content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Find the Numbers Games
Lost Numbers 20 is the 20th game in this hidden object style game series where you need to find all the numbers. Do to that, you need to use the magnifier and use your observation skills. Observe
each scene and try to find all the numbers and cones if you wish to complete the achievements. Good luck and enjoy the game! | {"url":"http://www.findthenumbersgames.com/tag/lost-numbers/","timestamp":"2014-04-18T03:02:45Z","content_type":null,"content_length":"45320","record_id":"<urn:uuid:2278fee6-2262-4c2e-a6bf-91aebf7d5417>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00460-ip-10-147-4-33.ec2.internal.warc.gz"} |
David Gabai
Spanning Laminations are Proper
Abstract: Suppose that H^3 is given the Riemannian metric r induced from a Riemannian metric on a closed hyperbolic 3-manifold.
Theorem If \sigma is a r-least area D^2-limit lamination spanning a smooth simple closed curve in S^2_\infty, then each leaf of \sigma is a proper plane. | {"url":"http://www.newton.ac.uk/programmes/SKG/gabai.html","timestamp":"2014-04-16T16:07:04Z","content_type":null,"content_length":"905","record_id":"<urn:uuid:1e478715-d563-43c8-935e-09fa2147dc5a>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00113-ip-10-147-4-33.ec2.internal.warc.gz"} |
Finally, LaTeX in HTML files
Writing a paper in LaTeX will always result in beautiful output, but if you’d like to put that document up on the web you’re limited to two reasonable options: serve the document as a .PDF (with the
horrors involves, although Chrome makes things much more palatable), or relying on third-party browser plugins like TeX The World. Now that [Todd Lehman] has finally cooked up a perl script to embed
LaTeX in HTML documents, there’s no reason to type e^i*pi + 1 = 0 anymore.
For those not in the know, LaTeX is a document typesetting language that produces beautiful output, usually in PDF form. Unfortunately, when [Tim Berners-Lee] was inventing HTML, he decided to roll
his own markup language instead of simply stealing it from [Don Knuth]. Since then, LaTeX aficionados have had to make do with putting TeX snippets into web pages as images or relying on the [; \
LaTeX ;] generated from the TeX The World browser extension.
[Todd Lehman]‘s perl script generates the PDF of his LaTeX file and pulls out all the weird font and math symbols into PNG files. These PNG files are carefully embedded into the HTML file generated
from the normal text pulled from the LaTeX file. It’s a ton of work to get these document systems working correctly, but at least there’s a reasonable way to put good-looking LaTeX on the web now.
1. Klemen Slavič says:
Actually, the accessibility of an image-based system is pretty lacking; investing in MathML with alternative rendering systems for non-supported browsers makes for a much better experience.
See http://www.mathjax.org/ for a rock-solid implementation. Also, that one’s scalable, images aren’t.
□ pascal says:
IMO the best way to do it is to convert the LaTeX-formulas (because they’re easy to write) into MathML (because it’s a standard format and Firefox even renders it, but it’s awful to write
manually) and embedd the MathJAX-script to render it in the browsers that don’t understand MathML.
2. Mythgarr says:
I’m confused – what was wrong with MathJax? I’ve been embedding LaTeX into HTML for a while now…
□ bkanuka says:
I agree. Why now use MathJax? it’s incredible what it can do.
and the “horrors of pdf” are pretty negligible. Sometimes I’m annoyed that it’s an end format but even then there’s a latex package to embed the tex file.
3. GaspingSpark says:
Or use the MathML support that already works in most browsers. http://www.w3.org/Math/
There are lots of converters from LaTex:
4. Aleksejs Popovs says:
In my opinion, this is a really stupid solution as the equations won’t be nor scalable nor selectable. I think that http://mathjax.com/ is a much more beautiful solution to this problem.
> Unfortunately, when [Tim Berners-Lee] was inventing HTML, he decided to roll his own markup language instead of simply stealing it from [Don Knuth]
Thank God! LaTeX is beautiful, but it’s absolutely not suited for web (unless you use it inline, like with MathJax).
5. Ivan says:
Having a public license I guess it wouldn’t had been “stolen” anyways.
6. jonored says:
I usually use latex2html to do essentially the same thing document structure – not perfect package handling, but otherwise pretty decent at its job.
7. Mohammed says:
PNG files? Why not SVG? I guess maybe bitmap is easier to work with during generation though.
8. Kalleguld says:
You need a paren around i*pi
□ Brian Benchoff says:
9. Brett W. (FightCube.com) says:
serve the document as a .PDF (with the horrors
What’s so bad about linking a file? PDFs open just fine for me.
I agree just embedding the equations into HTML directly is more elegant, but down and dirty gets it done as well.
Maybe you are referring to special fonts that need to be embedded into the PDF itself?
10. M4CGYV3R says:
If it’s difficult for you to read PDF files, you’re doing it (all) wrong. Acro Reader works just fine.
Especially on a Mac. You don’t even need plugins or a reader for that.
11. zokier says:
There is a very good reason TBL chose to create HTML instead of using TeX. Rendering simple HTML is much much faster than doing the perfect typography of TeX.
12. urdh says:
Isn’t this really old news? There’s been LaTeX-to-PNG renderers for online use for quite a while, and lots of LaTeX-to-MathML converters as well. And the quite excellent MathJAX.
13. svofski says:
I’m too in favour of MathJax, but, to all the naysayers — it’s a hack after all :D
14. Kemp says:
This seems great as a personal project looking into handling LaTeX/PDFs and generating images, but there have been existing solutions for a long time now. The article is pitched a little to far
to the “look at this great new technique” side of things.
15. cheshirekow says:
I’ve found LaTeXML to be a great way to get latex into an html format. Far better than latex2html and it’s not a javascript thing so there’s not processing delay in the page.
16. a.p. says:
This tool is very easy and beatiful to make math equation.
17. Antonio says:
As GaspingSpark already said ” the MathML support that already works in most browsers. http://www.w3.org/Math/ ”
To write easily formulas then export them in MathML
or Latex or Maxima, or other formats, just use
this applet which works also offline :
Also it’s not limited to a single browser,
like the chrome software cited before,
moreover the Licence is GPL !
export to clipboard, paste in a text editor and save
as .html or .tex or other as you like.
Bye ;-)
Come GaspingSpark ha già detto “il supporto MathML funziona già nella maggior parte dei browser. http://www.w3.org/Math/”
Per scrivere facilmente le formule e poi esportarle in MathML
o in latex o Maxima, o altri formati, potete usare
questa applet che funziona anche offline:
rilasciata con licenza GPL !
Non vincolata ad un solo programma di navigazione
come quella di Chrome.
esportare negli appunti, incollarlo in un editor di testo e salvare
come .html o .tex o altri formati che vi servano.
Ciao ;-)
18. Elliot says:
Also consider AsciiDoc. You can write (with embedded Latex math) into a single document that can be processed into DocBook, HTML, PDF, and etc.
19. Manoj says:
20. Manoj says:
other stuff
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MathGroup Archive: June 2006 [00113]
[Date Index] [Thread Index] [Author Index]
RE: Colored Tick Labels?
• To: mathgroup at smc.vnet.net
• Subject: [mg66941] RE: [mg66910] Colored Tick Labels?
• From: "David Park" <djmp at earthlink.net>
• Date: Sun, 4 Jun 2006 02:01:17 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com
With DrawGraphics one could use the CustomTicks command.
The following plots Sin[Pi t] from -1 to 1, then relabels the x axis so it
runs from -Pi to Pi and also multiplies the amplitude of the curve by 3 and
shifts it up by three so the y scale goes from 0 to 6. The x-axis labels are
in red and the y axis labels are in blue. One only has to enter StyleForm
once, in the CTNumberFunction option. We also obtain the unlabeled
subdivision ticks and for the top and right frame ticks we have the same
divisions with no tick labels. Just for fun, I also added very light grid
lines using CustomGridLines.
The pure functions take us from the tick label to the underlying plot value.
The list gives us {start, end, step, number of subdivisions} in terms of the
labeled tick values.
xticks = CustomTicks[#/Pi &, {-Pi, Pi, Pi/2, 5},
CTNumberFunction -> (StyleForm[#, FontColor -> Red] &)];
topticks =
CustomTicks[#/Pi &, {-Pi, Pi, Pi/2, 5}, CTNumberFunction -> ("" &)];
yticks = CustomTicks[(# - 3)/3 &, {0, 6, 1, 5},
CTNumberFunction -> (StyleForm[#, FontColor -> Blue] &)];
rightticks =
CustomTicks[(# - 3)/3 &, {0, 6, 1, 5}, CTNumberFunction -> ("" &)];
xgrids = CustomGridLines[#/Pi &, {-Pi, Pi, Pi/2}, {Gainsboro}];
ygrids = CustomGridLines[(# - 3)/3 &, {0, 6, 1}, {Gainsboro}];
{Draw[Sin[Pi t], {t, -1, 1}]},
Frame -> True,
FrameTicks -> {xticks, yticks, topticks, rightticks},
GridLines -> {xgrids, ygrids},
ImageSize -> 450];
David Park
djmp at earthlink.net
From: AES [mailto:siegman at stanford.edu]
To: mathgroup at smc.vnet.net
To create frames with different scales and tick labels on each axis, I
habitually use syntax like:
xTicks={{-1,"-Pi"}, {0.5,"0"}, {1,"Pi"}}
and so on, and then
FrameTicks->{xTicks, yTicks, topTicks, rightTicks}
Question: Any simple way to apply a different color to the set of ticks
on each axis ***without having to insert a syntax like
StyleForm["-Pi", FontColor->Red]
into each and every individual quoted "-Pi" label separately? | {"url":"http://forums.wolfram.com/mathgroup/archive/2006/Jun/msg00113.html","timestamp":"2014-04-21T04:41:56Z","content_type":null,"content_length":"36389","record_id":"<urn:uuid:6224e611-dfe9-4e9b-a875-ae5134018f65>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00184-ip-10-147-4-33.ec2.internal.warc.gz"} |
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4. CRACK EXTENSION FORCE: Crack extension force is the strain energy release rate per unit thickness of the material. It is usually represented by (G).
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Next: About this document ...
CURRICULUM VITAE
NAME: Ilias S. Kotsireas, Associate Professor CITIZENSHIP: Canadian
ADDRESS: Wilfrid Laurier University, Department of Physics and Computer Science, 75 University Avenue West Waterloo Ontario N2L 3C5, CANADA
CONTACT INFORMATION: Office Phone & Voice Mail: ++1-(519) 884-0710 ext. 2218, Fax: ++1-(519) 746-0677, e-mail: ikotsire@wlu.ca CARGO lab web page: http://www.cargo.wlu.ca/
• 1995-1998, Ph.D. Department of Computer Science, Université Paris 6, French National Bureau of Standards, (Bureau des Longitudes) Paris, France. Dissertation Title : ``Algorithms for solving
polynomial systems: application to central configurations in the N-body problem of celestial mechanics.''. Advisor : Professor Daniel Lazard
• 1994-1995, M.Sc. Department of Computer Science, Université Paris 6, French National Bureau of Standards, Paris, France. Dissertation Title : ``Central configurations in the N-body problem''.
Advisors : Professor Daniel Lazard, Professor Alain Albouy.
• 1992-1994 B.Sc. Department of Computer Science, Université Paris 6, Paris, France.
• 1986-1990 B.Sc. Department of Mathematics, University of Athens, Athens, Greece.
• December 2005 - present, Associate Professor, Wilfrid Laurier University, Department of Physics and Computer Science, Waterloo, Ontario, Canada.
• July 2001 - December 2005, Assistant Professor, Wilfrid Laurier University, Department of Physics and Computer Science, Waterloo, Ontario, Canada.
• October 1999 - June 2001, Post-Doctoral Fellow, Ontario Research Centre for Computer Algebra, (ORCCA) University of Western Ontario, London, Ontario, Canada.
• 1998-99 Lecturer (Attaché Temporaire Enseignement Recherche, ATER), Department of Computer Science, Université Paris 6, Paris, France.
• 1997-98 Laboratory Assistant (Travaux Dirigés, TD), Lycée Saint-Louis, Paris, France.
• 1994-98 Teaching Assistant (Formation Permanente) Department of Computer Science, Université Paris 6, Paris, France.
• 1995-96 Laboratory Assistant Travaux Dirigés, TD), Université de Versailles Saint-Quentin-en-Yvelines , Versailles, France.
• Merit Award, December 2005, Wilfrid Laurier University
• FTICA, Fellow of the Institute of Combinatorics and its Applications, January 28, 2004
• Best Poster Award, with D. Butcher, SHARCnet Power Partnership Performance event, January 2004, UWO, London ON, Canada
• ACM Web Assistant Award, ISSAC'2001 London, Ontario, Canada
• Best Poster Award, with A. Galligo, R. Corless, S. Watt, ISSAC'2001 London, Ontario, Canada
• Ontario Research Centre for Computer Algebra Post-Doctoral Fellowship, 1999-2001
• French Ministry of National Education Research and Technology Doctoral Scholarship, 1995-1998
i. Editorial Board
• Editor, (term of appointment: 2004-2007), Communications in Computer Algebra, published by the Association for Computing Machinery (ACM) Special Interest Group on Symbolic and Algebraic
Manipulation (SIGSAM)
• Mathematics and Computers in Simulation, Elsevier, Special Issue on Applications of Computer Algebra in Science, Engineering, Simulation and Special Software, 67, 2004, no. 1-2. Guest Editors:
Michael Wester, Elizabeth A. Arnold, Patrizia Gianni, Ilias S. Kotsireas, Eugenio Roanes-Lozano, Stanly Steinberg
• Journal of Symbolic Computation, Elsevier, Special Issue on Applications of Computer Algebra, 40, 2005, no. 4-5. Guest Editors: Ilias S. Kotsireas, Alkis G. Akritas, Stanly Steinberg, Michael
• Journal of Statistical Planning and Inference, Elsevier, Special Issue on Metaheuristics, Combinatorial Optimization and Design of Experiments, in progress. Guest Editors: Ilias S. Kotsireas,
Christos Koukouvinos
ii. Memberships
• ACM/SIGSAM (Association for Computing Machinery, Special Interest Group on Symbolic and Algebraic Manipulation)
• AMS (American Mathematical Society)
• HMS (Hellenic Mathematical Society)
• ICA (Institute of Combinatorics and its Applications)
iii. Journal Referee
• Journal of Symbolic Computation
• Mathematics and Computers in Simulation
• International Journal of Computers and Mathematics With Applications
• Applied Mathematics Letters
iv. External Grant Referee
National Science Foundation (NSF) ITR Panel, Numeric, Symbolic and Geometric program, May 2001, Arlington, VA, USA.
v. Selected research visits & stays
• Research Institute for Symbolic Computation, RISC-Linz, November 1999, Linz, Austria. (contact: Prof. Josef Schicho)
• Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, CENOLI, ULB, February 2001, Brussels, Belgium. (contacts: Professor G. Nicolis, Dr. K. Karamanos)
• Intensive Summer School in Computer Algebra, July 2001, Queen's University, Kingston, Ontario, Canada.
• ZIB-Berlin, Germany.
• MMRC, Beijing, P.R. China. (contact: Professor Xiao-shan Gao, Director)
• EAGER - EMS Summer School on Computational Algebraic Geometry and Applications Eilat, Israel, February 2002.
vi. Conference Organization
• Waterloo Workshop on Computer Algebra, WWCA, April 2006, co-organizer, Waterloo, Ontario, Canada
• Program Committee Chair, Maple Conference 2006, Waterloo Ontario, Canada
• Publicity Chair, International Symposium on Symbolic and Algebraic Computation 2006, ISSAC 2006, Genoa, Italy
• General Chair, Computer Algebra in Scientific Computing CASC 2005, Kalamata, Greece
• Program Committee Chair, Maple Conference 2005, Waterloo ON, Canada
• Publicity Chair, International Symposium on Symbolic and Algebraic Computation 2005, ISSAC 2005, Beijing, China
• Scientific Committee Chair and Steering Committee Member, High Performance Computing Systems and Applications 2005, HPCS 2005, hosted by SHARCnet, Guelph, Ontario, Canada
• Poster Committee, International Symposium on Symbolic and Algebraic Computation 2005, ISSAC 2004, University of Cantabria, Santander, Spain
• General Chair, East Coast Computer Algebra Day 2004, ECCAD 2004, Waterloo, Ontario, Canada
• General Chair, Applications of Computer Algebra 2002, ACA 2002 Volos, Thessaly, Greece
• Local Arrangements, ECCAD 2000/SONAD 2000, & ISSAC 2001, London, Ontario, Canada
• Permanent member of the Program Committee-Scientific Committee in the ACA (Applications of Computer Algebra) and CASC (Computer Algebra in Scientific Computing) conference series, since 2000.
STUDENT SUPERVISION (9 undergraduate students)
1. Dan Butcher, SHARCnet Round III Graduate Fellowship
2. Jason Cousineau, Research Assistant
3. Alexei Karpenko, Research Assistant
4. Edmond Lau, Research Assistant
5. Chris Odorjan, Research Assistant
6. Gil Pinheiro, Directed Research Course, Research Assistant
7. Dimitra Rentas, co-op Student
8. Michael Sukman, Research Assistant
9. Paul Walrath, Directed Research Course, Research Assistant
TEACHING (taught 7 different courses at Wilfrid Laurier University)
1. CP102 Information Processing with Microcomputer Systems, Fall 2004, Winter 2006, Winter 2007.
2. CP114 Data Structures I, Winter 2005.
3. CP315 Introduction to Scientific Computation, Fall 2004, Fall 2005.
4. CP363 Databases I, Winter 2002, Winter 2003, Winter 2004, Winter 2005, Winter 2006, Winter 2007.
5. CP411 Computer Graphics, Fall 2002, Fall 2005.
6. CP463 Discrete Event Simulation, Winter 2002, Winter 2003, Winter 2004, Winter 2007.
7. CP465 Databases II, Fall 2002, Winter 2007.
│ Year │ Source │ Type │ Amount │ Purpose │
│ 2002-2006 │ NSERC │ Individual Research Grant │ $ 64000 │ Research │
│ 2002 │ SHARCnet │ Round III Graduate Fellowship │ $ 22000 │ Grad. Fell. │
│ 2006-2011 │ NSERC │ Individual Research Grant │ $ 75000 │ Research │
│ 2007-2008 │ EU │ ENTER │ € 96000 │ Research │
│ Year │ Source │ Type │ Amount │ Purpose │
│ 2002 │ WLU │ Conference/Workshop Grant │ $ 3000 │ ACA 2002 │
│ Fall 2003 │ WLU │ Course Remission Grant │ $ 10000 │ Research │
│ 2004 │ WLU │ Conference/Workshop Grant │ $ 3000 │ ECCAD 2004 │
│ 2004 │ WLU │ Laurier Lecture co-sponsorship Fund │ $ 1100 │ CSASM │
│ 2004 │ WLU │ STEP │ $ 5000 │ CSASM │
│ 2004 │ WLU │ Academic Development Fund │ $ 1100 │ ICPSS 2004 │
│ 2005 │ WLU │ Academic Development Fund │ $ 1500 │ CASC 2005 │
│ 2005 │ WLU │ Merit Award │ $ 3000 │ Research │
│ 2006 │ WLU │ Academic Development Fund │ $ 3000 │ WWCA 2006 │
BOOKS EDITED (6)
(available from the TriUniversity Group of Libraries (TUG) TRELLIS service http://www.tug-libraries.on.ca/ )
1. Waterloo Workshop on Computer Algebra 2006, World Scientific, in progress, Editors: Ilias S. Kotsireas, Eugene Zima.
2. Maple Conference 2005, Waterloo, Canada, Proceedings, Editor: Ilias S. Kotsireas (with the assistance of I. J. Sinclair, J. Duketow, R. M. Kalbfleisch), 515 pages.
3. High Performance Computing Systems and Applications, HPCS 2005, Guelph, Canada, Conference Proceedings, IEEE, Editors: Ilias S. Kotsireas and Deborah Stacey, 362 pages.
4. International Symposium on Symbolic and Algebraic Computation, ISSAC 2004, University of Cantabria, Santander Spain. Collection of Poster Abstracts, Editor: Ilias S. Kotsireas, 55 pages.
5. East Coast Computer Algebra Day, ECCAD 2004, Waterloo, Canada, Book of Abstracts, Editors: Ilias S. Kotsireas, 22 pages.
6. Applications of Computer Algebra, ACA 2002, Volos, Greece, Book of Abstracts, Editors: Alkis G. Akritas, Ilias S. Kotsireas 148 pages.
CHAPTERS IN BOOKS (2)
1. Ilias Kotsireas. Central Configurations in the Newtonian N-body problem of Celestial Mechanics. Computer Algebra Handbook, Springer Verlag, 2002, J. Grabmeier, E. Kaltofen, V. Weispfenning eds pp
2. Ilias Kotsireas. Panorama of methods for exact implicitization of algebraic curves and surfaces. Geometric Computation, World Scientific, 2003, D. Wang, F. Chen eds pp 126-155.
PAPERS IN REFEREED JOURNALS (18)
1. Ilias S. Kotsireas and Daniel Lazard. Central Configurations of the 5-body problem with equal masses in three-dimensional space. J.Math. Sci. (New York), vol. 108, 2002, no. 6, pp. 1119-1138
2. Ilias S. Kotsireas. Central configurations in the Newtonian N-body problem of Celestial Mechanics. Contemporary Mathematics, AMS, vol. 286, 2000, pp. 71-98
3. K. Karamanos, I. Kotsireas, Thorough numerical entropy analysis by lumping of some substitutive sequences. Kybernetes 2002, Volume 31, no. 9/10, pp. 1409-1417
4. K. Karamanos, I. Kotsireas, Statistical analysis of the first digits of the binary expansion of Feigenbaum constants Journal of the Franklin Institute, Volume 342 (2005) pp. 329-340.
5. I. Kotsireas, K. Karamanos. Exact computation of the Bifurcation point B4 of the logistic map and the Bailey-Broadhurst conjectures. International Journal of Bifurcation and Chaos Volume 14, no.
7, 2004, pp. 2417-2423
6. H. Evangelaras, I. Kotsireas, C. Koukouvinos. Applications of Groebner bases to the analysis of certain two or three level factorial designs. Advances and Applications in Statistics 3, no. 1,
2003 pp. 1-13.
7. I. S. Kotsireas, C. Koukouvinos. J. Seberry. Hadamard ideals and Hadamard matrices with circulant core J. Combin. Math. Combin. Comput. 57, 2006, pp. 47-63.
8. I. S. Kotsireas, C. Koukouvinos, J. Seberry. Hadamard ideals and Hadamard matrices with two circulant cores. European Journal of Combinatorics 27, 2006, no. 5, pp. 658-668.
9. I. S. Kotsireas, C. Koukouvinos, Inequivalent Hadamard matrices with buckets, J. Discrete Math. Sci. Cryptogr. 7, 2004, no. 3, pp. 307-317.
10. I. S. Kotsireas, C. Koukouvinos, Genetic Algorithms for the construction of Hadamard matrices with two circulant cores J. Discrete Math. Sci. Cryptogr. 8, 2005, no. 2, pp. 241-250.
