough information about the shadow for your brain to subtract off the excessive blackness.
Been a long day, hope I'm making sense.
Comment author: 27 January 2010 10:08:24PM 4 points [-]
As others have pointed out, the difficulty here is more in the semantics of "color" than in the optics.
As a simplification, we can consider the projected color.P of a tile to be a product of its surface properties (color.S) and the intensity of the incident light. The illusion straightforwardly contrives one of these terms - the light intensity - so that the color.P of tile A equals the color.P of tile B. But the brain, interpreting the image as a 3D scene with light and shadow, reports the color.S-es of the tiles, which are different under that very reasonable and useful interpretation.
I'm sorry if this is a big distraction from the point of your post. I'm still interested in the point, so perhaps you can find another way of getting it across.
Comment author: 27 January 2010 10:45:56PM 2 points [-]
As others have pointed out, the difficulty here is more in the semantics of "color" than in the optics.
Yeah. I missed the semantic shift. All it took was someone pointing out that there were two uses of Color drifting around and almost all the comments snapped back into making sense.
I'm sorry if this is a big distraction from the point of your post. I'm still interested in the point, so perhaps you can find another way of getting it across.
The point is that an illusion generally gives off a sense of bizarreness because we are expecting X but the illusion gives us Y. In the case of the color example, I once expected boxes A and B to appear to be the same color (perceived) if and only if they were the same color (RGB). The illusion shows this is not the case. Being curious, I sought to understand the underlying principles behind why we perceive two different colors. Once this is understood, the illusion should no longer seem bizarre but a trivial example of the underlying principles.
In trying to find where I went wrong with the post, I come up with this:
• "Color" is an extremely ambiguous term. I should have seen this one coming.
• I think some people thought I was trying to give an explanation of the illusion in the post. I was not.
• I think some people thought I was saying that the visual system itself was stupid or broken and we needed to "fix" our brain to adjust for its shortcomings. I was not. I was trying to say that our feeling of "bizarre" was stupid because we are expecting something different from our visual system than what the visual system provides.
• I deliberately wrote this post more aggressively and concisely than I generally write. Perhaps this degraded its clarity even further.
I am half tempted to take this post down, rewrite it, and put it back up, but I don't know how much that would help.
Comment author: 27 January 2010 11:08:03PM 4 points [-]
Well, don't do anything that takes down the comment section. Many of the comments are insightful and, um, say things that should have been in your original post.
Demystifying optical illusions, and visual cognition in general, is a very good exercise in rationalist reduction.
Comment author: 27 January 2010 11:22:22PM 0 points [-]
Okay. Do you think it would be valuable to just edit the post in place?
As best as I can tell, these are the trouble paragraphs:
Today I looked at the above illusion and thought, "Why do I keep thinking A and B are different colors? Obviously, that is not what my visual system is trying to tell me." I am being stupid when my eye looks at this illusion and I interpret the data in such a way to determine distinct colors. That information is not being transmitted and received. If it were, the illusion wouldn't be an illusion.
An optical illusion is only bizarre if you are making a bad assumption about how your visual system is supposed to be working. It is a flaw in the Map, not the Territory. I should stop thinking that the system is reporting True Colors. It isn't. And, now that I know this, I am suddenly curious about what it is reporting. I have dropped a bad belief and am looking for a replacement. Once I have found the right answer, this optical illusion should become as uninteresting as questioning whether 1 is prime. It should stop being weird, bizarre, and incredible. It should highlight an obvious reality.
Is this better?:
Today I looked at the above illusion and thought, "Why do I keep thinking A and B are different colors? Obviously, something is wrong with how I am thinking about colors." I am being stupid when I look at this illusion and interpret the data in such a way to determine distinct colors. My expectations of Reality and the information being transmitted and received are not lining up. If it were, the illusion wouldn't be an illusion.
An optical illusion is only bizarre if you are making a bad assumption about how your visual system is supposed to be working. It is a flaw in the Map, not the Territory. I should stop thinking that the system is reporting True Colors. It isn't. And, now that I know this, I am suddenly curious about what it is reporting. I have dropped a bad belief and am looking for a replacement. In this case, the visual system is distinguishing between [what term goes here?], not individual RGB style colors. Now that I have right answer, this optical illusion should become as uninteresting as questioning whether 1 is prime. It stops being weird, bizarre, and incredible. It merely highlights an obvious reality.
Comment author: 28 January 2010 12:07:48AM * 1 point [-]
It seems to me that you are still using the word colour in a way that suggests you haven't really grasped the insight that makes this illusion seem not-bizarre. That insight is fundamentally that the statement "this ball is blue" is not equivalent to the statement "a digital photo of a scene containing this ball would have pixel values of 0, 0, 255 at pixel locations where light from the ball reached the sensor". It is a much more complex (and more useful) statement than that. The bad assumption is that 'colour' when used to refer to a property of objects in the world determined through visual perception has any simple relationship with RGB values recorded by a digital camera. You still seem to be talking as if RGB values are somehow 'true' colours.
Comment author: 28 January 2010 01:37:54AM * 1 point [-]
The bad assumption is that 'colour' when used to refer to a property of objects in the world determined through visual perception has any simple relationship with RGB values recorded by a digital camera.
Especially in the case of human tetrachromats.
Comment author: 28 January 2010 12:22:30AM * 0 points [-]
I am trying to find a way to say what you said with one phrase or word. I feel like I am struggling to find a term.
Comment author: 28 January 2010 12:35:30AM 4 points [-]
I think the key for me in understanding this type of illusion (and the general phenomenon of colour constancy) was to realize that 'colour' in common usage ("this ball is blue") is perceived as a property of objects and we infer it indirectly based on light that reaches our retinas. That light also has a 'colour' (subtly different meaning) but it is not something we perceive directly because it is not very useful in itself.
This makes perfect sense when you think about it from an evolutionary perspective - we evolved to recognize invariant properties of objects in the world (possibly fruit in trees for primates) under widely varying lighting conditions. Directly perceiving the 'colour' (RGB) of light would not tell us anything very useful about invariant object properties. There is enough overlap between the two meanings of colour for them to be easily confused however and that is really the root of this particular illusion.
In computer graphics we commonly use the term 'material' to describe the set of properties of a surface that govern how it responds to incident light. This encompasses properties beyond simple colour ("shiny blue ball", "matte blue ball", "metallic blue ball"). I don't know if that usage is well understood outside of the computer graphics field however.
Comment author: 28 January 2010 01:12:39AM 0 points [-]
I completely agree with you. At this point, I am just trying to clean up the article to help clarify the answer behind the illusion. Does the phrase, "I should stop thinking that the visual system is reporting RGB style colors" mesh okay? That is the only location of RGB as of this edit.
Comment author: 28 January 2010 12:32:49AM 0 points [-]
How is "Color gross of lighting conditions"?
Comment author: 27 January 2010 11:26:23PM 1 point [-]
It sounds like you're trying to come up with a sentence or two that captures all of the insight on color that the commenters have given. While I'm a big fan of summarizing, and a big critic of those who can't, I don't think you can get it to work here. Instead of your final bolded change (the others are good), just point to or quote a few good comments that show what the visual system is doing, and how the optical illusions trick it.
Comment author: 27 January 2010 11:33:58PM * 0 points [-]
In this case, the visual system is not trying to distinguish between individual RGB style colors (more details in the comments).
EDIT: I updated it with something similar. Hopefully it was an improvement. :) Thanks again for your help (which isn't to say that I wouldn't mind more help...)
Comment author: 27 January 2010 11:16:56PM * 0 points [-]
I am half tempted to take this post down, rewrite it, and put it back up, but I don't know how much that would help.
Taking this and SilasBarta's thoughts together: can you apply this same meta-principle to something substantially different in a new post, written with a recognition of these confusions? That post could cite this post with a "Followup to:" line, and elaborate on your discovery in some way.
Comment author: 27 January 2010 11:23:42PM 0 points [-]
Would it be better to just replace the content of this post? I can archive the original in a comment here for future context.
Comment author: 28 January 2010 03:06:52AM 4 points [-]
I would be disinclined to that course, but hard-pressed to justify it more effectively than by my idiosyncratic generalization of one of a number of principles I have heard - I quote from the post:
Don't try to rewrite history. Look, we make mistakes. We all do. Sometimes we post an essay and we get stuff wrong in it. Sometimes that stuff makes the whole essay wrong. Sometimes, we put up an essay innocently and it turns into a firestorm of controversy we never meant. Sometimes, we find ourselves in a crucible on all sides.
The temptation is to go back. Revise. Reword what we said. Take the essay down entirely.
It is never a good idea. Ever.
I don't think you have anything to be ashamed of in this post. It's not deep, it's not extraordinary in its conclusions, but it is correct and brief. The complaints seem to me best addressed by elaboration and discussion - things which require far more than a brief edit placed at the end of the post.
As SilasBarta mentioned, there's a lot of commentary on this post that is worth preserving, and should be preserved with the original post. It would be unfair to the commenters to render their comments incomprehensible - even briefly - by distortion of that to which they responded.
And, if I may be frank, if the idea which inspired this post is interesting, it is probably capable of generalization. The idea of my own which I promoted to a post I did so because I saw that it was applicable beyond the scope of its origination, and in a manner which was natural, elegant, and interesting. It proved of interest to a number of people here, despite its unabashedly algebraic treatment. If you can find a profitable extension of your concept, it will be likely to be worth reporting in a followup post (and if you are concerned about the appropriateness of it, I - as one remaining upvoter of the OP - will have sent my email to you in a PM, and be willing to comment on any draft you wish to send).
If you cannot find a profitable extension of your concept, it is probably not worth the time to revise. Consider your post dubiously successful (it is still in positive territory, is it not?) and leave it be.
Comment author: 28 January 2010 03:50:47AM 1 point [-]
I don't think you have anything to be ashamed of in this post. It's not deep, it's not extraordinary in its conclusions, but it is correct and brief. The complaints seem to me best addressed by elaboration and discussion - things which require far more than a brief edit placed at the end of the post.
It's not so much that I am ashamed; I am just frustrated. The behavior of this post caught me completely off-guard. It was upvoted to +5 within a few hours and people started asking questions. After my responses, the post dropped to +1. The karma itself doesn't mean much to me, but the feedback here was evidence of something greater than a non-interesting or incorrect post.
People were willing to talk about it, so I stuck it out for as much feedback as I could. The investment was completely worth it. I got several comments worth of extremely valuable insights to my writing style and how to better post here at LessWrong.
I think the post itself failed, but the whole experience has been a net gain.
As SilasBarta mentioned, there's a lot of commentary on this post that is worth preserving, and should be preserved with the original post. It would be unfair to the commenters to render their comments incomprehensible - even briefly - by distortion of that to which they responded.
I agree. My intent in the revisions has been to keep people from being distracted by my quirks and leading them into a wonderful discussion in the comments. This particular illusion has a lot more history behind it than I originally thought; I learned a lot.
And, if I may be frank, if the idea which inspired this post is interesting, it is probably capable of generalization. The idea of my own which I promoted to a post I did so because I saw that it was applicable beyond the scope of its origination, and in a manner which was natural, elegant, and interesting. It proved of interest to a number of people here, despite its unabashedly algebraic treatment. If you can find a profitable extension of your concept, it will be likely to be worth reporting in a followup post (and if you are concerned about the appropriateness of it, I - as one remaining upvoter of the OP - will have sent my email to you in a PM, and be willing to comment on any draft you wish to send).
Thank you very much. I have to sit on the events of today and ponder if there is a next step to take. If a followup is coming I will certainly take you up on your offer.
Comment author: 28 January 2010 03:19:53AM 0 points [-]
An addendum - as far as my recollection of the original goes, your edits appear reasonable, although I would not have risked them on my own post. I congratulate you on a successful revision, but my offer stands.
Comment author: 27 January 2010 08:51:28PM * 3 points [-]
In the spirit of not disputing definitons, may I suggest: A and B are the same colorRGB, but, interpreting the image as a picture (as the eye does), not the same colorALBEDO.
Edit: Correction - "as the visual system does".
Comment author: 27 January 2010 09:00:11PM 1 point [-]
This is perhaps beating a dead horse, but "albedo" is supposed to be a ratio between reflected and incident lights, and I would bet that the albedo of these two patches of screen is also identical, just as their RGB values are identical.
Comment author: 27 January 2010 09:05:58PM 2 points [-]
Not in the actual 3D scene that your brain interprets the picture to be of, only in the context of a 2D printout of the image (albedo is not really a relevant property for emissive display devices like LCDs or CRTs).
Comment author: 27 January 2010 09:04:59PM * 2 points [-]
When I said "interpreting the image as a picture", I meant, "interpreting the image as a picture of a checkerboard with a cylinder casting a shadow on it" - the albedos in question are of the squares A and B on the depicted board.
Comment author: 27 January 2010 09:10:58PM 3 points [-]
Ah, "inferred albedo". In that case we agree.
Comment author: 27 January 2010 08:55:08PM 0 points [-]
Thank you.
Comment author: 27 January 2010 09:11:56PM 2 points [-]
They are the same screen-color, but different inferred-colors.
Comment author: 27 January 2010 11:42:21PM 4 points [-]
I am being stupid when my eye looks at this illusion and I interpret the data in such a way to determine distinct colors.
Tell that to your ancestors who escaped from the saber-tooth cat hiding in the shadows at dusk.
Comment author: 28 January 2010 12:29:44AM 2 points [-]
A side note: The only reason that prime numbers are defined in such a way as to exclude 1 and negative numbers is because mathematicians found this way of defining them a bit more useful than the alternative possibilities. Mathematicians generally desire for important theorems to be stated in a manner that is as simple as possible, and the theorems about primes are generally simpler if we exclude 1. There is a more detailed analysis of this question here:
Comment author: 27 January 2010 10:31:23PM * 2 points [-]
if you use a poor definition such as, "Prime is a number that is only divisible by itself and 1."
I have a fondness for this particular definition, and like to think of 1 as a "very special" prime number. To the extent that I usually give a little speech whenever an opportunity arises that (ahem) the only reason I know of that '1' is excluded from the primes (more often than not) is because almost every theorem about prime numbers would have to be modified with an "except 1" clause. But a natural definition (anything along the lines of "already completely factored") would include it. If you disagree, which definition --- or the satisfaction of which theorem -- do you think is more compelling?
(Just in case you perceived you were getting too much heat about "colour"...)
Comment author: 27 January 2010 10:40:03PM 11 points [-]
I'd have to pretty strongly disagree. To me, the "essence" of primes is that you can factor any number into primes in a unique way. That's the most natural definition. They're the multiplicative building blocks of the natural numbers; everything can be reduced to them. If 1 were prime, you could no longer factor uniquely.
Comment author: 27 January 2010 10:45:32PM * 4 points [-]
Your comment and this comment were adjacent in my message folder which I found amusing.
Thomblake wrote:
Caring about what's right might be as arbitrary (in some objective sense) as caring about what's prime, but we do actually happen to care about what's right.
It's funny how we do care.
Comment author: 27 January 2010 11:52:09PM 5 points [-]
Your comment and this comment were adjacent in my message folder which I found amusing.
That's awesome. Thanks for sharing.
Comment author: 27 January 2010 10:50:06PM 1 point [-]
Possibly related: my comment about paperclips.
Comment author: 28 January 2010 12:41:00AM * 3 points [-]
Hmm... I agree this is compelling. However, since I'm resistant to updating my world view about 1-the-discriminated-prime-number, I'll continue to proffer counter-arguments:
• the Fundamental Theorem of Arithmetic is pretty important, but may still not be the "essence" of what prime is
• the FTA itself requires the "except 1" clause: "all natural numbers can be uniquely factored into primes except 1" -- which would make someone thing 1 ought to be prime
• the FTA already assumes 'modulo permutations', we could easily throw in 'modulo 1'
• Wikipedia -- the first and last authority on such things -- carefully writes in an entire sentence unto itself, "The number 1 is by definition not a prime number," suggesting just how arbitrary this is. (My own emphasis added.)
The best argument I came up with for not including 1 as prime, because I tend to worry about how things are constructed, was with the seive of Eratosthenes.
The seive of Eratosthenes says that you can find the primes by starting with all the natural numbers > 1; let 2 be the first prime number, and then begin eliminating all multiples of 2 and the multiples of subsequent primes as you find them. If you included '1' in the first step, then you would eliminate all the numbers in the first step.
Comment author: 28 January 2010 01:43:25AM * 14 points [-]
I think you're really failing to grasp the content of the unique factorization theorem here. Firstly we don't think about factored numbers as products of primes up to permutation, we think of them as products of distinct prime powers (up to permutation, I suppose - but it's probably better here to just take a commutative viewpoint and not regard "up to permutation" as worth specifying). But more importantly, you need to take a multiary view of multiplication here, not a binary one. 1 is the empty product, so in particular, it is the product of no primes, or the product of each prime to the 0th power. That is its unique prime factorization. To take 1 as a prime would be like having bases for vector spaces include 0. Almost exactly like it - if we take the Z-module of positive rationals under multiplication, the set of primes forms a free basis; 1 is the zero element.
Comment author: 28 January 2010 04:52:13PM * 2 points [-]
Information and expertise like this is why hanging out at Less Wrong is worth the time. I estimate that I value the information in your comment at about $35, meaning my present self would advise my former self to pay up to$35 to read it.
So, I get it. My brain is more wired for analysis than algebra; so this isn't the first time that linear algebra has been a useful bridge for me. I see that we could have a 'vector space' of infinite-dimensional vectors where each vector (a1, a2, ..., an, ...) represents a number N where N = (P1^a1)(P2^a2)...(Pn^an)... and Pi are the ordered primes. Clearly 1 is the zero element and would never be a basis element.
I should admit here that my background in algebra is weak and I have no idea how you would need to modify the notion of 'vector space' to make certain things line up. But I can already speculate on how the choice of the "scalar field" for specifying the a_i would have interesting consequences:
• non-negative integer 'scalar field' --> the positive integers,
• all integers 'scalar field' --> positive rational numbers,
• complex integers --> finally include the negative rationals.
I'd like to read more. What sub-field of mathematics is this?
Comment author: 08 February 2010 10:05:35AM 0 points [-]
I see that we could have a 'vector space' of infinite-dimensional vectors where each vector (a1, a2, ..., an, ...) represents a number N where N = (P1^a1)(P2^a2)...(Pn^an)... and Pi are the ordered primes.
Oh! And orthogonal vectors are relatively prime!
Comment author: 08 February 2010 11:15:17AM 1 point [-]
I'm not sure that the idea of orthogonality is defined for modules, is it? Is there a standard definition of an inner product for a Z-module?
Comment author: 08 February 2010 06:49:48PM 1 point [-]
I'm not sure that the idea of orthogonality is defined for modules, is it? Is there a standard definition of an inner product for a Z-module?
Yes; the same definition works. See here.
Comment author: 09 February 2010 12:05:14AM 0 points [-]
Yay! I actually got something right!
Comment author: 28 January 2010 05:07:58PM 0 points [-]
What sub-field of mathematics is this?
Number theory, ne? Or is that too general?
Comment author: 28 January 2010 05:57:34PM 2 points [-]
It looks like it's more abstract algebra (possibly applied to number theory) that byrnema is interested in. Check out Wikipedia on module.
Comment author: 28 January 2010 06:32:53PM 1 point [-]
Precisely! Thanks also.
Comment author: 28 January 2010 05:33:24PM * 1 point [-]
A second comment...
You've certainly convinced me that '1' should not be included in the set of things that are used to uniquely factor numbers. However, how I can I know if this set is the set of "primes"?
I guess I was thinking that the essence of primes was about their irreducibility/atomic-ness. The number 5 would be considered prime because you can't describe it multiplicatively in any way except by using the number 5. Using my preferred notion, the number 0 and the number -1 would also be "prime" (as Mr Hen guessed). Is there a different word for this concept?
Comment author: 28 January 2010 05:47:59PM * 1 point [-]
See wikipedia on natural generalizations of prime numbers. In particular note that most of the definitions say "units" instead of "1", like "Irreducible elements are ones which cannot be written as a product of two ring elements that are not units." which rules out 0 for the integers, +, x and includes the possibility of multiple units (-1 and 1).
I don't know offhand of any nice, commonly referenced property P(S,O) that is: A,x,y in a structure S with operation O: A is P just when if x O y = A then either x = A or y = A. Which I believe is the general property you're thinking about?
Edit: with O commutative I do believe
Comment author: 28 January 2010 05:56:55PM 0 points [-]
Thank you. And yes, that is the property.
Comment author: 28 January 2010 04:58:05AM * 0 points [-]
For some reason, I never imagined factors this way.
18 = 3^2 * 2^1
97,020 = 2^2 * 3^2 * 5 * 7^2 * 11
I suppose I have seen them printed out that way, but the deeper structure there never clicked. Cool.
Comment author: 28 January 2010 08:27:20AM * 1 point [-]
As it happens I'm partway through "An Introduction to the Theory of Numbers" by Niven, Zuckerman, and Montgomery at the moment. Lots of problems are incredibly easy to solve given this structure. The example that springs to mind is the very straightforward proof why the combinatorial formula n! / (r! (n-r)!) always gives you an integer.
Update: Well having been scored up I feel like I should give a hint on the actual proof: for any prime p and any n, the greatest power of p that divides n is
\sigma_{i=1}^{\infty} floor( \over{n}{p^i} )
and for any real numbers a, b, floor(a + b) >= floor(a) + floor(b).
Oh for real TeX markup!
Comment author: 28 January 2010 05:12:39PM 0 points [-]
Do you recommend the book? If I were interested in the subject, is this good to pick up or can you think of a better option?
Comment author: 28 January 2010 06:10:59PM 0 points [-]
I'm enjoying it, but it touches on abstract algebra as an alternative approach rather than leaning on it for everything; I'd kind of prefer the latter.
Comment author: 28 January 2010 06:40:17PM * 1 point [-]
You may be a good person to ask this question:
I was wondering if there was a function f(x, y, z) so that x and z represent the left and right sides of common mathematic operators and y represents the level of operation. So f(1, 2, 4) would be 1 + 4 and f(2, 2, 4) would be 2 * 4. Better versions of f(x, y, z) would have fewer end cases hardcoded into it.
The reason behind this is to handle operator levels greater than addition, multiplication, and exponents. The casual analysis from my grade school and undergrad level math shows the pattern that multiplication is repeated addition and exponents are repeated multiplication.
My quick attempts at coming up with such a function are spiraling into greater and greater complexities. I figured someone else has to have thought about this. Do you know of a place I can start reading up on ideas similar to this? Is what I am doing even plausible?
Quick thoughts based on me playing around:
• Addition may be level 0, not level 1
• The sequences never really look exactly like multiplication tables, but the patterns are similar enough to appease me
• Ideally, everything can be reduced to the simple concept of X + 1 so as to walk along the number line
• In practical terms, I have no idea how to express "negative" levels. Division and roots are unapproachable at this point in my playing around.
Comment author: 28 January 2010 06:49:26PM * 4 points [-]
Comment author: 28 January 2010 06:55:50PM * 1 point [-]
Hyper operators. You can represent even bigger numbers with Conway chained arrow notation. Eliezer's 3^^^^3 is a form of hyper operator notation, where ^ is exponentiation, ^^ is tetration, ^^^ is pentation, etc.
If you've ever looked into really big numbers, you'll find info about Ackermann's function, which is trivially convertable to hyper notation. There's also Busy Beaver numbers, which grow faster than any computable function.
Comment author: 28 January 2010 05:52:15AM 3 points [-]
Wikipedia -- the first and last authority on such things -- carefully writes in an entire sentence unto itself, "The number 1 is by definition not a prime number," suggesting just how arbitrary this is.
A measure of the arbitrariness is the history, which is that 1 was considered prime up to the 19th century and was a matter of fashion during the 19th century. That suggests that unique factorization is not, in itself, enough to motivate the definition. Perhaps its extension to the gaussian integers or the more radical version for general number rings prompted the definiton.
Comment author: 28 January 2010 08:01:47AM 0 points [-]
This reminds me. Pre-19th century it was thought that part of what it was to be a mammal was to give live birth, in addition to having mammary glands. 1 is the platypus of numbers.
Comment author: 27 January 2010 10:56:51PM 4 points [-]
natural definition: "A prime is a natural number with exactly two factors"
I'm not sure I quite understand your suggestion: we should define 1 as prime, but then write "except for 1" every time we use the word prime? Wouldn't it be quicker just to exclude 1 in the first place (even if there were some sense in which 1 was prime)?
Comment author: 27 January 2010 11:28:39PM 1 point [-]
But a natural definition (anything along the lines of "already completely factored") would include it.
How do you see 0 or -1, using this definition?
Comment author: 28 January 2010 01:07:01AM * 0 points [-]
A factor of a number M is a number that evenly divides M with no remainder. Zero has infinitely many factors, definitely not prime.
...regarding -1, I can't think of anything relevant that I know about the relationship between negative numbers and prime numbers.
Later edit: Then I completely changed my mind... and decided 0 and -1 should be prime relative to how I would define it's essence. I note that you intuited what I really meant by prime better than I did!
Comment author: 28 January 2010 01:13:38AM 0 points [-]
Yeah, I was just curious. I like toying around with the fundamentals behind the maths and seeing what happens. :)
Comment author: 27 January 2010 06:49:02PM 2 points [-]
Uh? What do you mean by "obvious" in that last sentence?
(Post otherwise interesting, and I for one like them short.)
Comment author: 27 January 2010 07:21:56PM * 0 points [-]
"Obvious" as in not "weird, bizarre, or incredible." Would "simple" be a better word there?
Comment author: 27 January 2010 07:33:23PM 2 points [-]
I'm just not seeing what obvious reality it highlights, so either I'm particularly dense or it's not in fact obvious.
So, rephrasing: what reality is being highlighted by the "illusion" ?
Your prime number analogy suggests that it's in fact the "both colors are the same" assertion which is an illusion. The perceptual reality is that the pixels in these areas are discriminated as different colors. The illusion consists of looking at pixel with identical RGB values and thinking "Oh, these have the same position in colorspace, I expect my brain to perceive them as identical."
The reality suggested by the "illusion" is that this expectation doesn't hold in general, it's a stupid model. A smarter model would take more things into account before it predicted what our brain will perceive as identical colors.
But this is very much non-obvious...
Comment author: 27 January 2010 07:52:22PM 1 point [-]
I'm just not seeing what obvious reality it highlights, so either I'm particularly dense or it's not in fact obvious. [...] But this is very much non-obvious...
The post is keying off of Think Like Reality.
Reality has been around since long before you showed up. Don't go calling it nasty names like "bizarre" or "incredible". The universe was propagating complex amplitudes through configuration space for ten billion years before life ever emerged on Earth. Quantum physics is not "weird". You are weird. You have the absolutely bizarre idea that reality ought to consist of little billiard balls bopping around, when in fact reality is a perfectly normal cloud of complex amplitude in configuration space. This is your problem, not reality's, and you are the one who needs to change.
Looking at the image it should be obvious that the colors do not look the same. This is reality. We think they should look the same even though it is obvious they don't. Once we find the right answer to why they don't look the same, the illusion should stop being bizarre.
If you find an explanation, return to the illusion, and still think the illusion is bizarre, than something is wrong. You fall into the category that EY is discussing in Think Like Reality.
I am convinced that most of what we consider to be fancy illusions will be considered obvious to future generations. They will look at this image and wonder why we thought it was so fascinating. When our optics system is solved it would completely ridiculous to assume that we would look at that image and think that the two squares should look like the same color.
Comment author: 27 January 2010 08:19:46PM * 4 points [-]
But your post hasn't offered an explanation. And I don't, in fact, look at that image and think that the two squares should look like the same color.
A and B are in fact different colors, for a value of "in fact" which takes into account that the picture is a picture of something - a checkerboard. My visual system makes the correct inference, conditioned on the assumption that I'm looking at a checkerboard.
EDIT: what I should say is that I'm still surprised, knowing what I know about my visual system and how it works, when you tell me that the pixels have the same RGB values. But that's not a "reality is weird" surprise, it's more like the surprise of learning some interesting bit of trivia.
To really be totally unsurprised, I'd have to enhance not just my knowledge of the visual system, but my visual system - to include an RGB calibration system.
Comment author: 27 January 2010 08:33:50PM * 1 point [-]
EDIT: Oh, okay, I read your edit and that makes much more sense. I agree that it may be difficult to get to the point of being unsurprised. Getting there isn't obvious. You know you are there when you are unsurprised by the illusion. Once Reality is unsurprising and obvious, you are there.
I feel like I have lost the point of this conversation. What, in the following, do you disagree with?
• The image is an optical illusion
• The squares marked A and B appear to be different colors
• In reality, the squares marked A and B are the same color
• It is more correct to say that A and B are the same color than to say they are different colors
• The reason behind the optical illusion explains why A and B appear to be different colors
• This reason is contained somewhere inside of the "visual system"
• It is better to not be surprised by Reality
• The squares A and B should appear to be different colors
• We should not be surprised when A and B look like different colors
• It is incorrect to call Reality bizarre as per Think Like Reality
• "Obvious" is not a good word for the opposite of "bizarre"
Comment author: 27 January 2010 08:37:45PM 2 points [-]
I disagree with #3 and #4. Also #6, mildly - it's not just our visual system that's at issue here, it's our color vocabulary and our "meta" thinking about our color system that explains the "illusion" - that explains why we might think it bizarre.
Comment author: 27 January 2010 08:53:16PM 1 point [-]
Oh, okay. Well... I guess I don't feel like debating the definition of color since that is completely irrelevant to the point of the post. I wish this was made clear earlier.
Uh? What do you mean by "obvious" in that last sentence?
It has nothing to do with the actual answer to the example illusion. What I mean is that once we have the answer to an illusion, the illusion should stop surprising us.
Comment author: 27 January 2010 09:09:08PM * 3 points [-]
Neither do I feel like debating the definition of color.
What I am is disappointed. You brought up the "color constancy" phenomenon as an instance where "think like reality" is applicable, and then failed to follow through with an analysis of what is actually going on. You sound as if you are content to know that the phenomenon is in principle explicable; as if the post has done its job by demonstrating your commitment to "thinking like reality". I would prefer you had gone deeper into the object-level analysis and offered your own explanation of what is going on in this particular case.
This is a little bit like parents who lecture their children about the importance of being truthful, vs. parents who demonstrate being truthful - and being OK with confronting unpleasant truths.
EDIT: I didn't mean to sound sanctimonious (I realize I do sound sanctimonious). My main intent is to express a wish regarding what I'd like to see in future posts of this type.
Comment author: 27 January 2010 09:38:53PM 2 points [-]
What I am is disappointed. You brought up the "color constancy" phenomenon as an instance where "think like reality" is applicable, and then failed to follow through with an analysis of what is actually going on.
The point of this post is not to debate, discuss, or analyze color constancy. The point of the post is to talk about illusions and how we think of them as bizarre when we shouldn't.
I have not once debated color or the theories behind the illusion. All I did was use a word one way when other people use it another way. I am not trying to offer some strange, new truth about a picture. I used it as an example because people recognize it, not because that particular example matters.
EDIT: I didn't mean to sound sanctimonious (I realize I do sound sanctimonious). My main intent is to express a wish regarding what I'd like to see in future posts of this type.
I apparently have completely missed with this post. I have watched its karma swing all over the place in just a few hours and all of the 36 comments so far are talking about something I consider completely irrelevant to the intended point. The same thing happened with my last post, too, so something is very off in my expectations regarding people's responses to the post. Something I did caused you to expect something that I had absolutely no intention of providing. It sucks for you and it sucks for me.
I don't really mind you sounding sanctimonious because I don't care about our relative moral status. I find it frustrating that we had to go back and forth so long to end up where we did. I am not fully convinced my point ever did get across, but at least now I know how you perceived the post. Hopefully by the next one I will have figured out what went wrong.
Comment author: 27 January 2010 07:46:04PM 0 points [-]
In other words, this is the visual version of "if a tree falls in the forest...", except that we already defined 'color' as qualia rather than wavelengths, right?
Comment author: 27 January 2010 09:55:01PM * 3 points [-]
Since you mention it, that's something I should have brought up in one of the Mitchell_Porter consciousness threads: the colors you see are not actually matched up one-to-one with the wavelengths hitting your retina. Rather, the visual system does something like subtracting away the average color.
Meaning, the color that you experience seeing depends on all the colors in the scene, not just the wavelength of the light coming off each specific object.
Some people were talking as if you were getting direct knowledge of (something equivalently expressible as) wavelengths, which is unfortunate, since part of the path to demystifying qualia is understanding this kind of processing.
Comment author: 27 January 2010 07:59:45PM 0 points [-]
Um, "we already defined" - the referent(s) of that phrase are very ambiguous, I'm afraid. Who's "we" and where was that definition ?
I definitely agree that color discriminations in the brain (the processes that eventually end up with color words coming out of our mouths) are about way more than wavelengths. I'd prefer the term "discrimination" to "qualia", the latter carries philosophical baggage that I'd rather do without.
Comment author: 27 January 2010 08:06:26PM 0 points [-]
Um, "we already defined" - the referent(s) of that phrase are very ambiguous, I'm afraid. Who's "we" and where was that definition ?
It's a royal 'we' in this case: Some subset of the group of commenters here at LW, and that subset doesn't include me. It was discussed at some length in the recent discussion of consciousness. I wasn't paying much direct attention to the conversation, though, so I can't be more specific than that. (I'm not even sure that the relevant bits are all in one post's comments.)
Comment author: 27 January 2010 07:38:44PM 1 point [-]
I think what you're trying to say is that the phenomenon becomes highly expected, not aberrant with respect to your world model.
Comment author: 27 January 2010 07:55:24PM 2 points [-]
Yes.
Comment author: 18 February 2010 08:07:51PM * 1 point [-]
There's a rather awesome colour constancy optical illusion in this American Scientist article - click on the enlarge image link on the rubik's cube image. I've mirrored the image here in case the link goes dead. The blue tiles in the left image are the same shade of grey in RGB terms as the yellow tiles in the right image. H/T to this article.
Comment author: 28 January 2010 05:16:08AM 1 point [-]
If I look at the picture without focusing (i.e. thereby seeing it as 2D), A and B look the same color.
If I look at the picture and focus but cover the green cylinder with my hand, they can look either the same or different (depending on whether or not I notice the shadow in that case.)
I agree with Andy Woods: this is no illusion at all, except in the sense that it is an "illusion" that the board is three dimensional.
Comment author: 27 January 2010 07:35:46PM * 0 points [-]
Since the topics are related (but I'll admit I'm biased toward seeing that) - maybe a "Related" link to the "Adaptive bias" post would make sense.
Comment author: 27 January 2010 07:13:41PM * 0 points [-]
"Why do I keep thinking A and B are different colors? Obviously, that is not what my eyes are trying to tell me." I am being stupid when my eye looks at this illusion and I interpret the data in such a way to determine distinct colors. That information is not being transmitted by my eye. If it were, the illusion wouldn't be an illusion.
I think this is a bit misleading. To the extent that your eyes can be thought to be trying to tell you anything, "these are different colors" is exactly what they are trying to say. It's fallacious to assume that there is some sort of "true" input that one's eyes are reporting, and which the post-processing stages corrupt. (I'm not saying you committed this mistake, but people might get that impression.) Instead the meaning emerges purely from the post-processing stages. In fact, they're doing a pretty darn good job, as one might note from the fact that optical illusions nearly never noticeably distort our judgment in natural conditions.
Comment author: 27 January 2010 07:21:07PM 1 point [-]
Yes, I completely glossed over the finer points of eye-brain interaction. I did not think it was needed to get the point across. I suppose I also used a bit of linguistic sleight-of-hand by implying the eye "tells" me things. I sort of just called everything from the occurrence of light leaving an object to me perceiving the light leaving the object as "The Eye."
If you can think of a better way to say it without adding a heck of a lot of complexity I am more than willing to edit the post.
Comment author: 27 January 2010 08:04:59PM 1 point [-]
I'm not sure if the eye-brain interaction is the relevant part. Even if you change "eyes" to "the visual system", the point of "these are of a different color is the very thing your visual system is trying to tell you" remains true.
Comment author: 27 January 2010 08:16:33PM 0 points [-]
What would you call it? The point remains that the colors are not the same. If we think the colors are the same, we are incorrect.
Comment author: 27 January 2010 10:10:28PM 0 points [-]
They are not the same, but our visual system is trying to tell us they are the same - and you can't really say it's wrong to make that judgment, as doing so leads to correct results the overwhelmingly vast majority of the time. (Basically, I'm saying the same thing as AndyWood's comment and the responses to it.)
Comment author: 27 January 2010 10:48:10PM 1 point [-]
They are not the same, but our visual system is trying to tell us they are the same.
Yeah, this makes sense. The trick is asking, "The same what?" The answer, "Color" is not descriptive enough.
I never meant to say it is right or wrong for the visual system to do whatever it is doing. I mean to say it is wrong to expect something different from the visual system than what it is doing.
Comment author: 27 January 2010 07:23:19PM 1 point [-]
"my optic system"? "my visual system"?
Comment author: 27 January 2010 07:27:41PM 0 points [-]
Optics system works for me. I changed a few of the sentences to use this instead of "eye."
Is it "optics system" or "optic system"?
Comment author: 27 January 2010 07:41:02PM 0 points [-]
*checks wikipedia*
Apparently it's either 'optical system' or 'visual system', and the latter seems to be preferred for organisms.
Comment author: 27 January 2010 07:54:25PM 0 points [-]
Updated to visual system. | 2013-05-22 23:14:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4561626613140106, "perplexity": 1141.6937867163874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368702454815/warc/CC-MAIN-20130516110734-00040-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://docs.cogdl.ai/en/0.4.1/trainer.html | # Trainer¶
In this section, we will introduce how to implement a specific Trainer for a model.
In previous section, we introduce the implementation of different tasks. But the training paradigm varies and is incompatible with the defined training process in some cases. Therefore, CogDL provides Trainer to customize the training and inference mode. Take NeighborSamplingTrainer as the example, this section will show how to define a trainer.
Design
1. A self-defined trainer should inherits BaseTrainer and must implement function fit to define the training and evaluating process. Necessary parameters for training need to be added to the add_args in models and can be obtained here in __init___.
class NeighborSamplingTrainer(BaseTrainer):
def __init__(self, args):
# ... get necessary parameters from args
def fit(self, model, dataset):
# ... implement the training and evaluation
@classmethod
def build_trainer_from_args(cls, args):
return cls(args)
2. All training and evaluating process, including data preprocessing and defining optimizer, should be implemented in fit. In other words, given the model and dataset, the rest is up to you. fit accepts two parameters: model and dataset, which usually are in cpu. You need to move them to cuda if you want to train on GPU.
def fit(self, model, dataset):
self.data = dataset[0]
# preprocess data
data=self.data,
sizes=self.sample_size,
batch_size=self.batch_size,
num_workers=self.num_workers,
shuffle=True,
)
)
# move model to GPU
self.model = model.to(self.device)
# define optimizer
# training
best_model = self.train()
self.model = best_model
# evaluation
acc, loss = self._test_step()
return dict(Acc=acc["test"], ValAcc=acc["val"])
3. To make the training of a model use the trainer, we should assign the trainer to the model. In Cogdl, a model must implement get_trainer as static method if it has a customized training process. GraphSAGE depends on NeighborSamplingTrainer, so the following codes should exsits in the implementation.
@staticmethod | 2022-08-20 01:32:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3943180739879608, "perplexity": 3852.914220153548}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573876.92/warc/CC-MAIN-20220820012448-20220820042448-00199.warc.gz"} |
https://forum.snap.berkeley.edu/t/pixel-list-from-a-list-of-integers/11283 | # Pixel list from a list of integers
I'm sure I've done this before but I can't remember how
So I have an RGBA bitmap that I've loaded into a variable tile image
Each value is the 32bit representation of a pixel
How can I convert it to the same format as you get from the pixels of costume reporter?
I feel there's a shortcut available to me somewhere to save me the trouble of decoding the integers
Here's what I would do. Convert the 32 bit to hex, then just convert that to rgba. Of course there's probably a much better way to do that though.
That's the sort of thing I'm trying to avoid the effort of doing
Something in the back of my mind says there's an easy way to do this
RGBA mod 256 to get the lowest byte
RGBA = round( RGBA/256) to shift bytes
Repeat 4 times for 32bit
Ta for that but it turns out my data isn't quite right - it's not actually integers
It's sort of in bytes (but not quite!)
The data length should be exactly 520000 but it looks like some some of the byes are being returned as 16bit values - very strange - I need to work out what's going on!
I'm down a deep rabbit hole
I'm getting my data via my MQTT extension and it turns out that JavaScript fails to convert it cleanly and gets confused due to unicode
I'm off out for the evening so I'll have to look at it later!
I see nothing wrong with Chrome@Win10
When I google stuff, there seems to be problems with large arrays using that approach (as shown in my project)
The use of the spread operator ("..."), which simulates calling a proc with zillions of parameters, usually causes a problem.
But it seems that "typed array".toString() is skewed on your platform.
In this case toString() should return text "53,89,137..." without any unicode consideration.
I found this solution and it seems to work
With HOF in JS
result = payload.reduce( (res, val) => res+String.fromCharCode( val), "")
Much neater thank you
This topic was automatically closed 30 days after the last reply. New replies are no longer allowed. | 2022-06-29 19:52:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38185301423072815, "perplexity": 1692.644130701773}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103642979.38/warc/CC-MAIN-20220629180939-20220629210939-00037.warc.gz"} |
https://www.nature.com/articles/s43246-020-00088-w | Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
# Spin-polarized Weyl cones and giant anomalous Nernst effect in ferromagnetic Heusler films
## Abstract
Weyl semimetals are characterized by the presence of massless band dispersion in momentum space. When a Weyl semimetal meets magnetism, large anomalous transport properties emerge as a consequence of its topological nature. Here, using in−situ spin- and angle-resolved photoelectron spectroscopy combined with ab initio calculations, we visualize the spin-polarized Weyl cone and flat-band surface states of ferromagnetic Co2MnGa films with full remanent magnetization. We demonstrate that the anomalous Hall and Nernst conductivities systematically grow when the magnetization-induced massive Weyl cone at a Lifshitz quantum critical point approaches the Fermi energy, until a high anomalous Nernst thermopower of ~6.2 μVK−1 is realized at room temperature. Given this topological quantum state and full remanent magnetization, Co2MnGa films are promising for realizing high efficiency heat flux and magnetic field sensing devices operable at room temperature and zero-field.
## Introduction
When electric and thermal currents flow through a ferromagnet, an electric field emerges orthogonally to the current path. The two effects are, respectively, called the anomalous Hall (AHE) and Nernst (ANE) effects and are exploited as operating mechanisms in various novel applications such as energy harvesting1,2, magnetic sensor3, and heat flux sensing4. The associated transverse voltage of the electric field is empirically proportional to its spontaneous magnetization. In contrast to the general belief, recent discoveries of both large AHE and ANE, which do not scale with magnetization, have elicited great surprise5,6,7,8,9,10. In particular, the observed ANE thermopower of single crystalline bulk Co2MnGa at room temperature, ~6.0 μV K−1 is an order of magnitude larger than that of other ferromagnets with similar magnetizations7,9. These transverse properties are postulated to arise from a Berry curvature emerging within band structures near the Fermi energy (EF)11,12.
Topologically non-trivial Weyl semimetals possessing spin-split massless fermions characterized by zero-gap and linear band dispersions are promising candidates featuring a large Berry curvature13,14,15. Weyl fermions in solids can be realized in materials that break inversion symmetry or time-reversal symmetry. With the breaking of such symmetries, Weyl nodes appear as pairs in momentum space and act as magnetic monopoles with positive and negative chiralities16. To date, Weyl fermions have been verified in experiments in non-centrosymmetric (e.g., TaAs-family) and magnetic materials (e.g., Mn3Sn) through angle-resolved photoelectron spectroscopy (ARPES) and magneto-transport measurements8,17,18,19,20,21,22,23,24,25,26.
Recently, a Co2MnGa Heusler alloy has also been theoretically predicted to be a ferromagnetic Weyl semimetal with a high Curie temperature and has been experimentally demonstrated in the bulk form to exhibit large anomalous transport properties under an external magnetic field7,9,27. The nature of this highly symmetric crystal (Fig. 1a) creates mirror-symmetry-protected Weyl nodal lines in the band structure as encountered by theory and experiments28,29. However, the nodal lines lead to vanishing Berry curvature when integrated over the whole Brillouin zone30,31 and cannot explain the observed phenomena. One way to obtain a large Berry curvature is to gap out their nodal lines using remanent magnetization or an external magnetic field (specifically, to break the mirror symmetry). Yet, the experimental evidence for broken mirror symmetry was not provided by the recent ARPES measurement on bulk Co2MnGa crystal because the remanent magnetization was negligible as applying external magnetic fields is not permitted in this measurement. For practical applications in which zero-field operation and gigantic outputs are a requirement, it is thus indispensable to truly understand the band structure responsible for the anomalous transport properties in films with full remanent magnetization.
Here, we experimentally and theoretically investigated the topological band structure and anomalous transport properties of ferromagnetic Co2MnGa thin films. Growth of high-quality thin films possessing full remanent magnetization and in situ spin-resolved ARPES (SARPES) measurements permit access to their non-trivial band structures modified by the broken mirror symmetry. We observed spin-polarized Weyl cones located mostly at a Lifshitz quantum critical point and a flat band of surface states. Furthermore, when the energy associated with the massive Weyl cone approaches EF, the anomalous Hall and Nernst conductivities systematically increase as the electron number rises. In particular, the ANE reaches thermopower of ~6.2 μV K−1 at room temperature, which is the highest amongst magnetic films to the best of our knowledge.
## Results and discussion
### AHE and ANE properties of epitaxial Co2MnGa films
To examine the influence of the relative location of the Weyl cone-associated energy with respect to EF on the transverse transport properties, we grew epitaxial Co2MnGa films with different compositions and investigated their AHE and ANE properties. Highly L21-ordered structure was confirmed in all Co2MnGa films by the quantitative analysis of X-ray diffraction patterns with taking the off-stoichiometry in each film into account (Supplementary Fig. 1 and the associated text). Figure 1b, c show the dependence on the perpendicular external magnetic field μ0Hz of the anomalous Hall resistivity $${\rho }_{yx}^{{\rm{A}}}$$ and the normalized transverse thermoelectric voltage $${E}_{xy}^{{\rm{A}}}$$ for Co2MnGa films with different compositions labeled H8, H7, H5, H1, and E2 (see Table 1), leading to a different number of valence electrons (Nv) ranging from 25.6 (H8) to 28.5 (E2). These curves clearly show that for all these films the respective magnitude of the AHE and the ANE changes significantly depending on the composition ratio of Co2MnGa regardless of its atomic ordering. It should be mentioned that here we apply the Tx and Hz for evaluating the thermopower of the ANE strictly, causing no ANE signal at μ0Hz = 0 as the spontaneous magnetization does not appear to the z-direction due to the strong demagnetization field. Thus, we also measured ANE signal by applying the Tz and sweeping the magnetic field in the plane and observed clear ANE voltage at zero-field due to the perfect spontaneous in-plane magnetization as shown in Fig. 1d, which is advantageous to thermoelectric applications such as heat flux sensor4.
Resistivity $${\rho }_{yx}^{{\rm{A}}}$$ and thermopower $${S}_{xy}^{{\rm{A}}}$$ (Fig. 1f, i) were estimated by finding the intercept of the linear-fitted curve for the saturation region in Fig. 1b, c, respectively. From Fig. 1e, f, ρxx monotonically decreases with increasing Nv, whereas $${\rho }_{yx}^{{\rm{A}}}$$ has a maximum value of 22.5 μΩ cm at around Nv = 27.3–27.4. $${\sigma }_{xy}^{{\rm{A}}}$$ evaluated from $${\sigma }_{xy}^{{\rm{A}}}$$ = $${\rho }_{yx}^{{\rm{A}}}$$/($${\rho }_{xx}^{2}$$ + $${{\rho }_{yx}^{{\rm{A}}}}^{2}$$) exhibits a nearly monotonic increase from 2 S cm−1 at Nv = 25.6 to 485 S cm−1 at 28.5 (Fig. 1g).
Figure 1h displays the Nv dependence of the Seebeck coefficient Sxx. In the small Nv region, Sxx is positive and gradually decreases with Nv changing sign at Nv = 26.7. A very tiny ANE is observed in the small Nv region (Fig. 1i); for instance, 0.3 μV K−1 at Nv = 25.6. Interestingly, $${S}_{xy}^{{\rm{A}}}$$ increases steeply with increasing Nv with the largest $${S}_{xy}^{{\rm{A}}}$$ achieved being 6.2 μV K−1 at Nv = 27.3, which is the highest in the ferromagnetic thin films32,33 and slightly smaller than the highest value in bulk Co2MnGa ~ 8.0 μV K−134. $${S}_{xy}^{{\rm{A}}}$$ is obtained from the linear response equation
$${S}_{xy}^{{\rm{A}}}={\rho }_{xx}{\alpha }_{xy}^{{\rm{A}}}-{\rho }_{yx}^{{\rm{A}}}{\alpha }_{xx}.$$
(1)
Here, αxx and $${\alpha }_{xy}^{{\rm{A}}}$$ are the longitudinal and transverse thermoelectric conductivities, respectively. This equation indicates that there are two different phenomenological contributions to ANE. To simplify our explanation, we denote the associated terms as SI = $${\rho }_{xx}{\alpha }_{xy}^{{\rm{A}}}$$ and SII = $$-{\rho }_{yx}^{{\rm{A}}}{\alpha }_{xx}$$. As mentioned in refs. 35,36, SII is regarded as the contribution of AHE to ANE induced by a Seebeck-driven longitudinal current. Similarly, SI stems from the direct conversion from a temperature gradient to a transverse current via $${\alpha }_{xy}^{{\rm{A}}}$$. SII can be converted to $$-{S}_{xx}{\rho }_{yx}^{{\rm{A}}}/{\rho }_{xx}$$, enabling a direct estimate from experimental data plotted in Fig. 1i. One clearly views the contribution of SII to $${S}_{xy}^{{\rm{A}}}$$ to be very limited. Therefore, SI$$(={S}_{xy}^{{\rm{A}}}-{S}_{{\rm{II}}})$$ dominates the observed $${S}_{xy}^{{\rm{A}}}$$ over the whole range of Nv. Transverse thermoelectric conductivity $${\alpha }_{xy}^{{\rm{A}}}$$ estimated from SI and ρxx increases enormously with Nv in a similar way to $${\sigma }_{xy}^{{\rm{A}}}$$ and has a maximal value of 3.3 A m−1K−1 at Nv = 28.5 (Fig. 1j). Figure 1k shows the relationship between $${\alpha }_{xy}^{{\rm{A}}}$$ and $${\sigma }_{xy}^{{\rm{A}}}$$ at 300 K for Co2MnGa films compared to other magnets. A previous study by Xu et al.34 clarified that many topological magnets have a universal relationship $${\alpha }_{xy}^{{\rm{A}}}/{\sigma }_{xy}^{{\rm{A}}} \sim {k}_{{\rm{B}}}/e$$. Thus, we examined this ratio in our present samples as shown in Fig. 1k. While both $${\alpha }_{xy}^{{\rm{A}}}$$ and $${\sigma }_{xy}^{{\rm{A}}}$$ are enhanced more than one order of magnitude by increasing Nv from 25.6 to 28.5, we confirmed that the $${\alpha }_{xy}^{{\rm{A}}}/{\sigma }_{xy}^{{\rm{A}}}$$ ratios of our Co2MnGa films also follow this universal behavior. It signifies that the main origin of AHE and ANE in all Co2MnGa films comes from the intrinsic topological nature34.
We also found that ρxx decreases with increasing Nv whereas $${S}_{xy}^{{\rm{A}}}$$ increases in our Co2MnGa films (Fig. 1e, i). This relation between ρxx and $${S}_{xy}^{{\rm{A}}}$$ is opposite to that reported in Co3Sn2S2, where $${S}_{xy}^{{\rm{A}}}$$ is enhanced by increasing ρxx37. It is worth mentioning that this discrepancy arises from a difference in an origin of enlargement of $${S}_{xy}^{{\rm{A}}}$$. Namely, Ding et al. prepared Co3Sn2S2 single crystals having different impurity concentrations and found that the impurity scatterings in Co3Sn2S2 enlarge ρxx but preserve $${\alpha }_{xy}^{{\rm{A}}}$$, resulting in the enlargement of $${S}_{xy}^{{\rm{A}}}$$ through SI term. On the other hand, this study tunes the position of EF by adjusting the composition of Co2MnGa film, leading to a drastic enlargement of $${\alpha }_{xy}^{{\rm{A}}}$$ with Nv which is large enough to overcome the reduction of ρxx, thus $${S}_{xy}^{{\rm{A}}}$$ is enhanced through SI term as well.
### Theoretical calculations
To understand the origin of the strong Nv dependence of AHE and ANE, we have performed ab initio calculations. Figure 2a shows the band structure of Co2MnGa along the X–K line at kz = 2π/a and along the K–Γ–W line at kz = 0 plane in the Brillouin zone (Fig. 2h). The red and blue dashed lines represent the majority- and minority-spin bands, respectively, in the absence of the spin–orbit interaction (SOI). A large minority-spin hole pocket is found around Γ, whereas majority-spin bands dominate near EF around X, K, and W and form crossing bands, labeled A through E. Previous studies have shown that such majority-spin band structures have three types of Weyl nodal loops in the Brillouin zone reflecting mirror symmetries with respect to the ki = 0 planes (i = xyz)9,15,28,38,39. For example, the nodal points of bands C and D in Fig. 2a are connected by the nodal loop shown in Fig. 2h.
When the SOI and the magnetization along the [100] direction are taken into account, only the mirror plane kx = 0 remains; the other planes disappear. This is because the mirror plane perpendicular to the magnetization conserves the direction of spins whereas that parallel to the magnetization does not9,15,38,39. Once the SOI is present, the energy gaps open at all points of the Weyl cones A–E, since the mirror symmetry is broken with respect to the kz = 0 and the kz = 2π/a planes (Fig. 2a, gray curves). Specifically, several Weyl nodal loops collapse and become massive (gapped) Weyl cones, and only those protected by kx = 0 mirror symmetry survive (Fig. 2i and Supplementary Fig. 7). Here, we emphasize that the gapped Weyl cones A and C are tilted, most being at a Lifshitz quantum critical point between type-I and type-II Weyl fermions.
From the calculated anomalous Hall conductivity $${\sigma }_{xy}^{{\rm{A}}}$$ (Fig. 2b), we see that $${\sigma }_{xy}^{{\rm{A}}}$$ has large values of ~103 S cm−1 and exhibit a peak near EF, consistent with previous results7,9. In Fig. 2c, we show the calculated transverse thermoelectric conductivity $${\alpha }_{xy}^{{\rm{A}}}$$. Around E = EF, $${\alpha }_{xy}^{{\rm{A}}}$$ increases with increasing E − EF (i.e., by electron doping) and takes a maximum at E − EF = 0.07 eV, near to where $${\sigma }_{xy}^{{\rm{A}}}$$ sharply drops (Fig. 2b). This is reasonable as $${\alpha }_{xy}^{{\rm{A}}}$$ is approximately proportional to $$-{\rm{{d}}}{\sigma }_{xy}^{{\rm{A}}}/{\rm{{d}}}E$$ (see “Methods” section, Eq. (4), and the Sommerfeld expansion40).
To discuss the correlation between the large $${\sigma }_{xy}^{{\rm{A}}}$$ around E = EF (Fig. 2b) and the band structures (Fig. 2a), we plotted the (kxky) dependences of the Berry curvature Ωz(k) (Fig. 2d–g); the nodal loops at the kz = 0 and 2π/a planes are marked as highlighted areas in Fig. 2i. At EEF = 0.05 and 0.03 eV (Fig. 2d and e), the Berry curvature has large values on the X–K line but has small values on the K–Γ–W line, because the gap of Weyl cone A mainly contributes to the Berry curvature at these energies (Fig. 2a). In contrast, at the lower energy of E − EF = −0.08 eV (Fig. 2g), large values of the Berry curvature are obtained close to the W point on the Γ–W line, which is determined by the gap of Weyl cone D (Fig. 2a). From these results, we conclude that both gaps open on two different nodal loops in the kz = 0 and kz = 2π/a planes and yield a large Berry curvature at E = EF (Fig. 2f), leading to a large anomalous Hall conductivity. The calculated $${\sigma }_{xy}^{{\rm{A}}}$$ and $${\alpha }_{xy}^{{\rm{A}}}$$ qualitatively explain the experimental results (Fig. 1g, j). Specifically, the electron-doped sample exhibits large anomalous transport properties. However, the simple EF shift that follows the rigid band model based on the stoichiometric Co2MnGa gives a quantitative discrepancy with the Nv dependence of $${\sigma }_{xy}^{{\rm{A}}}$$ and $${\alpha }_{xy}^{{\rm{A}}}$$ obtained from experiments. For example, the calculated negative $${\alpha }_{xy}^{{\rm{A}}}$$ in the E − EF < 0 region is not observed in the experimentally hole-doped samples. This discrepancy may be caused by an extrinsic mechanism or the formation of anti-site defects arising from off-stoichiometric compositions, which is not taken into account in the calculations.
### Band structures of an electron-doped Co2MnGa film
We performed ARPES experiments for the E2 (Nv = 28.5) sample as it exhibits the highest $${\sigma }_{xy}^{{\rm{A}}}$$ and $${\alpha }_{xy}^{{\rm{A}}}$$, to determine the EF location and the band structure with the mirror-symmetry breaking that yields a large Berry curvature as well as anomalous transport properties. Figure 3a shows the observed Fermi surface recorded at 80 eV photon energy after magnetization along [100] (Supplementary Movie 1 for a detailed continuous change in constant energy maps). Around the Γ point, a circular-shaped contour is evident. At each X point of the first Brillouin zone, we recognize a point-like structure. The calculated constant energy contour of the stoichiometric Co2MnGa (Nv = 28.0) with the SOI along [100] was also plotted (Fig. 3a). Except for the intensities in between Γ and K, the features observed in the experiments are well reproduced by the calculation when the chemical potential shifts by +70 meV (Supplementary Fig. 5 and the associated text). Figure 3b shows the ARPES images and the band dispersions calculated at several momentum cuts (Fig. 3a, white dashed lines). At cut 1 (Γ–K line), we confirmed the large hole-pocket crossing EF around Γ. One also finds distinct features that get closer to the K points with increasing E − EF and finally cross EF, indicated by a red inverted triangle. These features are consistent with the calculated minority-spin hole band at Γ point and one branch of the tilted majority-spin Weyl cone C (Fig. 2a). In going from cut 1 to cut 3, the slope of the Weyl cone evolves to be sharper for both experiment and calculation. Here we also emphasize that an almost non-dispersive band is observed just below EF around the Γ point represented by a gray inverted triangle at cut 1. This flat band corresponds to the Fermi surface between the Γ and K points (Fig. 3a) and is not reproduced by the calculations. At the cut 4 (X–X line), we can confirm the hole bands with energy maximum of EEF ~ −100 and −350 meV around $${k}_{| | }^{\prime}$$ = 0 and ±0.8 Å−1, respectively, in both experiment and calculation. We also realize that the bottom of the electron bands located near EF at the X points create prominent point-like structures on the Fermi surface (Fig. 3a).
Figure 3c, d shows the wide range ARPES images along the Γ–K–X line (cut 1) taken at 80 eV energy for incident photons with p- and s-polarization, respectively. With s-polarized light, a sharp electron-pocket is markedly enhanced around the X points, whereas the photoemission intensities of the tilted Weyl cone C and the flat band observed by p-polarization have mostly diminished. In Fig. 3e, f, we present the ARPES image and its second derivative with the calculated band structure. Here, to eliminate the effect of the light polarization-dependent matrix-element, these images acquired with p- and s-polarized light are mixed. The experimental result is well reproduced by the calculations with EF shifted upward by 70 meV (i.e., electron doping). This chemical potential shift is consistent with a higher Nv of 28.5 for this sample than the stoichiometric one (Nv = 28.0). Small discrepancies are noted between the observed and calculated band dispersions (Fig. 3f), for instance, the location of the bottom of the band of the sharp electron-pocket around the X points. The differences may arise through correlation or kz broadening effects41,42.
Figure 3g shows the magnified ARPES image in the frame shown in Fig. 3e. With suppression through the matrix-element effect, the band structure around the X point at the kz = 2π/a plane in the second Brillouin zone are clearly visualized. In a comparison with the calculation (Fig. 3g, lower panel), we realized that the observed band structure around the X point resembles the tilted and gapped Weyl cone A (Fig. 2a). Because the upper part of the Weyl cone cannot be seen, EF is probably located in the gap of the massive Weyl cone A.
Here, we turn our attention to the flat band observed around the Γ point. To clarify the origin of the flat band, we show the photon-energy-dependent energy distribution curves (EDCs) at k = −0.5 Å−1 taken after magnetization (Fig. 3h). The prominent peaks caused by the flat band do not show a clear photon-energy (kz) dependence. Details of the photon energy dependence are shown in Supplementary Fig. 6 and the associated text. We therefore conclude that the observed flat band just below EF belongs to a surface state.
To gain deeper insight into the Weyl fermion and the peculiar surface state in the Co2MnGa film, we performed spin-resolved measurements. Since the light spot size (>1 mm) is much larger than the magnetic domain, we carried out SARPES measurements before and after magnetizing the sample. Figure 4a and b shows the spin-resolved EDCs and spin-polarizations at θ = −4 (k ~ −0.3 Å−1) taken before and after magnetizations, respectively. Although the spin-polarization is negligible over the whole energy region before magnetization due to the formation of magnetic domains, it is enhanced enormously after magnetization. In particularly, the large negative spin-polarization (~40%) originating from the flat surface state has been observed at EF. Figure 4c shows the calculated band structures (left) and the experimentally observed spin-polarization map along the Γ–K–X line overlaid with the calculated band structures (right). Here the positive (negative) spin-polarizations are marked in red (blue). Also, the spin-resolved band structures in the majority (left) and minority (right) spin channels (Fig. 4d) feature a large hole band around the Γ point having a minority-spin component. There are bands having a strong majority-spin component around the K point near EF. From calculations, the Weyl cone C is characterized by the majority-spin component and tilting (see Fig. 2a). Furthermore, because Weyl cones A and C are mostly at Lifshitz quantum critical points, they have a large density of states at EF when the Weyl node approaches EF7. These features are confirmed by our ARPES and SARPES measurements, and therefore we conclude that the observed spin-polarized feature with positive spin-polarization near the K point can be ascribed to one flatter branch of the tilted spin-polarized Weyl cone C induced by broken mirror symmetry.
In addition, we find that the flat surface state has a minority-spin component (Fig. 4d, right), the sign of which is opposite to that of the Weyl cone (Fig. 4d, left). Figure 4e shows the spin-resolved EDCs taken from θ = 0 to 20, which corresponds to the k-line from Γ to X. The clear minority-spin peaks persist over a wide momentum region marked with inverted triangles. There are two main possibilities for the origin of this peculiar surface state. One is the topologically non-trivial Fermi-arc surface state17,18,19,20,23,25,26. The other is a trivial surface resonance state, which was predicted for half-metallic Co-based Heusler alloys43,44. Having considered the location of the Weyl node before magnetization, it seems that the minority-spin surface state connects the Weyl cones at positive and negative momenta although further study is needed to elucidate the origin of the surface state.
### Band structures of a hole-doped Co2MnGa film
In order to compare the band structures of electron-doped and hole-doped samples, we also performed SARPES experiments for the sample H3* (Nv = 27.3). Figure 5a, b show the observed Fermi surface and wide range ARPES image along Γ–K–X line recorded at 50 eV photon energy with p-polarized light. At X point, the electron-pocket crossing EF can be seen while the large hole-pocket is located at Γ point. These features are in good agreement with the electron-doped sample (Fig. 3). In Fig. 5c, d, we present the magnified ARPES image along the Γ–K–X direction and its second derivative along the momentum axis. Around −0.60 Å−1 near EF, one can see a tilted band, in which the node is slightly located above EF (see gray arrow in Fig. 5d). As we can see in Fig. 5f, this tilted band has a strong majority-spin component. These experimental results agree with the results of the calculation, shown in Fig. 5e with the position of EF shifted downward by 220 meV. Therefore, we conclude that the tilted cone close to K point belongs to the Weyl cone E (see Fig. 2a).
Finally, we discuss the role of the Fermi energy tuning in optimizing the anomalous transport properties. From ARPES measurements from two samples, we determined the relative EF shifts from the stoichiometric condition (Nv = 28.0), specifically, +70 meV for the sample E2 (Nv = 28.5) and −220 meV for the sample H3* (Nv = 27.3). The latter has lower $${\sigma }_{xy}^{{\rm{A}}}$$ and $${\alpha }_{xy}^{{\rm{A}}}$$ than the former. These results reasonably explain the observed and calculated large anomalous transport properties presented in Fig. 1 based on our calculations given in Fig. 2. In detail, because EF of sample E2 is closely located at the gapped region of the Weyl cones A and C through slight electron doping, it satisfies a criterion whereby a large Berry curvature in both kz = 0 and kz = 2π/a planes is generated and thereby results in a gigantic ANE because the $${\alpha }_{xy}^{{\rm{A}}}$$ is large. Therefore, we have clearly confirmed that enhancing the anomalous transport properties of topological ferromagnets involves two key features: (1) the creation of a massive (gapped) Weyl cone from the nodal loop by mirror-symmetry breaking and (2) the tuning of the Fermi energy.
## Conclusion
In summary, we have experimentally and theoretically investigated a one-to-one correspondence between the electronic structure and the anomalous transport properties in the film form of Co2MnGa topological ferromagnets. By in situ SARPES, we provided a direct visualization of the spin-polarized and massive Weyl cones and the peculiar surface state under mirror-symmetry breaking. When EF approaches the gap of the massive Weyl cones by the electron doping, we have recorded the highest anomalous Nernst thermopower (6.2 μV K−1) at room temperature among ferromagnetic films to the best of our knowledge. Our findings signify the insufficient EF position tuning against the Weyl cone is the most probable cause for the smaller anomalous Nernst thermopower in the Co2MnGa films reported in previous studies32,33 and provide the reliable guiding principle to maximize the Nernst thermopower by the band engineering utilizing the SARPES, transport measurements, and ab-initio calculations, for the first time. From an applications perspective, our work facilitates the implementation of various novel applications based on the thin film form of topological magnets; namely, the large transverse electric and thermoelectric conversions without an external magnetic field, which cannot be realized in the bulk form because of the formation of magnetic domains, are promising for novel thermoelectric applications such as heat flux sensor.
## Methods
### Thin film growth
Epitaxial thin films of (001)-oriented Co2MnGa having different compositions were deposited on a MgO(001) single crystalline substrate at 600 C using a co-sputtering technique with Co, Mn, and Co41.2Mn27.5Ga31.3 sputtering targets. The films were designed to be 50 nm thick. To prevent the film from oxidizing for transport measurements, a 1.5 nm-thick Al-capping layer was deposited by rf magnetron sputtering. The base pressure of the deposition chamber was near 2 × 10−7 Pa. The thickness and the composition of the film were evaluated by wavelength dispersive X-ray fluorescence analysis. Table 1 shows the result of the composition analysis and the evaluated total valence electron number for all prepared Co2MnGa films.
For SARPES measurements, uncapped films were deposited on a MgO substrate with a buffer layers of Cr (10 nm) and Ag (100 nm) to smooth the surface. To avoid surface contaminations, grown films were transferred from the magnetron-sputtering chamber to the preparation chamber of the SARPES instrument using a portable suitcase chamber to avoid exposure to air (<1 × 10−6 Pa). Note that sample H3* was deposited on the MgO(001) substrate with buffer layers at room temperature, and then post-annealed at 550 °C for 30 min.
### Measurement of transport and magnetic properties
The magnetic properties of the Co2MnGa films were measured with a superconducting quantum interference device-vibrating sample magnetometer (SQUID-VSM, Quantum Design Co. Ltd). The crystal structure was revealed through X-ray diffraction with a Cu Kα X-ray source and a two-dimensional detector (PILATUS 100K/R, Rigaku Co.). The substrate was cleaved to a size with lateral dimensions of ~7.0 × 10.0 mm2, and then the film was patterned into a Hall bar structure with a width of 2.0 mm and a length of 7.0 mm through photolithography and Ar ion milling. The AHE was measured at 300 K applying a 1 mA electric current along the [110] direction and a magnetic field was applied perpendicular to the [001] direction of the Co2MnGa film using a Physical Property Measurement System (PPMS, Quantum Design Co., Ltd.). The ANE was also measured at 300 K applying a temperature gradient Tin along the [110] direction and a magnetic field along the [001] direction of the Co2MnGa film in PPMS. To evaluate Tin, the Seebeck coefficient Sxx in the Co2MnGa film was obtained outside the PPMS by measuring the Tout using an infrared thermal camera and a black-body coating to calibrate the emissivity. Then, Tin can be estimated from the Seebeck voltage obtained in the PPMS and Sxx. The accuracy of this method to evaluate ANE has been confirmed in the previous studies35,36.
### Theoretical calculations
We calculated the electronic structure of L21-ordered Co2MnGa applying the full-potential linearized augmented plane-wave method including SOI, which is implemented in the WIEN2k program45. The lattice constant of the cubic unit cell was set to the experimentally determined value of 5.755 Å and the k-point number in the self-consistent-field calculation was chosen as 20 × 20 × 20 after confirming the convergence of the total energy. Using the electronic structure obtained, we calculated the anomalous Hall conductivity $${\sigma }_{xy}^{{\rm{A}}}$$ using46
$${\sigma }_{xy}^{{\rm{A}}}(\epsilon )=-\frac{{e}^{2}}{\hslash }\int \frac{{{\rm{{d}}}}^{3}k}{{(2\pi )}^{3}}\ \ {\Omega }^{z}({\bf{k}}),$$
(2)
$${\Omega }^{z}({\bf{k}})=-{\left(\frac{\hslash }{m}\right)}^{2}\ \sum _{n}f({E}_{n,{\bf{k}}},\epsilon )\sum _{n^{\prime} \ne n}\frac{2\ {\rm{Im}}\langle {\psi }_{n,{\bf{k}}}| {p}_{x}| {\psi }_{n^{\prime} ,{\bf{k}}}\rangle \langle {\psi }_{n^{\prime} ,{\bf{k}}}| {p}_{y}| {\psi }_{n,{\bf{k}}}\rangle }{{({E}_{n^{\prime} ,{\bf{k}}}-{E}_{n,{\bf{k}}})}^{2}},$$
(3)
where n and $$n^{\prime}$$ denote band indices, Ωz(k) denotes the Berry curvature, px(py) the x(y) component of the momentum operator, ψn,k the eigenstate with the eigenenergy En,k, and f(En,kϵ) the Fermi distribution function for the band n and the wave vector k at the energy ϵ relative to the Fermi energy. In the calculation of $${\sigma }_{xy}^{{\rm{A}}}$$, the direction of the magnetization was set along the [100] direction (Fig. 1a) and 90 × 90 × 90 k points were used for the Brillouin zone integration ensuring good convergence for $${\sigma }_{xy}^{{\rm{A}}}$$.
From Boltzmann transport theory, we calculated the transverse thermoelectric conductivity $${\alpha }_{xy}^{{\rm{A}}}$$ for a given temperature T by substituting the obtained $${\sigma }_{xy}^{{\rm{A}}}$$ into the following expression:
$${\alpha }_{xy}^{{\rm{A}}}=-\frac{1}{eT}\int \ {\mathrm{{d}}}\epsilon \left(-\frac{\partial f}{\partial \epsilon }\right)(\epsilon -\mu )\ {\sigma }_{xy}^{{\rm{A}}}(\epsilon ),$$
(4)
where $$f=1/[\exp ((\epsilon -\mu )/{k}_{{\rm{B}}}T)+1]$$ denotes the Fermi distribution function with μ the chemical potential. Here, μ = 0 corresponds to the Fermi level.
### Spin-resolved and angle-resolved photoelectron spectroscopy
ARPES and SARPES measurements were performed at the ESPRESSO end-station (BL-9B) in the Hiroshima Synchrotron Radiation Center47,48. The base pressure of the SARPES chamber was 5 × 10−9 Pa. The photoelectrons were acquired using a hemispherical electron analyser (R4000, Scienta-Omicron). The spin-polarization was measured using a very low-energy electron diffraction-type spin detector. The experimental geometry is shown in Supplementary Fig. 4. The energy and angular resolutions for ARPES (SARPES) were set to 45 meV (55 meV) and ±0.3° (±1.5°), respectively. The effective Sherman function was 0.28 for the SARPES measurements.
During all SARPES measurements, the temperature was maintained below 40 K. Before each ARPES and SARPES measurement, the sample was annealed at 550 °C for 30 min at the preparation chamber (base pressure ~ 4 × 10−8 Pa). The quality and cleanliness of the annealed sample was checked by low-energy electron diffraction (Supplementary Fig. 3 and the associated text). A magnetic field as large as ~0.1 T was applied to the samples along the [100] for sample E2 ([110] for sample H3*) easy-axis using a permanent magnet in the preparation chamber at room temperature. A 0.1 T magnetic field was sufficiently high to saturate the magnetization (Supplementary Fig. 2 and the associated text). The ratio between the remanent and saturation magnetizations along the [100] easy-axis of the sample E2 is 0.96. Therefore, a single magnetic domain was overall obtained for SARPES measurements.
## Data availability
The data presented in this paper are available from the authors on reasonable request.
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## Acknowledgements
The SARPES measurements were performed with the approval of the Proposal Assessing Committee of Hiroshima Synchrotron Radiation Center (Proposals Nos. 18BG038 and 19AG054). This work was financially supported by KAKENHI (Nos. 16H02114, 17H06152, 17H06138, and 18H03683). K.S. was financially supported by a Grant-in-Aid for JSPS Fellows (No. 19J00858). Y.S. was financially supported by a JSPS KAKENHI Grant-in-Aid for Young Scientists (A) (No. JP2670945) and PRESTO from the Japan Science and Technology Agency (No. JPMJPR17R5). We thank S. Kurdi, A. Sakuma, K. Nawa, K. Uchida, and K. Hono for valuable discussions and N. Kojima and B. Masaoka for a technical support.
## Author information
Authors
### Contributions
K.S., T.K., and M.K. performed the SARPES experiments with the assistance of K.Mi and T.O. Y.S., K.G., and W.Z. synthesized the thin films and performed the transport measurements. K.Ma and Y.M. performed the ab initio calculations. K.S., Y.S., and K.Ma wrote the manuscript with inputs from all authors. A.K. supervised the work.
### Corresponding authors
Correspondence to Kazuki Sumida or Yuya Sakuraba or Akio Kimura.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Primary handling editor: Aldo Isidori
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Reprints and Permissions
Sumida, K., Sakuraba, Y., Masuda, K. et al. Spin-polarized Weyl cones and giant anomalous Nernst effect in ferromagnetic Heusler films. Commun Mater 1, 89 (2020). https://doi.org/10.1038/s43246-020-00088-w
• Accepted:
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Physical Review B (2021) | 2021-05-14 14:48:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7914485335350037, "perplexity": 2989.616641408553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989526.42/warc/CC-MAIN-20210514121902-20210514151902-00372.warc.gz"} |
https://www.albert.io/learn/ap-physics-c-mechanics/question/satellite-orbit-determining-altitude | Limited access
Geostationary satellites have circular orbits with a 24-hour period that matches the rotation of the Earth. This causes them to remain over the same location on the equator at all times.
Mass of the Earth: $6.0$x${10}^{24}kg$
Radius of the Earth: $6.4$x${10}^{6}m$
Mass of the geostationary satellite: $5.2$x${10}^{2}kg$
How high above the surface of the Earth is the geostationary satellite?
A
$36$ x ${10}^{6}\text{ }m$
B
$43$ x ${10}^{6}\text{ }m$
C
$278$ x ${10}^{6}\text{ }m$
D
$284$ x ${10}^{6}\text{ }m$
E
The altitude can not be determined from the given information.
Select an assignment template | 2017-04-24 02:15:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34005874395370483, "perplexity": 1866.7184179130393}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917118950.30/warc/CC-MAIN-20170423031158-00412-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcds.2008.22.427 | # American Institute of Mathematical Sciences
January & February 2008, 22(1&2): 427-443. doi: 10.3934/dcds.2008.22.427
## Algebro-geometric methods for hard ball systems
1 Budapest University of Technology and Economics, Institute of Mathematics, Budapest, Egry J. u. 1, H–1111, Hungary
Received March 2007 Revised May 2007 Published June 2008
For the study of hard ball systems, the algebro-geometric approach appeared in 1999 --- in a sense surprisingly but quite efficiently --- for proving the hyperbolicity of typical systems (see [26]). An improvement by Simányi [22] also provided the ergodicity of typical systems, thus an almost complete proof of the Boltzmann--Sinai ergodic hypothesis. More than that, at present, the best form of the local ergodicity theorem for semi-dispersing billiards, [6] also uses algebraic methods (and the algebraicity condition on the scatterers). The goal of the present paper is to discuss the essential steps of the algebro-geometric approach by assuming and using possibly minimum information about hard ball systems. In particular, we also minimize the intersection of the material with the earlier surveys [29] and [20].
Citation: Domokos Szász. Algebro-geometric methods for hard ball systems. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 427-443. doi: 10.3934/dcds.2008.22.427
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2019 Impact Factor: 1.338 | 2020-12-02 16:09:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6841317415237427, "perplexity": 12448.58406556239}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141711306.69/warc/CC-MAIN-20201202144450-20201202174450-00690.warc.gz"} |
https://math.stackexchange.com/questions/457342/is-the-countable-sum-of-a-set-of-measurable-functions-also-measurable | # Is the countable sum of a set of measurable functions also measurable?
It is rather straightforward to show that the sum of two measurable functions is also measurable. Therefore we can extend the logic to say that $\sum\limits_{i=1}^n f_i$ is measurable providing $f_i$ is measurable. However can we take $n\rightarrow\infty$ and say that $\sum\limits_{i=1}^\infty f_i$ is a measurable function?
• Pointwise limits of measurable functions are measurable. Apply that to the partial sums. – Daniel Fischer Aug 1 '13 at 15:59
• Functions from where to where? And by sum of functions, do you mean like $(f+g)(x) = f(x) + g(x)$ or direct sum of functions on the direct sum of the domains? – Daniel Robert-Nicoud Aug 1 '13 at 16:00
• I am not sure if your first sentence is so straight forward; I would prefer to have the math for this and know about what type of function etc. you are talking? – al-Hwarizmi Aug 1 '13 at 16:02
• Functions from an arbitrary set to the extended real-domain and I mean $(f+g)(x)=f(x)+g(x)$. – Simon Aug 1 '13 at 16:03
## 2 Answers
As Daniel pointed out, the pointwise limit of measurable functions is measurable.
If we define $g_n=\sum_{i=1}^n f_i$, then as long as $\sum_{i=1}^{\infty} f_i = \lim_{n\to\infty}g_n$ converges, it defines a measurable function.
• It should be fine for the partial sums to converge to a value in the extended reals i.e. $\infty$ or $-\infty$, is that right? In which case the pointwise limit is the constant function $x \mapsto \infty$. – jII Oct 22 '18 at 0:04
You need some assumptions here, otherwise your infinite sum will not converge, think of $f_n(x)=1$. See https://en.wikipedia.org/wiki/Dominated_convergence_theorem for details.