11. I.S. Kotsireas, C. Koukouvinos, A computational algebraic approach for saturated D-optimal designs with Utilitas Mathematica 71, 2006, pp. 197-207.
12. I. Kotsireas, C. Koukouvinos and M.P. Rogantin, Inequivalent Hadamard matrices via indicator functions. Int. J. Applied Math. 16, 2004, no. 3, pp. 355-363.
13. J. Cousineau, I. Kotsireas, C. Koukouvinos, Genetic Algorithms for Orthogonal Designs Australasian Journal of Combinatorics 35, 2006, pp. 263-272.
14. S. Georgiou, I. Kotsireas, C. Koukouvinos, Inequivalent Hadamard matrices of order 2n from Hadamard matrices of order n. J. Combin. Math. Combin. Comput. to appear.
15. I. S. Kotsireas, C. Koukouvinos, G. Pinheiro, Metasoftware for Hadamard matrices. Int. J. Appl. Math. 18, 2005, no. 2, pp. 263-278.
16. I. S. Kotsireas, C. Koukouvinos Orthogonal designs via computational algebra. Journal of Combinatorial Designs 14, 2006, Issue 5, pp. 351-362.
17. I. Z. Emiris, I. S. Kotsireas, Implicitization exploiting sparseness. Geometric and algorithmic aspects of computer-aided design and manufacturing, pp. 281-297, DIMACS Ser. Discrete Math.
Theoret. Comput. Sci., 67, AMS Providence, RI, 2005.
18. I. S. Kotsireas, C. Koukouvinos Constructions for Hadamard matrices of Williamson type. J. Combin. Math. Combin. Comput. 59, 2006, pp. 17-32.
1. Jean-Charles Faugère and Ilias Kotsireas. Symmetry theorems for the Newtonian 4- and 5-body problems with equal masses. CASC 1999 Proceedings, Springer Verlag, LNCSE, V. Ganzha, et al. (Eds). pp.
2. Ilias Kotsireas. The Erdos-Straus conjecture on Egyptian Fractions. Paul Erdos and his mathematics (Budapest 1999) Janos Bolyai Math. Soc. A. Sali, M. Simonovits, V. Sos, eds. pp. 140-144
3. Robert M. Corless, Mark W. Giesbrecht, Ilias S. Kotsireas, Stephen M. Watt. Numerical implicitization of parametric hypersurfaces with linear algebra. AISC 2000 Proceedings, Springer Verlag, LNAI
1930, E. Roanes-Lozano, ed. pp. 174-183
4. Robert M. Corless, Mark W. Giesbrecht, Mark van Hoeij, Ilias S. Kotsireas, Stephen M. Watt. Towards Factoring Bivariate Approximate Polynomials. ISSAC 2001 Proceedings, ACM Press, B. Mourrain ed.
pp. 85-92
5. Robert M. Corless, André Galligo, Ilias S. Kotsireas, Stephen M. Watt. A Geometric-Numeric Algorithm for Absolute Factorization of Multivariate Polynomials. ISSAC'2002 Proceedings, ACM Press, T.
Mora ed. pp. 37-45
6. K. Karamanos, Ilias S. Kotsireas, Towards Large-Scale Entropy Computations CASYS 2003 Proceedings, AIP, pp. 385-391
7. Ilias S. Kotsireas, Edmond Lau. Implicitization of Polynomial Curves, IPCurves. ASCM 2003 Proceedings, Beijing, China, Z. Li, W. Sit (Eds) pp. 217-226
8. Ioannis Z. Emiris, Ilias S. Kotsireas. Implicit Polynomial Support Optimized for Sparseness ICCSA'2003, Proceedings, LNCS 2669 Montreal, Canada, V. Kumar et al. (Eds) pp. 397-406
9. Ilias S. Kotsireas, Edmond Lau, Richard Voino. Implicitization of Polynomial Surfaces, IPSurfaces. CASC 2003 Proceedings, Passau, Germany, E. W. Mayr et al. (Eds) pp. 241-247
10. Ilias S. Kotsireas, Gil Pinheiro, A Meta-Software System for the Discovery of Hadamard Matrices, HPCS 2005 Proceedings, IEEE Guelph ON, Canada, I. Kotsireas, D. Stacey (Eds) pp. 17-23
11. I. S. Kotsireas, C. Koukouvinos, K. E. Parsopoulos, M. N. Vrahatis Unified Particle Swarm Optimization for Hadamard Matrices of Williamson Type MACIS 2006 Proceedings, Beijing, China
12. I. S. Kotsireas, C. Koukouvinos, D. E. Simos Inequivalent Hadamard Matrices via Orthogonal Designs MACIS 2006 Proceedings, Beijing, China
TECHNICAL REPORTS (7)
1. Ioannis Z. Emiris, Ilias S. Kotsireas. On the Support of the Implicit Equation of Rational Parametric Hypersurfaces. August 2002, Technical Report TR-02-01 ORCCA
2. Ilias Kotsireas and Gregory Reid. Alternative Ways of Solving Polynomial Systems. 2001, Technical Report TR-01-03 ORCCA
3. Ilias S. Kotsireas. Homotopy and polynomial system solving. 2000, Technical Report TR-00-23 ORCCA
4. Robert M. Corless, Mark Giesbrecht, Ilias Kotsireas and Stephen Watt Symbolic-Numeric Algorithms for Polynomials 2000, Technical Report TR-00-21 ORCCA
5. Robert M. Corless, Mark W. Giesbrecht, Ilias S. Kotsireas and Stephen M. Watt. Numerical implicitization of parametric hypersurfaces with linear algebra. 2000, Technical Report TR-00-03 ORCCA
6. Ilias Kotsireas and Josef Schicho. A Computer Algebra solution to a planar newtonian 4-body problem with unequal masses. Technical Report 00-16/2000 RISC-Linz.
7. Ilias Kotsireas. Configurations centrales dans le problème des N Corps. M.Sc. Thesis, 1995, LIP6, Université Paris 6, (in french)
PAPERS NON-REFEREED (1)
1. I. S. Kotsireas. Homotopies and polynomial system solving I. Basic Principles. SIGSAM Bulletin, March 2001, vol. 35, no. 1, issue 135, pp. 19-32
1. Astronomy and Dynamical Systems seminar, French National Bureau of Standards, (Bureau des Longitudes), June 1998, Paris, France.
2. French National Days on Symbolic Computation, CIRM, October 1998, Marseille, France.
3. Computer Algebra and Complexity seminar, Rennes I University, February 1999, IRMAR, Rennes, France. (invited)
4. Summer School of young scientists in Algorithms and Symbolic Computation. March 1999, LaBRI, Université Bordeaux 1, Bordeaux, France.
5. FRISCO (an Esprit-LTR European Commission Project) Closing Workshop April, 1999, NAG Corporation, Oxford, England. (invited)
6. CASC'99, May 1999, TUM, Munich, Germany.
7. ACA'99, June 1999, Universidad Complutense de Madrid, Madrid, Spain.
8. Paul Erdos and his Mathematics, Paul Erdos Memorial Conference, (satellite conference of the UNESCO World Conference on Science) July 4-11, 1999, Budapest, Hungary. (poster and research
communication) (partial support)
9. Effective Methods in Algebraic Geometry, Méthodes Effectives en Géométrie Algébrique , MEGA 2000, June 20-24, 2000, University of Bath, Bath, England
10. Applications of Computer Algebra, ACA 2000, June 2000, Steklov Institute of Mathematics, St. Petersburg, Russia. (partial support)
11. Applications of Computer Algebra, ACA 2004, July 2004, Lamar University, Beaumont, Texas, USA
12. International Conference on Polynomial System Solving, ICPSS 2004, November 2004, Paris, France
13. American Institute of Mathematics, AIM, workshop on Computational Algebraic Statistics, December 14-18, 2003, Palo Alto, California, USA
14. International Conference On Design Of Experiments: theory and applications, ICODOE, May 13-15, 2005, Memphis TN, USA
15. Applications of Computer Algebra, ACA 2005, July 31 - August 3, 2005, Nara Women's University, Nara, Japan
16. Asian Symposium on Computer Mathematics ASCM 2005, December 2005, Korea Institute for Advanced Study, Seul, Korea (partial support)
17. Mathematical Aspects of Computer and Information Sciences, MACIS 2006, July 2006, Beihang University, Beijing, China
1. Web page Committee, 2005
2. DAP Committee, 2002
1. Ontario Universities Fair, Toronto, Faculty of Science kiosk
2. Teachers Science Day 2005, Presentation Title: "Working with 200 computers simultaneously, high-performance computing demonstration", February 2005
3. co-founder (with CRC Tier I Roderick Melnik) of the Laurier Seminar Series in Computational Science and Applied and Statistical Modelling (CSASM) 2004. web page: http://www.mmcs.wlu.ca/csasm/
4. Environment/occupational Health and Safety Committee, Emergency Warden, 2004-2005
1. Senate Committee on Information Technology, 2005-2006
2. Shared Hierarchical Academic Research Computing Network (SHARCnet) site leader for Wilfrid Laurier University, December 2005-
1. Promoting Women in Science, PROWIS 2003,
Workshop Title: "The Fractal Geometry of Nature" May 2003
2. Promoting Women in Science, PROWIS 2002,
Workshop Title: "Have fun with the computer while learning useful Mathematics" May 2002
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III Abstraction
16 Similarities Everywhere
Many of our data definitions and function definitions look alike. For example, the definition for a list of strings differs from that of a list of numbers in only two regards: the name of the class
of data and the words “string” and “number.” Similarly, a function that looks for a specific string in a list of strings is nearly indistinguishable from one that looks for a specific number in a
list of numbers.
Over the years, Seen this way, a program is just like an essay. The first version is just a draft, and drafts demand editing. people have come to realize that these kinds of similarities are
problematic. Therefore good programmers try to eliminate them as much as possible. “Eliminate” implies that programmers write down their first drafts of programs, spot similarities, and then rework
their drafts. Good programmers go through several such rounds of editing, but they also know when to stop because further rounds of editing eliminate only insignificant similarities.
This chapter is about eliminating similarities in function definitions and in data definitions, a process that is called abstraction. Our means of avoiding similarities are specific to “Intermediate
Student Language” or ISL for short. In DrRacket, choose “Intermediate Student Language” from the “How to Design Programs” submenu in the “Language” menu. Other programming languages provide similar
means, and in object-oriented languages you may find additional mechanisms. Nevertheless, all these mechanisms share the basic characteristics spelled out in this chapter, and thus the design ideas
explained here apply to these languages, too.
16.1 Similarities In Functions
The design recipe determines a function’s basic organization because the template is created from the data definition without regard to the purpose of the function. Not surprisingly then, functions
that consume the same kind of data look alike.
Consider the two functions in figure 50, which consume lists of strings and look for specific strings. The function on the left looks for "dog", the one on the right for "cat". The two functions are
nearly indistinguishable. Each consumes lists of strings; each function body consists of a cond expressions with two clauses. Each produces false if the input is empty; each uses an or expressions to
determine whether the first item is the desired item and, if not, uses recursion to look in the rest of the list. The only difference is the string that is used in the comparison of the nested cond
expressions: contains-dog? uses "dog" and contains-cat? uses "cat". To highlight the differences, the two strings are shaded.
Good programmers are too lazy to define several closely related functions. Instead they define a single function that can look for both a "dog" and a "cat" in a list of strings. This general function
consumes an additional piece of data—the string to look for—and is otherwise just like the two original functions:
If you really needed a function such as
now, you could define it as a one-line function, and the same is true for the
Figure 51
does just that, and you should briefly compare it with
figure 50
to make sure you understand how we get from there to here. Best of all, though, with
it is now trivial to look for
string in a list of strings and there is no need to ever define a specialized function such as
; Los -> Boolean ; Los -> Boolean
; does l contain "dog" ; does l contain "cat"
(define (contains-dog? l) (define (contains-cat? l)
(contains? "dog" l)) (contains? "cat" l))
Computing borrows the term “abstract” from mathematics. A mathematician refers to “6” as an abstract number because it only represents all different ways of naming six things. In contrast, “6 inches”
or “6 eggs” are concrete instances of “6” because they express a measurement and a count. What you have just seen is called functional abstraction. Abstracting different versions of functions is one
way to eliminate similarities from programs, and as you can see with this one simple example, doing so simplifies programs.
Exercise 198. Use contains? to define functions that search for "atom", "basic", and "zoo", respectively.
Then abstract over them. Define above two functions in terms of the abstraction as one-liners and use the existing test suites to confirm that the revised definitions work properly. Finally,
design a function that subtracts 2 from each number on a given list.
16.2 More Similarities In Functions
Abstraction looks easy in the case of contains-dog? and contains-cat?. It takes only a comparison of two function definitions, a replacement of a literal string with a function parameter, and a quick
check that it is easy to define the old functions with the abstract function. This kind of abstraction is so natural that it showed up in the preceding two parts of the book without much ado.
This section illustrates how the very same principle yields a powerful form of abstraction. Take a look at figure 52. Both functions consume a list of numbers and a threshold. The left one produces a
list of all those numbers that are below the threshold, while the one on the right produces all those that are above the threshold.
The two functions different in only one place: the comparison operator that determines whether a number from the given list should be a part of the result or not. The function on the left uses <, the
right one >. Other than that, the two functions look identical, not counting the function name.
Let us follow the first example and abstract over the two functions with an additional parameter. This time the additional parameter represents a comparison operator rather than a string:
To apply this new function, we must supply three arguments: a function
that compares two numbers; a list
of numbers, and a threshold
. The function then extracts all those items
for which
(R i t)
evaluates to
Stop! At this point you should ask whether this definition makes any sense. Without further ado, we have created a function that consumes a function—something that you probably have not seen before.
If you have taken a calculus course, you encountered the differential operator and the indefinite integral, both of which are functions that consume and produce a function. But we do not assume that
you have taken such a course and show you these wonderful tricks anyway. It turns out, however, that your simple little teaching language ISL supports these kinds of functions, and that defining such
functions is one of the most powerful tools of good programmers—even in languages in which function-consuming functions do not seem to be available.
Now that you have recovered from this surprise, let us see see how
actually works. Clearly, as long as the input list is
the result is
, too, no matter what the other arguments are:
So next we look at a one-item list:
The result should be
(cons 4 empty)
because the only item of this list is
(< 4 5)
is true. Here is the first step of the evaluation:
It generalizes the rule of application; the application is replaced with the body of the
function and all occurrences of
replaced by
(cons 4 empty)
, and
. The rest is straightforward:
The last step is the equation discussed above, meaning there is no need to spell out the reasoning again.
Our final example is an application of extract to a list of two items:
Step 1 is new and says that extract determines that the first item on the list is not less than the threshold and that it therefore is not added to the result of the natural recursion.
by hand. Show every step.
by hand. Show the new steps, rely on prior calculations where possible.
The calculations show that
(extract < l t)
computes the same result as
(small l t)
. Indeed, they suggest that the resulting expressions are nearly identical. Similarly,
(extract > l t)
produces the same output as
(large l t)
, which means that you can define the two original functions like this:
The important insight is not that small-1 and large-1 are one-line definitions. Once you have an abstract function such as extract, you can put it to good uses elsewhere:
1. (extract = l t): This expression extracts all those numbers in l that are equal to t.
2. (extract <= l t): This one produces the list of numbers in l that are less than or equal to t.
3. (extract >= l t): This last expression computes the list of numbers that are greater than or equal to the threshold.
As a matter of fact, the first argument for
need not be one of ISL’s predefined operations. Instead, you can use any function that consumes two arguments and produces a
. Consider this example:
That is, the function checks whether the claim x2 > c holds, and it is usable with extract:
(extract squared>? (list 3 4 5) 10)
This application extracts those numbers in
(list 3 4 5)
whose square is larger than
202. Evaluate
(squared>? 3 10)
(squared>? 4 10)
, and
(squared>? 5 10)
by hand. Then show that
(extract squared>? (list 3 4 5) 10)
So far you have seen that abstracted function definitions can be more useful than the functions you start from. For example, contains? is more useful than contains-dog? and contains-cat?, and extract
is more useful than small and large. These effects of abstraction are crucial for large, industrial programming projects. For that reason, programming language and software engineering research has
focused on how to create single points of control in large projects. Of course, the same idea applies to all kinds of computer designs (word documents, spread sheets) and organizations in general.
Another important aspect of abstraction is that you now have a single point of control over all these functions. If it turns out that the abstract function contains a mistake, fixing its definition
suffices to fix all other definitions. Similarly, if you figure out how to accelerate the computations of the abstract function or how to reduce its energy consumption, then all functions defined in
terms of this function are improved without further ado. The following exercises indicate how these single-point-of-control improvements work.
Both consume non-empty lists of numbers (
) and produce a single number. The left one produces the smallest number in the list, the right one the largest.
Define inf-1 and sup-1 in terms of the abstract function. Test each of them with these two lists:
(list 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1)
(list 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25)
Why are these functions slow on some of the long lists?
Modify the original functions with the use of max, which picks the larger of two numbers, and min, which picks the smaller one. Then abstract again, define inf-2 and sup-2, and test them with the
same inputs again. Why are these versions so much faster?
For a complete answer to the two questions on performance, see Local Function Definitions.
16.3 Similarities In Data Definitions
Now take a close look at the following two data definitions:
The one on the left introduces lists of numbers; the one on the right describes lists of strings. And the two data definitions are similar. Like similar functions, the two data definitions use two
different names, but this is irrelevant because any name would do. The only real difference concerns the first position inside of
in the second clause, which specifies what kind of items the list contains.
In order to abstract over this one difference, we proceed as if a data definition were a function. We introduce a parameter, which makes the data definition look like a function, and where there used
to be different references, we use this parameter:
We call such abstract data definitions parametric data definitions because of the parameter. Roughly speaking, a parametric data definition abstracts from a reference to a particular collection of
data in the same manner as a function abstracts from a particular value.
The question is, of course, what these parameters range over. For a function, they stand for an unknown value; when the function is applied, the value becomes known. For a parametric data definition,
a parameter stands for an entire class of values. The process of supplying the name of a data collection to a parametric data definition is called
; here are some sample instantiations of the
• [List-of Number] says that ITEM represents all numbers so the notation is just another name for List-of-numbers;
• [List-of String] defines the same class of data as List-of-String; and
• if we had identified a class of inventory records, like this:
(define-struct ir (name price))
; An IR is
; (make-ir String Number)
then [
List-of IR
]] would be a name for the class of lists of inventory records.
By convention, we use names with all capital letters for parameters of data definitions, while the arguments are (usually) the names of existing data collections.
Our way to validate that these shorthands really mean what we say they mean is to substitute the actual name of a data definition, e.g.,
, for the parameter
of the data definition and to use a plain name for the data definition:
; A List-of-numbers-again is one of:
; – empty
; – (cons Number List-of-numbers-again)
Since the data definition is self-referential, we copied the entire data definition. The resulting definition looks exactly like the conventional one for lists of numbers and truly identifies the
same class of data.
Let us take a brief look at a second example, starting with a structure type definition:
Here are two different data definitions that rely on this structure type definition:
; A CP-boolean-string is a structure: ; A CP-number-image is a structure:
; – (make-point Boolean String) ; – (make-point Number Image)
In this case, the data definitions differ in two places—both marked by highlighting. The differences in the hori fields correspond to each other, and so do the differences in the veri fields. It is
thus necessary to introduce two parameters to create an abstract data definition:
Here H is the parameter for data collections for the hori field, and V stands for data collections that can show up in the veri field.
To instantiate a data definition with two parameters, you need two names of data collections. Using Number and Image for the parameters of CP, you get [CP Number Image], which describes the
collections of points that combine a number with an image. In contrast [CP Boolean String] combines Boolean values with strings in a point structure.
Once you have parametric data definitions, you can even mix and match them to great effect. Consider this one:
The outermost notation is [
...], which means that you are dealing with a list. Question is what kind of data the list contains, and to answer that question, you need to study the inside of the
The inner part combines
in a
, naming yet another class of data that pairs up two different classes in structure instances. In short,
is an instantiation of
that combines one
with a list of
s. If this went too fast, tease apart this data expression, like above, and explain each pieces as you go.
Both data definitions exploit this structure type definition:
Both define nested forms of data’ one is about numbers and the other about strings. Make examples for both. Abstract over the two. Then instantiate the abstract definition to get back the
; A [Bucket ITEM] is
; (make-bucket N [List-of ITEM])
; interp. the n in (make-bucket n l) is the length of l
; i.e., (= (length l) n) is always true
When you instantiate
, and
, you get three different data collections. Describe each of them with a sentence and with two distinct examples.
Now consider this instantiations:
Construct three distinct pieces of data that belong to this collection.
; A [Maybe X] is one of:
; – false
; – X
Interpret the following data definitions:
What does the following function signature mean:
Design the function.
16.4 Functions Are Values
The functions of this section stretch our understanding of program evaluation. It is easy to understand how functions consume more than numbers, say strings, images, and Boolean values. Structures
and lists are a bit of a stretch, but they are finite “things” in the end. Function-consuming functions, however, are strange. Indeed, these kind of functions violate the BSL grammar of the first
intermezzo in two ways. First, the names of primitive operations and functions are used as arguments in applications. Second, parameters are used as if they were functions, that is, the first
position of applications.
Spelling out the problem tells you how the ISL grammar differs from BSL’s. First, our expression language should include the names of functions and primitive operations in the definition. Second, the
first position in an application should allow things other than function names and primitive operations; at a minimum, it must allow variables and function parameters. In anticipation of other uses
of functions, we agree on allowing arbitrary expressions in that position.
The changes to the grammar seem to demand changes to the evaluation rules, but they don’t change at all. All that changes is the set of values. To accommodate functions as arguments of functions, the
simplest change is to say that functions are values. Thus, we start using the names of functions and primitive operations as values; later we introduce another way to deal with functions as values.
207. Assume the definitions area in DrRacket contains
(define (f x) x)
. Identify the values among the following expressions:
Explain why they are values and why the remaining expressions are not values.
Explain your reasoning.
Exercise 209. Develop function=at-1-2-3-and-5.775?. The function determines whether two functions from numbers to numbers produce the same results for 1.2, 3, and -5.775.
Mathematicians say that two functions are equal if they compute the same result when given the same input—for all possible inputs.
Can we hope to define function=?, which determines whether two functions from numbers to numbers are equal f? If so, define the function. If not, explain why and consider the implication that you
have encountered the first easily definable idea for which you cannot define a function.
17 Designing Abstractions
In essence, to abstract is to turn something concrete into a parameter. We have this several times in the preceding section. To abstract similar function definitions, you add parameters that replace
concrete values in the definition. To abstract similar data definitions, you create parametric data definitions. When you will encounter other programming languages, you will see that their
abstraction mechanisms also require the introduction of parameters, though they may not be function parameters.
17.1 Abstractions From Examples
When you first learned to add, you worked with concrete examples. Your parents probably taught you to use your fingers to add two small numbers. Later on, you studied how to add two arbitrary
numbers; you were introduced to your first kind of abstraction. Much later still, you learned to formulate expressions that convert temperatures from Celsius to Fahrenheit or calculate the distance
that a car travels at a certain speed in a given amount of time. In short, you went from very concrete examples to abstract relations.