• In your example is it not the case that the series converges to the constant function $x\mapsto\infty$? In any case dominated convergence seems like overkill here, instead of e.g. $\forall n:f_n\geq0$ (and monotone convergence theorem). – Alp Uzman Mar 10 '16 at 7:50
• This answer does not seem correct to me. The dominated convergence theorem relates to interchanging an integral and a limit. The question is about whether the pointwise limit of a sequence of measurable functions is measurable and not whether it is integrable. Maybe I'm missing something about @MoisheCohen's answer. – jII Oct 21 '18 at 23:57 | 2020-08-10 21:19:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9564943909645081, "perplexity": 220.42431314589754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738699.68/warc/CC-MAIN-20200810205824-20200810235824-00060.warc.gz"} |
https://space.stackexchange.com/questions/19910/optimal-change-of-argument-of-periapsis?noredirect=1 | # Optimal change of argument of periapsis?
If I want to rotate an eccentric orbit around the central body - retain orbital plane, retain apoapsis and periapsis altitudes, but have the orbit rotated in its orbital plane - change the argument of periapsis - what is the optimal maneuver to that end?
I know an easy way to achieve this effect is performing a radial burn (towards the center of the central body) at periapsis, at thrust such that the craft retains altitude, against centripetal acceleration; moving in circular path around the body; "dragging the periapsis along" - the moment the engines are cut off, it enters the new trajectory. I'm also aware this method may be awfully costly, especially for highly eccentric orbits and large changes of argument of periapsis.
Another method is circularizing the orbit at apoapsis, and then returning to desired eccentricity back upon achieving the desired argument of periapsis. This one has a fixed cost, which will be excessive in case the orbit is very eccentric and the desired shift in angle is small.
There's also a method involving only tangential burns (pro/retrograde) at various points of the orbit, but I only have a rough hunch of how it works, no good solid recipe.
Is there an universal strategy to optimally perform this change?
This is an expensive operation. Suppose $\Delta \omega$ is the angle by which you wish to change the argument of periapsis. The instantaneous delta V needed to perform that optimal change is $$\Delta v = 2\sqrt{\frac{\mu}{a(1-e^2)}}\,\sin\left(\frac {\Delta \omega} 2\right)$$ Note that this is very similar in form to the $\Delta v$ needed to change the inclination by an angle $\Delta i$.
• Hmm, I think there's also a $e$ factor missing in the formula here. To change the argument of periapsis by angle $\Delta \omega$, one needs to reverse the radial component of the velocity at true anomaly $\Delta \omega/2$ and these equations in Wikipedia (and my calcluations too long to fit here) say that $\dot{r} = \sqrt{\mu/p}e \sin(\theta)$ where $p=a(1-e^2)$ and $\theta$ is the true anomaly. Then $\Delta v$ is $2\dot{r}$ at $\theta=\Delta \omega/2$. – JiK Oct 14 '17 at 19:34 | 2020-08-05 19:43:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8688973784446716, "perplexity": 594.6668096616745}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735964.82/warc/CC-MAIN-20200805183003-20200805213003-00066.warc.gz"} |
https://www.physicsforums.com/threads/finding-the-sampled-signal-fourier-series-and-integration-problems.676339/ | # Finding the sampled signal - Fourier series and integration problems.
1. Mar 5, 2013
### Evo8
1. The problem statement, all variables and given/known data
The signal g(t) is band limited to B Hz and is sampled by a periodic pulse train $PT_{s}(t)$ made up of a rectangular pulse of width $1/8B$ second (centered at the origin) repeating at the nyquist rate (2B pulses per second). Show that the sampled signal $\bar{g}(t)$ is given by
$\bar{g}(t)=\frac{1}{4} g(t)+\sum_{n=1}^\infty sin(\frac{n \pi}{4}) g(t) cos(4nBt)$
2. Relevant equations
$$a_0=\frac{1}{T_0} \int_{T_0} g(t) dt$$
$$a_n=\frac{2}{T_0} \int_{T_0} g(t) \cos(n\omega_0 t) dt$$
$$b_n=\frac{2}{T_0} \int_{T_0} g(t) \sin(n\omega_0 t) dt$$
3. The attempt at a solution
Ok so my fourier series and integration skills are weak at best. Ontop of that im rusty. I think i know how to do this but i get stuck on some simple but paramount steps.
In a similar problem from a previous edition text they give the following pulse train. $PT_s(t)=C_0+\sum_{n=1}^\infty \cos(n\omega_s t)$ I see how this helps as it is in the same form as the sampled signal im trying to find $\bar{g}(t)$.
My book had the following to work with
$$\bar{g}(t)=g(t)\delta T_s(t)=\sum_{n} g(nT_s) \delta(t-nT_s)$$
How do they get the pulse train from this? I just dont see it.
For the first coefficient $a_0$ I get something like this:$$a_0=\frac{1}{T_0} \int_{\frac{-1}{16B}}^\frac{1}{16B} g(t) dt$$
$$2B\int_{\frac{-1}{16B}}^\frac{1}{16B} g(t) dt$$
$$=2B[\frac{1}{16B}+\frac{1}{16B}]$$
$$=\frac{1}{4}$$
Which is the first term in $\bar{g}(t)$
Now i got the limits for the integral from the pulse width of $\frac{1}{8B}$ being centered at the origin. So the equation would be 1 between $\frac{-1}{16B} and \frac{1}{16B}$
Is this correct? Am I on the right track? I have my doubts....
Thanks for any help!
2. Mar 5, 2013
### rude man
Can you find out if the pulses forming the sampled signal $\bar{g}(t)$ have flat tops or do they follow the contour of g(t) for the duration of a pulse? If using finite-width sampling pulses this distinction is important.
Never mind, I think I know.
This is just multiplying your g(t) by the Fourier series of the described pulse train.
The pulse train has unit height, is centered at t = 0, has width w = 1/8B and period T = 1/2B. From this, can you write the Fourier series of the pulse train?
Last edited: Mar 5, 2013
3. Mar 6, 2013
### Evo8
Thanks for your response rude man.
Im not sure I understand how to find the pulse train equation from the description. I understand the characteristics that are described but am still unsure of how to get the $PT_s$ equation. How do i determine the form from the given information?
Thanks again
4. Mar 6, 2013
### rude man
OK, so let's look at the sampling pulse train by itself (no g(t) here):
One form of the Fourier series for any periodic function is
f(t) = 1/2 A_0 + ∑(A_n cos nwt + B_n sin nwt), n = 1 to ∞.
If your function is even, which is to say f(-t) = f(t), then you only need the A_n terms. This is the case here because the problem specifically states that the pulse is " ... centered at the origin". This is one of the important things you need to review about Fourier series.
So now you need to solve for the A_n coefficients. Dig up the integral for the coefficients, perform the integration over one period, and you have represented the sampling pulse train by its Fourier series. Multiply the series by g(t) and you're done.
5. Mar 6, 2013
### Evo8
Why do only th $A_n$ terms apply when my function is even?
If using my $A_n$ from above I get something like this.
$$A_n=\frac{2}{T_0} \int_{T_0} g(t) cos(n\omega_0 t)dt$$
$$=4B \int_{T_0} g(t) cos(4 \pi nB)$$
$$=4B cos(4 \pi nB)$$
Where does the sin term come from in my $\bar{g}(t)$? I would have assumed from the $A_b$ equation...
6. Mar 6, 2013
### rude man
Because if the function is even you can't have any sine terms because the sine function is odd. Basic Fourier series theory which you need to review.
I said not to include g(t) in this ... you are supposed to be deriving the series for the pulse train which has nothing to do with g(t).
Your expression for the A_n would apply if you were given g(t) & trying to express g(t) in a Fourier series.
You are not even given g(t) so how could you do this even if you wanted to?
7. Mar 6, 2013
### Evo8
Opps. Yes your correct you did mention not to include g(t). So what would my expression for A_n be?
Just $A_n=\frac{2}{T_0} \int_{T_0} dt$? In this case wouldnt I get the same result as I posted before? It seems I didnt include the g(t) there i just included it in the "math" which is weird i know.
I would still see$A_n=4B \cos(4 \pi n B)$
8. Mar 6, 2013
### rude man
No.
Look up the integral expression in your textbook for the A_n coefficients. It's got cosines in it.
9. Mar 6, 2013
### Evo8
Ok. My text doesnt really have fourier series covered but ive been reffering to this site. The Fourier Transform - Fourier series specifically this page of the site. Equation 7 is $A_n=\frac{2}{T} \int_0^{T} cos(\frac{2 \pi n t}{T})dt$
this has a cosine function in it...
After performing the integral I would end up with
$A_n=4B \int_0^{\frac{1}{2B}} cos(\frac{2 \pi nt}{\frac{1}{2B}})dt$
$=4B \int_0^{\frac{1}{2B}} cos(4B \pi nt)dt$
$=4B sin(2 \pi n)$
Am I on a better track here?
Thanks again for your help on this. I do appreciate it.
10. Mar 6, 2013
### rude man
Actually, eq. 7 has f(t) in it, does it not? However, you lucked out because in your case f(t) = 1 from t = -1/16B to 1/16B, repeating every 1/2B seconds. At all other times, f(t) = 0.
You therefore now have the right integral for computing the A_n coefficients of the Fourier series, but you are not using the correct limits.
First: fact: you can integrate from any point on the waveform to any other point as long as the total periodic interval is T = 1/2B. The limits do not have to be 0 to T. So, to make life easier, run the integration from t = 1/16B to t = 1/2B + 1/16B.
Look at your pulse train picture which you should have in front of you. For what values of t is your f(t) = 1? = 0? Then, choose your limits of integration accordingly.
(BTW I can't fathom why you are doing this problem without access to a Fourier series text. But, you did pick an excellent Website. I have it bookmarked myself.)
11. Mar 6, 2013
### Evo8
Ok,
Im a little confused by what to select for the limits. You first mention to try $\frac{1}{16B} to (\frac{1}{16B}+\frac{1}{2B})$ so the limits i evaluate my integral at are $\frac{1}{16B} to \frac{9}{16B}$
This yields $A_0=4B[sin(\frac{9 \pi n}{4}) - sin(\frac{\pi n}{4})]$ if I did it correctly.
Why not choose the limits of $+/-\frac{1}{16B}$?
Evaluating with these limits I get $8B sin(\frac{ \pi n}{4})$ this is of course if I did it correctly. I just re learned how to do basic integrals or "remembered" so I appologize for any blatant mistakes..
You also mention that I would solve for the coefficients of $A_n$ I was under the impression that the $A_n$ IS the coefficient? No?
I agree this is crazy without a fourier series text. I may stop by a book store this evening and see what I can find. However for now I'm "limited" to what i can find online.
12. Mar 6, 2013
### rude man
I meant the span of integration is from 1/16B to 1/2B + 1/16B. That does not mean the limits of your integral become those end-points. You are to integrate over those regions of t where your f(t) = 1 only.
Absolutely fine! I thought you might be confused by integrating from negative time but it's actually the easiest thing to do.
Sorry, that integration is incorrect. Way off. Need to check your math.
There is a lot of good stuff on-line but sometimes it's a bit heavy on the math. I would try to get an electrical engineering text where they apply the Fourier series to practical applications.
You also need to remember that ∫cos(ax)dx = (1/a)sin(ax)!
Another tricky part is when you solve for A_0. That coefficient will be of the form sin(x)/x with x = 0. Hint: = 1.
You have one final step: combining with g(t). Remember I said g_bar(t) = g(t) * sampling function.
13. Mar 7, 2013
### Evo8
Success!! Sort of....
Thanks for the integration help as well. After rereadeing a few intro to integral sections and the form you posted I understand now. Hopefully i wont make such mistakes in the future.
So now for my $A_n$ I end up with something like this. $A_n=\frac{2}{n \pi} Sin(\frac{\pi n}{4})$
For the $A_0$ I am a little confused as to what you mean by
are you saying that $\frac{sin(x)}{x} \ with\ x=0\ \ gives\ \frac{sin(0)}{0}\ which\ is\ \neq1$?
My integral goes something like this $$a_0=2B\int_{\frac{-1}{16B}}^{\frac{1}{16B}}1dt=\frac{1}{4}$$
However in a previous post
The 1/2 before the A_0 term makes me feel like something is wrong? I would end up with 1/8 not 1/4 in the pulse train..
I would end up with something like this $$\bar{g}(t)=\frac{1}{4}g(t)+\sum_{n=1}^\infty \frac{2}{n \pi} sin(\frac{n \pi}{4}) cos(4 \pi nBt)g(t)$$
So im left with two problems.... I "should" get 1/4 for that first term according to my texts exercise, and I have a $\pi$ in the cosine term where the text does not.. The quoted equation below is the expected "solution" that i am meant to show how its derived.
Where did the $\pi$ go?
14. Mar 7, 2013
### rude man
That is exactly right.
No. sin(x)/x = 1 if x = 0. cf. below.
If you evaluate the first term the way you did, it's the entire term, not the half. So you get 1/4 for the n = 0 coefficient as you should.
Never mind about the sin(x)/x for now. I'll just say that you can use the general expression for A_n to include n = 0 if you handle it right.
As to the missing π, you may tell your teach he made a typo or otherwise got it wrong. The π belongs!
Good going there in the stretch!
15. Mar 7, 2013
### Evo8
Thanks Rude man!
It is the text book that has the incorrect solution. Oddly enough I have an older edition of the same text with the "same" problem and the $\bar{g}(t)$ has the $\pi$.
I did some quick research on the sin(x)/x and I see what you mean. I dont fully understand/remember quite yet but i can at least buy it. Im still reading....
Thanks again for your help! I'm sure i'll have more questions in the near future... | 2017-08-20 06:59:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8247024416923523, "perplexity": 738.4711416543635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105976.13/warc/CC-MAIN-20170820053541-20170820073541-00599.warc.gz"} |
https://www.nevis.columbia.edu/~seligman/root-class/html/cpppath/WriteFile.html | Exercise 10: Writing histograms to a file (10 minutes)
In all the analysis macros we’ve worked with, we’ve drawn any plots in the Terminate method. Pick one of your analysis macros that creates histograms, and revise it so that it does not draw the histograms on the screen, but writes them to a file instead. Make sure that you don’t try to write the histograms to “experiment.root”; write them to a different file named “analysis.root”. When you’re done, open “analysis.root” in ROOT and check that your plots are what you expect.
Hint
In Saving your work, part 2, I described all the commands you’re likely to need.
Don’t forget to use the ROOT web site as a reference. Here’s a question that’s also a bit of a hint: What would be the difference between opening your new file with “UPDATE” access, “RECREATE” access, and “NEW” access? Why might it be a bad idea to open a file with “NEW” access? (A hint within a hint: what would happen if you ran your macro twice?) | 2022-12-07 21:16:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5207051634788513, "perplexity": 1416.727766285908}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711218.21/warc/CC-MAIN-20221207185519-20221207215519-00143.warc.gz"} |
http://math.stackexchange.com/questions/232854/transpose-of-the-differential-operator | # Transpose of the differential operator
Is the differential operator ${d\over dx}$ antisymmetric? If so, what does it even mean to take it's transpose? Thank you.
-
On what function space do you mean? – Berci Nov 8 '12 at 11:57
@Berci ,"complex inner product space"? – Henry Nov 8 '12 at 12:03
This is actually a quite large topic. Typically, for the inner product space you pick $L^2(a,b)$ with the inner product $$\langle u,v\rangle=\int_a^b u(x)\overline{v(x)}\,dx,$$ and you will find that $d/dx$ is at least formally skew symmetric in that partial integration $$\langle u',v\rangle=-\langle u,v'\rangle,$$ at least when $u$ and $v$ are suitably smooth.
But wait (you say), this is wrong, for partial integration also involves boundary terms! And herein lies the rub. You need homogenous boundary conditions to complete this argument, such as $u(a)=u(b)=0$, or you may require periodic boundary conditions on $u$ and $v$ both.
To complicate things further, differentiation is an unbounded operator on $L^2$, and you can extend the definition from the smooth functions in various ways, depending on the homogeneous boundary conditions you chose to begin with. In order to get a good theory, you need the resulting operator to be closed (in the sense of having a closed graph).
In general, the adjoint of a closed, densely defined operator $A$ is defined such that $$\langle u,A^*v\rangle=\langle Au,v\rangle,$$ for all $v$ such that $u\mapsto\langle Au,v\rangle,$ is a bounded linear functional. (Then $A^*v$ exists by the Riesz representation theorem.)
There is much more to be said, but I can't write an entire textbook into an answer!
-
Thank you, Harald! – Henry Nov 8 '12 at 13:20
The integration by parts rule can be interpreted as saying that the adjoint of $\frac{d}{dx}$ is $-\frac{d}{dx}$, which means $\frac{d}{dx}$ is anti-symmetric.
Here's the idea: let $V$ be the vector space of all $C^{\infty}$ functions from $\mathbb{R}$ to $\mathbb{R}$ with period $T$. $V$ is an inner produce space over $\mathbb{R}$ with inner product defined by $$\langle f, g \rangle = \int_0^T f(x) g(x) \, dx.$$
$\frac{d}{dx}$ is a linear operator on $V$, and integration by parts tells us that if $u,v \in V$ then $$\int_0^T \frac{du }{dx} v \, dx = - \int_0^T u \frac{dv}{dx} \, dx.$$ (The boundary term is $0$ because $u$ and $v$ are periodic.) In other words, $$\langle \frac{d}{dx} u, v \rangle = \langle u, - \frac{d}{dx} v \rangle$$ which shows the adjoint of $\frac{d}{dx}$ is $-\frac{d}{dx}$.
This is a very interesting result. Because $\frac{d}{dx}$ is anti-symmetric, it is also "normal", so one could hope a version of the spectral theorem applies. This leads us to expect that there is a complete orthonormal basis of eigenfunctions for $\frac{d}{dx}$, and this is where Fourier series comes from.
Note: Somebody better at functional analysis may be able to improve on my explanation.
-
Thank you, littleO! – Henry Nov 8 '12 at 13:21 | 2016-05-06 07:55:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.9509873390197754, "perplexity": 124.08861965252657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461861735203.5/warc/CC-MAIN-20160428164215-00133-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://pypi.org/project/zamba/ | Zamba is a tool to identify the species seen in camera trap videos from sites in central Africa.
# zamba - a command line interface for species classification
### DOCUMENTATION
Zamba means "forest" in the Lingala language.
Zamba is a command-line tool built in Python to automatically identify the species seen in camera trap videos from sites in central Africa. The tool makes predictions for 24 common species in these videos. For more information, see the documentation.
The zamba command will be the entry point for users (see example usage below).
## Installing zamba (for more detail, see the documentation)
### GPU or CPU
zamba is significantly faster when using a machine with a GPU instead of just a CPU. To use a GPU, you must be using an nvidia gpu, installed and configured CUDA, and installed and configured CuDNN per their specifications. Once this is done, you can select to install the version of zamaba that uses tensorflow compiled for GPU.
When a user installs zamba that user must specify to install the GPU or CPU version. If the user fails to make this specification, no version of tensorflow will be installed, thus everything will fail.
To install with tensorflow cpu (you do not have a GPU)
$pip install zamba[cpu] To install with tensorflow gpu $ pip install zamba[gpu]
## Example usage
Once zamba is installed, you can see the commands with zamba:
zamba
Usage: zamba [OPTIONS] COMMAND [ARGS]...
Options:
--help Show this message and exit.
Commands:
predict Identify species in a video.
train [NOT IMPLEMENTED] Retrain network from...
tune [NOT IMPLEMENTED] Update network with new...
And you can see the options you can pass to the predict command with:
zamba predict --help
Usage: zamba predict [OPTIONS] [DATA_PATH] [PRED_PATH]
Identify species in a video.
This is a command line interface for prediction on camera trap footage.
Given a path to camera trap footage, the predict function use a deep
learning model to predict the presence or absense of a variety of species
of common interest to wildlife researchers working with camera trap data.
Options:
--tempdir PATH Path to temporary directory. If not
specified, OS temporary directory is used.
--proba_threshold FLOAT Probability threshold for classification. if
specified binary predictions are returned
with 1 being greater than the threshold, 0
being less than or equal to. If not
specified, probabilities between 0 and 1 are
returned.
--output_class_names If True, we just return a video and the name
of the most likely class. If False, we return
a probability or indicator (depending on
--proba_threshold) for every possible class.
--model_profile TEXT Defaults to 'full' which is slow and
accurate; can be 'fast' which is faster and
less accurate.
during processing.
--help Show this message and exit.
Once zamba is installed, you can execute it on any directory of video files. The tool does not recursively search directories, so all of the files must be at the top level of the directory. The algorithm will work the best with 15 second videos since that is what it is trained on, though it will sample frames from longer videos, which may be less reliable.
NOTE: zamba needs to download the "weights" files for the neural networks that it uses to make predictions. On first run it will download ~1GB of files with these weights. Once these are downloaded, the tool will use the local versions and will not need to perform this download again.
zamba predict path/to/videos
By default the output will be written to the file output.csv in the current directory. If the file exists, it will be overwritten.
## Running the zamba test suite
The included Makefile contains code that uses pytest to run all tests in zamba/tests.
The command is (from the project root),
\$ make test
### Testing End-To-End Prediction With test_cnnensemble.py
The test tests/test_cnnensemble.py runs an end-to-end prediction with CnnEnsemble.predict(data_dir) using a video that automatically gets downloaded along with the input directory (this and all required directories are downloaded upon instantiation of CnnEnsemble if they are not already present in the project).
By default this test is skipped due to the pytest decorator
@pytest.mark.skip(reason="This test takes hours to run, makes network calls, and is really for local dev only.")
def test_predict():
data_dir = Path(__file__).parent.parent / "models" / "cnnensemble" / "input" / "raw_test"
manager = ModelManager('', model_class='cnnensemble', proba_threshold=0.5)
manager.predict(data_dir, save=True)
It is reccomended that the decorator be commented out in order to test end-to-end prediction locally. However, this change should never be pushed, as the lightweight machines on codeship will not be happy, or able, to complete the end-to-end prediction.
To test end-to-end prediction using make test on a different set of videos, simply edit data_dir.
## Project details
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https://rupress.org/jgp/article/146/6/509/43462/Understanding-the-physiology-of-the-asymptomatic | The diaphragm muscle of hyperkalemic periodic paralysis (HyperKPP) patients and of the M1592V HyperKPP mouse model rarely suffers from the myotonic and paralytic symptoms that occur in limb muscles. Enigmatically, HyperKPP diaphragm expresses the mutant NaV1.4 channel and, more importantly, has an abnormally high Na+ influx similar to that in extensor digitorum longus (EDL) and soleus, two hindlimb muscles suffering from the robust HyperKPP abnormalities. The objective was to uncover the physiological mechanisms that render HyperKPP diaphragm asymptomatic. A first mechanism involves efficient maintenance of resting membrane polarization in HyperKPP diaphragm at various extracellular K+ concentrations compared with larger membrane depolarizations in HyperKPP EDL and soleus. The improved resting membrane potential (EM) results from significantly increased Na+ K+ pump electrogenic activity, and not from an increased protein content. Action potential amplitude was greater in HyperKPP diaphragm than in HyperKPP soleus and EDL, providing a second mechanism for the asymptomatic behavior of the HyperKPP diaphragm. One suggested mechanism for the greater action potential amplitude is lower intracellular Na+ concentration because of greater Na+ K+ pump activity, allowing better Na+ current during the action potential depolarization phase. Finally, HyperKPP diaphragm had a greater capacity to generate force at depolarized EM compared with wild-type diaphragm. Action potential amplitude was not different between wild-type and HyperKPP diaphragm. There was also no evidence for an increased activity of the Na+–Ca2+ exchanger working in the reverse mode in the HyperKPP diaphragm compared with the wild-type diaphragm. So, a third mechanism remains to be elucidated to fully understand how HyperKPP diaphragm generates more force compared with wild type. Although the mechanism for the greater force at depolarized resting EM remains to be determined, this study provides support for the modulation of the Na+ K+ pump as a component of therapy to alleviate weakness in HyperKPP.
Hyperkalemic periodic paralysis (HyperKPP) is an autosomal-dominant disease with nearly complete penetrance (Gamstorp et al., 1957; Bradley et al., 1990). The disease manifests with periods of myotonic discharge and episodic paralytic attacks (Lehmann-Horn et al., 1983; Bradley et al., 1990; Miller et al., 2004). Weakness is prominent in the limbs and may completely incapacitate patients for hours at a time, with the frequency of paralytic attacks ranging from 3 to 28 per month (Miller et al., 2004). A key hallmark of the disease is the precipitation of paralytic attacks after potassium ingestion (Gamstorp et al., 1957; Poskanzer and Kerr, 1961; Streeten et al., 1971; Wang and Clausen, 1976; Bradley et al., 1990). Paralysis is in some cases associated with plasma [K+] increasing from the normal 4 mM to 6–8 mM (Gamstorp et al., 1957; Lehmann-Horn et al., 1983; Miller et al., 2004), whereas in other cases there is no increase (Poskanzer and Kerr, 1961; Rüdel and Ricker, 1985; Chinnery et al., 2002). Most of the available treatments for HyperKPP are limited either by partial effectiveness or declining efficacy over time (Clausen et al., 1980; Lehmann-Horn et al., 1983).
HyperKPP is caused by missense mutations in the SCN4A gene that encodes for the α subunit of the Nav1.4 sodium channel expressed in adult skeletal muscles (Cannon, 2006). Most HyperKPP cases (∼66%) result from two mutations: threonine to methionine at residue 704 (T704M) or methionine to valine at residue 1592 (M1592V); the remaining cases are related to seven other mutations (Miller et al., 2004). The mutations cause three primary functional defects in the Nav1.4 channel. First, the steady-state activation curve shifts toward more negative membrane potentials (EMs; Rojas et al., 1999), which lowers action potential threshold. Second, the steady-state slow inactivation shifts toward less negative EMs (Hayward et al., 1999). Third, mutant channels enter a non-inactivation mode upon membrane depolarization at elevated extracellular K+ concentration ([K+]e), an effect not observed in normal channels (Cannon et al., 1991).
As a consequence of these defects, Nav1.4 channels open at greater frequency at rest, resulting in large Na+ influx (Clausen et al., 2011; Lucas et al., 2014), which then depolarizes the cell membrane (Lehmann-Horn et al., 1983; Ricker et al., 1989; Clausen et al., 2011; Lucas et al., 2014). Myotonic discharges occur when the EM approaches a threshold for sustained firing. [K+]e increases as several action potentials are generated. In normal muscles, increases in [K+]e cause a membrane depolarization that inactivates NaV1.4 channels, thereby reducing action potential amplitude (Yensen et al., 2002). As a consequence of lower action potential amplitude, less Ca2+ is released from the sarcoplasmic reticulum, resulting in decreased force or sarcomere shortening as [Ca2+]i becomes submaximal (Lucas et al., 2014; Zhu et al., 2014). In HyperKPP limb muscles, the effects of increased K+ on EM and force are greater than in normal muscles (Wang and Clausen, 1976; Lehmann-Horn et al., 1983, 1987; Hayward et al., 2008; Clausen et al., 2011; Lucas et al., 2014), often causing complete loss of membrane excitability. In fact, this sensitivity to elevated K+ constitutes a key feature of HyperKPP.
HyperKPP patients primarily suffer from limb weakness or paralysis but, surprisingly, only ∼25% of patients experience respiratory distress (Charles et al., 2013). In one study (Lucas et al., 2014), only two of eight tested diaphragms from the M1592V HyperKPP knock-in mouse model completely stopped contracting upon stimulation while developing a prolonged contracture (in the absence of stimulation) when [K+]e was increased to 10 mM, whereas the remaining six HyperKPP diaphragms had similar tetanic force at 4.7 and 10 mM K+ when compared with their wild-type counterpart. Furthermore, in this study, neither a sudden loss in the ability to respond to stimuli nor a prolonged contracture was observed from the 44 tested HyperKPP diaphragms during an experiment at any [K+]e, in the presence of ouabain to inhibit the Na+–K+ ATPase pump (NKA) or ORM-10103 to inhibit the Na+–Ca2+ exchanger (NCX). Thus, the susceptibility to paralysis of HyperKPP diaphragm appears to be very low, at least in the context of the M1592V mutation. An asymptomatic diaphragm muscle is surprising for two reasons. First, diaphragm expresses the NaV1.4 channel protein at 75% of the level in extensor digitorum longus (EDL) and twice the level in soleus, being two muscles suffering from HyperKPP symptoms (Zhou and Hoffman, 1994; Lucas et al., 2014). Second, the tetrodotoxin (TTX)-sensitive Na+ influx in the diaphragm is the largest of these three HyperKPP muscles (Lucas et al., 2014). A better understanding of the mechanisms by which HyperKPP diaphragm maintains its force may help us develop better and more effective treatments for HyperKPP patients.
The objective of this study was to clarify the physiological mechanisms that render diaphragm muscle less vulnerable to the ionic perturbations produced by HyperKPP mutant Na channel expression. We compared how various [K+]e affect the resting EM, action potential, and tetanic force, as well as the protein content and electrogenic contribution of the α1 and α2 isoforms of the NKA in EDL, soleus, and diaphragm. The results showed that the HyperKPP diaphragm is resistant to weakness because (a) the NKA electrogenic contribution to resting EM was greater in HyperKPP than in wild-type diaphragm, HyperKPP EDL, and soleus; (b) at depolarized resting EM, HyperKPP diaphragm generated an action potential with greater amplitude than HyperKPP soleus and EDL; and (c) it generated greater tetanic force than its wild-type counterpart.
Animals and approval for animal studies
HyperKPP mice (strain FVB.129S4(B6)-Scn4atm1.1Ljh/J) were generated by knocking in the equivalent of human missense mutation M1592V into the mouse genome; i.e., at position 1585 as described previously by Hayward et al. (2008). The FVB strain was used as a wild-type mouse. All mice were 2–3-mo old and weighed 20–25 g. The homozygous mutants generally do not survive beyond postnatal day 5, so knock-in mice were maintained as heterozygotes by crossbreeding with FVB mice. Mice were fed ad libitum and housed according to the guidelines of the Canadian Council for Animal Care. The Animal Care Committee of the University of Ottawa approved all experimental procedures used in this study. Before muscle excision, 2–3-mo-old mice were anaesthetized with a single intraperitoneal injection of 2.2 mg ketamine/0.4 mg xylazine/0.22 mg acepromazine per 10 g of animal body weight, and sacrificed by cervical dislocation. EDL, soleus, flexor digitorum brevis (FDB), or diaphragm was then dissected out. For force measurements, 5–7-mm wide diaphragm strips were used.
Genotyping
A 2-mm tail piece was incubated overnight with 500 µl of tail digestion buffer (0.2 mM Na2EDTA and 25 mM NaOH, pH 12.3) and 50 µl proteinase K (1 mg/ml) at 56°C. DNA extraction involved the addition of 650 µl of 1:1 phenol/CIA and centrifuged at 12,000 g for 10 min. 650 µl of CIA was added twice to the pellet and centrifuged before suspending the resulting pellet in 750 µl isopropyl alcohol. After 10 min, the solution was centrifuged for 15 min at 15,000 g. The alcohol was removed and the pellet was suspended in 750 µl of 70% ethanol and centrifuged. After removing the alcohol, the pellet was left to dry for 30 min before the addition of 200 µl TE buffer (10 mM Tris and 1 mM EDTA, pH 8.0) and incubated at 65°C for 2 h. PCR was then completed using the previously extracted DNA and the following primers: NC1F (forward): 5′-TGTCTAACTTCGCCTACGTCAA-3′ and NC2R (reverse): 5′-GAGTCACCCAGTACCTCTTTGG-3′.
PCR products were digested for 6 h using the restriction digest enzyme NspI. The mutation that is knocked in to the HyperKPP mice causes the removal of one NspI cut site that is easily detected by agarose gel electrophoresis; two bands were visualized for wild-type mice, which carry the cut site on both alleles, and three bands were seen for heterozygous HyperKPP mice harboring one normal allele and one mutant allele.
Western blots of Na+ K+ ATPase α1 and α2
Muscles were homogenized in buffer containing (mM): 50 Tris, 150 NaCl, 1% Triton X-100, 0.5% sodium deoxycholate, 0.1% SDS, and protease inhibitor cocktail, pH 8.0. Samples were rotated end-over-end for 1 h. Homogenates were centrifuged for 30 min at 17,500 g and 4°C. Protein concentration was determined in supernatants using the BCA assay method (Thermo Fisher Scientific). 40 μg of protein aliquots of the muscle lysates was diluted with Laemmli sample buffer and heated for 20 min at 56°C. Proteins were separated on 8% acrylamide gel at 100 V and transferred onto nitrocellulose membranes (Mini-PROTEAN 3 apparatus; Bio-Rad Laboratories). Equal protein loading was verified with Ponceau S (MP Biomedicals). Membranes were blocked overnight at 4°C with 5% skim milk powder in PBS containing 0.1% Tween, washed three times (10 min each) with PBS, and incubated overnight with either rabbit anti-NKAα1 antibody (Cell Signaling Technology) diluted at 1:1,000 in PBS containing 5% BSA or rabbit anti-NKAα2 antibody (EMD Millipore) diluted at 1:5,000 in PBS containing 5% skim milk. Membranes were washed three times (10 min each) with PBS and incubated for 1 h with horseradish peroxidase–conjugated goat anti–rabbit antibody (Jackson ImmunoResearch Laboratories, Inc.) diluted at 1:10,000 in PBS containing 5% skim milk. Bands were visualized by chemiluminescence using the ECL kit (PerkinElmer) on Cl-Xposure film (Thermo Fisher Scientific). Cl-Xposure films were scanned (MP 600 PIXMA; Cannon) and quantified using ImageJ software (National Institutes of Health).
Physiological measurements
Solutions.
Control solution contained (mM): 118.5 NaCl, 4.7 KCl, 1.3 CaCl2, 3.1 MgCl2, 25 NaHCO3, 2 NaH2PO4, and 5.5 d-glucose. Solutions containing different K+ concentrations were prepared by adding the appropriate amount of KCl. Solutions containing ouabain, an NKA inhibitor (Sigma-Aldrich), were prepared by dissolving ouabain directly in the control solution. Solutions containing 2-[(3,4-dihydro-2-phenyl-2H-1-benzopyran-6-yl)oxy]-5-nitro-pyridine (ORM-10103; an inhibitor of the NCX; Sigma-Aldrich) were prepared by first dissolving ORM-10103 in DMSO before it was added to the control solution. For the experiments involving ORM-10103, the final DMSO concentration was 0.1% (vol/vol) including the control solution. Solutions were continuously bubbled with 95% O2–5% CO2 to maintain a pH of 7.4. Experimental temperature was 37°C. Total flow of solutions in the muscle chamber was 15 ml/min being split just above and below the muscle to prevent any buildup of reactive oxygen species, which is quite large at 37°C (Edwards et al., 2007).
Force measurement.
Muscle length was adjusted to give maximal tetanic force. Muscles were positioned horizontally in a Plexiglas chamber. One end of the muscle was fixed to a stationary hook, whereas the other end was attached to a force transducer (model 400A; Aurora Scientific Canada). The transducer was connected to a data acquisition system (KCP13104; Keithley), and data were recorded at 5 kHz. Tetanic force was defined as the force developed while muscles were electrically stimulated and was calculated as the difference between the maximum force during a contraction and the baseline force measured 5 ms before stimulation. Electrical stimulations were applied across two platinum wires (4 mm apart) located on opposite sides of the fibers. They were connected to a Grass S88 stimulator and a Grass SIU5 isolation unit (Grass Technologies). Tetanic contractions were elicited with 200-ms trains of 0.3-ms, 10-V (supramaximal voltage) pulses. Stimulation frequencies were set to give maximum tetanic force: 140 Hz for soleus and 200 Hz for EDL and diaphragm. Tetanic contractions were elicited every 5 min.
Resting EM and action potential measurements.
Resting EM and action potential were measured using glass microelectrodes. Microelectrode tip resistances were 7∼15 MΩ, and that of the reference electrode was ∼1 MΩ. All electrodes were filled with 2 M K-citrate. A recording was rejected when the change in potential upon penetration was not a sharp drop or when the microelectrode potential did not return to zero upon withdrawal from the fiber. Single action potentials were elicited using fine platinum wires placed along the surface fibers using a single 10-V, 0.3-ms square pulse.
Statistics
Data are expressed as mean ± SEM (SEM). For statistical differences, two-way ANOVAs were used for Western blot measurements. Statistical comparisons between wild-type and HyperKPP muscles involved different mice, and for such comparison, force and EMs were independent from one another. Force and EMs were also measured at different [K+]e or ouabain concentration using the same muscles; in this case, the measurements are not independent from one another. As a consequence of the experimental design, we had two error terms: (1) the population error from the variability between mice when comparing wild-type and HyperKPP, and (2) within muscle error when comparing the effects of K+ and ouabain. So, statistical analyses were performed using spit-plot ANOVA designs as described by Steel and Torrie (1980). For this, the comparison between wild-type and HyperKPP was part of the whole plot, which used the population error, whereas the comparison for the K+ and ouabain effect was part of the split plot, which used the within muscle error. Calculations were made using the general linear model procedures of Statistical Analysis Software (version 9.3; SAS Institute Inc.). When a main effect or an interaction was significant, the least square difference (LSD) was used to locate the significant differences. The word “significant” refers only to a statistical difference (P < 0.05).
[K+]e effects on tetanic force and resting EM
As mentioned in the Introduction, the K+-induced force depression starts with a membrane depolarization. So, we first analyzed the K+ effects on tetanic force and resting EM to construct tetanic force–resting EM relationships to document how much of the force losses in HyperKPP muscles compared with wild type are related to membrane depolarization. At 4.7 mM K+ (control), mean tetanic forces using the data from all tested HyperKPP soleus and EDL in this study were significantly lower than their wild-type counterparts. Wild-type and HyperKPP soleus, respectively, generated on average a tetanic force of 19.3 and 12.7 N/cm2; i.e., mean tetanic force of HyperKPP soleus was 66% of the wild-type force (Fig. 1 A). Wild-type and HyperKPP EDL, respectively, generated 29.8 and 22.7 N/cm2 for a HyperKPP force being 76% of wild type (Fig. 1 B). For the diaphragm, however, there was no significant difference between wild type and HyperKPP (Fig. 1 C). The relative decreases in mean tetanic force of HyperKPP soleus and EDL at 9–11 mM K+ were significantly greater in HyperKPP than in wild-type EDL and soleus (Fig. 2, A and B) but not in the diaphragm, as there was no difference between wild-type and HyperKPP (Fig. 2 C). Thus, HyperKPP diaphragms do not have a greater sensitivity to the K+-induced force depression as observed with EDL and soleus.
HyperKPP muscle fibers are known to have less negative resting EM than normal fibers (Lehmann-Horn et al., 1983, 1987; Ricker et al., 1989; Clausen et al., 2011). Furthermore, the K+-induced force depression is caused by a depolarization of the cell membrane, which then causes an inactivation of NaV1.4 channels (Renaud and Light, 1992; Cairns et al., 1997; Yensen et al., 2002). To better understand the importance of the membrane depolarization in the lower force generated by HyperKPP EDL and soleus or the conservation of force for the diaphragm, we measured resting EM in 10–15 fibers chosen at random from the muscle surface at different [K+]e (note here that action potentials were not measured to determine if a fiber was excitable or not). At 4.7 mM K+, mean resting EM was 15 mV lower in HyperKPP than in wild-type soleus, a difference that became smaller as [K+]e was increased because the K+-induced membrane depolarization per [K+]e decade was smaller in HyperKPP soleus (Fig. 3 A). For EDL, the difference in resting EM between wild-type and HyperKPP was smaller, being 5–9 mV among the different [K+]e as the slopes of the membrane depolarization were not different between wild type and HyperKPP (Fig. 3 B). The smallest difference in resting EM between wild type and HyperKPP was with the diaphragm, being only 2–4 mV and nonsignificant at all [K+]e (Fig. 3 C).
Resting EM varied tremendously among mammalian muscle fibers. Consequently, one cannot just use the mean resting EM to represent all fibers. The frequency distribution of resting EM was then documented by separating all measured values in bins of 5 mV. For soleus muscles, the frequency distribution for HyperKPP fibers was shifted toward less negative resting EM compared with wild-type fibers at both 4.7 and 13 mM K+ (Fig. 4, A and B). At 4.7 mM K+, 69% of all wild-type soleus fibers had a resting EM above −70 mV compared with only 21% for HyperKPP fibers. Muscles with mean resting EM of less than −55 mV lose their capacity to generate force (Renaud and Light, 1992; Cairns et al., 1997). At 4.7 mM K+, 27% of all HyperKPP soleus fibers had a resting EM below −55 mV compared with <1% for wild-type fibers. The proportion of HyperKPP soleus fibers with resting EM below −55 mV increases to 78% at 13 mM K+, whereas it increased only to 46% for wild-type fibers. A similar shift toward less negative resting EM in HyperKPP EDL fibers was also observed at 4.7 and 14 mM K+ (Fig. 4, C and D). There were, however, two major differences between HyperKPP soleus and EDL: at 4.7 mM K+, 47% of HyperKPP EDL fibers had a resting EM above −70 mV compared with only 21% for soleus fibers, whereas the proportion of fibers with a resting EM below −55 mV was just 11% for EDL compared with 27% for soleus. Contrary to hindlimb muscles, there was no major shift in the frequency distribution of resting EM between the wild-type and HyperKPP diaphragm (Fig. 4, E and F). Only small differences were observed where the number of fibers with a resting EM ranging between −65 and −75 mV was less in HyperKPP than in wild type, whereas the number of fibers with a resting EM between −55 and −60 mV was slightly higher in HyperKPP. These small differences explained the slightly less negative mean resting EM of HyperKPP fibers compared with wild-type fibers in Fig. 3 C.
Next, we ascertained how much of the lower forces in HyperKPP EDL and soleus and the conservation of force in the HyperKPP diaphragm are related to resting EM. To do this, we took into consideration that the tetanic force from a whole muscle is a function of the mean resting EM of all fibers, as reported previously (Renaud and Light, 1992; Cairns et al., 1997). To construct a tetanic force–resting EM relationship, we calculated all mean absolute forces at different [K+]e as a percentage of the force generated by wild-type muscles at 4.7 mM K+. The force generated by HyperKPP soleus at 4.7 and 9 mM K+ fell very close to the tetanic force–resting EM of wild-type soleus, suggesting that the lower tetanic forces of HyperKPP soleus at those [K+]e are largely caused by less negative mean resting EM (Fig. 5 A). Interestingly, although tetanic force in wild-type soleus is expected to reach zero at −54 mV, HyperKPP soleus still generated some force between −55 and −49 mV.
The tetanic force of HyperKPP EDL at 4.7 and 9 mM fell below the expected force from the force–EM relationship of wild-type EDL (Fig. 5 B). For example, HyperKPP EDL tetanic force at −71 mV (measured at 4.7 mM K+) was 76%, whereas at that EM, the expected tetanic force of wild-type EDL was 98%. It thus appears that the lower force in HyperKPP EDL is not just related to less negative resting EM. At resting EM smaller than −53 mV, HyperKPP EDL developed more force than wild type. For example, HyperKPP EDL force at −48 mV (measured at 14 mM K+) was 3%, whereas for wild-type, zero force was expected to occur at −52 mV. Remarkably, the tetanic force–resting EM was significantly shifted toward less negative resting EM in HyperKPP diaphragm (Fig. 5 C). The largest difference was observed at −59 mV, as the expected wild-type force was 41% compared with 86% for HyperKPP, representing a 45% difference.
The results so far revealed two major features in this model of HyperKPP that spare the diaphragm from altered function compared with the robust abnormalities observed in EDL and soleus. The first one is smaller membrane depolarization in HyperKPP diaphragm (this study) despite the fact that it has the largest TTX-sensitive Na+ influx of all three muscles (Lucas et al., 2014). The second factor is a greater capacity of the HyperKPP diaphragm to generate more force than wild type at depolarized resting EM. To further understand the lower membrane depolarization in the HyperKPP diaphragm, we determined how the content and electrogenic contribution of NKA differ between the HyperKPP diaphragm, EDL, and soleus. To further understand the capacity of the HyperKPP diaphragm to generate more force at depolarized EM, we tested (a) the possibility that the NCX works in the reverse mode when the membrane is depolarized, as has been suggested previously for the wild-type diaphragm (Zavecz and Anderson, 1992); and (b) whether the HyperKPP diaphragm fibers generate better action potentials than wild type when the membrane is depolarized by K+.
NKA
For the measurements of NKAα1 and NKAα2 protein content, we included the FDB because it is also an asymptomatic muscle, but contrary to the diaphragm, it has a very low TTX-sensitive Na+ influx even though its NaV1.4 channel protein content is comparable to that of soleus (Lucas et al., 2014). Here, we first determined whether the NKA protein content is greater in HyperKPP than in the wild-type diaphragm, as has been reported for HyperKPP EDL (Clausen et al., 2011). HyperKPP EDL was the only muscle with significantly greater NKAα1 protein content than in the wild-type counterpart (Fig. 6 A). Next, we determined how the large increase in NKAα1 protein content in HyperKPP EDL affected the relative differences between muscles in wild type and HyperKPP. Although wild-type EDL had two to three times less NKAα1 content compared with soleus, diaphragm, and FDB, the NKAα1 content in HyperKPP EDL was no longer different from the other three muscles (Fig. 6 B). The situation was the same for the NKAα2 protein content. That is, HyperKPP EDL had a greater NKAα2 content than in wild type (Fig. 7 A). As a consequence of the increase, the wild-type diaphragm had similar NKAα2 protein content to wild-type EDL, whereas for HyperKPP NKAα2, the content was less in the diaphragm than in EDL (Fig. 7 B). It was also noted that the 83% higher content in HyperKPP than in wild-type EDL as well as the 21% higher content in HyperKPP soleus were similar to the values reported from ouabain-binding studies (Clausen et al., 2011).
Next, we assessed how inhibiting NKA activity with ouabain affected tetanic force and resting EM. We first used ouabain at a concentration of 1 µM to reduce NKAα2 activity by 92% and that of NKAα1 by 6% according to the ouabain Ki values reported by Chibalin et al. (2012) that were measured from changes in rat diaphragm resting EM after exposure to various ouabain concentrations. At 4.7 mM K+, 1 µM ouabain reduced mean tetanic force of wild-type soleus by 7%, whereas resting EM depolarized by 8 mV (Fig. 8 A). The decrease in tetanic force for HyperKPP soleus was much larger at 42% despite a similar depolarization of 7 mV. Ouabain also caused a greater decrease in force at 9 mM K+ in HyperKPP soleus, even though the ouabain-induced membrane depolarization was 7 mV in wild type and 9 mV in HyperKPP. The apparent greater ouabain effect on HyperKPP soleus force despite a similar extent of depolarizations was because in the absence of ouabain, resting EM at 4.7 mM K+ was −75 and −60 mV in wild-type and HyperKPP soleus, respectively. As per the tetanic force–resting EM curve (Fig. 5 A), small 7–9-mV depolarization is expected to have small effects on wild-type force, whereas in HyperKPP soleus, the effects were greater because the starting resting EM was in the steepest portion of the curve. The ouabain and K+ effects in EDL (Fig. 8 B) resembled those of soleus.
The situation was very different with the diaphragm (Fig. 8 C). First, the decrease in force upon exposure to 1 µM ouabain at 4.7 mM K+ was not different between the wild-type and HyperKPP diaphragm because resting EM in the presence of ouabain did not decrease below −72 mV for wild type and −63 mV for HyperKPP; i.e., resting EM remained in a range for which there is little effect on tetanic force (Fig. 5 C). Second, although the decrease in tetanic force in the wild-type diaphragm upon raising [K+]e to 9 mM at 1 µM ouabain was similar to that of wild-type soleus and EDL, the decrease in the HyperKPP diaphragm was only 42% compared with 89–91% in soleus and EDL. For a complete loss of force in the HyperKPP diaphragm at 9 mM K+, the HyperKPP diaphragm had to be exposed to 10 µM ouabain (Fig. 9 A), which fully inhibits NKAα2 and NKAα1 by 37% (Chibalin et al., 2012). The large decrease in force in that condition was related to a membrane depolarization to −50 mV (Fig. 9 B), a EM at which tetanic force is expected to be zero in the HyperKPP diaphragm (Fig. 5 C).
At 100 µM ouabain, both NKAα1 and NKAα2 are fully inhibited (Chibalin et al., 2012). At 4.7 mM K+, 100 µM ouabain reduced the tetanic force of the HyperKPP diaphragm by 84% (Fig. 9 C; similar experiments were not performed with EDL and soleus because 10 µM ouabain is sufficient to completely abolish tetanic force in HyperKPP EDL and soleus; Lucas et al., 2014). Although the HyperKPP diaphragm had the most negative and the HyperKPP soleus had the least negative resting EM at 4.7 mM K+, an exposure to 100 µM ouabain depolarized the membrane to the same level in all three muscles, i.e., −47 mV (Fig. 9 D).
The total electrogenic contribution of NKAα1 and NKAα2, calculated from the differences in resting EM in the absence and presence of 100 µM ouabain, was 19–21 mV in wild-type muscles and was not significantly different from the 18–23-mV contribution in the HyperKPP soleus and EDL (Fig. 10 A). In the HyperKPP diaphragm, however, the total NKA contribution was significantly greater at 32 mV compared with only 19 mV in the wild-type diaphragm. The NKAα2 electrogenic contribution, calculated from the depolarization caused by 1 µM ouabain, was similar in wild-type and HyperKPP soleus and higher in the HyperKPP EDL and diaphragm than in their wild-type counterparts, even though the differences were not significant (Fig. 10 B). The NKAα1 electrogenic contribution, or the difference between total and NKAα2 contribution, was not different between wild-type and HyperKPP EDL and soleus, whereas it was significantly greater in HyperKPP than in the wild-type diaphragm (Fig. 10 C).
NCX
Increasing [Ca2+]e improves force generation of HyperKPP muscles (Creutzfeldt et al., 1963; Lucas et al., 2014). Furthermore, contrary to wild-type EDL and soleus, the diaphragm depends on Ca2+ influx to maintain twitch force (Viirès et al., 1988), and there is evidence that NCX plays a role in diaphragm contractility (Zavecz et al., 1991; Zavecz and Anderson, 1992). In cardiac muscle, membrane depolarization and increases in [Na+]i are two mechanisms by which [Ca2+]i is elevated during contraction, as NCX works in the reverse mode (Janvier and Boyett, 1996) and a similar mechanism has been proposed for the diaphragm (Zavecz and Anderson, 1992). We therefore investigated whether the greater force in the HyperKPP diaphragm at less negative resting EM involves higher NCX activity in the reverse mode. To test this, we measured tetanic force while exposing muscles to 3 µM ORM-10103, an NCX inhibitor (Jost et al., 2013).
The effects of ORM-10103 were first tested in wild-type EDL, as this muscle is the least dependent on extracellular Ca2+ (Viirès et al., 1988) and for which there is no evidence for a role of NCX during contractions (Blaustein and Lederer, 1999). As expected, ORM-10103 had no effect on tetanic force of that muscle at 4.7 and 12.5 mM K+ (Fig. 11 A). ORM-10103 had no effect on HyperKPP EDL while exposed to 4.7 mM K+, but the force loss at 11 mM K+ was greater in the presence of ORM-10103 (Fig. 11 B). Similarly, the decrease in tetanic force in the wild-type and HyperKPP diaphragm at 12.5 mM K+ was greater in the presence than in the absence of ORM-10103 (Fig. 11, C and D). However, the difference in force in the absence and presence of ORM-10103 at 12.5 mM K+ was the same for the wild-type and HyperKPP diaphragm, suggesting that an increased NCX activity in the reverse mode is not a mechanism that can explain the greater force in the HyperKPP diaphragm at depolarized resting EM (Fig. 5 C).
Action potential–resting EM relationship
Next, we measured action potentials in soleus, EDL, and diaphragm fibers. In general, at 4.7 mM K+, action potentials were easily triggered in wild-type muscles, for which >95% of fibers generated an action potential upon stimulation. The situation was very different in HyperKPP soleus and EDL. At 4.7 mM K+, 30% of HyperKPP soleus fibers failed to generate an action potential upon stimulation, while 27% of HyperKPP EDL did the same. Among the excitable fibers, action potential shapes were quite similar between wild-type and HyperKPP EDL fibers when resting EM was greater than −80 mV (measured at 4.7 mM K+; Fig. 12, A and B). At less negative resting EM, such as between −55 and −75 mV (8–10 mM K+), HyperKPP EDL fibers had action potentials with lower amplitude than their wild-type counterparts. Very few fibers generated action potential below a resting EM of −55 mV (12–15 mM K+), and for those that did, the amplitude was very small.
A major aim here was to test whether the shift in the tetanic force versus resting EM relationship in the HyperKPP diaphragm versus wild type (Fig. 5) can be explained by action potentials with greater amplitude in the HyperKPP than wild-type diaphragm at depolarized resting EM. To do this, we first took into consideration the large variability in resting EM (Fig. 4). That is, action potentials were measured in several fibers while being exposed at [K+]e varying between 4.7 and 14.5 mM. Fibers were then separated according to their resting EM in bins of 5 mV. For each bin, mean resting EM, action potential amplitude, and peak were averaged to construct action potential amplitude versus resting EM relationship.
For soleus and EDL, mean action potential amplitudes were similar between wild type and HyperKPP at resting EM above −70 mV (Fig. 12, C and D). At resting EM less negative than −70 mV, action potential amplitude became smaller in HyperKPP soleus and EDL compared with their wild-type counterpart. Contrary to the situation with soleus and EDL, mean action potential amplitudes at various resting EMs were not different between the wild-type and HyperKPP diaphragm (Fig. 12 E).
We then compared the action potential amplitude versus resting EM between muscles for each of wild type and HyperKPP. For wild type, mean action potential amplitudes were lower in soleus and EDL compared with the diaphragm, with significant differences at depolarized resting EM (Fig. 13 A). The differences between diaphragm and hindlimb muscles were more pronounced in HyperKPP (Fig. 13 B). Similar analyses for action potential peak versus resting EM relationships gave rise to similar differences between wild type and HyperKPP and between EDL, soleus, and diaphragm to those observed for the action potential amplitude versus resting EM relationships (not depicted).
Individuals with HyperKPP rarely suffer from respiratory distress, although this can occur more frequently after procedures or exposure to anesthetic agents (Charles et al., 2013). In two of our studies (this study and Lucas et al., 2014), only 2 of the 52 tested diaphragm muscles from the M1592V HyperKPP mouse model suddenly stopped contracting upon stimulation as if they had become paralyzed in the course of an in vitro experiment. The objective of this study was to identify physiological mechanisms that render HyperKPP diaphragm resistant to weakness triggered by elevated [K+]e. The major findings of this study were: (a) contrary to HyperKPP soleus and EDL fibers, which were highly depolarized compared with their wild-type counterparts, there was no significant difference in mean or in the frequency distribution of resting EM between diaphragm fibers from HyperKPP compared with wild type; (b) the entire tetanic force–resting EM relationship of the HyperKPP diaphragm was shifted to less negative resting EM when compared with the wild-type relationship, whereas only a partial shift was observed for HyperKPP soleus and EDL; (c) NKAα1 and NKAα2 protein content was the same in the wild-type and HyperKPP diaphragm; (d) the NKA electrogenic contribution, especially the α1 subunit, was greater in the HyperKPP diaphragm, whereas for soleus and EDL, the total NKAα1 and NKAα2 electrogenic contribution was not different between wild type and HyperKPP; (e) the action potential amplitude versus resting EM relationship of the HyperKPP diaphragm was similar to that of the wild-type diaphragm, whereas it was shifted toward more depolarized resting EM when compared with HyperKPP soleus and EDL.
Protection of resting EM by NKA as one mechanism
It is well known that resting EM is more depolarized in HyperKPP muscles from both mouse and patients, which likely is a major contributor to muscle weakness (Lehmann-Horn et al., 1983, 1987; Ricker et al., 1989; Clausen et al., 2011). The importance of less polarization of the resting EM contributing to the lower force is further confirmed in this study from three points of view. First, there was a major shift in the frequency distribution of resting EM toward lower EM values in HyperKPP soleus and EDL (Fig. 4, A and B). Second, many fibers had resting EM lower than −55 mV, an EM below which muscles failed to generate force (Renaud and Light, 1992; Cairns et al., 1997). In that regard, 30% of HyperKPP soleus fibers failed to generate an action potential upon stimulation, a value that was close to the 27% of fibers with a resting EM below −55 mV. Third, the tetanic force generated at 4.7 and 9 mM K+ by HyperKPP soleus was very close to the expected force from the tetanic force versus resting EM relationship of wild-type soleus.
At 4.7 mM K+, the relative difference in tetanic force between wild-type and HyperKPP was less for EDL (24%) than soleus (34%). We can again demonstrate that the difference between the two muscles is in part related to differences in resting EM. That is, HyperKPP EDL had a greater mean resting EM because of a larger proportion of fibers having resting EM above −70 mV and less below −55 mV than in HyperKPP soleus. However, contrary to HyperKPP soleus, the tetanic forces generated by HyperKPP EDL at 4.7 and 9 mM K+ was less than the expected forces from the mean resting EM and the tetanic force versus resting EM relationship of wild-type EDL, suggesting that perhaps other factors are involved in lowering force in HyperKPP EDL, as discussed in the section below, Importance of stronger action potential.
The situation in the HyperKPP diaphragm was completely different to that of hindlimb muscles. The differences in mean resting EM between wild type and HyperKPP at various [K+]e ranged between 2 and 4 mV in the diaphragm compared with 6–15 mV in the soleus and 5–9 mV in the EDL. Furthermore, contrary to HyperKPP soleus and EDL, there were only small differences in the frequency distribution of resting EM between the wild-type and HyperKPP diaphragm. The importance of better resting EM in the HyperKPP diaphragm than in the soleus and EDL especially at high [K+]e can be illustrated as follows. For HyperKPP soleus at 9 mM K+, resting EM was −56 mV and tetanic force was 40% of wild-type force at 4.7 mM K+ (used as the maximum force normal muscle can generate; Fig. 5). At the same [K+]e, resting EM of the HyperKPP diaphragm was −62 mV and tetanic force was 96% of maximum force, representing a 56% difference in force between the HyperKPP soleus and diaphragm. If, on the other hand, the HyperKPP diaphragm had depolarized to −56 mV, then according to the tetanic force–resting EM relationship shown in Fig. 5 C, tetanic force would have only been 50%, representing just a 10% difference with soleus. When similar calculations are performed for EDL, the differences in tetanic force at 9 mM K+ with the HyperKPP diaphragm is 39%, being reduced to 21% if the diaphragm-resting EM had depolarized to −59 mV as observed in EDL instead of the measured −62 mV. Thus, a better maintenance of resting EM (this study) despite large Na+ influx (Lucas et al., 2014) is one important mechanism that protects the HyperKPP diaphragm from the robust symptoms observed in the HyperKPP soleus and EDL.
This study now provides evidence for an increased electrogenic contribution of NKA, primarily its α1 isoform, to the resting EM of the HyperKPP diaphragm. We suggest that the increase in NKA electrogenic contribution is related to greater NKA activity in HyperKPP than in the wild-type diaphragm, as we did not observe a difference in the NKAα1 or NKAα2 protein content between the wild-type and HyperKPP diaphragm. In fact, it appears that changes in NKA content does very little, as the NKA electrogenic contribution in the HyperKPP EDL was similar to that in the wild-type EDL despite a 1.5- to 2.0-fold greater NKAα1 and NKAα2 content in HyperKPP (Figs. 6 A and 7 A; Clausen et al., 2011). Also, an increased electrogenic contribution was observed only in the HyperKPP diaphragm despite the fact that the NKA content was no longer different between the HyperKPP soleus, EDL, and diaphragm as it was for wild-type muscles (Figs. 6 B and 7 B). A possible mechanism for the difference in NKA activity between the diaphragm and limb muscles is the fact that the former is constantly active. Reducing the time interval between contractions to 1 min not only acutely increases NKA activity in wild-type and HyperKPP muscles, but it also allows for an increase in tetanic force in the HyperKPP soleus toward the wild-type level (Overgaard et al., 1999; Clausen et al., 2011). This is in agreement with the fact that HyperKPP patients sometimes can avoid a weakness or paralytic attack with mild exercise (Poskanzer and Kerr, 1961).
An activation of NKA by the β-adrenergic agonist salbutamol has been shown to eliminate muscle weakness in hindlimb muscles of patients and mice suffering from HyperKPP (Wang and Clausen, 1976; Clausen et al., 2011). Altogether it would appear that a salbutamol treatment would help in keeping a higher NKA activity in hindlimb muscles and thus prevent weakness. Unfortunately, the efficacy of salbutamol decreases over time in human patients (Clausen et al., 1980). The loss of effectiveness over time suggests that there is a fundamental difference in the regulation of NKA activity between HyperKPP hindlimb and diaphragm muscles, and a better understanding of this regulation in future studies will help us develop more effective and sustained pharmacological strategies to treat HyperKPP patients.
Importance of stronger action potential
As discussed above, the mean tetanic forces generated by the HyperKPP soleus at resting EM between −55 and −60 mV (or at 4.7 or 9 mM K+) were close to our expectation from the tetanic force versus resting EM relationship of wild-type soleus. In EDL, on the other hand, the mean tetanic forces between −60 and −70 mV were less than the expectation. One possible reason for this difference may be in the generation of action potentials. HyperKPP soleus fibers generated action potentials of similar amplitude as their wild-type counterpart when resting EM was greater than −70 mV, with just a small tendency for lower amplitude below −70 mV. For EDL, the differences between wild-type and HyperKPP were more pronounced and started when resting EM was less than −80 mV. Considering the shift in the resting EM frequency distribution toward a lower potential, we suggest that even at 4.7 mM K+, HyperKPP EDL has many fibers with resting EM at which it generates lower action potential amplitude than wild type, which then most likely results in lower Ca2+ release and force generation. This difference between the HyperKPP soleus and EDL cannot be explained from our results. Lower action potential amplitude can be related to higher [Na+]i, which are known to be higher in HyperKPP muscles (Lehmann-Horn et al., 1987; Ricker et al., 1989; Amarteifio et al., 2012). Higher [Na+]i then lowers action potential amplitude as the decreased Na+ concentration gradient is reduced (Cairns et al., 2003). However, this explanation is questionable if we consider the lack of any significant difference in NKA electrogenic contribution (Fig. 10) and Na+ influx between the HyperKPP soleus and EDL (Lucas et al., 2014). Another possibility is the difference in ClC-1 Cl channel activity between these two muscles, and varying this channel activity at elevated [K+]e significantly impacts the capacity of muscle to generate action potentials and force (Pedersen et al., 2005, 2009a, b).
Interestingly, HyperKPP EDL and soleus still generated a small amount of force when resting EM became below −55 mV, a potential at which no more force was generated by wild-type muscles as reported previously (Renaud and Light, 1992; Cairns et al., 1997). At those resting EMs, action potential measurements became very difficult because of a very large number of unexcitable fibers. Furthermore, although many fibers failed to generate single action potentials, the situation may be different during a tetanic stimulation train during which a few action potentials may eventually be generated considering that the steady-state slow inactivation never reaches zero (Hayward et al., 1999) and that K+ at a concentration as low as 10 mM provokes a non-inactivation mode in the mutant NaV1.4 channel (Cannon et al., 1991). So, at a resting EM below −55 mV, few action potentials are generated by mutant NaV1.4 channels during a tetanus giving rise to small force development.
The situation was again different with the diaphragm. Contrary to soleus and EDL, the HyperKPP diaphragm was slightly less sensitive to the K+-induced force depression than the wild-type diaphragm; although the effect was not significant, it was constantly observed in all experiments. The lower sensitivity to K+ was not accompanied by smaller membrane depolarizations when [K+]e was increased. In fact, the reverse was observed as the mean resting EMs were slightly lower in the HyperKPP than in the wild-type diaphragm. Moreover, there was little difference in the resting EM frequency distribution. As a consequence of this situation, the tetanic force versus mean resting EM was significantly shifted toward less negative resting EM so that at depolarized EM, the HyperKPP diaphragm generated greater tetanic force than the wild-type diaphragm.
The shift cannot be explained by the generation of an action potential with greater amplitude in HyperKPP than in wild type because there was no difference in the action potential amplitude versus resting EM relationship, at least in regards to the generation of single action potentials. Another possibility is greater increases in [Ca2+]i during a contraction in HyperKPP diaphragm fibers in the absence of any difference in action potential amplitude. An NCX inhibition resulted in lower force in both the wild-type and HyperKPP diaphragm at 12.5 mM K+ (Fig. 11), supporting a role for NCX working in the reverse mode to increase [Ca2+]i during contraction, which is in agreement with previous studies (Zavecz et al., 1991; Zavecz and Anderson, 1992; Blaustein and Lederer, 1999). However, the force reduction upon NCX inhibition was the same in wild type and HyperKPP. So, if there is a greater [Ca2+]i during contraction in the HyperKPP diaphragm, it is unlikely that the mechanism involves NCX. Another possibility is a greater Ca2+ entry via the store-operated Ca2+ entry to maintain high Ca2+ content in the sarcoplasmic reticulum as reported previously (Lucas et al., 2014) or greater Ca2+ release by the sarcoplasmic reticulum when CaV1.1 and RyR1 channels are activated by action potentials.
Although there was no difference in the action potential amplitude versus resting EM between the wild-type and HyperKPP diaphragm, we observed that this relationship in the diaphragm was shifted toward less negative resting EM when compared with EDL and soleus. The shift was small with few significant differences for wild-type muscles when resting EM was less than −70 mV. For HyperKPP muscles, on the other hand, the differences were much greater and became significant when resting EM was less than −80 mV. It thus appears that the HyperKPP diaphragm has a better capacity of generating normal action potentials than HyperKPP hindlimb muscles. One possible mechanism for the improved action potential amplitude is lower [Na+]i than in the hindlimb muscles because of greater NKA activity. The better action potentials also constitute a second mechanism that renders the HyperKPP diaphragm asymptomatic.
In conclusion, we provide evidence here that compared with the HyperKPP soleus and EDL, (a) the HyperKPP diaphragm muscle better maintains its resting EM at various [K+]e despite very large Na+ influx through defective NaV1.4 channels and (b) generates larger action potentials at depolarized EM, providing two essential mechanisms that minimize weakness in the HyperKPP diaphragm. The improved resting EM is related to significant increases in the electrogenic contribution (or activity) of the NKAα1 isoform rather than an increase in NKA protein content. The improved capacity of the HyperKPP diaphragm to generate action potentials may also be caused by the higher NKA activity that maintains low [Na+]i. Finally, compared with wild type, the HyperKPP diaphragm generated more force at depolarized resting EM. The mechanism for this improved force generation could not be discerned here as it was not related to the generation of stronger action potentials compared with wild type nor to a greater contribution of NCX working in the reverse mode to the increase in [Ca2+]i during contraction. Future studies will be necessary to understand the mechanisms responsible for the higher NKA activity in the HyperKPP diaphragm and to examine whether facilitating these pathways in hindlimb muscles could improve function for HyperKPP patients.
This study was supported by a grant from the Canadian Institute of Health Research to J.-M. Renaud.
The authors declare no competing financial interests.
Eduardo Rios served as editor.
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Abbreviations used in this paper:
• EDL
extensor digitorum longus
•
• EM
membrane potential
•
• FDB
flexor digitorum brevis
•
• HyperKPP
hyperkalemic periodic paralysis
•
• LSD
least square difference
•
• NCX
Na+–Ca2+ exchanger
•
• NKA
Na+–K+ ATPase pump
•
• TTX
tetrodotoxin | 2022-05-27 05:53:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5793042182922363, "perplexity": 9398.5802640572}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662636717.74/warc/CC-MAIN-20220527050925-20220527080925-00526.warc.gz"} |
https://mathoverflow.net/questions/269648/curious-fact-about-number-of-roots-of-mathfraksl-n | # Curious fact about number of roots of $\mathfrak{sl}_n$
The Lie algebra $\mathfrak{sl}_n$ has many special features which are not shared by other simple Lie algebras, for example all of its fundamental representations are minuscule.
I recently discovered another curious fact.
The number of positive roots of $\mathfrak{sl}_n$ is the same as the dimension of $Sym^2 \mathfrak h$ --- both are $\binom{n}{2}$. All other simple Lie algebras have more positive roots. Is this fact is connected to the special properties enjoyed by $\mathfrak{sl}_n$? Is there some a priori reason to expect this equality?
I came upon this fact because I was thinking about the following linear map (here $\mathfrak g$ is any semisimple Lie algebra and $\mathfrak h$ is its Cartan subalgebra): \begin{align*} Sym^2 \mathfrak h &\rightarrow U \mathfrak g \\ xy &\mapsto \sum_{\alpha \in \Delta_+} \langle \alpha, x \rangle \langle \alpha, y \rangle E_\alpha F_\alpha \end{align*} The set $\{ E_\alpha F_\alpha \}_{\alpha \in \Delta_+}$ in $U \mathfrak g$ is linearly independent and I was hoping that this map would be an isomorphism onto its span, but this can only be true for $\mathfrak{sl}_n$.
Has anyone seen this map before?
• I have not seen this map before. Up to some rescaling, I think it is something like $[xy, R]$, where $R$ is the R-matrix and the commutator takes place in $U\mathfrak g \otimes U\mathfrak g$? – Theo Johnson-Freyd May 13 '17 at 4:26
• Interesting. If I compute correctly, the map sends $(e_{ii}-e_{jj})^2$ to $e_{ij}e_{ji}$, so it must be as claimed. – მამუკა ჯიბლაძე May 13 '17 at 15:25
• You can identify $\operatorname{Sym}^2(\mathfrak{h})$ with the space of symmetric bilinear forms on $\mathfrak{h}$. For each root $\alpha$, you can define a symmetric bilinear form $\varphi_\alpha$ on $\mathfrak{h}$ via $\varphi_{\alpha}(h, h') = \alpha(h)\alpha(h')$ for all $h, h' \in \mathfrak{h}$. Then it seems in the case of $\mathfrak{sl}_n$ that $\{ \varphi_\alpha : \alpha \text { positive root} \}$ is linearly independent in $\operatorname{Sym}^2(\mathfrak{h})$. ... – spin May 13 '17 at 15:27
• I wonder if the same is true for other simple types? That would at least show that the dimension of $\operatorname{Sym}^2(\mathfrak{h})$ is $\geq$ number of positive roots, then maybe there is some reason why equality holds only for type $A$? – spin May 13 '17 at 15:27
• @spin I believe for all simples, $\sum_\alpha\varphi_\alpha$ is nondegenerate (and is, up to a scalar multiple, the only Weyl-group-invariant symmetric bilinear form on $\mathfrak h$), although I do not quite understand whether this helps in any way to answer your question – მამუკა ჯიბლაძე May 15 '17 at 5:37
I will mostly answer spin's approach and question here, but it is rather long to write as a comment only. I am not sure if it will answer the original OP's question, but it does seem like a good step.
Given a positive root $\alpha$, we associate to it, we associate to it (following spin's comment) a symmetric bilinear form $\varphi_\alpha$ on $\mathfrak{h}$, defined by:
$\varphi_\alpha(h,h') = \alpha(h)\alpha(h')$
for all $h,h'\in \mathfrak{h}$.
We claim that the set $\{\varphi_\alpha; \alpha \text{ is a positive root}\}$ spans $\operatorname{Sym}^2(\mathfrak{h})$ for simple algebras. This is equivalent to showing that any real homogeneous quadratic form $B(-,-)$ on $\mathfrak{h}^*$ which vanishes identically on the set of positive roots must vanish identically.
Lemma: if $V$ is a real vector space, with a symmetric bilinear form $B(-,-)$ on $V$, and if $v_1, v_2\in V$ are linearly independent, and suppose further that $B(v_i,v_i) = 0$ for $i=1,2$ and that $B(v_1+cv_2, v_1+cv_2)=0$ for some real nonzero constant $c$, then $B(-,-)$ must vanish identically on the span of $v_1$ and $v_2$.
The proof of the lemma is easy. Indeed it implies in particular that the $v_i$ are null, and that $B(v_1,v_2) = 0$.
We now apply this lemma inductively. We start on one end of the Dynkin diagram, and apply the lemma to $v_1$ being the first simple root, and $v_2$ being the second simple root, which is connected to the first simple root by either a simple, a double or triple edge. Applying reflection to $v_2$ about the hyperplane orthogonal to $v_1$, we get the required third vector of the form either $v_1+cv_2$ or $v_2+cv_1$ for some $c>0$, which is also a positive root. We thus get that $B(-,-)$ vanishes identically on the span of the first 2 simple roots, and then apply the lemma again to the second simple root, and a new simple root connected to the second simple root by a simple, double or triple edge, and so on.
This would show that $B(-,-) = 0$ identically, thus proving the claim.
Hence this would show that the number of positive roots is always greater or equal to the dimension of $\operatorname{Sym}^2(\mathfrak{h})$, answering spin's question at least. The proof is conceptual and does not rely on the classification result. I am not sure if it really answers the OP's original question though.
Edit 1: building up on my previous argument, we would like to show, in order to answer the OP's question, that if the $\varphi_\alpha$'s are linearly independent over $\mathbb{R}$ in $\operatorname{Sym}^2(\mathfrak{h})$, then the root system is of the A type. Thus we have to rule out the existence of multiple edges, and rule out the existence of Dynkin subdiagrams which are isomorphic to that of $SO(8)$ (here, note that I am unfortunately relying on the classification result). So let us assume that the $\varphi_\alpha$ are linearly independent. Then for any positive root $\alpha$, there exists a symmetric bilinear form $B(-,-)$ on $\mathfrak{h}^*$, such that $B(\beta,\beta) = 0$ for any positive root $\beta \neq \alpha$ and $B(\alpha,\alpha) \neq 0$. By restricting to a rank 2 Dynkin subdiagram, corresponding to 2 simple roots connected by a multiple edge, one can remove one positive root spanned by these 2 simple roots, and still get at least 3 positive roots satisfying the lemma above, so that $B$ must vanish identically on the span of these 2 simple roots, thus leading to a contradiction.
It remains to rule out the existence of rank 4 Dynkin subdiagrams isomorphic to that of $SO(8)$ under the linear independence assumption on the $\varphi_\alpha$. This can be done explicitly by removing a specific positive root, and showing that if $B$ vanishes on all other positive roots, then it must be 0 on this rank 4 subspace of $\mathfrak{h}^*$, thus leading to a contradiction too. Alternatively, one can simply count the dimensions in this case: $SO(8)$ has 12 positive roots, while $\operatorname{Sym}^2(\mathfrak{h})$ has dimension 10, thus ruling out the existence of such Dynkin subdiagrams under the linear independence hypothesis on the $\varphi_\alpha$.
Remark: the first part of my post does not rely on the classification result, while edit 1 does unfortunately rely on at least part of the classification result. Edit 1 consists in ruling out multiple edges and Dynkin subdiagrams isomorphic to that of $SO(8)$. I hope that over all, my post answers the OP's question (it could use some editing though). | 2019-03-22 11:09:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9432480335235596, "perplexity": 123.09382891970878}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202642.32/warc/CC-MAIN-20190322094932-20190322120932-00426.warc.gz"} |
https://brilliant.org/practice/projectile-motion-with-robin-hood/ | # Projectile Motion with Robin Hood
Robin Hood was a beguiling casanova who sometimes lived in a tree, but he was also a master archer who used his bow and arrow to challenge nemeses and rescue his friends from danger.
In this quiz, we'll model Robin Hood's shots using projectile motion and see how, precisely, he pulls it off.
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
Near the surface of Earth, the gravitational field is roughly constant and is equal to $$g$$. This causes all objects to accelerate downward with $$a_y = g$$ toward Earth, including Robin Hood's arrows.
Though not equal to the measured value, we'll take $$g$$ to be $$-\SI[per-mode=symbol]{10}{\meter\per\second\squared}$$ throughout this quiz for ease of calculation.
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
Robin Hood is known for stealing things from the rich to give them to the poor. He finds himself surrounded by some angry noblemen who demand he return their gold chest.
Outnumbered, he takes an arrow from his quiver and fires it straight into the air, scattering the noblemen who are afraid it will hit them on the way down.
How much time (in seconds) does he have before the arrow hits the ground?
Details and Assumptions:
• The arrow's launch velocity is $$v = \SI[per-mode=symbol]{120}{\meter\per\second}.$$
• $$g = \SI[per-mode=symbol]{10}{\meter\per\second\squared}.$$
• Assume that Robin Hood shoots the arrow from ground level.
• Neglect air resistance.
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
As we alluded at the end of the Derive Quiz, our results work in multiple dimensions. Recall, the vector position obeys $\mathbf{r}(t) = \mathbf{r}(0) + \mathbf{v}(0)t + \frac12 \mathbf{a}_0 t^2.$ Motion along each dimension is independent so that \begin{align} r_x(t) &= r_x(0) + v_x(0)t + \frac12 a_x t^2 \\ r_y(t) &= r_y(0) + v_y(0)t + \frac12 a_y t^2. \end{align} To visualize the relationship between a vector and its components, consider the velocity of the arrow below.
In the vector view, our arrow travels with speed $$\lvert\mathbf{v}\rvert$$ at an angle $$\theta$$ to the horizontal. Broken into components, it has speed $$\lvert \mathbf{v} \rvert \cos\theta$$ in the $$x$$ direction and speed $$\lvert \mathbf{v} \rvert \sin\theta$$ in the $$y$$ direction.
The independent relations for $$r_x$$ and $$r_y$$ are all there is to projectile motion.
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
Fresh off his daring escape, Robin Hood found a band of Merry Men. They agree to join him if he can shoot an arrow further than any of them can.
How far (in $$\si{\meter}$$) will his arrow go?
Details and Assumptions
• Ignore Robin Hood's height, e.g. he shoots the arrow while lying down.
• The arrows releases at $$v = \SI[per-mode=symbol]{120}{\meter\per\second}$$ at an angle of $$60^\circ.$$
• $$g = \SI[per-mode=symbol]{10}{\meter\per\second\squared}.$$
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
If Robin Hood wants to maximize the range of his arrow, at what angle $$\theta$$ (in degrees) should he aim his bow?
Details and Assumptions
• Ignore Robin Hood's height, e.g. he shoots the arrow while lying down.
• $$g = \SI[per-mode=symbol]{10}{\meter\per\second\squared}.$$
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
The Merry Men can't believe how far he shot the arrow. They insist that he one-up himself to prove he didn't cheat, so they place an apple high on a $$\SI{40}{\meter}$$ wall that's $$\SI{400}{\meter}$$ away.
To test his restraint, they force him to angle his bow at $$\theta =60^\circ$$.
Note: figure not drawn to scale
How fast (in $$\si[per-mode=symbol]{\meter\per\second}$$) should he shoot the arrow if he wants to hit the apple?
Details and Assumptions
• Ignore Robin Hood's height, e.g. he shoots the arrow while lying down.
• $$g = \SI[per-mode=symbol]{10}{\meter\per\second\squared}.$$
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
In the last problem, we solved for the case where Robin Hood's angle was constrained. Suppose we constrained him to fire at full speed, $$v=\SI[per-mode=symbol]{120}{\meter\per\second}$$.
At what angle should he release the arrow?
Hint: You may not be able to do this with analysis alone, so consider resorting to a plot. As this problem is quite tricky, we'll solve it together on the next pane.
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
Once again, we start with \begin{align} \ell &= v\cos\theta T \\ H &= v\sin\theta T - \frac12 gT^2. \end{align} We can use the first equation to eliminate $$T$$ in the second: \begin{align} H &= \ell\tan\theta - \frac12 g\frac{\ell^2}{v^2\cos^2\theta} \\\\ 2v^2\cos^2\theta H &= 2\ell\tan\theta v^2\cos^2\theta - g\ell^2 \\\\ 2v^2\cos^2\theta\left(\ell\tan\theta - H\right) &= g\ell^2. \end{align} This equation is a pain to solve analytically for $$\theta$$ and is easier to resolve by numerical means. We can plot the functions on either side and look for an intercept.
We find that the plots intersect at roughly $$\theta \approx 13.75^\circ$$ and $$81.36^\circ$$.
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
In the last problem we saw that setting a launch speed does not pick out a specific launch angle, despite the fact that setting a launch angle picks out a specific launch velocity.
Why is this? The answer is clear from a drawing.
If an arrow has enough speed to hit the apple on the way up (before its peak) then it has enough speed to hit it on the way down. Therefore, any speed $$v$$ will have two solutions, $$\theta_+$$ and $$\theta_-,$$ corresponding to a hit before and after the trajectory's peak. At the minimal velocity $$v_\textrm{min}$$ that's still capable of hitting the apple, the two angles coincide.
Here we argued for the existence of two solutions informally.
• Can you take it a step further and prove this result?
• Can you find $$v_\textrm{min}$$?
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
Having gained their allegiance, the Merry Men agree to help Robin Hood take down a corrupt noblewoman who reigns over the valley with onerous taxes. Each night she sleeps alone on her porch that overlooks the valley, as her guards tell each other jokes nearby.
What is the smallest angle, $$\gamma = \phi + \theta$$ (in degrees), the crew should aim their arrows to hit the noblewoman?
Details and Assumptions:
• Each archer shoots their arrow at full speed, $$v_0 = \SI[per-mode=symbol]{120}{\meter\per\second}$$.
• The bed is located a distance $$s = \SI{400}{\meter}$$ up the hill (measured along the hill) and the hill is pitched at $$\theta=60^\circ$$.
• Ignore the height of the archers, e.g. each person shoots their arrow while lying down.
• $$g = \SI[per-mode=symbol]{10}{\meter\per\second\squared}.$$
## Projectile Motion with Robin Hood
### Kinematics
# Projectile Motion with Robin Hood
In helping Robin Hood win over the Merry Men and take down the punitive nobles, we have solved some of the classic hard problems of projectile motion.
Though we considered the projectile motion of the point particle (as an idealization of our arrow), we used the same basic building blocks as the techniques used to put satellites into orbit, dock the space station, and chart the course of passenger planes.
By breaking the problem down into the horizontal and vertical dimensions, we were able to treat the arrow's complete motion as two independent, one-dimensional problems, a process known as vector decomposition. As you will see in the next chapter, this step reflects a deep principle of mechanics embedded in Newtons laws of motion—that motion in a given dimension is controlled solely by forces along that dimension.
## Projectile Motion with Robin Hood
### Kinematics
× | 2018-04-20 02:58:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8672448992729187, "perplexity": 2149.046458586007}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125937113.3/warc/CC-MAIN-20180420022906-20180420042906-00414.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/12708/prove-expressions-for-visibilities-operatornametr-rho-a2-rho-b2-operat | # Prove expressions for visibilities $\operatorname{Tr}(\rho_a^2\rho_b^2), \operatorname{Tr}[(\rho_a \rho_b)^2]$ in terms of SWAP operations
Given two states $$\rho_a$$ and $$\rho_b$$, and knowing that the SWAP gate swaps two qubits, how can one prove that visibilities
$$v_1=Tr[\rho_a^2 \rho_b^2] = Tr[S_{AB} S_{BC} S_{CD} \rho_a \otimes \rho_a \otimes \rho_b \otimes \rho_b ]$$
$$v_2=Tr[(\rho_a \rho_b)^2] = Tr[S_{BC} S_{CD} S_{AB} S_{BC} S_{AB} \rho_a \otimes \rho_a \otimes \rho_b \otimes \rho_b]$$
Reference: https://arxiv.org/pdf/1501.03099.pdf
Explicitly write out what the trace is supposed to be: $$\text{Tr}(S_{AB}S_{BC}S_{CD}\rho_a\otimes\rho_a\otimes\rho_b\otimes\rho_b)=\sum_{i,j,k,l}\langle ijkl|S_{AB}S_{BC}S_{CD}\rho_a\otimes\rho_a\otimes\rho_b\otimes\rho_b|ijkl\rangle.$$ Now you can apply the swaps to the basis elements $$=\sum_{i,j,k,l}\langle jikl|S_{BC}S_{CD}\rho_a\otimes\rho_a\otimes\rho_b\otimes\rho_b|ijkl\rangle$$ and keep going... $$=\sum_{i,j,k,l}\langle jkli|\rho_a\otimes\rho_a\otimes\rho_b\otimes\rho_b|ijkl\rangle$$ So, this is the same thing as $$\sum_{ijkl}\langle j|\rho_a|i\rangle\langle i|\rho_b|l\rangle\langle l|\rho_b|k\rangle\langle k|\rho_a|j\rangle.$$ It's worth a comment about how I chose the ordering of those terms. I just started with the first index ($$j$$). Because it's closing index was $$i$$, the next term I wrote down started with $$i$$, and so on. Now, knowing the result I want, I'll use the commutativity of multiplication to choose a slightly different order: $$\sum_{ijkl}\langle k|\rho_a|j\rangle\langle j|\rho_a|i\rangle\langle i|\rho_b|l\rangle\langle l|\rho_b|k\rangle.$$ Next, notice that there's lots of completeness relations appearing here: $$\sum_j|j\rangle\langle j|=I$$. Thus, this is the same as $$\sum_k\langle k|\rho_aI\rho_aI\rho_bI\rho_B|k\rangle=\text{Tr}(\rho_a^2\rho_b^2).$$
The other one will work just the same. | 2022-12-03 19:47:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 14, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6968151330947876, "perplexity": 222.2099003643227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710936.10/warc/CC-MAIN-20221203175958-20221203205958-00341.warc.gz"} |
https://www.azdictionary.com/definition/governmentalism | # governmentalism definition
• noun:
• The theory that the authority regarding the basic government cannot simply be exercised fully but extended.
## Related Sources
• Definition for "governmentalism"
• The theory that the authority regarding the basic…
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https://allendowney.github.io/ThinkBayes2/chap19.html | # MCMC¶
For most of this book we’ve been using grid methods to approximate posterior distributions. For models with one or two parameters, grid algorithms are fast and the results are precise enough for most practical purposes. With three parameters, they start to be slow, and with more than three they are usually not practical.
In the previous chapter we saw that we can solve some problems using conjugate priors. But the problems we can solve this way tend to be the same ones we can solve with grid algorithms.
For problems with more than a few parameters, the most powerful tool we have is MCMC, which stands for “Markov chain Monte Carlo”. In this context, “Monte Carlo” refers to to methods that generate random samples from a distribution. Unlike grid methods, MCMC methods don’t try to compute the posterior distribution; they sample from it instead.
It might seem strange that you can generate a sample without ever computing the distribution, but that’s the magic of MCMC.
To demonstrate, we’ll start by solving the World Cup problem. Yes, again.
## The World Cup Problem¶
In <<_PoissonProcesses>> we modeled goal scoring in football (soccer) as a Poisson process characterized by a goal-scoring rate, denoted $$\lambda$$.
We used a gamma distribution to represent the prior distribution of $$\lambda$$, then we used the outcome of the game to compute the posterior distribution for both teams.
To answer the first question, we used the posterior distributions to compute the “probability of superiority” for France.
To answer the second question, we computed the posterior predictive distributions for each team, that is, the distribution of goals we expect in a rematch.
In this chapter we’ll solve this problem again using PyMC3, which is a library that provide implementations of several MCMC methods. But we’ll start by reviewing the grid approximation of the prior and the prior predictive distribution.
## Grid Approximation¶
As we did in <<_TheGammaDistribution>> we’ll use a gamma distribution with parameter $$\alpha=1.4$$ to represent the prior.
from scipy.stats import gamma
alpha = 1.4
prior_dist = gamma(alpha)
I’ll use linspace to generate possible values for $$\lambda$$, and pmf_from_dist to compute a discrete approximation of the prior.
import numpy as np
from utils import pmf_from_dist
lams = np.linspace(0, 10, 101)
prior_pmf = pmf_from_dist(prior_dist, lams)
We can use the Poisson distribution to compute the likelihood of the data; as an example, we’ll use 4 goals.
from scipy.stats import poisson
data = 4
likelihood = poisson.pmf(data, lams)
Now we can do the update in the usual way.
posterior = prior_pmf * likelihood
posterior.normalize()
0.05015532557804499
Soon we will solve the same problem with PyMC3, but first it will be useful to introduce something new: the prior predictive distribution.
## Prior Predictive Distribution¶
We have seen the posterior predictive distribution in previous chapters; the prior predictive distribution is similar except that (as you might have guessed) it is based on the prior.
To estimate the prior predictive distribution, we’ll start by drawing a sample from the prior.
sample_prior = prior_dist.rvs(1000)
The result is an array of possible values for the goal-scoring rate, $$\lambda$$. For each value in sample_prior, I’ll generate one value from a Poisson distribution.
from scipy.stats import poisson
sample_prior_pred = poisson.rvs(sample_prior)
sample_prior_pred is a sample from the prior predictive distribution. To see what it looks like, we’ll compute the PMF of the sample.
from empiricaldist import Pmf
pmf_prior_pred = Pmf.from_seq(sample_prior_pred)
And here’s what it looks like:
from utils import decorate
pmf_prior_pred.bar()
decorate(xlabel='Number of goals',
ylabel='PMF',
title='Prior Predictive Distribution')
One reason to compute the prior predictive distribution is to check whether our model of the system seems reasonable. In this case, the distribution of goals seems consistent with what we know about World Cup football.
But in this chapter we have another reason: computing the prior predictive distribution is a first step toward using MCMC.
## Introducing PyMC3¶
PyMC3 is a Python library that provides several MCMC methods. To use PyMC3, we have to specify a model of the process that generates the data. In this example, the model has two steps:
• First we draw a goal-scoring rate from the prior distribution,
• Then we draw a number of goals from a Poisson distribution.
Here’s how we specify this model in PyMC3:
import pymc3 as pm
with pm.Model() as model:
lam = pm.Gamma('lam', alpha=1.4, beta=1.0)
goals = pm.Poisson('goals', lam)
After importing pymc3, we create a Model object named model.
If you are not familiar with the with statement in Python, it is a way to associate a block of statements with an object. In this example, the two indented statements are associated with the new Model object. As a result, when we create the distribution objects, Gamma and Poisson, they are added to the Model.
Inside the with statement:
• The first line creates the prior, which is a gamma distribution with the given parameters.
• The second line creates the prior predictive, which is a Poisson distribution with the parameter lam.
The first parameter of Gamma and Poisson is a string variable name.
PyMC3 provides a function that generates a visual representation of the model.
pm.model_to_graphviz(model)
In this visualization, the ovals show that lam is drawn from a gamma distribution and goals is drawn from a Poisson distribution. The arrow shows that the values of lam are used as parameters for the distribution of goals.
## Sampling the Prior¶
PyMC3 provides a function that generates samples from the prior and prior predictive distributions. We can use a with statement to run this function in the context of the model.
with model:
trace = pm.sample_prior_predictive(1000)
The result is a dictionary-like object that maps from the variables, lam and goals, to the samples. We can extract the sample of lam like this:
sample_prior_pymc = trace['lam']
sample_prior_pymc.shape
(1000,)
The following figure compares the CDF of this sample to the CDF of the sample we generated using the gamma object from SciPy.
from empiricaldist import Cdf
def plot_cdf(sample, **options):
"""Plot the CDF of a sample.
sample: sequence of quantities
"""
Cdf.from_seq(sample).plot(**options)
plot_cdf(sample_prior,
label='SciPy sample',
color='C5')
plot_cdf(sample_prior_pymc,
label='PyMC3 sample',
color='C0')
decorate(xlabel=r'Goals per game ($\lambda$)',
ylabel='CDF',
title='Prior distribution')
The results are similar, which confirms that the specification of the model is correct and the sampler works as advertised.
From the trace we can also extract goals, which is a sample from the prior predictive distribution.
sample_prior_pred_pymc = trace['goals']
sample_prior_pred_pymc.shape
(1000,)
And we can compare it to the sample we generated using the poisson object from SciPy.
Because the quantities in the posterior predictive distribution are discrete (number of goals) I’ll plot the CDFs as step functions.
def plot_pred(sample, **options):
Cdf.from_seq(sample).step(**options)
plot_pred(sample_prior_pred,
label='SciPy sample',
color='C5')
plot_pred(sample_prior_pred_pymc,
label='PyMC3 sample',
color='C13')
decorate(xlabel='Number of goals',
ylabel='PMF',
title='Prior Predictive Distribution')
Again, the results are similar, so we have some confidence we are using PyMC3 right.
## When Do We Get to Inference?¶
Finally, we are ready for actual inference. We just have to make one small change. Here is the model we used to generate the prior predictive distribution:
with pm.Model() as model:
lam = pm.Gamma('lam', alpha=1.4, beta=1.0)
goals = pm.Poisson('goals', lam)
And here is the model we’ll use to compute the posterior distribution.
with pm.Model() as model2:
lam = pm.Gamma('lam', alpha=1.4, beta=1.0)
goals = pm.Poisson('goals', lam, observed=4)
The difference is that we mark goals as observed and provide the observed data, 4.
And instead of calling sample_prior_predictive, we’ll call sample, which is understood to sample from the posterior distribution of lam.
options = dict(return_inferencedata=False)
with model2:
trace2 = pm.sample(500, **options)
Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [lam]
100.00% [3000/3000 00:01<00:00 Sampling 2 chains, 0 divergences]
Sampling 2 chains for 1_000 tune and 500 draw iterations (2_000 + 1_000 draws total) took 1 seconds.
Although the specification of these models is similar, the sampling process is very different. I won’t go into the details of how PyMC3 works, but here are a few things you should be aware of:
• Depending on the model, PyMC3 uses one of several MCMC methods; in this example, it uses the No U-Turn Sampler (NUTS), which is one of the most efficient and reliable methods we have.
• When the sampler starts, the first values it generates are usually not a representative sample from the posterior distribution, so these values are discarded. This process is called “tuning”.
• Instead of using a single Markov chain, PyMC3 uses multiple chains. Then we can compare results from multiple chains to make sure they are consistent.
Although we asked for a sample of 500, PyMC3 generated two samples of 1000, discarded half of each, and returned the remaining 1000. From trace2 we can extract a sample from the posterior distribution, like this:
sample_post_pymc = trace2['lam']
sample_post_pymc.shape
(1000,)
And we can compare the CDF of this sample to the posterior we computed by grid approximation:
posterior.make_cdf().plot(label='posterior grid',
color='C5')
plot_cdf(sample_post_pymc,
label='PyMC3 sample',
color='C4')
decorate(xlabel=r'Goals per game ($\lambda$)',
ylabel='CDF',
title='Posterior distribution')
The results from PyMC3 are consistent with the results from the grid approximation.
## Posterior Predictive Distribution¶
Finally, to sample from the posterior predictive distribution, we can use sample_posterior_predictive:
with model2:
post_pred = pm.sample_posterior_predictive(trace2)
100.00% [1000/1000 00:00<00:00]
The result is a dictionary that contains a sample of goals.
sample_post_pred_pymc = post_pred['goals']
sample_post_pred_pymc.shape
(1000,)
I’ll also generate a sample from the posterior distribution we computed by grid approximation.
sample_post = posterior.sample(1000)
sample_post_pred = poisson(sample_post).rvs()
And we can compare the two samples.
plot_pred(sample_post_pred,
label='grid sample',
color='C5')
plot_pred(sample_post_pred_pymc,
label='PyMC3 sample',
color='C12')
decorate(xlabel='Number of goals',
ylabel='PMF',
title='Posterior Predictive Distribution')
Again, the results are consistent. So we’ve established that we can compute the same results using a grid approximation or PyMC3.
But it might not be clear why. In this example, the grid algorithm requires less computation than MCMC, and the result is a pretty good approximation of the posterior distribution, rather than a sample.
However, this is a simple model with just one parameter. In fact, we could have solved it with even less computation, using a conjugate prior. The power of PyMC3 will be clearer with a more complex model.
## Happiness¶
Recently I read “Happiness and Life Satisfaction” by Esteban Ortiz-Ospina and Max Roser, which discusses (among many other things) the relationship between income and happiness, both between countries, within countries, and over time.
It cites the “World Happiness Report”, which includes results of a multiple regression analysis that explores the relationship between happiness and six potentially predictive factors:
• Income as represented by per capita GDP
• Social support
• Healthy life expectancy at birth
• Freedom to make life choices
• Generosity
• Perceptions of corruption
The dependent variable is the national average of responses to the “Cantril ladder question” used by the Gallup World Poll:
Please imagine a ladder with steps numbered from zero at the bottom to 10 at the top. The top of the ladder represents the best possible life for you and the bottom of the ladder represents the worst possible life for you. On which step of the ladder would you say you personally feel you stand at this time?
I’ll refer to the responses as “happiness”, but it might be more precise to think of them as a measure of satisfaction with quality of life.
In the next few sections we’ll replicate the analysis in this report using Bayesian regression.
We can use Pandas to read the data into a DataFrame.
import pandas as pd
filename = 'WHR20_DataForFigure2.1.xls'
df.head(3)
Country name Regional indicator Ladder score Standard error of ladder score upperwhisker lowerwhisker Logged GDP per capita Social support Healthy life expectancy Freedom to make life choices Generosity Perceptions of corruption Ladder score in Dystopia Explained by: Log GDP per capita Explained by: Social support Explained by: Healthy life expectancy Explained by: Freedom to make life choices Explained by: Generosity Explained by: Perceptions of corruption Dystopia + residual
0 Finland Western Europe 7.8087 0.031156 7.869766 7.747634 10.639267 0.954330 71.900826 0.949172 -0.059482 0.195445 1.972317 1.285190 1.499526 0.961271 0.662317 0.159670 0.477857 2.762835
1 Denmark Western Europe 7.6456 0.033492 7.711245 7.579955 10.774001 0.955991 72.402504 0.951444 0.066202 0.168489 1.972317 1.326949 1.503449 0.979333 0.665040 0.242793 0.495260 2.432741
2 Switzerland Western Europe 7.5599 0.035014 7.628528 7.491272 10.979933 0.942847 74.102448 0.921337 0.105911 0.303728 1.972317 1.390774 1.472403 1.040533 0.628954 0.269056 0.407946 2.350267
df.shape
(153, 20)
The DataFrame has one row for each of 153 countries and one column for each of 20 variables.
The column called 'Ladder score' contains the measurements of happiness we will try to predict.
score = df['Ladder score']
## Simple Regression¶
To get started, let’s look at the relationship between happiness and income as represented by gross domestic product (GDP) per person.
The column named 'Logged GDP per capita' represents the natural logarithm of GDP for each country, divided by population, corrected for purchasing power parity (PPP).
log_gdp = df['Logged GDP per capita']
The following figure is a scatter plot of score versus log_gdp, with one marker for each country.
import matplotlib.pyplot as plt
plt.plot(log_gdp, score, '.')
decorate(xlabel='Log GDP per capita at PPP',
It’s clear that there is a relationship between these variables: people in countries with higher GDP generally report higher levels of happiness.
We can use linregress from SciPy to compute a simple regression of these variables.
from scipy.stats import linregress
result = linregress(log_gdp, score)
And here are the results.
pd.DataFrame([result.slope, result.intercept],
index=['Slope', 'Intercept'],
columns=[''])
Slope 0.717738
Intercept -1.198646
The estimated slope is about 0.72, which suggests that an increase of one unit in log-GDP, which is a factor of $$e \approx 2.7$$ in GDP, is associated with an increase of 0.72 units on the happiness ladder.
Now let’s estimate the same parameters using PyMC3. We’ll use the same regression model as in Section <<_RegressionModel>>:
$y = a x + b + \epsilon$
where $$y$$ is the dependent variable (ladder score), $$x$$ is the predictive variable (log GDP) and $$\epsilon$$ is a series of values from a normal distribution with standard deviation $$\sigma$$.
$$a$$ and $$b$$ are the slope and intercept of the regression line. They are unknown parameters, so we will use the data to estimate them.
The following is the PyMC3 specification of this model.
x_data = log_gdp
y_data = score
with pm.Model() as model3:
a = pm.Uniform('a', 0, 4)
b = pm.Uniform('b', -4, 4)
sigma = pm.Uniform('sigma', 0, 2)
y_est = a * x_data + b
y = pm.Normal('y',
mu=y_est, sd=sigma,
observed=y_data)
The prior distributions for the parameters a, b, and sigma are uniform with ranges that are wide enough to cover the posterior distributions.
y_est is the estimated value of the dependent variable, based on the regression equation. And y is a normal distribution with mean y_est and standard deviation sigma.
Notice how the data are included in the model:
• The values of the predictive variable, x_data, are used to compute y_est.
• The values of the dependent variable, y_data, are provided as the observed values of y.
Now we can use this model to generate a sample from the posterior distribution.
with model3:
trace3 = pm.sample(500, **options)
Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sigma, b, a]
100.00% [3000/3000 00:04<00:00 Sampling 2 chains, 0 divergences]
Sampling 2 chains for 1_000 tune and 500 draw iterations (2_000 + 1_000 draws total) took 5 seconds.
The number of effective samples is smaller than 25% for some parameters.
When you run the sampler, you might get warning messages about “divergences” and the “acceptance probability”. You can ignore them for now.
The result is an object that contains samples from the joint posterior distribution of a, b, and sigma.
trace3
<MultiTrace: 2 chains, 500 iterations, 6 variables>
ArviZ provides plot_posterior, which we can use to plot the posterior distributions of the parameters. Here are the posterior distributions of slope, a, and intercept, b.
import arviz as az
with model3:
az.plot_posterior(trace3, var_names=['a', 'b']);
The graphs show the distributions of the samples, estimated by KDE, and 94% credible intervals. In the figure, “HDI” stands for “highest-density interval”.
The means of these samples are consistent with the parameters we estimated with linregress.
print('Sample mean:', trace3['a'].mean())
print('Regression slope:', result.slope)
Sample mean: 0.715698157714354
Regression slope: 0.717738495630452
print('Sample mean:', trace3['b'].mean())
print('Regression intercept:', result.intercept)
Sample mean: -1.174412246262264
Regression intercept: -1.1986460618088843
Finally, we can check the marginal posterior distribution of sigma
az.plot_posterior(trace3['sigma']);
The values in the posterior distribution of sigma seem plausible.
The simple regression model has only three parameters, so we could have used a grid algorithm. But the regression model in the happiness report has six predictive variables, so it has eight parameters in total, including the intercept and sigma.
It is not practical to compute a grid approximation for a model with eight parameters. Even a coarse grid, with 20 points along each dimension, would have more than 25 billion points. And with 153 countries, we would have to compute almost 4 trillion likelihoods.
But PyMC3 can handle a model with eight parameters comfortably, as we’ll see in the next section.
20 ** 8 / 1e9
25.6
153 * 20 ** 8 / 1e12
3.9168
## Multiple Regression¶
Before we implement the multiple regression model, I’ll select the columns we need from the DataFrame.
columns = ['Ladder score',
'Logged GDP per capita',
'Social support',
'Healthy life expectancy',
'Freedom to make life choices',
'Generosity',
'Perceptions of corruption']
subset = df[columns]
subset.head(3)
Ladder score Logged GDP per capita Social support Healthy life expectancy Freedom to make life choices Generosity Perceptions of corruption
0 7.8087 10.639267 0.954330 71.900826 0.949172 -0.059482 0.195445
1 7.6456 10.774001 0.955991 72.402504 0.951444 0.066202 0.168489
2 7.5599 10.979933 0.942847 74.102448 0.921337 0.105911 0.303728
The predictive variables have different units: log-GDP is in log-dollars, life expectancy is in years, and the other variables are on arbitrary scales. To make these factors comparable, I’ll standardize the data so that each variable has mean 0 and standard deviation 1.
standardized = (subset - subset.mean()) / subset.std()
Now let’s build the model. I’ll extract the dependent variable.
y_data = standardized['Ladder score']
And the dependent variables.
x1 = standardized[columns[1]]
x2 = standardized[columns[2]]
x3 = standardized[columns[3]]
x4 = standardized[columns[4]]
x5 = standardized[columns[5]]
x6 = standardized[columns[6]]
And here’s the model. b0 is the intercept; b1 through b6 are the parameters associated with the predictive variables.
with pm.Model() as model4:
b0 = pm.Uniform('b0', -4, 4)
b1 = pm.Uniform('b1', -4, 4)
b2 = pm.Uniform('b2', -4, 4)
b3 = pm.Uniform('b3', -4, 4)
b4 = pm.Uniform('b4', -4, 4)
b5 = pm.Uniform('b5', -4, 4)
b6 = pm.Uniform('b6', -4, 4)
sigma = pm.Uniform('sigma', 0, 2)
y_est = b0 + b1*x1 + b2*x2 + b3*x3 + b4*x4 + b5*x5 + b6*x6
y = pm.Normal('y',
mu=y_est, sd=sigma,
observed=y_data)
We could express this model more concisely using a vector of predictive variables and a vector of parameters, but I decided to keep it simple.
Now we can sample from the joint posterior distribution.
with model4:
trace4 = pm.sample(500, **options)
Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sigma, b6, b5, b4, b3, b2, b1, b0]
100.00% [3000/3000 00:04<00:00 Sampling 2 chains, 0 divergences]
Sampling 2 chains for 1_000 tune and 500 draw iterations (2_000 + 1_000 draws total) took 4 seconds.
Because we standardized the data, we expect the intercept to be 0, and in fact the posterior mean of b0 is close to 0.
trace4['b0'].mean()
-0.0009400028402880869
We can also check the posterior mean of sigma:
trace4['sigma'].mean()
0.5157546237813752
From trace4 we can extract samples from the posterior distributions of the parameters and compute their means.
param_names = ['b1', 'b3', 'b3', 'b4', 'b5', 'b6']
means = [trace4[name].mean()
for name in param_names]
We can also compute 94% credible intervals (between the 3rd and 97th percentiles).
def credible_interval(sample):
"""Compute 94% credible interval."""
ci = np.percentile(sample, [3, 97])
return np.round(ci, 3)
cis = [credible_interval(trace4[name])
for name in param_names]
The following table summarizes the results.
index = columns[1:]
table = pd.DataFrame(index=index)
table['Posterior mean'] = np.round(means, 3)
table['94% CI'] = cis
table
Posterior mean 94% CI
Logged GDP per capita 0.244 [0.085, 0.41]
Social support 0.225 [0.081, 0.377]
Healthy life expectancy 0.225 [0.081, 0.377]
Freedom to make life choices 0.187 [0.087, 0.289]
Generosity 0.054 [-0.039, 0.14]
Perceptions of corruption -0.100 [-0.197, 0.002]
It looks like GDP has the strongest association with happiness (or satisfaction), followed by social support, life expectancy, and freedom.
After controlling for those other factors, the parameters of the other factors are substantially smaller, and since the CI for generosity includes 0, it is plausible that generosity is not substantially related to happiness, at least as they were measured in this study.
This example demonstrates the power of MCMC to handle models with more than a few parameters. But it does not really demonstrate the power of Bayesian regression.
If the goal of a regression model is to estimate parameters, there is no great advantage to Bayesian regression compared to conventional least squares regression.
Bayesian methods are more useful if we plan to use the posterior distribution of the parameters as part of a decision analysis process.
## Summary¶
In this chapter we used PyMC3 to implement two models we’ve seen before: a Poisson model of goal-scoring in soccer and a simple regression model. Then we implemented a multiple regression model that would not have been possible to compute with a grid approximation.
MCMC is more powerful than grid methods, but that power comes with some disadvantages:
• MCMC algorithms are fiddly. The same model might behave well with some priors and less well with others. And the sampling process often produces warnings about tuning steps, divergences, “r-hat statistics”, acceptance rates, and effective samples. It takes some expertise to diagnose and correct these issues.
• I find it easier to develop models incrementally using grid algorithms, checking intermediate results along the way. With PyMC3, it is not as easy to be confident that you have specified a model correctly.
For these reasons, I recommend a model development process that starts with grid algorithms and resorts to MCMC if necessary. As we saw in the previous chapters, you can solve a lot of real-world problems with grid methods. But when you need MCMC, it is useful to have a grid algorithm to compare to (even if it is based on a simpler model).
All of the models in this book can be implemented in PyMC3, but some of them are easier to translate than others. In the exercises, you will have a chance to practice.
## Exercises¶
Exercise: As a warmup, let’s use PyMC3 to solve the Euro problem. Suppose we spin a coin 250 times and it comes up heads 140 times. What is the posterior distribution of $$x$$, the probability of heads?
For the prior, use a beta distribution with parameters $$\alpha=1$$ and $$\beta=1$$.
See the PyMC3 documentation for the list of continuous distributions.
# Solution
n = 250
k_obs = 140
with pm.Model() as model5:
x = pm.Beta('x', alpha=1, beta=1)
k = pm.Binomial('k', n=n, p=x, observed=k_obs)
trace5 = pm.sample(500, **options)
az.plot_posterior(trace5)
Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [x]
100.00% [3000/3000 00:00<00:00 Sampling 2 chains, 0 divergences]
Sampling 2 chains for 1_000 tune and 500 draw iterations (2_000 + 1_000 draws total) took 1 seconds.
Exercise: Now let’s use PyMC3 to replicate the solution to the Grizzly Bear problem in <<_TheGrizzlyBearProblem>>, which is based on the hypergeometric distribution.
I’ll present the problem with slightly different notation, to make it consistent with PyMC3.
Suppose that during the first session, k=23 bears are tagged. During the second session, n=19 bears are identified, of which x=4 had been tagged.
Estimate the posterior distribution of N, the number of bears in the environment.
For the prior, use a discrete uniform distribution from 50 to 500.
See the PyMC3 documentation for the list of discrete distributions.
Note: HyperGeometric was added to PyMC3 after version 3.8, so you might need to update your installation to do this exercise.
# Solution
k = 23
n = 19
x = 4
with pm.Model() as model6:
N = pm.DiscreteUniform('N', 50, 500)
y = pm.HyperGeometric('y', N=N, k=k, n=n, observed=x)
trace6 = pm.sample(1000, **options)
az.plot_posterior(trace6)
Multiprocess sampling (2 chains in 2 jobs)
Metropolis: [N]
100.00% [4000/4000 00:00<00:00 Sampling 2 chains, 0 divergences]
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 1 seconds.
The number of effective samples is smaller than 25% for some parameters.
Exercise: In <<_TheWeibullDistribution>> we generated a sample from a Weibull distribution with $$\lambda=3$$ and $$k=0.8$$. Then we used the data to compute a grid approximation of the posterior distribution of those parameters.
Now let’s do the same with PyMC3.
For the priors, you can use uniform distributions as we did in <<_SurvivalAnalysis>>, or you could use HalfNormal distributions provided by PyMC3.
Note: The Weibull class in PyMC3 uses different parameters than SciPy. The parameter alpha in PyMC3 corresponds to $$k$$, and beta corresponds to $$\lambda$$.
Here’s the data again:
data = [0.80497283, 2.11577082, 0.43308797, 0.10862644, 5.17334866,
3.25745053, 3.05555883, 2.47401062, 0.05340806, 1.08386395]
# Solution
with pm.Model() as model7:
lam = pm.Uniform('lam', 0.1, 10.1)
k = pm.Uniform('k', 0.1, 5.1)
y = pm.Weibull('y', alpha=k, beta=lam, observed=data)
trace7 = pm.sample(1000, **options)
az.plot_posterior(trace7)
Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [k, lam]
100.00% [4000/4000 00:01<00:00 Sampling 2 chains, 0 divergences]
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 2 seconds.
The acceptance probability does not match the target. It is 0.8819724175144361, but should be close to 0.8. Try to increase the number of tuning steps.
Exercise: In <<_ImprovingReadingAbility>> we used data from a reading test to estimate the parameters of a normal distribution.
Make a model that defines uniform prior distributions for mu and sigma and uses the data to estimate their posterior distributions.
Now estimate the parameters for the treated group.
data = responses['Treated']
# Solution
with pm.Model() as model8:
mu = pm.Uniform('mu', 20, 80)
sigma = pm.Uniform('sigma', 5, 30)
y = pm.Normal('y', mu, sigma, observed=data)
trace8 = pm.sample(500, **options)
Auto-assigning NUTS sampler...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [sigma, mu]
100.00% [3000/3000 00:01<00:00 Sampling 2 chains, 0 divergences]
Sampling 2 chains for 1_000 tune and 500 draw iterations (2_000 + 1_000 draws total) took 2 seconds.
# Solution
with model8:
az.plot_posterior(trace8)
Exercise: In <<_TheLincolnIndexProblem>> we used a grid algorithm to solve the Lincoln Index problem as presented by John D. Cook:
“Suppose you have a tester who finds 20 bugs in your program. You want to estimate how many bugs are really in the program. You know there are at least 20 bugs, and if you have supreme confidence in your tester, you may suppose there are around 20 bugs. But maybe your tester isn’t very good. Maybe there are hundreds of bugs. How can you have any idea how many bugs there are? There’s no way to know with one tester. But if you have two testers, you can get a good idea, even if you don’t know how skilled the testers are.”
Suppose the first tester finds 20 bugs, the second finds 15, and they find 3 in common; use PyMC3 to estimate the number of bugs.
Note: This exercise is more difficult that some of the previous ones. One of the challenges is that the data includes k00, which depends on N:
k00 = N - num_seen
So we have to construct the data as part of the model. To do that, we can use pm.math.stack, which makes an array:
data = pm.math.stack((k00, k01, k10, k11))
Finally, you might find it helpful to use pm.Multinomial.
I’ll use the following notation for the data:
• k11 is the number of bugs found by both testers,
• k10 is the number of bugs found by the first tester but not the second,
• k01 is the number of bugs found by the second tester but not the first, and
• k00 is the unknown number of undiscovered bugs.
Here are the values for all but k00:
k10 = 20 - 3
k01 = 15 - 3
k11 = 3
In total, 32 bugs have been discovered:
num_seen = k01 + k10 + k11
num_seen
32
# Solution
with pm.Model() as model9:
p0 = pm.Beta('p0', alpha=1, beta=1)
p1 = pm.Beta('p1', alpha=1, beta=1)
N = pm.DiscreteUniform('N', num_seen, 350)
q0 = 1-p0
q1 = 1-p1
ps = [q0*q1, q0*p1, p0*q1, p0*p1]
k00 = N - num_seen
data = pm.math.stack((k00, k01, k10, k11))
y = pm.Multinomial('y', n=N, p=ps, observed=data)
# Solution
with model9:
trace9 = pm.sample(1000, **options)
Multiprocess sampling (2 chains in 2 jobs)
CompoundStep
>NUTS: [p1, p0]
>Metropolis: [N]
100.00% [4000/4000 00:02<00:00 Sampling 2 chains, 0 divergences]
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 3 seconds.
The acceptance probability does not match the target. It is 0.5430480274605854, but should be close to 0.8. Try to increase the number of tuning steps.
The rhat statistic is larger than 1.05 for some parameters. This indicates slight problems during sampling.
The estimated number of effective samples is smaller than 200 for some parameters.
# Solution
with model9:
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http://math.jacobs-university.de/penkov/cv.php | Curriculum Vitae
IVAN B. PENKOV
Personal:
US citizen, born May 19, 1957, in Sofia, Bulgaria, wife Irina Sousokolova, son Boyan and daughter Anna
University Education:
• June 1982, Master Degree in Mathematics, Moscow State University. Thesis: The inverse Penrose transform
• Oct. 1987, Candidate of Physical and Mathematical Sciences (Ph.D.), Steklov Mathematical Institute. Thesis: Typical modules over classical Lie superalgebras and their geometric realizations
Research and Teaching Positions:
1982-1983 Research Assistant, Institute of Mathematics Bulgarian Academy of Sciences, Sofia May-June 1986 Visiting Professor at Institute Fourier, Grenoble, France 1987-1989 Research Fellow, Section of Algebra, Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria Oct. 1989 Visiting Professor at the University of Paris VII, France Jan.-Dec. 1990 Visiting Assistant Professor, Department of Mathematics, University of California at Berkeley June-July 1990 Visiting Scholar at the University of Göttingen, Germany Jan.-June 1991 Visiting Assistant Professor, Department of Mathematics, University of California at Davis 1991-June 93 Assistant Professor, Department of Mathematics, University of California at Riverside Sept.- Oct. 1993 Visiting Scholar at the Erwin Schrödinger Institute, Vienna, Austria July 1993-June 00 Associate Professor (tenured appointment), Department of Mathematics, University of California at Riverside Apr.–May 1996 Visiting Scholar and Program Organizer at The Erwin Schrödinger Institute, Vienna, Austria June 1996 Visiting Scholar at the Max-Planck Institute of Mathematics, Bonn, Germany June 1997 Visiting Professor at the University of Burgundy Dijon, France Sept. 1997 Visiting Professor at the University Louis Pasteur, Strasbourg, France Aug.-Sept. 1998 Visiting Scholar at the Max-Planck Institute of Mathematics, Bonn, Germany Nov. 1999 Visiting Professor at the University of Hamburg, Germany April-June 2000 Visiting Scholar at the Max-Planck Institute of Mathematics, Bonn, Germany July 2000 - Oct 04 Professor (tenured appointment), Department of Mathematics, University of California at Riverside Aug.-Sept. 2001 Visiting Scholar at the Max-Planck Institute of Mathematics, Bonn, Germany Jan.-March 2002 Program organizer and MSRI scholar, MSRI, Berkeley Aug.-Sept. 2002 Visiting Scholar at the Max-Planck Institute of Mathematics, Bonn, Germany Fall 2003 Visiting Professor at Yale University Spring 2004 Visiting Scholar at the Max-Planck Institute of Mathematics, Bonn, Germany Fall 2007 Visiting Professor at University of California at Berkeley Spring 2012 Visiting Professor at Yale University Fall 2012 Visiting Scholar at the Max-Planck Institute of Mathematics, Bonn, Germany Since 2004 Professor, Jacobs University Bremen, Germany
Main Research Interests:
Representations of finite and infinite dimensional Lie algebras, generalized Harish-Chandra modules, representations of Lie superalgebras and Lie supergroups, geometry of supermanifolds, homogeneous superspaces, super D-modules, homogeneous ind-spaces and geometry of ind-varieties.
Grants and Awards:
1991--1995 NSF research grant jointly with R.E.Block and V.Chari. 1995 NSF grant for organizing the conference in honor of R.E.Block (jointly with V.Chari). 1995 Grant from the Erwin Schr\"odinger Institute for the program Representation Theory with Applications to Mathematical Physics'' (jointly with Joseph A. Wolf). 1996 NSF grant for support of US graduate students at the Erwin Schr\"odinger Institute. 1995--1997 NSF research grant. 1997--2002 NSF Group Infrastructure Grant Lie Groups, Lie Algebras and Their Representations'' (jointly with Joseph A. Wolf and Geoffrey Mason). Jan--May, 2002 Coorganizer of the one-semester MSRI Program Representations of Infinite Dimensional Algebras and Mathematical Physics'' (jointly with Edward Frenkel, Victor Kac, Vera Serganova and Gregg Zuckerman). April 2002 Coorganizer of the MSRI Workshop Quantum field theory and supersymmetry''. Sept. 2003 Coorganizer of the workshop Locally finite Lie algebras'' at the Banff Research Institute, Canada. 2003--2006 NSF Grant Lie Groups, Lie Algebras and Their Representations'' (jointly with Joseph A. Wolf and Geoffrey Mason). 2009--2012 DFG grant (one postdoc position). 2009 Volkswagen Foundation grant for organizing the summer school "Structures in Lie Representation Theory" (jointly with A. Joseph and A. Melnikov). 2009--2015 DFG grant within the Schwerpunktprogramm 1388 "Darstellungstheorie" 2012--2013 DFG grant for German-Israeli cooperation 2012--2015 DFG grant within the priority program "SPP 1388 Darstellungstheorie" 2015--2016 AIM SQuaRE grant 2015--2018 DFG grant (3/4 postdoc position). 2018--2020 DFG grant (one postdoc position).
Other Professional Activities:
1991--2006: Founding coorganizer of the California Lie Theory Program, a cooperation program in Lie Theory between the campuses of the University of California.
2004-- : Coorganizer of the Program ”Darstellungstheorie, Transformationsgruppen und Mathematische Physik Eof the consortium of six universities: Jacobs University Bremen, University of Bochum, University of Erlangen, University of Hamburg, University of Koeln, and University of Paderborn.
2005: Coorganizer of the conference in honor of Yu. I. Manin at the Max Planck Institute for Mathematics, February 2005.
2005: Coorganizer of the International Conference in Honor of R.O. Wells, Jr., Jacobs University Bremen (formerly IUB), December 1-2, 2005.
2010: Coorganizer of the International Conference in "Infinite-dimensional Lie Algebras" in Newfoundland, Canada
2011: Coorganizer of the conference "Lie Groups" in honor of J.A. Wolf, Bochum, Germany
2012: Coorganizer of the German-Israeli workshop "Symplectic Geometry with applications to Representation Theory" at Jacobs University, Bremen, Germany
2012: Organizer of the meeting "7 1/2" in honor of Yu. I. Manin at IHES, Bures sur Yvette, France
2013: Organizer of a conference in honor of V. Tsanov, Sofia, Bulgaria.
2014: Coorganizer of the annual meeting in Soltau of the priority program SPP 1388 Darstellungstheorie''.
2014: Coorganizer of the meeting Hodge theory and unitary representations of reductive Lie groups'', Jacobs University Bremen.
2015: Coorganizer of the meeting Groups and Rings Theory and Applications'', July 15-22, Sofia, Bulgaria.
2019: Coorganizer of the meeting Perspectives on representations of semisimple Lie algebras: A workshop in honor of Gregg Zuckerman on the occasion of his 70th birthday'', May 22-24, Bochum, Germany.
1.The Penrose transform on general grassmannians, C.R. Acad. Sci. Bulg. 33 (1980), 1439-1442.
2.Null-geodesics of complex Einstein spaces, Funk. Anal. i Priloz. 16, No.1 (1982), 78-79 (Russian), (with Yu.I.Manin).
3.Linear differential operators and cohomology of analytic spaces, Uspekhi Mat. Nauk 37, No 4 (1982), 171-172 (Russian).
4.D-modules on supermanifolds, Invent. Math. 71 (1983), 501-512.
5.The formalism of left and right connections on supermanifolds, (1982). In: Lect. on supermanifolds, geom. methods and conf. groups, editors Döbner et al, World Scientific, Singapore, 1989, 3-13 (with Yu. I. Manin).
6.An introduction to geometric representation theory for complex simple Lie superalgebras. In: Proc. Conf. Diff. Geom. Math. in Physics, Shumen 1984, World Scientific 1986, 89-106.
7.Serre duality for projective supermanifolds, Funk. Anal. i Priloz. 18,No.1 (1984), 78-79, (Russian) (with O. Ogievetsky).
8.Projectivity and D-affinity of flag supermanifolds, Uspekhi Math. Nauk 40 No.1 (1985), 211-212, (Russian) (with I. Skornyakov).
9.A geometric approach to the linear Penrose transform, Transactions AMS 290 (1985), 555-575.
10.Cohomologie des D-modules typiques sur les supervariétés de drapeaux, C. R. Acad. Sci. Paris 199 (1984), 1005-1008 (with I. Skornyakov).
11.Characters of typical irreducible finite dimensional -modules, Funk. Anal. i Priloz. 20 (1986), No. 1, 37-45 (Russian).
12.Localisation des representations typiques d'une superalgèbre de Lie complexe classique, C. R. Acad. Sci. Paris 304 (1987), 163-166.
13.Borel-Weil-Bott theory for classical Lie supergroups. In: Curr. Probl. Math., Moscow, vol. 32 (1988), 71-124 (Russian).
14.Elements of supergeometry. In: Curr. Probl. Math., Moscow, vol. 32 (1988), 3-25, (Russian), (with Yu. I. Manin and A. Voronov).
15.Character formulas for some classes of atypical -and -modules, Lett. Math. Phys. 16 (1988), 251-261 (with V. Serganova).
16.Cohomology of for classical complex Lie supergroups and characters of some atypical G-modules, Annales Fourier 39 (1989), 845-873, (with V. Serganova).
17.Generic irreducible representations of classical Lie superalgebras. In: Lecture Notes in Physics, vol. 375 (1991), Springer Verlag, Berlin, 311-319 (with V. Serganova).
18.Representations of classical Lie superalgebras of type 1, Indag. Math., N.S. 3(2), (1992), 419-466 (with V. Serganova).
19.Generic representations of classical Lie superalgebras and their localisation, Monatsh. Math. 118 (1994), 267-313.
20.Generic irreducible representations of finite-dimensional Lie superalgebras, Intern. Journ. Math. 5 (1994), 389-419 (with V. Serganova).
21.Partially and fully integrable modules over Lie superalgebras. In: Studies in Advanced Mathematics (series editor S.-T. Yau), vol. 4, AMS and Internatl. Press 1997, 49-67 (with I. Dimitrov).
22.Characters of irreducible -modules and cohomology of for the Lie supergroup , Journ. of Math. Sci. 84 (1997), 1382-1412 (with V. Serganova).
23.Characters of finite-dimensional irreducible -modules, Lett. Math. Phys. 40 (1997), 147-158 (with V. Serganova).
24.Characters of strongly generic irreducible Lie superalgebra representations, Intern. Journ. Math. 9 (1998), 331-366.
25.Partially integrable highest weight modules, Transformation Groups 3 (1998), 241-253 (with I. Dimitrov).
26.The support of an irreducible Lie algebra representation, Journ. of Algebra 209 (1998), 129-142 (with V. Serganova).
27.Weight modules of direct limit Lie algebras, IMRN 1999, no. 5, 223-249 (with I. Dimitrov).
28.Weight representations of the Cartan type Lie algebras and , Math. Research Lett. 6 (1999), 397-416 (with V. Serganova).
29.On the structure of weight modules, Transactions AMS 352 (2000), 2857-2869 (with I. Dimitrov and O. Mathieu).
30. Generalized Harish-Chandra modules, Moscow Math. Journal 2 (2002), 753-767 (with V. Serganova).
34. Finite rank vector bundles on inductive limits of grassmannians, IMRN 2003, no. 34, 1871-1887 (with J. Donin).
35. Classically semisimple locally finite Lie superalgebras, Forum Math. 16 (2004), 431-446.
36.On the existence of (g,k)-modules of finite type, Duke Math. Journal 125 (2004), 329-349 (with V. Serganova and G. Zuckerman).
37.A Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras, Comm. Math. Physics 250(2004), no. 1, 47-53 (with I. Dimitrov and V. Futorny).
38.Generalized Harish-Chandra modules: a new direction of the structure theory of representations, Acta Applicandae Mathematicae, 81(2004), 311-326(with G. Zuckerman).
39. Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups, IMRN 2004, no. 55, 2935-2953 (with I. Dimitrov).
40. Borel subalgebras of , Resenhas IME-USP 6 (2004),111-119 (with I. Dimitrov).
41. Generalized Harish-Chandra modules with generic minimal k-type, Asian Journal of Mathematics 8(2004), 795-812 (with G. Zuckerman)
42. A construction of generalized Harish-Chandra modules with arbitrary minimal k-type, Canad. Math. Bull. 50 (2007), 603-609 (with G. Zuckerman).
43. Cartan subalgebras of root-reductive Lie algebras, Journ. of Algebra, 308(2007), 583-611 (with E. Dan-Cohen and N. Snyder).
44. A construction of generalized Harish-Chandra modules for locally reductive Lie algebras, Transformation Groups 13 (2008), 799-817 (with G. Zuckerman).
45. Locally semisimple and maximal subalgebras of the finitary Lie algebras $$\mathfrak{gl}(\infty), \mathfrak{sl}(\infty), \mathfrak{so}(\infty)$$, and $$\mathfrak{sp}(\infty)$$, Journal of Algebra 322 (2009), 2069-2081 (with I. Dimitrov)
46. Parabolic and Levi subalgebras of finitary Lie algebras, IMRN, International Mathematics Research Notices, Article ID rnp169, doi:10.1093/imrn/rnp169 (with E. Dan-Cohen)
47. Parabolic Subgroups of Real Direct Limit Lie Groups. In: Contemporary Mathematics, vol. 499, AMS 2009 (with E. Dan-Cohen and J. A. Wolf)
48. Rank 2 vector bundles on ind-grassmannians.In, Manin Festschrift, Progress in Mathematics 270, Birkhaeuser Boston, 2009, 555-572 (with A. S. Tikhomirov).
49. Bounded simple (g; sl(2))-modules for rkg = 2, Journal of Lie Theory 20 (2010), 581-615 (with V. Serganova).
50. Tensor representations of infinite-dimensional root-reductive Lie algebras. In: Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, Birkh\"auser, 2011, 127-150 (with K. Styrkas).
51. On bounded generalized Harish-Chandra modules, Annales de l'Institut Fourier, 62(2012), 477-496 (with V. Serganova).
52. Triviality of vector bundles on twisted ind-Grassmannians, Mat. Sbornik 202:1 (2011), 1-39 (with A. S. Tikhomirov).
53. A Bott-Borel-Weil theorem for diagonal ind-groups, Canadian Journal of Mathematics, doi:10.4153/CJM-2011-032-6 (with I. Dimitrov).
54. Categories of integrable $sl(\infty)$-, $o(\infty)$-, $sp(\infty)$-modules. In: Representation Theory and Mathematical Physics, Contemporary Mathematics 557, AMS 2011, 335-357 (with V. Serganova).
55. Levi components of parabolic subalgebras of finitary Lie algebras, arXiv:1008.0311v1, in Representation Theory and Mathematical Physics, Contemporary Mathematics, vol. 557, AMS 2011, 129-149 (with E. Dan-Cohen).
56. A Koszul category of representations of finitary Lie algebras, Advances of Mathematics 289 (2016), 250-278 (with E. Dan-Cohen and V. Serganova).
57. On the structure of the fundamental series of generalized Harish-Chandra modules, Asian Journal of Mathematics 16 (2012), 489-514 (with G. Zuckerman).
58. On Ideals in the enveloping algebra of a locally simple Lie algebra, Int Math Res Notices 2015 (2015), 5196-5228 (with A. Petukhov).
59. Linear ind-Grassmannians, Pure and Applied Math. Quarterly 10 (2014), 289-323 (with A. Tikhomirov).
60. Algebraic methods in the theory of generalized Harish-Chandra modules. In: Developments and Retrospectives in Lie Theory: Algebraic Methods, Developments in Mathematics, vol. 38, Springer Verlag, 2014, 331-350 (with G. Zuckerman).
61. Tensor representations of Mackey Lie algebras and their dense subalgebras. In: Developments and Retrospectives in Lie Theory: Algebraic Methods, Developments in Mathematics, vol. 38, Springer Verlag, 291-330 (with V. Serganova).
62. On the Barth-Van de Ven-Tyurin-Sato Theorem, Sbornik: Mathematics, 206:6 (2015), 814-848 (with A. Tikhomirov).
63. A categorification of the boson-fermion correspondence via representation theory of $sl(\infty)$, Comm. Math. Phys., 341:3 (2016), 911-931 (with I. Frenkel and V. Serganova).
64. Annihilators of highest weight $sl(\infty)$-modules, Transformation Groups 21 (2016), 821-849 (with A. Petukhov).
65. Infinite Konstant cascades and centrally generated primitive ideals of U(n) in types A_∞, C_∞, Journal of Algebra 447(2016), 109-134 (with M. Ignatyev).
66. Schubert decompositions for ind-varieties of generalized flags, Asian Journal of Mathematics 21 (2017), 599-630 (with L. Fresse).
67. On ideals in U(sl(∞)), U(o(∞)), U(sp(∞)). In: Representation theory - current trends and perspectives, EMS Series of Congress Reports, European Mathematical Society (EMS), 2016, 565-602 (with A. Petukhov).
68. Ordered tensor categories and representations of the Mackey Lie algebra of infinite matrices, Algebra and Representation Theory (2018), DOI:10.1007/s10468-018-9765-9, arXiv:1512.08157 (with A. Chirvasitu).
69. Real group orbits on flag ind-varieties of SL(∞,ℂ). In: Lie Theory and Its Applications in Physics, Springer Proceedings in Mathematics and Statistics, vol. 191 (2016), 111-135 (with M. V. Ignatyev and J. A. Wolf), corrected version.
70. On categories of admissible $$\big(\mathfrak{g},\mathrm{sl}(2)\big)$$-modules, Transformation Groups 23(2) (2018), 463-489(with V. Serganova and G. Zuckerman).
71. Primitive ideals of $$\mathrm{U}\big(\mathfrak{sl}(\infty)\big)$$, Bulletin LMS 50 (2018), 443-448 (with A. Petukhov).
72. Ind-varieties of generalized flags: a survey of results, arXiv:1701.08478, preprint 2016 (with M. Ignatyev).
72'. Ind-varieties of generalized flags: a survey of results. In:'' Itogi nauki i techniki, series Modern mathematics and its applications'', Russian Academy of Sciences 2018, Topic Surveys 147, 3-50, (with M. Ignatyev), Russian version.
73. Decomposition of cohomology of vector bundles on homogeneous ind-spaces, C. R. Acad. Sci. Bulg. 70:7 (2017), 907-916 (with E. Hristova).
74. Orbit duality in ind-varieties of maximal generalized flags, Transactions of the Moscow Mathematical Society, 2017, 131-160 (with L. Fresse).
74'. Orbit duality in ind-varieties of maximal generalized flags, Transactions of the Moscow Mathematical Society 78 (2017), no 1(with L. Fresse), Russian version, arXiv:1704.03671.
75. Representation categories of Mackey Lie algebras as universal monoidal categories, Pure and Applied Mathematics Quarterly 13 (2017), 77-121 (with A. Chirvasitu).
76. Primitive ideals of $$\mathrm{U}\big(\mathfrak{sl}(\infty)\big)$$ and the Robinson-Schensted algorithm at infinity, arXiv:1801.06692, to appear in Representation of Lie Algebraic Systems and Nilpotent orbits, Progress in Mathematics, Birkhauser (with A. Petukhov).
77. Integrable $$\mathfrak{sl}(\infty)$$-modules and category $$\mathcal{O}$$ for $$\mathfrak{gl}(m|n)$$, Journal LMS, DOI: 10.1112/jlms.12176, arXiv:1712.00664 (with C. Hoyt and V. Serganova).
78. On an infinite limit of BGG categories $$\mathcal{O}$$, Moscow Mathematical Journal, to appear; arXiv:1802.06343 (with K. Coulembier).
79. On the existence of infinite-dimensional generalized Harish-Chandra modules, Sao-Paulo Jour. Math. 12(2018), 290-294(with G. Zuckerman).
80. Simple bounded weight modules of $$\mathfrak{sl}_{\infty}$$, $$\mathfrak{o}_{\infty}$$, $$\mathfrak{sp}_{\infty}$$, arXiv:1807.01899, preprint 2018 (with D. Grantcharov).
81. Large annihilator category $$\mathcal{O}$$ for $$\mathfrak{sl}_{\infty}$$, $$\mathfrak{o}_{\infty}$$, $$\mathfrak{sp}_{\infty}$$, Journal of Algebra, to appear; arXiv:1809.09394 (with V. Serganova).
82. Examples of automorhism groups of ind-varieties of generalized flags, Journal of Geometry and Symmetry in Physics 50(2018), 71-77(doi:10.7546/jgsp-50-2018-71-77)
Books edited:
1. V. Chari, I. B. Penkov - editors. Modular Interfaces: Modular Lie Algebras, Quantum Groups, and Lie Superalgebras. Studies in Advanced Mathematics, vol. 4, 1997, AMS.
2. A. Joseph, A. Melnikov, I. Penkov - editors. Highlights in Lie Algebraic Methods. Progress in Mathematics, vol. 295, 2011, Birkhauser.
3. A. Huckleberry, I. Penkov, G. Zuckerman - editors. Lie Groups: Structure, Actions, and Representations. Progress in Mathematics, vol. 306, 2013, Birkhauser.
4. G. Mason, I. Penkov, J.A. Wolf - editors. Developments and Retrospectives in Lie Theory: Geometric and Analytic Methods, Developments in Mathematics, vol.37, 2014, Springer Verlag.
5. G. Mason, I. Penkov, J.A. Wolf - editors. Develpoments and Retrospectives in Lie Theory: Algebraic Methods, Developments in Mathematics, vol. 38, 2014, Springer Verlag.
Special journal issues edited:
Sao Paulo Journal of Mathematical Sciences, vol. 12(2018), issue 2 (jointly with A.Huckleberry). | 2019-07-22 12:30:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7041906118392944, "perplexity": 4967.80343300291}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195528013.81/warc/CC-MAIN-20190722113215-20190722135215-00305.warc.gz"} |
http://kinfit.r-forge.r-project.org/kinfit_static/kinerrmin.html | # Calculate the minimum error to assume in order to pass the variance test
## Usage
kinerrmin(kinfits, kinmodel = "SFO", alpha = 0.05)
## Arguments
kinfits
The list of kinetic model(s) from which to select. Usually this will have been generated by kinfit.
kinmodel
The kinetic model to be checked. Should be one of the names of the kinetic models used for generating kinfits.
alpha
The confidence level chosen for the chi-squared test.
## Description
This function uses optimize in order to iteratively find the smallest relative error still resulting in passing the chi-squared test as defined in the FOCUS kinetics report from 2006.
## Value
The relative error, expressed as a fraction.
## References
FOCUS (2006) “Guidance Document on Estimating Persistence and Degradation Kinetics from Environmental Fate Studies on Pesticides in EU Registration” Report of the FOCUS Work Group on Degradation Kinetics, EC Document Reference Sanco/10058/2005 version 2.0, 434 pp, http://focus.jrc.ec.europa.eu/dk
## Examples
data(FOCUS_2006_A)
kinfits <- kinfit(FOCUS_2006_A)
kinerrmin(kinfits)
[1] 0.08384814
Johannes Ranke | 2022-05-16 14:16:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5985628366470337, "perplexity": 4737.644496987384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510138.6/warc/CC-MAIN-20220516140911-20220516170911-00248.warc.gz"} |
https://www.officerentinfo.ro/article/officemarket-news/america-house-is-about-to-become-greener-with-atalian-energy-solutions-system | Following the BREEAM In-Use certification in 2014, its environmental footprint is about to be improved further thanks to the installation of the NEW ATALIAN Energy Solutions with Ergelis Inside, that follows the objective to reduce the energy consumption of the building progressively up to 10-15%.
America House will be connected to the ATALIAN Ergelis Control Center, where energy managers continuously control performance and support on-site local teams in the effective energy process, leading to large cost savings. (source: ATALIAN) | 2022-10-03 21:15:52 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8279640078544617, "perplexity": 3752.7414601237124}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337432.78/warc/CC-MAIN-20221003200326-20221003230326-00709.warc.gz"} |
http://math.stackexchange.com/questions/21832/log-log-plotting-question?answertab=active | # Log-log plotting question
I know this is rather simple, but despite my best efforts and quite a bit of searching through google I can't seem to find a satisfactory answer.
An example of what I'm trying to do:
Given a value $N= 6.8\cdot10^{-4}$ and upper and lower errors for that value of $N^{uperr} = 9.1\cdot 10 ^{-4}$ and $N_{lowerr} = 3.9\cdot 10 ^{-4}$
With an equation such as $1.6\cdot 10^{-9}Nx$
How would I go about plotting this point at $x=1$ on a log-log scale, with error-bars given by those I've specified?
Whenever I try to plot this I get error bars which are far too large, since from what I've read online simply adding $N+N^{uperr}$ doesn't work for log-log plots.
-
I'll only note that when performing an arbitrary transformation $f(u)$ on an uncertain number of the form $x\pm\epsilon$, the formula you'll need goes like
$f(x\pm\epsilon)=f(x)\pm f^{\prime}(x)\epsilon$ | 2015-01-30 20:31:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6300908923149109, "perplexity": 122.1498904083609}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115861872.41/warc/CC-MAIN-20150124161101-00071-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://www.whxb.pku.edu.cn/CN/Y2020/V36/I8/1911035 | ### 单原子分散的Au/Cu(111)表面合金的表面结构与吸附性质
• 收稿日期:2019-11-19 录用日期:2019-12-27 发布日期:2020-05-19
• 通讯作者: 邵翔 E-mail:shaox@ustc.edu.cn
• 基金资助:
国家自然科学基金(21872130);国家自然科学基金(91545128);国家重点研究开发项目(2017YFA0205003)
### Atomic Structure and Adsorption Property of the Singly Dispersed Au/Cu(111) Surface Alloy
Wenyuan Wang, Jiefu Zhang, Zhe Li, Xiang Shao()
• Received:2019-11-19 Accepted:2019-12-27 Published:2020-05-19
• Contact: Xiang Shao E-mail:shaox@ustc.edu.cn
• Supported by:
the National Natural Science Foundation of China(21872130);the National Natural Science Foundation of China(91545128);the National Key Research and Development Program of China(2017YFA0205003)
Au-Cu双金属合金纳米颗粒对包括CO氧化和CO2还原等在内的多个反应有较好的催化活性，然而关于其表面性质的研究却相当匮乏。在此工作中，我们通过对低覆盖度的Au/Cu(111)和Cu/Au(111)双金属薄膜退火，制备出了单原子级分散的Au/Cu(111)和Cu/Au(111)合金化表面，并利用高分辨扫描隧道显微镜(STM)和扫描隧道谱(STS)进一步研究了掺杂原子的电子性质及其对CO吸附行为的影响。研究发现，分散在Cu(111)表面的表层和次表层Au单原子在STM上表现出不同衬度。在−0.5 eV附近，前者表现出相较于Cu(111)明显增强的电子态密度，而后者则明显减弱。吸附实验表明表层Au单原子对CO的吸附能力并没有得到增强，甚至会减弱其周围Cu原子的吸附能力。与Au在Cu(111)表面较好的分散相反，Cu原子倾向于钻入Au(111)的次表层，并且形成多原子聚集体。且Cu原子受Au(111)衬底吸电子作用的影响，其对CO的吸附能力明显减弱。这个研究结果揭示了合金表面的微观结构与性质的关联，为进一步阐明Au-Cu双金属催化剂的表面反应机理提供参考。
Abstract:
Atomic-scale characterization of the atomic structure as well as molecular adsorption on an alloy surface plays a vital role in elucidating the catalytic mechanism of effective catalysts. Au-Cu alloy nanoparticles have important applications in catalyzing CO oxidation and CO2 reduction. However, the atomic-scale properties of Au-Cu alloy surfaces are rarely investigated. In particular, the physical and chemical properties of singly-dispersed doping atoms, either Au in Cu or vice versa, as well as their influence on the overall surface properties, have not been studied in detail. In response, we first prepared low-coverage bimetallic Au/Cu(111) and Cu/Au(111) films, which were then annealed at high temperature to realize single atomically-dispersed Au/Cu(111) and Cu/Au(111) surface alloys (SA). We characterized the surface structures and adsorption properties by low-temperature scanning tunneling microscopy and spectroscopy (LT-STM/STS). For the SA-Au/Cu(111) system, we found that Au atoms can be incorporated in both the skin and subsurface layer of the Cu(111) substrate. These species can be readily distinguished from the topography contrast in STM. Moreover, STS measurements showed clear differences between the electronic states of doped Au atoms and the Cu host. In particular, we found that Au in the skin layer was strengthened while the subsurface Au showed weakened filled states at approximately −0.5 eV compared with the Cu(111) surface, which corresponds to the characteristic Shockley state of an Au surface. These altered electronic properties at the sites of doped atoms are also reflected by changes in the interactions with probe molecules. Adsorption experiments showed that Au atoms in the top surface prevented the binding of CO molecules, causing various adsorption vacancies in the CO adlayer. In contrast, the subsurface Au atoms had little influence on surface binding with CO molecules. For the SA-Cu/Au(111) system, we found that Cu atoms tend to aggregate into small clusters in the subsurface region of the Au(111) substrate. Only few Cu atoms can be stabilized at the elbow positions of the reconstructed top surface of Au(111). Adsorption experiments showed that only Cu atoms in the skin layer can adsorb CO molecules at liquid nitrogen temperature, while the subsurface Cu atoms cannot. On the other hand, the Au atoms around the doped Cu atoms do not seem to be influenced at all, possibly because of the weak effect of Cu. These experimental results provide details on the atomistic aspects of Au-Cu alloy surfaces, which can improve our understanding of the catalytic mechanism of Au-Cu alloy catalysts.
MSC2000:
• O647 | 2023-01-29 21:54:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.518711507320404, "perplexity": 4383.291212498504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499768.15/warc/CC-MAIN-20230129211612-20230130001612-00353.warc.gz"} |
https://tex.stackexchange.com/questions/539122/why-do-i-get-a-an-error-when-using-parencite-inside-my-custom-environment-defi | # My Current Problem
I have a custom environment called revisions which underlines everything in a customisable way using soul (see the context for why), defined with the Environ. When I use \parencite inside this environment, I get the following errors (copied directly from the panel in vim). What am I doing wrong, and how can I make it work?
essay.tex|22 warning| LaTeX Warning: Citation '{McTaggart1908}' on page 0 undefined on input line 22.
essay.tex|22 error| Argument of \blx@citeargs@iii has an extra }.
essay.tex|22 error| Paragraph ended before \blx@citeargs@iii was complete.
essay.tex|22 error| You can't use end-group character }' after \the.
essay.tex|22 error| TeX capacity exceeded, sorry [main memory size=5000000]. ...}{}{}{\endgroup \blxcitecmd {parenci
essay.tex|22 error| ==> Fatal error occurred, no output PDF file produced!
|| [37] Utils.pm:209> ERROR - .auxfiles/essay.bcf is malformed, last biblatex run probably failed. Deleted .auxfiles/essay.bbl
A minimal working example:
\documentclass{article}
\usepackage{environ}
\usepackage{soul}
\usepackage{biblatex}
\NewEnviron{revision}{\expandafter\ul\expandafter{\BODY}}
\title{Some Title}
\author{}
\date{}
\begin{document}
\begin{revision}
\parencite[10]{McTaggart1908}
\end{revision}
\end{document}
# Context
I am including this in the post because it may be that I have taken the long way round, and actually there is a much easier way to solve the whole thing which avoids the above problem altogether. It may also be useful in general. I welcome all ideas.
I am a student and I use LaTeX mainly for writing essays. My university require that in the second, revised version of each essay has all revisions and edits underlined. Initially (in the spirit of WYSIWYM) I just had a command \rev:
\newcommand{\rev}[1]{\ul{#1}}
But this threw similar errors to above when I used \parencite inside it. I don't know much about LaTeX, but I thought that perhaps the 'right' way to do it (and thus a way LaTeX would deal with) would be as an environment, hence the above. But that still doesn't work!
• soul commands are quite fragile and there are a number of commands you shouldn't use inside their arguments (unless you protect them with an \mbox}, see the documentation. Consider to use lualatex and the new lua-ul package which is much more robust. Apr 17 '20 at 14:15
With lualatex and lua-ul (this requires a current tex system!) your example compiles without problems:
\documentclass{article}
\usepackage[soul]{lua-ul}
\usepackage{biblatex}
\newcommand{\rev}[1]{\ul{#1}}
\begin{document}
\rev{\parencite[10]{doody}}
\end{document}
It's not a problem of environ, but of soul. You can try
\ul{\parencite{McTaggart1908}}
and you'll get the same problem.
Solution: use \mbox. I have filecontents* just to make the example self-contained.
\begin{filecontents*}{\jobname.bib}
@article{McTaggart1908,
author={X. McTaggart},
title={Title},
journal={Journal},
year={1908},
}
\end{filecontents*}
\documentclass{article}
\usepackage{environ}
\usepackage{soul}
\usepackage{biblatex}
\NewEnviron{revision}{\expandafter\ul\expandafter{\BODY}}
\title{Some Title}
\author{}
\date{}
\begin{document}
\ul{\mbox{\parencite[10]{McTaggart1908}}}
\begin{revision}
\mbox{\parencite[10]{McTaggart1908}}
\end{revision}
\end{document}
` | 2022-01-21 03:07:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9717171788215637, "perplexity": 3444.8281977324473}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320302715.38/warc/CC-MAIN-20220121010736-20220121040736-00626.warc.gz"} |
https://physics.stackexchange.com/questions/110046/lippmann-schwinger-equation-with-outgoing-solutions/110053#110053 | # Lippmann-Schwinger Equation with Outgoing Solutions
I'm reading about Green's functions and how the Lippmann-Schwinger equation eventually leads to the textbook expression for the form of wavefunctions in the far radiation zone after scattering by a central potential.
Why does adding $i\epsilon$ to the energy in the Green's function ensure that the L-S equation will give us outgoing solutions? Where does that intuition come from?
One way to look at it is simply a mathematical trick that encodes the boundary conditions of the Schroedinger equation. An alternative and only slightly more intuitive view is the following. In order to obtain only outgoing solutions it is essential to assume that the scattering potential is slowly switched on adiabatically. Formally one can achieve this by treating the scattering potential as a function of time of the form $V(r,t) = V(r)e^{\epsilon t}$, so that $V\to 0$ for large negative $t$. Working through the algebra, you find that exactly the same effect is achieved by adding a small imaginary term $i\epsilon$ to the energies instead. This is discussed in Landau & Lifshitz QM, section 43, and also mentioned somewhere in the extensive sections on scattering theory in the same text. As you probably already know, this procedure shifts the pole of the Green function in the right direction so that you pick up the outgoing spherical wave after contour integration in the complex momentum plane.
So long as the interaction is switched on very slowly, the adiabatic theorem tells you that the system will remain in the same eigenstate, even though the form of that eigenstate itself changes as the Hamiltonian is altered. It is not so hard to believe that the adiabatic deformation of a coherent plane wave state at momentum $\mathbf{k}$ is again a wave at momentum $\mathbf{k}$, but now with a small outgoing spherical wave component. If you instead switch the scattering potential on abruptly, the sudden 'jolt' this gives to the system will excite modes at all possible momenta. The interference between these different modes can lead to both incoming and outgoing spherical wave components, in general.
Strictly speaking, adiabaticity requires the interaction to be switched on slower than any inverse frequency difference. For scattering in a large volume $L^3$, this is essentially impossible since the frequency differences are proportional to $1/L^2$. However, as long as the interaction is not switched on infinitely fast (which of course is also impossible), one expects the initial transients to eventually average to zero, leaving only the outgoing spherical wave. And of course this is indeed what happens in the experiment.
• I like this idea of encoding boundary conditions into Schrodinger's equation. Thanks for that reference in L&L's QM. I didn't tie the idea with the adiabatic theorem as you did when asking the question, so thank you for that additional point! :)
– eqb
Apr 28 '14 at 3:02
First, as I understand the Lippman-Schwinger equation, there are actually two different cases, denoted by the $\pm$'s in the standard form of the equation: $$| \psi^{(\pm)} \rangle = | \phi \rangle + \frac{1}{E - H_0 \pm i \epsilon} V |\psi^{(\pm)} \rangle. \$$
In this way, we're talking about not just the outgoing solution (the $+$ equations), but also an incoming solution (the $-$). Furthermore, the $i\epsilon$ term exists in the Lippman-Schwinger equation before you introduce the Green's function (you might introduce the Green's function while solving the equation).
As described on the Wikipedia page (here) the complex term you're referring to is simply introduced to avoid making the expression blow up for $(E-H_0)\rightarrow 0$. You can contour integrate around the pole then, in the usual fashion. Perhaps I'm wrong (and I'd be happy to understand this in a deeper way), but I don't think of the $i\epsilon$ term as being physically interesting; instead, I see it as more of a math trick to allow us to solve. | 2022-01-24 00:49:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9317391514778137, "perplexity": 264.93251429434895}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304345.92/warc/CC-MAIN-20220123232910-20220124022910-00089.warc.gz"} |
http://mymathforum.com/math/346208-calculate-customer-value.html | My Math Forum > Math Calculate customer value
Math General Math Forum - For general math related discussion and news
April 12th, 2019, 06:55 PM #1 Newbie Joined: Apr 2019 From: Australia Posts: 1 Thanks: 0 Calculate customer value Hi all, first time posting something! As part of work, I need to assign value to customer and was wondering if anyone can help to solve this maths problem. If the total customer spend is \$100/month and 25% of it is on fruits, how much does the customer need to spend per month in total to increase the spend of fruit to 35% assuming the \$ spent on all other product category stays the same? Last edited by skipjack; April 13th, 2019 at 07:19 AM.
April 13th, 2019, 06:10 AM #2 Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,344 Thanks: 1024 NOT clear! Is your problem same as: If a customer spends 100 dollars per month and 25 dollars of it is on fruits, how much does the customer need to spend per month in total in order to increase what he spends on fruits to 35% of total expenditure, assuming the dollars spent on all other product category stays the same as before. ?????????
April 13th, 2019, 07:30 AM #3 Global Moderator Joined: Dec 2006 Posts: 20,485 Thanks: 2041 If the customer spends \$115.38/month, 35% of that is approximately \$40.38, so \\$75 is still spent on other products.
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Contact - Home - Forums - Cryptocurrency Forum - Top | 2019-04-26 08:15:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21605582535266876, "perplexity": 4225.620465881834}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578762045.99/warc/CC-MAIN-20190426073513-20190426095513-00239.warc.gz"} |
http://math.stackexchange.com/questions/178286/calculation-of-x-yv | # Calculation of |[X,Y]^V|
I want to follow the proof of Theorem 3.1 in "On Eschenburg's Habilitation on Biquotients" - Wolfgang Ziller.
The situation is as follows: $Q$ is a biinvariant metric on $G$. So from the symmetric tensor $P : T_eG \rightarrow T_eG$, we have a left invariant metric $g\doteq Q(\cdot , P\cdot )$.
Consider the bigquotient $(G,g)// U$. Here we want to calculate
$sec(x,y) = sec_G(X,Y) + \frac{3}{4} | [ X,Y]^V|^2$ where $X$ and $Y$ are hozontal lifts of $x$ and $y$.
To calculate $| [ X,Y]^V|^2$, define one-form $\omega (A) \doteq g ( A, X_L\cdot p + p\cdot X_R)$ where $X_L\cdot p + p\cdot X_R$ is a vertical vector at $p\in G$
If we let $X$ and $Y$ to be horizontal vector fields, let $A$ and $B$ are left invariant vector fields such that $A(p) = X(p)$ and $B(p) = Y(p)$
Then we have $A \omega (B) - B \omega (A) - \omega( [ A,B]) = - \omega([X,Y])$
From $Z_p Q(W, Ad_{p^{-1}} V) = Q(W, - ad_u Ad_{p^{-1}} V)$ where $Z_p$ is a left invariant vector field wrt $g$ and $V$, $W\in T_eG$ (See 404 page in "An exotic sphere with nonnegative sectional curvature"- Gromoll and Meyer), we have
$| \omega([X,Y]) | = | g (Ad_{p^{-1}} X_L, L(A,B) ) - g(X_R,[A,B])|$ where $L(A,B) \doteq - (ad_A)^\ast B + (ad_B)^\ast A - [A,B]$
Question : But in the paper, $L(A,B) = (ad_A)^\ast B - (ad_B)^\ast A - [A,B]$. I do not know where there exists a mistake. Please, help me.
-
I haven't read either paper your reference in enough detail to comment on your actual question, but I know that when Gromoll and Meyer say that the $0$ curvature points in $\Sigma^7$ are of smaller dimension, they are wrong. See F. Wilhelm: An exotic sphere with positive curvature almost everywhere, J. Geom. Anal. 11 (2001), 519 - 560 for more details (I currently don't have access to the paper). – Jason DeVito Aug 3 '12 at 8:06
On second thought, here's Wilhelm's paper, from his own website mathdept.ucr.edu/pdf/publications/ladder/11-GM-alm.pdf – Jason DeVito Aug 3 '12 at 8:10 | 2013-12-13 05:31:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9151543974876404, "perplexity": 334.9554167633172}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164886464/warc/CC-MAIN-20131204134806-00011-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/75009/local-finiteness-and-coarse-bounded-geometry?sort=oldest | # Local finiteness and coarse bounded geometry
I've just started learning these things and so probably my questions will be very easy. Please forgive me.
A metric space $(X,d)$ is called locally finite if every bounded set is finite. A metric space is said to have coarse bounded geometry if there is $\Gamma\subseteq X$ such that
1) there exists $c>0$ such that the set of points $x\in X$ such that $d(x,\Gamma)\leq c$ is dense in $X$.
2) For all $r>0$, there exists $K_r$ such that, for all $x\in X$, $|\Gamma\cap B_r(x)|\leq K_r$, where $B_r(x)$ stands for the ball of radius $r$ about $x$.
Question 1: what is an example of metric space without coarse bounded geometry?
Well, infinite dimensional Banach spaces. But I would like something more handable.
Question 2: Is it true that locally finiteness implies coarse bounded geometry?
Maybe I have misunderstood, but in a published paper I have found a sentence that looks implicitly assume that the answer is positive. It might be trivial, but I am not quite convinced.
Valerio
-
Q2--no. Let $A_n$ have cardinality $n+1$ for $n=0,1,...$. Specify all distances between distinct points in the same $A_n$ to be one, and the distance between a point in $A_n$ to a point in $A_m$ to be $n+m$ when $n\not= m$.
This gives a simple example for Q1 as welll.
-
Thank you! I was just thinking to an example of that sort, while getting back home! – Valerio Capraro Sep 9 '11 at 16:12
The answer to question 2 is negative, but if you require quasi-homogeneity (i.e. you have a group of isometries with a $c-$dense orbit form some $c$) then it becomes affirmative. You typically have this.
Also, to construct examples as in question 1 you can consider non-quasi-homogeneous spaces. Hope this helps, I can be more explicit on this point if you need clarifications.
-
thank you - also for the specification about the density. – Valerio Capraro Sep 9 '11 at 16:14
@Alessandro re ps: What the OP wrote is equivalent even if your wording is the usual way of writing the condition. – Bill Johnson Sep 9 '11 at 20:45
@Bill: I think it has been edited. In any case, I removed the "ps" part, thanks for pointing this out. – Alessandro Sisto Sep 10 '11 at 0:00
I am sorry, what is quasi-homogeneity? – Valerio Capraro Sep 10 '11 at 2:19
As I mentioned, it means that there are many isometries, meaning that there exists some $c$ such that for each $x,y$ there exists an isometry mapping $y$ to a point at distance at most $c$ from $x$. Notice that being homogeneous is the same thing as being quasi-homogeneous with constant $c=0$. – Alessandro Sisto Sep 10 '11 at 14:04
One more comment (which also implies answers to your questions): bounded geometry implies that the space has a finite exponential growth rate (defined, say, with respect to covers by balls of a fixed radius).