This section introduces a design recipe for creating abstractions from examples. As the preceding section shows, creating abstractions is easy. We leave the difficult part to the next section where
we show you how to find and use existing abstractions.
Recall the essence of
Similarities Everywhere
. We start from two concrete function definitions or two concrete data definitions; we compare them; we mark the differences; and then we abstract. And that is mostly all there is to creating
1. Step 1 is to compare two items for similarities.
When you find two function definitions that are almost the same except for their namesThe recipe requires a substantial modification for abstracting over non-values. and some values at a few
analogous places, compare them, mark the differences. If the two definitions differ in more than one place, connect the corresponding differences with a line or a comment.
Here is a pair of similar function definitions:
; Number -> Number
; convert one Celsius
; temperature to Fahrenheit
(define (C2F c)
(+ (* 9/5 c) 32))
The two functions apply a function to each item in a list. They differ only as to which function they apply to each item. The two highlights emphasize this essential difference. They also differ
in two inessential ways: the names of the function and the names of the parameters.
2. Next we abstract. To abstract means to replace the contents of corresponding code highlights with new names and add these names to the parameter list. For our running example, we obtain the
following pair of functions after replacing the differences with g:
This first change eliminates the essential difference. Now each function traverses a list and applies some given function to each item.
The inessential differences—the names of the functions and occasionally the names of some parameters—are easy to eliminate. Indeed, if you have explored DrRacket, you know that check syntax
allows you to do this systematically and easily:
We choose to use map1 for the name of the function and k for the name of the list parameter. No matter which names you choose, the result is two identical function definitions.
Our example is simple. In many cases, you will find that there is more than just one pair of differences. The key is to find pairs of differences. When you mark up the differences on paper and
pencil, connect related boxes with a line. Then introduce one additional parameter per line. And don’t forget to change all recursive uses of the function so that the additional parameters go
along for the ride.
3. Now we must validate that the new function is a correct abstraction of the original pair of functions. To validate means to test, which here means to define the two original functions in terms of
the abstraction.
Thus suppose that one original function is called f-original and consumes one argument and that the abstract function is called abstract. If f-original differs from the other concrete function in
the use of one value, say, val, the following function definition
(define (f-from-abstract x)
(abstract x val))
introduces the function f-from-abstract, which should be equivalent to f-original. That is, for every proper value V, (f-from-abstract V) should produce the same answer as (f-original V). This
particularly true for all values that your tests for f-original use. So re-formulate and re-run those tests for f-from-abstract and make sure they succeed.
Let us return to our running example:
A complete example would include some tests, and thus we can assume that both cf* and names come with some tests:
To ensure that the functions defined in terms of map1 work properly, you can copy the tests and change the function names appropriately:
(make-IR "doll" 21.0)
(make-IR "bear" 13.0)))
(list "doll" "bear"))
4. To make a new abstraction useful, it needs a signature. As Using Abstractions, Part I explains, reuse of abstract functions start with their signatures. Finding useful signatures is, however, a
serious problem. For now we just use the running example to illustrate the problem. Similarities In Signatures below resolves the issue.
So consider the problem of finding a signature for map1. On the one hand, if you view map1 as an abstraction of cf*, you might think the signature is
That is, the original signature extended with one signature for functions: (
). Since the additional parameter for
is a function, the use of a function signature shouldn’t surprise you. This function signature is also quite simple; it is a “name” for all the functions from numbers to numbers. Here
is such a function, and so are
, and
On the other hand, if you view map1 as an abstraction of names, the signature is quite different:
This time the additional parameter is
, which is a selector function that consumes
s and produces
s. But clearly this second signature would be useless in the first case, and vice versa. To accommodate both cases, the signature for
must express that
, and
are coincidental.
Also concerning signatures, you are probably eager to use List-of by now. It is clearly easier to write [List-of IR] than spelling out a data definition for Inventory. So yes, as of now, we use
List-of when it is all about lists and you should too.
Once you have abstracted two functions, you should check whether there are other uses for the abstract function. If so, the abstraction is truly useful. Consider map1 for example. It is easy to see
how to use it to add 1 to each number on a list of numbers:
can also be used to extract the price of each item in an inventory. When you can imagine many such uses for a new abstraction, add it to a library of useful functions to have around. Of course, it is
quite likely that someone else has thought of it and the function is already a part of the language. For a function like
, see
Using Abstractions, Part I
is properly designed, use it to define a tabulation function for
Compare this exercise with
exercise 211
. Even though both involve the
function, this exercise poses an additional challenge because the second function,
, consumes a list of
s and produces an
. Still, the solution is within reach of the material in this section, and it is especially worth comparing the solution with the one to the preceding exercise. The comparison yields interesting
insights into abstract signatures.
Last but not least, when you are dealing with data definitions, the abstraction process proceed in an analogous manner. The extra parameters to data definitions stands for collections of values, and
to testing means to spell out a data definition for some concrete examples. All in all, abstracting over data definitions tends to be easier than abstracting over functions, and so we leave it to you
to adapt the design recipe appropriately.
17.2 Similarities In Signatures
As it turns out, a function’s signature is key to its reuse. Thus, to increase the usefulness of an abstract function, you must learn to formulate signatures that describes abstracts in their most
general terms possible. To understand how this works, we start with a second look at signatures and from the simple—though possibly startling—insight that signatures are basically data definitions.
Both signatures and data definitions specify a class of data; the difference is that data definitions also name the class of data while signatures don’t. Nevertheless, when you write down
; Number Boolean -> String
(define (f n b) "hello world")
your first line describes an entire class of data, and your second line states that
belongs to that class. To be precise, this signature describes the class of
all functions
that consume a
and a
and that produce a
In general, the arrow notation of signatures is like the List-of notation from Similarities In Data Definitions. The latter consumes (the name of) one class of data, say X, and describes all lists of
X items—without assigning it a name. The arrow notation consumes an arbitrary number of classes of data and describes collections of functions.
What this means is that the abstraction design recipe applies to signatures, too. You compare similar signatures; you highlight the differences; and then you replace those with parameters. But the
process of abstracting signatures feels more complicated than the one for functions, partly because signature are already abstract pieces of the design recipe and partly because the arrow-based
notation is much more complex than anything else we have encountered.
Let us start with the signatures of cf* and names:
The diagram is the result of the compare-and-contrast step. Comparing the two signatures shows that they differ in two places: to the left of the arrow, we see
and to its right, it is
If we replace the two differences with parameters, say X and Y, we already get one and the same signature:
The new signature starts with a sequence of variables, which demands an explanation. Drawing on the analogy between signatures and data definitions, these variables are the parameters of the
signature. To make this analogy concrete, the sequence of X and Y is like the ITEM in the definition of
or the X and Y in the definition of
Similarities In Data Definitions
. And just as in parametrized data definitions, these parameters range over class of values.
An instantiation of this parameter list is the rest of the signature with the parameters replaced by the data collections: either their names or other parameters or abbreviations such as
from above. Thus, if you replace X and Y above with
, you get back the signature for
and if you choose
instead, you get back the signature for
This last step explains why we may consider this parametrized signature as an abstraction of the original signatures for cf* and names.
Once we add the extra function parameter to these two functions we get map1 and the signatures are as follows:
Again, the signatures are in pictorial form and with arrows connecting the corresponding differences. These mark-ups suggest that the differences in the second argument—
a function—
are related to the differences in the original signatures. Specifically,
on the left of the new arrow refer to items on the first argument—
a list—
and the
on the right refer to the items on the result—
also a list.
If we now use the same variables as above, we get a signature that applies to map1:
Concretely, map1 consumes a list of items, all of which belong to some collection of data called X. It also consumes a function that consumes elements of X and produces elements of a second unknown
collection, called Y. The result of map1 are lists that contain items from Y.
As you may guess from our first example, abstracting over signatures requires practice. So here is a second pair of signatures:
They are the signatures for
exercise 212
. While the two signatures have some common organization, the differences are distinct. Let us first spell out the common organization in detail:
• both signatures describe one-argument functions;
• both argument descriptions employ the List-of construction;
The difference is that the first signature refers to
twice, and the second one refers to
s and
s. Putting similarities and differences together, the first occurrence of
corresponds to
and the second one to
To make more progress on a signature for the abstraction of the two functions in exercise 212, we need to take the first two steps of the design recipe:
Since the two functions differ in two pairs of values, the revised versions consume two additional values: one is an atomic value, to be used in the base case, and the other one is a function that
joins together the result of the natural recursion with the first item on the given list.
The key is to translate this insight into two signatures for the two new functions. When you do so for pr*, you get
because the result in the base case is a number and the function that combines the first item and the natural recursion is
in the original function. Similarly, for
the signature is
As you can see from the function definition for
, the base case returns an image and the combination function is
, which combines a
and an
into an
Now we take the diagram from above and extend it to the signatures with the additional inputs:
From this diagram, you can easily see that the two revised signatures share even more organization than the original two. Furthermore, the pieces that describe the base cases correspond to each other
and so do the pieces of the sub-signature that describe the combination function. All in all there are six pairs of differences but they boil down to just two:
So to abstract we need two variables, one per kind of correspondence.
Here, then, is the signature for
, the abstraction that
exercise 212
; [X Y] [List-of X] Y (X Y -> Y) -> Y
Stop! Make sure that replacing X with
and Y with
yields the signature for
and that replacing the same variables with
, respectively, yields the signature for
The two examples illustrate how to find general signatures. In principle the process is just like the one for abstracting functions. Due to the informal nature of signatures, however, it cannot be
checked with running examples the way code is checked. Here is step-by-step formulation:
1. Given two similar function definitions, f and g, compare their signatures for similarities and differences. The goal is to discover the organization of the signature and to mark the places where
one signature differs from the other. Connect the differences as pairs just like you do when you analyze function bodies.
2. Abstract f and g into f-abs and g-abs. That is, add parameters that eliminate the differences between f and g. Create signatures for f-abs and g-abs. Keep in mind what the new parameters
originally stood for; this helps you figure out the new pieces of the signatures.
3. Check whether the analysis of step 1 extends to the signatures of f-abs and g-abs. If so, replace the differences with variables that range over classes of data. Once the two signatures are the
same you have a signature for the abstracted function.
4. Test the abstract signature in two ways. First, ensure that suitable substitutions of the variables in the abstract signature yield the signatures of f-abs and g-abs. Second, check that the
generalized signature is in sync with the code. For example, if p is a new parameter and its signature is
then p should always be applied to two arguments, the first one from A and the second one from B. And the result of an application of p is going to be a C and should be used where elements of C
are expected.
As with abstracting functions, the key is to compare the concrete signatures of the examples and to determine the similarities and differences. With enough practice and intuition, you will soon be
able to abstract signatures without much guidance.
Describe these collections with at least one example from ISL.
□ sort-n, which consumes a list of numbers and a function that consumes two numbers (from the list) and produces a Boolean; sort-n produces a sorted list of numbers.
□ sort-s, which consumes a list of srings and a function that consumes two strings (from the list) and produces a Boolean; sort-s produces a sorted list of strings.
Then abstract over the two signatures, following the above steps. Also show that the generalized signature can be instantiated to describe the signature of a sort function for lists of
□ map-n, which consumes a list of numbers and a function from numbers to numbers to produce a list of numbers.
□ map-s, which consumes a list of srings and a function from strings to strings and produces a list of strings.
Then abstract over the two signatures, following the above steps. Also show that the generalized signature can be instantiated to describe the signature of the map-IR function above.
17.3 Single Point Of Control
In general, programs are like drafts of papers. Editing drafts is important to correct typos, to fix grammatical mistakes, to make the document consistent, and to eliminate repetitions. Nobody wants
to read papers that repeat themselves a lot, and nobody wants to read such programs either.
The elimination of similarities in favor of abstractions has many advantages. Creating an abstraction simplifies definitions. It may also uncover problems with existing functions, especially when
similarities aren’t quite right. But, the single most important advantage is the creation of single points of control for some common functionality.
Putting the definition for some functionality in one place makes it easy to maintain a program. When you discover a mistake, you have to go to just one place to fix it. When you discover that the
code should deal with another form of data, you can add the code to just one place. When you figure out an improvement, one change improves all uses of the functionality. If you had made copies of
the functions or code in general, you would have to find all copies and fix them; otherwise the mistake might live on or the only one of the functions would run faster.
We therefore formulate this guideline:
Creating Abstractions: Form an abstraction instead of copying and modifying any piece of a program.
Our design recipe for abstracting functions is the most basic tool to create abstractions. To use it requires practice. As you practice, you expand your capabilities to read, organize, and maintain
programs. The best programmers are those who actively edit their programs to build new abstractions so that they collect things related to a task at a single point. Here we use functional abstraction
to study this practice; in other courses on programming, you will encounter other forms of abstraction, most importantly inheritance in class-based object-oriented languages.
17.4 Abstractions From Templates
Over the course of the first two chapters, we have designed many functions using the same template. After all, the design recipe says to organize functions around the organization of the (major)
input data definition. It is therefore not surprising that many function definitions look similar to each other.
Indeed, you should abstract from the templates directly, you should do so automatically, and some experimental programming languages do so. Even though this topic is still a subject of research, you
are now in a position to understand the basic idea. Consider the template for lists:
It contains two gaps, one in each clause. When you use this template to define a list-processing function, you usually fill these gaps with a value in the first
clause and with a function
in the second clause. The
function consumes the first item of the list and the result of the natural recursion and creates the result from these two pieces of data.
Now that you know how to create abstractions, you can complete the definition of the abstraction from this informal description:
; [X Y] [List-of X] Y (X Y -> Y) -> Y
(define (reduce l base combine)
[(empty? l) base]
[else (combine (first l)
(reduce (rest l) base combine))]))
It consumes two extra arguments: base, which is the value for the base case, and combine, which is the function that performs the value combination for the second clause.
Using reduce you can define many plain list-processing functions as “one liners.” Here are definitions for sum and product, two functions used several times in the first few sections of this chapter:
For sum, the base case always produces 0; adding the first item and the result of the natural recursion combines the values of the second clause. Analogous reasoning explains product. Other
list-processing functions can be defined in a similar manner using reduce.
18 Using Abstractions, Part I
Many programming languages provide a number of looping constructs, or loop for short. A loop processes a compound piece of data, one piece at a time. In our terminology a loop abstracts over the
traversal of data and applies some given function to each of its pieces. You have encountered several such loops in the first two sections of this chapter: extract, fold1, map1, etc. These functions
consume a function and apply it to each item on some list.
Once you have such loop abstractions, you should use them when possible. They create single points of control data, and they are a work-saving device. To make this precise, the use of an abstraction
helps the reader of your code to understand your intentions, in particular if the abstraction is well-known and built into the language or comes with its standard libraries.
This chapter is all about the reuse of existing ISL abstractions. It starts with a section on existing ISL abstractions, some of which you have seen under false names. The remaining sections are
about re-using such abstractions. One key ingredient is a new syntactic construct, local, for defining functions and variables (and even structure types) locally within a function. An auxiliary
ingredient, introduced in the last section, is the lambda construct for creating nameless functions; lambda is a convenience but inessential to the idea of re-using abstract functions.
; N (N -> X) -> [List-of X]
; construct a list by applying f to 0, 1, ..., (sub1 n)
; (build-list f n) = (list (f 0) ... (f (- n 1)))
(define (build-list n f) ...)
; (X -> Boolean) [List-of X] -> [List-of X]
; produce a list from all those items on alox for which p holds
(define (filter p alox) ...)
; [List-of X] (X X -> Boolean) -> [List-of X]
; produce a variant of alox that is sorted according to cmp
(define (sort alox cmp) ...)
; (X -> Y) [List-of X] -> [List-of Y]
; construct a list by applying f to each item on alox
; (map f (list x-1 ... x-n)) = (list (f x-1) ... (f x-n))
(define (map f alox) ...)
; (X -> Boolean) [List-of X] -> Boolean
; determine whether p holds for every item on alox
; (andmap p (list x-1 ... x-n)) = (and (p x-1) ... (p x-n))
(define (andmap p alox) ...)
; (X -> Boolean) [List-of X] -> Boolean
; determine whether p holds for at least one item on alox
; (ormap p (list x-1 ... x-n)) = (or (p x-1) ... (p x-n))
(define (ormap p alox) ...)
; (X Y -> Y) Y [List-of X] -> Y
; compute the result of applying f from right to left to all of
; alox and base, that is, apply f to
; the last item in alox and base,
; the penultimate item and the result of the first step,
; and so on up to the first item
; (foldr f base (list x-1 ... x-n)) = (f x-1 ... (f x-n base))
(define (foldr f base alox) ...)
; (X Y -> Y) Y [List-of X] -> Y
; compute the result of applying f from left to right to all of
; alox and base, that is, apply f to
; the first item in alox and base,
; the second item and the result of the first step,
; and so on up to the last item:
; (foldl f base (list x-1 ... x-n)) = (f x-n ... (f x-1 base))
(define (foldl f base alox) ...)
; (X -> Real) [List-of X] -> X
; finds the (first) element in alox that maximizes f, that is:
; if (argmax f (list x-1 ... x-n)) = x-i,
; then (>= (f x-i) (f x-1)), (>= (f x-i) (f x-2)), and so on
(define (argmax f alox) ...)
; (X -> Real) [List-of X] -> X
; finds the (first) element in alox that minimizes f, that is:
; if (argmin f (list x-1 ... x-n)) = x-i,
; then (<= (f x-i) (f x-1)), (<= (f x-i) (f x-2)), and so on
(define (argmin f alox) ...)
WARNING: you cannot design build-list, build-string, or foldl with the design principles you know at this point; Accumulators covers the necessary ideas.
Figure 53: ISL's built-in abstract functions for list-processing
18.1 Existing Abstractions
ISL provides a number of abstract functions for processing natural numbers and lists.
Figure 53
collects the header material for the most important ones. The first one processes natural numbers and builds lists:
The next three process lists and produce lists:
> (filter odd? (list 1 2 3 4 5))
(list 1 3 5)
> (sort (list 3 2 1 4 5) >)
(list 5 4 3 2 1)
> (map add1 (list 1 2 2 3 3 3))
(list 2 3 3 4 4 4)
This kind of computation is called a reduction because a list of values is reduced to a single value.
Of all the functions in
figure 53
, the last two are the most powerful functions. Both reduce lists to values. The following two computations explain how to use the abstract examples in the headers of
to explain an application to
, and
(list 1 2 3 4 5)
(foldr + 0 '(1 2 3 4 5)) (foldl + 0 '(1 2 3 4 5))
== (+ 1 (+ 2 (+ 3 (+ 4 (+ 5 0))))) == (+ 5 (+ 4 (+ 3 (+ 2 (+ 1 0)))))
== (+ 1 (+ 2 (+ 3 (+ 4 5)))) == (+ 5 (+ 4 (+ 3 (+ 2 1))))
== (+ 1 (+ 2 (+ 3 9))) == (+ 5 (+ 4 (+ 3 3)))
== (+ 1 (+ 2 12)) == (+ 5 (+ 4 6))
== (+ 1 14) == (+ 5 10)
== 15 == 15
As you can see from these calculations,
processes the list values from right to left and
from left to right. While for functions
In mathematics such functions are called associative. And indeed, + is associative on integers and rational numbers in ISL. For inexact numbers, however, this is not true. See below.
such as
the direction seems to make no difference, this isn’t true in general as you can see soon.
(define-struct address (first-name last-name street))
; Addr is (make-address String String String)
; [List-of Addr] -> String
; a string of first names, sorted in alphabetical order,
; separated and surrounded by blank spaces
(define (listing l)
(foldr string-append-with-space
" "
(sort (map address-first-name l)
; String String -> String
; juxtapoint two strings and prefix with space
(define (string-append-with-space s t)
(string-append " " s t))
(define ex0
(list (make-address "Matthias" "Fellson" "Sunburst")
(make-address "Robert" "Findler" "South")
(make-address "Matthew" "Flatt" "Canyon")
(make-address "Shriram" "Krishna" "Yellow")))
(check-expect (listing ex0) " Matthew Matthias Robert Shriram ")
Figure 54 illustrates the power of composing the functions from figure 53. Its main function is listing. The purpose is to create a string from a list of addresses. More precisely, the expected
result is a string that represents a sorted list of first names, separated and surrounded by blank spaces.
A moment’s reflection suggests a straightforward design plan for this problem:
1. design a function that extracts the first names from the given list of Addr;
2. design a function that sorts these names in alphabetical order;
3. design a function that juxtaposes the strings from step 2.
Before you read on, you may wish to execute this plan. That is, design all three functions and then compose them in the sense of
Composing Functions
to obtain your own version of
In the new world of abstractions, it becomes unnecessary to design three separate functions. Take a close look at the innermost expression of
figure 54
(map address-first-name l)
By the purpose statement of
, it applies
to every single instance of
producing a list of first names as strings. Here is the immediately surrounding expression:
The dots represent the result of the
expression. Since the latter supplies a list of strings, the
expression produces a sorted list of first names. And that leaves us with the outermost expression:
(foldr string-append-with-space " " ..)
This one reduces the sorted list of first names to a single string, using a function named
. With such a suggestive name, you can easily imagine now that this reduction juxtaposes all the strings in the desired way—
even if you do not quite understand the details of how the combination of
18.2 Local Function Definitions
Take a second look at figure 54. The string-append-with-space function clearly plays a subordinate role and has no use outside of this narrow context. Almost all programming languages support some
way for stating this relationship as a part of the program. The idea is called a local definition, sometimes also a private definition. In ISL, local expressions introduce locally defined functions,
variables, and structure types, and this section introduces the mechanics of local.
; [List-of Addr] -> String
; a string of first names, sorted in alphabetical order,
; separated and surrounded by blank spaces
(define (listing.v2 l)
(local (; String String -> String
; juxtapoint two strings and prefix with space
(define (string-append-with-space s t)
(string-append " " s t)))
(foldr string-append-with-space
" "
(sort (map address-first-name l)
The body of the
function is now a
expression, which consists of two pieces: a sequence of definitions and a body expression. All local definitions are visible everywhere within the opening parenthesis of
and the closing one and only there.
In this example, the sequence of definitions consists of a single function definition, the one for string-append-with-space. The body of local is the body of the original listing function. Its
reference to string-append-with-space is now resolved locally, that is, there is no need to look in the global sequence of definitions. Conversely, outside of the local expression, it is impossible
to refer to string-append-with-space. Since there is no such reference in the original program, it remains intact and you can confirm this with the test suite.
In general, a
expression has this shape:
(local ( ... sequence of definitions ...) body-expression)
The evaluation of such an expression proceeds like the evaluation of a complete program. First, the definitions are set up, which may involve the evaluation of the right-hand side of a constant
definition. Just as with the top-level definitions that you know and love, the definitions in a
expression may refer to each other. They may also refer to parameters of the surrounding function. Second, the
is evaluated. The result of
is the result of the
figure 55
. The organization of this definition tells the reader that
needs two auxiliary functions:
. Note the reference to
in the latter, which tells the reader that the comparison function remains the same for the entire sorting process.
By making insert local, it also becomes impossible to abuse insert. Re-read its purpose statement. The adjective “sorted” means that a program should use insert only if the second argument is already
sorted. As long as insert is defined at the top-level, nothing guarantees that insert is always used properly. Once its definition is local to the sort-cmp function, the proper use is guaranteed—
because isort applies insert to a sorted version of (rest alon).
Exercise 216. Use a local expression to organize the functions for drawing a polygon in figure 47. If a globally defined functions is widely useful, do not make it local.
Exercise 217. Use a local expression to organize the functions for rearranging words from Rearranging Words.
For a second example, let us take a look at the
function from
exercise 203
Here the
expression shows up in the middle of a
expression. It also doesn’t define a function but a variable whose initial value is the result of a natural recursion. Now recall that the evaluation of a
expression evaluates the definitions once and for all before the body is evaluated. What this means here is that
(inf (rest l))
is evaluated once, but the body of the
expression refers to the result twice. Put differently, the use of
saves the re-evaluation of
(inf (rest l))
at each stage in the computation. To confirm this insight, re-evaluate the above version of
on the inputs suggested in
exercise 203
; Inventory -> Inventory
; to create an Inventory from an-inv for all
; those items that cost less than $1
(define (extract1 an-inv)
[(empty? an-inv) empty]
[else (cond
[(<= (ir-price (first an-inv)) 1.0)
(cons (first an-inv) (extract1 (rest an-inv)))]
[else (extract1 (rest an-inv))])]))
Both clauses in the nested
expression extract the first item from
and both compute
(extract1 (rest an-inv))
. Use a
expression to name the repeated expressions. Does this help increase the speed at which the function computes its result? Significantly? A little bit? Not at all?
Create a test suite for sort->.
Design the function sort-<, which sorts lists of numbers in ascending order.
Create sort-a, which abstracts sort-> and sort-<. It consumes the comparison operation in addition to the list of numbers. Define versions sort-> and sort-< in terms of sort-a.
Use sort-a to define a function that sorts a list of strings by their lengths, both in descending and ascending order.
Later we will introduce several different ways to sort lists of numbers, all of them faster than sort-a. If you then change sort-a, all uses of sort-a will benefit.
18.3 ... Add Expressive Power
The third and last example illustrates how local adds expressive power to BSL+. It starts from the code for simulating finite state machine in Finite State Machines. After a first analysis, this
section points out that it is impossible to organize the program in a natural way—in BSL+.
; FSM -> FSM-State
; interpret a given finite state machine
(define (simulate fsm0)
(local (; A SimulationState is FSM-State.
(define initial-world-state (fsm-current fsm0))
(define TRANSITION-TABLE (fsm-transitions fsm0))
; SimulationState -> Image
; render current state as colored square
(define (state-as-colored-square s)
(square 100 "solid" s))
; SimulationState KeyEvent -> SimulationState
; find the next state in the transition table of fsm0
(define (find-next-state s key-event)
(find TRANSITION-TABLE s))
; Transitions SimulationState -> SimulationState
; find the state matching current
; in the given transition table
(define (find transitions current)
[(empty? transitions) (error "not found")]
(local ((define s (first transitions)))
(if (state=? (transition-current s) current)
(transition-next s)
(find (rest transitions) current)))])))
; now launch the world
(big-bang initial-world-state
[to-draw state-as-colored-square]
[on-key find-next-state])))
Figure 56
shows an ISL solution that uses
variable and function definitions. Specifically, the finite-state machine simulator locally defines three functions: the rendering function, the key event handler, and the auxiliary function
. Because these locally defined functions can refer to the originally given finite state machine
, it is possible to formulate the world program in a natural manner:
1. a world state is just a finite-state machine state;
2. the transition table of fsm0 is like a constant with respect to the locally defined functions even though it is extracted from the given fsm0;
3. the key event handler and the rendering function consume only a SimulationState not the entire FSM;
4. but they may access fsm0’s transition table anyway if needed.
The need arises when find-next-state must look for the next state given that a key event took place. It performs this task by calling find on TRANSITION-TABLE.
18.4 Using Abstractions, By Examples
Now that you understand
, you are ready to use the abstractions from
figure 53
. Let us look at some examples, starting with this one:
Sample Problem: Design the function add-3-to-all. The function consumes a list of Posns and adds 3 to the x coordinates of each of them.
If we follow the design recipe and take the problem statement as a purpose statement, we can quickly step through the first three steps:
While you can run the program, doing so signals a failure in the one test case because the function returns the default value
At this point, we stop and ask what kind of function we are dealing with. Clearly, add-3-to-all is clearly a list-processing function. The question is whether it is like any of the functions in
figure 53. The signature tells us that add-3-to-all is a list-processing function that consumes and produces a list. In figure 53, we have several functions with similar signatures: map, filter, and
The purpose statement and example also tell you that add-3-to-all deals with each Posn separately and assembles the results into a single list. Some reflection says that also confirms that the
resulting list contains as many elements as the given list. All this thinking points to one function—map—because the point of filter is to drop items from the list and sort has an extremely specific
Here is
’s signatures again:
It tells us that
consumes a function from X to Y and a list of Xs. Given that
consumes a list of
s, we know that X stands for
. Similarly,
is to produce a list of
s, and this means we replace Y with
From the analysis of the signature we conclude that
could do the job of
if we can find an appropriate function from
s to
s. With
, we can express this idea as a template for
Doing so reduces the problem to defining a function on
Given the example for
and the abstract example for
, you can actually imagine how the evaluation proceeds:
And that shows how
is applied to every single
on the given list, meaning it is its job to add
to the
From here, it is straightforward to wrap up the definition:
We chose
as the name for the local function because the name tells you what it computes. It adds
to (the
coordinate of) one
as opposed to
, which adds
s. It is
’s task to apply
to each of the
s on
and to assemble the results into one list.
You may think now that using abstractions is hard. Keep in mind, though, that this first example spells out every single detail and that it does so because we wish to teach you how to pick the proper
abstraction. Let us take a look at a second example a bit more quickly:
Sample Problem: Design a function that eliminates all Posns from a list that have a y coordinate of larger than 100.
The first two steps of the design recipe yield this:
By now you may have guessed that this function is like
whose purpose is to separate “good” items from ones.
thrown in, the next step is also straightforward:
function definition introduces the helper function needed for
and the body of the
expression applies
to this local function and the given list. The local function is called
retains all those items of
for which
Before you read on, analyze the signature of filter and keep-good and determine why the helper function consumes individual Posns and produces Booleans.
Putting all of our ideas together yields this definition:
Explain the definition of good? and simplify it.
Before we spell out a design recipe, let us deal with one more example:
Sample Problem: Design a function that determines whether any of a list of Posns is close to some given position pt where “close” means a distance of at most 5 pixels.
This problem clearly consists of two distinct parts: one concerns processing the list and another one calls for a function that determines whether the distance between a point and pt is “close.”
Since this second part is unrelated to the reuse of abstractions for list traversals, we assume the existence of an appropriate function:
You should complete this definition on your own.
As required by the problem statement, the function consumes two arguments—
the list of
s and the “given” point
and produces a
The latter point immediately differentiates this example from the preceding ones.
The Boolean range also gives away a clue with respect to figure 53. Only two functions in this list produce Boolean values—andmap and ormap—and they must be primary candidates for defining close?’s
body. While the explanation of andmap says that some property must hold for every item on the given list, the purpose statement for ormap tells us that it looks for only one such item. Given that
close? just checks whether one of the Posns is close to pt, we should try ormap first.
Let us apply our standard “trick” of adding a
whose body uses the chosen abstraction on some locally defined function and the given list argument:
Following the description of
, the local function consumes one item of the list at a time. This accounts for the
part of its signature. Also, the local function is expected to produce
, and
checks these results until it finds
Here is a comparison of the signature of
, starting with the former:
In our case, the list argument is a list of
s. Hence
stands for
, which explains what
consumes. Furthermore, it determines that the result of the local function must be
so that it can work as the first argument to
The rest of the work requires just a bit more thinking. While
consumes one argument—
function consumes three: two
s and a “tolerance” value. While the argument of
is one of the two
s, it is also obvious that the other one is
, the argument of
itself. Naturally the “tolerance” argument is
, as stated in the problem:
; [List-of Posn] -> Boolean
(define (close? lop pt)
(local (; Posn -> Boolean
; is one shot close to pt?
(define (is-one-close? p)
(close-to p pt CLOSENESS)))
(ormap is-one-close? lop)))
(define CLOSENESS 5)
Note two properties of this definition. First, we stick to the rule that constants deserve definitions. Second, the reference to
signals that this
stays the same for the entire traversal of
18.5 Designing With Abstractions
Three examples suffice for formulating a design recipe for reusing abstractions:
1. Step 1 is to follow the design recipe for functions for three steps. Specifically, you should distill the problem statement into a signature, a purpose statement, an example, and a stub
Consider the problem of defining a function that places small red circles on a 200 by 200 canvas for a given list of
s. The first three steps design recipe yield this much:
2. Next we exploit the signature and purpose statement to find a matching abstraction. To match means to pick an abstraction whose purpose is more general than the one for the function to be
designed; it also means that the signatures are related. It is often best to start with the desired output and to find an abstraction that has the same or a more general output.
For our running example, the desired output is an
. While none of the available abstractions produces an image, two of them have a variable to the right of
; foldr : [X Y] (X Y -> Y) Y [List-of X] -> Y
; foldl : [X Y] (X Y -> Y) Y [List-of X] -> Y
meaning we can plug in any data collection we want. If we do use
, the signature o the left of
demands a helper function that consumes an
and an
and it produces an
. Furthermore, since the given list contains
does stand for the
3. Write down a template. For the reuse of abstractions a template uses local for two different purposes. The first one is to note which abstraction to use and how in the body of the local
expression. The second one is to write down a stub for the helper function: its signature, its purpose (optionally), and its header. Keep in mind that the signature comparison in the preceding
step suggests most of the signature for the auxiliary function.
Here is what this template looks like for our running example if we choose
description calls for a “base”
value, to be used if or when the list is empty. In our case, we clearly want the empty canvas for this case. Otherwise,
uses a helper function and traverses the list of
4. Finally, it is time to define the helper function inside local. In most cases, this auxiliary function consumes relatively simple kinds of data, like those encountered in Fixed-Size Data. You
know how to design those in principle. The only difference is that now you may not only use the function’s arguments and global constants but also the arguments of the surrounding function.
In our running example, the purpose of the helper function is to add one dot to the given scene, which you can (1) guess or (2) derive from the example:
5. The last step is to test the definition in the usual manner.
For abstract functions, it is occasionally possible to use the abstract example of their purpose statement to confirm their workings at a more general level. You may wish to use the abstract
example for foldr to confirm that dots does add one dot after another to the background scene.
In the third step, we picked
without further ado. Experiment with
to see how it would help complete this function. Functions like
are well-known and are spreading in usage in various forms. Becoming familiar with them is a good idea and the next section will help.
18.6 Exercises And Projects
Each of the following set of exercises suggests small practice problems for specific abstractions in ISL.
Exercise 220. Use map to define the function convert-euro, which converts a list of U.S. dollar amounts into a list of euro amounts based on an exchange rate of 1.22 euro for each dollar.
Also use map to define convertFC, which converts a list of Fahrenheit measurements to a list of Celsius measurements.
Finally, try your hands at translate, a function that translates a list of Posns into a list of list of pairs of numbers, i.e., [List-of [list Number Number]].
Exercise 221. A toy structure specifies the name of a toy, a description, the acquisition price, and the recommended sales price.
Define a function that sorts a list of toy structures by the difference between the two prices.
Exercise 222. Define eliminate-exp, which consumes a number, ua and a list of toy structures, and it produces a list of all those structures whose sales price is below ua.
Then use filter to define recall, which consumes the name of a toy, called ty, and a list of toy structures and which produces a list of toy structures that do not use the name ty.
In addition, define selection, which consumes two lists of names and selects all those from the second one that are also on the first.
223. Use
to define functions that
1. creates the list 0 ... (n - 1) for any natural number n;
2. creates the list 1 ... nfor any natural number n;
3. creates the list (list 1 1/10 1/100 ...) of n numbers for any natural number n;
4. creates the list of the first n even numbers;
5. creates a list of lists of 0 and 1 in a diagonal arrangement, e.g.,
Exercise 224. Use ormap to define find-name. The function consumes a name and a list of names. It determines whether any of the names on the latter are equal to or an extension of the former.
With andmap you can define a function that checks all names on a list of names start with the letter "a".
Should you use ormap or andmap to define a function that ensures that no name on some list exceeds some given width?
225. Recall that the
function in ISL juxtaposes the items of two lists or, equivalently, replaces
at the end of the first list with the second list:
to define
. What happens if you replace
Now use one of the fold functions to define functions that compute the sum and the product, respectively, of a list of numbers.
With one of the fold functions, you can define a function that horizontally juxtaposes a list of Images. Hint: look up beside and empty-image. Can you use the other fold function? Also define a
function that stacks a list of images vertically. Hint: check for above in the libraries.
Exercise 226. The fold functions are so powerful that you can define almost any list-processing functions with them. Use fold to define map-from-fold, which simulates map.
Now that you have some experience with the existing list-processing abstractions in ISL, it is time to tackle some real projects. Specifically, we are looking for two kinds of improvements. First,
inspect the game programs for functions that traverse lists. For these functions, you already have signatures, purpose statements, tests, and working definitions that pass the tests. Change the
definitions to use abstractions from figure 53. Second, also inspect the game programs for opportunities to abstract. Indeed, you might be able to abstract across games and provide a framework of
functions that helps you write additional game programs. The following exercises supply specific suggestions for the projects mentioned in this book. If you are using this book for a course, you may
have had to deal with other non-trivial exercises; adapt the exercises to those projects.
Exercise 227. Full Space War spells out a game of space war. In the basic version, a UFO descends and a player defends with a tank. One additional suggestion is to equip the UFO with charges that
it can drop at the tank; the tank is destroyed if a charge comes close enough.
Inspect the code of your project for places where it can benefit from existing abstraction, e.g., processing lists of shots or charges.
Once you have simplified the code with the use of existing abstractions look for opportunities to create abstractions. Consider moving lists of objects as one example where abstraction may pay
Exercise 228. Feeding Worms explains how another one of the oldest computer games work. The game features a worm that moves at a constant speed in a player-controlled direction. When it
encounters food, it eats the food and grows. When it runs into the wall or into itself, the game is over.
This project can also benefit from the abstract list-processing in ISL. Look for places to use them and replace existing code one piece at a time, relying on the tests to ensure the program
19 Nameless Functions
The use of abstract functions employs functions as arguments. On some occasions, these functions are existing primitive functions, library functions, or functions that you defined. For example,
• (build-list n add1) creates (list 1 ... n);
• (foldr cons another-list one-list) juxtaposes the items on one-list and another-list into a single list; and
• the expression (foldr above a-list-of-images) stacks the images on the given list.
On other occasions, the uses of abstract functions require the definition of simple helper functions, whose definition requires often a single line. Consider the following use of
It finds all toys on the inventory list whose price is below threshold. The auxiliary function is nearly trivial yet its definition takes up three lines and does not even have a purpose statement.
This situation calls for an improvement to the language. Programmers should be able to create such small and insignificant functions without much effort. In DrRacket, choose “Intermediate Student
Language with lambda” from the “How to Design Programs” submenu in the “Language” menu. This section presents lambda, the new feature, and introduces “Intermediate Student Language with lambda.” This
language, short: ISL+, is an extension of ISL with lambda. The history of lambda is interesting and intimately involved with the early history of programming and programming language design. When you
have time, you should chase down the origins of lambda.
The first two subsections focus on the design of abstractions using lambda. The remaining sections use lambda and local to introduce some basic programming language concepts, namely, scope and
abbreviations. In addition, they indicate with one use of lambda that the idea of functions as values has additional ramifications for program design.
19.1 Functions From lambda
expression creates a nameless function. Its syntax is straightforward:
(lambda (variable-1 ... variable-N) expression)
Its distinguishing characteristic is the keyword
. The keyword is followed by a sequence of variables, enclosed in a pair of parentheses. The last piece is an arbitrary expression, and it computes the result of the function when it is given values
for its parameters.
Here are three simple examples, all of which consume one argument:
1. (lambda (x) (expt 10 x)), which assumes that the argument is a number and computes the exponent of 10 to the number;
2. (lambda (name) (string-append "hello, " name ", how are you?")), which consumes a string and creates a greeting with string-append; and
3. (lambda (ir) (<= (ir-price ir) threshold)), which is a function on an IR struct that extracts the price field from the structure and compares it with some threshold value.
The last example should recall the
example from above.
To use a function created from
expression, you can apply it to the correct number of arguments. Abstractly, you evaluate such expression according to the following law,
Stated like this, the law of function application is known as Church’s beta axiom.
((lambda (x-1 ... x-n) exp)
val-1 ... val-n)
= exp ; with free occurrences of x-1 replaced by val-1, etc.
That is, the application of a
expression proceeds just like that of an ordinary function. We replace the parameters of the function with the actual argument values and compute the value of the function body. Concretely, it works
as expected:
> ((lambda (x) (expt 10 x)) 2)
> ((lambda (name rst) (string-append name ", " rst)) "Robby" "etc.")
"Robby, etc."
> ((lambda (ir) (<= (ir-price ir) threshold)) (make-ir "bear" 10))
Note how the second sample function requires two arguments and that the last example assumes a definition for threshold in the definitions window such as this one:
The result of the last example is
because the price field of the inventory record contains
is less than
Now the important point is that these nameless functions can be used wherever a function is required, including with the abstractions from
figure 53
(list 10 100 1000)
"Robby, Matthew, etc."
> (filter (lambda (ir) (<= (ir-price ir) threshold))
(list (make-ir "bear" 10) (make-ir "doll" 33)))
(list (ir ...))
Once again, the last example assumes a definition for threshold.
229. Decide which of the following phrases are legal
Explain why they are legal or illegal. If in doubt, experiment in the interactions area.
Check your results in DrRacket.
231. Write down a
expression that
1. consumes a number and decides whether it is less than 10;
2. consumes two numbers, multiplies them, and turns the result into a string;
3. consumes an Posn p and a rectangular Image and adds a red 3-pixel dot to the image at p;
4. consumes an inventory record and compares them by price; and
5. consumes a natural number and produces 0 if it is even and 1 if it is odd.
Demonstrate how to use these functions in the interactions area.
19.2 Abstracting With lambda I
The two interactive examples for
suggest a simplification for our
Instead of introducing a named, local function, this second version uses
to create the auxiliary function and immediately hands it to
to extract all “good” inventory records from the list. Assuming
came with a test suite, you can immediately ensure that this revision passes the same tests as the original version.
Although it may take you a bit to get used to
notation, you will soon notice that
makes such short functions much more readable than
definitions. Indeed, you will find that you can adapt step 4 of the design recipe from
Designing With Abstractions
to use
instead of
. Consider the running example from that section. Its template based on
is this:
If you spell out the parameters so that their names include signatures, you can easily pack all the information from
into a single
From here, you should be able to complete the definition as well as from the original template:
• the purpose of the first function is to add
to each
coordinate on a given list of
expects a function of one argument, we use
(lambda (p) ...)
. The function then de-structures the given
, adds
to the
coordinate, and repackages the data into a
• the second one eliminates
s whose
coordinate is larger than
Here we know that
needs a function of on one argument that produces a
. First, the
function extracts the
coordinate from the
to which
applies the function. Second, it checks whether it is less than or equal to
, the desired limit.
• and the third one determines whether any
is close to some given point:
Like the preceding two examples,
is a function that expects a function of one argument and applies this functional argument to every item on the given list. If any result is
, too; if all results are
It is probably best to compare the definitions from
Using Abstractions, By Examples
and the definitions above side by side. When you do so, you should notice how easy the transition from
is and how concise the
version is in comparison to the
version. Thus, if you are ever in doubt, design with
first and then convert this tested version into one that uses
. Keep in mind, however, that
is not a cure-all. The locally defined function comes with a name that explains its purpose and, if it is long, the use of an abstraction with a named function is much easier to understand than one
with a large
The following exercises request that you solve the problems from Exercises And Projects with lambda in ISL+ .
Exercise 232. Use map to define the function convert-euro, which converts a list of U.S. dollar amounts into a list of euro amounts based on an exchange rate of 1.22 euro for each dollar.
Also use map to define convertFC, which converts a list of Fahrenheit measurements to a list of Celsius measurements.
Finally, try your hands at translate, a function that translates a list of Posns into a list of list of pairs of numbers, i.e., [List-of [list Number Number]].
Exercise 233. A toy structure specifies the name of a toy, a description, the acquisition price, and the recommended sales price.
Define a function that sorts a list of toy structures by the difference between the two prices.
Exercise 234. Use filter to define eliminate-exp. The function consumes a number, ua and a list of toy structures (containing name and price), and it produces a list of all those structures whose
acquisition price is below ua.
Then use filter to define recall, which consumes the name of a toy, called ty, and a list of toy structures and which produces a list of toy structures that do not use the name ty.
In addition, define selection, which consumes two lists of names and selects all those from the second one that are also on the first.
235. Use
to define functions that
1. creates the list 0 ... (n - 1) for any natural number n;
2. creates the list 1 ... nfor any natural number n;
3. creates the list (list 1 1/10 1/100 ...) of n numbers for any natural number n;
4. creates the list of the first n even numbers;
5. creates a list of lists of 0 and 1 in a diagonal arrangement, e.g.,
Exercise 236. Use ormap to define find-name. The function consumes a name and a list of names. It determines whether any of the names on the latter are equal to or an extension of the former.
With andmap you can define a function that checks all names on a list of names start with the letter "a".
Should you use ormap or andmap to define a function that ensures that no name on some list exceeds some given width?
237. Recall that the
function in ISL juxtaposes the items of two lists or, equivalently, replaces
at the end of the first list with the second list:
to define
. What happens if you replace
Now use one of the fold functions to define functions that compute the sum and the product, respectively, of a list of numbers.
With one of the fold functions, you can define a function that horizontally juxtaposes a list of Images. Hint: look up beside and empty-image. Can you use the other fold function? Also define a
function that stacks a list of images vertically. Hint: check for above in the libraries.
Exercise 238. The fold functions are so powerful that you can define almost any list-processing functions with them. Use fold to define map-from-fold, which simulates map.
19.3 lambda, Technically
One way to understand
is to view it as an abbreviation for a
expression. For example,
is short for
This “trick” works in general as long as some-random-name does not appear in the body of the function.
What this means is that lambda creates a function without a name, an anonymous function. This function is a value like any other value. The name of the function is irrelevant. The only interesting
parts are the function parameters—the sequence of names in parentheses—and the function body—the expression in the third position.
This insight is also the missing piece that connects constant definitions and function definitions. Instead of viewing function definitions as given, we can take lambdas as the primitive concept and
say that a function definition abbreviates a plain constant definition with a lambda expression on the right-hand side.