-
I am sorry, what do you mean by "finite exponential growth"? – Valerio Capraro Sep 9 '11 at 18:50
In the discrete case the exponential growth rate is defined as $\limsup \frac1n \log|B_n(x)|$, where $B_n(x)$ is the ball of radius $n$ centered at a point $x$. In the continuous case instead of cardinality one takes the minimal number of balls of a fixed radius necessary to cover $B_n$. – R W Sep 9 '11 at 19:03
OK, at least in the discrete case it's the classical notion. I got a bit scared because of the "balls of fixed radius". – Valerio Capraro Sep 9 '11 at 19:43 | 2015-08-05 04:44:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.942919135093689, "perplexity": 326.76151175269285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438043060830.93/warc/CC-MAIN-20150728002420-00093-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/kelvin-scale-cant-make-sense-of-the-last-step-of-this-solution.266749/ | # Homework Help: KELVIN scale-Can't make sense of the last step of this solution
1. Oct 24, 2008
I do not understand how they went from $\ln T_c-\lnT_H=\theta_c-\theta_H$
to
$\theta=\ln T+C$
Why did the subscripts drop out? What just happened?
2. Oct 25, 2008
### physicsnoob93
exp means e^thethac if that is thetha
so you get e^(thethac-thethah) is Tc/Th.
If you ln both sides, you'll get ln(Tc/Th) = lnTc-lnTh and on the other side lne^(thethac-thethah) = (thethac-thethah)lne. lne is one.
Thus
lnTc-lnTh = thethac-thethah
3. Oct 25, 2008
### HallsofIvy
"C" and "H" just represent different measurements of the temperature. Since you take TH to be a fixed temperature, that just leaves one temperature- you don't need the "C" to distinguish it form "H".
4. Oct 26, 2008 | 2018-11-16 08:23:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7220097780227661, "perplexity": 5380.174946732348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742981.53/warc/CC-MAIN-20181116070420-20181116092420-00169.warc.gz"} |
https://cn.chinaproject.harvard.edu/corresponding-research-pages/renewable-electric-power-grid-integration | # Renewable and Low-Carbon Electric Power and Grid Integration
Xi Lu, Liang Cao, Haikun Wang, Wei Peng, Jia Xing, Shuxiao Wang, Siyi Cai, Bo Shen, Qing Yang, Chris P. Nielsen, and Michael B. McElroy. 2019. “Gasification of coal and biomass as a net carbon-negative power source for environment-friendly electricity generation in China.” Proceedings of the National Academy of Sciences, 116, 17, Pp. 8206-8213. Publisher's VersionAbstract
Realizing the goal of the Paris Agreement to limit global warming to 2 °C by the end of this century will most likely require deployment of carbon-negative technologies. It is particularly important that China, as the world’s top carbon emitter, avoids being locked into carbon-intensive, coal-fired power-generation technologies and undertakes a smooth transition from high- to negative-carbon electricity production. We focus here on deploying a combination of coal and biomass energy to produce electricity in China using an integrated gasification cycle system combined with carbon capture and storage (CBECCS). Such a system will also reduce air pollutant emissions, thus contributing to China’s near-term goal of improving air quality. We evaluate the bus-bar electricity-generation prices for CBECCS with mixing ratios of crop residues varying from 0 to 100%, as well as associated costs for carbon mitigation and cobenefits for air quality. We find that CBECCS systems employing a crop residue ratio of 35% could produce electricity with net-zero life-cycle emissions of greenhouse gases, with a levelized cost of electricity of no more than 9.2 US cents per kilowatt hour. A carbon price of approximately $52.0 per ton would make CBECCS cost-competitive with pulverized coal power plants. Therefore, our results provide critical insights for designing a CBECCS strategy in China to harness near-term air-quality cobenefits while laying the foundation for achieving negative carbon emissions in the long run. Fei Xiao, Tianguang Lu, Qian Ai, Xiaolong Wang, Xinyu Chen, Sidun Fang, and Qiuwei Wu. In Press. “Design and implementation of a data-driven approach to visualizing power quality.” IEEE Transactions on Smart Grid.Abstract Numerous underlying causes of power-quality (PQ) disturbances have enhanced the application of situational awareness to power systems. This application provides an optimal overall response for contingencies. With measurement data acquired by a multi-source PQ monitoring system, we propose an interactive visualization tool for PQ disturbance data based on a geographic information system (GIS). This tool demonstrates the spatio–temporal distribution of the PQ disturbance events and the cross-correlation between PQ records and environmental factors, leveraging Getis statistics and random matrix theory. A methodology based on entity matching is also introduced to analyze the underlying causes of PQ disturbance events. Based on real-world data obtained from an actual power system, offline and online PQ data visualization scenarios are provided to verify the effectiveness and robustness of the proposed framework. Peter Sherman, Xinyu Chen, and Michael B. McElroy. 2020. “Offshore wind: an opportunity for cost-competitive decarbonization of China’s energy economy.” Science Advances, 6, 8, Pp. eaax9571.Abstract China has reduced growth in its emissions of greenhouse gases, success attributable in part due to major investments in onshore wind. By comparison, investments in offshore wind have been minor, limited until recently largely by perceptions of cost. Assimilated meteorological data are used here to assess future offshore wind potential for China. Analysis on a provincial basis indicates that the aggregate potential wind resource is 5.4 times larger than current coastal demand for power. Recent experiences with markets both in Europe and the US suggest that potential offshore resources in China could be exploited to cost-competitively provide 1148.3 TWh of energy in a high-cost scenario, 6383.4 TWh in a low-cost option, equivalent to between 36% and 200% of the total coastal energy demand post 2020. The analysis underscores significant benefits for offshore wind for China, with prospects for major reductions greenhouse emissions with ancillary benefits for air quality. Shi Chen, Xi Lu, Yufei Miao, Yu Deng, Chris P. Nielsen, Noah Elbot, Yuanchen Wang, Kathryn G. Logan, Michael B. McElroy, and Jiming Hao. 2019. “The potential of photovoltaics to power the Belt and Road Initiative.” Joule, 3, Pp. 1-18. Publisher's VersionAbstract Construction of carbon-intensive energy infrastructure is well underway under the Belt & Road Initiative (BRI), challenging the global climate target. Regionally abundant solar power could provide an alternative for electricity generation. An integrative spatial model was developed to evaluate the technical potential of solar photovoltaic power. The influence of impacting factors was quantified systematically on an hourly basis. Results suggest that the electricity potential for the BRI region reaches 448.9 PWh annually, 41.3 times the regional demand for electricity in 2016. Tapping 3.7% of the potential through deploying 7.8 TW capacity could satisfy the regional electricity demand projected for 2030, requiring an investment of approximately 11.2 trillion 2017 USD and a commitment in land area of 88,426 km2, approximately 0.9% of China’s total. Countries endowed with 70.7% of the overall potential consume only 30.1% of regional electricity. The imbalance underscores the advantage of regional cooperation and investments in interconnected grids. Haikun Wang, Xi Lu, Yu Deng, Yaoguang Sun, Chris P. Nielsen, Yifan Liu, Ge Zhu, Maoliang Bu, Jun Bi, and Michael B. McElroy. 2019. “China’s CO2 peak before 2030 implied from diverse characteristics and growth of cities.” Nature Sustainability, 2, Pp. 748–754. Publisher's VersionAbstract China pledges to peak CO2 emissions by 2030 or sooner under the Paris Agreement to limit global warming to 2 °C or less by the end of the century. By examining CO2 emissions from 50 Chinese cities over the period 2000–2016, we found a close relationship between per capita emissions and per capita gross domestic product (GDP) for individual cities, following the environmental Kuznets curve, despite diverse trajectories for CO2 emissions across the cities. Results show that carbon emissions peak for most cities at a per capita GDP (in 2011 purchasing power parity) of around US$21,000 (80% confidence interval: US\$19,000 to 22,000). Applying a Monte Carlo approach to simulate the peak of per capita emissions using a Kuznets function based on China’s historical emissions, we project that emissions for China should peak at 13–16 GtCO2 yr−1 between 2021 and 2025, approximately 5–10 yr ahead of the current Paris target of 2030. We show that the challenges faced by individual types of Chinese cities in realizing low-carbon development differ significantly depending on economic structure, urban form and geographical location.
Hongjian Wei, Wenzhi Liu, Xinyu Chen, Qing Yang, Jiashuo Li, and Hanping Chen. 2019. “Renewable bio-jet fuel production for aviation: a review.” Fuel, 254. Publisher's VersionAbstract
Due to excessive greenhouse gas emissions and high dependence on traditional petroleum jet fuel, the sustainable development of the aviation industry has drawn increasing attention worldwide. One of the most promising strategies is to develop and industrialize alternative aviation fuels produced from renewable resources, e.g. biomass. Renewable bio-jet fuel has the potential to reduce CO2 emissions over their life cycle, which make bio-jet fuels an attractive substitution for aviation fuels. This paper provided an overview on the conversion technologies, economic assessment, environmental influence and development status of bio-jet fuels. The results suggested that hydrogenated esters and fatty acids, and Fischer-Tropsch synthesis can be the most promising technologies for bio-jet fuels production in near term. Future works, such as searching for more suitable feedstock, improving competitiveness for alternative jet fuels, meeting emission reduction targets in large-scale production and making measures for the indirect impact are needed for further investigation. The large-scale deployment of bio-jet fuels could achieve significant potentials of both bio-jet fuels production and CO2 emissions reduction based on future available biomass feedstock.
April 8, 2019
March 8, 2019
2019 Apr 11
# How Does Interprovincial Electricity Trade Respond to the Renewable Energy Surge in China?
3:30pm to 4:45pm
## Location:
Pierce Hall 100F, 29 Oxford St., Cambridge, MA
2019 Feb 20
# Book Talk: Will China Save the Planet?
12:00pm
## Location:
WCC Milstein West B, Harvard Law School, 1585 Massachusetts Avenue, Cambridge
A Harvard Law School Library Book Talk with Barbara Finamore '80... Read more about Book Talk: Will China Save the Planet?
2019 Mar 07
# China and Asia in a Changing Climate: Natural Science for the Non-Scientist
12:15pm to 1:45pm
## Location:
CGIS South S020, Belfer Case Study Room, 1730 Cambridge St., Cambridge, MA
Panelists:
• Professor John Holdren, Teresa and John Heinz Professor of Environmental Policy, Harvard Kennedy School (HKS) and Department of Earth and Planetary Sciences, Harvard University; Co-Director of Science, Technology, and Public Policy Program, HKS; former Science Advisor to President Barack Obama and former Director of the White House Office of Science and Technology Policy
• Professor Peter Huybers, Department of Earth and Planetary Sciences, Harvard University, and Harvard John A. Paulson School of Engineering and Applied Sciences
• Professor Elsie Sunderland, Gordon McKay Professor of Environmental Chemistry, Harvard John A. Paulson School of Engineering and Applied Sciences and Harvard T.H. Chan School of Public Health
• Professor Steve Wofsy, Abbott Lawrence Rotch Professor of Atmospheric and Environmental Science, Department of Earth and Planetary Sciences, Harvard University, and Harvard John A. Paulson School of Engineering and Applied Sciences
Chair: Professor Mike McElroy, Gilbert Butler Professor of Environmental Studies, Department of Earth and Planetary Sciences, Harvard University, and Harvard John A. Paulson School of Engineering and Applied Sciences; Chair, Harvard-China Project on Energy, Economy and Environment... Read more about China and Asia in a Changing Climate: Natural Science for the Non-Scientist
Xinyu Chen, Michael B. McElroy, Qiuwei Wu, Yinbiao Shu, and Yusheng Xue. 2018. “Transition towards higher penetration of renewables: an overview of interlinked technical, environmental and socio-economic challenges.” Journal of Modern Power Systems and Clean Energy, 7, 1, Pp. 1-8. Publisher's VersionAbstract
Investment for renewables has been growing rapidly since the beginning of the new century, and the momentum is expected to sustain in order to mitigate the impact of anthropogenic climate change. Transition towards higher renewable penetration in the power industry will not only confront technical challenges, but also face socio-economic obstacles. The connected between environment and energy systems are also tightened under elevated penetration of renewables. This paper will provide an overview of some important challenges related to technical, environmental and socio-economic aspects at elevated renewable penetration. An integrated analytical framework for interlinked technical, environmental and socio-economic systems will be presented at the end.
Qing Yang, Hewen Zhou, Xiaoyan Zhang, Chris P. Nielsen, Jiashuo Li, Xi Lu, Haiping Yang, and Hanping Chen. 2018. “Hybrid life-cycle assessment for energy consumption and greenhouse gas emissions of a typical biomass gasification power plant in China.” Journal of Cleaner Production, 205, Pp. 661-671. Publisher's VersionAbstract
Among biomass energy technologies which are treated as the promising way to mitigate critical energy crisis and global climate change, biomass gasification plays a key role given to its gaseous fuels especially syngas for distributed power plant. However, a system analysis for the energy saving and greenhouse gas emissions abatement potentials of gasification system has been directed few attentions. This study presents a system analysis that combines process and input-output analyses of GHG emissions and energy costs throughout the full chain of activities associated with biomass gasification. Incorporating agricultural production, industrial process and wastewater treatment which is always ignored, the energy inputs in life cycle are accounted for the first commercial biomass gasification power plant in China. Results show that the non-renewable energy cost and GHG emission intensity of the biomass gasification system are 0.163 MJ/MJ and 0.137 kg CO2-eq/MJ respectively, which reaffirm its advantages over coal-fired power plants in clean energy and environmental terms. Compared with other biomass energy processes, gasification performs well as its non-renewable energy cost and CO2 intensity are in the central ranges of those for all of these technologies. Construction of the plant is an important factor in the process’s non-renewable energy consumption, contributing about 44.48% of total energy use. Wastewater treatment is the main contributor to GHG emissions. The biomass gasification and associated wastewater treatment technologies have critical influence on the sustainability and renewability of biomass gasification. The results provide comprehensive analysis for biomass gasification performance and technology improvement potential in regulating biomass development policies for aiming to achieve sustainability globally. | 2020-05-26 11:43:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5657622218132019, "perplexity": 6987.47257102285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347390758.21/warc/CC-MAIN-20200526112939-20200526142939-00574.warc.gz"} |
https://dmoj.ca/problem/ecoo17r1p2 | ## ECOO '17 R1 P2 - Chocolate Chewsday
View as PDF
Points: 5 (partial)
Time limit: 30.0s
Memory limit: 64M
Problem type
Allowed languages
Ada, Assembly, Awk, Brain****, C, C#, C++, COBOL, CommonLisp, D, Dart, F#, Forth, Fortran, Go, Groovy, Haskell, Intercal, Java, JS, Kotlin, Lisp, Lua, Nim, ObjC, OCaml, Octave, Pascal, Perl, PHP, Pike, Prolog, Python, Racket, Ruby, Rust, Scala, Scheme, Sed, Swift, TCL, Text, Turing, VB, Zig
##### Author: Andrew Seidel
At the local candy factory, every Tuesday, there is a contest for whoever can come up with the best new chocolate. To make a decision on whether or not each chocolate is a winner, there is a panel of impartial judges that come in from the community. The judges are given the following criteria for judging:
• Packaging (), up to point
• Flavour (), up to points
• Minimal ingredients (), up to points
A score is given to each chocolate from each judge assigned to that chocolate . There will be a random number of judges , assigned to each chocolate .
Your task is to declare the winner based on the highest total score in the competition. If there is a tie for the highest total score, it can sometimes be broken using the total scores for , and (try first, then , then ).
The competition is not really fair because some chocolates get more judges than others. But that's life at the candy factory.
#### Input Specifications
The input will contain competitions. The first line of each competition will contain a single integer , to indicate the number of chocolates in the competition . For each of the chocolates, there will be lines in the file. The line is the name of the chocolate (a single word with no spaces) and the next lines will contain the judges' scores . Each score will be contained on a single line, starting with the letter J followed by the integers , , and separated by spaces. Each competitions ends with an asterisk * (ASCII ).
#### Output Specifications
Output the name of the winner. If there is more than one winner, print out all winners on a single line separated by commas (order does not matter — i.e., an output of A,B is the same as B,A).
#### Sample Input 1
2
C1
J 0 1 1
J 0 1 0
J 1 0 0
C2
J 1 2 3
*
#### Sample Output 1
C2
Note: Only case is shown in this sample.
#### Sample Input 2
4
ChocolateOfChocolates
J 0 2 2
J 0 1 2
J 1 2 0
Choco-Fun
J 1 2 3
J 1 2 0
ChocolateHaven
J 1 2 0
J 0 2 3
J 1 0 1
ChocolatesRock
J 1 2 1
J 1 2 0
J 1 2 0
*
1
ChocolateFilledCandy
J 0 0 0
*
#### Sample Output 2
ChocolateOfChocolates
ChocolateFilledCandy
Note: Only cases are shown in this sample.
#### Explanation of Sample 2
For the first competition, there is a tie between ChocolateOfChocolates, ChocolateHaven, and ChocolatesRock. We then had to look at the values, which were tied for ChocolateOfChocolates and ChocolateHaven. Consequently, we had to then check the value. At this point, ChocolateOfChocolates has the higher value.
#### ECOO 2017 Question Development Team
Kevin Forest ............................................... Sheridan College
John Ketelaars ....................................... ECOO-CS Communications
Stella Lau .......................................... University of Cambridge
Greg Reid .................. St. Francis Xavier Secondary School, Mississauga
Sam Scott .................................................... Mohawk College
Andrew Seidel ..................... John Fraser Secondary School, Mississauga
David Stermole ............................................ ECOO-CS President
Reyno Tilikaynen ..................................... University of Waterloo
Educational Computing Organization of Ontario - statements, test data and other materials can be found at ecoocs.org | 2020-09-25 01:50:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5148817896842957, "perplexity": 6251.60773358793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400221382.33/warc/CC-MAIN-20200924230319-20200925020319-00620.warc.gz"} |
https://crypto.stackexchange.com/tags/order-preserving/hot | # Tag Info
13
I do work in this area. OPE and ORE are important primarily because of their tremendous utility in building systems which can perform some computation on encrypted data. Contrary to general-purpose solutions like fully-homomorphic encryption, OPE and ORE can be used to provide drop-in (with no code change) security in applications like databases. They can do ...
10
If you know the order of the plaintext just possessing the correspondent ciphertext, then you can perform sorting, interval querying, and all the sort of algorithms based on neighborhoods on the ciphertext domain. This is why those schemes are used in practice. To see another example of the use of OPE, take a look at the cryptoDB: Queries of type "SELECT *...
9
Timely question, since attacks on the order preserving encryption in CryptDB were recently in the news. Quoting the research paper (pdf), there are two attacks they use on OPE: sorting attack: is an attack that decrypts OPE-encrypted columns. This folklore attack is very simple but, as we show, very powerful in practice. It is applicable to columns that are ...
8
For an easy to grasp explanation, you can have a look at the talk Obfuscation I at the Cryptography Bootcamp by Amit Sahai. Here's a link to youtube. In this context he also explains matrix branching programs, which are also used in the construction of indistuingishability obfuscation. He starts explaining them at the minute 40. In short: You're given $2k$ ...
6
No, an order preserving public key encryption scheme cannot be secure. Consider any PKE scheme for plaintext space $\mathbb{Z}_n$ for which there exists a public operation that given two ciphertexts (and possibly the public key) allows to test the relative order of the corresponding plaintexts. Given a ciphertext $c$, and the public key we can then recover ...
6
Homomorphic Encryption on Reals In theory, homomorphic encryption can be done on real numbers. This answer describes two options you have when dealing with real numbers or operations that will result in real numbers. Kristin Lauter is doing some of the cutting edge research in this area. In a recent paper, CryptoNets: Applying Neural Networks to Encrypted ...
5
If you want to know more about leakage in Order Preserving Encryption (OPE) and Order Revealing Encryption (ORE) Scheme, you can find some interesting findings in two papers: What Else is Revealed by Order-Revealing Encryption?, 2016 ACM SIGSAC. Leakage-Abuse Attacks against Order-Revealing Encryption, 2017 IEEE S&P In the first paper they explain how ...
4
The basic method is easily cracked: it is well known how to find a polynomial of degree at most $k$ from $k+1$ (input, output) pairs; that's the polynomial interpolation problem. There are numerous ways to efficiently carry it for high degree and large integers (one such method is to carry it modulo some medium primes, and use the Chinese Remainder Theorem ...
4
cryptdb has these implementations inside it . But their licensing is not Open sources as in GPL etc . They say its available for research purposes ! I have implemented Symmetric Searchable Encryption in Java, its LGPL
4
How to know how much space to reserve? There are two ways: Take an implementation of the scheme, encrypt a 32-bit plaintext, and see how long the resulting ciphertext is. This is the simplest approach. Understand the scheme at a conceptual level, and then use your understanding of the algorithm to predict how long the ciphertext will be. Since it sounds ...
4
I think you are confusing functional encryption and homomorphic encryption. In a functional encryption scheme, using a secret key for some function $f$ on a ciphertext $c$ which is an encryption of $m$ allows you to get $f(m)$ in clear. In an homomorphic encryption scheme, you can run some operation on ciphertexts, and get an encryption of the result, for ...
3
The question as currently stated is true if we assume the equation takes place in $\mathbb{Z}$, since all the values are small integers. Proof: If $x<x'$, then $x^3<(x')^3$ and $ax<ax'$, so $$E(x)= x^3 +ax+b < (x')^3 +ax'+b = E(x')$$ The problem with trying to answer the more general issue you appear to be considering is working out what it ...
3
In cryptography the notation of $x\stackrel{\\\$}{\gets}S$(also sometimes seen as$x\gets_{\\\$}S$) means that $x$ is chosen uniformly at random from the set $S$. If an algorithm is on the right side of the $\stackrel{\\\$}{\gets}$then it typically means that the algorithm is invoked and may use randomness, for algorithms the$\\\$$is also sometimes ... 2 Boldyreva et al.'s scheme is not randomized. However, there is a folklore way to "randomize" it by choosing randomly from the range gap in the last recursive step of the algorithm. It's not clear what security improvement this buys you, though, since it only really randomizes the last few bits of the ciphertext. There are schemes meeting a randomized ... 2 Paillier cryptosystem has the property that the product of 2 ciphertexts decrypt to the sum of the plaintexts. Strings are integers. Only that they are usually large. So this algorithm is also available for strings. This algorithm doesn't allow you to find encrypted string in a ciphertext. If you want an encryption scheme in which you can do any operation ... 1 The definition of OPE used in Boldyreva's work (section 3.1) is basically$$\forall m_0, m_1 \in \mathcal{M}, m_0 > m_1 \Leftrightarrow E(m_0) > E(m_1) and any scheme satisfying this definition is deterministic. To understand it, consider that $m_0$ and $m_1$ are equal messages. Then, $E(m_0) \not > E(m_1)$, otherwise we would have \$m_0 > ...
1
In cryptography it is common to reason about the probability of an event in the probability space of all the random choices made (i.e. the random bits generated) during an algorithm's execution. So, in this description, "over the random coins of HGD" means the probability is computed over the probability space defined by the random bits used during HGD ...
1
Your hunch is wrong because of the definition of CPA security: Assume that some knowing some kind of relation between two plaintexts would give the attacker an advantage. Now think of the INC-CPA game: Nothing stops the attacker from choosing exactly this kind of relationship. And if the scheme is IND-CPA secure, knowledge of such a relation does not break ...
1
In Paillier, the size of ciphertext is about the double of the plaintext. (Might be interesting for you to read: http://courses.engr.illinois.edu/cs598man/fa2011/slides/ac-f11-lect15.pdf) For Order-Preserving symmetric Encryption (OPE), check http://www.cc.gatech.edu/~aboldyre/papers/operev.pdf which describes "Choosing the Ciphertext Space Size" on page 9. ...
1
Ziv-Lempel is a data compression algorithm, so in general it doesn't protect your data. As for your question: More generally, how difficult is it for an adversary to distinguish two strings which have been Ziv-Lempel encoded but not encrypted? An adversary just can decode two strings and compare them. Due to the fact that Ziv-Lempel is an encoding ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2021-12-01 14:40:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7589713335037231, "perplexity": 861.8921694939785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964360803.6/warc/CC-MAIN-20211201143545-20211201173545-00494.warc.gz"} |
https://qubit.guide/5.10-appendices.html | ## 5.10Appendices
### 5.10.1 Tensor products in components
In our discussion of tensor products we have so far taken a rather abstract approach. There are, however, situations in which we have to put numbers in, and write tensor products of vectors and matrices explicitly. For example, here is the standard basis of two qubits written explicitly as column vectors: \begin{aligned} |00\rangle &\equiv |0\rangle\otimes|0\rangle = \begin{bmatrix}1\\0\end{bmatrix} \otimes \begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}1\\0\\0\\0\end{bmatrix} \\[1em] |01\rangle &\equiv |0\rangle\otimes|1\rangle = \begin{bmatrix}1\\0\end{bmatrix} \otimes \begin{bmatrix}0\\1\end{bmatrix} = \begin{bmatrix}0\\1\\0\\0\end{bmatrix} \\[1em] |10\rangle &\equiv |1\rangle\otimes|0\rangle = \begin{bmatrix}0\\1\end{bmatrix} \otimes \begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}0\\0\\1\\0\end{bmatrix} \\[1em] |11\rangle &\equiv |1\rangle\otimes|1\rangle = \begin{bmatrix}0\\1\end{bmatrix} \otimes \begin{bmatrix}0\\1\end{bmatrix} = \begin{bmatrix}0\\0\\0\\1\end{bmatrix} \end{aligned} Given |a\rangle = \alpha_0|0\rangle + \alpha_1|1\rangle and |b\rangle = \beta_0|0\rangle +\beta_1|1\rangle, we write |a\rangle\otimes|b\rangle as \begin{aligned} |a\rangle\otimes|b\rangle &= \begin{bmatrix}\alpha_0\\\alpha_1\end{bmatrix} \otimes \begin{bmatrix}\beta_0\\\beta_1\end{bmatrix} \\&= \begin{bmatrix}\alpha_0\begin{bmatrix}\beta_0\\\beta_1\end{bmatrix}\\\alpha_1\begin{bmatrix}\beta_0\\\beta_1\end{bmatrix}\end{bmatrix} \\&= \begin{bmatrix}\alpha_0\beta_0\\\alpha_0\beta_1\\\alpha_1\beta_0\\\alpha_1\beta_1\end{bmatrix}. \end{aligned} Note that each element of the first vector multiplies the entire second vector. This is often the easiest way to get the tensor products in practice.
The matrix elements of the tensor product operation A\otimes B
are given by (A\otimes B)_{ik,jl} = A_{ij}B_{kl} where ik\in\{00,01,10,11\} labels the rows, and kl\in\{00,01,10,11\} labels columns, when forming the block matrix:83 \begin{aligned} A\otimes B &= \begin{bmatrix}A_{00}&A_{01}\\A_{10}&A_{11}\end{bmatrix} \otimes \begin{bmatrix}B_{00}&B_{01}\\B_{10}&B_{11}\end{bmatrix} \\&= \begin{bmatrix}A_{00}B&A_{01}B\\A_{10}B&A_{11}B\end{bmatrix} \\&= \left[ \, \begin{array}{c|c} \begin{matrix}A_{00}B_{00}&A_{00}B_{01}\\A_{00}B_{10}&A_{00}B_{11}\end{matrix} & \begin{matrix}A_{01}B_{00}&A_{01}B_{01}\\A_{01}B_{10}&A_{01}B_{11}\end{matrix} \\\hline \begin{matrix}A_{10}B_{00}&A_{10}B_{01}\\A_{10}B_{10}&A_{10}B_{11}\end{matrix} & \begin{matrix}A_{11}B_{00}&A_{11}B_{01}\\A_{11}B_{10}&A_{11}B_{11}\end{matrix} \end{array} \, \right] \end{aligned}
The tensor product induces a natural partition of matrices into blocks. Multiplication of block matrices works pretty much the same as regular matrix multiplication (assuming the dimensions of the sub-matrices are appropriate), except that the entries are now matrices rather than numbers, and so may not commute.
1. Evaluate the following matrix product of (4\times 4) block matrices: \left[ \, \begin{array}{c|c} \mathbf{1}& X \\\hline Y & Z \end{array} \, \right] \left[ \, \begin{array}{c|c} \mathbf{1}& Y \\\hline X & Z \end{array} \, \right] (where X, Y, and Z are the Pauli matrices).
2. Using the block matrix form of A\otimes B expressed in terms of A_{ij} and B_{ij} (as described above), explain how the following operations are performed on the block matrix:
• transposition (A\otimes B)^T; partial transpositions A^T\otimes B and A\otimes B^T;
• trace \operatorname{tr}(A\otimes B); partial traces (\operatorname{tr}A )\otimes B and A\otimes (\operatorname{tr}B).
Consider the Hadamard transform H\otimes H\otimes H on three qubits, which is described by a (2^3\times2^3) matrix. We know that H = \frac{1}{\sqrt2}\begin{bmatrix}1&1\\1&-1\end{bmatrix} and so we can calculate that H\otimes H = \frac12 \left[ \, \begin{array}{c|c} \begin{matrix}1&1\\1&-1\end{matrix} & \begin{matrix}1&1\\1&-1\end{matrix} \\\hline \begin{matrix}1&1\\1&-1\end{matrix} & \begin{matrix}-1&-1\\-1&1\end{matrix} \end{array} \, \right] and thus that H\otimes H\otimes H = \frac12 \left[ \, \begin{array}{c|c|c|c} \begin{matrix}1&1\\1&-1\end{matrix} & \begin{matrix}1&1\\1&-1\end{matrix} & \begin{matrix}1&1\\1&-1\end{matrix} & \begin{matrix}1&1\\1&-1\end{matrix} \\\hline \begin{matrix}1&1\\1&-1\end{matrix} & \begin{matrix}-1&-1\\-1&1\end{matrix} & \begin{matrix}1&1\\1&-1\end{matrix} & \begin{matrix}-1&-1\\-1&1\end{matrix} \\\hline \begin{matrix}1&1\\1&-1\end{matrix} & \begin{matrix}1&1\\1&-1\end{matrix} & \begin{matrix}-1&-1\\-1&1\end{matrix} & \begin{matrix}-1&-1\\-1&1\end{matrix} \\\hline \begin{matrix}1&1\\1&-1\end{matrix} & \begin{matrix}-1&-1\\-1&1\end{matrix} & \begin{matrix}-1&-1\\-1&1\end{matrix} & \begin{matrix}1&1\\1&-1\end{matrix} \end{array} \, \right]. The rows and columns of H\otimes H\otimes H are labelled by the triples 000,001,\ldots,111. Now, suppose we apply H\otimes H\otimes H to the state |110\rangle:
1. The output state is a superposition of all binary strings: \sum_x c_x|x\rangle, with x\in\{0,1\}^3. Where in the H^{\otimes 3} matrix will you find the coefficients c_x?
Now, do you want to write down H\otimes H\otimes H\otimes H? I don’t think so. This is an exponentially growing monster and you may soon run out of space if you actually do try to write it down. Instead, let’s spot the pattern of the entries \pm1 in these matrices.
2. Consider the Hadamard gate matrix H_{ab}, where a,b=0,1 are the labels for the rows and the columns. Observe that H_{ab}=(-1)^{ab}/\sqrt{2}. (This may look like a fancy way of writing the entries of the Hadamard matrix but it will pay off in a moment). Using the fact that (A\otimes B)_{ik,jl} = A_{ij}B_{kl}, or any other method, analyse the pattern of the \pm1 in the tensor product of Hadamard matrices. What is the entry H^{\otimes 4}_{0101,1110}?
3. For any two binary strings a=(a_1,\ldots, a_n) and b =(b_1,\ldots , b_n) of the same length we can define their “scalar” product as a\cdot b = (a_1b_1\oplus \ldots \oplus a_n b_n). Show that, up to the constant (1/\sqrt{2})^n, the entry H^{\otimes n}_{a,b}, for any n and for any binary strings a and b of length n, is (-1)^{a\cdot b}.
4. Show that H^{\otimes n} acts as |a\rangle \longmapsto \left(\frac{1}{\sqrt 2}\right)^n \sum_{b\in\{0,1\}^n} (-1)^{a\cdot b}|b\rangle
5. A quantum register of 10 qubits holds the binary string 0110101001. The Hadamard Transform is then applied to this register yielding a superposition of all binary strings of length 10. What is the sign in front of the |0101010101\rangle term?
### 5.10.2 The Schmidt decomposition
An arbitrary vector in the Hilbert space \mathcal{H}_{\mathcal{A}}\otimes \mathcal{H}_{\mathcal{B}} can be expanded in a product basis as |\psi\rangle = \sum_{ij} c_{ij}|a_i\rangle|b_j\rangle. Moreover, for each particular joint state |\psi\rangle, we can find orthonormal bases, \{|\widetilde{a_i}\rangle\} in \mathcal{H}_{\mathcal{A}} and \{|\widetilde{b_j}\rangle\} in \mathcal{H}_{\mathcal{B}}, such that |\psi\rangle can be expressed as |\psi\rangle = \sum_{i} d_{i}|\tilde a_i\rangle|\tilde b_i\rangle, where the coefficients d_i are non-negative numbers. This is known as the Schmidt decomposition of |\psi\rangle. Any bipartite state can be expressed in this form, but remember that the bases used depend on the state being expanded. Indeed, given two bipartite states |\psi\rangle and |\phi\rangle, we usually cannot perform the Schmidt decomposition using the same orthonormal bases in \mathcal{H}_{\mathcal{A}} and \mathcal{H}_{\mathcal{B}}. The number of terms in the Schmidt decomposition is, at most, the minimum of \dim\mathcal{H}_{\mathcal{A}} and \dim\mathcal{H}_{\mathcal{B}}.
The Schmidt decomposition follows from the singular value decomposition or SVD): any (n\times m) matrix C can be written as C = UDV where U and V are (respectively) (n\times n) and (m\times m) unitary matrices and D is an (n\times m) diagonal matrix with real, non-negative elements in descending order d_1\geqslant d_2\geqslant\ldots\geqslant d_{\min{(n,m)}} (and with the rest of the matrix is filled with zeros). The elements d_k are called the singular values of C.
You can visualize the SVD by thinking of C as representing a linear transformation from m-dimensional to n-dimensional Euclidean space: it maps the unit ball in the m-dimensional space to an ellipsoid in the n-dimensional space; the singular values are the lengths of the semi-axes of that ellipsoid; the matrices U and V carry information about the locations of those axes and the vectors in the first space which map into them. Thus SVD tells us that the transformation C is composed of rotating the unit ball (transformation V), stretching the axes by factors d_k, and then rotating the resulting ellipsoid (transformation U).
Using the index notation C_{ij} = \sum_k U_{ik}d_k V_{kj}, we can thus apply SVD to c_{ij}, \begin{aligned} |\psi\rangle &= \sum_{ij} c_{ij}|a_ib_j\rangle \\&= \sum_{ij} \sum_k U_{ik}d_k V_{kj}|a_ib_j\rangle \\&= \sum_k d_k \left(\sum_i U_{ik}|a_i\rangle\right)\otimes\left(\sum_j V_{kj}|b_j\rangle\right). \end{aligned} The Schmidt decomposition of a separable state of the form |a\rangle\otimes|b\rangle is trivially just this state. The Bell states \Psi^+ and \Phi^+ are already written in their Schmidt form, whereas \Psi^- and \Phi^- can be easily expressed in the Schmidt form. For example, for |\Psi^-\rangle we have d_1 = d_2 = \frac{1}{\sqrt 2}, and the Schmidt basis is |\bar a_1\rangle =|0\rangle, |\bar a_2\rangle=|1\rangle, |\bar b_1\rangle = |1\rangle, |\bar b_2\rangle=-|0\rangle. The number of non-zero singular values of c_{ij} is called the rank of c_{ij}, or the rank of the corresponding quantum state, or sometimes, the Schmidt number. Clearly, all bipartite states of rank one are separable.
The Schmidt decomposition is almost unique. The ambiguity arises when we have two or more identical singular values, as, for example, in the case of the Bell states. Then any unitary transformation of the basis vectors corresponding to a degenerate singular value, both in \mathcal{H}_a and in \mathcal{H}_b, generates another set of basis vectors.
1. We always use the lexicographic order 00<01<10<11.↩︎ | 2022-05-19 21:44:35 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9746626615524292, "perplexity": 1872.6041870665233}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662530066.45/warc/CC-MAIN-20220519204127-20220519234127-00362.warc.gz"} |
https://www.markhagemann.com/post/dockerized-remote-r-futures-file-transfer/ | # Dockerized remote R futures + file transfer
The future R package makes it easy to delegate expensive computations to a separate R process, including parallel computing on cloud infrastructure. Best of all (and fundamentally), your local R process doesn’t get stuck waiting for the remote evaluation; you can happily chug along through your R script and deal with the fruits of remote-executed code when they are good and ripe. In this post I’ll briefly outline my current future-enabled workflow, and then detail how I shoehorned in the ability to transfer files produced in the cloud back to my local machine.
### Cloud-computing future Workflow
My personal choice for cloud-computing resources has been Google Compute Engine, which I access using the googleComputeEngineR package (in addition to gcloud and the cloud console). I like it because it has simple pricing and super-cheap preemptible images, and also because, well, it’s the one I chose and I’m sticking to it. Plenty of people swear by Digital Ocean, and of course half the internet runs on AWS, .
Because I often work with R packages that rely on external libraries (gdal, netcdf, stan, etc), and getting the installation correct on a new machine is often tedious and time consuming, I opt where possible to set up remote environments as docker containers. In addition to making things “just work” more readily, this setup has a couple added advantages. First, it means you have clearly documented and reproducible (and modular!) setups that you can modify and transfer to whatever hardware you have available. Second, it leverages other people’s dockerfiles and published images–I don’t need to figure out the hard way which libraries, apt repositiories, and compiler flags to use. And lastly, It’s a fine way to learn Docker if you want to see what all the fuss is about (as I did).
My workflow for a future + GCE + docker setup is roughly as follows:
1. Select a R- and parallel-ready docker image like this one or create your own, possibly by adding on to the one in the like using a FROM statement.
2. Spin up a container-optimized instance cluster using gce_vm_cluster() with the selected docker image (docker_image argument in gce_vm_cluster()). Even though this is a “cluster”, it need not be multiple instances; I typically use a “cluster” of size 1.
3. Turn this into a parallel-ready cluster (as.cluster()) and plan() using remote and supplying the cluster as the workers argument.
4. Go nuts running futures in the cloud!
This is a version of the recipe documented in this vignette from the googleComputeEngineR package. Check out that link for more examples.
### The file-transfer conundrum
This workflow works great for most situations, but occasionally I run into a problem for which it is not well equipped. The latest of these has been the following situation. Suppose you have a computationally expensive piece of R code that produces not an R object (which would automatically be sent back to your local R session when ready), but a file written to disk. For example:
f1 <- function(x, filename) {
# pretend this is computationally expensive R code:
towrite <- sum(x)
# Pretend this is difficult to return as an R object
write.csv(towrite, file = filename)
return(TRUE)
}
Here, the value of the return statement gets back to your local R session via the future protocol, but the file written stays in the docker container (which stops running once the future is resolved). Getting the file back seems like it should be easy to do, but alas, it is not. Remote futures are communicated via ssh, and come with no file transfer capabilities. How to transfer the file back to my local machine? Here are some things I tried:
1. Open a scp port when creating the docker container and somehow get the container to persist long enough to transfer the file(s) directly to my local machine using scp. This idea failed pretty much immediately and on several fronts. Just to name a few: the docker run call is buried deep in the future call stack; port mapping is not allowed because of host networking; and just how would I get the container to persist anyway?
2. Push the file to a Box folder using the boxr package, then download it from my local R session. This only got as far as authentication on the remote server, which failed because httr’s oauth apparently requires browser verification. However, I am no expert on this and it’s possible I just didn’t find the workaround.
3. Similar to 2, but using Google Cloud Storage instead of Box. Attempted using googleCloudStorageR package, but gave up because it requires a json file for authentication and I couldn’t think how to securely get that into the container without an ugly readLines()/writeLines() hack.
4. Similar to 2 and 3, but using S3 from AWS. For this I used the aws.s3 package, which authenticates using an access key and a secret–both passed as stings. This is easy to get from local to remote, and voila! it worked.
Here, then, is my solution to transfer files from the remote container to my local machine. It assumes you already have the cluster in place as described above, and have installed and setup aws.s3 as described in the package readme.
library(aws.s3)
# authentication credentials
awskey <- Sys.getenv("AWS_ACCESS_KEY_ID")
awssecret <- Sys.getenv("AWS_SECRET_ACCESS_KEY")
# Bucket to use for transferring
bucketname <- "myfavoritebuckettouse"
putbucket <- get_bucket(bucket = bucketname,
key = awskey,
secret = awssecret)
# Data in need of computationally expensive processing
data2run <- rnorm(100)
outfile <- "f1output.csv" # file name where results will be written
# Make sure aws.s3 is installed in the remote container.
installed %<-% {install.packages("aws.s3")}
# Run the code remotely using future
f1_output %<-% {
# generate the file on disk
f1(data2run, filename = outfile)
put_object(outfile, bucket = putbucket, key = awskey, secret = awssecret)
}
resolved(futureOf(f1_output)) # Check to see if it's ready to transfer
# Retrieve the object to local storage
save_object(outfile, bucket = bucketname)
That’s it! In a future post I’ll show the results of the project that led me down this rabbit hole: remote-animated (and remote rendered!) .gif/.mp4 visualizations using gganimate.
##### Mark Hagemann
###### Post-Doctoral Researcher
I use statistics to learn about rivers from space. | 2019-08-26 07:26:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25669237971305847, "perplexity": 2806.109286159164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027331228.13/warc/CC-MAIN-20190826064622-20190826090622-00103.warc.gz"} |
https://discovery.researcher.life/article/the-hit-and-run-version-of-top-to-random/ab1b4c10410236ada293c30974a701a4 | Journal of Applied Probability | VOL. 59
# The hit-and-run version of top-to-random
Publication Date Jul 12, 2022
## Abstract
AbstractWe study an example of a hit-and-run random walk on the symmetric group $\mathbf S_n$ . Our starting point is the well-understood top-to-random shuffle. In the hit-and-run version, at each single step, after picking the point of insertion j uniformly at random in $\{1,\ldots,n\}$ , the top card is inserted in the jth position k times in a row, where k is uniform in $\{0,1,\ldots,j-1\}$ . The question is, does this accelerate mixing significantly or not? We show that, in $L^2$ and sup-norm, this accelerates mixing at most by a constant factor (independent of n). Analyzing this problem in total variation is an interesting open question. We show that, in general, hit-and-run random walks on finite groups have non-negative spectrum.
## ConceptsPowered By Our platform integrates UNSILO’s semantic concept extraction, with power of ML, to identify the most relevant papers for you.
Top Card Non-negative Spectrum Random Walk Symmetric Group Constant Factor Total Variation Finite Groups Single Step Problem In Variation Point Of Insertion
## Introducing Weekly Round-ups!Beta
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Introduction: As of early March 2022, the COVID-19 pandemic has killed more 5.9 million people worldwide, and infected more than 437 million. | 2022-10-04 03:43:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26653891801834106, "perplexity": 3406.3252749588255}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337473.26/warc/CC-MAIN-20221004023206-20221004053206-00097.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/elementary-and-intermediate-algebra-concepts-and-applications-6th-edition/chapter-2-equations-inequalities-and-problem-solving-review-exercises-chapter-2-page-149/33 | ## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)
Published by Pearson
# Chapter 2 - Equations, Inequalities, and Problem Solving - Review Exercises: Chapter 2 - Page 149: 33
#### Answer
$140$
#### Work Step by Step
In the given problem, $P=49$ and $R=0.35.$ Using $P=BR$ or equivalently $B=\dfrac{P}{R},$ then \begin{array}{l}\require{cancel} B=\dfrac{P}{R} \\\\ B=\dfrac{49}{0.35} \\\\ B=\dfrac{\cancel7(7)}{\cancel7(0.05)} \\\\ B=\dfrac{7}{.05} \\\\ B=\dfrac{7}{.05}\cdot\dfrac{100}{100} \\\\ B=\dfrac{700}{5} \\\\ B=140 .\end{array}
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-12-15 06:50:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9817154407501221, "perplexity": 7945.029354750949}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376826800.31/warc/CC-MAIN-20181215061532-20181215083532-00460.warc.gz"} |
https://www.gamedev.net/forums/topic/601966-first-person-shooter-camera-question/ | • 13
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# First Person Shooter Camera Question
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I apologize if this is a frequently asked question. I'm working on an augmented reality project that uses sensors to orient and position the camera. The GPS isn't always accurate and that can be annoying while testing! Anyways, I'm working a d'pad solution so that I can move the camera through the augmented world to test various objects and interactions. I have a forward and d'pad direction vector, but I'm not sure how to use them to update the position of the camera.
Orientation is represented as a quaternion. I can easily pull a normalized forward vector from the quaternion. If I add the forward vector to the camera position the camera moves in the direction the camera is oriented (aka forward). This is great, but I want to strafe and move backwards.
I implemented a touch screen d'pad that outputs a normalized vector (x = -1 to 1, y = 0, z = -1 to 1). If I add the d'pad direction vector the camera position the camera moves along the x and z axis. This is good, but I want to move in the direction of the camera.
How do I use the forward vector and d'pad vector to update only the position of the camera (not orientation)? Am I missing something?
##### Share on other sites
I apologize if this is a frequently asked question. I'm working on an augmented reality project that uses sensors to orient and position the camera. The GPS isn't always accurate and that can be annoying while testing! Anyways, I'm working a d'pad solution so that I can move the camera through the augmented world to test various objects and interactions. I have a forward and d'pad direction vector, but I'm not sure how to use them to update the position of the camera.
Orientation is represented as a quaternion. I can easily pull a normalized forward vector from the quaternion. If I add the forward vector to the camera position the camera moves in the direction the camera is oriented (aka forward). This is great, but I want to strafe and move backwards.
I implemented a touch screen d'pad that outputs a normalized vector (x = -1 to 1, y = 0, z = -1 to 1). If I add the d'pad direction vector the camera position the camera moves along the x and z axis. This is good, but I want to move in the direction of the camera.
How do I use the forward vector and d'pad vector to update only the position of the camera (not orientation)? Am I missing something?
What you need to do is exactly what you are doing in number 1. Get the forward vector from your orientation(quats in your case), and use the value from your d pad to scale that vector to add to your position. Same thing for strafe
position+= forward * dpad.z + right * dpad.x
now if you push up you move forward, and if you push down, dpad.z is negative and you move backwards. If you dont push dpad at all, then z==0 & x==0 and your position wont change | 2018-03-18 08:32:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20292828977108002, "perplexity": 628.5496964572183}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645550.13/warc/CC-MAIN-20180318071715-20180318091715-00235.warc.gz"} |
https://dsp.stackexchange.com/questions/44500/blackman-tukey-periodogram/44545 | # Blackman-Tukey periodogram?
I have been reading about the non-parametric estimation of the spectral density, and I'm particularly interested in the Blackman-Tukey method. If we take the rectangular window we will have the truncated periodogram, computed on a reduced covariance sequence length $M$, where $M<N$: $$\hat{P}(\omega)=\sum\limits_{k=0}^{M-1}\hat{r}(k)e^{-jk\omega}$$ where $\hat{r}(k)$ is the estimate of the covariance of the sample and $\omega$ is the frequency of interest. What I'm having trouble is: first, how do we choose our new sample of length $M$? could someone please explain how it does actually work without being too technical, like this explanation on how does the Bartlett method works
The resolution of the Blackman-Tukey periodogram estimator is $\mathcal{O}(1/M)$, whereas its variance is on the $\mathcal{O}(M/N)$. This compromise between resolution and variance is to be considered when choosing the window’s length. | 2020-09-26 11:22:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7724459767341614, "perplexity": 305.21246294291404}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400241093.64/warc/CC-MAIN-20200926102645-20200926132645-00469.warc.gz"} |
http://pma.caltech.edu/news?p=30 | Gravitational Waves Detected a Second Time
06/15/2016
## Gravitational Waves Detected a Second Time
The LIGO Scientific Collaboration and the Virgo collaboration identify a second gravitational wave event in the data from Advanced LIGO detectors.
Live Webcast: Ligo, Virgo Scientists to Discuss Continued Search for Gravitational Waves
06/14/2016
## Live Webcast: Ligo, Virgo Scientists to Discuss Continued Search for Gravitational Waves
The latest research in the effort to detect gravitational waves will be discussed in a press briefing on Wednesday, June 15.
Live webcast: LIGO, Virgo scientists to discuss continued search for gravitational waves
06/14/2016
## Live webcast: LIGO, Virgo scientists to discuss continued search for gravitational waves
The latest research in the effort to detect gravitational waves will be discussed in a press...
Prebiotic Molecule Found in Interstellar Cloud
06/14/2016
## Prebiotic Molecule Found in Interstellar Cloud
Lorinda Dajose
For the first time, a chiral molecule has been detected outside of our solar system. The discovery is an important step to understanding the origins of life.
Shou Receives Fellowship for Studies in Germany
06/09/2016
## Shou Receives Fellowship for Studies in Germany
Lorinda Dajose
Senior Laura Shou has received a Graduate Study Scholarship to pursue a master's degree in Germany.
The Next Big Thing
06/08/2016
## The Next Big Thing
Nehaly Shah
We talked to a handful of undergraduate and graduate students prior to commencement to find out what they think will be the next big thing in science and engineering and how their plans after graduation reflect those ideas.
Distinguished Alumnus: Eric Betzig (BS '83)
06/06/2016
## Distinguished Alumnus: Eric Betzig (BS '83)
Betzig received the award for his groundbreaking contributions to microscopy. He pioneered a method known as single-molecule microscopy, or “nanoscopy,” which allows cellular structures at the nanoscale to be observed using an optical microscope, for which he shared the Nobel Prize in Chemistry in 2014.
Kavli Prize Goes to LIGO Founders
06/02/2016
Whitney Clavin | 2020-06-07 00:50:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3126367926597595, "perplexity": 4744.118959622274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348521325.84/warc/CC-MAIN-20200606222233-20200607012233-00120.warc.gz"} |
http://www.jb.man.ac.uk/DARA/unit4/Workshops/3C277_eMERLIN_part1_casa5.html | # 3C277.1 e-MERLIN: Part 1 Calibration
This is the CASA v5 version of this tutorial! If you don't mean to be here, got back to the Unit 4 workshop page to get the CASA v6 tutorials.
This guide provides a basic example for calibrating e-MERLIN C-Band (5 GHz) continuum data in CASA. It has been tested in CASA 4.3-5.5. It is designed to run using an averaged data set all_avg.ms (1.7 G), in which case you will need at least 5.5 G disk space to complete this tutorial.
If you have a very slow laptop, you are advised to skip some of the plotms steps (see the plots linked to this page instead) and especially, do not use the plotms options to write out a png. This is not used in this web page, but if you run the script as provided, you are advised to comment out the calls to plotms or at least the part which writes a png, see tutors for guidance.
1. Guidance for calibrating 3C277.1 in CASA
2. Data inspection and flagging
3. Calibration
## 1. Guidance for calibrating 3C277.1 in CASA
### 1A. The data and supporting material
For this part of the workshop, we only need four files in total. Please double check that you have the following in your current directory:
• all_avg.ms - the data (after conversion to a measurement set).
• 3C286_C.clean.model.tt0 - used for fluxscaling of the data set.
• all_avg.flags.txt - pre-made flagging file to save time
• 3C277.1_cal_outline.py - the script that we shall input values into to calibrate these data.
Note that some of these maybe tarred (i.e. end in .tar.gz or .tar) so you need to use tar -xzvf <filename> to untar these.
### 1B. How to use this guide
This User Guide presents inputs for CASA tasks e.g. gaincal to be entered interactively at a terminal prompt for the calibration of the averaged data and initial target imaging. This is useful to examine your options and see how to check for syntax errors. You can run the task directly by typing gaincal at the prompt; if you want to change parameters and run again, if the task has written a calibration table, delete it (make sure you get the right one) before re-running. You can review these by typing:
# In CASA
!more gaincal.last
This will output something like:
taskname = "gaincal"
vis = "all_avg.ms"
caltable = "bpcals_precal.ap1"
...
...
#gaincal(vis="all_avg.ms",caltable="bpcals_precal.ap1",field="1407+284",spw="",intent="",
selectdata=True,timerange="",uvrange="",antenna="",scan="",observation="",msselect="",solint="120s",
combine="",preavg=-1.0,refant="Mk2",minblperant=2,minsnr=2,solnorm=True,gaintype="G",smodel=[],
calmode="ap",append=True,splinetime=3600.0,npointaver=3,phasewrap=180.0,docallib=False,callib="",
gaintable=['all_avg.K', 'bpcals_precal.p1'],gainfield=[''],interp=[],spwmap=[],parang=False)
As you can see in the script, the second format (without '#') is the form to use in the script, with a comma after every parameter value. When you are happy with the values which you have set for each parameter, enter them in the script 3C277.1_cal_outline.py (or whatever you have called it). You can leave out parameters for which you have not set values; the defaults will work for these (see e.g. help(gaincal) for what these are), but feel free to experiment if time allows.
The script already contains most plotting inputs in order to save time. Make and inspect the plots; change the inputs if you want, zoom in etc. There are links to copies of the plots (or parts of plots) but only click on them once you have tried yourself or if you are stuck.
The parameters which should be set explicitly or which need explanation are given below, with ** if they need values.
If you have already plotted the antenna positions with plotants you will have seen that Mk2 has many short baselines which makes it more likely to give good calibration solutions and it makes a good reference antenna (refant).
The input data set all_avg.ms contains five fields:
eMERLIN name Common name Role
1252+5634 3C277.1 target
1302+5748 J1302+5748 phase reference source
1407+284 OQ208 bandpass cal. (usually about 2 Jy at 5-6 GHz)
0319+415 3C84 bandpass, polarisation leakage cal. (usually 10-20 Jy at 5-6 GHz)
1331+305 3C286 flux, polarisation angle cal. (flux density known accurately, see Step 7)
These data were taken in full polarization in 0.512 GHz bandwidth centred on 5072 MHz, in 4 adjacent spectral windows (spw's, also known as IFs). Each 128-MHz spw contains 512 channels 0.25 MHz wide and 1 s integrations were used (the data are later averaged).
The preprocessing of the original raw fits idi files involved (see the full script e-MERLIN-CASA_v0.1.py steps 1-3 for details):
1. Load into CASA as measurement sets.
2. Sort and adjust $uvw$'s for Earth rotation, concatenate all data and adjust scan numbers to increment with each source change.
3. Average every 8 channels and 4s
## 2. Data Inspection and Flagging
### 2A. Check data: listobs and plotants (script step 1)
Check that you have all_avg.ms in a directory with enough space and start CASA.
Enter the parameter in step 4 of the script to specify the measurement set and, optionally, the parameter to write the listing to a text file. Important: To make it easier, the line numbers corresponding to the script 3C277.1_cal_outline.py are included in the excerpts from the script shown in this tutorial.
listobs(vis=**,
overwrite=True,
listfile=**)
Because e-MERLIN stores data for each source in a separate fitsidi file, the sources are listed one by one even though the phase-ref and target scans interleave. So selected entries from listobs, re-ordered by time, show:
Timerange (UTC) Scan FldId FieldName nRows
SpwIds Average Interval(s) #(i.e. RR RL LR LL integration time) 05-May-2015
...
20:02:08.0 - 20:05:08.5 7 0 1302+5748 2760 [0,1,2,3] [3.91, 3.91, 3.91, 3.91] # phase-ref
20:05:11.0 - 20:12:33.5 72 1 1252+5634 6660 [0,1,2,3] [3.98, 3.98, 3.98, 3.98] # target
20:12:36.0 - 20:15:34.5 8 0 1302+5748 2700 [0,1,2,3] [3.96, 3.96, 3.96, 3.96] # phase-ref
...
22:02:04.0 - 22:47:00.5 132 2 1331+305 40500 [0, 1, 2, 3] [3.99, 3.99, 3.99, 3.99] # flux scale/pol angle calibrator
22:47:03.0 - 23:29:59.5 133 3 1407+284 38700 [0, 1, 2, 3] [3.99, 3.99, 3.99, 3.99] # bandpass calibrator
06-May-2015
07:30:03.0 - 08:29:59.5 134 4 0319+415 54000 [0, 1, 2, 3] [4, 4, 4, 4] # bandpass/leakage calibrator
There are 4 spw:
SpwID Name #Chans Frame Ch0(MHz) ChanWid(kHz) TotBW(kHz) CtrFreq(MHz) Corrs
0 none 64 TOPO 4817.000 2000.000 128000.0 4880.0000 RR RL LR LL
1 none 64 TOPO 4945.000 2000.000 128000.0 5008.0000 RR RL LR LL
2 none 64 TOPO 5073.000 2000.000 128000.0 5136.0000 RR RL LR LL
3 none 64 TOPO 5201.000 2000.000 128000.0 5264.0000 RR RL LR LL
And 6 antennas:
ID Name Station Diam. Long. Lat. ITRF Geocentric coordinates (m)
0 Mk2 e-MERLIN:02 24.0 m -002.18.08.9 +53.02.58.7 3822473.365000 -153692.318000 5085851.303000
1 Kn e-MERLIN:05 25.0 m -002.59.44.9 +52.36.18.4 3859711.503000 -201995.077000 5056134.251000
2 De e-MERLIN:06 25.0 m -002.08.35.0 +51.54.50.9 3923069.171000 -146804.368000 5009320.528000
3 Pi e-MERLIN:07 25.0 m -002.26.38.3 +53.06.16.2 3817176.561000 -162921.179000 5089462.057000
4 Da e-MERLIN:08 25.0 m -002.32.03.3 +52.58.18.5 3828714.513000 -169458.995000 5080647.749000
5 Cm e-MERLIN:09 32.0 m +000.02.19.5 +51.58.50.2 3919982.752000 2651.982000 5013849.826000
Enter one parameter to plot the antenna positions (optionally, a second to write this to a png).
plotants(vis=**, figfile=**)
Consider what would make a good reference antenna. Although Cambridge has the largest diameter, it has no short baselines.
Plot the uv coverage of the data for the phase-reference source. See annotated listobs output above to identify this. You need to enter several parameters but some have been done for you:
plotms(vis=**,
xaxis=**,yaxis=**,
coloraxis='spw',avgchannel='8',
field='1302+5748', plotfile='')
Note that we averaged in channels a bit so that this doesn't destroy your computer. If this takes ages, move on and look at the plot below. The $u$ and $v$ coordinates represent (wavelength/projected baseline) as the Earth rotates whilst the source is observed. Thus, as each spw is at a different wavelength, it samples a different part of the uv plane, improving the aperture coverage of the source, allowing Multi-Frequency Synthesis (MFS) of continuum sources.
If you are happy with your entries, execute step 1 and you should see these two plots in your current directory. Remember we execute a step using the following syntax in the CASA prompt:
mysteps=[1]
execfile('3C277.1_cal_outline.py')
### 2B. Identify 'late on source' bad data (step 2)
Use plotms to plot the phase-reference source amplitudes as a function of time. Just a few central channels are selected and averaged; enough to give good S/N but not so many that phase errors across the band cause decorrelation (in real life, you might have to experiment with different selections or plot phase vs. channel to make the selection). Some hints on the proper parameter selection are given in the code blocks below.
plotms(vis='***',
field='***', # phase ref
spw='***', # plot just the inner 3/8 of the channels for each spw (see listobs output)
avgchannel=***, # average these channels
xaxis='***',yaxis='***',
antenna=antref+'&**', # This uses Mk2 as refant for ease of comparison but another with many short baselines could be used.
correlation='**,**', # Just select the parallel hands, since the cross hands (polarised intensity) are much fainter.
coloraxis='corr', # Color by baseline or correlation as you prefer
showgui=True,
overwrite=True,plotfile='')
Once you have inputted the parameters, execute step 2. Note that you can change the parameters interactively once plotms has started.
Roughly the same interval at the start of each scan is bad for all sources and antennas, so the phase reference is a good source to examine. (In a few cases there is extra bad data, ignore that for now). Use the Zoom and display to estimate the time interval, and we shall enter that value in next part.
### 2C. Flag the bad data at the start of each scan (step 3)
Remember from the EVN continuum tutorial that we need to always back-up our flags before we do any more flagging. This is so that we don't have to go back to the start in case we accidentally flag or un-flag large proportions of good data. We use flagmanager to do this.
# Back up original flag state
flagmanager(vis='***',
mode='***',
versionname='pre_quack')
If the parameters are set correctly, then we should have a backup named pre_quack in the all_avg.ms.flagversions file. You can check that it exists by running flagmanager again using mode='list' instead.
Next, we need to actually do the quacking or flagging of these channels. We use the CASA task flagdata to do this along with the special mode called quack. Fill in the flagdata parameters, inserting the time that the antennas were off source that we found earlier.
# Flag first 40 s of each scan for all sources
flagdata(vis='***',
mode='***',quackinterval=***)
Re-plot in plotms (tick reload option and replot if you still have it open) to check that the bad data at the start of each scan are gone. There are still some irregular periods of bad data which will be flagged later.
### 2D. Flag the bad end channels (step 4)
In this step, we are going to do the plotting and flagging all at once. To obtain the correct parameters for the flagging, it is advisable to copy and paste the plotms commands (with the correct parameters) into your CASA prompt and then input these parameters in the 3C277.1_cal_outline.py code for future reference.
Let's continue. The edges of the bandpass for each spw are low sensitivity and must be flagged. This is the same for all sources, polarizations and antennas but different spectral windows may be different. Just look at the brightest source, 0319+415.
plotms(vis='***',
field='***',
xaxis='***',yaxis='***', # plot amplitude against channel
antenna=***, # select one baseline
correlation='**,**', # just parallel hands
avgtime='***', # set to a large number
avgscan=****, # average all scans
iteraxis='***', coloraxis='***', # view each spectral window in turn, and color
showgui=True, # change to False if you want ,
overwrite=True,plotfile='')
Take note of the channel ranges which are significantly less sensitive at each end of each spw.
Back up the flags and enter the appropriate parameters in flagdata. Note the odd CASA notation for selecting spectral windows and channels. We have provided the "bad"/insensitive channels in spw 0 with the correct syntax, so use that as a guide for the other spectral windows.
# Back up flags
flagmanager(vis='***',
mode='***',
versionname='pre_endchans')
# end chans
flagdata(vis='***',mode='***',
spw='0:0~6;60~63,***') # enter the ranges to be flagged for the other spw
Execute step 3 and then re-plot in plotms (the code should already do this replotting for you!)
### 2E. Identify and flag remaining bad data (step 5)
Finally, you usually have to go through all baselines, averaging channels and plotting amplitude against time. To save time (and sanity!), you don't have to do it all here. Have a look at 0319+419 and write down a few commands. This is done in the form of a list file, which has no commas and the only spaces must be a single space between each parameter for a given flagging command line. Some errors affect all the baselines to a given antenna; others (e.g. correlator problems) may only affect one baseline. Usually all spw will be affected so you can just inspect one but check afterwards that all are 'clean'. It can be hard to see what is good or bad for a faint target but if the phase-ref is bad for longer than a single scan then usually the target will be bad for the intervening time. Because the data are taken over two days, it is safest to use the full date for any time range.
Before we begin to flag, remember we should start by backing up the flags again!
flagmanager(vis='all_avg.ms',
mode='save',
versionname='pre_timeflagging')
Use plotms to inspect 0319+415, plotting amplitude against time, RR,LL only, averaging all channels, all baselines to Mk2, paging (iterating) through the baselines.
Here are some of the flagging commands in list format for the data shown below
mode='manual' field='0319+415' timerange='2015/05/06/07:30:00~07:35:00'
mode='manual' field='0319+415' antenna='Da' timerange='2015/05/06/07:30:00~07:56:22'
mode='manual' field='0319+415' antenna='Mk2&Kn' timerange='2015/05/06/07:30:00~07:35:34'
mode='manual' field='0319+415' antenna='Mk2&Kn' timerange='2015/05/06/07:48:55~07:48:58'
mode='manual' field='0319+415' antenna='Mk2,De' timerange='2015/05/06/07:35:00~07:35:18'
mode='manual' field='0319+415' timerange='2015/05/06/08:28:30~08:30:00'
Note that, by paging through all the baselines, you can figure out what bad data affect which antennas. Have a look at 1331+305 and write its commands.
The commands are applied in flagdata using mode='list', inpfile='***' where inpfile is the name of the flagging file. Write a file all_avg_1.flags with all the commands to flag bad data. Apply it in flagdata using mode='list'. Look carefully at the messages in the logger and terminal in case any syntax errors are reported.
Alternatively, if you are running out of time and don't want to flag, add instead all_avg.flags.txt as the inpfile command and it will flag all the data once you execute step 5.
Step 5 will also re-plot the data so check that it is now clean!
## 3. Calibration
### 3A. Delay calibration
Begin by plotting the raw phase against frequency for a bright source. This is usually a bandpass calibrator.
plotms(vis='all_avg.ms', field='0319+415', xaxis='frequency',
gridrows=5,gridcols=1,iteraxis='baseline', # Plot multiple baselines on one page
yaxis='phase',ydatacolumn='data',antenna=antref+'&*', # Plot all baselines to the reference antenna
avgtime='600',avgscan=True,correlation='LL,RR', # Average 10 min
coloraxis='corr',
plotfile='',
overwrite=True)
The y-axis spans -200 to +200 deg and some data has a slope of about a full 360 deg across the 512-MHz bandwidth.
QUESTION 1 What is the apparent delay error this corresponds to? What will be the effect on the amplitudes?
QUESTION 2 Delay corrections are calculated relative to a reference antenna which has its phase assumed to be zero, so the corrections factorised per other antenna include any correction due to the refant. Roughly what do you expect the magnitude of the Cm corrections to be? Do you expect the two hands of polarisation to have the same sign?
Plot amplitude against time for just one channel per spw.
# amp against time, longest baseline, single channel
plotms(vis='all_avg.ms', field='0319+415',
spw='0~3:55', ydatacolumn='data', # Just one channel per spw
yaxis='amp', xaxis='time', plotrange=[-1,-1,0,0.018], # Plot amplitude v. time with fixed y scale
avgtime='60',correlation='RR', antenna=antref+'&Cm', # 60-sec average, one baseline/pol
coloraxis='spw',plotfile='',
showgui=True,
overwrite=True)
Repeat with all channels averaged
# amp against time, longest baseline, average all channels
plotms(vis='all_avg.ms', field='0319+415',
xaxis='time', yaxis='amp',ydatacolumn='data', # Plot amplitude v. time with fixed y scale
spw='0~3',avgchannel='64', # Average all unflagged channels
antenna=antref+'&Cm', avgtime='60',correlation='RR', # 60-sec average, one baseline/pol
plotfile='',
plotrange=[-1,-1,0,0.018],coloraxis='spw',
showgui=True,
overwrite=True)
For the purposes of the same y axis scaling, these plots have been drawn in python. However, your plotms outputs should be semi-identical.
QUESTION 3 In the plots, which has the higher amplitudes - the channel-averaged data or the single channel? Why?
#### 3A i. Derive delay corrections
The CASA task gaincal is used to derive most calibration. Type inp gaincal or help gaincal to find out more. Fill in the **.
# Derive gain solutions
os.system('rm -rf all_avg.K') # Remove any old version
gaincal(vis='all_avg.ms',
gaintype='**', # Delay calibration (see help gaincal)
field=calsources, # All calibration sources
caltable='all_avg.K', # Output table containing corrections
spw='0~3',solint='**', # 600s solution interval
refant=**, # Reference antenna (phase set to 0)
minblperant=2, # For each antenna and solint, require solutions on
minsnr=2) # at least 2 baselines with S/N > 2
Inspect the solutions. Are they of the magnitude you expect?
plotcal(caltable='all_avg.K',
xaxis='**',
yaxis='**', # Plot delay against time
subplot=321,
iteration='antenna', # Each antenna in a separate pane, coloured by spw
figfile='all_avg.K.png')
The solutions are stable in time, showing that the phase ref solutions can be applied to the target.
### 3B. Setting the flux of the primary flux scale calibrator (script step 7)
1331+305 (= 3C286) is an almost non-variable radio galaxy and its total flux density is very well known as a function of time and frequency (Perley & Butler 2013) . It is somewhat resolved by e-MERLIN, so we use a model made from a ~12-hr observation centred on 5.5 GHz. This is scaled to the current observing frequency range (around 5 GHz) using setjy which derives the appropriate total flux density, spectral index and curvature using parameters from Perley & Butler (2013).
However, a few percent of the flux is resolved-out by e-MERLIN; the simplest way (at present) to account for this is later to scale the fluxes derived for the other calibration sources by 0.938 (based on calculations for e-MERLIN by Fenech et al.).
The model is a set of clean components (see the imaging steps of the EVN continuum tutorial) and setjy enters these in the Measurement Set so that their Fourier transform can be used as a uv-plane model to compare later with the actual visibility amplitudes. If you make a mistake (e.g. set a model for the wrong source) use task delmod to remove it.
setjy(vis='**',
field='**',
standard='Perley-Butler 2013',
model='**') # You should have downloaded and untar'd this model
By default, setjy scales the model for each channel, as seen if you plot the model amplitudes against $uv$ distance, coloured by spectral window. If you have plotms running, go to the axes and select the model, or see the script for inputs. The model only appears for the baselines present for the short time interval for which 1331+305 was observed here. If not, then copy the following for plotms.
os.system('rm -rf 3C286_model.png')
plotms(vis='all_avg.ms', field='1331+305', xaxis='uvwave',
yaxis='amp', coloraxis='spw',ydatacolumn='model',
correlation='LL,RR',
showgui=True,
symbolsize=4, plotfile='')
### 3C. Derive the initial bandpass calibration and inspect effects (script step 8)
We have a Catch 22 situation!. In order to derive good time-dependent corrections, we need to average amplitude as well as phase in frequency, but in order to average in frequency we need the phase to be flat in time, and the amplitude to have the correct spectral index!
So we derive initial time-dependent solutions for the bandpass calibration sources, but discard these after they have been applied to derive the initial bandpass solutions. These, in turn, are replaced in script step 11.
#### 3C i. Time-dependent pre-calibration to prepare for bandpass calibration
Estimate the time-averaging interval for phase by plotting phase against time for a single channel near the centre of the band for a bandpass calibration source. If the results for the first channel and antenna you try look bad, it may be a partly flagged channel; try a different one.
plotms(vis='all_avg.ms',
field='**', # Plot bandpass calibrator
spw='0~3:**', correlation='**' # This gives all spw, choose channel in centre of spw in each and one correlation
antenna='**', # Plot the longest baseline to the refant, likely to have the fastest-changing phase
xaxis='time', yaxis='phase',ydatacolumn='data', # Plot raw phase against time, no averaging
coloraxis='spw',plotfile='',
showgui=gui,
overwrite=True)
Zoom to help decide what phase solution interval to use. The integration-to-integration scatter is noise (mostly instrumental) which cannot be removed but the longer-term wiggles are mainly due to the atmosphere.
We want to derive corrections to reduce these to about the level of the noise or at least to less than 5-10 deg. The longer a solution interval, the better the S/N, but the interval has to be short enough not to average over the wiggles. To illustrate this, we have taken the phase fluctuations on spw 0, channel 55 and used different averaging intervals. This replicates the approximate solutions we would obtain when deriving the phases.
As you can see, the various solution intervals will track the long term slopes fairly effectively, however, the short 60s wiggles are only effectively tracked by the 30s solution interval. If we used the >60s intervals, we would still have errors ~20 deg whereeas, with the 30s intervals, the residuals will be nearer 5 deg.
QUESTION 4 What other effect, aside from atmospheric phase errors, is solved by calibrating phase against a point model at the phase centre?
Let's continue. Using the solution interval, which can track the wiggles, we want to derive the phase solutions using the task gaincal:
os.system('rm -rf bpcals_precal.p1')
gaincal(vis='all_avg.ms',
calmode='**', # Solve for phase only as function of time
field='**', # Two sources suitable for bandpass calibration were observed
caltable='bpcals_precal.p1', # The output solution table
solint='**', # Enter a suitable solution interval
refant='**',
minblperant=2,
gaintable='all_avg.K', # Apply the delay solutions to allow averaging phase across the band
minsnr=2)
Check the solutions. Each spw should have a coherent set of solutions, not just noise.
plotcal(caltable='bpcals_precal.p1',
xaxis='**',
yaxis='**',
subplot=321,
plotrange=[-1,-1,-180,180],
iteration='antenna',
figfile='bpcals_precal.p1.png')
Note that the above plot is zoomed in on 0319+415 so we can see the phase solutions clearly. The plot of amplitude against time in Step 6 also showed that the uncorrected amplitudes have only a slow systematic slope with time, so a longer solution interval can be used, minimising phase noise.
Now derive amplitude solutions. We have not yet derived flux densities for the bandpass calibration sources so we have to normalise the gains (i.e. individual solutions differ but the product is unity so that the overall flux scale is not affected). Thus we have to derive the solutions separately for each of the two bandpass calibrators, but write them to the same solution table.
os.system('rm -rf bpcals_precal.ap1')
gaincal(vis='all_avg.ms',
field='0319+415', # First bpcal
caltable='bpcals_precal.ap1', # Output solution table
calmode='**' # Solve for amplitude and phase v. time
solint='**', # Longer solution interval
solnorm=**, # Normalise the amplitude solutions,
refant='Mk2',
minblperant=2;minsnr=2,
gaintable=['all_avg.K','bpcals_precal.p1']) # Apply the existing delay and phase solutions
# Repeat for the other source!!!
gaincal(vis='all_avg.ms',
field='1407+284', # First bpcal
caltable='bpcals_precal.ap1', # Output solution table
calmode='**' # Solve for amplitude and phase v. time
solint='**', # Longer solution interval
solnorm=**, # Normalise the amplitude solutions,
refant='Mk2',
minblperant=2;minsnr=2,
append=True,
gaintable=['all_avg.K','bpcals_precal.p1']) # Apply the existing delay and phase solutions
Alternatively, if are doing this in the CASA prompt, you can enter the parameters for the first source and then just adjust the field and append parameters.
Let's inspect the solutions in plotcal:
plotcal(caltable='bpcals_precal.ap1',
xaxis='**',
yaxis='**',
subplot=321,
iteration='antenna',
figfile='')
#### 3C ii. Bandpass calibration
Use bandpass to derive solutions for phase and amplitude as a function of frequency (i.e., per channel). In order to allow averaging in time, apply the time-dependent corrections which you have just derived.
os.system('rm -rf bpcal.B1')
bandpass(vis='all_avg.ms',
caltable='bpcal.B1', # New table with bandpass solutions
field='**,**', # Bandpass calibration sources
fillgaps=16, # Interpolate over flagged channels
solint='inf',combine='scan', # Average the whole time for each source
solnorm=**, # Normalise solutions
refant='**',
minblperant=2,
gaintable=['all_avg.K','bpcals_precal.p1','bpcals_precal.ap1'], # Apply all the tables derived for these sources
minsnr=3)
You may see messages about failed solutions but these should all be for the end channels which were flagged anyway.
Inspect the solutions in plotcal. The phase corrections are small as the delay solutions were already applied. The amplitude solutions will remove local deviations from the bandpass but the corrections so far assume that the source flux density is flat in each spw.
Firstly, lets check phases:
plotcal(caltable='bpcal.B1',
xaxis='**', yaxis='**', # Plot phase solutions v. frequency
plotrange=[-1,-1,-180,180],
iteration='antenna',subplot=321,
figfile='')
plotcal(caltable='bpcal.B1',
xaxis='**', yaxis='**', # Plot amp solutions v. frequency
iteration='antenna',subplot=321,
figfile='')
Apply the delay and bandpass solutions to the calibration sources as a test, to check that they have the desired effect of producing a flat bandpass for the phase-reference. There is no need to apply the time-dependent calibration as we have not derived this for the phase-reference source anyway.
Some notes about applycal:
• The first time applycal is run, it creates the CORRECTED data column in the MS, which is quite slow.
• Each time applycal is run, it replaces the entries in the CORRECTED column, i.e. the corrected column is not cumulative, but you can give applycal a cumulative list of gain tables.
• applymode can be used to flag data with failed solutions but we do not want that here as this is just a test.
• The interp parameter can take two values for each gaintable to be applied, the first determining interpolation in time and the second in frequency. The times of data to be corrected may coincide with, or be bracketed by the times of calibration solutions, in which case linear interpolation can be used, the default, normally used if applying solutions derived from a source to itself, or from phase ref to target in alternating scans. However, if solutions are to be extrapolated in time from one source to another which are not interleaved, then nearest is used. A similar argument covers frequency. There are additional modes not used here (see help applycal).
The delay correction ('K') table contains entries for all sources except the target. The phase reference scans bracket the target scans so it is OK to use the default 'linear' interpolation.
The bandpass ('B1') table is derived from just two bandpass calibration observation and there may be data for other sources outside these times, so extrapolation in time is needed, using 'linear'. 'linear' interpolation is OK for the bandpass calibrator in frequency.
If you are finding your laptop is slow, you can skip this applycal and the plotms immediately after and just look at the figures here and go to Step 9.
applycal(vis='all_avg.ms',
field='0319+415,1302+5748,1331+305,1407+284',
calwt=False,
applymode='calonly', # Test, so don't change the weights or flag data with failed solutions
gaintable=['**','**'], # Enter the names of the tables used to calibrate the bandpass
interp=['**','**,**']) # The values depend on which table you listed first.
Check that the bandpass corrections are good for all data by plotting the 'before and after' phase reference phase and amplitude against frequency. First let's check the uncorrected phase (in the interactive CASA prompt this time!)
# In CASA:
default(plotms)
vis='all_avg.ms'; field='1302+5748' # Phase reference
xaxis='frequency';yaxis='phase';ydatacolumn='data' # First, plot uncorrected phase
gridrows=5;gridcols=1;avgtime='600';coloraxis='scan'
iteraxis='baseline';antenna='Mk2&*';correlation='LL,RR'
plotms()
And next, the corrected phase,
ydatacolumn='corrected'
plotms()
Then, the uncorrected amplitude,
yaxis='amp'
ydatacolumn='data'
plotms()
And finally, the corrected amplitude,
ydatacolumn='corrected'
plotms()
The scripted versions of these commmands are also included in the 3C277.1_cal_outline.py script on lines 452-483. The plot below shows the uncorrected and corrected amplitudes and phases for the Mk2-Cm baseline of the phase reference source (1302+5748) that is coloured by scan. It has been replotted in python for clarity!
In the figure, each scan is coloured separately and shows a random offset in phase and some discrepancies in amplitude compared to other scans, but for each individual time interval it is quite flat. So we can average in frequency across each spw and plot the data to show remaining, time-dependent errors.
Plot the corrected phase against time with all channels averaged for the phase reference source. Zoom in on a few scans to deduce a suitable averaging timescale for correction of phase errors as a function of time, i.e. where atmospheric fluctuations dominate over noise. Again, we shall use the interactive CASA prompt.
# In CASA:
default(plotms)
vis='all_avg.ms'; field='1302+5748'
xaxis='time';yaxis='phase';ydatacolumn='corrected'
antenna='Mk2&*';iteraxis='baseline'
avgchannel='64'; coloraxis='spw'
correlation='LL' # Just one pol as previous plots show both are similar
plotms()
And repeat for amplitudes
### 3D. Gain calibration (script step 9)
#### 3D i. Time-dependent phase calibration of all calibration sources
gaincal(vis='all_avg.ms',
calmode='p', # Phase only first
caltable='calsources.p1',
field='**', # All calibration sources
solint='**', # Insert solution interval
refant=antref,
minblperant=2,minsnr=2,
gaintable=['all_avg.K','bpcal.B1'], # Existing correction tables to apply
interp=['**','**,**']) # Insert suitable interpolation modes
When you run gaincal, check the messages in the logger and the terminal. The logger shows something like:
2019-03-22 06:47:45 INFO Writing solutions to table: calsources.p1
2019-03-22 06:47:46 INFO calibrater::solve Finished solving.