Here is a concrete example:
(define (f x) ; the plain version
(+ (sin x) x (* 3 x)))
This function definition is just short for
You actually do not need define to create recursive functions; lambda is enough. If you are curious, read up on Church’s Y combinator.
on the right-hand side creates a function of one argument. It is
that actually gives the
expression a name, and that is mostly relevant for recursive functions.
You may wish to test this explanation with experiments in DrRacket. Try the above example. Also add
to the definitions window after renaming the first f to f-plain and the second one to f-lambda. Run
a few times to make sure the two functions agree on all kinds of random inputs.
19.4 Abstracting With lambda II
Because functions are first-class values in ISL+, we may think of them as another form of data and use them for data representation. This section provides a taste of this idea; the next few chapters
do not rely on it. Its title uses “abstracting” because people consider functional data representation abstract.
As always, we start from a representative problem: This problem is easily solvable with a self-referential data representation that says a shape is either a circle, a rectangle, or a combination of
two shapes. We will study such representations in the next chapter.
Sample Problem: Navy strategists represents fleets of ships as rectangles (the ship itself) and circles (their reach). The coverage of a fleet of ships is the combination of all these shapes.
Design a data representation for rectangles, circles, and combinations of shapes. Then design a function that determines whether some point is within a shape.
The problem comes with all kinds of concrete interpretations, which we leave out here. A slightly more complex version was the subject of a programming competition in the mid 1990s.
One mathematical approach considers shapes as sets of points. In turn, a set is a predicate on points, that is, a function that maps a position to Boolean. One particular advantage is that this
representation allows the inclusion of infinitely large shapes though we don’t exploit this possibility here.
So here is the data definition:
; Shape is a function:
; [Posn -> Boolean]
; interpretation:
; if s is a shape and p a Posn,
; (s p) is true if the Posn is inside the shape
; and false if the Posn is outside
Its interpretation part is extensive because this data representation is so unusual. Although such an unusual representation calls for examples, we delay this step for a moment and instead define a
function that checks whether a point is inside some shape:
The definition is trivial. Since an element of
is a function that computes whether some
is inside the shape, the function merely applies the representation of the shape to the point.
Now let us return to the problem of elements of
. Here is a simplistic element of the class:
As required, it consumes a
Posn p
. Its body, however, merely compares
to the representation of the point (3
4), meaning this function represents a single point. While the data representation of a point as a
might seem silly, it suggests how we can represent shapes such as circles:
; Number Number Number -> Shape
; create a data representation for a circle of radius r
; located at (center-x, center-y)
(define (make-circle center-x center-y r)
; Posn -> Boolean
(lambda (p)
(<= (distance-between center-x center-y p) r)))
That is, we define a constructor-like function. Instead of a record, the function creates a function via a
, and this function computes whether the distance between some given
and the circle’s center is smaller than the radius
. For now we assume that you can design the function
, and since it is not critical to the point of this section, we ignore it.
The data representation of a rectangle is located in a similar manner:
; Number Number Number Number -> Shape
; create a data representation of a rectangle
(define (make-rectangle upperleft-x upperleft-y width height)
; Posn -> Boolean
(lambda (p)
(and (between upperleft-x (posn-x p) width)
(between (- upperleft-y height) (posn-y p) height))))
Its constructor receives four numbers: the position of the upper, left corner, its width, and its height. The result is again a
expression, i.e., a function. As for circles, this function consumes
s and produces a
, checking whether the
coordinates of the
are in the proper intervals. The only minor surprise concerns the calculation for the
coordinate; recall that the geometry of a computer canvas interprets increasing coordinate values as going downwards.
239. Design the function
. It consumes two numbers and a
, and
. The function computes the distance between the points (
) and
. Hint: the distance between (
0) and (
1) is
i.e., the distance of (x0 - y0,x1 - y1) to the origin.
Exercise 240. Design the function between. It consumes three numbers: left, x, and width. The function produces true if x is between left and (+ left width) inclusive.
Let us create some examples and test the examples with the inside? function:
(define circle1 (make-circle 3 4 5))
(define rectangle1 (make-rectangle 0 3 10 3))
The creation of a circle calls for three numbers, that of a rectangle needs four. If you were to run this program now, you could observe the following interactions in the interactions area:
> circle1
> rectangle1
Just as a reminder, this means that both circle1 and rectangle1 are functions. And this interaction also explains why people call a functional data representation “abstract.” Unlike a structural
representation, it is impossible for you to see the pieces of a function.
Here are some more interesting interactions:
These two interactions show that the origin of the coordinate system is inside both rectangle1 and circle1.
Exercise 241. Draw the shapes represented by circle1 and rectangle1 on grid paper. Check that the origin belongs to both.
The interactions suggest that we can formulate test cases now. Specifically, we call inside? on shapes-as-functions and some points, like this:
Note how (-1,3) is inside our sample circle but outside of the rectangle.
Exercise 242. Confirm these tests with the drawing from exercise 241.
At this points, we have only one task left, namely, the design of function that maps two
representations to their combination. The signature and the header are easy:
Indeed, even the default value is straightforward. We know that a shape is represented as a function from
, so we write down a
that consumes some
and produces
, meaning it says no point is in the combination.
So suppose we wish to combine the circle and the rectangle from above:
(define combination1 (make-combination circle1 rectangle1))
While we cannot inspect this value and compare it to some expected result for equality, we know that some points are inside and others are outside of this combination:
(make-posn 0 0)
is inside both, there is no question that is inside the combination of the two. In a similar vein,
(make-posn 0 -1)
is in neither shape, and so it isn’t in the combination. Finally,
(make-posn -1 3)
is in
but not in
. But the point must be in the combination of the two shapes because every point that is in one or the other shape is in their combination.
This analysis of examples implies the obvious revision for the function body of make-combination:
expression says that the result is
if one of two expression produces
(inside? s1 p)
(inside? s2 p)
. The first expression determines whether
is in
and the second one whether
is in
. And that is precisely a translation into ISL+ of our explanation above.
; Shape = [Posn -> Boolean]
; Shape Posn -> Boolean
(define (shape-inside? s p)
(s p))
; Number Number Number -> Shape
(define (make-circle center-x center-y r)
(lambda (p)
(local (; Number Number Posn -> Boolean
(define (distance-between x y p)
(+ (sqr (- x (posn-x p)))
(sqr (- y (posn-y p)))))))
; — IN —
(<= (distance-between center-x center-y p) r))))
; Number Number Number Number -> Shape
(define (make-rectangle northwest-x northwest-y width height)
(lambda (p)
(local (; Number Number Number -> Boolean
; is z between left and (+ left interval)
(define (between left z interval)
(<= left z (+ left interval))))
; — IN —
(and (between northwest-x (posn-x p) width)
(between (- northwest-y height) (posn-y p) height)))))
; Shape Shape -> Shape
(define (make-combination s1 s2)
(lambda (x)
(or (shape-inside? s1 x) (shape-inside? s2 x))))
Figure 58: A functional representation of geometric shapes, examples
Figure 57 collects all the program fragments into one complete program; figure 58 supplements it with a test suite. Copy and paste all the code into DrRacket and experiment. Most importantly, confirm
that all tests pass.
Exercise 243. Design a data representation for finite and infinite sets so that you can represent the sets of all odd numbers, all even numbers, all numbers divisible by 10, etc. Hint:
Mathematicians sometimes interpret sets as functions that consume a potential element e and produce true if the e belongs to the set and false if it doesn’t.
Design the functions
1. add-element, which adds an element to a set;
2. union, which combines the elements of two sets; and
3. intersect, which collects all elements common to two sets;
Keep in mind the analogy between sets and shapes.
20 Summary
This third part of the book is about the role of abstraction in program design. Abstraction has two sides: creation and use. It is therefore natural if we summarize the chapter as two lessons:
1. Repeated code patterns call for abstraction. To abstract means to factor out the repeated pieces of code—the abstraction—and to parameterize over the differences. With the design of proper
abstractions, programmers save themselves future work and headaches because mistakes, inefficiencies, and other problems are all in one place. One fix to the abstraction can thus improve all
uses. In contrast, the duplication of code means that a programmer must find all copies and fix all of them when a problem is found.
2. Most languages come with a large collection of abstractions. Some are contributions by the language design and implementation team; others are added by programmers who used the language. To
enable effective reuse of these abstractions, their creators must supply the appropriate pieces of documentation: a purpose statement, a signature, and good examples. These pieces should suffice
to help programmers figure out how to use the abstraction.
All programming languages come with the means to build abstractions though some means are better than others. All programmers must get to know the means of abstractions and the abstractions that a
language provides. A discerning programmer will learn to distinguish programming languages along these axes.
In addition on creating and using abstractions, this third part of the book introduces another idea, namely,
the idea of functions as data and their role in representing information as data.
While the idea is old for programming language specialists, dating back to the 1960s, it has only recently gained wide acceptance with additions to most modern mainstream languages—C#, C++, Java,
JavaScript, Perl, Python. The last chapters of this book will therefore return to the theme of using functions in data representations. In the meantime, the next few chapters will first deal with
some more ordinary topics concerning the design of programs. | {"url":"http://www.ccs.neu.edu/home/matthias/HtDP2e/Draft/part_three.html","timestamp":"2014-04-20T18:53:19Z","content_type":null,"content_length":"854729","record_id":"<urn:uuid:71938205-3afa-4bd3-93d4-a4645c918c7d>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00265-ip-10-147-4-33.ec2.internal.warc.gz"} |
ith Algebra
Fall 2010
Calculus with Algebra
Listed in: Mathematics and Statistics, as MATH-05
John D. Condon (Section 01)
Mathematics 05 and 06 are designed for students whose background and algebraic skills are inadequate for the fast pace of Mathematics 11. In addition to covering the usual material of beginning
calculus, these courses will have an extensive review of algebra and trigonometry. There will be a special emphasis on solving word problems.
Mathematics 05 starts with a quick review of algebraic manipulations, inequalities, absolute values and straight lines. Then the basic ideas of calculus--limits, derivatives, and integrals--are
introduced, but only in the context of polynomial and rational functions. As various applications are studied, the algebraic techniques involved will be reviewed in more detail. When covering related
rates and maximum-minimum problems, time will be spent learning how to approach, analyze and solve word problems. Four class hours per week.
Note: While Mathematics 05 and 06 are sufficient for any course with a Mathematics 11 requisite, Mathematics 05 alone is not. However, students who plan to take Mathematics 12 should consider taking
Mathematics 05 and then Mathematics 11, rather than Mathematics 06. Students cannot register for both Mathematics 05 and Chemistry 11 in the same semester.
Fall semester. Visiting Professor Condon.
Science & Math for non-majors
Offered in
Fall 2013Other years:
Offered in
Fall 2007
Fall 2008
Fall 2009
Fall 2010
Fall 2011
Fall 2012
Fall 2014 | {"url":"https://www.amherst.edu/academiclife/departments/courses/1011F/MATH/MATH-05-1011F","timestamp":"2014-04-19T07:20:36Z","content_type":null,"content_length":"33725","record_id":"<urn:uuid:062d2248-5c73-49df-be87-36fd8e554cf8>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00474-ip-10-147-4-33.ec2.internal.warc.gz"} |
Number of items: 3.
Ibort, A. and Martínez Montalba, Celia (1996) Arnold’s conjecture and symplectic reduction. Journal of geometry and physics, 18 (1). pp. 25-37. ISSN 0393-0440
Abellanas, L. and Martínez Montalba, Celia (1993) Lie symmetries versus integrability in evolution equations. Journal of physics A: Mathematical and general, 26 (23). pp. 1229-1232. ISSN 0305-4470
Martínez Montalba, Celia and Muñoz Masqué, Jaime and Valdés Morales, Antonio (2003) On the structure of the moduli of jets of G-structures with a linear connection. Differential Geometry and Its
Applications, 18 (3). pp. 271-283. ISSN 0926-2245 | {"url":"http://eprints.ucm.es/view/people/Mart=EDnez_Montalba=3ACelia=3A=3A.html","timestamp":"2014-04-18T00:51:12Z","content_type":null,"content_length":"10554","record_id":"<urn:uuid:95f001ec-d614-4f29-8740-9521c408a4b2>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00098-ip-10-147-4-33.ec2.internal.warc.gz"} |
Explicit Casting?
July 14th, 2013, 04:54 PM #1
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Reading my text book and it says that implicit casting will work if the result of a casting fits into the target variable. If it doesn't I'm told to use explicit tasking.
EXAMPLE (IMPLICIT)
byte b = 'a';
int i = 'a';
EXAMPLE (EXPLICIT)
byte b = (byte)'\uFFF4';
Data types have a range. Therefore, if a value is not within this range how on earth can it be assigned explicitly at all? I just do not understand this completely.
Range and size can be (usually are?) related. The type byte allows values from -128 to 127. 'a' = 97 and 'z' = 122 so both will fit in a byte variable using implicit casting. Try to implicitly
cast the char = 128 into a type byte, and you'll get an error. Explicitly cast char = 128 int a byte, and what do you get?
It tells me there is a loss of precision. So is this the same as rounding???
Screen Shot 2013-07-14 at 4.39.19 PM copy.jpg
Last edited by syregnar86; July 14th, 2013 at 05:42 PM. Reason: info
It's not quite the same as rounding, though it is somewhat related because you just can't represent some values in a "smaller" type.
The fact is modern computers represent numbers in binary. A "smaller" type simply has less bits. According to the 5.1.3 of the Java Language Specifications, down-casting for integral types (byte,
char, short, int, long) takes the n least significant bits, where n is the number of bits in the target value.
// this is hexedecimal so I don't have to write 32-bit values in binary :P
int(0x12345678) = byte(0x78)
int(0x12345678) = short(0x5678)
In addition to this, Java uses two's compliment to represent signed types.
So even though int(0x7FFF) = byte(0xFF), the integral value of the int is 32767, and the integral value of the byte is -1!
In Java you must make an explicit case from int to byte because of this. The reasoning is that this behavior is usually not expected (in what world does 32767 even come close to equal -1?), but
it is very important to be able to make this case because the int value may not necessarily be out of range (int(1) = byte(1)), and there are many useful things you can do using bit-fields and
bit-twiddling (mucking with the bits), though I have qualms with using Java with bit-fields and bit-twiddling.
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Blog Entries | {"url":"http://www.javaprogrammingforums.com/whats-wrong-my-code/30454-explicit-casting.html","timestamp":"2014-04-16T05:18:20Z","content_type":null,"content_length":"60973","record_id":"<urn:uuid:aeaea200-dd44-4d5f-810a-8277e353fea0>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00366-ip-10-147-4-33.ec2.internal.warc.gz"} |
Positive definite
March 22nd 2010, 05:51 AM #1
Sep 2007
Positive definite
Hi guys, can you please help me out??
A and B are nonsingular matrices and positive definite. Show that:
If $A-B>0$ , then $B^{-1}-A^{-1}>0$
What I have tried
I tried showing $B^{-1}-A^{-1}$ is a quadratic form but firstly I'm guessing that doesn't even show it's positive definite, only positive semi-definite. And secondly, I couldn't get it to work
Please help!!
The proof that I know relies on two facts. First, if $A\geqslant0$ and T is a matrix with Hermitian adjoint T*, then $T^*AT\geqslant0$. Second, if $T^*T\leqslant I$ then $TT^*\leqslant I$. I'll
also need the fact that a positive matrix $A$ has a positive square root $A^{1/2}$.
Given positive definite (therefore invertible) matrices A and B with A – B > 0, it follows that $0\leqslant A^{-1/2}(A-B)A^{-1/2} = I - A^{-1/2}BA^{-1/2}$. Therefore $(B^{1/2}A^{-1/2})^*(B^{1/2}A
^{-1/2})\leqslant I$. Hence $B^{1/2}A^{-1}B^{-1}B^{1/2} = (B^{1/2}A^{-1/2})(B^{1/2}A^{-1/2})^*\leqslant I$, from which $A^{-1}\leqslant B^{-1}$.
March 22nd 2010, 12:50 PM #2 | {"url":"http://mathhelpforum.com/advanced-algebra/135023-positive-definite.html","timestamp":"2014-04-19T20:16:59Z","content_type":null,"content_length":"36706","record_id":"<urn:uuid:37386993-b5a3-4509-b9f0-b7ffd4f6870f>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00359-ip-10-147-4-33.ec2.internal.warc.gz"} |
Cardiff By The Sea Calculus Tutor
Find a Cardiff By The Sea Calculus Tutor
...I was able to help them pass their classes and exams and ultimately go to various Universities. I learned the different learning styles of each of the students, and did my best to work with
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42 Subjects: including calculus, reading, English, Spanish
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11 Subjects: including calculus, geometry, public speaking, GMAT
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and I would be comfortable well into college math. During the learning process, small knowledge gaps from past courses tend to reappear as roadblocks down the line.
14 Subjects: including calculus, physics, geometry, statistics
...Luckily, they are my specialty. I am an expert at seeing these kinds of problems from many different angles, and my students have benefited greatly. I graduated with a Bachelor of Science in
theoretical mathematics, which uses logic as its backbone.
26 Subjects: including calculus, Spanish, chemistry, physics
...I recently just retook calculus 1-3 and received an A as well as reviewed this subject for the GRE. I have run study groups and have tutored other math subjects. My major is currently in math,
and I'm working to become a math teacher My strength is breaking down what seems like big concepts and relating it to stuff students have seen before.
13 Subjects: including calculus, chemistry, geometry, statistics | {"url":"http://www.purplemath.com/cardiff_by_the_sea_ca_calculus_tutors.php","timestamp":"2014-04-18T06:05:25Z","content_type":null,"content_length":"24323","record_id":"<urn:uuid:75025f85-4c11-4064-828a-45e8448e84f6>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00459-ip-10-147-4-33.ec2.internal.warc.gz"} |
Static Tension Question
1. A 120 ft tower is being held by three cables in tension. Cable AO is in 1000
lb of tension. Find the tension in the other cables and the vertical force from
the tower if the whole system is at rest.
3. So using lots and lots of trig I was able to find most of the magnitudes, thetas, and phi's as you can see from the last 2 images. But now I'm lost from there. During office hrs. my prof.
mentioned something about using unit vectors and then labeling the pts. A, B & C as coordinates A(-30,0,-20), b(-20,0,20), & c(40,0,10), but I don't see how those can help me. Any suggestions? | {"url":"http://www.physicsforums.com/showthread.php?t=336253","timestamp":"2014-04-19T19:45:58Z","content_type":null,"content_length":"23011","record_id":"<urn:uuid:be2b6803-fd1a-4d8e-a599-46b6ef70e791>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00302-ip-10-147-4-33.ec2.internal.warc.gz"} |
Daily Peloton - Pro Cycling News
Welcome to live coverage of Paris-Nice!
Today, the riders face the 166.5km drag from Chaville, just outside of Paris, to Montargis. The ride will be south over five rolling Cat 3 hills in the first 69km. Then for the rest of the day, it's
still south over roads flatter than a Texas road armadillo. There will be two intermediate sprints along the way, and in the end, this should be a day for the sprinters to rub elbows a bit at the
CSC will be riding in defense of the Yellow and White Jersey of Jorg Jaksche, who smoked everyone yesterday to show his outstanding early-season form on a day when most riders were a bit cautious
because of the slippery road.
So as we begin coverage, there are 36 men on the front about 20km from the finish. The winds have split the peloton to hell, with CSC burning up the roads to put the hurt on their rivals.
Most of the leaders are in this main group. Americans Levi Leipheimer (Rabobank) and Tyler Hamilton (CSC) are not in this lead group, however, and are in a big chasing bunch about 2' 30" back.
The "Big Rig," George Hincapie (US Postal -Berry Floor) has made this lead group. His early season form is outstanding.
17km to go. Bjarne Riis and his boys from CSC are pushing hard on the front, and look to have made a conscious effort to blow the race up. Ekimov and Pena from Postal have also made the lead group.
Rebellin (Gerolsteiner) made the split. Vinokourov (T-Mobile) did not. Bobby Julich is driving the CSC train. They have seven men in this lead group. Frank Vandenbroucke is in this group in the lead.
Jose Azevedo (US Postal) and Thor Hushovd (Credit Agricole) are in the lead group.
Wow, Baden Cooke (FDJeux.com) is leading a bunch of men chasing who look to have their tails between their legs, riding like a bunch of whipped dogs. Poor Erik Dekker (Rabobank), who rode such a
great time trial yesterday, looks to be in this chasing group.
11km left to go. CSC as all seven men on the front of the lead group still, driving it, trying to put their rivals who missed the break away once and for all. The closest chasers are over 3' back
Chocolade Jacques Wincor has a few men in this lead group as well. Alex Zuelle (Phonak) has made the split. Tyler Hamilton is way off the back, riding easily at the front of a small group. Tyler is
more worried about getting over his flu and getting some racing legs under him right now.
This split is really, really blowing the race to pieces as far as the GC is concerned. There are no big sprinters that I can see in this lead group... that means that men like Hincapie will have a
good shot at taking the stage. CSC is keeping the pace high, and this will likely discourage attacks until the final couple of kilometers.
6km left. Here goes an attack by a Rabobank rider. It's Mathew Hayman, the Australian!
He has a good gap, but the CSC train is working well and will likely chase him back.
The Posties have several men near the front, right behind CSC. They have now caught Hayman... nice try, though.
Here comes an attack by Fassa Bortolo... He's pursued by a few men. Fabian Cancellara is the Fassa rider, and the front of the race has broken up a bit in the pursuit.
Jaksche himself has covered this small break of five men. They are in the tight corners in the town. The break is over, and men are looking over their shoulders to see who will attack next.
2km left. Here comes the 36 leaders, flying along towards the finish.
There is now no organization at the front of the pack. Riders from several different teams are mixed in, but the the pace is still high.
Another attack! It's Horrillo of Quick Step!
He's being chased, but he has it! Horrillo wins it!
The big man, formerly of Mapei, jumped at just the right time, and nobody could match him. Hushovd pulled his foot out of the pedal, and crossed the finish line with only one foot clipped in.
Hincapie and a few others were mixed in trying to chase Horrillo, but he was far enough clear to look over his shoulder, celebrate a bit, and enjoy the victory.
Beat Zberg (Gerolsteiner) was 2nd, Michele Bartoli (CSC) was 3rd.
I didn't see Postal's top GC man, Floyd Landis, in the lead group. Several of his teammates were there and attentive. The GC will be interesting to see once everyone crosses the line.
The main chasing pack crosses over 5' behind the leaders. They come through at 5' 15", a major time loss.
The GC is still 1. Jaksche (CSC), 2. Rebellin (Gerolsteiner) @ 6", 3. Julich (CSC) @ 18".
"Gorgeous" Jorg Jaksche now only seems to really have Rebellin to worry about.
Hincapie finished in 9th on the stage. He should be well up the GC by the end of the day...
More General Classification:
4. F. Vandenbroucke (Fassa Bortolo) @ 22", 5. Jens Voigt (CSC) @ 22".
Jaksche now picks up his second Yellow and White Jersey. He's really flourishing now that he is out of the shadow of Manolo Saiz and Joseba Beloki.
Rebellin picks up the Green Jersey as leader of the points competition. He came across in 6th today to take the lead in this competition.
Stage 2 Results
1 HORRILLO Pedro QSD 3h 47' 55"
2 ZBERG Beat GST s.t.
3 BARTOLI Michele CSC s.t.
4 GILBERT Philippe FDJ s.t.
5 VIERHOUTEN Aart LOT s.t.
6 REBELLIN Davide GST s.t.
7 LÖWIK Gerben CHO s.t.
8 HUSHOVD Thor C.A s.t.
9 HINCAPIE George USP s.t.
10 VOIGT Jens CSC s.t.
11 HIEKMANN Torsten MOB s.t.
12 VANDENBROUCKE Frank FAS s.t.
13 PEREIRO SIO Oscar PHO s.t.
14 VERHEYEN Geert CHO s.t.
15 JAKSCHE Jorg CSC s.t.
16 HULSMANS Kevin QSD s.t.
17 LE MEVEL Christophe C.A s.t.
18 JOERGENSEN René ALB s.t.
19 ROGERS Michael QSD s.t.
20 LOTZ Marc RAB s.t.
21 SCHLECK Frank CSC s.t.
22 TANKINK Bram QSD s.t.
23 VAN DE WALLE Jurgen CHO s.t.
24 JULICH Bobby CSC s.t.
25 CANCELLARA Fabian FAS s.t.
26 VALJAVEC Tadej PHO s.t.
27 CHAVANEL Sylvain BLB s.t.
28 HAYMAN Mathew RAB s.t.
29 LEQUATRE Geoffroy C.A s.t.
30 AZEVEDO José USP s.t.
31 PENA Victor Hugo USP s.t.
32 NOVAL GONZALEZ Benjamin USP s.t.
33 BASSO Ivan CSC s.t.
34 EKIMOV Vjatceslav USP s.t.
35 BLAUDZUN Michaël CSC s.t.
36 PIIL Jakob CSC 00' 17"
37 MIHOLJEVIC Vladimir ALB 05' 15"
38 BROCHARD Laurent A2R s.t.
39 LJUNGQUIST Marcus ALB s.t.
40 KRIVTSOV Yuriy A2R s.t.
41 DUMOULIN Samuel A2R s.t.
42 GOUBERT Stephane A2R s.t.
43 MOOS Alexandre PHO s.t.
44 GONZALEZ GALDEANO Alvaro LST s.t.
45 BUFFAZ Mickaël RAG s.t.
46 BENETEAU Walter BLB s.t.
47 PEERS Chris CHO s.t.
48 LEIPHEIMER Levi RAB s.t.
49 DETILLOUX Christophe LOT s.t.
50 CIONI Dario FAS s.t.
51 EISEL Bernhard FDJ s.t.
52 CAUCCHIOLI Pietro ALB s.t.
53 VASSEUR Cédric COF s.t.
54 OSA Aitor IBB s.t.
55 CONTADOR Alberto LST s.t.
56 REYNES Vicente IBB s.t.
57 LASTRAS Pablo IBB s.t.
58 COOKE Baden FDJ s.t.
59 DAVIS Allan LST s.t.
60 HALGAND Patrice C.A s.t.
61 POILVET Benoit C.A s.t.
62 POLLACK Olaf GST s.t.
63 GALPARSORO Dionisio EUS s.t.
64 TOTSCHNIG Georg GST s.t.
65 ZIEGLER Thomas GST s.t.
66 LARSSON Gustav FAS s.t.
67 AUGER Guillaume RAG s.t.
68 NIERMANN Grischa RAB s.t.
69 GONZALEZ Gorka EUS s.t.
70 ETXEBARRIA Unai EUS s.t.
71 WERNER Christian MOB s.t.
72 SINKEWITZ Patrik QSD s.t.
73 SERPELLINI Marco GST s.t.
74 SANCHEZ Samuel EUS s.t.
75 RENIER Franck BLB s.t.
76 SUNDERLAND Scott ALB s.t.
77 MOLLER Claus Michael ALB s.t.
78 PIATEK Zbigniew CHO s.t.
79 SALMON Benoit C.A s.t.
80 SANCHEZ Luis LST s.t.
81 DE CLERCQ Hans LOT s.t.
82 EDALEINE Christophe COF s.t.
83 BESSY Frédéric COF s.t.
84 VAN BON Leon LOT s.t.
85 VIRENQUE Richard QSD s.t.
86 LÖVKVIST Thomas FDJ s.t.
87 ARRIZABALAGA Gorka EUS s.t.
88 RAMIREZ Javier LST s.t.
89 WEGMANN Fabian GST s.t.
90 SERRANO Marcos LST s.t.
91 BARANOWSKI Dariusz LST s.t.
92 KESSLER Matthias MOB s.t.
93 DESSEL Cyril PHO s.t.
94 BRARD Florent CHO s.t.
95 BOONEN Tom QSD s.t.
96 CRETSKENS Wilfried QSD s.t.
97 VERBRUGGHE Rik LOT s.t.
98 NOZAL Isidro LST s.t.
99 FRIGO Dario FAS s.t.
100 AERTS Mario MOB s.t.
101 PELLIZOTTI Franco ALB s.t.
102 ONGARATO Alberto FAS s.t.
103 FOFONOV Dmitriy COF s.t.
104 MUTSCHLER Klaus RAG s.t.
105 GUSTOV Volodymir FAS s.t.
106 DA CRUZ Carlos FDJ s.t.
107 CASAR Sandy FDJ s.t.
108 VINOKOUROV Alexandre MOB s.t.
109 KARPETS Vladimir IBB s.t.
110 ROUS Didier BLB s.t.
111 SCHRECK Stephan MOB s.t.
112 RINERO Christophe RAG s.t.
113 NAZON Jean-Patrick A2R s.t.
114 LANDALUZE Inigo EUS s.t.
115 ARTETXE Mikel EUS s.t.
116 ZUELLE Alex PHO s.t.
117 MONCOUTIE David COF s.t.
118 DE GROOT Bram RAB s.t.
119 JALABERT Nicolas PHO s.t.
120 PORTAL Nicolas A2R s.t.
121 MILLAR David COF s.t.
122 MENCHOV Denis IBB s.t.
123 BRUYLANDTS Dave CHO s.t.
124 CUESTA Inigo COF s.t.
125 KIRCHEN Kim FAS s.t.
126 BOUYER Franck BLB s.t.
127 KASHECHKIN Andrey C.A s.t.
128 WHITE Matthew COF s.t.
129 BOTERO Santiago MOB s.t.
130 DE WEERT Kevin RAB s.t.
131 BELOHVOSCIKS Raivis CHO s.t.
132 SEIGNEUR Eddy RAG s.t.
133 ZABRISKIE David USP s.t.
134 HUNTER Robert RAB s.t.
135 DEKKER Erik RAB s.t.
136 JOACHIM Benoit USP s.t.
137 RADOCHLA Steffen IBB s.t.
138 GUESDON Frédéric FDJ s.t.
139 THIBOUT Bruno RAG s.t.
140 LE BOULANGER Yoann RAG s.t.
141 LANDISFloyd USP s.t.
142 MERCKX Axel LOT s.t.
143 BERTHOU Eric RAG s.t.
144 PINEAU Jérôme BLB s.t.
145 MC EWEN Robbie LOT s.t.
146 SCHOLZ Ronny GST 13' 55"
147 SCHNIDER Daniel PHO s.t.
148 HAMILTON Tyler PHO s.t.
149 YAKOVLEV Serguei MOB s.t.
150 ZANDIO Xabier IBB s.t.
151 BECKE Daniel IBB s.t.
152 SCANLON Mark A2R s.t.
153 RASTELLI Ellis ALB s.t.
154 IRIZAR Markel EUS s.t.
155 ASTARLOZA Mikel A2R s.t.
156 GESLIN Anthony BLB s.t.
157 WIGGINS Bradley C.A s.t.
158 WILSON Matthew FDJ s.t.
159 MOERENHOUT Koos LOT s.t.
160 VOECKLER Thomas BLB s.t.
General Classification after Stage 2
1 JAKSCHE Jorg CSC 4h 05' 12"
2 REBELLIN Davide GST 00' 06"
3 JULICH Bobby CSC 00' 18"
4 VANDENBROUCKE Frank FAS 00' 22"
5 VOIGT Jens CSC 00' 22"
6 PEREIRO SIO Oscar PHO 00' 25"
7 HORRILLO Pedro QSD 00' 31"
8 ZBERG Beat GST 00' 32"
9 ROGERS Michael QSD 00' 32"
10 CANCELLARAF abian FAS 00' 34"
11 HINCAPIE George USP 00' 37"
12 HIEKMANN Torsten MOB 00' 41"
13 NOVAL GONZALEZ Benjamin USP 00' 45"
14 HUSHOVD Thor C.A 00' 47"
15 TANKINK Bram QSD 00' 50"
16 BARTOLI Michele CSC 00' 50"
17 BASSO Ivan CSC 00' 51"
18 AZEVEDO José USP 00' 54"
19 VAN DE WALLE Jurgen CHO 00' 55"
20 VALJAVEC Tadej PHO 00' 59"
21 LOTZ Marc RAB 01' 00"
22 LÖWIK Gerben CHO 01' 00"
23 EKIMOV Vjatceslav USP 01' 01"
24 SCHLECK Frank CSC 01' 02"
25 GILBERT Philippe FDJ 01' 03"
26 HULSMANS Kevin QSD 01' 04"
27 CHAVANEL Sylvain BLB 01' 05"
28 LEQUATRE Geoffroy C.A 01' 09"
29 PENA Victor Hugo USP 01' 10"
30 BLAUDZUN Michaël CSC 01' 13"
31 LE MEVEL Christophe C.A 01' 23"
32 VIERHOUTEN Aart LOT 01' 24"
33 PIILJ akob CSC 01' 39"
34 JOERGENSEN René ALB 01' 47"
35 VERHEYEN Geert CHO 01' 49"
36 HAYMAN Mathew RAB 01' 52"
37 DEKKER Erik RAB 05' 19"
38 MILLAR David COF 05' 30"
39 CONTADOR Alberto LST 05' 30"
40 ZUELLE Alex PHO 05' 31"
41 NIERMANN Grischa RAB 05' 39"
42 BROCHARD Laurent A2R 05' 39"
43 GONZALEZ Gorka EUS 05' 41"
44 KARPETS Vladimir IBB 05' 42"
45 ETXEBARRIA Unai EUS 05' 42"
46 VINOKOUROV Alexandre MOB 05' 43"
47 DE GROOT Bram RAB 05' 45"
48 KIRCHEN Kim FAS 05' 46"
49 LANDIS Floyd USP 05' 46"
50 LARSSON Gustav FAS 05' 48"
Points Classification after Stage 2
1 REBELLIN Davide GST 37pts
2 JAKSCHE Jorg CSC 33pts
3 HORRILLO Pedro QSD 25pts
4 VOIGT Jens CSC 25pts
5 VANDENBROUCKE Frank FAS 22pts
6 ZBERG Beat GST 22pts
7 DEKKER Erik RAB 22pts
8 BARTOLI Michele CSC 20pts
9 GILBERT Philippe FDJ 18pts
10 MILLAR David COF 18pts
Best Climber Classification after Stage 2
1 BELOHVOSCIKS Raivis CHO 16pts
2 ZIEGLER Thomas GST 11pts
3 PIIL Jakob CSC 4pts
4 DA CRUZ Carlos FDJ 2pts
5 LÖWIK Gerben CHO 1pts
6 BLAUDZUN Michaël CSC 1pts
Best Young Rider after Stage 2
1 ROGERS Michael AUS 4h 05' 44"
2 CANCELLARA Fabian SUI 00' 02"
3 HIEKMANN Torsten GER 00' 09"
4 NOVAL GONZALEZ Benjamin ESP 00' 13"
5 SCHLECK Frank LUX 00' 30"
6 GILBERT Philippe BEL 00' 31"
7 CHAVANEL Sylvain FRA 00' 33"
8 LEQUATRE Geoffroy FRA 00' 37"
9 LE MEVEL Christophe FRA 00' 51"
10 CONTADOR Alberto ESP 04' 58"
Teams Classification after Stage 2
1 Team CSC 12h 16' 20"
2 Quick Step - Davitamon 01' 09"
3 US Postal - Berry Floor 01' 09"
4 Credit Agricole 01' 38"
5 Chocolade Jacques Wincor 01' 52"
6 Rabobank 05' 31"
7 Phonak Hearing Systems 05' 51"
8 Fassa Bortolo 06' 00"
9 Gerolsteiner 06' 05"
10 T-Mobile Team 11' 48"
11 Fdjeux.com 11' 57"
12 Brioches La Boulangere 12' 20"
13 Alessio - Bianchi 12' 41"
14 Lotto - Domo 13' 56"
15 Euskaltel - Euskadi 16' 40"
16 Cofidis Credit Par Telephone 16' 42"
17 Illes Balears - B. Santander 16' 47"
18 AG2R Prevoyance 16' 58"
19 Liberty Seguros 17' 03"
20 R.a.G.T. Semences - MG Rover 17' 50" | {"url":"http://www.dailypeloton.com/displayarticle.asp?pk=5781","timestamp":"2014-04-20T00:59:10Z","content_type":null,"content_length":"24151","record_id":"<urn:uuid:a88767a4-584d-4b8b-b67c-0088e3c637e2>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00076-ip-10-147-4-33.ec2.internal.warc.gz"} |
Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 079.06304
Autor: Erdös, Paul; Shapiro, Harold N.
Title: On the least primitive root of a prime. (In English)
Source: Pac. J. Math. 7, 861-865 (1957).
Review: Let g(p) be the least positive primitive root of a prime p. The authors prove that g(p) = O(m^c p^ ½) where c is a constant and m is the number of distinct prime factors of p-1. As m large,
it is an improvement of a result of the reviewer: g(p) \leq 2^m+1 p^ ½. The authors introduce a lemma and then apply Brun's method to obtain the result. The lemma runs as following: Let S and T be
two sets with distinct integers, mod p. Then for any non-principal character \chi, we have
|sum[u in S, v in T] \chi (u+v)|^2 \leq p sum[u in S] 1 sum[v in T] 1.
Reviewer: L.K.Hua
Classif.: * 11N69 Distribution of integers in special residue classes
11A07 Congruences, etc.
Index Words: Number Theory
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag
│Books │Problems │Set Theory │Combinatorics │Extremal Probl/Ramsey Th. │
│Graph Theory │Add.Number Theory│Mult.Number Theory│Analysis │Geometry │
│Probabability│Personalia │About Paul Erdös │Publication Year│Home Page │ | {"url":"http://www.emis.de/classics/Erdos/cit/07906304.htm","timestamp":"2014-04-16T10:12:16Z","content_type":null,"content_length":"3831","record_id":"<urn:uuid:878868de-237e-45c0-9087-278ea909f223>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00371-ip-10-147-4-33.ec2.internal.warc.gz"} |
How do you write an equation in standard form when given two points which the line passes through?
A number is a mathematical object used to count, label, and measure. In mathematics, the definition of number has been extended over the years to include such numbers as 0, negative numbers, rational
numbers, irrational numbers, and complex numbers.
Mathematical operations are certain procedures that take one or more numbers as input and produce a number as output. Unary operations take a single input number and produce a single output number.
For example, the successor operation adds 1 to an integer, thus the successor of 4 is 5. Binary operations take two input numbers and produce a single output number. Examples of binary operations
include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.
A repeating or recurring decimal is a way of representing rational numbers in arithmetic. The decimal representation of a number is said to be repeating if it becomes periodic (repeating its values
at regular intervals) and the infinitely-repeated digit is not zero. The decimal representation of ⅓ becomes periodic just after the decimal point, repeating the single-digit sequence "3" forever. A
more complicated example is 3227/[555], whose decimal becomes periodic after the second digit following the decimal point and then repeats the sequence "144" forever. At present, there is no
universally-accepted notation or phrasing for repeating decimals.
If the repeated digit is a zero, the rational number is called a terminating decimal, since the number is said to "terminate" before these zeros. Instead of taking any note of the repeated zeroes,
they are simply omitted. All terminating decimals can be written as a decimal fraction whose divisor is a power of 10 (1.585 = 1585/[1000]); they may also be written as a ratio of the form k/[2n5m] (
1.585 = 317/[2352]). However, every terminating decimal also has a second representation as a repeating decimal. This is obtained by decreasing the final non-zero digit by one and appending an
infinitely-repeating sequence of nines, a non-obvious phenomenon that many find puzzling. 1 = 0.999… and 1.585 = 1.584999… are two examples of this. (This type of repeating decimal can be obtained by
long division if one uses a modified form of the usual division algorithm.)
In algebra, which is a broad division of mathematics, abstract algebra is a common name for the sub-area that studies algebraic structures in their own right. Such structures include groups,
rings, fields, modules, vector spaces, and algebras. The specific term abstract algebra was coined at the beginning of the 20th century to distinguish this area from the other parts of algebra.
The term modern algebra has also been used to denote abstract algebra.
Two mathematical subject areas that study the properties of algebraic structures viewed as a whole are universal algebra and category theory. Algebraic structures, together with the associated
homomorphisms, form categories. Category theory is a powerful formalism for studying and comparing different algebraic structures.
Elementary arithmetic is the simplified portion of arithmetic which includes the operations of addition, subtraction, multiplication, and division.
Elementary arithmetic starts with the natural numbers and the written symbols (digits) which represent them. The process for combining a pair of these numbers with the four basic operations
traditionally relies on memorized results for small values of numbers, including the contents of a multiplication table to assist with multiplication and division.
Related Websites: | {"url":"http://answerparty.com/question/answer/how-do-you-write-an-equation-in-standard-form-when-given-two-points-which-the-line-passes-through","timestamp":"2014-04-19T01:48:22Z","content_type":null,"content_length":"29513","record_id":"<urn:uuid:1d734760-8677-451f-913e-b4024dc4fb74>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00282-ip-10-147-4-33.ec2.internal.warc.gz"} |
Principal Component Analysis (PCA)
Principal Component Analysis (.pdf)
Principal component analysis (also known as principal components analysis) (PCA) is a technique from statistics for simplifying a data set. It was developed by Pearson (1901) and Hotelling (1933),
whilst the best modern reference is Jolliffe (2002). The aim of the method is to reduce the dimensionality of multivariate data whilst preserving as much of the relevant information as possible. It
is a form of unsupervised learning in that it relies entirely on the input data itself without reference to the corresponding target data (the criterion to be maximized is the variance).
PCA is a linear transformation that transforms the data to a new coordinate system such that the new set of variables, the principal components, are linear functions of the original variables, are
uncorrelated, and the greatest variance by any projection of the data comes to lie on the first coordinate, the second greatest variance on the second coordinate, and so on. In practice, this is
achieved by computing the covariance matrix for the full data set. Next, the eigenvectors and eigenvalues of the covariance matrix are computed, and sorted according to decreasing eigenvalue. Note
that PCA's bias is not always appropriate; features with low variance might actually have high predictive relevance, it depends on the application. | {"url":"http://www.stats.org.uk/pca/","timestamp":"2014-04-17T09:36:19Z","content_type":null,"content_length":"43505","record_id":"<urn:uuid:11601263-5c1c-454b-b44b-be0cbb2be261>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00457-ip-10-147-4-33.ec2.internal.warc.gz"} |
Gravity Homework
Physics 152: Gravity Homework (and some Exam) Problems
Michael Fowler 6/1/07
1. Warm-up exercise: deriving acceleration in circular motion from Pythagoras’ theorem.
Imagine a cannon on a high mountain shoots a cannonball horizontally above the atmosphere at the right speed for it to go in a circular orbit. In one second, the ball will fall 5 meters below a
horizontal line, at the same time traveling v meters horizontally, as in the diagram below (where the distances traveled are grossly exaggerated to make clear what’s going on).
Apply Pythagoras’ theorem to the right-angled triangle to establish that the appropriate speed for a circular orbit just above the earth’s atmosphere is given by []
(Use the approximation that the distance traveled in one second is tiny compared to the radius of the earth.)
2. Properties of the Ellipse.
Take a point [] on the ellipse very close to P, and draw lines from the new point to the two foci.
Use the fact that the “rope” is the same length for P, [] to prove that a light ray from one focus to P will be reflected to the other focus.
3. Kepler’s Third Law states that [] has the same numerical value for all the sun’s planets.
For circular orbits, how are R, T related if the gravitational force is proportional to 1/R? to 1/R^3? To R? What can we conclude from Kepler’s Third Law about the gravitational force?
4. Television signals are relayed by synchronous satellites, placed in orbits such that they hover above the same spot on Earth. Use Kepler’s Laws and data about the Moon’s orbit to find how far
above the Earth’s surface the synchronous satellites are. Could one be placed directly above Charlottesville? If you say no, explain your reasoning.
5. An evil genius puts a spherical rock (made of ordinary stone) in the earth’s orbit, but moving around the sun the other way. It collides with the earth, landing in the desert. It is estimated
that the crater is about the same as would have been caused by a one-megaton hydrogen bomb. How big was the rock?
6. Halley’s comet follows an elliptical orbit, its closest approach to the Sun is observed to be 0.587AU. Given that the orbital period is 76 years, what is its furthest distance from the Sun?
What is the ellipticity of this orbit?
7. Halley’s Comet simplified.
(a) A comet having a period of 64 years has closest approach to the Sun 0.5 AU. Use Kepler’s Third Law, and comparison with the Earth, to figure out its farthest distance from the Sun.
(b) What is the ratio of its kinetic energy when nearest to the Sun to its kinetic energy at the farthest point?
(c) How does its kinetic energy at the closest approach to the Sun compare with that of an equal mass in a circular orbit around the Sun at that distance? (An approximate answer will do.)
8. Galileo discovered four satellites of Jupiter:
Satellite Orbital Radius in 10^6 km Orbital Period in Days
Io 0.422 1.77
Europa 0.671 3.55
Ganymede 1.070 7.16
Callisto 1.880 16.7
The orbits are all very close to circular.
(Data from http://www.ifa.hawaii.edu/~sheppard/satellites/jupsatdata.html , where 59 other satellites of Jupiter are listed!)
Check that Kepler’s Third Law is satisfied in this system, and use these data to find the mass of Jupiter.
9. The galaxy NGC 4258 contains a disk of matter, like a huge version of Saturn’s rings. The disk is not rigid, but is made up of rocks, etc., all going in approximately circular orbits. The disk
has inner radius 0.14 pc (parsec), outer radius 0.28 pc. The inmost part is orbiting with a period of 800 yrs, the outer edge with a period of 2200 yrs.
(a) Show that these data indicate the disk is in a gravitational field dominated by a central massive object (rather than, say, the field of the disk itself).
(b) Find the approximate mass of the central object. The densest known star cluster is about 10^5 solar masses/pc^3. Could the central object be a star cluster? If not, what?
10. Plotting the Gravitational Field.
The diagram below shows how to find the gravitational force at a particular point from a system of two masses.
(a) Draw the field vector at several other points, then construct a picture of the field by drawing field lines: continuous lines which at each point are in the direction of the field at that
point. (The same as “lines of force” in magnetism.)