2019-03-22 06:47:46 INFO gaincal:::: Calibration solve statistics per spw: (expected/attempted/succeeded):
2019-03-22 06:47:46 INFO gaincal:::: Spw 0: 1400/935/930
2019-03-22 06:47:46 INFO gaincal:::: Spw 1: 1400/935/930
2019-03-22 06:47:46 INFO gaincal:::: Spw 2: 1400/935/930
2019-03-22 06:47:46 INFO gaincal:::: Spw 3: 1400/935/930
This means that although there are 1400 16s intervals in the calibration source data, about a third ($1400-935=465$) of them have been partly or totally flagged. But almost all ($99.4%$) of the intervals which do have enough unflagged ('attempted') data gave good solutions ('succeeded'). There are no further messages about completely flagged data but in the terminal you see a few messages when there are fewer than the requested 2 baselines per antenna unflagged:
Insufficient unflagged antennas to proceed with this solve.
(time=2015/05/06/06:32:26.5 field=0 spw=0 chan=0)
Here, there are very few, predictable failures. If there were many or unexpected failures, investigate - perhaps there are unflagged bad data, or pre-calibration has not been applied, or an inappropriate solution interval was used.
Plot the solutions, separately for L and R for clarity:
plotcal(caltable='calsources.p1', xaxis='time',
plotrange=[-1,-1,-180,180],
iteration='antenna',yaxis='phase',subplot=321,
poln='L',figfile='') # Note that solutions are per antenna, hence 'L' not 'LL'
The phase corrections have a similar time-dependence as the uncorrected data phase, and are not random (if they look just like noise something is wrong).
#### 3D ii. Time-dependent amplitude calibration of all calibration sources
Derive time-dependent solutions for all the calibration sources, applying the delay and bandpass tables and the time-dependent phase solution table.
At this point, 1331+305 has a realistic model set in the MS (Step 7). The other calibration sources have the default of a 1 Jy point source at the phase centre. If (and only if) you are repeating a failed attempt, reset any previous models for sources other than 1331+305.
# In CASA:
default(delmod)
vis='all_avg.ms'
field='0319+415,1407+284,1302+5748,1252+5634'
delmod()
Insert a suitable solution interval (from inspection of the previous amp plots and listobs output), the tables to apply and interpolation modes. Note that if you set a solint longer than a single scan on a source, the data will still only be averaged within each scan unless an additional 'combine' parameter is set, not needed here.
gaincal(vis='all_avg.ms',
calmode='ap', # Amplitude and phase
caltable='calsources.ap1',
field='**', # All calibration sources
solint='**', # Insert solution interval.
solnorm=False, # Solutions will be used to determine the flux scale - don't normalise!
refant='Mk2',
minblperant=2,minsnr=2,
gaintable=['all_avg.K','bpcal.B1','calsources.p1'],
interp=['**','**,**','**']) # Insert suitable interpolation modes
The logger shows a similar proportion of expected:attempted solution intervals as for phase, but those with failed phase solutions are not attempted so all the attempted solutions work.
Plot the solutions separately for L and R for clarity:
plotcal(caltable='calsources.ap1',xaxis='time',
iteration='antenna',yaxis='amp',subplot=321,
poln='L',figfile='calsources.ap1_L.png')
plotcal(caltable='calsources.ap1',xaxis='time',
iteration='antenna',yaxis='amp',subplot=321,
poln='R',figfile='calsources.ap1_R.png')
The plot above shows the time dependent amplitude solutions on just the L polarisation and coloured by spw. The solutions for each separate source look similar but there are big differences between sources, since they have all (apart from 1331+305) been compared with a 1 Jy model but they have different flux densities.
### 3E. Determining flux densities of calibration sources (script step 10)
#### 3E i. Derive calibrator flux densities
The solutions for 1331+305 in calsources.ap1 contain the correct scaling factor to convert the raw units to Jy as well as removing time-dependent errors. This is used to calculate the scaling factor for the other calibration sources in the CASA task fluxscale. We run this task in a different way because the calculated fluxes are returned in the form of a python dictionary. They are also written to a text file. Only the best data are used to derive the flux densities but the output is valid for all antennas. This method assumes that all sources without starting models are points, so it cannot be used for an extended target.
os.system('rm -rf calsources.ap1_flux calsources_flux.txt')
calfluxes=fluxscale(vis='all_avg.ms', # calfluxes will hold the calculated fluxes in memory
caltable='calsources.ap1', # Input table containing amplitude solutions
fluxtable='calsources.ap1_flux', # Scaled output table
listfile='calsources_flux.txt', # Text file on disk listing the calculated fluxes
gainthreshold=0.3, # Exclude solutions >30% from mean
antenna='!De', # Exclude least sensitive antenna
reference='1331+305') # Source with known flux density and previous model
If you type calfluxes this will show you the python dictionary, but if you don't yet know how to parse this, look at the logger or the listfile, calsources_flux.txt using:
!more calsources_flux.txt # Use of ! to invoke shell command
The individual spw estimates may have uncertainties up to $\sim20\%$ but the bottom lines give the fitted flux density and spectral index for all spw which should be accurate to a few percent. These require an additional scaling of $0.938$ to allow for the resolution of 1331+305 by e-MERLIN (see Setting the flux of the primary flux scale calibrator, step 7). The python dictionary calfluxes is used to do this:
eMcalfluxes={}
for k in calfluxes.keys():
if len(calfluxes[k]) > 4:
a=[]
a.append(calfluxes[k]['fitFluxd']*0.938)
a.append(calfluxes[k]['spidx'][0])
a.append(calfluxes[k]['fitRefFreq'])
eMcalfluxes[calfluxes[k]['fieldName']]=a
This makes a new python dictionary named eMcalfluxes, which we can print using print(eMcalfluxes) to produce:
{'0319+415': [22.931930617690497, 1.3882377807343991, 5069980155.629359],
'1302+5748': [0.40434240363584933, -0.36545354992840479, 5069980155.629359],
'1407+284': [1.9012878707051455, 0.30684503924055434, 5069980155.629359]}
where the first column or 'key' is the source name and the values are [flux density (Jy), spectral index and reference frequency (Hz)]. You may get slightly different values if you have chosen different solution intervals but if they are more than a few percent different something is probably wrong.
#### 3E ii. Setting the calibrator flux densities
If you exit and restart CASA before running this step, you will have to re-define eMcalfluxes by copying the three lines starting with 'eMcalfluxes = ....' into the CASA prompt.
Nexe we shall use setjy in a loop to set each of the calibration source flux densities. The logger will report the values being set.
for f in eMcalfluxes.keys(): # For each source in eMcalfluxes:
setjy(vis='all_avg.ms',
field=f,
standard='manual', # Use our values, not a built-in catalogue
fluxdensity=eMcalfluxes[f][0],
spix=eMcalfluxes[f][1],
reffreq=str(eMcalfluxes[f][2])+'Hz')
The default in setjy is to scale by the spectral index for each channel and plotting the models for the bandpass calibrators shows a slope corresponding to the spectral index (zoom in on the fainter source to see this). Both sources have positive spectral indices in this frequency range; this is less unusual for compact, bright QSO than for radio galaxies in general.
Plot the models for the bandpass calibrators, amplitude against frequency, to see the spectral slopes.
plotms(vis='all_avg.ms', field='0319+415,1407+284', # Plot the bandpass calibrators
xaxis='***',
yaxis='***',ydatacolumn='***', # Fill in axis and data types
coloraxis='spw',correlation='LL', # Just LL for speed; model is unpolarized
symbolsize=5, customsymbol=True, symbolshape='circle') # Bigger symbols
### 3F. Improved bandpass calibration (script step 11)
Use the models including spectral indices to derive a more accurate bandpass correction. You can use the first use of bandpass as a guide but there are three major differences (apart from writing a different table).
os.system('rm -rf bpcal.B2')
bandpass(vis='all_avg.ms',
caltable='bpcal.B2',
field='**,**',
fillgaps=**,
solint='**',combine='**',
refant=antref,
minblperant=2,
solnorm=**, # There are now accurate flux densities and spectral indices so?
gaintable=['all_avg.K','**','**'], # and updated time-dependent solutions
gainfield='**,**', # but the new solutions contain entries for a number of sources
minsnr=3)
And now we want to plot the solutions for both phase and amplitude:
plotcal(caltable='bpcal.B2',xaxis='freq',
plotrange=[-1,-1,-180,180],
iteration='antenna',yaxis='phase',subplot=321,
figfile='bpcal.B2_phase.png')
plotcal(caltable='bpcal.B2',xaxis='freq',
iteration='antenna',yaxis='amp',subplot=321,
figfile='bpcal.B2_amp.png')
These solutions are similar to the initial bandpass approximation but with a different slope. To illustrate this, the initial bandpass and improved bandpass solutions for Knockin (Kn) are plotted below. You can see the same wiggles as in B1, but there is a different overall slope.
### 3G. Derive solutions for all cal sources with improved bandpass table (script step 12)
#### 3G i. Derive amplitude solutions
For each calibration source, there is now a flux density and spectral index set for the MS MODEL column, as well as the delay, phase and improved bandpass calibration. These are used to derive time-dependent amplitude solutions which will both correct for short-term errors and contain an accurate flux scale factor.
Look at the previous time that you ran gaincal, and identify the one thing which you need to change now (apart from the output table name).
os.system('rm -rf calsources.ap2')
gaincal(vis='all_avg.ms',
calmode='ap',
caltable='calsources.ap2',
field='**',
solint='**',
refant='**',
minblperant=**,minsnr=**,
gaintable=['**','**','**'],
interp=['**','**,**','**'])
Plot the solutions:
plotcal(caltable='calsources.ap2',xaxis='time',
iteration='antenna',yaxis='amp',subplot=321,
plotrange=[-1,-1,0.01,0.06],
poln='L') # And repeat for the other polarisation.
You expect to see a similar scaling factor for all sources. 1331+305 is somewhat resolved, giving slightly higher values for Cm (the antenna giving the longest baselines).
#### 3G ii. Derive scan-averaged phase solutions to be applied to the target source
Finally, we need phase solutions that can be applied to the target source. Because we will interpolate solutions between phase calibrator scans, we just need one phase solution per phasecal scan. So solint should include all the data from each scan to form a solution.
os.system('rm -rf calsources.p2')
gaincal(vis='all_avg.ms',
calmode='**',
caltable='calsources.p2',
field='**',
solint='**',
refant='**',
minblperant=**,minsnr=**,
gaintable=['**','**'],
interp=['**','**,**'])
And again, we should plot the solutions:
plotcal(caltable='calsources.p2',xaxis='time',
plotrange=[-1,-1,-180,180],
iteration='antenna',yaxis='phase',subplot=321,
poln='L') ## And repeat for other polarisation
### 3H. Apply final calibrator source solutions and plot phase-ref and target (script step 13)
#### 3H i. Apply solutions
Apply the bandpass correction to all sources. For the other gaintables, for each calibrator (field), the solutions for the same source (gainfield) are applied. gainfield='' for the bandpass table means apply solutions from all fields in that gaintable. applymode='calflag' will flag data with failed solutions which is ok as we know that these were due to bad data. We use the non-interactive format to allow us to use a loop over all calibration sources.
Cut and paste this all together; press return a few times afterwards till it runs.
cals = ['0319+415', '1302+5748', '1331+305', '1407+284']
for c in cals:
applycal(vis='all_avg.ms',
field=c,
gainfield=[c, '',c,c],
calwt=False,
applymode='calflag',
gaintable=['all_avg.K','bpcal.B2','calsources.p1','calsources.ap2'],
interp=['linear','linear,linear','linear','linear'],
flagbackup=False) # Don't back up flags multiple times here
Apply the phase-ref solutions to the target. We assume that the phase reference scans bracket the target scans so linear interpolation is used except for bandpass. Enter the parameters in interactive mode.
applycal(vis='all_avg.ms',
field='1252+5634',
gainfield=['**','','**','**'], # Enter the field whose solutions are applied to the target
calwt=False,
applymode='**', # Not too many failed solutions - OK to flag target data
gaintable=['**','**','**','**'],
interp=['**','**,**','**','**'],
flagbackup=True)
#### 3H ii. Plot phase-ref and target amps against uv distance
Next we want to visualise our target and phase reference source. We want to ensure that the phase reference source is a true point source and if there is structure on the target. We can do this by looking at the amp vs uv distance plots. Remember a point source will be flat across uv distance!
plotms(vis='all_avg.ms', field='1302+5748', xaxis='uvdist',
yaxis='amp',ydatacolumn='corrected',
avgchannel='64',correlation='LL,RR',
overwrite=True,
showgui=True,
coloraxis='spw')
This looks pretty pointlike!. Let's repeat the above for the target.
Hurray, there's some structure here!
### 3I. Split out target (script step 14)
Split out corrected target data. By default, this will take the corrected column.
os.system('rm -rf 1252+5634.ms*')
split(vis='all_avg.ms',
field='1252+5634',
outputvis='1252+5634.ms',
keepflags=True)
Now make sure that you save 1252+5634.ms for the imaging stages.
## Script with solutions
If you had problems in any of the steps, you should use the calibration script 3C277.1_cal_all.py
## Continue with imaging tutorial
You are ready to image the target source using the splitted calibrated data in 1252+5634.ms | 2022-01-20 04:50:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6237454414367676, "perplexity": 3646.097935434294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301720.45/warc/CC-MAIN-20220120035934-20220120065934-00263.warc.gz"} |
https://www.physicsforums.com/threads/poissons-ratio.334905/ | # Poissons Ratio ?
1. Sep 5, 2009
### Stacyg
A steel bar of rectangular cross section 120mm x 60mm is compressed along its longitudinal direction by a force of 1500kN. Do the cross sectional dimensions increase or decrease ? Calculate and write down the resulting dimensions for both sides for both sides of the cross section. Youngs modulus E=200GPa, and Poissons rato of v=0.3
I am not sure what equations to use to do this so I havent showed any attempts.
Any help would be great.
2. Sep 5, 2009
### Hootenanny
Staff Emeritus
HINT: How is Poisson's ratio related to the transverse and longitudinal strains? | 2017-12-12 02:54:40 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.817121684551239, "perplexity": 2492.19041624046}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948514250.21/warc/CC-MAIN-20171212021458-20171212041458-00335.warc.gz"} |
http://www.gamedev.net/topic/608061-understanding-1d-perlin-noise/ | • Create Account
## Understanding 1d Perlin Noise
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### #1Alaror Members
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Posted 08 August 2011 - 10:33 AM
I've been using this article as a starting point for 1d Perlin Noise in my terrain generation algorithm, but the author seems to have skipped parts of the explanation. In his example he assigns a float value from 0 to 1 for every point on the X axis. What I'm not understanding is if I need to assign a value for the points between the X axis points. If 0 on the X axis is given a value of 0.3 and 1 on the X axis is given a value of 0.9, don't I have to have points between the 0 and 1 on the X axis for it to be smoothed? How would I go about doing that in relation to his example code at the bottom?
### #2JTippetts Moderators
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Posted 08 August 2011 - 10:37 AM
Those intermediate points are generated by interpolating the values at the integer points.
For your example, if the point at 0 is assigned 0.3 and the point at one is assigned at 0.9, then the point at 0.5 would be equal to 0.3 + 0.5*(0.9-0.3) or 0.6.
A more general explanation:
If the input coordinate is called P, you can calculate the two bounding integral boundaries by:
P1=floor(P)
P2=P1+1
Then you call the function to obtain the values for P1 and P2. The interpolator is calculated as:
interp=P-P1
And the final interpolation (using a linear interpolation) is done as:
value=P1 + interp*(P2-P1)
### #3Alaror Members
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Posted 08 August 2011 - 02:34 PM
Thanks for the answer, that clears up most of it.
To look at it in a way that's more applicable to what I'm trying to do, if I have a map that's 1000 units wide would I start out by assigning a random value to points at a specific distance apart (let's say every 100 spaces), then use an interpolation function to find the value of every point between those (1 to 99, 101 to 199, etc), and then finally plug every single point (0 to 1000) into the main Perlin Noise function? The code structure would look something like the following:
• Assign a random variable every 100 units, starting with 0 and working toward 1000.
• Loop through every single point and give it a value based on the interpolation of the value immediately before and after it. Do this until the maximum width is reached.
• Plug every integer value (0 to 1000) into the main Perlin Noise function. The value returned by this replaces the previous value for this point.
### #4JTippetts Moderators
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Posted 08 August 2011 - 02:52 PM
Your way seems to me to indicate a fundamental misunderstanding. The typical application of Perlin noise is as a function that operates separately from whatever you are applying it to. You supply the function with a set of coordinates, you get a value in return that you can store into an array. There is, and should be, no correlation between the function and the grid that you are filling from it.
The typical means I use to fill an Array from a given Perlin function is as follows:
RegionWidth: The width of the 2D region of the function to map to the grid
RegionHeight: The height of the 2D region of the function to map to the grid
MapWidth: The width of the 2D grid
MapHeight: The height of the 2D grid
PerlinFunc: 2D Perlin noise function
for x=0,MapWidth-1,1 do
for y=0,MapHeight-1,1 do
local nx=x/MapWidth
local ny=y/MapWidth
nx=nx*RegionWidth
ny=ny*RegionHeight
Array[x][y]=PerlinFunc:get(nx,ny)
end
end
Doing the mapping in this way allows me to easily specify a different size of area to map to the grid, and doesn't require multiple iterating of the grid to generate intermediate values.
I think that one of your misconception is that the values for the grid integer locations of the Perlin noise function need to be stored somewhere. They do not, but are simply calculated on the fly whenever they are needed. The Perlin noise function is basically a purely mathematical entity that doesn't store state, but merely calculates values on the fly as it is called.
### #5Alaror Members
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Posted 08 August 2011 - 03:51 PM
Your way seems to me to indicate a fundamental misunderstanding.
Yep, and that's putting it lightly haha!
Going off of your example, in the case of a 1d array, would the X input for the Perlin Noise function simply be its point in the array (0 to 1000)? If you're trying to create terrain for an entire map why would the region width be different from the map width (you may just be using the function differently than I am)?
Now what lead me to believe I needed to assign a value to each point along the way was the fact that in the example code on the page I linked above he has X as a float. I assumed this to be the random value from 0 to 1. Would it be more accurate then to say that the following is true, or is that random value assigned elsewhere in the code?
Array[x]=PerlinFunc:get(RandomNoiseFunc(nx))
If this is the case it brings me back to my earlier question since from my understanding at this point you need to run the Perlin Noise function for every single point on the X axis, and each of these points is given a unique value from 0 to 1. My terrain is going to be block based, so I'm concerned that by randomly assigning heights to every single point (representing the location of a block) it will vary wildly rather than be a smooth transition. Will the combined result of the different amplitudes/frequencies result in a smooth transition?
### #6JTippetts Moderators
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Posted 08 August 2011 - 04:09 PM
The reason for my approach is that if you use the array coordinate as the input to the Perlin function then you get exactly what you are afraid of getting: a wildly varying terrain where every point is different. It is a matter of scale. Perlin noise provides what is called "continuously varying" noise, but the scale of it is such that the features (hills, valleys, whatever) are 1-unit in size. Every 1 unit in the function, there is a new value. So if you use the array coordinate unmodified, you are essentially creating one feature (hill or valley) for every array element. The result of this looks very much like white noise, or regular random noise.
By using a different region width, one that is much smaller than array width, you are in essences zooming into the function so that the features of it will be spread out across several array elements.
Array[x] = Perlin:get(RandomNoiseFunc(nx))
shows that you still don't quite get it. A Perlin function is self-contained; the random noise generation is part of it.
for x=0,MapWidth-1,1 do
local nx=x/MapWidth
nx=nx*RegionWidth
Array[x]=Perlin(nx)
end
So if you had a RegionWidth of 3, in essence this would fill your array with what amounts to approximately three hills or valleys across. The reason the noise function input is a float is because the function is defined for any input point, not just integral input coordinates. Inputs that include a fractional part are evaluated by interpolating the enclosing integral-boundary values.
### #7Alaror Members
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Posted 08 August 2011 - 08:15 PM
The pieces seem to be fitting together finally thanks to your explanation; I actually have it at a point where I can use the data it outputs to create the blocks. However, there are a few things that just don't seem to be lining up for me still.
In the following function, what is the purpose of changing the values into integers?
function InterpolatedNoise_1 (float x)
integer_X = int(x)
fractional_X = x - integer_X
v1 = SmoothedNoise1(integer_X)
v2 = SmoothedNoise1(integer_X + 1)
return Interpolate(v1, v2, fractional_X)
end function
In the random noise function, does it matter what the seed is? In particular, I'm not familiar with the meaning of this bit of code: x = (x<<13) ^ x; Is it just converting the X value somehow, or does it relate to setting the seed? In my tests when I set the seed to the X value that gets plugged into the function the RegionWidth has an effect on the final output, but when I set it to use a random seed the output is *completely* random, with each point not appearing to relate to the others. I've uploaded a screenshot that has a comparison of the two. The top screenshot is using a random seed, and the bottoms screenshot is using the X input as the seed, with the RegionWidth set to 50 (as an aside, when I have it at 3 it's more or less a strait line; is this to be expected?). The problem with this is that every single time I run the program the values are all the same.
### #8JTippetts Moderators
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Posted 08 August 2011 - 09:58 PM
The pieces seem to be fitting together finally thanks to your explanation; I actually have it at a point where I can use the data it outputs to create the blocks. However, there are a few things that just don't seem to be lining up for me still.
In the following function, what is the purpose of changing the values into integers?
function InterpolatedNoise_1 (float x)
integer_X = int(x)
fractional_X = x - integer_X
v1 = SmoothedNoise1(integer_X)
v2 = SmoothedNoise1(integer_X + 1)
return Interpolate(v1, v2, fractional_X)
end function
The point is that pseudo-random values are generated for integral coordinate locations. What the above function does is it takes an arbitrary input point and finds the enclosing integer coordinates. For each of those coordinates, a pseudo-random value is calculated. Then the fractional part is used to interpolate those positions.
So imagine that you call the function with x=5.5; taking the int of x gives us the value of 5, so we know that the coordinates X1=5 and X2=6 are the integer coordinates that enclose 5.5. Now, we feed these values of X1 and X2 to the SmoothedNoise function, which generates a pseudo-random value based on the coordinate. For example sake, assume that SmoothedNoise(5) returns 0.26 and SmoothedNoise(6) returns -0.95. We calculate the interpolant as the fractional part of x. This is simply done by subtracting X1 from x, since X1 is the nearest integer that is less than x. interpolant = x - X1. The result with x=5.5 and X1=5 is interpolant=0.5. Now, we use this interpolant to interpolate between the values of 0.26 and -0.95: value = 0.26 + interpolant * (-0.95 - 0.26) = 0.345. So the value of the function at x=5.5 is equal to 0.345.
In the random noise function, does it matter what the seed is? In particular, I'm not familiar with the meaning of this bit of code: x = (x<<13) ^ x; Is it just converting the X value somehow, or does it relate to setting the seed? In my tests when I set the seed to the X value that gets plugged into the function the RegionWidth has an effect on the final output, but when I set it to use a random seed the output is *completely* random, with each point not appearing to relate to the others. I've uploaded a screenshot that has a comparison of the two. The top screenshot is using a random seed, and the bottoms screenshot is using the X input as the seed, with the RegionWidth set to 50 (as an aside, when I have it at 3 it's more or less a strait line; is this to be expected?). The problem with this is that every single time I run the program the values are all the same.
All of the stuff that you see in the noise function is just a series of operations intended to hash the input integer coordinate to a pseudo-random value. A pseudo-random value is a value that is seemingly random, and apparently has no perceptible correlation to the value it is generated from, and yet is repeatable; ie, everytime the function is called with a given input it will provide the same output. The noise function provided there just does some magical trickery, including a multiplication by 8192 (which is what << 213 does ) then XORs the result with the original value of x. Then it does some additional trickery. The actual mechanics of this really aren't important; there are a thousand ways that you could perform this hash. The result is what is important: a seemingly random, repeatable value. The random-part of pseudo-random is what ensures that subsequent values (ie the function at x=5 and the function at x=6) have no relation to one another and thus will generate no visible patterns in the output.
What provides the "magic" of this technique is the deterministic aspect of it: that plugging in the coordinate to the noise function nets you the same output each time. Imagine the chaos that results if you evaluate the point x=4.5 where X1=4 and X2=5, and the randomized output of Noise(X2=5) was 0.26. But then you evaluated the point 5.5 where X1=5 and X2=6, but this time the value of Noise(X1=5) outputted 0.45 instead of 0.26 as it did initially. There would be not continuity, and the final output of the function would be chaos; or, basically, what you got in your top screenshot.
Now, if you do want to be able to provide a random seed to change the behavior of the function, you have to make it a part of the IntNoise function, and fold it into the hash somehow. And, of course, you have to ensure that you always provide the same seed for a given mapping or evaluation of the function across a map or array; if the seed changes all the time, then the result would be the same kind of chaos as using a random input to IntNoise.
### #9Alaror Members
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Posted 09 August 2011 - 01:48 PM
Well, I now have a working Perlin Noise function. Just need to spend some time playing with the values until I get the results I want. On the topic of using a seed value, I ended up just plugging another value into the random noise function right before it multiplies x * x, so it's now var_SeedValue * x * x. I also need to figure out the best way to convert the output values into something I can use, though I think some simple cross-multiplication will do the trick.
A huge thank you for all your help! I like understanding the code I put into my program and your thorough explanations (and patience) got me to that point
### #10JTippetts Moderators
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Posted 09 August 2011 - 04:50 PM
You're welcome. I'm actually in the process of putting together several more articles about various uses of noise at http://accidentalnoise.sourceforge.net/ where my noise library is available for download. Currently, I've got a rework of my article on using 3D noise to generate cube-style worlds. It's based on an earlier set of journal posts I had made, as well as an article I wrote for the April 2011 issue of Game Developer Magazine. The article discusses generating 3D Minecraft-style worlds as well as 2D side view Minecraft-alikes as a composition of various functions. Might be of interest to you, and I have more stuff on the way as I find time for it.
### #11Alaror Members
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Posted 10 August 2011 - 10:30 AM
Wow, looks like a fantastic source of info! Will definitely spend some time digging through there. Looking forward to more in the future too.
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Posted 10 August 2011 - 08:28 PM
You probably won't see this comment, but in case you do, I'm not sure my Perlin Noise is functioning correctly. Despite me having the persistence value set at 1/4, the transition from one to another doesn't seem to be smoothed out (here's the image). Do you happen to know what's likely to cause this?
### #13JTippetts Moderators
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Posted 10 August 2011 - 09:18 PM
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Posted 11 August 2011 - 10:52 PM
Thanks for the reply. I'm using DarkBasic Pro, but hopefully you can make sense of it. The value for glob_PerlinNoise_Persistance# is set at 0.25, and the value of glob_PerlinNoise_OctaveMax is set at 3.
Code that calls the function.
FOR var_PositionX = 0 TO var_WorldGen_WorldWidth - 1
var_X# = var_PositionX / var_WorldGen_WorldWidth#
var_X# = var_X# * var_WorldGen_Regions
arr_WorldGen_WidthData(var_PositionX) = var_WorldGen_Dirt_BorderStart + ((((func_PerlinNoise1D_Combined(var_X#) + 1.0) * var_WorldGen_HeightVariation) / 2.0) - (var_WorldGen_HeightVariation / 2))
NEXT var_PositionX
Perlin Noise code.
FUNCTION func_PerlinNoise1D_Combined(var_X#)
var_Total# = 0
FOR var_Octave = 0 TO glob_PerlinNoise_OctaveMax
var_Frequency = 2 ^ var_Octave
var_Amplitude# = glob_PerlinNoise_Persistance# ^ var_Octave
var_Total# = var_Total# + func_PerlinNoise1D_Interpolate(var_X# * var_Frequency) * var_Amplitude#
NEXT var_Octave
ENDFUNCTION var_Total#
FUNCTION func_PerlinNoise1D_Interpolate(var_X#)
var_X_Integer = INT(var_X#)
var_X_Fraction# = var_X# - var_X_Integer
var_SmoothNoise_1# = func_PerlinNoise1D_Smooth(var_X_Integer)
var_SmoothNoise_2# = func_PerlinNoise1D_Smooth(var_X_Integer + 1)
var_Interpolate# = func_PerlinNoise1D_InterpolateCosine(var_SmoothNoise_1#, var_SmoothNoise_2#, var_X_Fraction#)
ENDFUNCTION var_Interpolate#
FUNCTION func_PerlinNoise1D_InterpolateCosine(var_SmoothNoise_1#, var_SmoothNoise_2#, var_X_Fraction#)
var_Value1# = var_X_Fraction# * 3.1415927
var_Value2# = (1 - COS(var_Value1#)) * 0.5
var_InterpolateCosine# = var_SmoothNoise_1# * (1 - var_Value2#) + var_SmoothNoise_2# * var_Value2#
ENDFUNCTION var_InterpolateCosine#
FUNCTION func_PerlinNoise1D_Smooth(var_X)
var_SmoothNoise# = func_PerlinNoise1D_Random(var_X) / 2.0 + func_PerlinNoise1D_Random(var_X - 1) / 4.0 + func_PerlinNoise1D_Random(var_X + 1) / 4.0
ENDFUNCTION var_SmoothNoise#
FUNCTION func_PerlinNoise1D_Random(var_X)
var_X = (var_X << 13) ~~ var_X
var_Random# = (1.0 - ((var_X * (glob_WorldGen_Seed * var_X * var_X * 15731 + 789221) + 1376312589) && 0x7fffffff) / 1073741824.0)
ENDFUNCTION var_Random#
### #15JTippetts Moderators
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Posted 12 August 2011 - 09:09 PM
In the image that you posted, how many array samples are represented? Does one height sample represent a column of pixels, or a column of larger blocks? I'm not seeing anything in the posted code that jumps out at me, but it would be useful to see what you are doing to between the posted code and drawing to the screen.
### #16Alaror Members
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Posted 13 August 2011 - 11:47 AM
That's determined using your "region" method. This is the code that converts the output from the Perlin Noise (which will be a value from -1.0 to 1.0) into something that's usable.
arr_WorldGen_WidthData(var_PositionX) = var_WorldGen_Dirt_BorderStart + ((((func_PerlinNoise1D_Combined(var_X#) + 1.0) * var_WorldGen_HeightVariation) / 2.0) - (var_WorldGen_HeightVariation / 2))
To break it down a bit, there are three main values being combined in the code, but only this one is relevant to my problem: ((((func_PerlinNoise1D_Combined(var_X#) + 1.0) * var_WorldGen_HeightVariation) / 2.0)". Here you see the value "var_WorldGen_HeightVariation" appear; this is the maximum amount of blocks I want the border to be able to vary, which is set at 0.05% of the total map height. The Perlin Noise function is called, and immediately after being called the value it returns (again, -1.0 to 1.0) is increased by 1.0. This gets multiplied by "var_WorldGen_HeightVariation" and then divided by 2.0. I'm making it sound more complicated then it really is, since It's pretty much just cross multiplication.
Hopefully I answered your question, but if not let me know. Off to work for now, but I'll be able to elaborate further after I'm back.
### #17JTippetts Moderators
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Posted 13 August 2011 - 01:16 PM
What I'm asking is how are you converting the array full of noise values to the image that is shown on screen? Does each element in the array correspond to the height of one column of pixels in the output image? If that is the case, then it seems you have an interpolation problem. Or is one element in the array correspond to a column of blocks, where the blocks are comprised of some MxN pixels? If this is the case, then it is possible you don't really have a problem, since "magnifying" what is essentially a pixel array would serve to also magnify the stepped jaggedness of a function.
### #18Alaror Members
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Posted 14 August 2011 - 04:45 PM
Oh sorry, I understand now. One element in the array is equivalent to one column of blocks. I was using the Cosine interpolation method (which was probably unnecessary to begin with since I'm dealing with blocks, not smooth lines), but I switched it to Linear and it's working wonderfully now. Thanks yet again!
### #19JTippetts Moderators
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Posted 14 August 2011 - 05:19 PM
You're welcome, glad that you got it working.
Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic. | 2016-12-03 18:06:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.474417120218277, "perplexity": 684.0974634345865}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541066.79/warc/CC-MAIN-20161202170901-00408-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=60391 | by newcool
P: 42 Thanks for the help
HW Helper
P: 4,124
Quote by newcool Consider this problem: http://www.physics.harvard.edu/probweek/prob35.pdf The solution is here: http://www.physics.harvard.edu/probweek/sol35.pdf Why is N and $$mg *\cos\theta$$both positive? I thought N point away from the hoop
No N is inward. The beads try to go outward, but the loop pulls them inward.
P: 42 Thanks, I got the first part, anyone have any idea bout the second?
HW Helper
P: 4,124 | 2014-08-28 09:16:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19598914682865143, "perplexity": 4972.234509459331}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830721.40/warc/CC-MAIN-20140820021350-00175-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://compgroups.net/comp.text.xml/problem-with-child-text-node-when-constraining/1839351 | f
#### problem with child text node when constraining other child node types
I'm struggling with whether or not it is possible to represent the
following construct in a dtd.
I have an element X that I want to contain 3 types of child elements.
Child Element A should have a 0 or 1 constraint, Child element B should
have a 0-n constraint. I also want X to be able to contain text,
resulting in xml that the following
<X>
<A/>
<B/>
<B/>
child text of element X also possible
</X?>
I've got the following, but can't seem to figure out a way to also
allow for a text child node. Any help appreciated
<!ELEMENT X (A?, B*)>
Bryan
0
bax (3)
5/17/2006 2:24:06 PM
comp.text.xml 8781 articles. 0 followers.
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Bryan Ax wrote:
> I've got the following, but can't seem to figure out a way to also
> allow for a text child node. Any help appreciated
Declare X as having mixed content.
http://www.w3.org/TR/2004/REC-xml11-20040204/#sec-mixed-content
0
5/17/2006 2:27:57 PM
Looking at this, I don't see a way to constrain it so that A only can
appear once, whereas B can appear multiple times...I think I'm missing
something.
0
bax (3)
5/17/2006 2:36:00 PM
Upon further reading, there doesn't appear to be a way to do this.
Either I go to mixed-content, which is less constraining, or I need to
add a child node to hold the text element.
http://www.devguy.com/fp/XML/dtd.htm
0
bax (3)
5/17/2006 2:56:16 PM
Bryan Ax wrote:
> I have an element X that I want to contain 3 types of child elements.
> Child Element A should have a 0 or 1 constraint, Child element B should
> have a 0-n constraint. I also want X to be able to contain text,
All you can do with mixed contents is described here
<http://www.w3.org/TR/REC-xml/#sec-mixed-content>
So you could have
<!ELEMENT X (#PCDATA | A | B)*>
which would define the possible child elements (A, B) but does not allow
you to constrain their number.
--
Martin Honnen
http://JavaScript.FAQTs.com/
0
mahotrash (2052)
5/17/2006 3:29:54 PM
Bryan Ax wrote:
> Looking at this, I don't see a way to constrain it
That's correct. Mixed content in DTDs doesn't allow constraining the
number or order of instances of child elements, only their types. Live
with that and constrain it in your application code, or switch from DTDs
to schemas, or (as you suggest) move the text into an element.
0
5/17/2006 4:11:50 PM
Bryan Ax wrote:
> I'm struggling with whether or not it is possible to represent the
> following construct in a dtd.
>
> I have an element X that I want to contain 3 types of child elements.
> Child Element A should have a 0 or 1 constraint, Child element B should
> have a 0-n constraint. I also want X to be able to contain text,
> resulting in xml that the following
>
> <X>
> <A/>
> <B/>
> <B/>
> child text of element X also possible
> </X?>
>
> I've got the following, but can't seem to figure out a way to also
> allow for a text child node. Any help appreciated
>
> <!ELEMENT X (A?, B*)>
You can't do this in XML, only in SGML.
The content model in SGML would be
<!element x - - (a?,b*,#pcdata)>
but the XML spec says in Mixed Content, #PCDATA must come first,
and in any case cannot be used in a sequence group.
Is there a reason why this model should use Mixed Content? It's
normally only used for text documents, where the separator is
the vertical bar, eg <!ELEMENT X (#PCDATA|A|B)*> (like HTML
paragraphs). As you need to provide 0/1 and 0/+ constraints,
it looks like you are modelling data, not text. In that case,
put the text in a container, eg
<!ELEMENT X (A?,B*,C)>
<!ELEMENT C (#PCDATA)>
This will be much more robust, easier to process, and avoids
any unpleasantness with line-ends in pernicious mixed content.
///Peter
--
XML FAQ: http://xml.silmaril.ie/
0
Peter
5/17/2006 9:08:55 PM
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Cheers, Greg =================== <xhtml:html xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" > <xhtml:head> <xhtml:title> Intermingled XHTML and SVG </xhtml:title> <xhtml:script type="text/javascript" language="JavaScript"><![CDATA[ function gogo(evt){ var targetObj = evt.target; //The object that received the event var targetDoc = targetObj.ownerDocument; //Owner document var wg = document.getElementById("SVGroot"); var lg = document.getElementById("labelz"); //label labelBox= targetDoc.createElementNS("http://www.w3.org/2000/svg", "svg:rect"); labelBox.setAttributeNS(null, "id", "label1"); labelBox.setAttributeNS(null, "fill", "red"); labelBox.setAttributeNS(null, "fill-opacity", 1); labelBox.setAttributeNS(null, "x", 700); labelBox.setAttributeNS(null, "y", 400); labelBox.setAttributeNS(null, "width", 200); labelBox.setAttributeNS(null, "height", 20); labelBox.setAttributeNS(null, "visibility", "visible"); lg.appendC... Getting kind of abstract text snippets from text nodes Hi everybody, I am about implementing a little search engine that searches a phrase over xml text nodes. I got that all working fine but what I want as the results is not the complete text of the textnode, I would like to make an abstract like result list (such output that you get with google searches. For eg .... I am the <b>substring</b> from a complete text node ... where "substring" is the search term. The problem is simple (I think): I want to extract all the text parts of the complete text node, where search searchterm is highlighted, surrounded by the text like 30 characters. I found an intersting post "cut down text" which is almost that what I am looking for, but there the text is just trimmed by x characters. Is anybody here, that has an "elegant" way to solve that or some hints that get me to the solution? I am not able to use regex (would be nice though) My parser is Sablotron so I am restricted to the functions that I get. (1.0). Any help is greatly appreciated. regards, Andreas W Wylach Think about dividing the text into three parts: before your target, the target itself, and after the target. Process each appropriately. If you want to report multiple instances within the same block of text, look at the standard examples of recursive text processing. -- () ASCII Ribbon Campaign | Joe Kesselman /\ Stamp out HTML e-mail! | System architexture and kinetic poetry "Andreas W. Wylach" <aw@ioc... XSL select only nodes which contain a specific child node Given the following extremely simplified XML... <AA> <BB></BB> <BB><CC>foo</CC></BB> <BB></BB> <BB><CC>bar</CC></BB> </AA> ....is there an easy way to use and XSL select to get only the <BB> nodes which have a <CC> child node? <xsl:for-each select="AA/BB"> Gives me 4 nodes: <BB></BB> <BB><CC>foo</CC></BB> <BB></BB> <BB><CC>bar</CC></BB> <xsl:for-each select="AA/BB/CC"> Gives me 2 nodes, but it's at the CC level and I need it to be at the BB level because there's a bunch of other nodes there that need to be processed as well. <CC>foo</CC> <CC>bar</CC> I'm thinking that there must be some sort of conditional select XPATH that will get me what I need: <BB><CC>foo</CC></BB> <BB><CC>bar</CC></BB> Any tips? William Krick wrote: > Given the following extremely simplified XML... > > <AA> > <BB></BB> > <BB><CC>foo</CC></BB> > <BB></BB> > <BB><CC>bar</CC></BB> > </AA> > > ...is there an easy way to use and XSL select to get only the <BB> > nodes which have a <CC> child node? > > <xsl:for-each select="AA/BB"> ... text-text Wondering how what I input to my UTF-8 terminal gets passed along through my patched [1] trn ... Cyrillic: А Б В Г Д Е Ж З И Й К Л М Н О П а б в г д е ж з и й к л м н о п IPA: ᴀ ᴁ ᴂ ᴃ ᴄ ᴅ ᴆ ᴇ ᴈ ᴉ ᴊ ᴋ ᴌ ᴍ ᴎ ᴏ ɀ Ɂ ɂ Ƀ Ʉ Ʌ Ɇ ɇ Ɉ ɉ Ɋ ɋ Ɍ ɍ Ɏ ɏ [1] https://groups.google.com/d/msg/comp.sys.raspberry-pi/7Z37Hdrm0DM/6aqD-reXFzAJ ... XML Schema for Node with attribute and text I've tried to find how to write a schema for this type of node everywhere. <Param name="FirstParam">ABCD</Param> How do i arrange a complex type that contains an attribute and a text. This : <xs:complexType name="Param"> <xs:attribute name="name" /> </xs:complexType> Result as : <invalid char="13" code="cvc-complex-type.1.2" line="3086" resource="file:///C:/WINNT/Profiles/svaillan/Desktop/XML Schema/test.xml">element Param must be empty but is not</invalid> This : <xs:complexType name="Param"> <xs:all> <xs:element name="text" type="xs:string" /> </xs:all> <xs:attribute name="name" /> </xs:complexType> Result as : <invalid char="13" code="cvc-complex-type.1.2.3" line="3086" resource="file:///C:/WINNT/Profiles/svaillan/Desktop/XML Schema/test.xml">text not allowed: |ABCD|</invalid> I just dont know where to add the unnamed <xs:element ... > tag. Anyone can help me figure this out? Thx In article <de3c2ad2.0311281339.e74b69b@posting.google.com>, Simon Vaillancourt <svaillancourt@mediagrif.com> wrote: % I've tried to find how to write a schema for this type of node % everywhere. % % <Param name="FirstParam">ABCD</Param> % % How do... text + text What is "text + text" supposed to do right now? It doesn't seem very useful to me. What about making "text + text" as an equivalent for "text || text"? Most strongly-typed programming languages do this. And MS SQL Server too, I think (CMIIW). -- dave ---------------------------(end of broadcast)--------------------------- TIP 1: subscribe and unsubscribe commands go to majordomo@postgresql.org Am Freitag, 8. Oktober 2004 12:57 schrieb David Garamond: > What is "text + text" supposed to do right now? Nothing. > What about making "text + text" as an equivalent for "text > || text"? Most strongly-typed programming languages do this. And MS SQL > Server too, I think (CMIIW). What would this gain except for bloat? It's not like SQL is utterly compatible with any programming language; users will still have to learn all the operators anyway. -- Peter Eisentraut http://developer.postgresql.org/~petere/ ---------------------------(end of broadcast)--------------------------- TIP 9: the planner will ignore your desire to choose an index scan if your joining column's datatypes do not match Peter Eisentraut wrote: >>What is "text + text" supposed to do right now? > > Nothing. Then are these bugs? (7.4.5 and 8.0.0beta1 give same results). Frankly, the current behaviour is quite strange to me. ------------------ =... Problems with getting text from node (xalan) I have a node that I'd call a text node, but the parser disagrees. At any rate, I'd like to get the text out of it. (It is some text with infrequent markup. I'm using Xalan and Java. getNodeValue() returns null, and isTextNode(n) returns false. Here is the only code I've been able to use to get the text out: ByteArrayOutputStream baotc = new ByteArrayOutputStream(); StreamResult out = new StreamResult(new OutputStreamWriter(baotc)); serializer.transform(new DOMSource(n), out); String tt = baotc.toString(); It actually gives the text but has a jarbled(to me) footer and header. I'm doing something wrong of course. Thanks for your help. LNMEgo (btw, I did post this on the forums at sun, but didn't get a response. so, yes I did cross post, but that forum is pretty unresponsive. thanks) I realized that it was due to another line of code that I have since fixed. Thanks to anyone who had tried to think of a solution. -LNMEgo LNMEgo@hotmail.com (Jeff) wrote in message news:<ce9e21d9.0309202308.25c42a2a@posting.google.com>... > I have a node that I'd call a text node, but the parser disagrees. At > any rate, I'd like to get the text out of it. (It is some text with > infrequent markup. > > I'm using Xalan and Java. getNodeValue() returns null, and > isTextNode(n) returns false. Here is the only code I've been able to > use to get the text out: > > ByteArrayOutputStream baotc = new ByteArr... editor for editing xml text nodes We have xml documents which contain text imbedded as text content in some xml structures. These documents need to be translated by human translators from English into, for example, Chinese. We would like the translator to use an editor which shows him only the text content of the xml document, or gives him only editing access to the text content. The translator should be unable to touch or modify the xml tags, so that it is guaranteed that the xml-tags are unchanged when the translator returns the translated document. The translator simply replaces all English text accessible to him into Chinese text. Does anyone know an xml editing tool which has this feature, of allowing only modification to the text node content? Alois Zreindl wrote: > We have xml documents which contain text imbedded as text content in > some xml structures. > > These documents need to be translated by human translators from English > into, for example, Chinese. > > We would like the translator to use an editor which shows him only the > text content of the xml document, or gives him only editing access to > the text content. I am working on exactly this right now. It leaves the elements in element content inviolable, but allows element markup in mixed content (the only thing a translator should be touching). Please contact me by email to discuss this further. > The translator should be unable to touch or modify the xml tags, so that > it is guara... Web resources about - problem with child text node when constraining other child node types - comp.text.xml Are you constraining your back office performance? 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https://www.groundai.com/project/a-geometric-approach-to-the-distribution-of-quantum-states-in-bipartite-physical-systems/ | A geometric approach to the distribution of quantum states in bipartite physical systems
# A geometric approach to the distribution of quantum states in bipartite physical systems
J Batle Departament de Física, Universitat de les Illes Balears, 07122 Palma de Mallorca, Balearic Islands, Europe
July 5, 2019July 5, 2019
July 5, 2019July 5, 2019
###### Abstract
Any set of pure states living in an given Hilbert space possesses a natural and unique metric –the Haar measure– on the group of unitary matrices. However, there is no specific measure induced on the set of eigenvalues of any density matrix . Therefore, a general approach to the global properties of mixed states depends on the specific metric defined on . In the present work we shall employ a simple measure on that has the advantage of possessing a clear geometric visualization whenever discussing how arbitrary states are distributed according to some measure of mixedness. The degree of mixture will be that of the participation ratio and the concomitant maximum eigenvalue . The cases studied will be the qubit-qubit system and the qubit-qutrit system, whereas some discussion will be made on higher-dimensional bipartite cases in both the -domain and the -domain.
###### pacs:
03.65.Ud; 03.67.Bg; 03.67.Mn; 89.70.Cf
## I Introduction
The amount of entanglement and the purity of quantum states of composite systems exhibit a dualistic relationship. As the degree of mixture increases, quantum states tend to have a smaller amount of entanglement. In the case of two-qubits systems, states with a large enough degree of mixture are always separable ZHS98 (). A detailed knowledge of the relation between the degree of mixture and the amount of entanglement is essential in order to understand the limitations that mixture imposes on quantum information processes such as quantum teleportation or quantum computing. To study the relationship between entanglement and mixture we need quantitative measures for these two quantities. The entanglement of formation provides a natural quantitative measure of entanglement with a clear physical motivation. As for mixedness, there are several measures of mixture that can be useful within the present context. The von Neumann measure
S=−Tr(^ρln^ρ), (1)
is important because of its relationship with the thermodynamic entropy. On the other hand, the so called participation ratio,
R(^ρ)=1Tr(^ρ2), (2)
is particularly convenient for calculations ZHS98 (); MJWK01 (). Another measure for mixedness can be found in the maximum eigenvalue of a density matrix , which is in turn a monotonically increasing function for the Renyi entropy .
Given a particular way to explore the space of both pure and mixed states in bipartite systems, it is possible to provide a clear physical geometric insight into the problem of how states distribute according to their degree of mixture, which is the main subject of the present work.
This paper is organized as follows: in Section II we introduce the nature of the measures defined on the set of eigenvalues and unitary matrices. Section III analyzes 2x2 systems and how their concomitant states are distributed according to and . Section IV provides a further analytic approach for 2x3 and additional bipartite systems. In Section V we discuss the implications of the non-uniqueness of a general measure for mixed states and the corresponding geometric implications into our problem. Finally, some conclusions are drawn in Section VI.
## Ii Measures on the set of eigenvalues and unitary matrices
In order to perform a survey of the properties of arbitrary (pure and mixed) states of the concomitant state-space , it is necessary to introduce an appropriate measure on this space. Such a measure is needed to compute volumes within , as well as to determine what is to be understood by a uniform distribution of states on . The natural measure that we are going to adopt here was first considered in Refs. ZHS98 (); Z99 (). An arbitrary (pure or mixed) state of a quantum system described by an -dimensional Hilbert space can always be expressed as the product of three matrices,
ρ=UD[{λi}]U†. (3)
Here is an unitary matrix and is an diagonal matrix whose diagonal elements are , with , and . The group of unitary matrices is endowed with a unique, uniform measure: the Haar measure PZK98 (). On the other hand, the -simplex , consisting of all the real -uples appearing in (3), is a subset of a -dimensional hyperplane of . Consequently, the standard normalized Lebesgue measure on provides a natural measure for . The aforementioned measures on and lead then to a natural measure on the set of all the states of our quantum system ZHS98 (); Z99 (); PZK98 (), namely,
μ=ν×LN−1. (4)
All our present considerations are based on the assumption that the uniform distribution of states of a quantum system is the one determined by the measure (4). Thus, in our numerical computations, we are going to randomly generate states according to the measure (4). The quantities computed with a Monte Carlo procedure have an associated error which is on the type , where is the number of generated states, is the value corresponding to the Student distribution with degrees of freedom, computed with a certain desired accuracy , and is the usual computed standard deviation. Therefore, if we seek a result with an error say less than units, we have to generate a number of points around 10 or 100 million. If not stated explicitly, from now on all quantities computed are exact up to the last digit.
The applications that have appeared so far in quantum information theory, in the form of dense coding, teleportation, quantum cryptography and specially in algorithms for quantum computing (quantum error correction codes for instance), deal with finite numbers of qubits. A quantum gate which acts upon these qubits or even the evolution of that system is represented by a unitary matrix , with being the dimension of the associated Hilbert space . The state describing a system of qubits is given by a hermitian, positive-semidefinite () matrix, with unit trace. In view of these facts, it is natural to think that an interest has appeared in the quantification of certain properties of these systems, most of the times in the form of the characterization of a certain state , described by matrices of finite size. Natural applications arise when one tries to simulate certain processes through random matrices, whose probability distribution ought to be described accordingly.
### ii.1 The Haar measure
As stated before, in the space of pure states, with , there is a natural candidate measure, induced by the Haar measure on the group of unitary matrices. In mathematical analysis, the Haar measure Haar33 () is known to assign an “invariant volume” to what is known as subsets of locally compact topological groups. In origin, the main objective was to construct a measure invariant under the action of a topological group Mehta90 (). Here we present the formal definition Conway90 (): given a locally compact topological group (multiplication is the group operation), consider a -algebra generated by all compact subsets of . If is an element of and is a set in , then the set also belongs to . A measure on will be letf-invariant if for all and . Such an invariant measure is the Haar measure on (it happens to be both left and right invariant). In other words Haarsimetria (), the Haar measure defines the unique invariant integration measure for Lie groups. It implies that a volume element d is identified by defining the integral of a function over as , being left and right invariant
∫Gf(g−1x)dμ(x)=∫Gf(xg−1)dμ(x)=∫Gf(x)dμ(x). (5)
The invariance of the integral follows from the concomitant invariance of the volume element d. It is plain, then, that once d is fixed at a given point, say the unit element , we can move performing a left or right translation. Suppose that the map defines the action of a left translation. We have , with being the coordinates in the vicinity of . Assume, also, that dd defines the volume element spanned by the differentials d, d, …, d at point . It follows then that the volume element at point is given by ddd, where is the Jacobian of the previous map evaluated at the unit element : . In a right or left translation, both dd and are multiplied by the same Jacobian determinant, preserving invariance of d. The Lie groups also allow an invariant metric and d is just the volume element dd.
We do not gain much physical insight with these definitions of the Haar measure and its invariance unless we identify with the group of unitary matrices , the element with a unitary matrix and with subsets of the group of unitary matrices , so that given a reference state and a unitary matrix , we can associate a state to . Physically what is required is a probability measure invariant under unitary changes of basis in the space of pure states, that is,
P(N)Haar(U|Ψ⟩)=P(N)Haar(|Ψ⟩). (6)
These requirements can only be met by the Haar measure, which is rotationally invariant.
## Iii Analytical approach for 2x2 systems. Generation of states
The two-qubits case () is the simplest quantum mechanical system that exhibits the feature of quantum entanglement. The relationship between entanglement and mixedness has been described intensively in the literature. One given aspect is that as we increase the degree of mixture, as measured by the so called participation ratio Tr[], the entanglement diminishes (on average). As a matter of fact, if the state is mixed enough, that state will have no entanglement at all. This is fully consistent with the fact that there exists a special class of mixed states which have maximum entanglement for a given MJWK01 () (the maximum entangled mixed states MEMS). These states have been recently reported to be achieved in the laboratory MEMSexp () using pairs of entangled photons. Thus for practical or purely theoretical purposes, it may happen to be relevant to generate mixed states of two-qubits with a given participation ratio . It may represent an excellent tool in the simulation of algorithms in a given quantum circuit: as the input pure states go through the quantum gates, they interact with the environment, so that they become mixed with some . This degree of mixture , which varies with the number of iterations, can be used as a probe for the evolution of the degradation of the entanglement present between any two qubits in the circuit. Different evolutions of the degree of mixture on the output would shed some light on the optimal architecture of the circuit that has to perform a given algorithm.
Here we describe a numerical recipe to randomly generate two-qubit states, according to a definite measure, and with a given, fixed value of . Suppose that the states are generated according to the product measure (4), where is the Haar measure on the group of unitary matrices and the Lebesgue measure on provides a reasonable measure for the simplex of eigenvalues of . In this case, the numerical procedure we are about to explain owes its efficiency to the following geometrical picture which is valid only if the states are supposed to be distributed according to measure (4). We shall identify the simplex with a regular tetrahedron of side length 1, in , centred at the origin. Let stand for the vector positions of the tetrahedron’s vertices. The tetrahedron is oriented in such a way that the vector points towards the positive -axis and the vector is contained in the -semiplane corresponding to positive -values. The positions of the tetrahedron’s vertices correspond to the vectors
r1 = (−12√3,−12,−14√23) r2 = (1√3,0,−14√23) r3 = (−12√3,12,−14√23) r4 = (0,0,34√23). (7)
The mapping connecting the points of the simplex (with coordinates ) with the points within tetrahedron is given by the equations
λi = 2(r⋅ri)+14i=1,…,4, (8) r = 4∑i=1λiri (9)
The degree of mixture is characterized by the quantity . This quantity is related to the distance to the centre of the tetrahedron by
r2=−18+124∑i=1λ2i. (10)
Thus, the states with a given degree of mixture lie on the surface of a sphere of radius concentric with the tetrahedron . To choose a given is tantamount to define a given radious of the sphere. There exist three different possible regions (see Fig. 1):
• region I: (), where is the radius of a sphere tangent to the faces of the tetrahedron . In this case the sphere lies completely within the tetrahedron . Therefore we only need to generate at random points over its surface. The cartesian coordinates for the sphere are given by
x1 = rsinθcosϕ x2 = rsinθsinϕ x3 = rcosθ, (11)
Denoting rand_u() a random number uniformly distributed between 0 an 1, the random numbers rand_u() and rand_u() (its probability distribution being ) define an arbitrary state on the surface inside . The angle is defined between the centre of the tetrahedron (the origin) and the vector , and any point aligned with the origin. Substitution of in (8) provides us with the eigenvalues of , with the desired as prescribed by the relationship (10). With the subsequent application of the unitary matrices we obtain a random state distributed according to the usual measure .
• region II: (), where denotes the radius of a sphere which is tangent to the sides of the tetrahedron . Contrary to the previous case, part of the surface of the sphere lies outside the tetrahedron. This fact means that we are able to still generate the states as before, provided we reject those ones with negative weights .
• region III: (), where is the radius of a sphere passing through the vertices of . The generation of states is a bit more involved in this case. Again rand_u(), but the available angles now range from to . It can be shown that results from solving the equation . Thus, , with rand_u(). Some states may be unacceptable () still, but the vast majority are accepted.
Combining these three previous regions, we are able to generate arbitrary mixed states endowed with a given participation ratio .
### iii.1 R-domain
In this case the degree of mixture is characterized by the quantity . This quantity is related to the distance to the centre of the tetrahedron by
r2=−18+12ω2. (12)
Thus, the states with a given degree of mixture lie on the surface of a sphere of radius concentric with the tetrahedron .
The volume associated with states endowed with a value of lying within a small interval is clearly associated with the volume of the subset of points in whose distances to the centre of are between and , with . Let denote the sphere of radius concentric with . The volume is then proportional to the area of the part of which lies within . In order to compute the aforementioned area, it is convenient to separately consider three different ranges for the radius .
Let us first consider the range of values , where is the radius of a sphere tangent to the faces of the tetrahedron . In this case the sphere lies completely within the tetrahedron . Thus, the area we are interested in is just the area of the sphere,
AI(r)=4πr2. (13)
We now consider a second range of values of the radius, , where denotes the radius of a sphere which is tangent to the sides of the tetrahedron . In this case, the area of the portion of which lies within is
AII(r)=4π[r2−2r(r−h1)]. (14)
Finally, we consider the range of values , where is the radius of a sphere passing through the vertices of . This cas is, by far, the most intricate one, where many methods borrowed from spherical trigonometry are employed. In this case the area of the part of the sphere lying within is
AIII(r)=4(SA−3SB), (15)
where
SA = r2(3β−π), (16) SB = r2⎡⎢⎣h(−π+2sin−1(C1C2))+2sin−1 ⎷1−C21C221+C22⎤⎥⎦. (17)
The quantities appearing in the right hand sides of the above expressions are defined by
β=cos−1[cosA−cos2Asin2A];A=2sin−1(D1/r);D1=12(12−√r2−18), (18)
and
h = h1/r; C1=h√1−h2; C2=CB√1−C2B; (19) CB = ⎷D22−D21r2−D21; D2=r√1−h2. (20)
Using the relation between and the participation rate ,
r2=−18+12R, (21)
we analytically obtained the probability of finding a quantum state with a participation rate ,
F(R)=f(r)∣∣∣drdR∣∣∣, (22)
where , and is given by equations (13-15). The distribution was first determined numerically by Zyczkowski et al. in ZHS98 (). Here we compute analytically BCPP02b and, as expected, the calculations coincide with the concomitant numerical results and the ones reported in ZHS98 ().
### iii.2 λm-domain
Coming back to two-qubits, the quantity is not appropriately suited to discuss the limit . However, does exhibit a nice behaviour when . Indeed, we have
limq→∞(Trρq)1/q=limq→∞(∑ipqi)1/q=λm, (23)
where
λm=maxi{pi} (24)
is the maximum eigenvalue of the density matrix . Hence, in the limit , the -entropies (when properly behaving) depend only on the largest eigenvalue of the density matrix. For example, in the limit , the Rényi entropy reduces to
S(R)∞=−ln(λm). (25)
It is worth realizing that the largest eigenvalue itself constitutes a legitimate measure of mixture. Its extreme values correspond to (i) pure states () and (ii) the identity matrix (). It is also interesting to mention that, for states diagonal in the Bell basis, the entanglement of formation is completely determined by (This is not the case, however, for general states of two-qubits systems).
In terms of the geometric representation of the simplex , the set of states with a given value of their maximum eigenvalue is represented by the tetrahedron determined by the four planes
λm=2(r⋅ri)+14,i=1,…,4. (26)
The four vertices of this tetrahedron are given by the intersection points of each one of the four possible triplets of planes that can be selected among the four alluded to planes.
For the accessible states with a given degree of mixture are on the surface of a small tetrahedron concentric with the tetrahedron . We are going to characterize each tetrahedron (representing those states with a given value of ) by the distance between (i) the common centre of and and (ii) each vertex of . The volume associated with states with a value of belonging to a given interval is proportional to the area of the portion of lying within .
Following a similar line of reasoning as the one pursued in the case , we consider three ranges of values for . The first range of -values is given by . The particular value corresponds to a tetrahedron whose vertices are located at the centres of the faces of . Within the aforementioned range of -values, is lies completely within . Consequently, coincides with the area of ,
AI(l)=24√3l2. (27)
The second range of -values corresponds to . The area of the part of lying within is now
AII(l)=3√3[8l2−32(3l−h1)2] (28)
Finally, the third range of -values we are going to consider is . In this case we have
AIII(l)=32√3(3h1−l)2 (29)
In a similar way as in the case, the above expressions for lead to the analytical form of the probability (density) of finding a two-qubits state with a given value of its greatest eigenvalue,
F(λm)=A(l)Volume[TΔ]∣∣∣dldλm∣∣∣. (30)
Remarkably enough, as tends to infinity all discontinuities in the derivative of disappear. In the -domain the distribution is completely smooth, as opposed to the -domain.
## Iv Analytical approach for 2x3 systems and higher systems
### iv.1 R-domain
Previously, we obtained the distribution vs. for two-qubits (), under the assumption that they are distributed according to measure (4). Through a useful analogy, we have mapped the problem into a geometrical one in regarding interior and common sections of two geometrical bodies. In the previous case we saw that the main difficulty lies in the third region, where the region of the growing sphere inside the tetrahedron is not described by a spherical triangle. The extension to higher dimensions, however, requires a thorough account of the geometrical tools required, but still it is in principle possible. So, one can find the distribution of states according to basically by computing the surface area of a growing ball of radius in dimensions (sphere) that remains inside an outer regular -polytope (tetrahedron) of unit length, excluding the common regions. A -dimensional sphere can be parameterized in cartesian coordinates
x1 = rsin(ϕ1)sin(ϕ2)sin(ϕ3)...sin(ϕN−3)sin(ϕN−2) x2 = rsin(ϕ1)sin(ϕ2)sin(ϕ3)...sin(ϕN−3)cos(ϕN−2) x3 = rsin(ϕ1)sin(ϕ2)sin(ϕ3)...cos(ϕN−3) ... xN−2 = rsin(ϕ1)cos(ϕ2) xN−1 = rcos(ϕ1), (31)
with the domains for and . The definition of the -polytope then is required. This problem is not trivial at all, because new geometrical situations appear in the intersection of these two bodies. In point of fact there are intermediate regimes between and appearing at integer values of (recall the previous two-qubits case), where a change in the growth of interior hyper-surfaces occurs (at the values ). In any case we can always generate random states in arbitrary dimensions and compute the corresponding distributions. This is done in Fig. 2 for several cases. The relation (21) is generalized to dimensions in the form
r2=−12N+12R. (32)
The distribution can be obtained analytically
FN(R)∼1R2[1R−1N]N−32 (33)
which has been numerically checked. The particular form of for arbitrary is difficult to obtain, but nevertheless one can obtain quantitative results for asymptotic values of . It may be interesting to know the position of the maximum of or the mean value , which turns to be Zyckasimp () for states generated according to (4). There is the so called Borel lemma Borel () in discrete mathematics that asserts that (translated to our problem) when you grow a -ball inside , from the moment that it swallows, say, 1/2 of the volume of it, then the area outside drops very quickly with further grow. So the maximum intersection with the sphere should be approximately for the radious where the volume of the ball equals that of the -polytope . The usual formulas for the volumes of ()-dimensional spheres and regular unit -polytopes are and , respectively.
It is then that we can assume that the position of such that is maximal. Substituting in (32), and after some algebra, we obtain the beautiful result
limN→∞1/R(r∗)1/N=2π+e2π≃1.43. (34)
In other words, for large .
We must emphasize that this type of distributions are “degenerated” in some cases, that is, different systems may present identical distributions (for instance, there is nothing different from this perspective between and systems). We do not know to what extend these distributions are physically representative of such cases, as far as entanglement is concerned. What is certain is that all states with possess a positive partial transpose. In point of fact, they are indeed separable, as shown in balls (). We merely mean by this that a state close enough to the maximally mixed state is always separable. In other words, states lying on -spheres with radius are always separable.
### iv.2 λm-domain
When regarding the maximum eigenvalue as a proper degree of mixture one is able to find a geometrical picture analogue to the one of the growing sphere. In that case a nested inverted tetrahedron grows inside the outer tetrahedron representing the simplex of eigenvalues . The generalization to higher bipartite systems is similar to the -case, but far much easier to implement mathematically. As in that case, we have a high degree of symmetry in the problem. The advantage is that one does not deal with curved figures but perfectly flat and sharp surfaces instead. This fact makes the general problem more approachable.
We have seen that the problem of finding how the states of a bipartite quantum mechanical system are distributed according to their degree of mixedness can be translated to the realm of discrete mathematics. If we consider our measure of mixedness to be the maximum eigenvalue of the density matrix and the dimension of our problem to be , we compute the distribution of states in arbitrary dimensions by letting an inner regular -polytope to grow inside an outer unit length -polytope , the vertices of the former pointing towards the centre of the faces of the latter. In fact, it can be shown that the radius of the maximum hypersphere that can be inscribed inside the inner polytope is directly related to .
By computing the surface area of strictly inside , we basically find the desired probability (density) of finding a state with maximum eigenvalue in dimensions.
To fix ideas, it will prove useful first to define the vertices of and . In fact it is essential, because we need to deal with elements of cartesian geometry in -dimensions. This vectors are given as
→r1 = (−12,−12√3,−14√23,...,−1N−1√N−12N) →r2 = (12,−12√3,−14√23,...,−1N−1√N−12N) →r3 = (0,1√3,−14√23,...,−1N−1√N−12N) →r4 = (0,0,34√23,..,−1N−1√N−12N) ... →rN = (0,0,0,...,√N−12N), (35)
with being the distance from the center to any vertex of this regular -polytope of unit length. One can easily check that , as required. This particular choice for the position of the vertices of this -simplex is such that it simplifies going from one dimension to the next by adding a new azimuthal axis each time. This vectors comply with the relations
→ri⋅→rj = −12N+12δij, (36) λm = 2(→r⋅→ri)+1N,i=1...N, (37)
where the last equation is the general form of (26).
Once we have a well defined , to know the coordinates of is straightforward. In fact, is the reciprocication (see Sommer ()) of . This means that the coordinates of are obtained by reversing the sign of the ones of , multiplied by a suitable factor (which can be shown to be , with defined as the length between the centre of to the centre of any of its faces, which in turn points towards the vertices of ). Thus, we can relate with through a general (26)-relation , such that .
Several distributions are obtained numerically by generating random states according to (4) in Fig. 3. It becomes apparent that as grows, the distributions are biased towards , in absolute agreement with the result (34).
As in the -case, is distributed into regions separated at fixed values of . The general recipe for obtaining is tedious and long, but some nice general results are obtained. The -distributions for the ranges a) and b) are general and read
FI(λm) = κN(N−2)!√N−12N−2[√2N(N−1)l(λm)]N−2, (38) FLast(λm) = κN(N−2)!√N−12N−2[√N−12N−l(λm)√N2(N−1)]N−2, (39)
respectively, where is introduced for convenience.
For the qubit-qutrit system, we have . Defining and , in addition to the previous regions (38) we obtain
FII(λm) = κ[FI(λm)/κ−(N−1)N(N−2)!√N−12N−2[yi=1]N−2]; (40) FIII(λm) = κ[FII(λm)/κ+2954(N−1)N(N−2)!√N−12N−2[yi=2]N−2]; (41) FIV(λm) = κ[FIII(λm)/κ−23454(N−1)N(N−2)!√N−12N−2[yi=3]N−2], (42)
for , and , respectively. From the previous formulas one can infer a general induction procedure. Analytical results are in excellent agreement with numerical generations.
## V Non-uniqueness of a general measure for mixed states. Geometric implications
In Refs. ZHS98 (); Z99 (), a basic question regarding a natural measure for the set of mixed states was debated. As described in Secs. (7.1) and (9.1), it is know, the set of all states can be regarded as the cartesian product , where stands for the family of all complete sets of ortonormal projectors , ( being the identity matrix), and is the set of all real -tuples , with and . It is universally accepted to assume the Haar measure to be the one defined over , because of its rotationally-invariant properties. But when it turns to discuss an appropriate measure over the simplex , some controversy arises. In all previous considerations here, we have regarded the Lebesgue measure as being the “natural” one. But one must mention that Slater has argued sla1 (); sla2 () that, in analogy to the classical use of the volume element of the Fisher information metric as Jeffreys’ prior jef () in Bayesian theory, a natural measure on the quantum states would be the volume element of the Bures metric. The problem lies on the fact that there is no unique probability distribution defined over the simplex of eigenvalues of mixed states . In point of fact, the debate was motivated by the fact that the volume occupied by separable two-qubits states was found in ZHS98 () to be greater than () using the measure , something which is surprising.
One such probability distribution that is suitable for general considerations is the Dirichlet distribution Z99 ()
Pη(λ1,…,λN)=Cηλη−11λη−12...λη−1N, (43)
with being a real parameter and the normalization constant. This is a particular case of the more general Dirichlet distribution. The concomitant probability density for variables with parameters is defined by
Pη(λ1,…,λN)=Cηλη1−11λη2−12...ληN−1N, (44)
with , and , and . Clearly, distribution (44) generalizes (43). This distribution admits a clear interpretation. As known, the multinomial distribution provides a probability of choosing a given collection of items out of a set of items with repetitions, the probabilities being . These probabilities are the parameters of the multinomial distribution. The Dirichlet distribution is the conjugate prior of the parameters of the multinomial distribution.
A new measure then can be defined as , where denotes the simplex of eigenvalues distributed according to (43) (The Haar measure remains untouched). Thus, one clearly recovers the Lebesgue measure for (uniform distribution), and Slater’s argumentation reduces to take in (43). For one obtains a singular distribution concentrated on the pure states only, while for , the distribution peaks on the maximally mixed state . We will see shortly that changing the continuous parameter indeed modificates the average purity (as expressed in terms of ) of the generated mixed states.
In what follows we numerically generate mixed states whose eigenvalues are distributed following (43). This is done is order to tackle the dependence of relevant quantities on the parameter . Let us consider the way mixed states are distributed according to . We focus our attention on the two-qubits instance, but similar studies can be extended to arbitrary bipartite dimensions. As shown in Fig. 4, the distributions vs. are shown for (from left to right in this order) while Fig. 5 shows analogous distributions for the maximum eigenvalue for (from right to left). Notice the different shapes. We can no longer attribute a geometrical description to except for . In Z99 () for was first derived. Here we can provide different distributions for arbitrary -values.
A way to devise a certain range of reasonable -values is to study the average induced for every -distribution. This is performed in Fig. 6. The average -value and are plotted versus . (solid line) can only be computed numerically, but luckily (dashed line) is obtained in analytical fashion for all N
⟨Trρ2⟩N(η) = Cη∫10dλ1λη−11∫1−λ10dλ2λη−12...∫1−∑N−2i=1λi0dλN−1λη−1N−1 (46) (1−N−1∑i=1λi)η−1[N∑j=1λ2j]=[N−N−1η+1]−1.
The fact that matches exact results validates all our present generations. The actual value is slightly larger than for all values of , but both of them coincide for low and high values of the parameter . It is obvious from Fig. 6 that we cannot choose distributions that depart considerably from the uniform one , because in that case we induce probability distributions that favor high or low already.
Perhaps the best way is to go straight to the question that originated the controversy on the -measures: what is the dependency of the a priori probability of finding a two-qubits mixed state being separable? In Fig. 7 we depict vs. for states complying with PPT (lower curve) and those which violate the -entropic inequalities (upper curve). It seems reasonable to assume that a permissible range of -distributions belong to the interval , within which remains around the reference point .
However, in view of the previous outcomes we believe that the results obtained considering the uniform -distribution for the simplex (the one that allow us to exploit a simple geometric analogy) remains the most natural choice possible, independent of any form that one may adopt for a generic probability distribution.
## Vi Conclusions
We have introduced a geometric picture to obtain the probability of finding quantum states of two-qubits with a given degree of mixture (as measured by an appropriate function of ) is analytically found for and . In the latter case, the -entropies become functions of the statistical operator’s largest eigenvalue . In point of fact, itself constitutes a legitimate measure of mixture. During the derivation of the probability (density) distributions of finding a bipartite mixed state in arbitrary dimensions with a given degree of mixture, we saw that it is more convenient to use instead of . Finally, we have derived explicitly the distribution vs. for the physical meaningful case of a qubit-qutrit system ().
## Acknowledgements
J. Batle acknowledges partial support from the Physics Department, UIB. J. Batle acknowledges fruitful discussions with J. Rosselló, Maria del Mar Batle and Regina Batle.
## References
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• (6) M. L. Mehta, Random Matrices (Academic, New York, 1990).
• (7) J. Conway, Course in Functional Analysis (Springer-Verlag, New York, 1990).
• (8) M. Chaichian and R. Hagedorn, Symmetries in quantum mechanics, (Inst. of Phys. Publ., Bristol).
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https://math.stackexchange.com/questions/2359726/find-the-equation-of-the-polar | # Find the equation of the polar
Let $K$ be a circle with center $C=(x_0,y_0)$ and radius $r$. For each point $P=(p_1, p_2)$ outside the circle let $g_P$ be the line that passes through the intersection points of the tangent from $P$ at the circle and the circle.
I want to find the equation of the line $g_p$ (polar).
I have done the following:
Let $A=(a_1, a_2)$ and $B=(b_1, b_2)$ be the two intersection points.
Since $g_p$ passes through these two points the line has the equation of the form: $$g_P : y=\frac{b_2-a_2}{b_1-a_1}(x-a_1)+a_2$$
The points $A,B$ are on the circle, so we have the following two relations: $$(a_1-x_0)^2+(a_2-y_0)^2=r^2 \\ (b_1-x_0)^2+(b_2-y_0)^2=r^2$$
We also have that the triangle MBP and AMP are both right triangles so we could use the Pythagorian Theorem, or not?
EDIT:
The equation that we are looking for is \begin{equation*}y-a_2=\frac{b_2-a_2}{b_1-a_1}(x-a_1) \Rightarrow y=\frac{b_2-a_2}{b_1-a_1}(x-a_1)+a_2\end{equation*} Since from $A$ and $B$ the tangent passes through we have that \begin{equation*}(x-x_0)(a_1-x_0)+(y-y_0)(a_2-y_0)=r^2 \ \text{ und } \ (x-x_0)(b_1-x_0)+(y-y_0)(b_2-y_0)=r^2\end{equation*}
Since $P$ is on that tangent we have that
\begin{equation*}(p_1-x_0)(a_1-x_0)+(p_2-y_0)(a_2-y_0)=r^2 \ \text{ und } \ (p_1-x_0)(b_1-x_0)+(p_2-y_0)(b_2-y_0)=r^2\end{equation*}
Is everything correct so far?
• Do you have an idea how we could find the equation of the line? @Bernard – Mary Star Jul 15 '17 at 16:27
• Please seen my hint. You may check your answer from the answer provided by projective geometry. – Bernard Jul 15 '17 at 16:37
• Related: Construct tangent to a circle. Translating into algebra should be easy, particularly with Bernard's hint. – ccorn Jul 15 '17 at 16:50
• A higher-level approach would be to find the $3\times3$ coefficient matrix $M$ of the circle and to multiply it with $P$'s homogenous coordinate vector. The result is the polar's coefficient vector. There is a quickstart intro by user ja72 on that. – ccorn Jul 15 '17 at 16:53
• @MaryStar: Yes, it's correct, but the last to second equation should be better justified. – Bernard Jul 15 '17 at 19:14
To continue from where you’ve left off, you’re going to have to eliminate the $a$’s and $b$’s from your equations somehow. One (tedious) way to go might be to explicitly compute the two points of tangency and substitute the answer into the equations you have so far, and then see where that takes you. However, your observation in a comment that $\overline{CP}\perp\overline{AB}$ leads to a fairly straightforward solution that doesn’t involve explicitly computing the tangent points.
We know from your observation that the vector $P-C$ is normal to $\overline{AB}$, so an equation for the polar line is $$(P-C)\cdot(x,y)+c=0 \tag1$$ with $c$ some as-yet-to-be-determined constant. Using the formula for the distance from a line to a point, we have $$PM = {|(P-C)\cdot P+c|\over\|P-C\|}.\tag2$$ Now, $\triangle{PAC}\sim\triangle{PMA}$, so $PM:PA::PA:PC$, and thus (using the Pythagorean theorem) $$PM={PA^2\over PC}={\|P-C\|^2-r^2\over\|P-C\|}.\tag3$$ Combining equations (2) and (3) we have $${|(P-C)\cdot P+c|\over\|P-C\|}={\|P-C\|^2-r^2\over\|P-C\|}.\tag4$$ When the normal to the line points toward the point to which we’re measuring the distance, as we have here, the quantity inside the absolute value bars in (2) is nonnegative, so taking the absolute value on the left-hand side of (4) is superfluous. All that’s left is to solve for $c$ and plug that back into (1), which I’ll leave to you.
• An additional geometric observation: the point $M$ is the inversion of $P$ in $K$. – amd Jul 17 '17 at 2:42
• Thank you for your answer!! – Mary Star Jul 26 '17 at 23:39
Hint:
Solve the problem first for a circle centred at origin, and a point $P$ on the $x$-axis, with coordinates $(p,0)$.
Then deduce the general case by a change of coordinates.
Some details :
Consider a line through $P$ with (variable) slope $m$ and seek for the intersection points of this line with the circle. For that, write the equation in standard form: $$y=m(x-p_1)+p_2.$$ The line will be tangent to the circle if the equation for the abscissae of these intersection points has a double roots. As it is a quadratic equation, this means is discriminant is $0$. AS the discriminant is a quadratic polynomial in $m$, you'll get two roots, i.e. two tangents. Further more, as when an equation $ax^2+bx+c=0$ has a double root, this double root is $-b/2a$, so you obtain the abscissae of $A$ and $B$.
• Is the line that we are looking for always perpendicular to $PC$ ? I had now an idea, I will post in my initial post. – Mary Star Jul 15 '17 at 17:08
• I editet my initial post. Could you take a look at it? – Mary Star Jul 15 '17 at 17:13
Short answer: Here’s a different approach that might be beyond what you’ve studied so far. This problem is quite easy to solve using homogeneous coordinates—projective geometry—and the same method applies to any conic: If the equation of the conic is $\mathbf x^TC\mathbf x=0$, then the polar line of a point $\mathbf p$ is $C\mathbf p$.
For your circle, $$C=\begin{bmatrix}1&0&-x_0\\0&1&-y_0\\-x_0&-y_0&x_0^2+y_0^2-r^2\end{bmatrix}$$ so the polar line of $P$ is $$C[p_1,p_2,1]^T=[p_1-x_0,p_2-y_0,x_0^2-p_1x_0+y_0^2-p_2y_0-r^2]^T,$$ i.e., $$(p_1-x_0)x+(p_2-y_0)y=r^2-x_0^2-y_0^2+p_1x_0+p_2y_0.$$
Longer answer with derivation: Using homogeneous coordinates for points in $\mathbb R^2$, we can write the equation $ax+by+c=0$ of a line as $\mathbf x^T\mathbf l=0$, where $\mathbf l=(a,b,c)^T$ and $\mathbf x=(wx,wy,w)^T$, $w\ne0$. (Usually, we just set $w=1$ when converting from inhomogeneous (Cartesian) to homogeneous coordinates.) Similarly, the general equation of a conic $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ can be written in homogeneous coordinates as $\mathbf x^TC\mathbf x=0$, where $$C=\begin{bmatrix}A & \frac B2 & \frac D2 \\ \frac B2 & C & \frac E2 \\ \frac D2 & \frac E2 & F\end{bmatrix}.$$ Notice that $C$ is symmetric.