(b) Draw the field line diagram for two unequal masses, such as the Earth and the Moon. In particular, make clear how the field behaves along the direct line from the Earth to the Moon.
11. (a) Give a brief explanation, with a diagram, of why the gravitational field inside a uniform spherical shell of matter is zero.
(b) Suppose a very deep tunnel is drilled vertically down. What is the gravitational force felt by a mass of 1 kg in the tunnel at a distance r from the center of the earth, given that it is 10
Newtons at distance R[E] = 6400 km., that is, at the earth’s surface? (Assume the Earth’s density is uniform.)
* The rest of this question requires knowledge of Simple Harmonic Motion.
(c) Now suppose in a massive engineering project the tunnel is drilled in a straight line through the center of the earth and reemerges near Australia. The air is pumped out of the tunnel, leaving a
vacuum. A 1 kg package is dropped from rest at one end. How long does it take to reach the other end?
(d) Suppose there is an asteroid of 64 km radius, made of material with the same density as the earth. If an exactly similar tunnel is drilled through this asteroid, how long would it take a
package to “fall” from one end to the other?
12. In the year 3000, a group of bad guys fond of living in caves have excavated a huge spherical cave inside the Moon. (But it’s not centered at the center of the Moon!) Assuming the Moon is a
sphere of rock of uniform density, prove that the gravitational field inside the cave is the same everywhere. (Hint: figure out the field for Moon with no cave, then think of the cave as a uniform
sphere of negative mass density, and add the two contributions.)
13. Imagine a tunnel bored straight through the Earth emerging at the opposite side of the globe. The gravitational force on a mass m in the tunnel is [].
(a) Find an expression for the gravitational potential in the tunnel. Take it to be zero at the center of the Earth.
(b) Now sketch a graph of the potential as a function of distance from the Earth’s center, beginning at the center but continuing beyond the Earth’s radius to a point far away. This curve must be
(c) Conventionally, the potential energy is defined by requiring it to be zero at infinity. How would you adjust your answer to give this result?
14. Draw a plot of the gravitational potential along a straight line from the surface of the Earth to the surface of the Moon. What is the minimum speed of a rocket fired directly from the Earth to
the Moon to reach it? What speed will it be moving on reaching the Moon’s surface? (Ignore the Earth’s rotation and the Moon’s orbital speed—just consider two fixed masses.)
15. For this question, take the mass of Mars to be 0.1 Earth masses, and the radius of Mars to be 0.5 Earth radii.
(a) Given that g = 10 m/sec^2 at the Earth’s surface, what is the acceleration due to gravity at the surface of Mars? (Show your working.)
(b) A satellite in low Earth orbit travels at 8 km/sec. Use the value of g on Mars you found in part (a) to work out how fast a satellite in a low Mars orbit will travel.
(c) Calculate the escape velocity from Mars.
(d) A synchronous satellite is in a circular orbit (around the Earth) with radius 42,000 km. It happens that the length of a Martian day is close to 24 hours. What would be the orbit radius of a
synchronous satellite circling Mars?
16. Phobos, a satellite of Mars, has a radius of 11 km and a mass of 10^16 kg. It’s a bit lumpy, but let’s assume it’s spherical to get a doable problem.
(a) What is g on Phobos?
(b) If you can jump to a height of one meter on earth, how high could you jump on Phobos? (Think carefully about this.)
(c) Could an astronaut on a bicycle reach orbital speed on Phobos? (Guesses don’t count, I need to see a derivation). What about reaching escape velocity?
17. Uranus has a radius four times Earth’s radius, but gravity at the surface is only 0.8g[earth]. Escape velocity from Earth is 11.2 km/sec. Using these facts, and nothing else, find the escape
velocity from Uranus.
18. (a) Find the orbital speed of a spaceship in low orbit around the Moon, just skimming the mountain tops.
(b) Suppose the pilot suddenly increases the speed by a factor of [], but during the brief acceleration keeps the spaceship pointing the same way, that is, horizontally. Describe the path
the spaceship will take after the engine cuts out—does this curve have a name?.
19. In deep space, an astronaut is marooned ten meters from his four-ton spacecraft. If he is exactly at rest relative to the craft, and there are no other gravitational fields close by, estimate how
long it will be before he’s back on board. How fast will he be moving when he hits the craft (which is 5 meters in diameter)?
20. The escape speed from the moon is 2.38 km/sec. Suppose you had on the moon a cannon that could fire shells at 2.4 km/sec. Obviously, if you fired a shell vertically upwards, it would escape the
moon’s gravity. But what if you fired it almost horizontally, just elevated enough so it cleared the mountains? Describe its trajectory in this situation.
21. The escape velocity from Earth is 11.2 km./sec. What is the escape velocity from the Solar System starting in a high parking orbit several Earth radii from Earth. (Hint: what is the Earth’s
speed in orbit?) On the basis of this, estimate roughly how much more fuel energy is needed to reach the outer planets compared with going to the Moon. Is there a way around this problem?
22. Imagine a fictitious moon, which we’ll call Moon1, a sphere with the same density as the earth, but with radius exactly one-quarter the earth’s radius:
r[Moon1] = r[earth], R[Moon1]=0.25R[earth].
(a) Taking the acceleration due to gravity g[earth] to be 10 m.sec^-2 at the earth’s surface, what is the acceleration g[Moon1] due to Moon1’s gravity at Moon1’s surface?
(b) If the escape velocity from the earth’s surface is 11 km.sec^-1, what is the escape velocity from Moon1’s surface?
(c) If it takes 90 minutes for a satellite in low earth orbit (orbit radius approximately equal to earth radius) to go around once, how long will it take a satellite in a low Moon1 orbit (skimming
the surface of the airless moon) to go around once?
(d) The real Moon has a radius close to that of Moon1 above (our Moon’s radius is 10% bigger than one-quarter the earth’s, we’ll neglect that difference here). However, the real Moon has a density
only 60% that of the earth. In this part, use the real Moon’s density (but Moon1’s radius) to recalculate the answers to (a), (b) and (c) above.
23. Saturn’s satellite Titan has an orbit of radius 1.22 x 10^6 km., and a period of 15.9 days. Use this information to find the mass of Saturn, then use its radius of 60,300 km to deduce
(a) Saturn’s average density
(b) the value of g at the surface of Saturn
(c) the escape velocity from Saturn.
24. The furthest planet, Pluto, has a radius 20% of the Earth’s radius, and a mass only 0.2% that of the Earth. (Both figures are within about 5%.)
(a) Suppose an astronaut, in full insulated gear, can jump 0.5 m high on Earth. How high can she jump on Pluto? (You don’t need to know G to answer this!)
(b) Assuming “air” resistance is negligible, what speed would a (rocket driven) car racing over a flat plane (a frozen sea) on Pluto need to be traveling to attain escape velocity? (Escape
velocity from Earth is 11.2 km per sec: use this fact.)
(c) Would it in fact have left the ground before reaching that speed? Explain your answer.
25. The escape velocity from a certain planet is 10 km per sec. The planet has a moon having radius one-quarter that of the planet, and density one-half that of the planet. What is the escape
velocity from the planet’s moon?
26. In an imaginary universe, the gravitational force decreases with distance as 1/R instead of 1/R^2. Suppose in that universe there is a planet the same size as Earth and also having the same
value of g near the surface. Would the period of a satellite in low circular orbit (just above an atmosphere of negligible depth) be the same? Would the escape velocity be the same?
*27. Somewhere on the line from the Earth to the Sun there is a point, called a Lagrange point, such that a satellite placed there will orbit around the Sun in sync with the Earth. In fact, there’s
already a satellite there, it monitors the Sun continuously. Come up with some estimate of how far from Earth this Lagrange point is (the Web might be helpful). []
28. On a Moon mission, a spaceship is fired from Earth with just enough speed to reach the Moon, but aimed so that it just misses the Moon, and loops behind it, closest approach being near the point
on the Moon’s surface furthest from Earth. At that point, a small distance above the Moon’s surface, the ship fires a rocket to put it into low circular orbit around the Moon. What is
(approximately) the change in speed needed for this maneuver?
Elliptic Paths to Planets and Asteroids
29. The asteroid Gaspra is twice as far from the sun as we are. Assume it is in a circular orbit, and you are planning an expedition there.
The most economical trajectory is along an elliptical orbit, whose closest approach to the
sun, call it r[1], is at the earth’s orbit, and furthest distance from the sun, r[2], is at Gaspra’s orbit.
Suppose that after leaving the atmosphere, the spaceship is rapidly speeded up to v[1], then the engines cut out, and it follows the assigned elliptic path, arriving at Gaspra’s orbit with speed v
[2]. (Neglect the earth’s gravitational pull on the spaceship.)
(a) What quantities are conserved on the elliptic orbit?
(b) Find two equations for v[1], v[2] in terms of r[1], r[2] and GM, where M is the mass of the sun.
(c) Solve the two equations to find v[1].
(d) Find the speed of the earth in orbit in terms of r[1] and GM.
(e) Given that the earth’s speed in orbit is 30 km per sec, how much does the spaceship need to be speeded up relative to the earth to get to Gaspra along this ellipse?
(f) Show on a diagram the earth in orbit, and the direction in which the spaceship needs to be moving just after leaving the earth to reach Gaspra. Approximately, what path would the spaceship take
if fired in the opposite direction?
30. Suppose we are sending a space probe of mass m from Earth to Jupiter by the most economical elliptical route. Take the radius of Jupiter’s orbit around the sun to be 5 AU.
(a) What is the total energy of the probe in the elliptical orbit?
(b) Assume it is fired from a parking orbit circling the earth far above, so the earth’s own gravity has a negligible effect. Given that the earth moves in orbit at 30 km/sec, what is the speed of
the probe relative to earth as it enters the elliptical orbit?
(c) What is its speed when it reaches Jupiter’s orbit?
31. We plan to send a probe to an asteroid which has a circular orbit of radius three times that of the earth’s orbit (assumed also circular).
(a) Sketch the most efficient path, showing on your diagram the earth’s orbit and the asteroid’s.
(b) If the earth travels in its orbit at 30 km per sec, at what speed relative to the earth must the probe be moving after it has cleared essentially all the earth’s gravitational field?
32. Suppose a satellite is in low earth orbit, that is, in a circular orbit at a height of 200 km., so the radius of the circle is 6600 km., say. We want to raise it to a circular orbit of twice
that radius (so it will now be going in a circle at a height of 6800 km above the earth’s surface.)
The technique is to give it two quick boosts: boost1 puts it into an elliptical orbit, where its furthest point from the earth’s center is exactly twice its distance of closest approach, boost2,
delivered at the topmost point of the orbit, transfers it to a circular orbit at that radius.
Use conservation of angular momentum and energy in the elliptical orbit to answer these two questions:
(a) By what percentage did boost1 increase its speed?
(b) By what percentage did boost2 increase its speed?
(b) Give a qualitative explanation of how you would fire a rocket to get back to Earth from a parking orbit near Mars (so you neglect Mars’ own gravity).
33. A “Binary” System.
A very recently discovered “earthlike” planet—we’ll call it P—orbits the red dwarf star Gleise 581, which is 20 light years from us.
P’s sun (Gleise 581) has a mass one-third the mass of our sun.
The planet P’s presence was established by detecting a wobble in the motion of Gleise 581 with a period of 13 days. (The wobble being caused by the orbiting planet’s gravity: think binary system.)
(3) (a) How far is P from its sun?
Do this as follows: for any solar type system with circular planetary orbits [], M being the mass of that sun.
Use as units earth years and A.U. (distance of Earth from Sun), so for our solar system in these units [].
What is [] in the same units (earth years and earth A.U.’s) in the Gleise 581 system?
The wobble means P orbits its sun once in 13 days. Write that in earth years, and deduce how far P is from its sun, in A.U.
(b) Given that 1 A.U = 1.5x 10^8 km, how fast is P moving in its (assumed circular) orbit?
(c) From detecting the Doppler shift, it is found that the maximum speed of the sun Gleise 581 in its 13-day wobble is 3 m/sec. From this, thinking of the planet P and the sun Gleise 581 as a
“binary star” system, what is the ratio of the planet P mass to the sun (Gleise) mass?
(d) Our sun’s mass is about 300,000 earth masses. How does the “earthlike” planet P’s mass compare with the earth’s mass? (Recall Gleise has a mass one-third of our sun’s mass.)
General Relativity
34. (a) State the Equivalence Principle.
(b) Explain how shining light across an elevator can lead to the conclusion that light is deflected by gravity. (Include a diagram.)
(c) Given that light is deflected of order 1 second of arc on passing by the sun, and that the order of magnitude is correctly given by a simple classical approximation, how much would you estimate
light to be deflected (order of magnitude) passing the surface of a neutron star, having twice the mass of the Sun and a radius of 10 km (the sun’s radius being 700,000 km)? State what
approximations you’re making.
35. The first experimental test of General Relativity was an observation of the deflection of starlight by the Sun’s gravitational field (observed during a Solar eclipse). Classically, regarding
light as tiny particles, the deflection can be estimated within 20% or so by approximating the Sun’s gravitational effect as equal to gravity at the Sun’s surface acting for a period of time equal to
that needed for the particles to travel a distance equal to the Sun’s diameter. Calculate what angular deflection that would give, in seconds of arc. General Relativity predicts that the actual
deflection should be twice the classical value—and that was observed.
36. The GPS satellites are at an altitude of about 20,000 km. Find their speed, and figure out the necessary correction factors for their clocks from both Special and General Relativistic effects.
Are these corrections important for the functioning of the system, or can they be neglected in practice?
37. At http://www.enchantedlearning.com/subjects/astronomy/activities/coloring/Solarsystem.shtml you will find the following image:
What’s wrong with the orbit of Pluto as shown here?
38. Mercury can be observed as a small black dot crossing the face of the Sun, this occurs about every ten years on average. Since Mercury goes around in 88 days, why is this so rare? Also, it
only ever happens in May or November. How would you explain this pattern?
39. Some future explorer decides to fly by a neutron star, following a free-fall trajectory in which the spaceship loops behind the neutron star and comes back—so within the spaceship, the astronaut
will be “weightless”. However, if g varies significantly between the head and foot of the astronaut, this could have disastrous consequences.
(a) Estimate at what rate of variation of g the astronaut is unsafe.
(b) Assume the neutron star has a mass of two solar masses, and a radius of 10 km. How close to the surface is it safe for the ship to approach?
40. The asteroid Icarus was only four million miles from earth on a recent pass. If a collision took place, and Icarus fell to earth, give a ballpark estimate of the energy released in the inelastic
collision. Compare it with a one megaton hydrogen bomb.
Flashlet and Applet Exercises
41. (a) Activate the Mars trip flashlet. The initial launch speed you enter is from a high parking orbit (say at ten Earth radii) so that the Earth’s own gravitational field has negligible effect.
Find the minimum speed needed to reach Mars, sketch a picture of this most economical orbit, and estimate how long the trip would take.
(b) Give a qualitative explanation of how you would fire a rocket to get back to Earth from a parking orbit near Mars (so you neglect Mars’ own gravity).
42. (a) From the Fact Sheet, find the speed of Jupiter in orbit.
(b) Open the Jupiter slingshot flashlet. Note that you can adjust both the initial speed of the rocket approaching Jupiter, and how closely you begin to Jupiter’s orbit. Imagine your
rocket barely makes it out to Jupiter’s orbit, so has no speed left—but you’ve perfectly timed it to derive maximum benefit from the slingshot effect. Could Jupiter give it enough of a boost to get
it thrown out of the solar system? Justify your answer: what does the rocket’s orbit look like as seen by someone on Jupiter?
43. Open the Newtonian Mountain applet. The height of the mountain (Newton’s own drawing) is about 10% of the Earth’s diameter. This happens to be approximately the maximum height reached by an ICBM
on a trajectory going half way around the world, so the cannonball path is the second half of an ICBM trajectory.
(a) Experiment with the applet to find the speed at the top of an ICBM trajectory compared with speed in a circular orbit at the same height.
(b) If the ICBM is launched with an engine that cuts out as it leaves the atmosphere, what is the approximate speed as the engine cuts out? (Answer in km/sec – I apologize for the applet being in
mph. Take the radius of the Earth to be 6400 km., and neglect the thickness of the atmosphere.)
(c) When the engine cuts out, what is the angle between the trajectory and the horizontal?
(Hint: use conservation of angular momentum.) | {"url":"http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/Gravity%20_HW_File.htm","timestamp":"2014-04-19T11:56:21Z","content_type":null,"content_length":"86196","record_id":"<urn:uuid:c4602ada-258f-474b-9966-bbe464c5a5e0>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00191-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Short and Tweet
October 26, 2012
Proofs that are about the length of a tweet.
Jack Dorsey is the creator of Twitter. We thank him for the creation of of tweets, which are of course messages of up to 140 characters in length.
Today Ken and I wish to talk about very short proofs—proofs that could almost fit into a single tweet.
It’s a fun topic, but also touches on a serious one: what is a proof really supposed to do?
These proofs must be hard to find but quick to verify, which is the essential idea of ${\mathsf{NP}}$.
Euler’s Example
Let’s look at problems whose solutions fit into a single tweet, or almost.
Trying to generalize Fermat’s Last Theorem, the great Leonhard Euler conjectured that an ${n}$th power cannot be written as the sum of fewer than ${n}$ nontrivial powers for ${n \ge 3}$.
Leon Lander and Thomas Parkin used a computer to solve the problem for ${n=5}$ and found a counterexample. Here is their whole paper:
Note, if they had just sent the answer it would have been:
$\displaystyle 27^5 + 84^5 +110^5 + 133^5 = 144^5.$
Writing this as a LaTeX formula would have easily fit into a single Tweet—too bad they did their work 46 years ago.
Two decades later, Noam Elkies found a method to construct an infinite series of counterexamples for the ${n=4}$ case. He showed that
$\displaystyle (85v^{2}+484v-313)^{4} + (68v^{2}+586v+10)^{4} + (2u)^{4} = (357v^{2}-204v+363),$
$\displaystyle u^{2} = 22030+28849v + 56158v^{2} + 36941v^{3} + 31790v^{4}.$
In 1988, Roger Frye found the smallest counterexample based on Elkies’ ideas:
$\displaystyle 95800^{4} + 217519^{4} + 414560^{4} = 422481^{4}.$
Note that ${x^2 - P^2 = (x+P)(x-P)}$ always has the integer roots ${\pm P}$, which are distinct when ${P eq 0}$. And
$\displaystyle (x^2 - P)^2 - Q^2 = (x^2 - a^2)(x^2 - b^2)$
is solvable in integers by ${P = 5}$, ${Q = 4}$, ${a = 1}$, ${b = 3}$ among many other possibilities.
That is,
$\displaystyle (x^2 - 5)^2 - 4^2 = x^4 - 10x^2 + 9 = (x^2 - 1)(x^2 - 9)$
has four distinct integer roots in pairs ${\pm 1}$ and ${\pm 3}$.
Going from degree ${d = 4}$ to ${d = 8}$, we can solve
$\displaystyle ((x^2 - P)^2 - Q)^2 - R^2 = (x^2 - a^2)(x^2 - b^2)(x^2 - c^2)(x^2 - d^2)$
with ${P = 85}$, ${Q= 4,176}$, and ${R = 2,880}$. These induce ${a = 1}$, ${b=7}$, ${c=11}$, and ${d=13}$, again giving ${d}$-many distinct integer roots.
Can we do this with ${d=16}$ for
$\displaystyle (((x^2 - P)^2 - Q)^2 - R)^2 - S^2 = (x^2 - a^2)(x^2 - b^2)(x^2 - c^2)\cdots(x^2 - h^2)?$
There had been a conjecture—indeed, a claimed proof—of “no,” but Dominic Symes found an example—or counterexample:
$\displaystyle \begin{array}{rcl} P &=& 67,405 \\ Q &=& 3,525,798,096 \\ R &=& 533,470,702,551,552,000 \\ S &=& 469,208,209,191,321,600. \end{array}$
This gives distinct roots ${\pm 11,\pm 77,\pm 101,\pm 131,\pm 343,\pm 353,\pm 359,\pm 367}$, as you can verify. Andrew Bremner found two infinite families of solutions.
Can we take it a step further to degree ${d = 32}$ with ${P,Q,R,S,T}$ and ${a,\dots,p}$? Nobody knows. This is related to Stephen Smale’s Fourth Problem which we just blogged about.
Namely, the left-hand sides ${f}$ are straight-line programs of length ${\tau(f) = 2\log_2 d}$. If there are ${Z(f) = d}$ distinct integer zeroes, this violates the conjecture ${\tau(f) \geq Z(f)^{\
A degree-${d}$ polynomial ${f(x)}$ with ${d}$ distinct integer roots and a short straight-line program is called a gem by Bernd Borchert, Pierre McKenzie, and Klaus Reinhardt.
They give ${d}$-gems for ${d}$ up to ${55}$, but skipping ${d=32}$. See Borchert’s project page for more, including relevance to factoring.
Short Proofs and Interaction
Ken and I believe this issue of very short proofs highlights the real reason we prove theorems in mathematics.
A proof is not just a thing that we write out and then feel happy with the knowledge that something is proved. No. A proof must be something that can be understood by others.
To be understood it must be checkable. The above are extreme examples—you can just do the arithmetic. The existence of the proof is what was hard.
In some areas of computer science proofs are viewed in a different way. One area where I hear statements like “the proof is long and boring” is cryptographic protocols.
That some proofs of protocols are tedious is exactly what I do not like about them. I do not trust a long and boring proof, exactly because it is unlikely to be checked by others.
Of course protocols themselves are often proofs, of knowledge or identity or privilege or authority. The individual steps required of the user can be short.
The idea of interaction takes proofs beyond ${\mathsf{NP}}$. The interaction can be short, or have short rounds. Still, the proof that it is a proof can be long.
Even for problems like factoring, interaction can shorten the proof that one has a proof. Suppose I have a long proof of an efficient algorithm. What can I do?
I can say, “Tweet me a number.” Using ASCII for base 128, a 1,024-bit RSA modulus ${N}$ can fit into 140 characters. Then I tweet back ${p,q}$ and you verify ${N = pq}$.
With interaction you can’t accuse me of pre-computing results or exploiting holes like Arjen Lenstra’s project. This is a short proof of my proof.
Most protocols cannot self-prove this way because you have to prove security against possible attacks. But there as in math, it’s good to ask how far short proofs can go.
Open Problems
Could there be a very short proof of your favorite open problem? Can we make more proofs like tweets?
Is a post easier to read with Tweet-length paragraphs, sometimes with equations in-between, each giving one idea? Or our usual longer ones?
End Note
our congratulations to Leonid Levin, himself known for very short papers, on his
2012 Knuth Prize
. Ken heard his lecture at FOCS this past Monday—here is a nice
by Thomas Vidick on it.
1. October 26, 2012 5:21 pm
21519 should read 217519
2. October 26, 2012 6:01 pm
I sympathize with much of what you write. However, I do not believe that every “interesting” theorem in CS or mathematics is amenable to a short and elegant proof. For instance, proofs of
correctness for non-trivial programs or (cryptographic) protocols might need long proofs and perhaps we should accept this fact. What is boring/tedious is in the eye of the beholder and the
import of the correctness proof for a cryptographic protocols might just be that the protocol works.