Let $\mathbf x$ be a point on the conic $C$. Then the tangent line $\mathbf l$ to the conic at $\mathbf x$ is given by $\mathbf l=C\mathbf x$. The point $\mathbf x$ clearly lies on $\mathbf l$, since $\mathbf x^T\mathbf l=\mathbf x^TC\mathbf x=0$. Now suppose $\mathbf y$ is another point of intersection of $C$ and $\mathbf l$. Then $\mathbf y^TC\mathbf y=0$ and $\mathbf y^T\mathbf l=\mathbf y^TC\mathbf x=0$, from which we have $(\mathbf x+\lambda\mathbf y)^TC(\mathbf x+\lambda\mathbf y)=0$ for all $\lambda$, which means that the entire line joining $\mathbf x$ and $\mathbf y$ lies on $C$, and so the conic is degenerate.
With the above lemma, we can show that for any point $\mathbf x$, the tangents to $C$ at its intersection with the line $\mathbf l=C\mathbf x$ intersect at $\mathbf x$, i.e., that $\mathbf l$ is the polar line of $\mathbf x$. To see this, let $\mathbf y$ be a point on $C$. Then the tangent to $C$ at $\mathbf y$ passes through $\mathbf x$ if $\mathbf x^TC\mathbf y=0$, but because of $C$’s symmetry this condition is equivalent to $\mathbf y^T(C\mathbf x)=0$, i.e., that $\mathbf y$ lies on the line $C\mathbf x$.
The equation of your circle is $(x-x_0)^2+(y-y_0)^2=r^2$, which expands into $$x^2+y^2-2x_0x-2y_0y+x_0^2+y_0^2-r^2=0$$ from which $$C=\begin{bmatrix}1&0&-x_0\\0&1&-y_0\\-x_0&-y_0&x_0^2+y_0^2-r^2\end{bmatrix}.$$ The rest is just a matter of expanding the equation $(x,y,1)\,C\,(p_1,p_2,1)^T=0$.
• @ccorn Thanks! Fixed. – amd Jul 16 '17 at 18:36 | 2019-07-19 01:24:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9300496578216553, "perplexity": 143.82826233990187}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525973.56/warc/CC-MAIN-20190719012046-20190719034046-00031.warc.gz"} |
http://math.stackexchange.com/questions/114021/cardinality-of-the-power-set-of-natural-numbers | # Cardinality of the power set of natural numbers
I was reading an article on infinite sets and I came across a proof about how the power set of natural numbers has a greater cardinality than the set of natural numbers. I know that both given that proof and the proof of Cantor's theorem, that it must be true. So, my question is why does encoding the each set of natural numbers in the power set using Gödel numbering not work?
Thanks.
-
How do you propose to assign Gödel numbers to arbitrary subsets of $\mathbb N$? The standard arithmization schemes only handle finite subsets of $\mathbb N$. – Henning Makholm Feb 27 '12 at 15:36
Not all subsets of $\mathbb N$ are definable. – azarel Feb 27 '12 at 15:38
You can find encodings, though not fully satisfactory ones, of the recursively enumerable subsets of the natural numbers, via the index of a Turing machine that recognizes the set. But encoding all subsets is ruled out by cardinality considerations. – André Nicolas Feb 27 '12 at 16:39
You can find encodings in the natural numbers of certain collections of subsets of $\mathbb{N}$. For example, as was pointed out by Henning Makholm, there is a very nice encoding of the collection of finite subsets of $\mathbb{N}$.
More generally, let $A$ be a recursively enumerable subset of $\mathbb{N}$. Let $T_A$ be a Turing machine that, on input $n$, halts if $n\in A$, and does not halt otherwise. Then we can encode $A$ by using the usual index of the machine $T_A$.
This encoding is not fully satisfactory. There are infinitely many Turing machines that halt on input $n$ iff $n\in A$. Thus every recursively enumerable set has infinitely many encodings. Moreover, there is no algorithm for telling, given two natural numbers, whether they encode the same set.
For more modest collections than the collection of recursively enumerable sets, there are far more satisfactory encodings. As a small and not very interesting example, consider the collection of subsets of $\mathbb{N}$ that are either finite or cofinite (their complement is finite). A small modification of the encoding of finite subsets will take care of the finites and cofinites. Basically, just use the even natural numbers for the finites. If you have used $2k$ to encode a finite, use $2k+1$ to encode its complement.
In addition, the elements of any countably infinite subset of the power set of $\mathbb{N}$ can, by the definition of countably infinite, be encoded using the natural numbers. However, by cardinality considerations, we cannot encode all the subsets of $\mathbb{N}$ using elements of $\mathbb{N}$. | 2015-05-26 14:24:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9129090309143066, "perplexity": 174.96381882151795}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928864.16/warc/CC-MAIN-20150521113208-00225-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/226922/how-do-i-modify-the-placement-of-the-prime-symbol | # How do I modify the placement of the prime symbol?
Some colleagues and I are writing a series of math notes and the house style (not up to us) is to use the Gotham Book family of fonts. So our math italic font is Gotham Book Italic.
We're noticing that some of the placement of symbols is less than optimal. See for instance the MWE below:
\documentclass{article}
\usepackage{amsmath}
\usepackage{mathspec}
\setmainfont{Gotham Book}
\setmathrm{Gotham Book}
\setmathfont(Digits,Latin){Gotham Book}
\begin{document}
\begin{align*}
f'(x) &= x^2 \\
f\,'(x) &= x^2 \\
\end{align*}
\end{document}
In the first line the prime symbol is crashing into the f. Adding a thin space (second line) makes it look almost right, but obviously we don't want to go adding \,s throughout the code.
Is there a way to automatically add space to the f in this font to prevent the crashing?
• Define your own prime \newcommand\myprime{\,'} Feb 6 '15 at 18:35
## 2 Answers
You're forgetting the proper mathspec way:
\documentclass{article}
\usepackage{amsmath}
\usepackage{mathspec}
\setmainfont{Gotham Book}
\setmathrm{Gotham Book}
\setmathfont(Digits,Latin){Gotham Book}
\begin{document}
\begin{align*}
"f'(x) &= x^2 \\
f'(x) &= x^2 \\
f\,'(x) &= x^2
\end{align*}
\end{document}
With the italic font (that I managed to find in the meanwhile):
\documentclass{article}
\usepackage{amsmath}
\usepackage{mathspec}
\setmainfont{Gotham Book}
\setmathrm{Gotham Book}
\setmathfont(Digits,Latin){Gotham Book}
\begin{document}
\begin{align*}
"f'(x) &= x^2 \\
"h'(x) &= x^2 \\
"g'(x) &= x^2
\end{align*}
\end{document}
• Ah, no I didn't forget; I've never known that syntax! I had trouble finding it in mathspec's documentation, too. But there it is, in the code comments: put the " before a character, e.g $"f$, and the character is printed with kerns on either side. Feb 6 '15 at 18:45
• @egreg -- matthew's visual shows the italic "f"; yours is upright. i think something has been omitted from matthew's code. one could remedy the spacing by creating a virtual font for the gotham book italic, with "proper" italic corrections, but it would be a real pain. Feb 6 '15 at 19:26
• @barbarabeeton I only found the upright version. Now it's with italics. Feb 6 '15 at 20:33
Use ^{\prime}, e.g.
\documentclass{article}
\usepackage{amsmath}
\usepackage{mathspec}
\begin{document}
\begin{align*}
&f^{\prime}(x) = x^2\\
&f^{\prime \prime}(x) = 2 x
\end{align*}
\end{document}
• Typing $f'(x)$ or $f^{\prime}(x)$ produces exactly the same result. Feb 7 '15 at 10:20 | 2021-09-25 03:41:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9981547594070435, "perplexity": 3309.2525015582464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057589.14/warc/CC-MAIN-20210925021713-20210925051713-00184.warc.gz"} |
https://www.physicsforums.com/threads/derivative-of-e-2.389173/ | # Derivative of e^2
## Homework Statement
Derivative of f(x) = x3 + e2
Dex = ex
D constant = 0
## The Attempt at a Solution
f'(x) = 3x2 + 0?
Is e2 treated as a constant?
## Homework Statement
Derivative of f(x) = x3 + e2
Dex = ex
D constant = 0
## The Attempt at a Solution
f'(x) = 3x2 + 0?
Is e2 treated as a constant?
Yes. Or you can use the chain rule. if u = f(x) = 2 and y = g(u) = $e^u$ then
$$\frac {dy} {dx} = \frac {dy} {du} \frac {du} {dx}$$
since $$\frac {du} {dx} = 0$$ $$\frac {dy} {dx} = 0$$.
Mark44
Mentor
## Homework Statement
Derivative of f(x) = x3 + e2
Dex = ex
D constant = 0
## The Attempt at a Solution
f'(x) = 3x2 + 0?
Is e2 treated as a constant?
Not only is it treated as a constant, it is a constant. The derivative of any constant is zero. Period.
Using the chain rule certainly works, but it's definitely overkill, so not recommended. | 2021-04-18 00:41:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6437012553215027, "perplexity": 2020.5174347382645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038464065.57/warc/CC-MAIN-20210417222733-20210418012733-00274.warc.gz"} |
http://mathhelpforum.com/calculus/100654-cant-evaluate-integral-print.html | # Can't evaluate this integral
• September 5th 2009, 11:05 AM
synclastica_86
Can't evaluate this integral
I can't follow a proof in my physics class. It involves this:
$\int^{2pi}_0 e^{i(n+1)x}dx$
I am suppose to get 2pi for n=-1 and 0 otherwise.
I got:
$\frac{e^{i(n+1)x}}{i(n+1)}$
Using euler's formula I got:
$\frac{sin[(n+1)x]}{n+1}-i \frac{[cos(n+1)x]}{n+1}$
Pluging in the limits I only got 0 regardless of what n is.
What am I missing?
• September 5th 2009, 11:12 AM
Sampras
For n = -1, the integral is $\int_{0}^{2 \pi} 1 \ dx = 2 \pi$.
• September 5th 2009, 11:27 AM
synclastica_86
OMG........thank.... I spent so long working blind....... I was trying to plug in n after I do the work. | 2015-08-28 06:29:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.969959020614624, "perplexity": 3357.566601977256}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644060413.1/warc/CC-MAIN-20150827025420-00181-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://mathoverflow.net/feeds/user/3074 | User mnf - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T01:47:55Z http://mathoverflow.net/feeds/user/3074 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11074/partial-sums-of-multiplicative-functions/11174#11174 Answer by mnf for Partial sums of multiplicative functions mnf 2010-01-08T19:43:03Z 2010-01-08T19:43:03Z <p>Here is a largely historical remark: </p> <p>Littlewood showed that there exists $x$ so that $\pi(x)$ is greater than the log-integral $\mathrm{li}(x)$. In fact, $\pi(x) - \mathrm{li}(x)$ is (more or less) a linear combination of factors $x^{1/2+it}$, where $1/2+it$ are zeroes of $\zeta$. Form the multiplicative convolution with a suitable smooth function: you can thereby make the sum over only finitely many zeroes. Now find (by Dirichlet) some $x$ for which all the numbers $i t \log(x)$ are near multiples of $2 \pi$, etc. </p> <p>This shows that $\pi(x) - \mathrm{li}(x)$ is not bounded by $C \sqrt{x}$ for any $C$; in the case of Mobius, you get some $C$ but not any $C$, because it's harder to understand the coefficients of $x^{1/2+it}$. </p> <p>Finally, there are some results valid for any completely multiplicative function: See the paper by A. Granville and K. Soundararajan entitled "The spectrum of multiplicative functions." </p> | 2013-05-25 01:47:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9630633592605591, "perplexity": 682.3210359175492}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368705310619/warc/CC-MAIN-20130516115510-00051-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://mathhelpforum.com/latex-help/123042-we-now-seem-have-fraktur-font-print.html | # We now seem to have fraktur font!
• Jan 9th 2010, 11:44 AM
CaptainBlack
We now seem to have fraktur font!
It seems we now have \mathfrak. I have wanted this font for the symbol for the FT for some time and seem to recall it was not available. So when did we acquire it?
$\mathfrak{F}$, $\mathfrak{c}$
CB
• Jan 9th 2010, 12:02 PM
pomp
Quote:
Originally Posted by CaptainBlack
It seems we now have \mathfrak. I have wanted this font for the symbol for the FT for some time and seem to recall it was not available. So when did we acquire it?
$\mathfrak{F}$, $\mathfrak{c}$
CB
I was only yesterday trying to work out how to produce this font in Latex to use on another forum, gave up in the end.
$\mathfrak{L}_0$
I've also just realised that all these years of lecture notes I've read a mathfrak 'A' as a 'U'
$\mathfrak{A}$
• Jan 9th 2010, 12:15 PM
Drexel28
Quote:
Originally Posted by CaptainBlack
It seems we now have \mathfrak. I have wanted this font for the symbol for the FT for some time and seem to recall it was not available. So when did we acquire it?
$\mathfrak{F}$, $\mathfrak{c}$
CB
I am not sure. It has been on since I have come back! I do love it though!
$\text{ABCEDFHIJKLMNOPQRSTUVWXYZ}\mapsto$ $\mathfrak{ABCEFGHIJKLMNOPQRSTUVWXYZ}$
$\text{abcde}\text{fghijklmnopqrstuvwxzy}\mapsto$ $\mathfrak{abcde}\mathfrak{fghijklmnopqrstuvwxyz}$
• Jan 9th 2010, 12:37 PM
Drexel28
We still don't have the mathscript font though :/. $\mathsrc{A}$
• Jan 10th 2010, 09:08 AM
wonderboy1953
LOL
Quote:
Originally Posted by CaptainBlack
It seems we now have \mathfrak. I have wanted this font for the symbol for the FT for some time and seem to recall it was not available. So when did we acquire it?
$\mathfrak{F}$, $\mathfrak{c}$
CB
You're the moderator and you didn't know?
• Jan 10th 2010, 11:57 PM
Moo
frak has been set for a while (Surprised) I can still remember my signature with frak which I created months and months ago ! | 2016-10-01 15:36:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9331695437431335, "perplexity": 2498.295645734325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738663010.9/warc/CC-MAIN-20160924173743-00179-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://www.reference.com/browse/Extremal+graph+theory | Definitions
# Extremal graph theory
Extremal graph theory is a branch of mathematics.
In the narrow sense, extremal graph theory studies the graphs which are extremal among graphs with a certain property. There are various meanings for the word extremal: with the largest number of edges, the largest minimum degree, the smallest diameter, etc. In a broader sense, various other related questions can be included into extremal graph theory. In that case, the term extremal graph theory can encompass a large part of graph theory.
A typical result in extremal graph theory is Turán's theorem. It answers the following question. What is the maximum possible number of edges in an undirected graph G with n vertices which does not contain K3 (three vertices A, B, C with edges AB, AC, BC; i.e. a triangle) as a subgraph? The bipartite graph where the partite sets differ in their size by at most 1, is the only extremal graph with this property. It contains
$leftlfloor frac\left\{n^2\right\}\left\{4\right\} rightrfloor$
edges. Similar questions have been studied with various other subgraphs H instead of K3; for instance, the Zarankiewicz problem concerns the largest graph that does not contain a fixed complete bipartite graph as a subgraph. Turán also found the (unique) largest graph not containing Kk which is named after him, namely Turán graph. This graph has
$leftlfloor frac\left\{\left(k-2\right) n^2\right\}\left\{2\left(k-1\right)\right\} rightrfloor = leftlfloor left\left(1- frac\left\{1\right\}\left\{k-1\right\} right\right) frac\left\{n^2\right\}\left\{2\right\} rightrfloor$
edges. For C4, the largest graph on n vertices not containing C4 has
$left\left(frac\left\{1\right\}\left\{2\right\}+o\left(1\right)right\right) n^\left\{3/2\right\}$
edges.
## References
1. Béla Bollobás. Extremal Graph Theory. New York: Academic Press, 1978.
2. Béla Bollobás. Modern Graph Theory, chapter 4: Extremal Problems. New York: Springer, 1998.
3. M. Simonovits, Slides from the Chorin summer school lectures, 2006.
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• Please Login or Sign Up to use the Recent Searches feature | 2014-09-17 03:37:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2769119441509247, "perplexity": 704.102186586792}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657120974.20/warc/CC-MAIN-20140914011200-00286-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://webwork.maa.org/moodle/mod/forum/discuss.php?d=5008 | ## Installation
### mpm_prefork and RAM
by Jason Cowell -
Number of replies: 1
Im sorry if this has been answered before
My system is 64GB RAM, and is completely dedicated to WW
my mpm_prefork file is like so
<IfModule mpm_prefork_module>
StartServers 5
MinSpareServers 64
MaxSpareServers 128
MaxRequestWorkers 256
MaxConnectionsPerChild 100
</IfModule>
I hammer the system by browsing and loading many problems from the Library browser, using many web browsers (on the same computer)
it fills right up the 64GB of RAM, and then does give some back eventually.
But should my settings be something different, so it doesn't max out right at 64GB ? it would be nice to have a little room
thanks so much for helping , I am new to WW
### Re: mpm_prefork and RAM
by Nathan Wallach -
The library browser is a memory hog as it renders many problems in a single Apache worker. Gateway quizes also have similar issues.
You can reduce the memory it will use somewhat by:
• lowering MaxSpareServers (don't waste too much space on idle workers)
• lowering MaxRequestWorkers (do not allow to many to run, to avoid swapping and performance degredation)
• lowering MaxConnectionsPerChild (meant to kill workers before they get very large)
Adding in /etc/apache2/conf-enabled/webwork.conf (or wherever the real webwork.conf is on your system)
# size limiter for Apache2 use Apache2::SizeLimit; $Apache2::SizeLimit::MAX_PROCESS_SIZE = 340000;$Apache2::SizeLimit::MAX_UNSHARED_SIZE = 340000; \$Apache2::SizeLimit::CHECK_EVERY_N_REQUESTS = 5; | 2022-09-28 12:45:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40052706003189087, "perplexity": 10394.451000964424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335254.72/warc/CC-MAIN-20220928113848-20220928143848-00486.warc.gz"} |
https://aakashdigitalsrv1.meritnation.com/ask-answer/question/limx-tends-to-a-x-2-5-3-a-2-5-3-x-2/limits-and-derivatives/9774301 | limx tends to a ((x+2)^5/3-(a+2)^5/3)/x-2
Dear Student,
I think there should be (x-a) in the denominator. | 2022-05-28 06:43:52 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9695918560028076, "perplexity": 5667.258334288631}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663013003.96/warc/CC-MAIN-20220528062047-20220528092047-00323.warc.gz"} |
https://docs.oscar-system.org/stable/Hecke/quad_forms/integer_lattices/ | Integer Lattices
An integer lattice $L$ is a finitely generated $\mathbb{Z}$-submodule of a quadratic vector space $V = \mathbb{Q}^n$ over the rational numbers. Integer lattices are also known as quadratic forms over the integers. We will refer to them as $\mathbb{Z}$-lattices.
A $\mathbb{Z}$-lattice $L$ has the type ZLat. It is given in terms of its ambient quadratic space $V$ together with a basis matrix $B$ whose rows span $L$, i.e. $L = \mathbb{Z}^r B$ where $r$ is the ($\mathbb{Z}$-module) rank of $L$.
To access $V$ and $B$ see ambient_space(L::ZLat) and basis_matrix(L::ZLat).
Creation of integer lattices
From a gram matrix
ZlatticeMethod
Zlattice([B::MatElem]; gram) -> ZLat
Return the Z-lattice with basis matrix $B$ inside the quadratic space with Gram matrix gram.
If the keyword gram is not specified, the Gram matrix is the identity matrix. If $B$ is not specified, the basis matrix is the identity matrix.
Examples
julia> L = Zlattice(matrix(QQ, 2, 2, [1//2, 0, 0, 2]));
julia> gram_matrix(L) == matrix(QQ, 2, 2, [1//4, 0, 0, 4])
true
julia> L = Zlattice(gram = matrix(ZZ, [2 -1; -1 2]));
julia> gram_matrix(L) == matrix(ZZ, [2 -1; -1 2])
true
latticeMethod
lattice(V::AbsSpace, basis::MatElem ; check::Bool = true) -> AbsLat
Given an ambient space V and a matrix basis, return the lattice spanned by the rows of basis inside V. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.
By default, basis is checked to be of full rank. This test can be disabled by setting check to false.
Special lattices
root_latticeMethod
root_lattice(R::Symbol, n::Int)
Return the root lattice of type R given by :A, :D or :E with parameter n.
hyperbolic_plane_latticeMethod
hyperbolic_plane_lattice(n::RationalUnion = 1)
Return the hyperbolic plane with intersection form of scale n, that is, the unique (up to isometry) even unimodular hyperbolic $\mathbb Z$-lattice of rank 2, rescaled by n.
Examples
julia> L = hyperbolic_plane_lattice(6);
julia> gram_matrix(L)
[0 6]
[6 0]
julia> L = hyperbolic_plane_lattice(ZZ(-13));
julia> gram_matrix(L)
[ 0 -13]
[-13 0]
ZlatticeMethod
Zlattice(S::Symbol, n::RationalUnion = 1) -> Zlat
Given S = :H or S = :U, return a $\mathbb Z$-lattice admitting $n*J_2$ as Gram matrix in some basis, where $J_2$ is the 2-by-2 matrix with 0's on the main diagonal and 1's elsewhere.
leech_latticeFunction
leech_lattice()
Return the Leech lattice.
leech_lattice(niemeier_lattice::ZLat) -> Leech, neighbor vector, index
Return a triple L, v, h where L is the Leech lattice.
L is an h-neighbor of the Niemeier lattice N with respect to v. This means that L / L ∩ N ≅ ℤ / h ℤ. Here h is the Coxeter number of the Niemeier lattice.
This implements the 23 holy constructions of the Leech lattice in J. H. Conway, N. J. A. Sloane (1999).
Examples
julia> R = Zlattice(gram=2 * identity_matrix(ZZ, 24));
julia> N = maximal_even_lattice(R) # Some Niemeier lattice
Quadratic lattice of rank 24 and degree 24 over the rationals
julia> minimum(N)
2
julia> det(N)
1
julia> L, v, h = leech_lattice(N);
julia> minimum(L)
4
julia> det(L)
1
julia> h == index(L, intersect(L, N))
true
We illustrate how the Leech lattice is constructed from N, h and v.
julia> Zmodh = ResidueRing(ZZ, h);
julia> V = ambient_space(N);
julia> vG = map_entries(x->Zmodh(ZZ(x)), inner_product(V, v, basis_matrix(N)));
julia> LN = transpose(lift(kernel(vG)[2]))*basis_matrix(N); # vectors whose inner product with v is divisible by h.
julia> lattice(V, LN) == intersect(L, N)
true
julia> gensL = vcat(LN, 1//h * v);
julia> lattice(V, gensL, isbasis=false) == L
true
From a genus
Integer lattices can be created as representatives of a genus. See (representative(L::ZGenus))
rescaleMethod
rescale(L::ZLat, r::RationalUnion) -> ZLat
Return the lattice L in the quadratic space with form r \Phi.
Examples
This can be useful to apply methods intended for positive definite lattices.
julia> L = Zlattice(gram=ZZ[-1 0; 0 -1])
Quadratic lattice of rank 2 and degree 2 over the rationals
julia> shortest_vectors(rescale(L, -1))
2-element Vector{Vector{fmpz}}:
[0, 1]
[1, 0]
Attributes
ambient_spaceMethod
ambient_space(L::AbsLat) -> AbsSpace
Return the ambient space of the lattice L. If the ambient space is not known, an error is raised.
basis_matrixMethod
basis_matrix(L::ZLat)
Return the basis matrix $B$ of the integer lattice $L$.
The lattice is given by the row span of $B$ seen inside of the ambient quadratic space of $L$.
gram_matrixMethod
gram_matrix(L::ZLat) -> fmpq_mat
Return the gram matrix of $L$.
Examples
julia> L = Zlattice(matrix(ZZ, [2 0; -1 2]));
julia> gram_matrix(L)
[ 4 -2]
[-2 5]
rational_spanMethod
rational_span(L::ZLat) -> QuadSpace
Return the rational span of $L$, which is the quadratic space with Gram matrix equal to gram_matrix(L).
Examples
julia> L = Zlattice(matrix(ZZ, [2 0; -1 2]));
julia> rational_span(L)
Rational Field
with Gram matrix
[4 -2; -2 5]
base_ringMethod
base_ring(M::PMat)
The PMat $M$ defines an $R$-module for some maximal order $R$. This function returns the $R$ that was used to defined $M$.
base_ring(L::AbsLat) -> Ring
Return the order over which the lattice L is defined.
base_ring(I::MPolyIdeal)
Return the ambient ring of I.
Examples
julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"])
(Multivariate Polynomial Ring in x, y over Rational Field, fmpq_mpoly[x, y])
julia> I = ideal(R, [x, y])^2
ideal(x^2, x*y, y^2)
julia> base_ring(I)
Multivariate Polynomial Ring in x, y over Rational Field
source
base_ring(X::AbsSpec)
On an affine scheme $X/𝕜$ over $𝕜$ this returns the ring $𝕜$.
Examples
julia> X = affine_space(QQ,3)
Spec of Multivariate Polynomial Ring in x1, x2, x3 over Rational Field
julia> base_ring(X)
Rational Field
source
base_fieldMethod
base_field(E::EllCrv) -> Field
Return the base field over which E is defined.
base_field(C::HypellCrv) -> Field
Return the base field over which C is defined.
base_field(L::AbsLat) -> Field
Return the algebra over which the rational span of the lattice L is defined.
Invariants
rankMethod
rank(L::AbsLat) -> Int
Return the rank of the underlying module of the lattice L.
detMethod
det(L::ZLat) -> fmpq
Return the determinant of the gram matrix of L.
scaleMethod
scale(L::ZLat) -> fmpq
Return the scale of L.
The scale of L is defined as the positive generator of the $\mathbb Z$-ideal generated by $\{\Phi(x, y) : x, y \in L\}$.
normMethod
norm(L::ZLat) -> fmpq
Return the norm of L.
The norm of L is defined as the positive generator of the $\mathbb Z$- ideal generated by $\{\Phi(x,x) : x \in L\}$.
isevenMethod
iseven(L::ZLat) -> Bool
Return whether L is even.
An integer lattice L in the rational quadratic space $(V,\Phi)$ is called even if $\Phi(x,x) \in 2\mathbb{Z}$ for all $x in L$.
is_integralMethod
is_integral(L::AbsLat) -> Bool
Return whether the lattice L is integral.
is_primary_with_primeMethod
is_primary_with_prime(L::ZLat) -> Bool, fmpz
Given a $\mathbb Z$-lattice L, return whether L is primary, that is whether L is integral and its discriminant group (see discriminant_group) is a p-group for some prime number p. In case it is, p is also returned as second output.
Note that for unimodular lattices, this function returns (true, 1). If the lattice is not primary, the second return value is -1 by default.
is_primaryMethod
is_primary(L::ZLat, p::Union{Integer, fmpz}) -> Bool
Given an integral $\mathbb Z$-lattice L and a prime number p, return whether L is p-primary, that is whether its discriminant group (see discriminant_group) is a p-group.
is_elementary_with_primeMethod
is_elementary_with_prime(L::ZLat) -> Bool, fmpz
Given a $\mathbb Z$-lattice L, return whether L is elementary, that is whether L is integral and its discriminant group (see discriminant_group) is an elemenentary p-group for some prime number p. In case it is, p is also returned as second output.
Note that for unimodular lattices, this function returns (true, 1). If the lattice is not elementary, the second return value is -1 by default.
is_elementaryMethod
is_elementary(L::ZLat, p::Union{Integer, fmpz}) -> Bool
Given an integral $\mathbb Z$-lattice L and a prime number p, return whether L is p-elementary, that is whether its discriminant group (see discriminant_group) is an elementary p-group.
The Genus
For an integral lattice The genus of an integer lattice collects its local invariants. genus(::ZLat)
massMethod
mass(L::ZLat) -> fmpq
Return the mass of the genus of L.
genus_representativesMethod
genus_representatives(L::ZLat) -> Vector{ZLat}
Return representatives for the isometry classes in the genus of L.
Real invariants
signature_tupleMethod
signature_tuple(L::ZLat) -> Tuple{Int,Int,Int}
Return the number of (positive, zero, negative) inertia of L.
is_positive_definiteMethod
is_positive_definite(L::AbsLat) -> Bool
Return whether the rational span of the lattice L is positive definite.
is_negative_definiteMethod
is_negative_definite(L::AbsLat) -> Bool
Return whether the rational span of the lattice L is negative definite.
is_definiteMethod
is_definite(L::AbsLat) -> Bool
Return whether the rational span of the lattice L is definite.
Isometries
automorphism_group_generatorsMethod
automorphism_group_generators(E::EllCrv) -> Vector{EllCrvIso}
Return generators of the automorphism group of $E$.
automorphism_group_generators(L::AbsLat; ambient_representation::Bool = true)
-> Vector{MatElem}
Given a definite lattice L, return generators for the automorphism group of L. If ambient_representation == true (the default), the transformations are represented with respect to the ambient space of L. Otherwise, the transformations are represented with respect to the (pseudo-)basis of L.
automorphism_group_orderMethod
automorphism_group_order(L::AbsLat) -> Int
Given a definite lattice L, return the order of the automorphism group of L.
is_isometricMethod
is_isometric(L::AbsLat, M::AbsLat) -> Bool
Return whether the lattices L and M are isometric.
is_locally_isometricMethod
is_locally_isometric(L::ZLat, M::ZLat, p::Int) -> Bool
Return whether L and M are isometric over the p-adic integers.
i.e. whether $L \otimes \Z_p \cong M\otimes \Z_p$.
Root lattices
root_lattice_recognitionMethod
root_lattice_recognition(L::ZLat)
Return the ADE type of the root sublattice of L.
Input:
L – a definite and integral $\mathbb{Z}$-lattice.
Output:
Two lists, the first one conaining the ADE types and the second one the irreducible root sublattices.
For more recognizable gram matrices use root_lattice_recognition_fundamental.
Examples
julia> L = Zlattice(gram=ZZ[4 0 0 0 3 0 3 0;
0 16 8 12 2 12 6 10;
0 8 8 6 2 8 4 5;
0 12 6 10 2 9 5 8;
3 2 2 2 4 2 4 2;
0 12 8 9 2 12 6 9;
3 6 4 5 4 6 6 5;
0 10 5 8 2 9 5 8])
Quadratic lattice of rank 8 and degree 8 over the rationals
julia> R = root_lattice_recognition(L)
([(:A, 1), (:D, 6)], ZLat[Quadratic lattice of rank 1 and degree 8 over the rationals, Quadratic lattice of rank 6 and degree 8 over the rationals])
root_lattice_recognition_fundamentalMethod
root_lattice_recognition_fundamental(L::ZLat)
Return the ADE type of the root sublattice of L as well as the corresponding irreducible root sublattices with basis given by a fundamental root system.
Input:
L – a definite and integral $\mathbb Z$-lattice.
Output:
• the root sublattice, with basis given by a fundamental root system
• a Vector consisting of the irreducible root sublattices.
Examples
julia> L = Zlattice(gram=ZZ[4 0 0 0 3 0 3 0;
0 16 8 12 2 12 6 10;
0 8 8 6 2 8 4 5;
0 12 6 10 2 9 5 8;
3 2 2 2 4 2 4 2;
0 12 8 9 2 12 6 9;
3 6 4 5 4 6 6 5;
0 10 5 8 2 9 5 8])
Quadratic lattice of rank 8 and degree 8 over the rationals
julia> R = root_lattice_recognition_fundamental(L);
julia> gram_matrix(R[1])
[2 0 0 0 0 0 0]
[0 2 0 -1 0 0 0]
[0 0 2 -1 0 0 0]
[0 -1 -1 2 -1 0 0]
[0 0 0 -1 2 -1 0]
[0 0 0 0 -1 2 -1]
[0 0 0 0 0 -1 2]
ADE_typeMethod
ADE_type(G::MatrixElem) -> Tuple{Symbol,Int64}
Return the type of the irreducible root lattice with gram matrix G.
See also root_lattice_recognition.
Examples
julia> Hecke.ADE_type(gram_matrix(root_lattice(:A,3)))
(:A, 3)
coxeter_numberMethod
coxeter_number(ADE::Symbol, n)
Return the Coxeter number of the corresponding ADE root lattice.
If $L$ is a root lattice and $R$ its set of roots, then the Coxeter number $h$ is $|R|/n$ where n is the rank of $L$.
julia> coxeter_number(:D, 4)
6
highest_rootMethod
highest_root(ADE::Symbol, n) -> fmpz_mat
Return coordinates of the highest root of root_lattice(ADE, n).
julia> highest_root(:E, 6)
[1 2 3 2 1 2]
Module operations
Most module operations assume that the lattices live in the same ambient space. For instance only lattices in the same ambient space compare.
==Method
Return true if both lattices have the same ambient quadratic space and the same underlying module.
is_sublatticeMethod
is_sublattice(L::AbsLat, M::AbsLat) -> Bool
Return whether M is a sublattice of the lattice L.
is_sublattice_with_relationsMethod
is_sublattice_with_relations(M::ZLat, N::ZLat) -> Bool, fmpq_mat
Returns whether $N$ is a sublattice of $M$. In this case, the second return value is a matrix $B$ such that $B B_M = B_N$, where $B_M$ and $B_N$ are the basis matrices of $M$ and $N$ respectively.
+Method
+(L::AbsLat, M::AbsLat) -> AbsLat
Return the sum of the lattices L and M.
The lattices L and M must have the same ambient space.
*Method
*(a::RationalUnion, L::ZLat) -> ZLat
Returns the lattice $aM$ inside the ambient space of $M$.
intersectMethod
intersect(L::AbsLat, M::AbsLat) -> AbsLat
Return the intersection of the lattices L and M.
The lattices L and M must have the same ambient space.
inMethod
Base.in(v::Vector, L::ZLat) -> Bool
Return whether the vector v lies in the lattice L.
inMethod
Base.in(v::fmpq_mat, L::ZLat) -> Bool
Return whether the row span of v lies in the lattice L.
primitive_closureMethod
primitive_closure(M::ZLat, N::ZLat) -> ZLat
Given two $\mathbb Z$-lattices M and N with $N \subseteq \mathbb{Q} M$, return the primitive closure $M \cap \mathbb{Q} N$ of N in M.
Examples
julia> M = root_lattice(:D, 6);
julia> N = lattice_in_same_ambient_space(M, 3*basis_matrix(M)[1,:]);
julia> basis_matrix(N)
[3 0 0 0 0 0]
julia> N2 = primitive_closure(M, N)
Quadratic lattice of rank 1 and degree 6 over the rationals
julia> basis_matrix(N2)
[1 0 0 0 0 0]
julia> M2 = primitive_closure(dual(M), M);
julia> is_integral(M2)
false
is_primitiveMethod
is_primitive(M::ZLat, N::ZLat) -> Bool
Given two $\mathbb Z$-lattices $N \subseteq M$, return whether N is a primitive sublattice of M.
Examples
julia> U = hyperbolic_plane_lattice(3);
julia> bU = basis_matrix(U);
julia> e1, e2 = bU[1,:], bU[2,:]
([1 0], [0 1])
julia> N = lattice_in_same_ambient_space(U, e1 + e2)
Quadratic lattice of rank 1 and degree 2 over the rationals
julia> is_primitive(U, N)
true
julia> M = root_lattice(:A, 3);
julia> f = matrix(QQ, 3, 3, [0 1 1; -1 -1 -1; 1 1 0]);
julia> N = kernel_lattice(M, f+1)
Quadratic lattice of rank 1 and degree 3 over the rationals
julia> is_primitive(M, N)
true
is_primitiveMethod
is_primitive(L::ZLat, v::Union{Vector, fmpq_mat}) -> Bool
Return whether the vector v is primitive in L.
A vector v in a $\mathbb Z$-lattice L is called primitive if for all w in L such that $v = dw$ for some integer d, then $d = \pm 1$.
divisibilityMethod
divisibility(L::ZLat, v::Union{Vector, fmpq_mat}) -> fmpq
Return the divisibility of v with respect to L.
For a vector v in the ambient quadratic space $(V, \Phi)$ of L, we call the divisibility of v with the respect to L the non-negative generator of the fractional $\mathbb Z$-ideal $\Phi(v, L)$.
Embeddings
Orthogonal sublattices
orthogonal_sumMethod
orthogonal_sum(L1::ZLat, L2::ZLat)
Return the orthogonal direct sum of the lattices L1 and L2.
It lives in the orthogonal direct sum of their ambient spaces.
orthogonal_submoduleMethod
orthogonal_submodule(L::ZLat, S::ZLat) -> ZLat
Return the largest submodule of L orthogonal to S.
direct_sumMethod
direct_sum(x::Vararg{ZLat}) -> ZLat, Vector{AbsSpaceMor}, Vector{AbsSpaceMor}
direct_sum(x::Vector{ZLat}) -> ZLat, Vector{AbsSpaceMor}, Vector{AbsSpaceMor}
Given a collection of $\mathbb Z$-lattices $L_1, \ldots, L_n$, return their complete direct sum $L := L_1 \oplus \ldots \oplus L_n$, together with the injections $L_i \to L$ and the projections $L \to L_i$ (seen as maps between the corresponding ambient spaces).
irreducible_componentsMethod
irreducible_components(L::ZLat)
Return the irreducible components $L_i$ of the positive definite lattice $L$.
This yields a maximal orthogonal splitting of L as
$$$L = \bigoplus_i L_i.$$$
Dual lattice
dualMethod
dual(L::AbsLat) -> AbsLat
Return the dual lattice of the lattice L.
Overlattices
glue_mapMethod
glue_map(L::ZLat, S::ZLat, R::ZLat; check=true)
-> Tuple{TorQuadModMor, TorQuadModMor, TorQuadModMor}
Given three integral $\mathbb Z$-lattices L, S and R, with S and R primitive sublattices of L and such that the sum of the ranks of S and R is equal to the rank of L, return the glue map $\gamma$ of the primitive extension $S+R \subseteq L$, as well as the inclusion maps of the domain and codomain of $\gamma$ into the respective discriminant groups of S and R.
Example
julia> M = root_lattice(:E,8);
julia> f = matrix(QQ, 8, 8, [-1 -1 0 0 0 0 0 0;
1 0 0 0 0 0 0 0;
0 1 1 0 0 0 0 0;
0 0 0 1 0 0 0 0;
0 0 0 0 1 0 0 0;
0 0 0 0 0 1 1 0;
-2 -4 -6 -5 -4 -3 -2 -3;
0 0 0 0 0 0 0 1]);
julia> S = kernel_lattice(M ,f-1)
Quadratic lattice of rank 4 and degree 8 over the rationals
julia> R = kernel_lattice(M , f^2+f+1)
Quadratic lattice of rank 4 and degree 8 over the rationals
julia> glue, iS, iR = glue_map(M, S, R)
(Map with following data
Domain:
=======
Codomain:
=========
TorQuadMod [2//3 0; 0 2//3], Map with following data
Domain:
=======
Codomain:
=========
TorQuadMod [4//3 2//3; 2//3 2//3], Map with following data
Domain:
=======
Codomain:
=========
julia> is_bijective(glue)
true
overlatticeMethod
overlattice(glue_map::TorQuadModMor) -> ZLat
Given the glue map of a primitive extension of $\mathbb Z$-lattices $S+R \subseteq L$, return L.
Example
julia> M = root_lattice(:E,8);
julia> f = matrix(QQ, 8, 8, [ 1 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0;
1 2 4 4 3 2 1 2;
-2 -4 -6 -5 -4 -3 -2 -3;
2 4 6 4 3 2 1 3;
-1 -2 -3 -2 -1 0 0 -2;
0 0 0 0 0 -1 0 0;
-1 -2 -3 -3 -2 -1 0 -1]);
julia> S = kernel_lattice(M ,f-1)
Quadratic lattice of rank 4 and degree 8 over the rationals
julia> R = kernel_lattice(M , f^4+f^3+f^2+f+1)
Quadratic lattice of rank 4 and degree 8 over the rationals
julia> glue, iS, iR = glue_map(M, S, R);
julia> overlattice(glue) == M
true
local_modificationMethod
local_modification(M::ZLat, L::ZLat, p)
Return a local modification of M that matches L at p.
INPUT:
• $M$ – a \mathbb{Z}_p-maximal lattice
• $L$ – the a lattice isomorphic to M over \QQ_p
• $p$ – a prime number
OUTPUT:
an integral lattice M' in the ambient space of M such that M and M' are locally equal at all completions except at p where M' is locally isometric to the lattice L.
maximal_integral_latticeMethod
maximal_integral_lattice(L::AbsLat) -> AbsLat
Given a lattice L, return a lattice M in the ambient space of L which is maximal integral and which contains L.
Sublattices defined by endomorphisms
kernel_latticeMethod
kernel_lattice(L::ZLat, f::MatElem;
ambient_representation::Bool = true) -> ZLat
Given a $\mathbf{Z}$-lattice $L$ and a matrix $f$ inducing an endomorphism of $L$, return $\ker(f)$ is a sublattice of $L$.
If ambient_representation is true (the default), the endomorphism is represented with respect to the ambient space of $L$. Otherwise, the endomorphism is represented with respect to the basis of $L$.
invariant_latticeMethod
invariant_lattice(L::ZLat, G::Vector{MatElem};
ambient_representation::Bool = true) -> ZLat
Given a $\mathbf{Z}$-lattice $L$ and a list of matrices $G$ inducing endomorphisms of $L$, return the lattice $L^G$, consisting of elements fixed by $G$.
If ambient_representation is true (the default), the endomorphism is represented with respect to the ambient space of $L$. Otherwise, the endomorphism is represented with respect to the basis of $L$.
Computing embeddings
embedFunction
embed(S::ZLat, G::Genus, primitive=true) -> Bool, embedding
Return a (primitive) embedding of the integral lattice S into some lattice in the genus of G.
julia> G = genera((8,0), 1, even=true)[1];
julia> L, S, i = embed(root_lattice(:A,5), G);
embed_in_unimodularMethod
embed_in_unimodular(S::ZLat, pos, neg, primitive=true, even=true) -> Bool, L, S', iS, iR
Return a (primitive) embedding of the integral lattice S into some (even) unimodular lattice of signature (pos, neg).
For now this works only for even lattices.
julia> NS = orthogonal_sum(Zlattice(:U), rescale(root_lattice(:A, 16), -1))[1];
julia> LK3, iNS, i = embed_in_unimodular(NS, 3, 19);
julia> genus(LK3)
ZGenus
Signature: (3, 19)
Genus symbol at 2: 1^22
julia> iNS
Quadratic lattice of rank 18 and degree 22 over the rationals
julia> is_primitive(LK3, iNS)
true
LLL, Short and Close Vectors
LLL and indefinite LLL
lllMethod
lll(L::ZLat, same_ambient::Bool = true) -> ZLat
Given an integral $\mathbb Z$-lattice L with basis matrix B, compute a basis C of L such that the gram matrix $G_C$ of L with respect to C is LLL-reduced.
By default, it creates the lattice in the same ambient space as L. This can be disabled by setting same_ambient = false. Works with both definite and indefinite lattices.
Short Vectors
short_vectorsFunction
short_vectors(L::ZLat, [lb = 0], ub, [elem_type = fmpz]; check::Bool = true)
-> Vector{Tuple{Vector{elem_type}, fmpq}}
Returns all tuples (v, n) such that n = v G v^t satisfies lb <= n <= ub, where G is the Gram matrix of L and v is non-zero.
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is positive definite.
See also short_vectors_iterator for an iterator version.
shortest_vectorsFunction
shortest_vectors(L::ZLat, [elem_type = fmpz]; check::Bool = true)
-> fmpq, Vector{elem_type}, fmpq}
Returns the list of shortest non-zero vectors. Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is positive definite.
See also minimum.
short_vectors_iteratorFunction
short_vectors_iterator(L::ZLat, [lb = 0], ub,
[elem_type = fmpz]; check::Bool = true)
-> Tuple{Vector{elem_type}, fmpq} (iterator)
Returns an iterator for all tuples (v, n) such that n = v G v^t satisfies lb <= n <= ub, where G is the Gram matrix of L and v is non-zero.
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is positive definite.
See also short_vectors.
minimumMethod
minimum(L::ZLat)
Return the minimum squared length among the non-zero vectors in L.
kissing_numberMethod
kissing_number(L::ZLat)
Return the Kissing number of the sphere packing defined by L.
This is the number of non-overlapping spheres touching any other given sphere.
Close Vectors
close_vectorsMethod
close_vectors(L:ZLat, v:Vector, [lb,], ub; check::Bool = false)
-> Vector{Tuple{Vector{Int}}, fmpq}
Return all tuples (x, d) where x is an element of L such that d = b(v - x, v - x) <= ub. If lb is provided, then also lb <= d.
If filter is not nothing, then only those x with filter(x) evaluating to true are returned.
By default, it will be checked whether L is positive definite. This can be disabled setting check = false.
Both input and output are with respect to the basis matrix of L.
Examples
julia> L = Zlattice(matrix(QQ, 2, 2, [1, 0, 0, 2]));
julia> close_vectors(L, [1, 1], 1)
3-element Vector{Tuple{Vector{fmpz}, fmpq}}:
([2, 1], 1)
([0, 1], 1)
([1, 1], 0)
julia> close_vectors(L, [1, 1], 1, 1)
2-element Vector{Tuple{Vector{fmpz}, fmpq}}:
([2, 1], 1)
([0, 1], 1) | 2023-03-26 10:03:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6755026578903198, "perplexity": 2098.3549885867837}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945440.67/warc/CC-MAIN-20230326075911-20230326105911-00658.warc.gz"} |
https://jeremy9959.net/Math-5800-Spring-2020/notebooks/Convolution.html | ### Setting up¶
In [1]:
from bokeh.plotting import figure
from bokeh.io import output_notebook, show,curdoc
from bokeh.themes import Theme
import numpy as np
output_notebook()
theme=Theme(json={})
curdoc().theme=theme
Loading BokehJS ...
## Convolution¶
Convolution is a fundamental operation that appears in many different contexts in both pure and applied settings in mathematics.
#### Polynomials: a simple case¶
Let's look at a simple case first. Consider the space of real valued polynomial functions $\mathcal{P}$.
If $f\in \mathcal{P}$, we have
$$f(z) = \sum_{i=0}^{\infty} a_i z^i$$
where $a_i=0$ for $i$ sufficiently large. There is an obvious linear map
$$a:\mathcal{P}\to c_{00}(\mathbf{R})$$
where $c_{00}(\mathbf{R})$ is the space of sequences that are eventually zero.
For each integer $i$, we have a projection map $a_{i}:c_{00}(\mathbf{R})\to\mathbf{R}$ so that $a_{i}(f)$ is the coefficient of $z^{i}$ for $f\in\mathcal{P}$.
Since $\mathcal{P}$ is a ring under addition and multiplication of functions, these operations must be reflected on the other side of the map $a$.
Clearly we can add sequences in $c_{00}(\mathbf{R})$ componentwise, and since addition of polynomials is also componentwise we have: $$a(f+g)=a(f)+a(g)$$
What about multiplication? We know that
$$a_{n}(fg) = \sum_{r+s=n}a_{r}(f)a_{s}(g) = \sum_{r=0}^{n} a_{r}(f)a_{n-r}(g) = \sum_{r=0}^{\infty}a_{r}(f)a_{n-r}(g)$$
where we adopt the convention that $a_i(f)=0$ if $i<0$.
Given $(a_0,a_1,\ldots)$ and $(b_0,b_1,\ldots)$ in $c_{00}(\mathbf{R})$, define $$a*b = (c_0,c_1,\ldots)$$ where $$c_n = \sum_{j=0}^{\infty} a_j b_{n-j}$$ with the same convention that $a_i=b_i=0$ if $i<0$.
This operation is called convolution; together with addition it makes the vector space $c_{00}(\mathbf{R})$ into a commutative ring with identity element $(1,0,0,\ldots)$. Furthermore by construction we have
$$a(fg) = a(f)*a(g).$$
#### Some generalizations - infinite series¶
There are many generalizations of this.
• we may consider infinite series, not just polynomials. If $a$ and $b$ are such series, we can set $c=(c_r)$ where $c_r=\sum_{r=0}^{\infty}a_r b_{n-r}$ and as usual set $a_i=b_i=0$ if $i<0$. Then this sum only involves finitely many terms.
• we can consider infinite series that have a given radius convergence, or that have non-zero radius of convergence. Since the product of convergent series is convergent and the coefficients are computed as if the series were (infinite) polynomials, we can extend convolution to this vector space.
#### Fourier Series¶
A function $f(z)$ in $L^{2}(S^{1})$ has a Fourier Series $\sum_{n=-\infty}^{\infty} a_{n}e^{2\pi i nz}$ where
$$a_n = \int_{S^{1}} f(z)e^{-2\pi inz}dz.$$
These coefficients can be assembled to give a linear map
$$F: L^{2}(S^{1})\to \ell^{2}(\mathbf{Z})$$
where $\ell^{2}(\mathbf{Z})$ is the space of functions $h:\mathbf{Z}\to\mathbf{C}$ where $\sum_{n=-\infty}^{\infty} \|h(n)\|^2<\infty$.
Then $F(fg)=F(f)*F(g)$. Here, the sum giving the Fourier coefficient of the product
$$c(n) = \sum_{r=-\infty}^{\infty} a(r)b(n-r)$$
is infinite and you need the $L^2$ condition to show that it converges.
Proposition: The coefficients of the Fourier series of the product of periodic functions is the convolution of the original functions' Fourier coefficients.
The convolution of two functions $f$ and $g$ on the real line is given by the integral
$$(f*g)(\theta) = \int f(\phi)g(\theta-\phi) d\phi.$$
This is a continuous version of the fourier series situation, and you can show that, with the right convergence conditions, that the fourier transform of the product of functions is the convolution of their fourier transforms.
#### Inverse transform¶
One last remark. Suppose that $f$ and $g$ are functions on $S^{1}$. Their convolution is given by the integral
$$(f*g)(z) = \int_{S^{1}} f(\theta)g(z-\theta)d\theta.$$
The fourier coefficients of the convolution are:
$$a_n(f*g)=\int_{S^{1}}\int_{S^{1}} f(\theta)g(z-\theta)dz\ e^{-2\pi i n\theta}d\theta.$$
Make the substitution $\theta=z-u$ and the integral splits up to give $$a_n(f*g)=a_n(f)a_n(g).$$
So the Fourier transform of the convolution is the pointwise product of the fourier transforms.
This holds in the case of the Fourier transform on $\mathbf{R}$ as well, with appropriate convergence assumptions.
## An example¶
Suppose that $f(x)$ is a function on the real line, and $g(x)$ is a square bump $$g(x) = \begin{cases} 1 & 0\le x\le 1 \cr 0 & \hbox{otherwise}\end{cases}$$
Then $$(f*g)(y) = \int f(x)g(y-x)dx = \int_{x=y-1}^{y} f(x) dx.$$
In other words the value of $f*g$ at $y$ is the average value of $f$ over the preceeding unit interval -- $f*g$ is a "moving average" of $f$.
Let's see how convolution acts like a filtering operation. First we construct a "noisy" sine wave in python and plot it.
In [2]:
x = np.linspace(-20,20,1000)
y = np.sin(x)+np.random.normal(0,.3,size=x.shape[0])
In [3]:
F=figure(height=300,width=300)
F.line(x,y,legend_label='noisy')
show(F)
The np.linspace(a,b,n) function constructs an array $(x_i)$ where $x_0=a$, $\Delta=(b-a)/n$, and
$$x_i = x_0+i\Delta$$
for $i=0,\ldots, n-1$. We can approximate the convolution of $f$ and $g$ by the Riemann sum
$$(f*g)(x_j) = \sum_{i=0}^{n-1} f(x_i)g(x_j-x_i)\Delta = \sum_{i=0}^{n-1}f(x_i)g((j-i)\Delta)\Delta.$$
Here we are helped by the fact that $g$ is compactly supported and for fixed $j$, $g((j-i)\Delta)=1$ only when $i$ is between $j-\Delta^{-1}$ and $j$.
However we have the problem that when $j<\Delta^{-1}$ then we don't have some of the values $f(x_i)$ when $i<0$. So we have to make some kind of decision about what to do.
We can:
• assume $f(x_i)=0$ when $i<0$; or
• limit the computation only to $j$ where $j\ge \Delta^{-1}$.
The numpy convolve function for two arrays u and v computes
$$(u*v)[n]=\sum_{j=-\infty}^{\infty} u[j]v[n-j]$$
but it has to make some assumptions about what values to assign $u[j]$ and $v[j]$ when $j$ is outside the dimensions of the (finite) array.
Setting that aside for now, if we have an array $v$ with $v[i]=g(i\Delta)$ and an array $u$ with values $u[i]=f(x_i)$ then the (approximate) convolution of the functions is
$$(u*v)[n] = \sum_{i=0}^{n-1} u[i]v[n-i]\Delta$$
Exercise: Look at the documentation for the numpy convolve function. What is the significance of the 'mode' options 'valid', 'same', and 'full'?
In the code below, I choose the option 'same' which means that the arrays are padded with zeros so that the convolution has the same length as the original arrays.
In [4]:
f = np.vectorize(lambda x: 1 if x>=0 and x<=1 else 0)
g=f(np.linspace(0,1,1000//40))/25
# 1/sum(g) is delta for the riemann sum
fstarg = np.convolve(y,g,'same')
In [5]:
F = figure(height=400,width=400)
F.line(x,y,legend_label='noisy')
F.line(x,fstarg,color='green',legend_label='smoothed',line_width=2)
F.legend.background_fill_alpha=0
F.legend.click_policy='hide'
F.title.text="A noisy signal and convolution with a square wave"
show(F)
The convolution smooths out the noisy sine wave, but also introduces a small phase shift. (Why?)
## Convolution in two dimensions¶
Just like in one dimension, convolution is a type of filtering operation in two dimensions. The most direct generalization of the example at the beginning of this note would be to consider a function $f(x,y)$ of two variables (our "image"), and take $g(x,y)$ to be the function taking the value $1$ on a square centered at the origin and zero elsewhere. Then the convolution $f*g$ would make a new image whose value at a point is the average of the values at the nearby points in a square region.
Let's grab a few MNIST images and do some experiments.
In [6]:
import pandas as pd
from scipy.signal import convolve2d
from bokeh.layouts import row, column
theme = Theme(json={'attrs':{'Figure':{'toolbar_location':None},'Grid':{'visible': False}, 'Axis':{'visible':False}}})
curdoc().theme=theme
images = pd.read_csv('../data/MNIST/train.csv',nrows=50)
Let's also make three "stop functions", one of which is just a 3x3 bump, and the other two are vertical and horizontal lines.
In [7]:
K = np.array([[1,1,1],[1,1,1],[1,1,1]])
KV = np.array([[0,1,0],[0,1,0],[0,1,0]])
KH = np.array([[0,0,0],[1,1,1],[0,0,0]])
In [8]:
F = figure(width=300,height=300)
pic = images.iloc[1,1:].values.reshape(28,28)
F.image(image=[pic],x=0,y=0,dw=1,dh=1,palette='Spectral11')
F.title.text='An MNIST 0 (unfiltered)'
F1 = figure(width=300,height=300)
F1.image(image=[convolve2d(pic,K)/9],x=0,y=0,dw=1,dh=1,palette='Spectral11')
F1.title.text="Averaging Filter"
F2 = figure(width=300,height=300)
F2.image(image=[convolve2d(pic,KV)/3],x=0,y=0,dw=1,dh=1,palette='Spectral11')
F2.title.text="Vertical line filter"
F3 = figure(width=300,height=300)
F3.image(image=[convolve2d(pic,KH)/3],x=0,y=0,dw=1,dh=1,palette='Spectral11')
F3.title.text="Horizontal line filter"
show(column(row(F,F1),row(F2,F3)))
# Color Images¶
We'll take a pokemon image which is an RGBA image -- meaning it has 4 channels, one each for Red, Green, Blue, and Alpha (or transparency). Since the image is 120x120, it gives an array of dimensions 120x120x4. We will mostly ignore the last (alpha) layer when working with image.
In [9]:
import imageio
image = imageio.imread('../data/pokemon/delcatty.png')
from scipy.ndimage.filters import convolve
from scipy.ndimage.filters import gaussian_filter
K = np.array([[1,1,1],[1,1,1],[1,1,1]])
H = np.stack([K,K,K,K])
H=H.transpose(1,2,0)/H.sum()
filtered=convolve(image,H)
F = figure(width=300,height=300)
F.image_rgba(image=[image[::-1,::-1,:]],x=0,y=0,dw=1,dh=1)
F.title.text = "original pokemon"
F2 = figure(width=300,height=300)
F2.image_rgba(image=[filtered[::-1,::-1,:]],x=0,y=0,dw=1,dh=1,)
F2.title.text="pokemon with an averaging filter"
F.xgrid.visible=False
F.ygrid.visible=False
F2.xgrid.visible=False
F2.ygrid.visible=False
show(row(F,F2))
A gaussian filter means that instead of convolving our image with a "step function" we convolve with a Gaussian. This gives a weighted average for the pixel value at each point. The variance of the Gaussian affects the range over which the averaging is conducted. In the example below, we image our image sits in an infinite plane of zeros for purposes of computing the convolution.
In [10]:
def filter_image():
ims = []
for sigma in np.arange(0,5,1):
F = figure(width=200,height=200)
F.image_rgba(image=[gaussian_filter(image[::-1,::-1,:],sigma=sigma,mode='constant',cval=0.0)],x=0,y=0,dw=1,dh=1)
F.title.text="Gaussian Filter sigma={:.2f}".format(sigma)
ims.append(F)
return ims
ims=filter_image()
show(row(ims))
## Padding and stride¶
Padding: One question with filtering (or convolution) is how to handle boundary cases. Let's suppose for the moment that we are dealing with a discrete convolution kernel, like our 3x3 examples earlier, rather than a continuous one like a Gaussian. What happens when our 3x3 kernel hits an edge of our matrix?
1. Only consider situations where the kernel fits entirely within the field of the image. If the image is NxM, and the kernel is HxW, then you can fit M-W+1 copies of the kernel in each row and N-H+1 copies of the kernel in each column. So the result of the convolution would have size (N-H+1)x(M-W+1).
1. Pad the image (on each side, or top and bottom or both) by zeros -- in other words, allow the kernel to overlap the edge of the image, but assume the image is zeros outside of the original bounds. If we add p columns of zeros to each side of the image, then our padded image has M+2p columns and so we can fit M-W+2p+1 copies of the kernel horizontally; and similarly N-H+2p-1 vertically. Note: Commonly we can choose W=H, both odd, and then p=(W-1)/2, and we end up with our convolution producing output of the same size as the original image.
Stride: What if we move our filter by more than one step each time? The step size is called the stride s. In this case the output size is going to be (M-W+2p)/s+1.
### Discrete Fourier Transform¶
Let $G=\mathbf{Z}/\mathbf{NZ}$. The characters of $G$ are the homomorphisms $G\to \mathbf{C}^{*}$ where $\mathbf{C}^{*}$ is the multiplicative group of nonzero complex numbers. In other words, a character of $G$ is a function $f:G\to \mathbf{C}^{*}$ with $f(x+y)=f(x)f(y)$.
For $j=0,\ldots, N-1$, the function $$f_j(x)=e^{2\pi i jx/N}$$ is a character -- it is well defined because of you replace $x$ by $x+N$ the value $f_j(x)$ doesn't change. If $h$ is any function from $G\to \mathbf{C}$, then there is a unique expansion $$h(x) = \sum_{i=0}^{N} \hat{h}(j)f_{j}(x).$$
The $\hat{h}(j)$ are called the Fourier coefficients of $h$. To prove this, introduce the Hermitian inner product: $$<h,k> = \sum_{i=0}^{N-1} h(x)\overline{k}(x)$$
On the one hand the vector space of functions on $G$ is $N$ dimensional; on the other hand you can show that the $N$ functions $f_j$ form an orthogonal basis of the space.
The Fourier coefficients are the projection of the function $h$ onto the corresponding orthonormal basis: $$\hat{h}(j) = \frac{<h,f_{-j}>}{N}$$
### Fast Multiplication¶
Suppose you want to multiply two polynomials of degree $d-1$. If $f(x)=\sum_{i=0}^{d-1} a(i)x^i$ and $g(x)=\sum_{i=0}^{d-1} b(i)x^i$ then the coefficients $c(k)$ of the product are given by $$c(k) = \sum_{i+j=k}a(i)b(j) = \sum_{i=0}^{2d}a(i)b(k-i)$$ for $k=0,\ldots, 2d$. Notice that the sum on the right is the convolution of the two functions $a$ and $b$; so thinking of these as functions on the positive integers we see that the $k^{th}$ coefficient of $fg$ is $(a * b)(k)$. Also notice this requires $O(d^2)$ multiplications.
Let's think of $a$, $b$, and $c$ as functions $\mathbf{Z}/(2d\mathbf{Z})\to \mathbf{C}$.
We can:
• compute the Fourier Transforms $\hat{a}$ and $\hat{b}$ of the functions $a$ and $b$.
• compute the pointwise product $\hat{a}\hat{b}=\hat{c}$.
• compute the inverse Fourier transform of $\hat{c}$ to recover the function $c$.
This is only a good idea if it is more efficient then the direct multiplication. However, there is an algorithm known as the Fast Fourier Transform which computes $\hat{a}$ and $\hat{b}$, and then $\hat{\hat{c}}$, in time $O(d\log d)$, which is better than the $O(d^2)$ of the original method. For $d$ large this gives a much more efficient way to multiply polynomials.
Applied to integers (which are sort of polynomials in powers of 10, or powers of 2) one gets a fast algorithm for multiplying large integers.
In [ ]: | 2020-07-15 05:36:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9239113330841064, "perplexity": 288.68689964610877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657155816.86/warc/CC-MAIN-20200715035109-20200715065109-00058.warc.gz"} |
https://math.stackexchange.com/questions/1492497/projection-theorem-understanding-two-parts-of-the-proof | # Projection Theorem - understanding two parts of the proof
Projection Theorem. If $M$ is a closed subspace of the Hilbert space $H$ and $x\in H$, then (i) there is a unique element $x'\in M$ such that $$\lVert x-x'\rVert=\inf_{y\in M}\lVert x-y\rVert,$$ and (ii) $x'\in M$ and $\lVert x-x'\rVert=\inf_{y\in M}\lVert x-y\rVert$ if and only if $x'\in M$ and $(x-x')\in M^{\bot}$.
I have two questions, one concerning the proof of (i) and another concerning the proof of (ii).
(i) The proof of $(i)$ starts the following way:
If $d=\inf_{y\in M}\lVert x-y\rVert^2$, then there is a sequence $\left\{y_n\right\}$ of elements of $M$ such that $\lVert y_n-x\rVert^2\to d$. [...]
Why does such a sequence exist? Is it since we can associate a decreasing sequence $(\lVert y_n-x\rVert^2)$ which is bounded and therefore has to converge, say to $K$ and $K$ has to be identical to $d$? - Since if $K<d$, then $d$ isn't the infimum and if $K>d$, then for a sufficient small neighborhood of K we could have an $N$ such that there can be at least one element $\lVert y_n-x\rVert^2$ not in this neighborhood for $n\geq N$?
(ii)
One direction of (ii) is proven as follows:
If $x'\in M$ and $(x-x')\notin M^{\bot}$ then $x'$ is not the element of $M$ closest to $x$ since $$x''=x'+ay/\lVert y\rVert^2$$ is closer, where $y$ is any element of $M$ such that $\langle x-x',y\rangle\neq 0, a=\langle x-x',y\rangle$. To see this, we write $$\lVert x-x''\rVert^2=\langle x-x'+x'-x'',x-x'+x'-x''\rangle\\ =\lVert x-x'\rVert^2+\lvert a\rvert^2/\lVert y\rVert^2 +2\Re\langle x-x',x'-x''\rangle\\ =\lVert x-x'\rVert^2-\lvert a\rvert^2/\lVert y\rVert^2\\ <\lVert x-x'\rVert^2.$$
I do not understand the third equality, i.e. $$\lVert x-x'\rVert^2+\lvert a\rvert^2/\lVert y\rVert^2 +2\Re\langle x-x',x'-x''\rangle=\lVert x-x'\rVert^2-\lvert a\rvert^2/\lVert y\rVert^2\$$
• Considering (i): Basically you got it. This follows from the definition of the infimum as seen here. Had exactly the same question when seeing this proof for the first time. :) – Piwi Oct 22 '15 at 16:17
• Considering (ii): To see that this is true, note that this equality is equivalent to $\frac{|a|^2}{\lVert y\rVert ^2}+\mathfrak{R}\langle x-x',x'-x''\rangle=0$. Replace $x''$ in the product by its definition $x''=x'+ay/\lVert y\rVert^2$, pull $-\frac{a}{\lVert y\rVert^2}$ out of the second argument of the product, note that the remaining product reads $\langle x-x',y\rangle$ which is by definition the same as $a$. So in the end you have $\frac{|a|^2}{\lVert y\rVert^2}-\frac{|a|^2}{\lVert y\rVert^2}$ on the left side which vanishes. A bit sketchy but I'm confident this should put you on track. – Piwi Oct 22 '15 at 16:30
• I get $2\Re\langle x-x',x'-x''\rangle=2\Re(\overline{\left(-\frac{a}{\lVert y\rVert^2}\right)}\cdot a)=2\Re (\overline{-\frac{1}{\lVert y\rVert^2}}\cdot\overline{a}\cdot a)=2\Re (\overline{-\frac{1}{\lVert y\rVert^2}}\cdot\lvert a\rvert^2)$ – M. Meyer Oct 22 '15 at 17:23
• $-\frac{1}{\lVert y\rVert^2}$ is real, isn't it? Same for $|a|^2$. So $2\mathfrak{R} (\overline{-\frac{1}{\lVert y\rVert^2}}\cdot\lvert a\rvert^2)=-2\frac{|a|^2}{\lVert y\rVert^2}$. – Piwi Oct 22 '15 at 17:48
• Just a small remark/reminder: If you know that $\lVert\cdot \rVert$ is a norm, you know that it maps to $\mathbb{R}$ (even $\mathbb{R}_{\geq 0}$). This should be clearly stated in any definition of "norm" (see Wikipedia for example). So technically you don't need to check again that $\lVert \cdot \rVert$ is real. You already did this by verifying that it indeed is a norm. – Piwi Oct 23 '15 at 0:17 | 2020-08-03 09:26:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9545133709907532, "perplexity": 100.56563760287173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735792.85/warc/CC-MAIN-20200803083123-20200803113123-00550.warc.gz"} |
https://www.emse.fr/~yukna/prototypefactory/projects/proba/Loi_de_Poisson.htm | ## POISSON DISTRIBUTION
_______________________________________________________________________________________________________________________________________________________________
The Poisson distribution arises in many situations. It is safe to say that it is one of the three most important discrete probability distributions (the other two being the uniform and the binomial distributions). The Poisson distribution can be viewed as arising from the binomial distribution or from the exponential density.
This law was introduced by Poisson into his treaty " Search on the probability of the judgements out of criminal matter and civil matter " published in 1837. His goal was to estimate the influence of the size of the jury and the rule of majority on the reliability of the verdicts.
We will take an " industrial " example to introduce this law which has been used in very varied fields (communications, astrophysics, nuclear power, queues...). A factory manufactures a textile of which the density of defects (an average number of defects to m2) is lambda. By supposing the defects distributed independently from each other, we propose to determine the law of the number of defects in a part of textile of 1 m2.
We note N the number of defects in the piece of textile , we proposes the following modelisation
We divide the part of textile into n pieces of same size and sufficiently small, and by noting Ni the number of defects in the ith piece, we can suppose that the random variables N1... Nn are independents, with values in { 0,1 }, of hope l/n. In other words, N1.., Nn are variables of same law B(l/n) and per continuation N=å (Ni) ( for i=1…n) follows a law B(n, l/n).
Proposal on the convergence of the binomial distribution towards the Poisson distribution
Given (pn) n> 1 a series of reals in ]0, 1[ and l >0 such as
So if Sn follows a law B(n,pn) we have :
#### Proof :
We just have to write :
and to pass to the limit in each term of the expression. š
### II Definition of the Poisson distribution
A real random variable follows a Poisson distribution with parameter l >0 ( P(l)) when we have : In this case we have E(N)=Var(N)=l. ( The proof is let to the reader.)
The law of Poisson meets however to model phenomena of a type different from the repetitions of rare events that we have just approached here; we go with this intention carry out the first probabilistic modeling " dynamic " noncommonplace of this course.
Suppose that we have a situation in which a certain kind of occurrence happens at random over a period of time. For example, the occurrences that we are interested in might be incoming telephone calls to a police station in a large city. We want to model this situation so that we can consider the probabilities of events such as more than 10 phone calls occurring in a 5-minute time interval. Presumably, in our example, there would be more incoming calls between 6:00 and 7:00 P.M. than between 4:00 and 5:00 A.M., and this fact would certainly affect the above probability. Thus, to have a hope of computing such probabilities, we must assume that the average rate, i.e., the average number of occurrences per minute, is a constant. This rate we will denote by l. (Thus, in a given 5-minute time interval, we would expect about 5¸ occurrences.) This means that if we were to apply our model to the two time periods given above, we would simply use different rates for the two time periods, thereby obtaining two different probabilities for the given event.
Our next assumption is that the number of occurrences in two non-overlapping time intervals are independent. In our example, this means that the events that there are j calls between 5:00 and 5:15 P.M. and k calls between 6:00 and 6:15 P.M. on the same day are independent.
We can use the binomial distribution to model this situation. We imagine that a given time interval is broken up into n subintervals of equal length. If the subintervals are sufficiently short, we can assume that two or more occurrences happen in one subinterval with a probability which is negligible in comparison with the probability of at most one occurrence. Thus, in each subinterval, we are assuming that there is either 0 or 1 occurrence. This means that the sequence of subintervals can be thought of as a sequence of Bernoulli trials, with a success corresponding to an occurrence in the subinterval.
To decide upon the proper value of p, the probability of an occurrence in a given subinterval, we reason as follows. On the average, there are lt occurrences in a time interval of length t. If this time interval is divided into n subintervals, then we would expect, using the Bernoulli trials interpretation, that there should be np occurrences. Thus we want lt=np so p=lt / n.
We now wish to consider the random variable X, which counts the number of occurrences in a given time interval. We want to calculate the distribution of X. For ease of calculation, we will assume that the time interval is of length 1; for time intervals of arbitrary length t.
We know that P(X = 0) =B(n; p; 0) = (1 - p)n =(1-l/n)n
For large n, this is approximately e-l . It is easy to calculate that for any fixed k, we have
b(n; p; k)/ b(n; p; k - 1) = (l - (k - 1)p)/ kq
which, for large n (and therefore small p) is approximately l/k.
Thus, we have P(X = 1) » l e-l ,
And in general :
The above distribution is the Poisson distribution. We note that it must be checked that the distribution really is a distribution, i.e., that its values are non-negative and sum to 1.
The Poisson distribution is used as an approximation to the binomial distribution when the parameters n and p are large and small. However, the Poisson distribution also arises in situations where it may not be easy to interpret or measure the parameters n and p.
#### Examples
Example 1: In his book, Feller discusses the statistics of flying bomb hits in the south of London during the Second World War. Assume that you live in a district of size 10 blocks by 10 blocks so that the total district is divided into 100 small squares. How likely is it that the square in which you live will receive no hits if the total area is hit by 400 bombs? We assume that a particular bomb will hit your square with probability 1/100. Since there are 400 bombs, we can regard the number of hits that your square receives as the number of successes in a Bernoulli trials process with n = 400 and p = 1/100. Thus we can use the Poisson distribution with l= 400 * 1/100 = 4 to approximate the probability that your square will receive j hits. This probability is p(j) =e-4 ( 4)j / j!.
The expected number of squares that receive exactly j hits is then 100 * p(j).
If the reader would rather not consider flying bombs, he is invited to instead consider an analogous situation involving cookies and raisins. We assume that we have made enough cookie dough for 500 cookies. We put 600 raisins in the dough, and mix it thoroughly. One way to look at this situation is that we have 500 cookies, and after placing the cookies in a grid on the table, we throw 600 raisins at the cookies.
Example 2: Suppose that in a certain fixed amount A of blood, the average human has 40 white blood cells. Let X be the random variable which gives the number of white blood cells in a random sample of size A from a random individual. We can think of X as binomially distributed with each white blood cell in the body representing a trial. If a given white blood cell turns up in the sample, then the trial corresponding to that blood cell was a success. Then p should be taken as the ratio of A to the total amount of blood in the individual, and n will be the number of white blood cells in the individual. Of course, in practice, neither of these parameters is very easy to measure accurately, but presumably the number 40 is easy to measure. But for the average human, we then have 40=np, so we can think of X as being Poisson distributed, with parameter l= 40. In this case,it is easier to model the situation using the Poisson distribution than the binomial distribution.
_________________________________________________________________________________________________________________________________________________________
Back to the introduction:
Some exercices :
A little bit of linguistics... : exo6
A story of TGV : exo7
_________________________________________________________________________________________________________________________________________________________ | 2018-06-23 23:39:55 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8428691625595093, "perplexity": 373.0482361421882}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267865438.16/warc/CC-MAIN-20180623225824-20180624005824-00349.warc.gz"} |
https://deepai.org/publication/super-resolving-star-clusters-with-sheaves | DeepAI
# Super-resolving star clusters with sheaves
This article explains an optimization-based approach for counting and localizing stars within a small cluster, based on photon counts in a focal plane array. The array need not be arranged in any particular way, and relatively small numbers of photons are required in order to ensure convergence. The stars can be located close to one another, as the location and brightness errors were found to be low when the separation was larger than 0.2 Rayleigh radii. To ensure generality of our approach, it was constructed as a special case of a general theory built upon topological signal processing using the mathematics of sheaves.
• 13 publications
• 1 publication
03/19/2021
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## 1. Introduction
This article explains an optimization-based approach for counting and localizing stars wthin a small cluster, based on photon counts in a focal plane array. The array need not be arranged in any particular way, and relatively small numbers of photons are required in order to ensure convergence. The stars can be located close to one another, and good performance was obtained when the separation was larger than Rayleigh radii. To ensure generality of our approach, it was constructed as a special case of a general theory built upon topological signal processing using the mathematics of sheaves. While familiarity with sheaves is assumed in Section 5, it is not essential to understand the algorithm derived using them that is discussed in Section 3.
### 1.1. Historical context
Perhaps the most famous algorithm for separating nearly coincident stars in the CLEAN algorithm and its generalizations (see [1, 3, 5] for instance, among many others). These greedy algorithms rely on the geometric structure of stars, namely that they are point sources or nearly so.
While greedy algorithms can yield good performance, they also can fail in dramatic ways. This can be especially pronounced if photon counts are low. This article proposes that an optimization-based approach can lend some robustness to an algorithm. Optimization-based imaging using the Wasserstein metric has recently led to improvements in molecular microscopy [2]. Our approach is a generalization of this idea.
## 2. Detailed problem statement
We can model a scene of stars illuminating a focal plane by using a simplistic Dirac point-mass model,
(1) s(x;a1,b1,a2,b2,…,aN,bn)=N∑n=1anδ(|x−bn|),
where is the brightness of each star, is the location of each star in the focal plane, and can be taken to be the location of an arbitrary pixel.
The response of an actual sensor to the incoming photons from a set of stars in the scene will not be a sum of Dirac point-masses because of dispersion, diffraction, and other physical effects. Suppose that there are pixels in the focal plane, each collecting an integer number of photons. This means that the measurements lie in (rather than ). It is useful to record the photon counts as an integer-valued function . The problem to be solved in this article is therefore the determination of , and the sets and from .
## 3. Algorithmic implementation and results
The algorithm we used to count, locate, and characterize stars begins with a proposed maximum number of stars . Convergence is not ensured by Proposition 1 and Corollary 2 if the true number of stars is larger than . The algorithm proceeds through the following steps, which ultimately minimize the local consistency radii of the sheaf :
1. Collect photon counts from all pixels into a vector
2. For in ,
1. Initial guess:
1. If , use the centroid of the photon count distribution as the initial guess for and the total photon count as
2. Otherwise, use the previous step’s set of star locations and magnitudes as an initial guess with and chosen randomly.
2. Iteration: Solve the minimization problem Equation 5 given stars using the centroids as initial guesses for .
3. Store: the residual error, along with the parameters for the stars.
3. Return the proposed number of stars, locations, and brightnesses corresponding to the smallest residual.
We found that the minimization problem Equation 5 was solvable using the standard Matlab fmincon function, though the initial guess stage was necessary to aid in ensuring convergence under repeated Monte Carlo runs.
As a test of this approach, star fields consisting of between and stars were simulated. These star fields were propagated through a simulated optical system, resulting in photon counts on an array of pixels. These photon counts were then used to determine star locations and brightnesses using the sheaf approach discussed above, a multiple centroiding approach, and the standard CLEAN algorithm. Figure 2 shows a typical result for photons.
Assuming that the correct brightness and location of star is given by
, and that the algorithm’s estimate of the same is
, an algorithm can be scored by its efficiency, given by
(2) Efficiency:=100−minϕ ⎷(1−#image ϕP)2+N∑i=1P∑j=1|aϕ(j)−a′i|2+N∑i=1P∑j=1|bϕ(j)−b′i|2,
where ranges over all functions. Note that
is the Jaccard index of the two sets of stars, given the function
used as a matching. The efficiency is closely related to a conical metric, but does not assume that the number of stars is the same in both sets. Higher efficiency means that the algorithm is doing a better job of determining the correct number of stars, their brightnesses, and locations. The best possible efficiency is , while the worst is .
In order to compare the performance of the sheaf algorithm with the standard CLEAN algorithm [1], we ran a set of Monte Carlo runs of different configurations of stars, as shown in Table 1. Using the efficiency Equation 2 to rate the performance of both algorithms, the results from all runs of all tests are shown in Figure 7. The sheaf algorithm did better than CLEAN in of the runs. In of those runs the difference in efficiency was greater than .
Delving more deeply into the performance differences, Figure 7 shows a breakdown of efficiency for each Test shown in Table 1. For Tests A and B the sheaf algorithm performs nearly optimally. For Tests F and G (the hardest tests), the sheaf algorithm outperforms the CLEAN algorithm by a wide margin – nearly twice the efficiency, as shown in Figure 8. For the other tests, the sheaf algorithm’s performance is substantially more variable than CLEAN.
The location errors for both algorithms and all tests are shown in Figure 9. As should be expected, location errors increase in both algorithms as the number of stars increase. Generally, the sheaf algorithm yields better location estimates than CLEAN for a smaller number of stars, while CLEAN yields better location estimates than the sheaf for a larger number of stars.
Brightness estimate errors for both algorithms and all tests are shown in Figure 10. Both tests yield highly variable performance on brightness estimates regardless of test conditions. At least in the sheaf algorithm case, this may be a direct result of the use of the Euclidean metric on , since – in contrast to the Wasserstein metric [2] – the convergence in brightness is not guaranteed.
## 4. Conclusion
The results we obtained from our algorithm typically exceeded the performance of the CLEAN algorithm, especially with more challenging scenes, which indicates that this method shows promise. The fact that it converges very well for small numbers of stars and less well for larger numbers of stars suggests there is room for improvement. In particular, the use of a general optimization solver is not particularly efficient, and the solver converges slowly.
Computational considerations were not central to our approach on this problem, but could be important in future work. We have already developed a general library for sheaf algorithms [4], and could port this library to a high performance computing platform. We expect that high-performance sheaf computational tools would have broad applicability, providing numerous avenues for improvements.
Finally, the sheaf-based approach discussed in Section 5 is extremely general, and applies without change to many other imaging techniques, not just focal plane arrays. Quantum imaging with small number of photons [6] fits neatly into this perspective and deserves to be explored in future work.
## Acknowledgements
The authors thank Tom Ruekgauer and his team for the use of the simulated optical data presented in this article.
The views, opinions and/or findings expressed are those of the author and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government. This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) under Agreement No. HR00112090125.
## 5. Appendix: Modeling starfields sheafily
Consider that each star is determined by its magnitude (an element of ) and location (an element of ). These properties have different units and thereby suggest that the correct space for the parameters defining a star is
I=[0,∞)×R2⊂R×R.
Conventionally, we use for the brightness (or magnitude) of the star, and will use for its location.
If there are stars present in the scene, this would suggest that is the correct space of parameters defining all stars. Since the order in which the stars are listed in such a product does not matter, if there are stars present in the scene, the correct space of parameters for defining all of the stars is the quotient space , where is the group of permutations on items.
### 5.1. Sheaf model warmup: known star count
If we know the number of stars is , we can express the relationship between the observations and the star parameters by a sheaf diagram
(3)
The diagram expresses the fact that each observation is determined by Equation 1, and that no observation functionally determines any other. One should be aware that the lack of functional dependence does not imply that the observations are independent. Specifically, it may happen that one or more measurements (values in the second row of the diagram) suffice to determine a unique value in , after which point all of the other measurements can be determined from this unique value. This is exactly the situation in which an adaptive method is warranted, since later measurements are not actually required!
Suppose we have a set of measurements at , , , , for which we have observations , , , respectively. These observations form a partial assignment to the above sheaf, which is supported on the second row.
The partial assignment is not supported at the top row, since that would be tantamount to knowing all of the star parameters. Nevertheless, assuming that no noise is present and that we have correctly found the parameters , , , , , then it should be the case that
zm=s(xm;a1,b1,…,aN,bN)
for all .
If real measurements are taken from the scene determined by stars, then the previous discussions led to a simple sheaf diagram
\xymatrixIN/SN\ar[d]fRM
where
(4) fN(a1,b1,…,aN,bN):=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝s(x1;a1,b1,…,aN,bN)s(x2;a1,b1,…,aN,bN)⋮s(xM;a1,b1,…,aN,bN)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,
simply aggregates all of the pixel values into a single vector. One can consider this as a function taking specific values on regions within the plane , so we can interpret .
In the (unlikely) situation that the pixels have taken exactly the values specified by Equation 1, this corresponds to a global section of the sheaf diagram in Equation 3. Since global sections have zero consistency radius and the stalks of the sheaf are all metric spaces, any assignment which is not a global section will have positive consistency radius.
On the other hand, if the function defined in Equation 4 is injective then there is only one global section. This means that we can recast the problem of obtaining the stars from the observations as the minimization of consistency radius. With essentially no further work, we obtain the following result.
###### Corollary 1.
If the function defined in Equation 4 is injective then the solution to
(5) argmin{an,bn}Nn=1(M∑m=1|s(xm;a1,b1,…,an,bn)−zm|2).
is the correct star decomposition for the scene if one exists.
The reader may correctly argue that the optimization problem in Equation 5 can be obtained easily – though not solved – by inspection! However, if the number of stars is not known, then the correct optimization problem that obtains the stars parameters is difficult to state explicitly. However, the correct optimization problem is still a consistency radius minimization, but for a more elaborate sheaf. This sheaf accounts both for the unknown number of stars and the fact that the number of measurements that should be taken depends on the (unknown) number of stars.
### 5.2. Sheaf model encore: unknown star count
Now let us assume that the scene is built from an unknown number of stars. We cannot use a single map, because we do not know . Therefore, we need to consider many possibilities for the number of stars. For instance, let us consider that there may be as many as stars in the scene.
###### Definition 1.
The sheaf is defined by the diagram
(6)
in which each of the restriction maps are given by the map defined in Equation 4 with different numbers of stars.
Notice that the in is the maximum number of stars we will attempt to consider, which may not have much (if anything) to do with the actual number of stars . Of course, we hope that so that the actual number of stars is considered in our analysis. The global consistency radius of an assignment to this sheaf is given by
P∑i=1∥fi(wi,1,…,wi,i)−z∥,
where is the measurement and the is the proposed -th star parameter when we are considering a representation of the scene with stars. The global consistency radius is simply the sum of consistency radii of each subproblem, in which we consider a given number of stars. On the other hand, each one of these subproblems is the local consistency radius for an open set in the base space topology for . If the functions are injective, then it is clear that if , the only set upon which the local consistency radius can vanish is the set consisting of only the observation (ie. the bottom row of the sheaf diagram). Conversely, if the functions are injective and , then the minimum local consistency radius possible on the open set constructed as the star over the element with stalk is zero.
One particular drawback of though, is that values of an assignment on the different stalks in the top row of Equation 6 have nothing to do with one another. Another expression of this fact is that the base space topology for is quite large.
### 5.3. Pivoting to an algorithm
If we combine the sheaf methodology in the previous sections (Sections 5.1 and 5.2) with observation that the collection of photon counts from each pixel are best thought of as a (single) function on the plane, we obtain a way to recover star parameters from a distributional measurement.
To formalize this idea, consider the space of functions on star locations. In brief, a measurement specifies the amount of mass on a set of star positions. The map reinterprets stars as measures
iN((a1,b1),…(aN,bN)):=N∑n=1anδbn(x).
We can use these maps as the restriction maps in , which still works structurally, namely
(7)
Algorithmically, if we obtain a measurement as an element (which need not be a sum of a small number of Dirac distributions), then the best star decomposition is still obtained by minimizing local consistency radii:
∥iP(w1,…,wP)−z∥2,
and selecting the smallest for which this quantity is zero. In cases where the source magnitudes are widely varying, it is likely that a greedy approach is possible (and may be preferable). In Section 3, we employ a compromise approach by reusing solutions with smaller number of proposed stars when solving for larger numbers of proposed stars.
###### Proposition 1.
Suppose that the number of sources is fixed at and that is given by
z=S(a1,b1,…,aN,bN),
where is given by Equation 4.
If we assign to represent the measurement in the sheaf (see Equation 6) then there is an extension to a global assignment in which the local consistency radius vanishes for all sufficiently small open sets provided that .
###### Proof.
To that end, is assigned in the bottom row of the diagram in Equation 6.
Let us name the values in the assignment to the top row of the sheaf diagram
{(a1,1,b1,1)},{(a2,1,b2,1),(a2,2,b2,2)},{(a3,1,b3,1),(a3,2,b3,2),(a3,3,b3,3)},…,{(aP,1,bP,1),…,(aP,P,bP,P)}.
Notice that these values define many possibilities for the source parameters.
If the number of sources is not known, but decompositions with sources are encoded in the sheaf, then the consistency radius of any assignment supported on the star open set (minimal open set in the sense of inclusion) whose stalk is is given by
(8) P∑i=N∥S(ai,1,bi,1…,ai,i,bi,i)−z∥.
Define
aP,k={akif k≤N,0otherwise
and
bP,k={bkif k≤N,arbitraryotherwise.
If we then declare that and whenever they are both defined, then this definition clearly results in the sum of Equation 8 being zero. ∎
###### Corollary 2.
If the function is injective, then the assignment for which the local consistency radius of vanishes occurs precisely at the number of sources .
Therefore, from a methodological perspective, the number of sources can be determined from the consistency filtration given enough samples. If there is noise, look for the “knee” in local consistency radius as the open set size increases. This is precisely our algorithmic approach.
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Quantifying accuracy and heterogeneity in single-molecule super-resolution microscopy.
Nature Communications, 11(6353), December 2020.
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• [4] M. Robinson. Pysheaf: the python sheaf library, https://github.com/kb1dds/pysheaf/, 2020.
• [5] I. M. Stewart, D. M. Fenech, and T. W. B. Muxlow. A multiple-beam CLEAN for imaging intra-day variable radio sources. Astronomy and Astrophysics, 535(A81), 2011.
• [6] Mankei Tsang. Resolving starlight: a quantum perspective. Contemporary Physics, 60(4):279–298, 2019. | 2022-12-03 05:20:10 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8093032240867615, "perplexity": 470.2515937981384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710924.83/warc/CC-MAIN-20221203043643-20221203073643-00515.warc.gz"} |
http://forums.codeblocks.org/index.php?topic=16967.75 | ### Author Topic: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project... (Read 39731 times)
#### dmoore
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #75 on: November 01, 2012, 06:56:20 pm »
Fine with me. I won't have time to test anything new for at least several days.
#### rickg22
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #76 on: November 03, 2012, 03:30:25 am »
GOOD NEWS EVERYONE!
I'm very close to fixing the bug!
I made a relative path cache: If a file is in an already-tested directory, its relative path is calculated based on its directory's relative path. When a file is not found in the cache, its paths are calculated using MortenMacFly's slow-but-sure path calculation via wxFileName.
Here are the preliminary results:
WITHOUT THE PATH CACHE:
Code: [Select]
C:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():394 took : 50 msProject's common toplevel path: C:\htdocs\SIG\SGPG\C:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():460 took : 13496 msC:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():461: Num Files: 1927; Num Cache Misses: 1927
WITH THE PATH CACHE:
Code: [Select]
C:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():404 took : 55 msProject's common toplevel path: C:\htdocs\SIG\SGPG\C:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():464 took : 2437 msC:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():465: Num Files: 1927; Num Cache Misses: 318
Now, since the files in the project are not in order (because we're using a HashMap and not an ordered map), many directories still trigger cache misses. Even so, execution time has disminished by 80%.
I was able to diminish the load time by a further 2% by aditionally checking the parent directory in the cache; Then again, the randomized file order doesn't help. So I'm rewriting the algorithm by using a cleaned-up recursive version of the cache so that we'll have to use wxFileName's functions only twice: Once for the working directory, and once for the project's common top level path; the rest will be mere string comparisons and concatenations. I'll keep you updated when I get the algorithm working.
« Last Edit: November 03, 2012, 03:37:24 am by rickg22 »
#### rickg22
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #77 on: November 03, 2012, 08:51:38 am »
UPDATE:
I finished polishing the Path Cache. There are some edge cases I haven't tested, mainly at root directories or the top directories of other drives, both for filenames and for the project file.
For standard projects with all files under the project file's directory, things work Okay.
Code: [Select]
Loading project files...1927 files loadedDone loading project in 2042msProject's base path: C:\htdocs\SIG\SGPG\C:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():530 took : 50 msProject's common toplevel path: C:\htdocs\SIG\SGPG\C:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():570 took : 120 msC:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():571: Num Files: 1927; Num Cache Misses: 0
Ta-da!
I'm attaching the patch. It needs some styling (you know, braces, tabs, etc), but I hope you like it.
#### dmoore
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #78 on: November 03, 2012, 02:29:35 pm »
Attached is an untested refinement to Morten's patch that uses the old fast method in the more normal case where files are below the project dir, and uses MakeRelativeTo otherwise. This could be combined with Rick's patch to reduce the amount of cache usage.
Here's the changed code:
Code: [Select]
if ( (prjHasUNCName && fileHasUNCName) || ( !prjHasUNCName && !fileHasUNCName && vol.IsSameAs(f->file.GetVolume()) ) ) { if (!fileName.StartsWith(m_CommonTopLevelPath)) { wxFileName relFileCTLP(f->file); relFileCTLP.MakeRelativeTo( m_CommonTopLevelPath ); f->relativeToCommonTopLevelPath = relFileCTLP.GetFullPath(); } else f->relativeToCommonTopLevelPath = fileName.Right(fileName.Length() - m_CommonTopLevelPath.Length()); if (!fileName.StartsWith(GetBasePath())) { wxFileName relFileBase(f->file); relFileBase.MakeRelativeTo( GetBasePath() ); f->relativeFilename = relFileBase.GetFullPath(); } else f->relativeFilename = fileName.Right(fileName.Length() - GetBasePath().Length()); }
I am a little bit confused by this:
Code: [Select]
+ // The commented (old) method to obtain the relativeToCommonTopLevelPath is fast, but does *not* work, if you save+ // the project on a different drive in a sub-folder of an existing source file on that (different) drive:+ // I.e.: Project on C:\Folder\Project.cbp has file C:\Folder\SubFolder\foo.cpp and D:\Folder\bar.cpp+ // Saved the project under D:\Folder\SubFolder\ProjectNew.cbp would cause a wrong computation of bar.cpp otherwise!!
In this case should D:\Folder be the new common top level path? I'm not sure why the old method doesn't work unless the common top level path isn't being calculated correctly. Maybe the problem is really that f->relativeFilename isn't being calculated correctly?
#### rickg22
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #79 on: November 03, 2012, 03:33:25 pm »
Attached is an untested refinement to Morten's patch that uses the old fast method in the more normal case where files are below the project dir, and uses MakeRelativeTo otherwise. This could be combined with Rick's patch to reduce the amount of cache usage.
Here's the changed code:
Code: [Select]
...
Hmm... maybe that code will make my patch redundant, but I like having a separate cache for relative paths. It could become useful later, and it does separate all the path checking.
Quote
In this case should D:\Folder be the new common top level path? I'm not sure why the old method doesn't work unless the common top level path isn't being calculated correctly. Maybe the problem is really that f->relativeFilename isn't being calculated correctly?
Perhaps. Anyway I got another idea to speed up project loading, I'll post it in the next reply.
#### rickg22
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #80 on: November 03, 2012, 03:40:04 pm »
Code: [Select]
ProjectFile* cbProject::AddFile(int targetIndex, const wxString& filename, bool compile, bool link, unsigned short int /*weight*/){... if (!wxFileExists(fullFilename)) pf->SetFileState(fvsMissing); else if (!wxFile::Access(fullFilename.c_str(), wxFile::write)) // readonly pf->SetFileState(fvsReadOnly);
In other words, C::B checks for the existence of ALL the project files on load! Just to set the little "broken" or readonly flags on the project tree. Imagine doing this for a ten-thousand-files java project where you only got to modify the configuration -_-.
I changed it to this:
Code: [Select]
namespace cbProjectAuxVariables { bool check_for_existence_of_files = true;};...ProjectFile* cbProject::AddFile(int targetIndex, const wxString& filename, bool compile, bool link, unsigned short int /*weight*/){... // Check for the file's status if the User chose to in the Environment settings dialog; // or if the file is being added manually and not as part of the project loading. if(!m_CurrentlyLoading || cbProjectAuxVariables::check_for_existence_of_files) { if (!wxFileExists(fullFilename)) pf->SetFileState(fvsMissing); else if (!wxFile::Access(fullFilename.c_str(), wxFile::write)) // readonly pf->SetFileState(fvsReadOnly); }
cbProjectAuxVariables::check_for_existence_of_files is modified on project load by checking a new Environment setting:
Code: [Select]
<object class="sizeritem"> <object class="wxCheckBox" name="chkMissingFilesOnLoad"> <label>Check for missing files on project load</label> <tooltip>When checked, Code::Blocks will check that all the project files exist,
and update their status on the Project Tree.
Useful at times, but slower for very large projects.</tooltip> <checked>1</checked> </object> <flag>wxTOP|wxLEFT|wxEXPAND|wxALIGN_LEFT|wxALIGN_TOP</flag> <border>8</border></object>
And here's the result, with my previous patch and the file checking on load disabled:
Code: [Select]
1927 files loadedDone loading project in 365msProject's base path: C:\htdocs\SIG\SGPG\C:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():539 took : 50 msProject's common toplevel path: C:\htdocs\SIG\SGPG\C:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():579 took : 130 msC:\projects\cb\src\sdk\cbproject.cpp::void cbProject::CalculateCommonTopLevelPath():580: Num Files: 1927; Num Cache Misses: 0
I'm attaching the full patch, including my previous changes. Enjoy.
« Last Edit: November 03, 2012, 03:47:20 pm by rickg22 »
#### dmoore
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #81 on: November 03, 2012, 06:31:22 pm »
Hmm... maybe that code will make my patch redundant, but I like having a separate cache for relative paths. It could become useful later, and it does separate all the path checking.
Even if we do use the cache, it would make sense to use the logic to create the path (and add it to the cache) when there are cache missses to reduce the number of MakeRelativeTo calls.
#### dmoore
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #82 on: November 03, 2012, 06:34:56 pm »
Another good catch. But instead of having an environment setting, why not make this a project setting and/or offer an annoying dialog for projects with a file count above a certain size to disable the check for the project? In general, I would think that if you have one big project open and some smaller ones open, you would still want the checks on the smaller projects.
#### rickg22
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #83 on: November 03, 2012, 06:45:10 pm »
Actually, I was thinking on moving the whole file checking stuff to an event triggered by opening a node in the project tree. In the meantime, I think that having a user-modifiable setting will suffice. Users accustomed to having small projects won't bother changing the setting (which defaults to 'always check'), and users accustomed to large projects are free to trade between load speed and file checking.
We could also add a treshold number to define small and large projects (i.e. 100, 200 files?). In any case, at least now we have a choice. We can refine it later in further commits.
#### rickg22
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #84 on: November 03, 2012, 06:51:38 pm »
Even if we do use the cache, it would make sense to use the logic to create the path (and add it to the cache) when there are cache missses to reduce the number of MakeRelativeTo calls.
Yes, that's exactly what the cache does. Everytime it doesn't find a path, it creates it in memory and adds it to the list; (with the exception of files outside the base path i.e. the Common Top Level Path, because otherwise we could run in the risk of adding a parent path to the whole tree; actually, I think that extra check is unnecessary; we could cache external paths without problem, but we'd need to do some tests after disabling that check.)
#### dmoore
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #85 on: November 03, 2012, 07:12:49 pm »
Even if we do use the cache, it would make sense to use the logic to create the path (and add it to the cache) when there are cache missses to reduce the number of MakeRelativeTo calls.
Yes, that's exactly what the cache does.
Sorry, wasn't clear. What I meant was whenever you have a cache miss, instead of using MakeRelativeTo in all cases, use the faster method when possible.
#### rickg22
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #86 on: November 04, 2012, 12:08:30 am »
Sorry, wasn't clear. What I meant was whenever you have a cache miss, instead of using MakeRelativeTo in all cases, use the faster method when possible.
Ah, my apologies. I forgot to explain how my Cache works, and what I consider a "cache miss".
It's not really a full path cache; it's a PARENT PATH Cache. And it's recursive
Let's say that your filename has a full path + name like this:
What my cache does is searching first for the parent path, and if it doesn't find it, go to its own parent path, and so on, like this:
Code: [Select]
C:\myProjects\myProject1\src\plugins\myplugin\includesC:\myProjects\myProject1\src\plugins\mypluginC:\myProjects\myProject1\src\pluginsC:\myProjects\myProject1\srcC:\myProjects\myProject1
At that point, C:\myProjects\myProject1 is actually our project common top level path. So it's found (which is not considered a cache miss, because we can construct the relative path without resorting to wxFilename's functions). Then, as the cache returns from its call stack, adds the items in the reverse order in which they were searched:
Code: [Select]
C:\myProjects\myProject1\src => srcC:\myProjects\myProject1\src\plugins => src\pluginsC:\myProjects\myProject1\src\plugins\myplugin => src\plugins\mypluginC:\myProjects\myProject1\src\plugins\myplugin\includes => src\plugins\myplugin\includesC:\myProjects\myProject1\src\plugins\myplugin\includes\myheader.h => src\plugins\myplugin\includes\myheader.h (not added to the cache, just calculated)
The actual filename, C:\myProjects\myProject1\src\plugins\myplugin\includes\myheader.h is never added to the cache, because it will only be requested ONCE. However,
Code: [Select]
C:\myProjects\myProject1\src\plugins\myplugin\includes\myheader2.hC:\myProjects\myProject1\src\plugins\myplugin\includes\myheader3.hC:\myProjects\myProject1\src\plugins\myplugin\includes\myheader4.h
WILL be requested. Their common denominator is their path, which is already cached. And so are their parent directories.
This means that at most, ONE cache miss will be present: The project's common top level path itself, but that's already added by default!
The real cache misses will be files outside the project's common top level path, which require using wxFilename anyway, so they're not added to the cache. BUT I might add them, I just wasn't sure if adding them would trigger some misbehavior in the cache...
« Last Edit: November 04, 2012, 12:15:52 am by rickg22 »
#### rickg22
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #87 on: November 04, 2012, 12:33:24 am »
I just realized something.
With minor modifications to my cache's constructor, we can add all the parent paths from the project's CTLP to the root directory, and we will cover ALL cases!
Code: [Select]
C:\projects\myProject1 => ''C:\projects => '..'C:\ => '..\..'
which means that
Code: [Select]
C:\somepathoutside\someotherpath\whatever\myexternalfile.cpp => ..\..\somepathoutside\someotherpath\whatever\myexternalfile.cppAt this point, using wxFilename won't be necessary because it's already certain that the file we're searching for and our project reside in the same unit (that's the big "if" that was added in the latest SVN, for cases with different units, the absolute paths are added verbatim). And since they're in the same unit, we have no cache misses AT ALL! For ANY FILE, either inside or outside our CTLP.
(The only caveat is if we have a symlink circular reference around... I'd need to think about how to solve that)
« Last Edit: November 04, 2012, 12:44:50 am by rickg22 »
#### rickg22
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##### Re: Bug Report: [#18755] C::B hangs for 20 seconds while opening large project...
« Reply #88 on: November 04, 2012, 02:32:49 am »
OK, I'm adding a *disabled* version of the path cache to trunk (to enable it, you have to #define use_path_cache_for_ctlp). I moved the path cache code to a separate file, so the code changes can be more obvious.
I'm also adding the "check for missing files" option.
Should I commit, or just post another patch? The code's ready.
« Last Edit: November 04, 2012, 02:41:39 am by rickg22 » | 2020-01-27 03:40:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5535789132118225, "perplexity": 11951.75600581567}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251694176.67/warc/CC-MAIN-20200127020458-20200127050458-00195.warc.gz"} |
http://mymathforum.com/calculus/339409-finding-points-within-multi-variable-calculus-function.html | My Math Forum Finding points within a multi-variable calculus function
Calculus Calculus Math Forum
March 6th, 2017, 06:25 PM #1
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Finding points within a multi-variable calculus function
Consider the following function.
f (x, y) = [(y+6) ln x]−xe^2y−x(y−5)5f (x, y) = [(y+6) ln x]−xe^2y−x(y−5)5
(a) Find fx(1, 0)fx(1, 0) .
(b) Find fy(1, 0)fy(1, 0) .
I know I need to take the partial derivatives of ff and evaluate them at the same point. I added a pic of the question for verification. Please help me find the answer.
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q7.jpg (11.7 KB, 2 views)
March 6th, 2017, 06:48 PM #2 Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs We are given: $\displaystyle f(x,y)=(y+6)\ln(x)-xe^{2y}-x(y-5)^5$ And so: $\displaystyle f_x(x,y)=(y+6)\cdot\frac{1}{x}-e^{2y}-(y-5)^5$ $\displaystyle f_y(x,y)=\ln(x)-2xe^{2y}-5x(y-5)^4$ Can you continue? Thanks from puppypower123
March 6th, 2017, 07:19 PM #3 Newbie Joined: Mar 2016 From: Canada Posts: 24 Thanks: 0 So then do I plug in my points? so f_x = 3130 and f_y = -3127? because fx(x,y)=(0+6)⋅1(1)−e^2(0)−(y−5)^5 =3130 and fy(x,y) = ln(1)−2(1)e^2(0)−5(1)(0−5)^4 = -3127 Last edited by puppypower123; March 6th, 2017 at 07:22 PM.
March 6th, 2017, 07:55 PM #4
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From: St. Augustine, FL., U.S.A.'s oldest city
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Math Focus: Calculus/ODEs
Quote:
Originally Posted by puppypower123 So then do I plug in my points? so f_x = 3130 and f_y = -3127? because fx(x,y)=(0+6)⋅1(1)−e^2(0)−(y−5)^5 =3130 and fy(x,y) = ln(1)−2(1)e^2(0)−5(1)(0−5)^4 = -3127
Yes, that's correct.
Tags calculus, finding, function, multivariable, points
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http://math.stackexchange.com/questions/180220/riemann-siegel-formula-modification?answertab=oldest | # Riemann Siegel formula modification?
$$Z(t)= \sum_{n=1}^{\lfloor\sqrt t/2\pi\rfloor }\frac{\cos(N(t)-t\log p_{n})}{\sqrt n}$$
here $N(t)$ is the smooth part of the zeros and $p_{n}$ are the primes since $p_{n} =n\log n$ then $\log p_{n}=\log n \log\log n$ for big $n$ so the convergence should be pretty similar
but is this approximaiton valid , here the sum over $n$ is made so $p_{n} \le \lfloor\sqrt t/2\pi\rfloor$
-
Did I correctly assume that "$int$" meant the integer part? – martini Aug 8 '12 at 9:06
yes intx is integer part (or floor function) of 'x' i did not know how to put it – Jose Garcia Aug 8 '12 at 9:22
You can write $\lfloor x \rfloor$ for $\lfloor x \rfloor$ – martini Aug 8 '12 at 9:23 | 2015-01-30 00:24:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9732928276062012, "perplexity": 1540.6426246479405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122192267.50/warc/CC-MAIN-20150124175632-00195-ip-10-180-212-252.ec2.internal.warc.gz"} |
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Okay, So the question we're being asked to verify that the function of satisfying three high positive roles and begin interval. And then we're asked to find the number of sees the status by the conclusions of the rules doom. So the function here is after practical ex Cubans to X squared minus four plus two. So here we know that a Mexican ex humanity experiment for experts to is a polynomial. So we know that all polynomial is continuous and also are all defensible because those air properties upon our muse. So we know automatically now these first two conditions of roast and the satisfied and this is the last condition is where the end points are equal to each other. And if you actually plug in the values of of negative too, you get negative six. And if you plug it in for half of to, you also get negative six and you can check those values yourself and that again. Now that they're all satisfied, we can assume that the conclusion is also food in the conclusion there is a number. See, such that that there's a number c in the derivative of the function, so that it is. It is equal to zero. So we can actually solve for this function because we confined the derivative of affect. This is a very easy wanted to is very easy to do because it is a problem for me. So f Crime of X is equal to three x squared minus for X miners for and all we're doing now is asked to find where dysfunction equals zero. And this is a polynomial. So, uh, you can yes, you have to factor this. And so there are numerous ways to factory was function. I do it with the expended. You may have your own methods. So all you do is multiply ity and numbers are four time negative. Three times negative forgives us. Ah, negative twelve. And then we take the middle number, which is negative for. And then we find two numbers that multiply to give us negative phone. But add to give his naked for and in this case, in his negative six and positive too. And since this is a has a leading coefficient in front of the ah of ah, first number and and this polynomial we divide that way we divide that by the leading number. And then this gives us a simplified fraction of negative dory over one. That means we take the bottom number, which is one so be one X minus three times and this function is the most simplified fraction. So you just take the bottom for number in four acts in front of three X plus two vehicle two zero. This is the factored form of there are derivative. And now all we do is fine where these individuals terms equal dear, and they equal Geo when X is equal to three. I could go in the stall for this function in history, and then he saw for three exposed to it because three which gives us well, it gives us, I believe, negative two thirds police. So those are too, uh see or X values that give us where the derivatives equal to zero. And no, this is a very, very important point about rolls term. We're all storms only tells us that there is at least one see, so we know that there's a least one see for sure. But in this case, we have more than once that we have to, in fact, and this is an important point because all roads there tells us that there is one that there's at least one there could be more. Well, that could be one, two, three, four and infinitely many. And that is just an important point that should be understood of that role. Stone is that it proved that there's on ly that there's at least one if these conditions are satisfied.
#### Topics
Derivatives
Differentiation
Volume
##### Catherine R.
Missouri State University
##### Kristen K.
University of Michigan - Ann Arbor
##### Samuel H.
University of Nottingham
##### Michael J.
Idaho State University
Lectures
Join Bootcamp | 2021-10-18 02:15:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7398477792739868, "perplexity": 690.5974036866496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585186.33/warc/CC-MAIN-20211018000838-20211018030838-00532.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=130&t=30331&p=94331 | Standard enthalpy change
$\Delta U=q+w$
Moderators: Chem_Mod, Chem_Admin
Angel Gomez 1K
Posts: 36
Joined: Fri Sep 29, 2017 7:04 am
Standard enthalpy change
Can someone explain why the deltaH was multiplied by 2 mols?
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Michele 2C
Posts: 23
Joined: Fri Sep 29, 2017 7:06 am
Re: Standard enthalpy change
The reaction you need to calculate ∆H for has 4 moles NO, but in the reaction given, the stoichiometric coefficient of NO is 2, so in order to calculate the ∆H that is asked of you, you need to multiply the q (which is for the stoichiometric coefficients in the given reaction) by 2.
Gwyneth Huynh 1J
Posts: 30
Joined: Fri Sep 29, 2017 7:05 am
Re: Standard enthalpy change
So when you find q and divide that by 4 moles, that's the delta H per one mole of NO. You multiply that by 2 moles because in the overall reaction, 2 moles of NO is used.
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https://math.stackexchange.com/questions/2050471/the-value-of-hard-nonstandard-analysis | # The value of “hard” nonstandard analysis
First of all, I want to make clear what I'm NOT asking. I'm not hoping to do a rehash of the implications of nonstandard analysis on calculus. Rather, I'm interested in its use in "harder" math. I'm currently reading through Goldblatt's Lectures on the Hyperreals and working on the later sections, wherein he discusses ways of rephrasing other areas of math in nonstandard language (e.g. Loeb measures). I'm trying to understand what the purpose of this is.
I understand that nonstandard doesn't get us new results, that is there's nothing we can prove in a nonstandard framework that we can't prove over old-fashioned ZFC. I also understand that generally nonstandard allows us to see the spaces we work in "more intuitively", e.g. Loeb measures allow us to see Lebesgue measure in a more finitary light, but I don't have much of a sense for what this more intuition looks like when we're actually trying to prove statements.
So what is the use of nonstandard analysis in its broadest sense? To those of y'all who study/use/teach it, what do you see it as buying you over "standard" analysis?
• When I saw your subject line I thought you meant "hard" as opposed to "soft", and that I might find out what you mean by that by reading your question. Now I suspect you mean "hard" as opposed to "easy". – Michael Hardy Dec 9 '16 at 0:41
• "over old-fashioned ZFC" is not what NSA is opposed to. NSA is "over old-fashioned ZFC". Rather, the thing to contrast with NSA is not "old fashioned ZFC", but rather epsilon-delta methods and the like, i.e. the old-fashioned way of making calculus logically rigorous. $\qquad$ – Michael Hardy Dec 9 '16 at 0:43
• @MichaelHardy I meant hard to be closer to "not soft" than "not easy", but I was looking for a way to basically distinguish from your basic analysis of maps from $\mathbb{R}$ to $\mathbb{R}$ and whether they're continuous or differentiable, which seemed to be the topic of many posts I found, and more about how the general concept of taking a universe and enlarging it works. As for the "old-fashioned ZFC" comment, that point came up frequently in other threads so it seemed it mightn't be a bad idea to repeat one more time. – AJY Dec 9 '16 at 2:10
Robinson's framework does give you new results. For example see this recent result by Terry Tao and Van Vu in Discrete analysis, where the authors use the language of infinitesimals because a paraphrase in an epsilontic framework would have been nearly unmanageable.
Robinson did prove a theoretical result that a theorem proved in his framework can be proved without infinitesimals, as well. However, practically speaking such a translation may be unreadable because of an explosion of complexity. A related point was discussed in detail in an article by Keisler and Henson in 1986:
Henson, C. Ward; Keisler, H. Jerome. On the strength of nonstandard analysis. J. Symbolic Logic 51 (1986), no. 2, 377–386.
• So although technically a result of nonstandard will have an analogue in standard with a standard proof, in practice it may be the case that this translation would require so much "epsilon management" (which nonstandard techniques tend to conceal) that we might as well just let it stay nonstandard? Also, what was the Keisler & Henson article? – AJY Jan 5 '17 at 16:13
• Exactly. I will add the Keisler-Henson reference. – Mikhail Katz Jan 5 '17 at 16:16
I'm just going to give a fragment of an answer. "what do you see it as buying you over 'standard' analysis?" Here's one small insight that I think might never have come about without Robinson's NSA:
• (This part was there before Robinson's NSA, in the form of an intuitive statement.) Suppose $f:\mathbb R \to \mathbb R.$ Then continuity of $f$ at $a$ means if $\varepsilon$ is infinitely small, then so is $f(a+\varepsilon) - f(a)$. Thus $f$ is everywhere continuous if that holds for every real number $a$.
• But $f$ is uniformly continuous if the same is true not just at every real number $a$, but also every nonstandard real number $a$, including those infinitely close to some real number, and also including those that are infinitely large. For example, suppose $f(x)=e^x$. Then if $a>0$ is infinite, then you can have $f(a+\varepsilon) -f(a)=1$ even though $\varepsilon$ is infinitely small, since the growth rate of $a\mapsto e^a$ is infinitely large when $a$ is infinitely large. Thus $a\mapsto e^a$ is not uniformly continuous. Likewise, suppose $f(x) = \sin(1/x).$ Then when $a>0$ is infinitely close to $0$, you can have $f(a+\varepsilon)-f(a) = 2$ even though $\varepsilon$ is infinitely small. Thus $f$ is not uniformly continuous.
• I'm familiar with this. I'm trying to move past basic calculus and undergrad analysis to the "wider world" of nonstandard. – AJY Dec 9 '16 at 2:12
In the 1960's Bernstein and Robinson used NSA to prove for the first time that a polynomially compact operator on Hilbert space has an invariant subspace.
Halmos subsequently rewrote their proof in standard terms. In the abstract he says
The purpose of this paper is to show that by appropriate small modifications the Bernstein-Robinson proof can be converted (and shortened) into one that is expressible in the standard framework of classical analysis.
PACIFIC JOURNAL OF MATHEMATICS
Vol. 16, No. 3, 1966
INVARIANT SUBSPACES OF POLYNOMIALLY COMPACT OPERATORS
http://msp.org/pjm/1966/16-3/pjm-v16-n3-p05-p.pdf
... but the nonstandard proof to be modified came first.
• I wasn't looking for "historical" reasons, I was looking for why you use it here and now. – AJY Dec 10 '16 at 2:08
• @AJY I knew that but thought the story worth telling. – Ethan Bolker Dec 10 '16 at 2:46
• @EthanBolker, you might want to take a look at this article on Halmos and his antics. – Mikhail Katz Jan 5 '17 at 16:19 | 2019-10-23 02:08:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7977957725524902, "perplexity": 605.0732112275756}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987828425.99/warc/CC-MAIN-20191023015841-20191023043341-00054.warc.gz"} |
http://www.ima.umn.edu/2014-2015/SW5.28-31.15/abstracts.html | HOME » SCIENTIFIC RESOURCES » Volumes
SCIENTIFIC RESOURCES
Abstracts and Talk Materials
Workshop for Women in Analysis and PDE
May 28 - 31, 2015
The theory of boundary-value problems for the Laplacian in Lipschitz domains is by now very well developed. Furthermore, many of the existing tools and known results for the Laplacian have been extended to the case of second-order linear equations of the form div (A grad u), where A is a matrix of variable coefficients.
However, at present there are many open questions in the theory of higher-order elliptic differential equations. I will describe a generalization of layer potentials to the case of higher-order operators in divergence form; layer potentials are a common and very useful tool in the theory of second-order equations. I will then describe some applications of layer potentials to the theory of boundary-value problems.
This is joint work with Steve Hofmann and Svitlana Mayboroda.
The Bilinear Hilbert Transform(BHT) exhibits a one-dimensional modulation invariance, which places it half way between singular (multi-)linear operators and Carleson’s operator. The study of BHT in the 90s was the beginning of time-frequency analysis.
A question regarding a specific system of ODEs led to an operator resembling Rubio de Francia’s square function: that is, an operator given by Fourier projections of BHT associated with arbitrary intervals. This further led to vector-valued extensions for BHT, for which some partial results were already known. Similarly, one can get vector-valued extensions for paraproducts, and as a consequence, estimates for tensor products of paraproducts on L^p spaces with mixed norms.
Keywords of the presentation: decay estimate, water wave equation
In this talk, we discuss recent work concerning a decay estimate for solutions to the linear dispersive equation iu_t-(-Delta)^{1/4}u=0, for (t,x)in R times R, which corresponds to a factorization of the linearized two-dimensional water wave equation. Our arguments are based on a Littlewood-Paley decomposition and stationary phase estimates, and we obtain optimal decay of order |t|^{-1/2} for solutions, assuming only bounds in L^2-based spaces (with weights).
Keywords of the presentation: Variational inequalities, free boundary problems, Signorini problem, semi-permeable walls
In this talk we will present an overview of regularity results (both for the solution and for the free boundary) in a class of problems which arises in permeability theory. We will mostly focus on the parabolic Signorini (or thin obstacle) problem, and discuss the modern approach to this classical problem, based on several families of monotonicity formulas. In particular, we will present the optimal regularity of the solution, the classi?cation of free boundary points, the regularity of the regular set, and the structure of the singular set. These results have been obtained in joint work with N. Garofalo, A. Petrosyan, and T. To.
We will also discuss the regularity of solutions in a related model arising in problems of semi-permeable walls and of temperature control. This is joint work with T. Backing.
Keywords of the presentation: elliptic theory, parabolic theory, Carleman estimates
Experts have long realized the parallels between elliptic and parabolic theory of partial differential equations. It is well-known that elliptic theory may be considered a static, or steady-state, version of parabolic theory. And in particular, if a parabolic estimate holds, then by eliminating the time parameter, one immediately arrives at the underlying elliptic statement. Producing a parabolic statement from an elliptic statement is not as straightforward. In this talk, we demonstrate a method for producing parabolic theorems from their elliptic analogues. Specifically, we show that an L^2 Carleman estimate for the heat operator may be obtained by taking a high-dimensional limit of L^2 Carleman estimates for the Laplacian. Other applications of this technique will be indicated.
Keywords of the presentation: Free boundary problems, regularity theory.
In this talk we will provide an overview of the regularity theory for a one-phase (Bernoulli) free boundary problem. We will describe several results which parallel the regularity theory of minimal surfaces, including flatness theorems, monotonicity formulae, regularity in low dimensions, a priori gradient bounds. We will then talk about the thin one-phase problem, that can be viewed as a non-local version of the classical one-phase problem. We will highlight the differences in the strategy developed to investigate regularity issues with respect to the classical case.
We study the convergence of a semidiscrete scheme for the forward-backward parabolic equation $u_t = (W'(u_x))_x$ with periodic boundary conditions in one space dimension, where $W$ is a standard double-well potential. We characterize the equation satisfied by the limit of the discretized solutions as the grid size goes to zero. We also characterize the long-time behavior of the limit solutions. Using an approximation argument, we show that it is possible to flow initial data with derivative in the concave region ${ W''< 0}$ of $W$, where the backward character of the equation manifests.
Keywords of the presentation: T1 theorem, singular integral, non-homogeneous, haar functions
We prove the $T1$ theorem for singular integrals with general Calderon-Zygmund kernels in the non-homogeneous setting by modifying the dyadic representation theorem in a convenient manner. This presentation is based on a joint work with T. Hytonen.
Keywords of the presentation: NLS, probability, statistical mechanics
Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss progress with Gerard Ben Arous and Benjamin Schlein on a central limit theorem for the quantum many-body dynamics, a step towards large deviations for Bose-Einstein condensation.
The point of interest are so-called high contrast composite materials. High contrast means that the ratio between highest and lowest values of the parameter, that describes materials property, is very high, even infinite. Another key aspect is a complex geometry of materials of interest. Mathematical modeling of such materials yields in a PDE that has roughly oscillating coefficients with large variability in the domain. Solving such problems by traditional numerical methods is expensive due to a fine mesh needed in thin regions of the computational domain. The main goal is a numerical treatment of problems associated with described materials. The novelty idea is to take advantage of properties of structured materials to build new numerically efficient schemes. In particular, one of the focuses of proposed research is on the domain decomposition methods for the problems that describe media whose parts might have high contrast constituents. The key step here is to split a large domain into subdomains in a natural way to deal with homogeneous and high contrast parts. With that, we obtain a coupled problem where subdomains are bridged though the interface. Then, one can build an iterative method based on the resulted partition.
This presentation addresses certain notions of convexity for strain-limiting theories of elasticity in which the Green-St.Venant strain tensor is written as a nonlinear response function of the second Piola-Kirchhoff stress tensor. Previous results on strong ellipticity for special strain-limiting theories of elasticity required invertibility of the Frechet derivative of the response function as a fourth-order tensor. The present contribution generalizes the theory to cases in which the Frechet derivative of the response function is not invertible, by studying a weaker rank-1 convexity notion, monotonicity, applied to a general class of nonlinear strain-limiting models. It is shown that the generalized monotonicity holds for Green St. Venant strains with sufficiently small norms, and fails (through demonstration by counterexample) when the small strain constraint is relaxed.
Keywords of the presentation: elliptic equations, polyhedral domain, finite element method
I discuss well-posedness and regularity for strongly elliptic, linear systems, like the system of elasticity, on polyhedral domains using weighted Sobolev spaces. The domain need not be convex and can have curvilinear sides. If time permits, I will discuss how the regularity estimates for the solution can be used to study convergence of the Finite Element Method, a widely used numerical method to solve PDEs.
Liquid crystal phases are understood to be intermediate states between a liquid and a crystalline. Possessing both flow-like properties of liquids and molecular packing of solids, liquid crystals are widely used in optical devices. In addition to having a specific orientation, molecules in the Smectic-C phase tend to organize in layered structures. In a surface-stabilized cell however, the uniform layers deform into V-shaped layers, referred to as a chevron structure, causing distortions. Moreover, the helical structure of the chiral Smectic-C is suppressed in the thin cell which allows for two stable states.
In this poster, we present molecular reorientation dynamics between these stable states under the influence of an applied electric field. Our model is based on the Chen-Lubensky energy involving the molecular director and a general complex-valued layer order parameter. We construct a discrete-in-time gradient flow through an iterative minimization procedure. To establish the existence of a solution to this system of parabolic equations, we use elliptic estimates and prove discrete-to-continuous convergence of the gradient flow. This implicit time-discretization method, called Rothe method, has a strong numerical aspect and gives an insight into the structure of the solution unlike some abstract methods for the analysis of existence problems.
Keywords of the presentation: Periodic NLS, supercritical regime, random data
In this talk we first review recent progress in the study of certain evolution equations from a non-deterministic point of view (e.g. the random data Cauchy problem) which stems from incorporating to the deterministic toolbox, powerful but still classical tools from probability as well. We will explain some of these ideas and describe in more detail joint work with Gigliola Staffilani on the almost sure well-posedness for the periodic 3D quintic nonlinear Schrodinger equation in the supercritical regime; that is, below the critical space $H^1(mathbb T^3)$.
Keywords of the presentation: Homogenization, Bernoulli Problem, Plateau Problem
In this talk, we present some recent results on the regularity and the asymptotic behavior of some geometric variational problems. First analysis is about a free boundary value problem in random media that is motivated by the one-phase Bernoulli problem. Second analysis is about the classical problem of Plateau with a highly oscillatory boundary condition. This part of the talk will present mostly open problems that are emerged from some interesting behaviors of solutions in different settings.
In this poster, we survey several recent results on the Calder'on-Zygmund theory of multi-parameter singular integrals, with one of the most classical examples being the tensor product of Hilbert transforms. We develop a mixed type characterization of the multi-parameter Calder'on-Zygmund operator class that has been studied by Journ'e, and prove for this class of operators a dyadic representation theorem. As an application, new results on iterated commutators of singular integrals with BMO symbols are obtained. And a T(b) theorem on product spaces is proved in a similar fashion as well.
Keywords of the presentation: quantum many body systems, nonlinear dispersive PDE
The derivation of nonlinear dispersive PDE, such as the nonlinear Schr"{o}dinger (NLS) or nonlinear Hartree equations, from many body quantum dynamics is a central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is a coupled system of linear non-homogeneous PDE that describes the dynamics of a gas of infinitely many interacting bosons, while at the same time retains some of the features of a dispersive PDE.
In this talk we will discuss the process of going from a quantum many body system of bosons to the NLS via the GP. Also we will look into what a nonlinear PDE such as the NLS can teach us about the GP hierarchy and quantum many body systems.
The talk is based on joint works with T. Chen, C. Hainzl, R. Seiringer and N. Tzirakis.
There has been recent interest in the use of PDEs to gain understanding of various complex systems in the social sciences. In this talk I will introduce a system modeling social segregation. We analyze this system to understand the effect of social preference, economic disparity, and the environment on segregation. We discuss the existence of steady-state solutions as well as the local and global well-posedness of the system. These results are of broader interest as similar systems arise in the modeling opinion dynamics and rioting activity among other phenomena.
Keywords of the presentation: Signorini problem, thin obstacle problem, free boundary, regularity
We will describe the Signorini, or lower-dimensional obstacle problem, for a uniformly elliptic, divergence form operator L = div(A(x)nabla) with Lipschitz continuous coefficients. We will give an overview of this problem and discuss some recent developments, including the optimal regularity of the solution and the $C^{1,alpha}$ regularity of the regular part of the free boundary. These are obtained by proving a new monotonicity formula for a generalization of the celebrated Almgren’s frequency functional, a new Weiss type monotonicity formula and an appropriate version of the epiperimetric inequality. This is joint work with Nicola Garofalo and Arshak Petrosyan.
In the first part of these lectures I will present certain, by now classic, results on local and global well-posedness for the NLS via Strichartz estimates. In the second part I will show how one can use randomization of the initial data to prove well-posedness almost surely even when the problem lack enough regularity for a more deterministic approach. In this context I will also introduce certain Gibbs type measures and how their invariance can be used in certain cases to extend local solutions to a global ones.
In the 1920's Besicovitch studied linearly measurable sets in the plane, that is sets with locally finite "length". The basic question he addressed was whether the infinitesimal properties of the "length" of a set E in the plane yield geometric information on E itself. This simple question marks the beginning of the study of the geometry of measures and the associated field known as Geometric Measure Theory (GMT).
In this series of lectures we will present some of the main results in the area concerning the regularity of the support of a measure in terms of the behavior of its density or in terms of its tangent structure. We will discuss applications to PDEs, free boundary regularity problem and harmonic analysis. The aim is that the mini-course will be self contained.
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https://atdotde.blogspot.com/2009/ | Friday, October 09, 2009
Upon popular request, I wrote a small script to download all tweets of a given twitter id. Have fun!
#!/usr/bin/perl
$|=1; unless(@ARGV){ print "Usage:$0 twitter_id [sleep_seconds]\n";
exit 0;
}
my ($follow,$sleeper) = @ARGV;
# No account needed for this.
my $twit = Net::Twitter->new(username => 'MYNAME', password => 'XXX');$p=1;
while(1){
my $result =$twit->user_timeline({id => $follow, page =>$p});
foreach my $tweet (@{$result}){
print "At ", $tweet->{'created_at'},"\n"; print$tweet->{'text'},"\n\n";
}
++$p; sleep$sleeper if $sleeper; } You might have to install the Net::Twitter module. This is most easily done as sudo perl -MCPAN -e shell and then (possibly after answering a few questions) install Net::Twitter Monday, October 05, 2009 Not so canonical momentum Two weeks ago, I was on Corfu where I attended the conference/school/workshop on particles, astroparticles, strings and cosmology. This was a three week event, the first being on more conventional particle physics, the second on strings and the last on loops and non-commutative geometry and the like. I was mainly there for the second week but stayed a few days longer into the loopy week. I think it was a clever move by the organisers of the last week to give five hours to the morning lecturers rather than one or two as in the string week. So they had the time to really develop their subjects rather than just mention a few highlights. John Baez has already reported on some of the lectures. I would like to mention something I learned about elementary classical mechanics and quantum mechanics which was just a footnote in Ashtekar's first lecture but which was new to me: One canonical variable can have several canonical conjugates! In the loopy context, this appears as both the old and the new connection variables have the same canonical momentum although they differ by the Imirzi parameter times the second fundamental form (don't worry if you don't know what this is in detail, what's important that the 'positions' are different in the two sets of variables although they have the same canonical momentum). How can this be? I always thought that if is a canonical variable the conjugate variabel is determined by . What I had not realized is that you could for example take and obtain the same fundamental Poisson brackets (and consequently commuation relations after quantization). Similarly, you could add any function to the momentum without changing the commutation relations. The origin of this abiguity can be found in the fact that also the Lagrangian is not unique: You can always add a total derivative without changing the action (at least locally, see below). For example, to obtain by the derivative formula, add to the action. The most general change would be to add . What about the quantum theory? This is most easily seen by realising that upon a gauge transformatio , the action of a charge particle changes by . Thus our change in Lagrangian (with a corresponding change in the canonical momentum) can be viewed as a gauge transformation (even if no gauge field is around one could add a trivial one). Correspondingly, the wave function would have to be changed to as acting on by a canocially quantized is the same as acted on by . So, it seems as if you would get exactly the same physics in the primed variables as in the unprimed ones. But we know that not all total derivatives have no influence on the qunatum theory the -angle being the most prominent example. How would that appear in our much simpler quantum mechanics example? Here, it is important to remember that one should only use gauge transformations that are trivial at infinity. Here, if you change the phase of the wave function too wildly at you might leave the good part of the Hilbert space: For example the kinetic energy being an unbounded operator is not defined on all of Hilbert space but only on a dense subspace (most often taken to be some Sobolev space). And that you might leave by adding a wild phase and end up in a different self adjoint extension of the kinetic energy. I have no idea if all this is relevant in the loopy case and the old and new variables or the variables are related by a (generalized) gauge transformation but at least I found in amusing to learn that the canonical conjugate is not canonical. Saturday, August 08, 2009 Jazz makes you age faster Before I head off for the travel season (first vacation: St. Petersburg, Moscow, Transsiberian Railway, Irkutsk, Baikal Lake, Ulan Ude and Mongolian Border, then two weeks of workshop in Corfu, then meeting collaborators in Erlangen and finally lecuring in Nis, Yugoslawia) - you won't notice any change in posting frequency - I would like to leave you with the latest statistic I learned about in "Sueddeutsche Zeitung" today: In 1982, the average age of the audience of jazz concerts (don't know if in Germany or worldwide or whatsoever) was 29, today it is 64. So, even assuming immortality of improvised music enthusiasts, in 27 years, they got older by 35 years! Note well that for an average of 64, if I attend a jazz concert being 36, we need 78ers to get back to the average. Tuesday, July 14, 2009 Thermodynamic (in)stability of hydrogen The interview season for the "Theoretical and Mathematical Physics" master programme at LMU is approaching quickly. We have to come up with new questions and problems that help us judge our applicants. It turns out to be easy to find questions in quantum mechanics and those easily lead over to mathematics questions. However, we were always short on good stat mech problems. One possibility is to have an easy start with the harmonic oscillator and then couple that to a heat bath and compute the partition function (with geometric series featuring). But this time, we thought we could vary this a bit and came to a surprising realization: Hydrogen is unstable! This was news to me but google finds a number of pages where this is discussed. Often wrongly, but the good explanation is in a 2001 paper by Miranda. The idea is the following: Everybody knows that the energy of the -th level of the hydrogen atom has energy proportional to . This level has degeneracy since runs from 1 to and then runs from to . So the partition function is . First, we thought that this might be a function named after some 19th century mathematician but mathematica told us its name is actually since the exponent quickly approaches 1 for every positive . The conclusion seems to be that there is something wrong with the hydrogen atom. And we have not even started to consider the positive energy scattering states. Obviously, this problem has an IR divergence and it is probably better to embed it in some cavity of finite radius. But still, you would think that then for a large cavity, most of the statistical weight would be in the highly excited states and the probability to be in the ground state would go to zero as the cavity gets larger. The conclusion would be that a hydrogen atom at any temperature would almost never be in its ground state but always highly excited or even ionized. And all this only because the density of states diverges at 0. This looks like a situations worse than the Hagedorn transition that strings experience due to the exponentially growing density of states. The solution in the above mentioned paper is quite simple: Rather than these scaling arguments one should put in some numbers! Let us start with the Bohr radius, which is m and the radius grows like . This means in ameter sized cavity we can only fit states up to roughly . However, at room temperature, Boltzmann exponent and . Thus, to balance the Boltzmann suppression of the higher levels compared to the ground state one has to take into account at the order of states and not just the first . Or put differently, one should use and exponentially large cavity. Otherwise the partition function is essentially cut off at and the probablility to find the ground state is very very very close to 1. Monday, July 06, 2009 Wrocław summary So, I did not get around to live blog from the XXVth Max Born meeting "The Planck Scale". The main reason was, that there were no hot news or controversial things presented, rather people from the different camps talked about findings that a daily reader of hep-th had in some form or the other already noticed. I don't want to create the impression that it was boring, by no means. There were many interesting talks, there were just no breathtaking revelations. I myself am not an exception: I took the opportunity of having several loop-people in the audience to talk once more on the loop string, this time focussing on spontaneous breaking of diffeomorphism invariance. By now, the PDFs are online and in a few days you will also find video footage. To get an idea what people discussed, the organizers had the idea to assemble tag clouds from the slides, some are above. Let me mention a few presentations and speakers nevertheless. Steve Carlip talked about the notion of space-time being two dimensional at very short distances in several unrelated approaches. Related was a nice presentation of Silke Weinfurter on her papers with Visser on the scalar mode not decoupling in Horava gravity. That talk was probably on the most recent and hottest results and I had the impression that many other approaches still have to digest the lesson that it is non-trivial to modify gravity and still not throw out the baby with the bath tub. Hermann Nicolai presented his work (together with Meissner) on a classically scale invariant version of the standard model in which the only dimensionful coupling (the Higgs squared term) arises from an anomaly. They claim that their model is compatible with the current data and would imply that LHC sees the Higgs and only the Higgs. Daniel Litim gave a nice overview over the asymptotic safety scenario for gravity. Bergshoeff and Skenderis talked about models related to 3d topologically massive gravity and Jose Figueroa-O-Farrill presented a summary of algebraic structures relevant for M2 theories. Mavromatos discussed possible observations of time delays in gamma ray bursts and implications for bounding modifications of dispersion relations in quantum gravity. Steve Giddings talked about locality and unitarity in connection with black hole information loss and Catherine Meuseburger explained how in 3d gravity observers can make geometrical measurements with light rays to find the gauge invariant information on in which Ricci-flat world they are living. I was surprised how many people still work on non-commutative geometry (in the various forms). The Moyal-plane, however, seems to be out of fashion (not so much because of UV-IR-mixing which I think is the main reason to be careful but many people seem to think they can work around that but are worried about unitarity on the other hand). Kappa-Minkowski is a space many people care about and Dopplicher explained why we live in quantum space-time. The general attitude seemed to be (surprisingly) that Lorentz-breaking in those theories is not an issue. However, Piacitelli, showed a calculation that should have been done quite a while ago: People say that although Lorentz invariance is broken that is not a problem since there is a twisted co-product version that preserves at least some related quantum symmetry. Piacitelly now spelled out what that means in everyday's terms: When you do a boost or rotation, twisting the co-product is equivalent to treating theta as a tensor and rotating that as well. Great, that explains why the formalism shows that rotational symmetry is preserved while the physics clearly says that a tensor background field singles out preferred directions. I had for a long time the suspicion that this is what is behind this Hopf-algebra approach but could never motivate myself enough to understand that in detail so I could confirm it. In addition, there were many talks from loop-related people (also on spin foams, BF-type theories etc) about which I would like to mention just one: Modesto applied the reasoning found in the loop approach to cosmology (I would like to say more about this in a future posting) to a spherically symmetric space-time (i.e. what is Schwarzschild in the classical theory). What he finds is indeed Schwarzschild at large distances but the discretization inherent in that approach produces a solution that has a T-duality like R <--> l_p^2/R symmetry. A great opportunity for meetings of this style with people coming from different approaches are always extended discussion sessions. Once more, those were a great plus (although not as controversial as a few years back in Bad Honnef), there were two, one on quantum gravity and one on non-commutative geometry. There, once more, people complained that it is hard to do this kind of physics without new experimental input. Of course to a large degree, this is true. But to me it seems that also misses an important point: By no means, everything goes! At least you should be able to make sure you are really talking about gravity in the sense that in not so extreme regimes you recover well known physics (Newton's law for example). Above, I mentioned Horava gravity apparently failing that criterion and it seems many other approaches are not even there to be tested in that respect. We often say, we work on strings because it is the only game in town. On that meeting you could have a rather different impression: It seemed more like everybody was playing more or less their on game and many didn't even know the name of their game. Another example of such a trivial non-trivial test is what your theory says about playing snooker: The kinetics of billard balls tests tensor products of Poincare representations of objects with trans-planckian momenta and energies. If your approach predicts weird stuff because it does not allow for trans-planckian energies my interpretation would be that you face hard times phenomenologically, even if your model agrees with CMB polarizations. Monday, June 29, 2009 More conference blogging Instead of String '09 I decided to attend this year the Born Symposium on the Planck scale. There are a number of stringy speakers as well as quite a few people from the loop camp. Watch this space for some reports. The talks (including video) will be online as well (as opposed to Strings '09). Wednesday, May 06, 2009 More quality journals by Elsevier After all the fun re the El Nashie fan board "Chaos, Solitons and Fractals" it seems Elsevier has put another nail in their coffin by allowing the pharma company Merck to run their pseudo scientific marketing journal under their flag: Merck Makes Phony Peer-Review Journal | blog.bioethics.net . Once more, John Baez has more details: The Foibles of Science Publishing | The n-Category Café Great, I think nobody in the world can claim anymore that our libraries should throw big money at these commercial publishing houses because they provide the quality control that open access publication cannot provide. Monday, April 27, 2009 Journal club cgi For our journal club, I wrote a small cgi script that provides a web page where people can dump arxiv.org identifies of papers they are interested in and then everybody can see title, abstract and authours as well as a link to the pdf. You can copy the file to your cgi-bin directory, make it world executable and create a world writeable directory /opt/journal . You might need to install the XML::TreeBuilder module (as root run 'perl -MCPAN -e shell' and then do 'install XML::TreeBuilder'. #!/usr/bin/perl # # Journal club cgi # Written by Robert C. Helling (helling@atdotde.de) # Published under the Gnu Public License (most recent version) # use XML::TreeBuilder; use LWP::Simple; use CGI; my$g = CGI::new();
print "Content-type: text/html\n\n";
my @ids = ();
system "touch /opt/journal/numbers";
open (IN,'/opt/journal/numbers') ||die "Cannot open /opt/journal/numbers:$!!";; while(){ chomp; if(/(\d\d\d\d\.\d\d\d\d)/){ push @ids,$1;
}
elsif(/^\-\-\-/){
@ids = ();
}
}
close IN;
if($new_paper =$g->param('paper_id')){
my ($new_id) =$new_paper =~ /(\d\d\d\d\.\d\d\d\d)/;
push @ids,$new_id; open (OUT, ">>/opt/journal/numbers") || die "Cannot write to /opt/journal/numbers:$!";
print OUT "$new_id\n"; print "\nNEW$new_id\n";
close OUT;
}
my %seen = ();
@ids = grep { ! $seen{$_} ++ } @ids;
dbmopen %papers, '/opt/journal/papers', 0666;
print join '
<hr>', map {&show($_)} @ids; print ($g->start_form(),
$g->textfield('paper_id'),$g->submit(-name => "add paper"),
$g->end_form()); sub show{ my$id = shift;
my $data; unless($data = $papers{$id}){
$data = get("http://export.arxiv.org/oai2?verb=GetRecord\&identifier=oai:arXiv.org:$id\&metadataPrefix=arXivRaw");
$papers{$id} = $data; } my$all = XML::TreeBuilder->new;
$all->parse($data);
my $authors =$all->find('authors')->as_text;
my $title =$all->find('title')->as_text;
my $abstract =$all->find('abstract')->as_text;
return "$authors\n<a href=\"http://arxiv.org/pdf/$id\">$title</a>\n$abstract\n";
}
dbmclose %papers;
Sunday, March 01, 2009
Posts disappearing
As I have noted earlier, some posts by John Baez and others on certain crackpots having their own Elsevier journal to dump their E-infinity weirdness have disappeared from the net. Luckily, one brave soul keeps copies of this stuff.
Now, I had to learn, that they completely overtook the discussion section of the online version on a Scientific American article on dynamical triangulations and what is even more disturbing, made the weekly quality newspaper "Die Zeit" take down the online version of an article reporting these issues (in German). I would have thought it would take a lot to make a respectable paper with a law department to commit this kind of self censorship.
Tuesday, February 17, 2009
Initial data for higher order equations of motion
Over lunch, today, we had a discussion on higher order quantum corrections in the effective action. You start out with a classical action that only contains terms with up to two derivatives. This corresponds to equations of motion that are second order in time. As such, for the physical degrees of freedom (but I want to ignore a possible gauege freedom here) you then have to specify the field and its time derivate on a Cauchy surface to uniquely determine the solution.
Loop corrections, however, tyically lead to terms with any number of derivatives in the effective action. Corresponding equations of motion allow then for more initial data to be specified. The question then is what to do with the unwanted solutions. If you want this is the classical version of unitarity.
Rather than discussing higher derivative gravity (where our lunch discussion took off) I would like to discuss a much simpler system. Say, we have a one dimensional mechanical system and the classical equation of motion is as simple as it can get, just . To simplify things, this is only first order in time and I would like to view a second order term already as "small" correction. The higher order equation would then be with small .
To find solutions, one uses the ansatz and finds . For small , the two exponents behave as which blows up and which approaches the solution of the "classical equation".
The general solution is a linear combination of the two exponential functions. We see that the solution blows up over a time-scale of unless the initial data satisfies the classical equation .
We can turn this around and say that if the classical equation is satisfied initially, we are close to the classical solution for long time (it's not exactly the same since differs from the "classical exponent" by order terms. For other initial data, the solution blows up exponentially on a "quantum time" inversely proportional to the small parameter .
This plot shows for . On the axis that goes into the picture there is a parameter for the initial conditions which is for data satisfying the classical equation initially. You can see that this parameter determines if goes to or over short time. Only the classical initial data stays small for much longer.
Unfortunately, this still leaves us with the question of why nature chooses to pick the "classical" initial data and seems not to use the other solutions. In the case of higher order gravity there is of course an anthropic argument that suggests itself but I would rather like to live without this. Any suggestions? | 2021-09-23 14:04:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6748787760734558, "perplexity": 1016.3006122342701}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057424.99/warc/CC-MAIN-20210923135058-20210923165058-00279.warc.gz"} |
https://direct.mit.edu/tacl/article/doi/10.1162/tacl_a_00390/102844/Pretraining-the-Noisy-Channel-Model-for-Task | ## Abstract
Direct decoding for task-oriented dialogue is known to suffer from the explaining-away effect, manifested in models that prefer short and generic responses. Here we argue for the use of Bayes’ theorem to factorize the dialogue task into two models, the distribution of the context given the response, and the prior for the response itself. This approach, an instantiation of the noisy channel model, both mitigates the explaining-away effect and allows the principled incorporation of large pretrained models for the response prior. We present extensive experiments showing that a noisy channel model decodes better responses compared to direct decoding and that a two-stage pretraining strategy, employing both open-domain and task-oriented dialogue data, improves over randomly initialized models.
## 1 Introduction
Task-oriented dialogue agents provide a conversational interface to assist users in accomplishing specific goals, such as finding a restaurant or booking a hotel (Seneff and Polifroni, 2000; Raux et al., 2005; Budzianowski et al., 2018; Peng et al., 2020a). Increasing demand from industry for natural language assistants and scalable customer service solutions has recently been driving a renaissance in the development of task-oriented dialogue models. In addition, the specification of explicit dialogue agent goals, afforded by the task-oriented paradigm, makes such research easier to ground and evaluate than open-domain chatbots.
Current research on task-oriented dialogue is dominated by monolithic sequence-to-sequence models that directly parameterize the conditional distribution of the response given the prior dialogue context. However, this monolithic approach conflates the task-specific and language-general aspects of dialogue, and adversely favors short and generic responses (Bao et al., 2020) due to the explaining-away effect (Klein and Manning, 2002).
Here we pursue an alternative to the direct model. Using Bayes’ rule allows us to factorize the probability of the response given the context $p(R|C)$ into a language model $p(R)$ and a context model $p(C|R)$.1 Within natural language processing (NLP), this approach is traditionally known as the noisy channel model (Shannon, 1948), and has recently seen renewed interest with its successful application to neural machine translation (Yu et al., 2017, 2020; Yee et al., 2019).
We hypothesize that the noisy channel reformulation is advantageous for dialogue because the factorization enables each sub-module to specialize in a dialogue sub-task. In particular, the context conditional model can help to discount short and generic responses and mitigate the explaining-away effect, while the language model helps ensure that responses are natural. We find that a noisy channel model with the same number of parameters as a direct model achieves better accuracy on three task-oriented dialogue datasets. Moreover, a larger noisy channel model can be trained with the same hardware, by training the sub-modules separately, yielding additional improvements.
It has become common in recent years to pretrain dialogue models on large text data, either general text (Peng et al., 2020b; Budzianowski and Vulić, 2019; Wu et al., 2020a) or dialogue-structured data (Roller et al., 2020; Adiwardana et al., 2020), such as tweets and Reddit posts. We utilise a similar strategy with Reddit data and find that the benefits of pretraining to the noisy channel model are similar to those for the direct model. Further, we evaluate transfer across task-oriented dialogue datasets by implementing a second pretraining stage using Taskmaster (Byrne et al., 2019) and Schema-Guided Dialogue (Rastogi et al., 2020) as training data, before fine-tuning on our final tasks.
We evaluate the algorithm on three datasets, MultiWOZ 2.0 (Budzianowski et al., 2018), CamRest676 (Wen et al., 2017a), and SMCalFlow (Andreas et al., 2020), demonstrating that the noisy channel approach is robust to different dialogue schema annotations used across datasets. Further analysis demonstrates that the noisy channel models can decode responses with similar lengths and Zipf scores compared to ground-truth responses and reduce the likelihood of falling into repetition loops (Holtzman et al., 2019).
## 2 A Seq-to-Seq Dialogue Model
In this section, we introduce a discriminative sequence-to-sequence model for task-oriented dialogue. The traditional sequence of steps needed to produce a system turn in a task-directed dialogue is shown in Figure 1, with an example from MultiWOZ 2.0 (Budzianowski et al., 2018). Given a dialogue context containing previous user and system utterances, the dialogue system first predicts a belief state, consisting of a set of slot- value pairs (e.g., destination: Cambridge), to capture user intent. To ground the system with external information, the belief state can be converted into a database query in order to retrieve relevant information, such as the number of matches and booking information. Next, the system predicts a set of dialogue acts, representing the abstract meaning of the proposed dialogue response (Austin, 1975). Finally, a delexicalized dialogue response is generated, where slot values are replaced by generic placeholders, such as value_time for a train departure time, in order to reduce lexical variation. The delexicalized response can be converted to a lexicalized response in post-processing by filling in the slot values based on belief states and database information.
Figure 1:
The data flow of one turn in a task-oriented dialogue for train booking from MultiWOZ.
Figure 1:
The data flow of one turn in a task-oriented dialogue for train booking from MultiWOZ.
We use the MultiWOZ schema for illustration in Sections 2 and 3, but our models easily generalize to different schema annotations (e.g., datasets without annotated dialogue acts [Andreas et al., 2020]).
Because it is well known that pipelined models tend to suffer from error propagation, many NLP tasks have been reformulated in recent years as end-to-end text-to-text transformations (Raffel et al., 2020; Brown et al., 2020). State- of-the-art task-oriented dialogue systems have followed this approach (Hosseini-Asl et al., 2020; Peng et al., 2020b). We represent the example from Figure 1 as follows, serializing turns and using special start and end tokens to encapsulate each data field:
Given this text representation, the direct discriminative approach models $p(B,A,R|C)$, where $C$, ℬ, $A$, and $R$ represent dialogue context, belief state, dialogue act, and delexicalized response, respectively.2 We use the serialized text of the dialogue context as input, and the concatenation of belief state, dialogue act, and response as target output, making the task amenable to the application of an autoregressive sequence-to-sequence model. ℬ, $A$, and $R$ can be generated sequentially with direct decoding methods, such as greedy decoding and beam search. We use a sequence-to-sequence Transformer (Vaswani et al., 2017) to implement $p(B,A,R|C)$. This distribution will also be used to build the noisy channel model in Section 3.
## 3 Noisy Channel Model for Dialogue
While direct decoding is an effective approach for decoding belief states (Hosseini-Asl et al., 2020), it may be sub-optimal for generating responses. First, it favors short and generic responses (Bao et al., 2020). As a result, the decoded responses are bland and lack diversity (Li et al., 2016). Second, it suffers from the explaining-away effect (Klein and Manning, 2002), where inputs are “explained-away” by highly predictive output prefixes. For example, if there is one hotel matching the user’s intent as encoded in the belief state, the model is nevertheless prone to decoding “no” given the output prefix “there is”, ignoring the input information.
In this work, we propose using the neural noisy channel model (Yu et al., 2017) to mitigate the above problems for response generation. Given an input sequence x and output sequence y, the noisy channel formulation (Shannon, 1948) uses Bayes’ rule to rewrite the model p(y|x) as $p(x|y)p(y)p(x)∝p(x|y)p(y)$. It was originally applied to speech recognition, where p(y|x) is a conditional model of the source text given a noisy observation. The channel modelp(x|y) estimates the probability of the observation given the source, while p(y) is an unconditional language model (or source model), which can be trained on unpaired data. More recently it has been applied to machine translation, where y is a translation of input text x.
Abstracting away from belief states and dialogue acts, for task-oriented dialogue we want to estimate $p(R|C)$, the probability of a response given a context. The channel model $p(C|R)$, given a response, predicts a distribution over contexts which might have elicited that response. The source model $p(R)$ is an unconditional language model. In this extension of the noisy channel approach to task-oriented dialogue, the “channel” can be understood as connecting dialogue contexts with suitable responses.
For the full task, we develop a noisy channel model for $p(B,A,R|C)$. Using the chain rule, $p(B,A,R|C)=p(B|C)⋅p(A,R|C,B)$. Following Hosseini-Asl et al. (2020), we use the direct model described in Section 2 to parameterize $p(B|C)$ and decode ℬ, which our preliminary experiments confirmed to be advantageous.
We use the noisy channel formulation to parameterize $p(A,R|C,B)$. Using Bayes’ rule, $p(A,R|C,B)∝p(C,B|A,R)⋅p(A,R)$. The channel model $p(C,B|A,R)$ and source model $p(A,R)$ are implemented as Transformers.
We choose to use the noisy channel formulation for decoding $A$ based on preliminary experiments that showed improved overall accuracy over direct decoding, possibly because poor dialogue act prediction by the direct model led to worse quality responses. The serialized text of $A$ and $R$ are concatenated during training, and the decoded sequence is split into $A$ and $R$ with the special start/end tokens during decoding.
We suggest that the noisy channel model has three advantages over the direct model for response generation: (1) The channel model can penalize short and generic responses. Such responses can be mapped to a large number of contexts, resulting in a flat distribution over contexts. This leads to a lower channel model score for short and generic responses (Zhang et al., 2020b). (2) The channel model ensures that $(A,R)$ must explain the corresponding $(C,B)$, alleviating the explaining-away effect (Yu et al., 2017). (3) The source model, an unconditional distribution over $A$ and $R$, can make use of abundant non-dialogue textual data for pretraining, further improving the fluency of generated sequences (Brants et al., 2007). We leave exploration of this last advantage for future work, as we pretrain all sub-modules with the same data.
### 3.1 Decoding
Because exact decoding from the noisy channel model $arg maxA,Rp(C,B|A,R)⋅p(A,R)$3 is computationally intractable, we experiment with two approximation methods, noisy channel reranking and noisy channel online decoding. Since these methods rely on $p(A,R|C,B)$ as a proposal distribution for approximation, and both $p(A,R|C,B)$ and $p(B|C)$ are parameterized with the direct model introduced in Section 2, our noisy channel model therefore has three sub-modules: a direct model $p(B,A,R|C)$, a channel model $p(C,B|A,R)$, and a source model $p(A,R)$.
Noisy Channel Reranking: Noisy channel reranking first decodes ℬ and then continues decoding a list $S$ of $(A,R)$ pairs by beam search with the direct model, prior to utilizing the noisy channel model to rerank $(A,R)$ pairs. In particular, during beam search, partial sequences are expanded and pruned with $p(A,R|C,B)$ (from the direct model in Section 2). The pairs after decoding are reranked using the following model combination:
$(A′,R′)=arg max(A,R)∈Slogp(A,R|C,B)+λ1⋅logp(C,B|A,R)+λ2⋅logp(A,R)+λ3⋅|A,R|,$
(1)
where $|A,R|$ denotes the length of $(A,R)$, and λ1, λ2 and λ3 are hyperparameters. Besides the channel model $p(C,B|A,R)$ and the source model $p(A,R)$, we additionally use the direct model $p(A,R|C,B)$ and a length bias $|A,R|$ to encourage responses with high direct model likelihood and discourage short responses, respectively.
Noisy Channel Online Decoding: In contrast to reranking, online decoding applies the noisy channel model during beam search for pruning partial sequences, thus exploring a larger search space.
As shown in Algorithm 1, we first decode the belief state with $p(B|C)$, which comes from the direct model in Section 2. Then, starting with a beam $S$ containing a single sequence [a] (the dialogue act start token), we continuously expand the sequences in $S$ until end($S$) is met, namely, all sequences in $S$ either end with [/r] or have lengths larger than l. In each iteration, we first expand the sequences in the beam, then prune the expanded beam. To expand a partial act and response sequence (denoted as $O$ in Algorithm 1), a naive way is to use the noisy channel model to score |V | (the vocabulary size) possible expansions, which is computationally expensive. Instead, we use the probability of the next token $p(O|O|+1|C,B,O)$ (where $|O|$ denotes the length of $O$) to select k1 candidates to be scored by the noisy channel model. This next token probability is from the direct model introduced in Section 2. One straightforward way to select k1 expansions from $p(O|O|+1|C,B,O)$ is using the top-k maximization, but we can also take advantage of the advances in sampling from a categorical distribution for text generation (e.g., top-k sampling Fan et al., 2018 and nucleus sampling [Holtzman et al., 2019]). After the expansion, we prune the expanded beam $S′$ to obtain a smaller beam with k2 partial sequences based on the model combination in Eq. 1. Compared to noisy channel reranking, online decoding applies the noisy channel model during beam search, which is potentially less biased towards the direct model.
In summary, we note that beam search for both the direct model and the online decoding for our noisy channel model decodes ($B,A,R$) autoregressively. Thus both approaches are end-to-end models for task-oriented dialogue. The key difference is that noisy channel online decoding uses Eq. 1 for pruning, while the direct model uses $p(A,R|C,B)$.
## 4 Model and Pretraining
We use three Transformer (Vaswani et al., 2017) networks to parameterize the direct model $p(B,A,R|C)$, the channel model $p(C,B|A,R)$ and the source model $p(A,R)$, respectively. The input to each Transformer is the sum of four embeddings: word embeddings, position embeddings, role embeddings (user/system), and turn embeddings (each word corresponds to a turn number). Cross entropy is used as the loss function.
Given training samples $(C,B,A,R)$, if we train the channel model using complete $(A,R)$ pairs as input, a significant discrepancy arises between training and decoding for noisy channel online decoding. Since the channel model is used to score partial act and response pairs, that is, $p(C,B|O)$ in Algorithm 1, the channel model trained with complete $(A,R)$ pairs is unsuited to scoring partial sequences. In order to manually create partial sequences during training that are better matched for online decoding, we truncate the $(A,R)$ pairs with a truncation length uniformly sampled from 1 to the sequence length (inclusive). The direct model and the source model are trained with complete sequences, as partial sequences occur naturally in their standard autoregressive training procedure.
As in-domain dialogue data are usually scarce, we use a two-stage pretraining strategy to enhance the noisy channel model. Although the effectiveness of pretraining with Reddit data has been validated for open-domain dialogue (Zhang et al., 2020b; Bao et al., 2019; Adiwardana et al., 2020), relatively little work has applied such data to task-oriented dialogue.4 In the first stage, we explore Reddit pretraining (where the Reddit data is pre-processed into $(C,R)$, i.e., context-response, pairs as described below). In the second stage, we use two task-oriented dialogue datasets, Taskmaster5 (Byrne et al., 2019) and Schema-Guided Dialogue6 (Rastogi et al., 2020), to specialize the Reddit-pretrained models. Because the Reddit data consists of open-domain-style dialogues (where belief states and dialogue acts are missing), pretraining on these datasets can familiarize the models with the sequence-to-sequence representation of task-oriented dialogue. Three models, a context-to-response model, a response-to-context model and a response language model, are pretrained to initialize the direct model, the channel model and the source model, respectively.
### 4.1 Implementation Details
Models: All models are implemented with JAX (Bradbury et al., 2018) and Haiku (Hennigan et al., 2020). For the direct model introduced in Section 2, we use a Transformer model with hidden size 512, 12 encoder-decoder layers, and 16 self-attention heads. The model has 114M parameters. For the noisy channel model, we use a base setting and a large setting. The base setting reduces the number of layers to 5, hidden size to 384, and self-attention heads to 12. Its sub-modules, a direct model, a reverse model and a language model, have 43M, 43M, and 30M parameters, respectively. We employ the base setting for a fair comparison with a single direct model using roughly the same number of parameters (116M vs. 114M). For the large setting, we use the same hyperparameters as the direct model (114M), so that its sub-modules, a direct model, a reverse model, and a language model, have 114M, 114M, and 64M parameters, respectively. We use this large setting to explore the limits of the noisy channel model. The large noisy channel model (292M) is 2.56 times larger compared to the direct model (114M). This illustrates another advantage of the noisy channel model during training. While training a direct model with 292M parameters will overflow the memory of 16GB TPUs (v3) without using model parallelism, training the sub-modules of the large noisy channel model can easily fit into 16GB TPUs, as these modules are independently trained with no need to load three modules for training. This enables us to train a noisy channel model with more parameters compared to training a direct model using the same hardware. For inference, we still need to load the sub-modules into a TPU. Because gradients are not required during inference, we are able to load the three sub-modules of the large noisy channel model (292M) into a single TPU with 16GB memory for decoding. The large noisy channel model (292M) still consumes more memory than the direct model (114M) during inference.
Pretraining Settings: The maximum sequence length l is set to 1024, and sequences with longer lengths are truncated. We reuse the vocabulary from GPT-2 (Radford et al., 2019), which contains 50,257 BPE tokens. We use PreNorm (Nguyen and Salazar, 2019) for faster convergence. GELU (Hendrycks and Gimpel, 2016) is applied as the activation function. Following ALBERT (Lan et al., 2020), dropout is disabled during pretraining. We use the normal distribution truncated to the range [−0.01,0.01] to initialize the input embeddings, while other parameters are initialized using the normal distribution with zero mean and standard deviation 0.1. The batch size is set to 256. The LAMB optimizer (You et al., 2020) (b1 = 0.9 and b2 = 0.999) is employed for optimization. The initial learning rate is 1e-7, and we apply 4000 warmup steps to increase the learning rate to 1e-3, before utilizing cosine annealing to decay the learning rate. Gradient clipping with clipping value 1 is applied to avoid gradient explosion. We use gradient accumulation with accumulation step 20.
Pretraining: For Reddit pretraining, we download a Reddit dump (with Reddit posts ranging from 2005-12 to 2019-09) from PushShift.7 Since the comments of a Reddit post are organized into a tree, we extract paths from a tree as dialogue turns. The last comment of each comment path is regarded as the response, while the others are used as the dialogue context. We pretrain each model for 400,000 steps, consuming 102,400,000 (400,000 × 256) comment paths in total. For the task-oriented pretraining, we combine the two datasets, Taskmaster and Schema-Guided Dialogue, and pretrain for 1e5 steps. The statistics of the task-oriented dialogue datasets are shown in Table 1.
Table 1:
Statistics of task-oriented dialogue datasets. We define a multi-task dialogue as a dialogue involving multiple tasks (e.g., hotel and restaurant booking) while its counterpart handles a single task (e.g., hotel booking). Taskmaster and CamRest676 do not contain any multi-task dialogues.
Dataset# Dialog# TurnAvg. Turn/DialogAvg. Token/Turn# DomainMulti-Task# Unique Slot# Unique Value
Taskmaster 17,304 341,801 19.75 7.87 ✗ 281 66,659
Schema 22,825 463,284 20.3 9.86 17 ✓ 123 23,889
CamRest676 676 5,488 8.12 10.71 ✗ 89
MultiWOZ 10,438 143,048 13.7 15.03 ✓ 46 11,828
SMCalFlow 41,517 170,590 4.11 8.77 ✓ – –
Dataset# Dialog# TurnAvg. Turn/DialogAvg. Token/Turn# DomainMulti-Task# Unique Slot# Unique Value
Taskmaster 17,304 341,801 19.75 7.87 ✗ 281 66,659
Schema 22,825 463,284 20.3 9.86 17 ✓ 123 23,889
CamRest676 676 5,488 8.12 10.71 ✗ 89
MultiWOZ 10,438 143,048 13.7 15.03 ✓ 46 11,828
SMCalFlow 41,517 170,590 4.11 8.77 ✓ – –
We train each model using 64 TPU chips with 16GB memory each. The pretraining takes around 4 days to complete.
## 5 Experiments
We fine-tune and evaluate the pretrained models on three dialogue datasets: MultiWOZ 2.0, CamRest676 and SMCalFlow (Andreas et al., 2020). In this section we describe the datasets (Section 5.1), fine-tuning (Section 5.2), decoding (Section 5.3), and evaluation metrics (Section 5.4). Results are presented in Section 6, and analysis and ablation studies in Section 7.
### 5.1 Datasets
MultiWOZ8 is a multi-domain dataset consisting of dialogues annotated with $C,B,A,R$ in the following seven domains: attraction, hotel, hospital, police, restaurant, train, and taxi. Since its release, MultiWOZ has been one of the most commonly used task-oriented dialogue datasets.
CamRest6769 is annotated similarly to MultiWOZ and consists of dialogues in a single domain: restaurant reservations. Though CamRest676 is smaller than MultiWOZ and predates it, it still provides a widely used benchmark for evaluating task-oriented dialogue models.
SMCalFlow consists of dialogues in four domains: calendar, weather, places, and people. Unlike MultiWOZ and CamRest676, SMCalFlow uses dataflow graphs instead of slot-value pairs to represent belief states and does not annotate dialogue acts. We refer readers to Andreas et al. (2020) for a detailed description of the dataflow representation. We follow Andreas et al. (2020) to convert dataflow graphs into sequences to apply seq2seq models. This dataset is newer and offers fewer prior models to compare with, but we use this dataset to study the robustness of the noisy channel model under different annotation schemas.
We use the public splits for these datasets, where MultiWOZ, CamRest676 and SMCalFlow are split to 8438/1000/1000, 404/136/136, and 32647/3649/5211 dialogues for training, development, and testing, respectively. However, because SMCalFlow’s test set has not been publicly released, we randomly select 500 dialogues from its training set to tune hyperparameters and use its development set for testing.
Preprocessing: We use the standard preprocessing procedures for each dataset in order to facilitate fair comparison with previous methods.10,11,12 In particular, for MultiWOZ and CamRest676, delexicalization is used to reduce lexical variation, while SMCalFlow does not use delexicalization. During delexicalization, slot values are replaced by generic placeholders based on a pre-defined dictionary. During decoding, following prior work, our dialogue models generate delexicalized responses. These delexicalized responses are re-lexicalized in post-processing by replacing placeholders with their corresponding slot values based on belief states and database information. Since there is no public code for lexicalization,13 we implement our own functions for lexicalization with regular expressions, for the purpose of displaying example responses. However, this does not affect reported results, as the standard metrics for MultiWOZ and CamRest676 that we adopt here are calculated using delexicalized responses.
### 5.2 Fine-Tuning
We apply label smoothing with parameter 0.1. Dropout is used on input embeddings and hidden representations, with dropout rate 0.1. The Adam optimizer (Kingma and Ba, 2015) (b1 = 0.9 and b2 = 0.999) is adopted. We use a fixed learning rate 1e-4 with gradient clipping for fine-tuning.
### 5.3 Decoding
We use direct decoding for belief state. For dialogue act and response, we study three decoding methods: direct decoding, noisy channel reranking, and noisy channel online decoding. Since all of these decoding methods require choosing k1 tokens from a categorical distribution during expansion, we compare four methods, top-k maximization, sampling without replacement, top-k sampling, and nucleus sampling. Nucleus sampling with cumulative probability 0.98 performs marginally better and is adopted. We perform a range search with the range [1,20] on development sets for the beam sizes k1 and k2, and we set k1, k2 = 4, k1, k2 = 15, and k1, k2 = 4 for MultiWOZ, CamRest676, and SMCalFlow, respectively. For noisy channel reranking and noisy channel online decoding, a grid search with range [0,2] is performed for λ1, λ2, and λ3. We set (λ1 = 0.8, λ2 = 1, λ3 = 0.8), (λ1 = 1.2, λ2 = 1.2, λ3 = 0.8), and (λ1 = 0.4, λ2 = 1, λ3 = 0.2) for MultiWOZ, CamRest676, and SMCalFlow, respectively.
### 5.4 Evaluation Metrics
For MultiWOZ and CamRest676, following previous work, we adopt three automatic evaluation metrics: inform, success, and BLEU score. Peng et al. (2020a) showed that these metrics are well correlated to human evaluation. The evaluators14 ,15 provided with the datasets are used for calculating these metrics. To calculate the inform score for a dialogue, the evaluator first checks whether certain placeholders (e.g., [restaurant_name]) appear in decoded responses. If so, decoded belief states are converted to database queries to retrieve database records. These database records are compared with the records retrieved with ground-truth belief states. The inform score is one if these two sets of database records match. The success score takes all the requestable slots (e.g., postcode, phone number, and address) from a decoded response and compares these requestable slots with the ones in the ground-truth response. The success score is one if generated requestable slots coincide with the ground-truth ones. BLEU score (BLEU-4) compares the n-grams of generated responses and human responses, and is a widely used metric in NLP for evaluating text quality. Following Budzianowski et al. (2018), we also calculate a combined score, which is (Inform + Success) / 2 + BLEU. For SMCalFlow, inform and success scores are not applicable because calculation of these scores relies on delexicalization placeholders, and this dataset does not use delexicalization. We use SacreBLEU16 and TER17 to directly measure the quality of responses. As prior work on this dataset has focused on belief tracking rather than end-to-end response generation, we are the first to use these metrics on this dataset.
We perform significance tests, where we use t-test for inform, success, and TER scores and use permutation test for BLEU.
## 6 Results
MultiWOZ: Results on the MultiWOZ test set are shown in Table 2. We observe several trends. First, the base noisy channel model (116M) performs better than direct decoding (114M), despite having a similar number of parameters, showing that the noisy channel factorization is beneficial for task-oriented dialogue. The large noisy channel setting improves further over the base setting. Second, Reddit pretraining provides benefits over random initialization, validating the use of large open-domain dialogue-genre pretraining for task-oriented dialogue, while the models with a second stage of task-oriented pretraining obtain further improvements. This effect is consistent across both direct and noisy channel decoding. Finally, we observe that online decoding consistently outperforms reranking, indicating the benefits of tighter model integration during decoding.
Table 2:
MultiWOZ test results (end-to-end modeling with generated beliefs) with seq2seq approaches. Results are significant (p < 0.01) comparing noisy channel decoding and direct decoding. †Yang et al. (2021) also report a combined score of 105.1 with an alternative context and evaluation setting, contributions orthogonal to our work and the other benchmarks reported here.
ModelInformSuccessBLEUCombined
Sequicity (Lei et al., 201866.4 45.3 15.54 71.39
HRED-TS (Peng et al., 201970.0 58.0 17.50 81.50
DSTC8 Track 1 Winner (Ham et al., 202073.0 62.4 16.00 83.50
DAMD (Zhang et al., 2020a76.4 60.4 16.60 85.00
SimpleTOD (Hosseini-Asl et al., 202084.4 70.1 15.01 92.26
SOLOIST (Peng et al., 2020a85.5 72.9 16.54 95.74
UBAR (Yang et al., 2021) 88.2 79.5 16.43 100.28
Randomly Initialized
Direct decoding (114M) 81.0 54.7 15.12 82.97
Noisy channel reranking (116M) 82.7 57.1 15.29 85.19
Noisy channel online decoding (116M) 82.9 58.9 15.33 86.23
Noisy channel reranking (292M) 82.1 58.1 15.37 85.47
Noisy channel online decoding (292M) 83.9 60.9 15.57 87.97
Reddit Pretraining
Direct decoding (114M) 81.0 69.2 17.06 92.16
Noisy channel reranking (116M) 81.3 70.1 19.01 94.71
Noisy channel online decoding (116M) 81.6 71.1 19.31 95.66
Noisy channel reranking (292M) 82.2 70.9 19.89 96.44
Noisy channel online decoding (292M) 82.4 71.7 20.49 97.54
Direct decoding (114M) 85.2 72.9 17.00 96.05
Noisy channel reranking (116M) 85.6 73.8 19.38 99.08
Noisy channel online decoding (116M) 85.9 74.8 19.76 100.11
Noisy channel reranking (292M) 86.5 74.9 20.31 101.01
Noisy channel online decoding (292M) 86.9 76.2 20.58 102.13
ModelInformSuccessBLEUCombined
Sequicity (Lei et al., 201866.4 45.3 15.54 71.39
HRED-TS (Peng et al., 201970.0 58.0 17.50 81.50
DSTC8 Track 1 Winner (Ham et al., 202073.0 62.4 16.00 83.50
DAMD (Zhang et al., 2020a76.4 60.4 16.60 85.00
SimpleTOD (Hosseini-Asl et al., 202084.4 70.1 15.01 92.26
SOLOIST (Peng et al., 2020a85.5 72.9 16.54 95.74
UBAR (Yang et al., 2021) 88.2 79.5 16.43 100.28
Randomly Initialized
Direct decoding (114M) 81.0 54.7 15.12 82.97
Noisy channel reranking (116M) 82.7 57.1 15.29 85.19
Noisy channel online decoding (116M) 82.9 58.9 15.33 86.23
Noisy channel reranking (292M) 82.1 58.1 15.37 85.47
Noisy channel online decoding (292M) 83.9 60.9 15.57 87.97
Reddit Pretraining
Direct decoding (114M) 81.0 69.2 17.06 92.16
Noisy channel reranking (116M) 81.3 70.1 19.01 94.71
Noisy channel online decoding (116M) 81.6 71.1 19.31 95.66
Noisy channel reranking (292M) 82.2 70.9 19.89 96.44
Noisy channel online decoding (292M) 82.4 71.7 20.49 97.54
Direct decoding (114M) 85.2 72.9 17.00 96.05
Noisy channel reranking (116M) 85.6 73.8 19.38 99.08
Noisy channel online decoding (116M) 85.9 74.8 19.76 100.11
Noisy channel reranking (292M) 86.5 74.9 20.31 101.01
Noisy channel online decoding (292M) 86.9 76.2 20.58 102.13
Our model performs better on combined score than SOLOIST (Peng et al., 2020a), a closely related baseline that pretrains a GPT2-initialized Transformer with Taskmaster and Schema-Guided Dialogue and decodes with nucleus sampling.
CamRest676: Results on the CamRest676 test set are shown in Table 3. We observe that the base noisy channel model (116M) obtains better results compared to direct decoding (114M), again demonstrating the effectiveness of the noisy channel model. Reddit pretraining again provides a large benefit over random initialization for both direct decoding and noisy channel decoding, while task-oriented pretraining provides a further boost. Our model again performs better than SOLOIST.
Table 3:
CamRest676 test results (end-to-end modeling with generated beliefs) with seq2seq approaches. Noisy channel reranking performs comparable with noisy channel online decoding, and the results are not shown. Results are significant (p ¡ 0.01) comparing noisy channel decoding and direct decoding.
ModelInformSuccessBLEUCombined
Sequicity (Lei et al., 201892.3 85.3 21.40 110.20
GPT-2 fine-tuned (Wu et al., 2019b– 86.2 19.20 –
ARDM (Wu et al., 2019b– 87.1 25.20 –
SOLOIST (Peng et al., 2020a94.7 87.1 25.50 116.40
Randomly Initialized
Direct decoding (114M) 78.1 83.5 21.58 102.38
Noisy channel online decoding (116M) 79.8 84.1 22.83 104.78
Noisy channel online decoding (292M) 80.9 84.9 23.19 106.09
Reddit Pretraining
Direct decoding (114M) 93.3 83.9 23.41 112.01
Noisy channel online decoding (116M) 93.7 84.5 25.14 114.24
Noisy channel online decoding (292M) 93.9 84.7 25.38 114.68
Direct decoding (114M) 93.4 84.3 24.92 113.77
Noisy channel online decoding (116M) 94.3 85.2 25.98 115.73
Noisy channel online decoding (292M) 95.4 85.3 26.89 117.24
ModelInformSuccessBLEUCombined
Sequicity (Lei et al., 201892.3 85.3 21.40 110.20
GPT-2 fine-tuned (Wu et al., 2019b– 86.2 19.20 –
ARDM (Wu et al., 2019b– 87.1 25.20 –
SOLOIST (Peng et al., 2020a94.7 87.1 25.50 116.40
Randomly Initialized
Direct decoding (114M) 78.1 83.5 21.58 102.38
Noisy channel online decoding (116M) 79.8 84.1 22.83 104.78
Noisy channel online decoding (292M) 80.9 84.9 23.19 106.09
Reddit Pretraining
Direct decoding (114M) 93.3 83.9 23.41 112.01
Noisy channel online decoding (116M) 93.7 84.5 25.14 114.24
Noisy channel online decoding (292M) 93.9 84.7 25.38 114.68
Direct decoding (114M) 93.4 84.3 24.92 113.77
Noisy channel online decoding (116M) 94.3 85.2 25.98 115.73
Noisy channel online decoding (292M) 95.4 85.3 26.89 117.24
SMCalFlow: Results on the SMCalFlow development set are shown in Table 4. As end-to-end models have not previously been tested on this dataset, we use it to demonstrate that the noisy channel model, which we developed primarily on MultiWOZ, continues to be effective on task-oriented dialogue datasets with different annotation schema. The results are consistent with MultiWOZ and CamRest676. The noisy channel model outperforms the direct model by a large margin, demonstrating that dialogue act annotations are not essential for the noisy channel model, and that it remains effective across diverse dialogue representations.
Table 4:
SMCalFlow results. Reranking performs worse than online decoding, and the results are not shown. Results are significant (p < 0.01) comparing noisy channel decoding and direct decoding.
ModelSacreBLEUTER
Randomly Initialized
Direct decoding (114M) 51.30 89.13
Online decoding (116M) 53.66 74.18
Online decoding (292M) 54.39 73.18
Reddit Pretraining
Direct decoding (114M) 60.68 61.99
Online decoding (116M) 63.29 47.16
Online decoding (292M) 63.91 46.43
Direct decoding (114M) 61.02 59.84
Online decoding (116M) 63.72 46.27
Online decoding (292M) 64.29 45.81
ModelSacreBLEUTER
Randomly Initialized
Direct decoding (114M) 51.30 89.13
Online decoding (116M) 53.66 74.18
Online decoding (292M) 54.39 73.18
Reddit Pretraining
Direct decoding (114M) 60.68 61.99
Online decoding (116M) 63.29 47.16
Online decoding (292M) 63.91 46.43
Direct decoding (114M) 61.02 59.84
Online decoding (116M) 63.72 46.27
Online decoding (292M) 64.29 45.81
Reddit pretraining confers a similar large benefit on SMCalFlow as on the other datasets, but we observe that task-oriented pretraining brings only marginal further improvements. This may be due to differences in domain or format between our pretraining datasets and SMCalFlow. Alternatively, task-oriented pretraining may help more on task-specific metrics, such as inform and success scores, than on text quality metrics such as BLEU and TER scores. This hypothesis is further supported by the MultiWOZ results in Table 2.
## 7 Analysis
In this section, we use MultiWOZ and CamRest676 to perform ablation studies on the effects of model combination, large-scale pretraining, and sample efficiency; as well as analyzing the runtime requirements of our model and the reasons for its success.
### 7.1 Ablation on Model Combination
Noisy channel decoding involves a combination of four sub-modules, as in Eq. 1: the direct model, channel model, language model, and length bias. We perform an ablation study to determine whether all model components are important to the result, using the large model. Results on the development sets of CamRest676 and MultiWOZ are presented in Table 5. Note that the ablation is performed after applying the direct model to obtain k1 expansions at each beam search step for noisy channel online decoding. We find that the combination of all four sub-modules performs the best, followed by combinations of three and then two sub-modules. The results are significant when comparing ‘All’ and the baselines (p < 0.01). This result demonstrates the effectiveness of the noisy channel factorization, and the importance of each model component.
Table 5:
Ablation results for model combination on development sets (combined score). Results for reranking are similar and are not shown. ‘All’, ‘Direct’, ‘Source’, and ‘Channel’ denote no ablation, direct model, source model and channel model, respectively. Rows with ‘+’ are combinations of two sub-modules, while the rows with ‘-’ are combinations of three sub-modules.
ModelCamRest676MultiWOZ
Direct decoding 115.17 96.73
Noisy Channel Online Decoding
Direct + Channel 115.63 98.54
Direct + Source 115.91 99.12
Direct + Length 115.56 97.57
Channel + Source 115.82 99.18
Channel + Length 115.60 98.71
Source + Length 115.62 99.19
All - Direct 115.96 100.89
All - Channel 116.56 100.93
All - Source 116.38 99.92
All - Length 116.52 101.11
All 116.91 102.62
ModelCamRest676MultiWOZ
Direct decoding 115.17 96.73
Noisy Channel Online Decoding
Direct + Channel 115.63 98.54
Direct + Source 115.91 99.12
Direct + Length 115.56 97.57
Channel + Source 115.82 99.18
Channel + Length 115.60 98.71
Source + Length 115.62 99.19
All - Direct 115.96 100.89
All - Channel 116.56 100.93
All - Source 116.38 99.92
All - Length 116.52 101.11
All 116.91 102.62
### 7.2 Effect of Pretraining Scale
We investigate the importance of scale for both our pretraining stages. We select different checkpoints for Reddit pretraining, and truncate the two task-oriented dialogue datasets for task-oriented pretraining. We fine-tune these models using the full training data of CamRest676 or MultiWOZ. The results of three decoding methods (with the large noisy channel model) on the development sets are shown in Figure 2. In Figure 2(a) and (c), the combined scores of all three decoding methods improve with more Reddit pretraining steps, demonstrating the advantage of increasing amounts of data in the open-domain dialogue pretraining stage. In Figure 2(b) and (d), the combined scores further increase with more task-oriented data, confirming that additional task-oriented pretraining data is useful.
Figure 2:
Results showing the effect of pretraining scale.
Figure 2:
Results showing the effect of pretraining scale.
### 7.3 Sample Efficiency of Fine-Tuning
We investigate whether pretraining can improve sample efficiency during fine-tuning. We gradually increase the amount of fine-tuning data and evaluate the randomly-initialized, Reddit pretrained and task-oriented pretrained models. The results on the development sets are shown in Figure 3. Combined scores increase with more training data under all conditions. Crucially, Reddit pretrained models show better performance with a smaller amount of fine-tuning data than randomly initialized models, and task-oriented pretrained models better still. We conclude that both our pretraining stages can improve sample efficiency, which is especially important when the target task has little training data.
Figure 3:
Pretraining improves sample efficiency during fine-tuning.
Figure 3:
Pretraining improves sample efficiency during fine-tuning.
### 7.4 Decoding Runtime
In Table 6, we report the average clock time for decoding one turn (including its belief state, dialogue act and response). Noisy channel reranking is slightly slower compared to direct decoding, with overhead due to the reranking step in Eq. 1. Noisy channel online decoding is significantly slower, since it needs to apply Eq. 1 at each beam search step. In future work we will investigate ways to improve the efficiency of online decoding.
Table 6:
Average decoding time (in seconds) for each turn with different decoding methods.
ModelCamRest676MultiWOZ
Direct decoding 4.89 6.48
Reranking 5.43 6.92
Online decoding 8.73 10.97
ModelCamRest676MultiWOZ
Direct decoding 4.89 6.48
Reranking 5.43 6.92
Online decoding 8.73 10.97
### 7.5 Decoding Properties
In this section we analyze why the noisy channel model performed better than direct decoding.
Length: In Table 7 we show the average length of generated responses. Direct decoding produces shorter responses than the ground truth, confirming that the direct model prefers short and generic responses. Adding a length bias to direct decoding (with lambda tuned on the development sets) produces responses longer than the ground truth, which may be a disadvantage. The noisy channel models produce responses with average length closest to the ground truth.
Table 7:
The average length of responses with different decoding methods (on test set). The value closest to the ground truth is bold.
ModelCamRest676MultiWOZ
Ground truth 14.50 16.91
Direct decoding 12.07 12.85
Direct decoding + Length 15.98 17.73
Reranking 15.09 17.47
Online decoding 15.14 17.32
ModelCamRest676MultiWOZ
Ground truth 14.50 16.91
Direct decoding 12.07 12.85
Direct decoding + Length 15.98 17.73
Reranking 15.09 17.47
Online decoding 15.14 17.32
Zipf: Table 8 shows the Zipf scores of responses. We find that the word distributions of responses generated by the noisy channel models are closer to the word distribution of ground-truth responses.
Table 8:
The Zipf scores of responses with different decoding methods (on test set). The value closest to the ground truth is bold.
ModelCamRest676MultiWOZ
Ground truth 1.07 1.22
Direct decoding 0.84 0.91
Reranking 0.87 0.99
Online decoding 0.89 1.03
ModelCamRest676MultiWOZ
Ground truth 1.07 1.22
Direct decoding 0.84 0.91
Reranking 0.87 0.99
Online decoding 0.89 1.03
Repetition: In Table 9 we examine the likelihood of falling into repetition loops (Holtzman et al., 2019) for different decoding methods. Repetition loops are rare for all decoding methods, but noisy channel decoding can further decrease their likelihood. The channel model can discount a sequence with a repetition loop, since it conveys less information than a natural sequence of the same length, making it harder to “explain” the context.
Table 9:
The likelihood (%) of falling into repetition loops for different decoding methods (on test set).
ModelCamRest676MultiWOZ
Direct decoding 0.24 0.31
Reranking 0.12 0.14
Online decoding 0.08 0.11
ModelCamRest676MultiWOZ
Direct decoding 0.24 0.31
Reranking 0.12 0.14
Online decoding 0.08 0.11
Examples: Some examples of responses are shown in Table 10. We observe that noisy channel models decode longer responses compared to direct decoding, and that the responses can explain their dialogue contexts well to meet users’ requirements.
Table 10:
Case study on the responses decoded by direct decoding, noisy channel reranking, and noisy channel online decoding. The large noisy channel model is used.
ModelDelexicalized ResponseSacreBLEU
Dialogue: MUL1624, Turn Number: 4
Ground truth Sure. The trainid is [train_id]. It leaves at [value_time] and arrives at [value_time]. Would you like to make a reservation? –
Direct decoding The travel time is [value_count] minutes. 2.03
Reranking [train_id] leaves [value_place] at [value_time] and arrives in [value_place] at [value_time]. Would you like me to book that for you? 46.33
Online decoding [train_id] leaves at [value_time] and arrives at [value_time]. Would you like me to book that? 60.90
Dialogue: MUL1276, Turn Number: 6
Ground truth The [restaurant_name] has a postal code of [restaurant_postcode]. Will this work for you? I can book it if you please. –
Direct decoding There are [value_count] places. What type of food would you like? 1.63
Dialogue: MUL1898, Turn Number: 2
Ground truth Yes, [attraction_name] is on [attraction_address] and is in the [value_area] side of town. Is there anything else you need to know? –
Direct decoding [attraction_name] is located in the [value_area] part of town and has free admission. 27.53
Reranking [attraction_name] is located in the [value_area] of town at [attraction_address]. The entrance fee is free. Can I help you with anything else? 41.66
Online decoding [attraction_name] is located in the [value_area] part of town at [attraction_address]. Can I help you with anything else? 42.38
ModelDelexicalized ResponseSacreBLEU
Dialogue: MUL1624, Turn Number: 4
Ground truth Sure. The trainid is [train_id]. It leaves at [value_time] and arrives at [value_time]. Would you like to make a reservation? –
Direct decoding The travel time is [value_count] minutes. 2.03
Reranking [train_id] leaves [value_place] at [value_time] and arrives in [value_place] at [value_time]. Would you like me to book that for you? 46.33
Online decoding [train_id] leaves at [value_time] and arrives at [value_time]. Would you like me to book that? 60.90
Dialogue: MUL1276, Turn Number: 6
Ground truth The [restaurant_name] has a postal code of [restaurant_postcode]. Will this work for you? I can book it if you please. –
Direct decoding There are [value_count] places. What type of food would you like? 1.63
Dialogue: MUL1898, Turn Number: 2
Ground truth Yes, [attraction_name] is on [attraction_address] and is in the [value_area] side of town. Is there anything else you need to know? –
Direct decoding [attraction_name] is located in the [value_area] part of town and has free admission. 27.53
Reranking [attraction_name] is located in the [value_area] of town at [attraction_address]. The entrance fee is free. Can I help you with anything else? 41.66
Online decoding [attraction_name] is located in the [value_area] part of town at [attraction_address]. Can I help you with anything else? 42.38
## 8 Related Work
Task-Oriented Dialogue Models: Most task-oriented dialogue systems break down the task into three components: belief tracking (Henderson et al., 2013; Mrkšić et al., 2016; Rastogi et al., 2017; Nouri and Hosseini-Asl, 2018; Wu et al., 2019a; Zhang et al., 2019; Zhou and Small, 2019; Heck et al., 2020), dialogue act prediction (Wen et al., 2017a; Tanaka et al., 2019), and response generation (Chen et al., 2019; Budzianowski et al., 2018; Lippe et al., 2020). Traditionally, a modular approach is adopted, where these components are optimized independently (i.e., a pipeline design) or learned via multi-task learning (i.e., some parameters are shared among the components) (Wen et al., 2017b; Neelakantan et al., 2019; Zhao et al., 2019; Mehri et al., 2019; Tseng et al., 2020; Lee et al., 2020). However, it is known that improvements in one component do not necessarily lead to overall performance improvements (Ham et al., 2020), and the modular approach suffers from error propagation in practice (Liu and Lane, 2018). These observations gave rise to the sequence-to-sequence approach (Lei et al., 2018; Pei et al., 2019; Budzianowski and Vulić, 2019; Wu et al., 2019b; Zhang et al., 2020a; Ham et al., 2020; Hosseini-Asl et al., 2020; Peng et al., 2020a; Yang et al., 2021), where dialogue beliefs and acts are represented as text spans, and a sequence-to-sequence model is applied to subsume the three components. Our work is situated within this general approach. In contrast to previous work, however, which uses a direct model for decoding, we introduce the noisy channel model to improve task-oriented dialogue.
Pretraining Models for Dialogue: Recent work has applied pretraining (Peters et al., 2018; Devlin et al., 2019; Radford et al., 2019) to dialogue. For open-domain dialogue, DialoGPT (Zhang et al., 2020b) and CGRG (Wu et al., 2020b) extend GPT-2 (Radford et al., 2019) for response generation. PLATO (Bao et al., 2019) and PLATO-2 (Bao et al., 2020) pretrain a latent variable model with social media data for diversified response generation. Meena (Adiwardana et al., 2020) collects a large-scale social media corpus for pretraining and proposes a metric named sensibleness and specificity average for evaluation. Roller et al. (2020) study various strategies for building an open-domain chatbot with Reddit for pretraining. For task-oriented dialogue, ToD-BERT (Wu et al., 2020a) fine-tunes BERT (Devlin et al., 2019) for four tasks, including intention detection, belief tracking, dialogue act prediction, and response selection. SC-GPT (Peng et al., 2020b) fine-tunes GPT-2 for few-shot response generation with given dialogue acts. Ham et al. (2020) fine-tune GPT-2 for belief tracking and context-to-response generation. SimpleTOD (Hosseini-Asl et al., 2020) proposes a method to serialize dialogue beliefs and acts into text spans and fine-tunes GPT-2 for end-to-end dialogue modeling. SOLOIST (Peng et al., 2020a) uses a series of task-oriented dialogue datasets to further pretrain GPT-2 before fine-tuning it on final tasks for evaluation. Unlike these BERT- or GPT-initialized task-oriented dialogue models, which are essentially pretrained with general text, such as Wikipedia and BookCorpus, we use a Reddit dump to pretrain the models to learn from open-domain dialogues.
## 9 Conclusion
We introduced two noisy channel models, noisy channel reranking and noisy channel online decoding, for task-oriented dialogue. Large-scale pretraining was further adopted to tackle data scarcity in downstream tasks. Extensive experiments on MultiWOZ, CamRest676, and SMCalFlow demonstrated that (1) the noisy channel models significantly outperform direct decoding; (2) models with pretraining improve over randomly-initialized models; (3) the models are robust to different dialogue schema annotations; and (4) the noisy channel models can decode responses closer to ground-truth responses than direct decoding.
## Acknowledgments
We would like to thank the action editors (Maggie, Wenjie Li, and Eneko Agirre) and three anonymous reviewers for their insightful comments. We also thank Angeliki Lazaridou, Gábor Melis, Nando de Freitas, Chris Dyer, and the DeepMind language team for their helpful discussions.
## Notes
1
Here we abstract away from the prediction of belief states and dialogue acts, which also form part of our generative model; see Section 3 for details.
2
We do not model the probabilities of database state or lexicalized response, as these are deterministic given the belief state and delexicalized response, respectively.
3
Although exact decoding is also computationally intractable for the direct model, approximating $arg maxBp(B|C)$ is well-studied, e.g., beam search. The decoding for ℬ is therefore omitted here.
4
One exception is Henderson et al. (2019), who use Reddit data to improve response retrieval and selection. We focus on response generation in this work.
13
We confirmed this with the dataset authors by email.
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## Author notes
*
Work completed during an internship at DeepMind.
This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For a full description of the license, please visit https://creativecommons.org/licenses/by/4.0/legalcode | 2021-07-26 01:54:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 72, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5766547918319702, "perplexity": 2892.5820874482747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046151972.40/warc/CC-MAIN-20210726000859-20210726030859-00563.warc.gz"} |
https://www.whitman.edu/academics/departments-and-programs/mathematics-and-statistics/resources-for-students/senior-project-archive/2010 | Titles are hyperlinked to pdf copies of the final project write-up. Course coordinator: Patrick Keef
• TITLE: THE USE OF LINEAR ALGEBRA IN MODELING THE PROBABILITIES OF PREDICTED FUTURE OCCURRENCES (selected outstanding senior project)
AUTHOR: Gabrielle Boisrame
ABSTRACT: Singular Value Decomposition (SVD) and similar methods can be used to factor matrices into subspaces which describe their behavior. In this paper we review the SVD and generalized singular value decomposition (GSVD) and some of their applications. We give particular attention to how these tools can be used to isolate important patterns in a dataset and provide predictions of future behavior of these patterns. A major focus of this project is the examination of a component resampling method described by Michael Dettinger which provides estimates of probability distributions for small sets of data. We tested the results of using both the SVD and the GSVD for Dettinger's method. Similarly to Dettinger, we found that the method had a tendency to give probability distributions a Gaussian shape even when this did not seem to be represented in the original data. For some data sets, however, both using the SVD and GSVD provided what appear to be reasonable probability distributions. There was not a significant difference in how well original probability distributions were estimated when using Dettinger's original method or the modifications with the reduced SVD or the GSVD. Using Dettinger's method rather than a simple histogram always provided a higher resolution of information, and was sometimes capable of matching the shape of the original probability distributions more closely.
• TITLE: COMMUTATIVITY IN NON-ABELIAN GROUPS
AUTHOR: Cody Clifton
ABSTRACT: Let $P_2(G)$ be defined as the probability that any two elements selected at random from the group $G$, commute with one another. If $G$ is an Abelian group, $P_2(G) = 1$, so our interest lies in the properties of the commutativity of non-Abelian groups.
Particular results include that the maximum commutativity of a non-Abelian group is 5/8, and this degree of commutativity only occurs when the order of the center of the group is equal to one-fourth the order of the group. Explicit examples will be provided of arbitrarily large non-Abelian groups that exhibit this maximum commutativity, along with a proof that there are no 5/8 commutative groups of order 4 mod 8.
Further, we prove that no group exhibits commutativity 0, there exist examples of groups whose commutativity is arbitrarily close to 0. Then, we show that for every positive integer $n$ there exists a group $G$ such that $P_2(G) = 1/n$. Finally we prove that the commutativity of a factor group $G/N$ of a group $G$ is always greater than or equal to the commutativity of $G$.
• TITLE: THE EUCLIDEAN ALGORITHM AND A GENERALIZATION OF THE FIBONACCI SEQUENCE
AUTHOR: Ian Cooper
ABSTRACT: This paper will explore the relationship between the Fibonacci numbers and the Euclidean Algorithm in addition to generating a generalization of the Fibonacci Numbers. It will also look at the ratio of adjacent Fibonacci numbers and adjacent generalized Fibonacci numbers. Finally it will explore some fun applications and properties of the Fibonacci numbers.
• TITLE: THE GAME SET AS ${\bf F}^4_3$
AUTHOR: Hillary Fairbanks
ABSTRACT: In this paper we construct an isomorphism between the card game Set and the four-dimensional vector space over the three element field, ${\bf F}_3$, to draw various results about the game. By creating a one-to-one and onto correspondence between the cards and points in ${\bf F}_3^4$, we find that a collectable 3 set is in fact a line in the vector space. Using this, we are able to determine the total number of collectable sets that exist in the game, as well as find the maximum number of cards that can be played without having a collectable set. Furthermore, we can simulate the game using the Monte Carlo method to find the probability of having a collectable set in a random selection of cards from the deck.
• TITLE: CURVES OF CONSTANT WIDTH AND THEIR SHADOWS
AUTHOR: Lucia Paciotti
ABSTRACT: In this paper we will investigate curves of constant width and the shadows that they cast. We will compute shadow functions for the circle, Reuleaux Triangle, and the curves of constant width described by Stanley Rabinowitz. From these functions we will prove that you can distinguish the different curves from their shadows. A result about the perimeter and area of these curves is also presented.
• TITLE: MULTIPLICATIVE GROUPS IN ${\bf Z}_m$
AUTHOR: Brian Sloan
ABSTRACT: Our goal will be to find subsets of ${\bf Z}_m$ that form groups under the operation of multiplication modulo $m$. By utilizing the isomorphism ${\bf Z}_m = {\bf Z}_n \oplus {\bf Z}_k$ , we will find multiplicative groups contained in ${\bf Z}_n \oplus {\bf Z}_k$ and then map these back to ${\bf Z}_m$. In particular, if $m = nk$ with $\gcd(n, k) = 1$, our objective is to find particular multiplicative subsets of ${\bf Z}_n \times {\bf Z}_k$ that are groups and whose first coordinate is a projection onto $U(n)$. We will give a method to calculate the total number of these subsets, and identify the elements of which they are composed.
• TITLE: SYSTEMS OF PYTHAGOREAN TRIPLES
AUTHOR: Chris Tobin-Campbell
ABSTRACT: This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines which numbers can be the sides of primitive right triangles and how many primitive right triangles those numbers can be a side of, and finally explores systems of three and four right triangles that fit together in three dimensions.
ABSTRACT: This paper examines two methods of determining whether a positive integer can correspond to the semiperimeter of a number of Pythagorean triangles. For all positive integers $k$, using Bertrand's Postulate, we can find semiperimeters corresponding to $k$ or more isoperimetrical triangles, and using the Prime Number Theorem, we can find exactly $k$ generator pairs which correspond to a semiperimeter. | 2018-12-19 02:11:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6921398639678955, "perplexity": 283.4012855099922}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376830305.92/warc/CC-MAIN-20181219005231-20181219031231-00036.warc.gz"} |
https://ems.press/books/standalone/235/4488 | # The Affleck–Kennedy–Lieb–Tasaki (AKLT) model
• ### Ian Affleck
University of British Columbia, Vancouver, Canada
A subscription is required to access this book chapter.
## Abstract
The Haldane conjecture (1983), which led to a Nobel prize for Duncan Haldane, was that half-integer spin chains are gapless while integer spin chains are gapped. Haldane made this observation by a mapping onto a non-linear $\sigma$-model at large spin. Gapless behavior for half-integer spin was proven rigorously by the Affleck–Lieb–Schultz–Mattis (ALSM) theorem. Gapped behavior for integer spin was demonstrated by an exactly solvable model known as Affleck–Kennedy–Lieb–Tasaki (AKLT) model (1987, 1988). In this review article, I will review the ALSM theorem and present results on the AKLT model. An important aspect of this model is that it has decoupled spin-$1/2$’s at the two ends with open boundary conditions. This was shown, by Kennedy (1990), to extend to the regular antiferromagnetic model. The first experimental demonstration of this behavior was in a paper by Hagiwara, Katsumata, Affleck, Halperin, and Renard (1990). I will review this experiment. | 2023-03-30 10:51:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5193015336990356, "perplexity": 1972.9709743279855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949181.44/warc/CC-MAIN-20230330101355-20230330131355-00705.warc.gz"} |
https://codereview.stackexchange.com/questions/210341/form-a-string-consisting-of-the-common-letters-that-appear-in-two-input-strings | # Form a string consisting of the common letters that appear in two input strings
The idea is to write a function that takes two strings and returns a new string that repeats in the previous two: examples:
'ABBA' & 'AOHB' => 'AB'
'cohs' & 'ohba' => 'oh'
A brute force solution would be nested for loops like so:
const x = 'ABCD'
const y = 'AHOB'
function subStr(str1, str2) {
let final = ''
for (let i = 0; i < str1.length; i++) {
for (let j = 0; j < str2.length; j++) {
if (str1[i] === str2[j]) {
final += str1[i]
}
}
}
return final
}
console.log(subStr(x, y)) // => AB
• Could you clarify, with additional examples, what should happen when letters appear in different orders within the two input strings? Dec 26 '18 at 5:41
• How is this supposed to behave with repeated characters? For example, your subStr function returns "ABBA" on the input ("ABBA", "AOHB"), which isn't the output you specified ("AB") Jun 2 '19 at 2:13
## 3 Answers
I know the root of all evil is premature optimization, but I would first ask if this is expected to be either a hotspot or usef with large strings.
Because it can be made much more readable using the array methods, but doing so obviously comes with a performance cost. OTOH, your method has its own performance issues.
If this isn’t expected to be particularly performance sensitive, I think that a straightforward method that turned the two strings into arrays then used the filter or map method to generate an array that is then turned into a string would be much more readable, and I prefer readable over performant as long as the performance isn’t a problem.
One additional point, what should your function return for “a”, “aa”? currently it returns “aa”?
• That would make a nice one-liner: common = (str1, str2) => str1.split('').filter(s => str2.contains(s)).join('') Dec 28 '18 at 9:30
One option to simplify the procedure is use Set
const subStr = (a, b) => {
const set = s => new Set(s);
const [x, y, s = set([...x, ...y])] = [set(a), set(b)];
for (let z of s)
if (!x.has(z) || !y.has(z)) s.delete(z)
return [...s].join('')
}
console.log(subStr('ABBA', 'AOHB'), subStr('cohs', 'ohba'), subStr('aa', 'a'));
If both string are equal length and sort is allowed, performance can be reduced to linear o(n).
const x = 'ABCD'
const y = 'AHOB'
function subStr(str1, str2) {
str1 = str1.split('');
str2 = str2.split('');
str1.sort();
str2.sort();
let final = ''
for (let i = 0; i < str1.length; i++) {
if (str1[i] === str2[i]) {
final += str1[i]
}
}
return final
}
console.log(subStr(x, y)) // => AB
• What exactly do you mean with $n$, and how can sorting be $\mathcal O(n)$? Jun 2 '19 at 9:10 | 2021-10-25 16:44:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40179580450057983, "perplexity": 3794.2010732446365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587719.64/warc/CC-MAIN-20211025154225-20211025184225-00705.warc.gz"} |
https://cs.stackexchange.com/tags/nondeterminism/hot | # Tag Info
24
Suppose a DFA was allowed to have missing transitions. What happens if you encounter a symbol which has no transtion defined for it? The result is undefined. That would seem to violate the "deterministic" characteristic of a DFA. However, it's trivial to transform such an incomplete DFA into a complete DFA. Simply add a new state, illegal, and map any ...
23
"Deterministic" means "if you put the system in the same situation twice, it is guaranteed to make the same choice both times". "Non-deterministic" means "not deterministic", or in other words, "if you put the system in the same situation twice, it might or might not make the same choice both times". A non-deterministic finite automaton (NFA) can have ...
16
There are two answers, depending on how you define efficient. Compactness of representation Telling more with less: NFAs are more efficient. Converting a DFA to an NFA is straightforward and does not increase the size of the representation. However, there are regular languages for which the smallest DFA is exponentially bigger than the smallest NFA. A ...
16
Yes, you are correct computers are deterministic automate. Non-deterministic models are more useful for theoretical purpose, sometime the deterministic solution is not as obvious to the definition(or say problem statement) and so little hard to find solution. Then one approach is that first design a non-deterministic model that may be comparatively easy to ...
13
Excellent question! Nondeterminism first appears (so it seems) in a classical paper of Rabin and Scott, Finite automata and their decision problems, in which the authors first describe finite automata as a better abstract model for digital computers than Turing machines, and then define several extensions of the basic model, including nondeterministic finite ...
12
There are a few reasons I think we put less effort into the Halting problem for non-deterministic models. The first is that there are, in fact, two relevant halting problems for a ND model. Given an input $x$ and a non-deterministic machine $M$: Does there exist a valid run of $M$ on $x$ which halts? Does there exist a valid run of $M$ on $x$ which doesn'...
12
A DFA is specified by the following data: An alphabet $\Sigma$. A set of states $Q$. An initial state $q_0 \in Q$. A set of final states $F \subseteq Q$. A transition function $\delta\colon Q \times \Sigma \to Q$. As you can see from the signature of $\delta$, it specifies a transition at every state for every symbol.
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For DFA there is a nice algebraic structure that determines which states can be equivalent, the Myhill-Nerode equivalence on strings is related to minimization of DFA. For NFA the situation is complicated as there is no unique minimal NFA in general. Here is an example for the finite language $\{ ab, ac, bc, ba, ca, cb\}$. The two automata are both state-...
11
It doesn't decide. Nondeterminism isn't intended to be a realistic model of computation. Check the definition: a nondeterministic automaton accepts if there's any valid sequence of transitions that reach an accepting state.
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Consider a two state automaton for the language $a^*b$, two transitions from the initial state, one looping with label $a$, the other with label $b$ to the final state. Making the initial state final, would also accept $a^*$.
10
I looked up Hopcroft and Ullman 1979 and it say on page 281 that it is not closed under reversal. But I found no proof in my very fast look at the relevant chapter. Searching the web does also give a negative answer, with counter example, on stackoverflow by a member of CS (notation adapted): $(a+b+c)^*WcW^R$, where $W \in (a+b)^+$; this is non-...
10
Every time you are in a state which has a $\epsilon$ transition, it means you automatically are in BOTH states, to simplify this to you: If the string is $\epsilon$ then your automata ends both in $q_0$ and $q_1$ If your string is '0' it'll be again in $q_0$ and $q_1$ If your string is '1', it'll be only in $q_2$, because if you look from the point of $... 10 The notion of a PDA can be generalized to an$S(n)$auxiliary pushdown automaton ($S(n)$-AuxPDA). It consists of a read-only input tape, surrounded by endmarkers, a finite state control, a read-write storage tape of length$S(n)$, where$n$is the length of the input string, and a stack In "Hopcroft/Ullman (1979) Introduction to Automata Theory, Languages, ... 10 The NFA accepts strings where the fourth letter from the end is 1. Your DFA doesn't accept 11000. A DFA doesn't know how much input is left, so the property "the fourth character from the end" is difficult. You need to remember the last four characters to know whether it was a 1 or a 0 once you reach the end of the string. To do so you need a state for each ... 10 You are confusing$\epsilon$with a letter. It's not a letter! It's just the empty string. Let us consider a slightly more general model, "word-NFA". A word-NFA is like an NFA, but each transition is labeled with an arbitrary word. We say that the word-NFA accepts a word$w$if there is a walk from an initial state to a final state such that if we ... 9 An NP-complete problem can be transformed into another NP-complete problem. There's an abundance of known NP-complete problems, in fact, one could even say that any really interesting problem is NP-complete. So if you know of a way of solving any NP-complete problem$X$quickly, you can take any other NP-complete problem, transform it into an instance of$X$,... 9 It is more the other way around: automata arose first, as mathematical models. And nondeterminism is quite natural, you often have several paths open before you. Instead of some messy way of specifying that all paths must be followed to the end in some order, and perhaps getting bogged down by infinite branches, and... just use nondeterminism. And while ... 9 A non-deterministic automaton runs in all possible sequence of states on a given input. It is not a random run. An input symbol is accepted only if in at least one of the runs it has reached an accepting state after reading the given input. Hope this helps :D 9 If$\mathsf{NTIME}(n^k) \subseteq \mathsf{TIME}(n^\ell)$for any$k,\ell$then$\mathsf{P} = \mathsf{NP}$. Indeed, any problem$L \in \mathsf{NP}$can be solved in non-deterministic time$O(n^r)$for some$r$. Consider now the problem$L' = \{0^{|x|^{r/k}}1x : x \in L\}$. Clearly this problem is still in$\mathsf{NP}$, and furthermore the previous algorithm ... 9 You can show that$\mathsf{NL}[2] \subseteq \mathsf{NL}$as follows. We are given an$\mathsf{NL}[2]$machine$M$, and we want to simulate it with an$\mathsf{NL}$machine$M'$. The first that$M$does is to guess the state$\sigma$of$M'$after it finishes reading the witness tape for the first time. It then simulates two copies of$M$, one starting at$M$'... 9 Here are several ways of thinking about non-determinism (copied from this answer). The genie. Whenever the machine has a choice, a genie tells it which way to go. If the input is in the language, then the genie can direct the machine in such a way that it eventually accepts. Conversely, if the input is not in the language, whatever the genie tells the ... 9 Take this automaton for instance, it's an NFA and it accepts the string$0110$. To be more pedantic, it accepts strings that end in$10. To see that we just need to check whether it reaches an accept state. \begin{align*} q_0 & \rightarrow 1\\ q_0 & \rightarrow 0\\ \color{red}{q_1} &\rightarrow \color{red}{1}\\ q_2 &\rightarrow 0\\ \end{... 8 The two terms randomized algorithms and probabilistic algorithms are used in two different contexts. Randomized algorithms are algorithms that use randomness, in contradistinction with deterministic algorithms that do not. Probabilistic algorithms, for example probabilistic algorithms for primality testing, are algorithms that use randomness and could make ... 8 This is elaborated nicely in Dexter Kozen's Theory of Computation textbook, in chapter 2. Savitch's Theorem (Theorem 1 in his paper) says: ifS(n) \ge \log n$, then$\text{NSPACE}(S(n)) \subseteq \text{DSPACE}(S(n)^2)$. Space-constructibility often seems to be assumed in a proof, but this requirement can be removed by restarting the search with a fixed ... 8 You want to know why "the others" obtained a different expression? When contructing a regular expression there is no unique correct answer. There are usually several expressions "that make sense". In this case you use state elimination method (I learned this under the name of Brzozowski and McCluskey). Here the order of removal determines the expression ... 8 Quite simply, the mechanism is magic. The idea of non-determinism is that it simply knows which way it should take in order to accept the word, and it goes that way. If there are multiple ways, it goes one of them. Non-determinism can't be implemented as such in real hardware. We simulate it using techniques such as backtracking. But it's primarily a ... 8 Non-determinism is the same concept in all contexts – the machine is allowed several options to proceed at any given point. However, the semantics are a bit different since DFAs/NFAs and PDAs always define total functions, while Turing machines (deterministic or non-deterministic) in general define partial functions. A partial function is one defined only ... 8 It makes perfect sense. Non-deterministic automata and non-deterministic algorithms in general are useful in many situations. The best known situation is when one designs algorithms (or strategies or recipes or ..) and analyzes them. For example, an algorithm for computing the maximum element of a set of numbers will have a loop of the form "as long as the ... 8 Nondeterministic automata would make perfect sense in a predetermined universe, because, in the sense it is used in computer science, "nondeterministic" does not mean "not predetermined." In particular, the acceptance criterion of nondeterministic machines is defined in terms of the existence of a path of a particular kind through the state transition graph.... 8 Please refer Does Cook Levin Theorem relativize?. Also refer to Arora, Implagiazo and Vazirani's paper: Relativizing versus Nonrelativizing Techniques: The Role of local checkability. In the paper by Baker, Gill and Solovay (BGS) on Relativizations of the P =? N P question (SIAM Journal on Computing, 4(4):431–442, December 1975) they give a language$B\$ ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2021-01-26 02:15:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8309614658355713, "perplexity": 580.3733551516024}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704795033.65/warc/CC-MAIN-20210126011645-20210126041645-00601.warc.gz"} |
http://www.koreascience.or.kr/article/JAKO198424457071989.page | # 꽃사슴(Formosan deer)의 Hemoglobin형(型)에 관한 연구(硏究)
The hemoglobin phenotype and the gene frequencies of 44 Formosan deer(Cervus nippon) in Kyung-Gi area were examined by using cellulose acetate and starch gel electrophoresis. 1. The method of cellulose acetate electrophoresis was simplier, more clear and preserative than starch gel electrophoresis. 2. The hemoglobin phenotype was appeared 3 types as $Hb^F$ $Hb^{FS}$ and $Hb^S$. The frequencies of appearance were $Hb^F$ 47.7%, $Hb^{FS}$ 47.7% and $Hb^S$ 4.5%, respectively. 3. The genetic factors of hemoglobin were observed as $Hb^F$ and $Hb^S$ and the rates of gene frequencies were 71.6% and 28.4%, respectively. | 2020-08-06 08:16:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.659274697303772, "perplexity": 13972.920748767176}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439736883.40/warc/CC-MAIN-20200806061804-20200806091804-00146.warc.gz"} |
https://www.physicsforums.com/threads/variance-in-random-walk.350264/ | # Variance in Random Walk
## Main Question or Discussion Point
Hi,
I know that the expectation E(Sn) for a one-dimensional simple random walk is zero. But what about the variance?
I read in http://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk" that the variance should be E(Sn2) = n.
Why is that? Can anyone prove it?
Thank you very much!
Last edited by a moderator:
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Just write down the definition of $S_n$ and you will be able to answer your question yourself.
Var(Sn) = E(Sn2) = E(Z12 + Z22 + Z32 + ... + Zn2) =* E(Z12) + E(Z22) + ... + E(Zn2) = 1 + 1 + ... + 1 (n times) = n
*variables are independent and uncorrelated
Is this correct then?
This is almost correct. $S_n$ is defined to be $Z_1+\ldots +Z_n$, where the $Z_i$ are independent (or at least uncorrelated) with mean zero and variance one. It follows that
$$S_n^2 = \sum_{i,j=1}^n{Z_i Z_j}$$
and not, as you wrote,
$$S_n^2 = \sum_{i=1}^n{Z_i^2}$$
However, using independence of the $Z_i$ you can still do a similar computation to prove $$\mathbb{E}\left[S_n^2\right]=n$$.
Thank you!
You're welcome | 2020-08-05 08:33:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8108587265014648, "perplexity": 898.8717250532811}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735916.91/warc/CC-MAIN-20200805065524-20200805095524-00294.warc.gz"} |
https://electronics.stackexchange.com/questions/287944/how-does-the-commutator-work-right-here | How does the commutator work right here?
I understand that we use the commutator as a rectifier using two segments that alternatively touch the negative and positive brush. That is when we only have one coil. What happens when we have more than one? In the above diagram we have 4 coils and 4 commutator bars. What are the bars? Shouldn't we have 2 segments for each coil ? That means 8 bars. Maybe the bars and the segments are not the same thing. If they are not , what are the bars here? Because the coil has two sides and it has to get to both brushes. The textbook I'm studying also says that the voltage in the first picture below is eb+ec and in the second picture 7+18+20+18+7=70 volts. Now here is another part I don't understand. In the simple example of one coil each brush is always in touch with one coil. So I figured, if we have 2 coils we have four segments and we take advantage only of the coil which has the highest voltage each moment. However, that is not true. Here we add every coil's voltage. How can we take advantage of all the coils if only two segments touch the brushes each time? And how come the voltage in the first picture is eb+ec? Which coils touch the brushes? (I imagine that two coils could be touching a brush at the same time but only one contributes) I'm very confused and I can't find an answer. I also have this image for the actual physical construction of the first image .
• Commutator "bar" and "segment" refer to the same thing. Do you realize that these coils are connected in series? – user28910 Feb 21 '17 at 16:04
Commutator segments = Commutator bars. Segments are made of copper bars separated by mica.
Coil B is connected to commutator segment b and c. C to c & d. 4 segments connected to 4 coils. Each segment connects two coils. All coils are connected in series around commutator. Brushes connect commutator in parallel.
As your rotor spins, Faraday's Law applies. Whenever the flux linked or associated with a circuit changes, a voltage is induced in the circuit.
So in Figure 4.7, coil A and C are moving parallel to the flux. No flux lines are cut, so induced voltage is 0. Coils B and D are moving perpendicular to the flux, so maximum voltage is induced. The text says 20V. $E_B = E_D = 20V$.
And how come the voltage in the first picture is eb+ec?
This is not correct.
Bottom of p74.
Consequently, the voltage induced in these coils is at it maximum possible value (20V, say). That is also the voltage across the brushes at this particular instant.
So in Figure 4.11b, we have the same size coils, producing 20V at maximum. Coil A and B produce 0V. Coil C and D produce maximum or 20V. The 18V coil is $20\ sin (60°) = 17.3V$. The coil is cutting the flux lines at roughly 60°. The 7V coil is $20\ sin (30°) = 10V$. So: $$10V + 17.3V + 20V + 17.3V + 10V = 74.6V$$
The coils are not fully at 30° and 60°, so voltages are less. Or 70V. But this illustrates where we are.
I disagree with the authors Figure 4.8, which covers Figure 4.10, but this has more to do with understanding the theory of how it works. Two coils at 45° will produce more than 20V. No DC Generator commutator has 4 segments.
• I see you know the textbook. Is it good for a first study on electric machines ? And yes,that diagram confused me more because I thought that one coil stops interacting and the other takes its place because it has a greater voltage . I commented on the previous answer explaining one more little problem I have. Can you check it out ? – John Katsantas Feb 22 '17 at 10:34
• Any textbook has advantages and disadvantages. Read, throw stuff at the wall and see what sticks. There is no easy path to learning. I googled the text. But I'm no expert. I found the other answer was not at your level of understanding, so I tried to clarify your misconceptions. – StainlessSteelRat Feb 22 '17 at 11:02
The individual physical coils of the armature are connected in series around the ring. When you make contact with two of the segments of the commutator, you are essentially forming two parallel "virtual" coils that are composed of the individual physical coil segments on either of the two paths between the contacts. These virtual coils have the desired alignment with the stator field in order to achieve the desired result. Each physical coil segment experiences an EMF that is related to its actual physical angle with respect to the stator field, which explains the numbers in your Figure 4.11b.
As the armature turns, the virtual coil turns with it, until you get to the point at which a different set of commutator contacts is reached by the brushes. At this, point, you get a different combination of physical coils, and the resulting virtual coils have an alignment that is reset back to the beginning of the desired alignment angle.
In this way, the armature experiences the maximum amount of torque available at all times. More physical coils and commutator segments means that the torque ripple is reduced and the efficiency is increased, at least to a point.
• I believe that it is not correct to relate the number of commutator segments to "poles." A 4-pole DC motor has 4 field poles, 4 armature poles and 2 pairs of brush assemblies. At any instant in time, half of a larger number of armature coils would be serving to form each pair of poles. – Charles Cowie Feb 21 '17 at 21:03
• @CharlesCowie: Yes, strictly speaking, you're correct. Edited. – Dave Tweed Feb 21 '17 at 21:21
• I understand the diagram now but I'm confused as to what happens in the physical construction. Can you help me with the connections there ? I can't understand the voltage here , meaning what is point A and what is point B if Vab is the total voltage induced . Because I try to follow coil B which starts from the bottom segment of the brush on the left and ends up on the bottom segment of the brush on the right. So why isn't the total voltage only coil's B voltage VB ? Similarly on the other side with coil's D voltage VD. The V across the bottom segments is VB and the top ones VD. – John Katsantas Feb 22 '17 at 10:24
• I know I'm wrong . I just explain how I see it so you can find out what's wrong in my thinking . – John Katsantas Feb 22 '17 at 10:25
• @JohnKatsantas All the coils are connected in series and the 2 brushes make two sets of coils in parallel. So each induced voltage adds together to give a total. Segment a to Coil A to segment b to Coil B, etc. Currents are moving in different directions so both voltages are the same. – StainlessSteelRat Feb 22 '17 at 10:56 | 2021-06-23 15:15:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6751073002815247, "perplexity": 684.350044087087}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488539480.67/warc/CC-MAIN-20210623134306-20210623164306-00279.warc.gz"} |
http://mathoverflow.net/revisions/85165/list | 2 deleted 121 characters in body
Since you allow virtual characters you should definitely expect such a thing (due to the general philosophy of writing down Eisenstein series as linear combinations of theta series after Siegel, Weil and others).
Here is an explicit construction. Take the simplest affine Kac-Moody Lie algebra, namely $A_1^{(1)}$, and take the level to be $1$. Then there are (essentially) two integrable highest weight $A_1^{(1)}$-modules of this level. Let's denote them by $V_1$ and $V_2$ for simplicity.
As usual,let $$\theta_{00}(q)=\sum_{n \in \mathbb{Z}} q^{n^2}, \ \ \theta_{10}(q)=\sum_{n \in \mathbb{Z}} q^{(n+1/2)^2}.$$ Then the corresponding (homogeneous) characters are $\chi(V_1)(q)=\frac{\theta_{00}}{\eta}$ and $\chi(V_2)(q)=\frac{\theta_{10}}{\eta}$.
You can easily show that
$$E_6=-33 \theta_{00}^4 \theta_{10}^8+ \theta_{00}^{12}+\theta_{10}^{12}-33 \theta_{00}^8 \theta_{10}^4$$
Now $\frac{E_6}{\eta^{12}}$ is just a linear combination of level $12$ integrable $A_1^{(1)}$-modules (view each summand as a tensor product of 12 level one modules).If you believe in "level-rank duality" similar construction should be possible for $A_{12}^{(1)}$ at level one, I think.
One more thing. Your quotient reminds me of Serre's paper "Sur la lacunarite des puissances de $\eta$", on the lacunarity of even powers of the $\eta$-function. In the case of $\eta^{14}$, he uses a nice identity
$$\frac{E_6}{\eta^{12}}=\frac{\varphi_{K,c_+}+\varphi_{K,c_-}}{\eta^{14}},$$ where $\varphi_{K,c_{\pm}}$ are certain CM modular forms of weight $7$ (the field is $K=\mathbb{Q}(\sqrt{-3})$ and $c_\pm$ are Hecke characters). I wonder if the right-hand side can be linked to anything in representation theory.
1
Since you allow virtual characters you should definitely expect such a thing (due to the general philosophy of writing down Eisenstein series as linear combinations of theta series after Siegel, Weil and others).
Here is an explicit construction. Take the simplest affine Kac-Moody Lie algebra, namely $A_1^{(1)}$, and take the level to be $1$. Then there are (essentially) two integrable highest weight $A_1^{(1)}$-modules of this level. Let's denote them by $V_1$ and $V_2$ for simplicity.
As usual,let $$\theta_{00}(q)=\sum_{n \in \mathbb{Z}} q^{n^2}, \ \ \theta_{10}(q)=\sum_{n \in \mathbb{Z}} q^{(n+1/2)^2}.$$ Then the corresponding (homogeneous) characters are $\chi(V_1)(q)=\frac{\theta_{00}}{\eta}$ and $\chi(V_2)(q)=\frac{\theta_{10}}{\eta}$.
You can easily show that
$$E_6=-33 \theta_{00}^4 \theta_{10}^8+ \theta_{00}^{12}+\theta_{10}^{12}-33 \theta_{00}^8 \theta_{10}^4$$
Now $\frac{E_6}{\eta^{12}}$ is just a linear combination of level $12$ integrable $A_1^{(1)}$-modules (view each summand as a tensor product of 12 level one modules). If you believe in "level-rank duality" similar construction should be possible for $A_{12}^{(1)}$ at level one, I think.
One more thing. Your quotient reminds me of Serre's paper "Sur la lacunarite des puissances de $\eta$", on the lacunarity of even powers of the $\eta$-function. In the case of $\eta^{14}$, he uses a nice identity
$$\frac{E_6}{\eta^{12}}=\frac{\varphi_{K,c_+}+\varphi_{K,c_-}}{\eta^{14}},$$ where $\varphi_{K,c_{\pm}}$ are certain CM modular forms of weight $7$ (the field is $K=\mathbb{Q}(\sqrt{-3})$ and $c_\pm$ are Hecke characters). I wonder if the right-hand side can be linked to anything in representation theory. | 2013-05-20 02:33:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9586999416351318, "perplexity": 254.42680095123373}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368698207393/warc/CC-MAIN-20130516095647-00062-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://msl.stanford.edu/bibliography/haksar_controlling_2018 | Controlling Large, Graph-based MDPs with Global Control Capacity Constraints: An Approximate LP Solution
@inproceedings{haksar_controlling_2018,
address = {Miami Beach, FL},
title = {Controlling {Large}, {Graph}-based {MDPs} with {Global} {Control} {Capacity} {Constraints}: {An} {Approximate} {LP} {Solution}},
isbn = {978-1-5386-1395-5},
shorttitle = {Controlling {Large}, {Graph}-based {MDPs} with {Global} {Control} {Capacity} {Constraints}},
url = {https://ieeexplore.ieee.org/document/8618745/},
abstract = {We consider controlling graph-based MDPs (GMDPs) with two special properties: (i) Anonymous Influence and (ii) Symmetry. Large-scale spatial processes such as wildfires, disease epidemics, opinion dynamics, and robot swarms are well modeled by GMDPs with these properties. We derive two efficient and scalable algorithms for computing approximately optimal control policies for large GMDPs with Anonymous Influence and Symmetry and derive sub-optimality bounds for these policies. Unlike prior work, our algorithms explicitly enforce a global control capacity constraint. Our methods scale linearly in the number of equivalence classes in the GMDP rather than the total number of MDPs in the graph. We demonstrate our methods in simulations of controlling a wildfire with a global fire retardant constraint and controlling an Ebola outbreak with a global medicine constraint. Our Ebola model is derived from data from the 2014 West Africa outbreak.},
language = {en},
urldate = {2020-09-15},
booktitle = {2018 {IEEE} {Conference} on {Decision} and {Control} ({CDC})},
publisher = {IEEE},
author = {Haksar, Ravi N. and Schwager, Mac},
month = dec,
year = {2018},
keywords = {optimal\_control},
pages = {35--42},
month_numeric = {12}
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https://plus.tuni.fi/graderS/static/compcs140-f2022/modules/07/poikkeustakuut_en.html | # Error Situations in Interfaces¶
Preparing for different kinds of error situations is a tricky element of programming that can require even more code that the actual solution to the problem at hand. Reacting to error situations is often also difficult. Due to different execution paths the code easily becomes messy. All kinds of clean up can be needed when an error occurs and it is not rare for it to be left undone. Recovery from an error in a way that the program execution could still continue requires that operations left unfinished can somehow be cancelled which may not even be possible.
In programming, it is a reasonable starting point to prepare for others acting incorrectly. When an error is caught early it is possible to adapt to it and in the best case scenario even the program execution can be continued so that the error is not visible in the external behaviour of the program. When an error occurs in a software component, there are several different ways to react to it. It depends on the situation, which is the most suitable.
• Abrupt termination is the most extreme way to react to an error occuring in a program. Abrupt termination literally terminates the execution immediately when an error occurs. While it typically is the default in runtime environments it should be avoided.
• Abort, exit aims to clear the resources allocated by the software and to notify the user of the error. In addition the error situation is logged somewhere before terminating the execution.
• Continuation ignores the error. This is a really rare situation as an error is by nature something that requires handling in the program.
• Rollback returns the program state to where it was before the operation that caused the error. Implementing rollback is challenging and can be impossible.
• Recovery is aborting some local part of the program. Some part of the program is not able to handle the error on its own but it aims to clear its own state i.e. to free the resources it has allocated and then to notify some other part of the program of the error.
It is not possible to recover from or rollback all error situations as the situation may not be fixable even if it is tried. As an example of this is running out of memory: if there is no memory available any more, it is really difficult to do any recovery operations.
## Error¶
It is also good to define what is meant by an error in the first place. An error can be either external or internal. In an external error the program is asked to do someting it cannot or is not able to do. In an internal error the implementation itself runs into a situation where something goes wrong. The easiest step is typically noticing the error situation. Information hiding and encapsulating the implementation behind an interface also cause that the location of the error is not known by the caller of the service. A way to notify the caller of the service of the error as well as to offer a possibility to react to is suitable is needed. Exceptions are a suitable mechanism for this purpose.
## Exception Safety¶
In object oriented programming encapsulation hides the implementation of the service from the user of the method. This applies also to possible error situations. The implementation can also be changed without the user noticing. Furthermore, polymorphism means that the user doesn’t even know what object they are using at each point in the program. Inheritance also moves the implementation of the services to several levels in the class hierarchy. The documentation of the interface must hence also depict those error situations and exceptions that the user may need to react to.
In inheritance the subclasses are not allowed to break the promises of the baseclass i.e. they are not allowed to cause error situations or exceptions not documented for the base class. A baseclass must thus not promise too much on the error or lack of them. The baseclass is not aware of what kind of subclasses will be inherited from it. Documenting error situations is easier if common terms can be used for different situations.
In design by contract error situations are a part of the conditions. Documenting them can be done by using so called exception guarantees that in addition to the exception itself depict the state the object is in once the exception has occured.
### Exception Guarantees¶
Exception guarantees originate from C++ but are general purpose concepts for any programming language. Their idea is to categorize reacting to errors into simple basic cases. The guarantees depict that the user can expect from the object in situations where a service from the interface throws an exception out of it.
• Minimal guarantee guarantees that the object does not waste resources when an exception is thrown and the the object can be destroyed or reset in the program but is not otherwise usable. The class invariant does not necessarily hold after the exception.
• Basic guarantee guarantees the an exception thrown out of the service of the interface leaves the object into an unpredictable but as such sensible state. The class invariant still holds and the object is usable even if its state is not predicatable.
• Strong guarantee guarantees that the service is either completed without errors or in case of an exception the state of the object remains as it was before the service was called. Strong guarantee is commit or rollback as an operation. While it is nice for the user, it is often difficult to implement.
• Nothrow guarantee is the so called programmer’s heaven where the service guarantees never to leak an exception out of the interface. The operation always succeeds and if an exception does occur, the method handles it by itself.
In addition but separate to these there is exception neutrality which means that a software component leaks the exceptions thrown by its internal components as is without changing them. The exception neutral services can of course temporarily catch the exception, it is just thrown again forward. For example forEach() in ArrayList is one such service that relays exceptions thrown by the action to the caller.
In Java the finally block helps with implementing basic and strong quarantees as with is local cleanup operations can be made.
Exception Safety (kesto 24:20)
Preparing for error situations in programming
In exception safety
In exception guarantees | 2023-03-22 19:26:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19748036563396454, "perplexity": 870.4943691448252}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944452.74/warc/CC-MAIN-20230322180852-20230322210852-00650.warc.gz"} |
https://ltwork.net/which-of-the-following-is-true-of-solids-a-all-solids-have--10100848 | # Which of the following is true of solids? a. all solids have equal melting points. b. ionic solids have
###### Question:
Which of the following is true of solids?
a. all solids have equal melting points.
b. ionic solids have higher melting points than molecular solids.
c. molecular solids have higher melting points than all other types of solids.
d. it is impossible for solids to melt; therefore solids do not have melting points.
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### Status Exam An object, with a final position of x = 100.1 meters, initially started at x = -12.13 meters. What was its displacement?
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### Here is 20 FREE pointsNext give away is at 10:30
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Tasty Doughnuts has computed the net present value for capital expenditure at two locations. Relevant data related to the computation are as follows: Des Moines Cedar Rapids Total present value of net cash flow $712,500$848,000 Amount to be invested (750,000) (800,000) Net present value $(37,500)$... | 2023-01-27 07:13:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19523178040981293, "perplexity": 2895.8211366421606}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494974.98/warc/CC-MAIN-20230127065356-20230127095356-00541.warc.gz"} |
http://www.pudn.com/Download/item/id/3942150.html | exp3
(Vehicle-mounted inertial/satellite integrated navigation experiment: the vehicle stops at the initial moment, but there is some disturbance and shaking of the vehicle body. After 10 minutes, the vehicle moves, and IMU data of the inertial navigation system, vehicle odometer data, GPS data and so on are collected. The first 2 minutes' data were used for rough alignment, and the last 8 minutes' data were used for fine alignment. The navigation results of inertial navigation, inertial /GPS combined navigation, and inertial/odometer combined navigation were obtained respectively.)
exp3, 0 , 2018-07-28
exp3\coarse_align.m, 751 , 2018-06-09
exp3\dcm2eul.m, 274 , 2018-06-09
exp3\dcm2qua.m, 241 , 2018-06-09
exp3\eul2dcm.m, 489 , 2018-06-09
exp3\eul2qua.m, 439 , 2018-06-09
exp3\initpara.m, 901 , 2018-06-19
exp3\ins_navi.m, 1379 , 2018-07-27
exp3\KF_correct.m, 1533 , 2018-06-27
exp3\KF_correct_gps.m, 1557 , 2018-07-03
exp3\KF_correct_odo.m, 1545 , 2018-07-10
exp3\main.m, 3585 , 2018-07-27
exp3\main_ins_gps.m, 4123 , 2018-07-27
exp3\main_ins_odo.m, 4141 , 2018-07-27
exp3\pre_data.m, 1045 , 2018-07-10
exp3\qua2dcm.m, 310 , 2018-06-09
exp3\qua_multiply.m, 225 , 2018-06-09 | 2020-01-18 23:41:34 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8060049414634705, "perplexity": 10390.835761605658}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250593994.14/warc/CC-MAIN-20200118221909-20200119005909-00091.warc.gz"} |
https://www.zbmath.org/?q=ut%3Aspace+of+degree+%5C%28k%5C%29+holomorphic+self+maps+of+the+Riemann+sphere | # zbMATH — the first resource for mathematics
The cohomology of holomorphic self maps of the Riemann sphere. (English) Zbl 0814.55003
Let $$\text{Hol}_ k$$ denote the space of degree $$k$$ holomorphic self maps of the Riemann sphere, $$\mathbb{P}^ 1$$, and let $$\text{Rat}_ k \subset \text{Hol}_ k$$ denote the subspace of based maps. The cohomology groups $$H^*(\text{Rat}_ k;\mathbb{Z}_ p)$$ ($$p$$ prime) have been computed by F. R. Cohen, R. L. Cohen, B. M. Mann and R. J. Milgram [Acta Math. 166, No. 3/4, 163-221 (1991; Zbl 0741.55005)] and the algebra structure has been given by B. Totaro [The cohomology ring of the space of rational functions (preprint MSRI 1990)] for $$p$$ odd. In this note we compute the cohomology algebra $$H^*(\text{Hol}_ k;\mathbb{Z}_ p)$$ when $$p$$ does not divide $$k$$. We also determine the cohomology groups and a graded version of the cohomology algebra when $$k = pm$$. Direct analysis of the Leray-Serre spectral sequence for the standard bundle $$\text{Rat}_ k \to \text{Hol}_ k \to \mathbb{P}^ 1$$ leads to difficulties, and so we make use of the principal bundle $$\text{SO}(3) \to \text{Hol}_ k \to \text{Rat}_ k/S^ 1$$. Our computations rely heavily on Milgram’s calculation of the groups $$H^*(\text{Rat}_ k/S^ 1;\mathbb{Z}_ p)$$.
##### MSC:
55N99 Homology and cohomology theories in algebraic topology 55R20 Spectral sequences and homology of fiber spaces in algebraic topology 58D15 Manifolds of mappings
Full Text:
##### References:
[1] [CCMM] Cohen, F.R., Cohen, R.L., Mann, B.M., Milgram, R.J.: The topology of rational functions and divisors of surfaces. Acta Math.166, 163–221 (1991) · Zbl 0741.55005 · doi:10.1007/BF02398886 [2] [CLM] Cohen, F.R., Lada, T.J., May, J.P.: The homology of iterated loop spaces. (Lect. Notes Math., vol. 533) New York Berlin Heidelberg: Springer 1976 · Zbl 0334.55009 [3] [CS] Cohen, R.L., Shimamoto, D.H.: Rational functions, labelled configurations, and Hilbert schemes. J. London Math. Soc.43, 509–528 (1991) · Zbl 0756.55005 · doi:10.1112/jlms/s2-43.3.509 [4] [DL] Dyer, E., Lashof, R.K.: Homology of iterated loop spaces. Am. J. Math.84, 35–88 (1962) · Zbl 0119.18206 · doi:10.2307/2372804 [5] [G] Guest, M.A.: Topology of the space of absolute minima of the energy functional. Am. J. of Math.106, 21–42 (1984) · Zbl 0564.58014 · doi:10.2307/2374428 [6] [K] Kirwan, F.C.: On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles. Ark. Mat.24, 221–275 (1986) · Zbl 0625.14026 · doi:10.1007/BF02384399 [7] [MaM1] Mann, B.M., Milgram, R.J.: Some spaces of holomorphic maps to complex Grassmann manifolds. J. Differ. Geom.33, 301–324 (1991) · Zbl 0736.54008 [8] [MaM2] Mann, B.M., Milgram, R.J.: On the moduli of SU(n) monopoles and holomorphic maps to flag manifolds. Preprint, University of New Mexico and Stanford University 1991 [9] [May] May, J.P.: The geometry of iterated loop spaces. (Lect. Notes Math., vol. 271) New York Berlin Heidelberg: Springer 1972 · Zbl 0244.55009 [10] [M1] Milgram, R.J.: Interated loop spaces. Ann. of Math.84, 386–403 (1966) · Zbl 0145.19901 · doi:10.2307/1970453 [11] [M2] Milgram, R.J.: The structure of spaces of Toeplitz matrices. Preprint, Stanford University and the University of New Mexico 1992 [12] [MiS] Milnor, J.W., Stasheff, J.D.: Characteristic classes. (Ann. of Math. Studies, no. 76) Princeton University Press 1974 · Zbl 0298.57008 [13] [S] segal, G.: The topology of spaces of rational functions. Acta Math.143, 39–72 (1979) · Zbl 0427.55006 · doi:10.1007/BF02392088 [14] [T] Totaro, B.: The coholomogy ring of the space of rational functions. Preprint, MSRI 1990
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. | 2021-05-16 14:45:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6897482872009277, "perplexity": 3217.715114143357}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991224.58/warc/CC-MAIN-20210516140441-20210516170441-00578.warc.gz"} |
https://graphicmaths.com/a-level/pure/complex-numbers/imaginary-numbers/ | # Imaginary numbers
Martin McBride
2022-01-30
The unit imaginary number, usually called i, is defined as a solution to the equation:
$$x^2 = -1$$
In other words:
$$i = \sqrt{-1}$$
Of course there is no real number than can be squared to give a negative result. That is because any positive real number multiplied by itself will be positive, and any negative real number multiplied by itself will also be positive.
However, the mathematical concept of a number i, the square root of -1, is valid and useful. There is no real number that satisfies that description, but if we define the imaginary number i to have such a property, we can use mathematical rules to manipulate it. And it results in some interesting and useful maths.
The term imaginary number was originally coined by Descartes, to distinguish them from real numbers. It isn't necessarily a helpful name because ultimately all numbers are figments of our imagination, so i is no more imaginary than any other number. Still, that is what i is called.
## Surds
You will most likely already be familiar with surds. A surd is an unresolved root of a number.
For example consider the square roots 2. It is an irrational number - it cannot be expressed exactly as a fraction, and therefore cannot be expressed exactly as a decimal number. It is approximately 1.41421356, but that is not its exact value.
If we want to express the square root of 2 exactly, we must use a surd. We simply call it $\sqrt{2}$.
We can manipulate surds. For example we can add them:
$$\sqrt{2} + \sqrt{2} = 2 \sqrt{2}$$
We can sometimes simplify surds, for example:
$$\sqrt{2}(1 + \sqrt{2}) = \sqrt{2} + \sqrt{2} \sqrt{2} = \sqrt{2} + 2$$
In this case we use the fact that if we multiply root 2 by root 2 we get 2. However, we can't simplify the expression any further. We can't combine root 2 and 2 unless we resolve root 2 to an approximate number, to a certain number of decimal places. But as soon as we do that, our result is no longer exact.
## Imaginary arithmetic
We can perform basic arithmetic on i, in a similar way to handling surds. The difference is that i is not a real number. So while it is possible to approximate root 2 by 1.41421356, we cannot do a similar thing with i. There is no approximate real number for i. We must always keep it as i.
Here are some examples. We can add terms in i:
$$i + i + i = 3i$$
We can multiply i by a real number
$$2i + 1.5i = 3.5i$$
We can raise i to an integer power. Every i pair resolves to -1:
\begin{align} i^2 = i \times i = -1 \newline i^3 = i \times i \times i = -1 \times i = -i \newline i^4 = i \times i \times i \times i = -1 \times -1 = 1 \newline i^5 = i \times i \times i \times i \times i = -1 \times -1 \times i = i \end{align}
We can go further, to handle expressions like these (but this will have to wait until we cover complex numbers):
• $\sqrt{i}$ - we can find the square root of i.
• Powers - can can find $i^x$ where x need not be an integer. We can also find $x^i$ or even $i^i$.
• We can apply other standard functions to imaginary numbers, such as $\sin i$ or $\ln i$. | 2022-05-26 18:22:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9986575245857239, "perplexity": 361.02538314446684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662619221.81/warc/CC-MAIN-20220526162749-20220526192749-00626.warc.gz"} |