Aschbacher’s “Highly complex proofs and implications of such proofs” offers an interesting discussion of very long and complicated proofs in mathematics. Here are two excerpts:
“Conventional wisdom says the ideal mathematical proof should be short, simple
and elegant. However, there are now examples of very long, complicated proofs,
and as mathematics continues to mature, more examples are likely to appear.”
“My guess is that we will begin to encounter many more such problems,
theorems, and proofs in the near future. As a result we will need to re-examine
what constitutes a proof, and what constitutes a good proof. Elegance and
simplicity should remain important criteria in judging mathematics, but the
applicability and consequences of a result are also important, and sometimes
these criteria conflict. I believe that some fundamental theorems do not admit
simple elegant treatments, and the proofs of such theorems may of necessity be
long and complicated. Our standards of rigor and beauty must be sufficiently
broad and realistic to allow us to accept and appreciate such results and their
proofs. As mathematicians we will inevitably use such theorems when it is
necessary in the practice our trade; our philosophy and aesthetics should reflect
this reality.”
I recommend reading the essay.
3. October 26, 2012 6:07 pm
Proof that n^{1/n} -> 1 as n -> oo:
By Bernoulii’s inequality, (1+n^{-1/2})^n >= 1+n^{1/2} > n^{1/2}.
Raising to the 2/n power,
n^{1/n} < (1+n^{-1/2})^2 = 1 + 2n^{-1/2}+n^{-1} < 1 + 3 n^{-1/2}.
q e d
4. October 27, 2012 5:13 am
In the view of theorems as a shortcut statements in the different proves, long proves are actually good, when used many times, similar to subroutines in programming. They loose aesthetics, but
should be practical.
5. October 29, 2012 11:51 am
Two appreciations of tweet-length proofs from Michael Spivak’s well-regarded Calculus on Manifolds: (1965)
(page ix): “There are good reasons why theorems should all be easy and the definitions hard … Definitions serve a twofold purpose: they are rigorous replacements for vague notions, and
machinery for elegant proofs.”
(page 103): “Stokes’ Theorem shares three important attributes with many fully evolved major theorems: (1) It is trivial. (2) It is trivial because the terms appearing in it have been
properly defined. (3) It has significant consequences.”
Spivak’s remarks find an apt counterpoint in Saunders Mac Lane’s survey article Hamiltonian mechanics and geometry (1970):
“Often a book on physics will contrast a mathematical presentation of an idea (using coordinates) with a physical presentation (not using coordinates); in such cases the contrast is really
between two mathematical presentations, one with coordinates and one coordinate-free.”
Theorems having “trivial” proofs that are founded upon multiple levels of carefully crafted abstraction are emerging (it seems to me) as a hallmark of a 21st century STEM enterprise in which
mathematical naturality is appreciated as dual to experimental physicality.
□ October 29, 2012 3:28 pm
But then John, one needs exponentially many definitions if P!=NP ;), still unreadable.
☆ October 29, 2012 4:44 pm
mkatkov says “But then John, one needs exponentially many [S: definitions :S] oracles if P!=NP” ;)
To some folks (including me) oracles seem intrinsically more mysterious than definitions! :)
6. October 30, 2012 1:20 pm
Is a counterexample really the same as a proof? Or in other words: I guess I’m more impressed by a short proof of a “for all” statement than I am of a “there exists” statement.
□ October 30, 2012 3:53 pm
About Delta’s question “Is a counterexample really the same as a proof?”: It depends if the value of a theorem being true is the same as the value of a theorem being false. It seems to me, in
general, that it is more work to prove a theorem than to find a counterexample, so my answer to the question is “No.”
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The physicist, in his study of natural phenomena, has two methods of making progress: (1) the method of experiment and observation, and (2) the method of mathematical reasoning. The former is just
the collection of selected data; the latter enables one to infer results about experiments that have not been performed. There is no logical reason why the second method should be possible at all,
but one has found in practice that it does work and meets with reasonable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would
not suspect, but which nevertheless plays an important role in Nature's scheme.
One might describe the mathematical quality in Nature by saying that the universe is so constituted that mathematics is a useful took in its description. However, recent advances in physical science
show that this statement of the case is too trivial. The connection between mathematics and the description of the universe goes far deeper than this, and one can get an appreciation of it only from
a thorough examination of the various facts that make it up. The main aim of my talk to you will be to give you such an appreciation. I propose to deal with how the physicist's views on this subject
have been gradually modified by the succession of recent developments in physics, and then I would like to make a little speculation about the future.
Let us take as our starting-point that scheme of physical science which was generally accepted in the last century - the mechanistic scheme. This considers the whole universe to be a dynamical system
(of course an extremely complicated dynamical system), subject to laws of motion which are essentially of the Newtonian type. The role of mathematics in this scheme is to represent the laws of motion
by equations, and to obtain solutions of the equations referring to observed conditions.
The dominating idea in this application of mathematics to physics is that the equations representing the laws of motion should be of a simple form. The whole success of the scheme is due to the fact
that equations of simple form do seem to work. The physicist is thus provided with a principle of simplicity, which he can use as an instrument of research. If he obtains, from some rough
experiments, data which fit in roughly with certain simple equations, he infers that if he performed the experiments more accurately he would obtain data fitting in more accurately with the
equations. The method is much restricted, however, since the principle of simplicity applies only to fundamental laws of motion, not to natural phenomena in general. For example, rough experiments
about the relation between the pressure and volume of a gas at a fixed temperature give results fitting in with a law of inverse proportionality, but it would be wrong to infer that more accurate
experiments would confirm this law with greater accuracy, as one is here dealing with a phenomenon which is not connected in any very direct way with the fundamental laws of motion.
The discovery of the theory of relativity made it necessary to modify the principle of simplicity. Presumably one of the fundamental laws of motion is the law of gravitation which, according to
Newton, is represented by a very simple equation, but, according to Einstein, needs the development of an elaborate technique before its equation can even be written down. It is true that, from the
standpoint of higher mathematics, one can give reasons in favour of the view that Einstein's law of gravitation is actually simpler than Newton's, but this involves assigning a rather subtle meaning
to simplicity, which largely spoils the practical value of the principle of simplicity as an instrument of research into the foundations of physics.
What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined,
any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating. The theory of relativity introduced mathematical beauty to an
unprecedented extent into the description of Nature. The restricted theory changed our ideas of space and time in a way that may be summarised by stating that the group of transformations to which
the space-time continuum is subject must be changed from the Galilean group to the Lorentz group. The latter group is a much more beautiful thing than the former - in fact, the former would be called
mathematically a degenerate special case of the latter. The general theory of relativity involved another step of a rather similar character, although the increase in beauty this time is usually
considered to be not quite so great as with the restricted theory, which results in the general theory being not quite so firmly believed in as the restricted theory.
We now see that we have to change the principle of simplicity into a principle of mathematical beauty. The research worker, in his efforts to express the fundamental laws of Nature in mathematical
form, should strive mainly for mathematical beauty. He should still take simplicity into consideration in a subordinate way to beauty. (For example Einstein, in choosing a law of gravitation, took
the simplest one compatible with his space-time continuum, and was successful.). It often happens that the requirements of simplicity and of beauty are the same, but where they clash the latter must
take precedence.
Let us pass on to the second revolution in physical thought of the present century - the quantum theory. This is a theory of atomic phenomena based on a mechanics of an essentially different type
from Newton's. The difference may be expressed concisely, but in a rather abstract way, by saying that dynamical variables in quantum mechanics are subject to an algebra in which the commutative
axiom of multiplication does not hold. Apart from this, there is an extremely close formal analogy between quantum mechanics and the old mechanics. In fact, it is remarkable how adaptable the old
mechanics is to the generalization of non-commutative algebra. All the elegant features of the old mechanics can be carried over to the new mechanics, where they reappear with an enhanced beauty.
Quantum mechanics requires the introduction into physical theory of a vast new domain of pure mathematics - the whole domain connected with non-commutative multiplication. This, coming on top of the
introduction of new geometries by the theory of relativity, indicates a trend which we may expect to continue. We may expect that in the future further big domains of pure mathematics will have to be
brought in to deal with the advances in fundamental physics.
Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying that the mathematician plays a game in which he
himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds
interesting are the same as those which Nature has chosen. It is difficult to predict what the result of all this will be. Possibly, the two subjects will ultimately unify, every branch of pure
mathematics then having its physical application, its importance in physics being proportional to its interest in mathematics. At present we are, of course, very far from this stage, even with regard
to some of the most elementary questions. For example, only four-dimensional space is of importance in physics, while spaces with other numbers of dimensions are of about equal interest in
It may well be, however, that this discrepancy is due to the incompleteness of present-day knowledge, and that future developments will show four-dimensional space to be of far greater mathematical
interest than all the others.
The trend of mathematics and physics towards unification provides the physicist with a powerful new method of research into the foundations of his subject, a method which has not yet been applied
successfully, but which I feel confident will prove its value in the future. The method is to begin by choosing that branch of mathematics which one thinks will form the basis of the new theory. One
should be influenced very much in this choice by considerations of mathematical beauty. It would probably be a good thing also to give a preference to those branches of mathematics that have an
interesting group of transformations underlying them, since transformations play an important role in modern physical theory, both relativity and quantum theory seeming to show that transformations
are of more fundamental importance than equations. Having decided on the branch of mathematics, one should proceed to develop it along suitable lines, at the same time looking for that way in which
it appears to lend itself naturally to physical interpretation.
This method was used by Jordan in an attempt to get an improved quantum theory on the basis of an algebra with non-associative multiplication. The attempt was not successful, as one would rather
expect, if one considers that non-associative algebra is not a specially beautiful branch of mathematics, and is not connected with an interesting transformation theory. I would suggest, as a more
hopeful-looking idea for getting an improved quantum theory, that one take as basis the theory of functions of a complex variable. This branch of mathematics is of exceptional beauty, and further,
the group of transformations in the complex plane, is the same as the Lorentz group governing the space-time of restricted relativity. One is thus led to suspect the existence of some deep-lying
connection between the theory of functions of a complex variable and the space-time of restricted relativity, the working out of which will be a difficult task for the future.
Let us now discuss the extent of the mathematical quality in Nature. According to the mechanistic scheme of physics or to its relativistic modification, one needs for the complete description of the
universe not merely a complete system of equations of motion, but also a complete set of initial conditions, and it is only to the former of these that mathematical theories apply. The latter are
considered to be not amenable to theoretical treatment and to be determinable only from observation.
The enormous complexity of the universe is ascribed to an enormous complexity in the initial conditions, which removes them beyond the range of mathematical discussion.
I find this position very unsatisfactory philosophically, as it goes against all ideas of the unity of Nature. Anyhow, if it is only to a part of the description of the universe that mathematical
theory applies, this part ought certainly to be sharply distinguished from the remainder. But in fact there does not seem to be any natural place in which to draw the line. Are such things as the
properties of the elementary particles of physics, their masses and the numerical coefficients occurring in their laws of force, subject to mathematical theory? According to the narrow mechanistic
view, they should be counted as initial conditions and outside mathematical theory. However, since the elementary particles all belong to one or other of a number of definite types, the members of
one type being all exactly similar, they must be governed by mathematical law to some extent, and most physicists now consider it to be quite a large extent. For example, Eddington has been building
up a theory to account for the masses. But even if one supposed all the properties of the elementary particles to be determinable by theory, one would still not know where to draw the line, as one
would be faced by the next question - Are the relative abundances of the various chemical elements determinable by theory? One would pass gradually from atomic to astronomic questions.
This unsatisfactory situation gets changed for the worse by the new quantum mechanics. In spite of the great analogy between quantum mechanics and the older mechanics with regard to their
mathematical formalisms, they differ drastically with regard to the nature of their physical consequences. According to the older mechanics, the result of any observation is determinate and can be
calculated theoretically from given initial conditions; but with quantum mechanics there is usually an indeterminacy in the result of an observation, connected with the possibility of occurrence of a
quantum jump, and the most that can be calculated theoretically is the probability of any particular result being obtained. The question, which particular result will be obtained in some particular
case, lies outside the theory. This must not be attributed to an incompleteness of the theory, but is essential for the application of a formalism of the kind used by quantum mechanics.
Thus according to quantum mechanics we need, for a complete description of the universe, not only the laws of motion and the initial conditions, but also information about which quantum jump occurs
in each case when a quantum jump does occur. The latter information must be included, together with the initial conditions, in that part of the description of the universe outside mathematical
The increase thus arising in the non-mathematical part of the description of the universe provides a philosophical objection to quantum mechanics, and is, I believe, the underlying reason why some
physicists still find it difficult to accept this mechanics. Quantum mechanics should not be abandoned, however, firstly, because of its very widespread and detailed agreement with experiment, and
secondly, because the indeterminacy it introduces into the results of observations is of a kind which is philosophically satisfying, being readily ascribable to an inescapable crudeness in the means
of observation available for small-scale experiments. The objection does show, all the same, that the foundations of physics are still far from their final form.
We come now to the third great development of physical science of the present century - the new cosmology. This will probably turn out to be philosophically even more revolutionary than relativity or
the quantum theory, although at present one can hardly realize its full implications. The starting-point is the observed red-shift in the spectra of distance heavenly bodies, indicating that they are
receding from us with velocities proportional to their distances.* The velocities of the more distant ones are so enormous that it is evident we have here a fact of the utmost importance, not a
temporary or local condition, but something fundamental for our picture of the universe.
If we go backwards into the past we come to a time, about 2 x 109 years ago, when all the matter in the universe was concentrated in a very small volume. It seems as though something like an
explosion then took place, the fragments of which we now observe still scattering outwards. This picture has been elaborated by LemaÏtre, who considers the universe to have started as a single very
heavy atom, which underwent violent radioactive disintegrations and so broke up into the present collection of astronomical bodies, at the same time giving off the cosmic rays.
With this kind of cosmological picture one is led to suppose that there was a beginning of time, and that it is meaningless to inquire into what happened before then. One can get a rough idea of the
geometrical relationships this involves by imagining the present to be the surface of a sphere, going into the past to be going in towards the centre of the sphere, and going into the future to be
going outwards. There is then no limit to how far one may go into the future, but there is a limit to how far one can go into the past, corresponding to when one has reached the centre of the sphere.
The beginning of time provides a natural origin from which to measure the time of any event. The result is usually called the epoch of that event. Thus the present epoch is 2 x 109 years.
Let us now return to dynamical questions. With the new cosmology the universe must have been started off in some very simple way. What, then, becomes of the initial conditions required by dynamical
theory? Plainly there cannot be any, or they must be trivial. We are left in a situation which would be untenable with the old mechanics. If the universe were simply the motion which follows from a
given scheme of equations of motion with trivial initial conditions, it could not contain the complexity we observe. Quantum mechanics provides an escape from the difficulty. It enables us to ascribe
the complexity to the quantum jumps, lying outside the scheme of equations of motion. The quantum jumps now form the uncalculable part of natural phenomena, to replace the initial conditions of the
old mechanistic view.
One further point in connection with the new cosmology is worthy of note. At the beginning of time the laws of Nature were probably very different from what they are now. Thus we should consider the
laws of Nature as continually changing with the epoch, instead of as holding uniformly throughout space-time. This idea was first put forward by Milne, who worked it out on the assumptions that the
universe at a given epoch is roughly everywhere uniform and spherically symmetrical. I find these assumptions not very satisfying, because the local departures from uniformity are so great and are of
such essential importance for our world of life that it seems unlikely there should be a principle of uniformity overlying them. Further, as we already have the laws of Nature depending on the epoch,
we should expect them also to depend on position in space, in order to preserve the beautiful idea of the theory of relativity there is fundamental similarity between space and time. This goes more
drastically against Milne's assumptions than a mere lack of uniformity in the distribution of matter.
We have followed through the main course of the development of the relation between mathematics and physics up to the present time, and have reached a stage where it becomes interesting to indulge in
speculations about the future. There has always been an unsatisfactory feature in the relation, namely, the limitation in the extent to which mathematical theory applies to a description of the
physical universe. The part to which it does not apply has suffered an increase with the arrival of quantum mechanics and a decrease with the arrival of the new cosmology, but has always remained.
This feature is so unsatisfactory that I think it safe to predict it will disappear in the future, in spite of the startling changes in our ordinary ideas to which we should then be led. It would
mean the existence of a scheme in which the whole of the description of the universe has its mathematical counterpart, and we must suppose that a person with a complete knowledge of mathematics could
deduce, not only astronomical data, but also all the historical events that take place in the world, even the most trivial ones. Of course, it must be beyond human power actually to make these
deductions, since life as we know it would be impossible if one could calculate future events, but the methods of making them would have to be well defined. The scheme could not be subject to the
principle of simplicity since it would have to be extremely complicated, but it may well be subject to the principle of mathematical beauty.
I would like to put forward a suggestion as to how such a scheme might be realized. If we express the present epoch, 2 x 109 years, in terms of a unit of time defined by the atomic constants, we get
a number of the order 1039, which characterizes the present in an absolute sense. Might it not be that all present events correspond to properties of this large number, and, more generally, that the
whole history of the universe corresponds to properties of the whole sequence of natural numbers? At first sight it would seem that the universe is far too complex for such a correspondence to be
possible. But I think this objection cannot be maintained, since a number of the order 1039 is excessively complicated, just because it is so enormous. We have a brief way of writing it down, but
this should not blind us to the fact that it must have excessivly complicated properties.
There is thus a possibility that the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realized. To do so physics will have to develop a long
way to establish the details of how the correspondence is to be made. One hint for this development seems pretty obvious, namely, the study of whole numbers in modern mathematics is inextricably
bound up with the theory of functions of a complex variable, which theory we have already seen has a good chance of forming the basis of the physics of the future. The working out of this idea would
lead to a connection between atomic theory and cosmology.
* The recession velocities are not strictly proved, since one may postulate some other cause for the spectral red-shift. However, the new cause would presumably be equally drastic in its effect on
cosmological theory and would still need the introduction of a parameter of the order 2 x 109 years for its mathematical discussion, so it would probably not disturb the essential ideas of the
argument in the text. | {"url":"http://www.damtp.cam.ac.uk/events/strings02/dirac/speach.html","timestamp":"2014-04-18T18:31:44Z","content_type":null,"content_length":"22947","record_id":"<urn:uuid:37c8fd46-04e9-4a0c-8aa3-b0b34328b8bc>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00448-ip-10-147-4-33.ec2.internal.warc.gz"} |
[SciPy-Dev] Numpy special methods and NotImplemented vs. TypeError
Hoyt Koepke hoytak@stat.washington....
Tue Mar 8 15:57:33 CST 2011
I've been working on implementing a matrix-like class of symbolic
objects that have to interact seamlessly with numpy arrays. It's a
wrapper for CPlex's Concert library, which effectively represents
variables in an optimization problem as symbolic C++ objects that can
be used to build the model. I'm trying to wrap this to work with
numpy -- I'd be happy to share my code when it's done -- and I'm
running into a few problems.
The main issue I'm working against is that when ndarray's special
methods, e.g. __add__, __gt__, etc. encounter a type they don't
understand, they raise a TypeError instead of a NotImplemented
exception. This is a problem as the corresponding right-operator
methods (__radd__, __le__) are only called under three conditions
1. When the left object doesn't implement the left operator.
2. When the left object raises a NotImplemented exception.
3. When the right object is a subclass of the left object. In this
case, the left operator method is never called.
Consider working with A + X, where A is an ndarray and X is an array
representing my symbolic objects, and let's suppose my matrix class
implements __radd__ sensibly. Obviously, 1. doesn't apply here. If I
go with 2., a TypeError is raised, which blows things apart. So the
only solution I have is 3.
However, it isn't possible for me to use the underlying array
machinery by, for example, keeping a numpy array of python objects,
each wrapping a single symbolic variable. This is due to space /
speed constraints and the fact that what I'm really doing is wrapping
a given CPlex array class of these symbolic objects. Thus, if I
subclass ndarray, the underlying array is meaningless -- while I do
implement slicing and various other array-like methods, the only
reason I'm subclassing ndarray is (3). The big problem with this is
that many of the attributes and methods of ndarray don't make sense
for my class, so I have to basically cover them up by overriding them
with methods that simply raise exceptions. Doable, but far from
So my main question is this: Is there any reason that ndarrays throw
a TypeError in this case instead of NotImplemented? It seems like the
latter would be more standards compliant.
+ Hoyt Koepke
+ University of Washington Department of Statistics
+ http://www.stat.washington.edu/~hoytak/
+ hoytak@gmail.com
More information about the SciPy-Dev mailing list | {"url":"http://mail.scipy.org/pipermail/scipy-dev/2011-March/016087.html","timestamp":"2014-04-17T10:09:37Z","content_type":null,"content_length":"5294","record_id":"<urn:uuid:4fe1b7e3-e303-4ef2-a5fe-bc94c1b3aead>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00362-ip-10-147-4-33.ec2.internal.warc.gz"} |
A very disturbing question...
If you choose an answer to this question at random, what is the chance you will be correct?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: A very disturbing question...
Hmmm, looks like a paradox.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: A very disturbing question...
Oh that's a great one! It seems to me that this is caused by the self referential nature of the question. It's a bit like the question "Am I lying now?"
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Re: A very disturbing question...
Agnishom wrote:
If you choose an answer to this question at random, what is the chance you will be correct?
I would want to change this to:
If you choose an answer to this question at random, what is the chance you will be correct?
Even better
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Super Member
Re: A very disturbing question...
maybe 50% because I am very good at guessing but at times it maybe 0% too...I am not sure..
Jake is Alice's father, Jake is the ________ of Alice's father?
Why is T called island letter?
think, think, think and don't get up with a solution...
Re: A very disturbing question...
That made it much easier to calculate. According to Weakipedia it is all of the above.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: A very disturbing question...
Maybe because it is Weakipedia?
Bobbym, do you think that #4 makes the paradox more stronger ruling out none of the above?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: A very disturbing question...
It is possible to have an UNSOUND multiple choice rather than simply one that could be had to have "a non 100% percent chance
of you being considered correct by an infinitely* intelligent (imagined) person". (* = impossible in practice)
I have encountered this concept before, but for obvious reasons I shall not give a real example.
However I could easily make up a deliberately flawed multiple choice question about a factual subject:
What is the capital of France:
(1) Red
(2) Blue
(3) Purple
(4) Orange
Well none of them because all of those are colours, and we wanted a town for strarters, and we know that Paris is correct.
(Maybe there is a very very small chance that one could be argued to be correct due to a highly eccentric pseudo-course,
but if it were an ordinary thing like an exam then it would be an accidental error and considered unsound.
Bad for an exam because no student could ever be certain of getting the answer right about a matter of fact,
but on the other hand perhaps a strange history course might have said that a colour were the answer at some
point in history. Obviously this is NOT true, but if it were then it might be possible if that insight were both known and true.)
The question is not simply difficult, but impossible, because none of the choices is a satisfactory answer.
I would say that although you have created a very clever and amusing brain teaser in mathematics my answer would be:
Just like the above answer, although there is a paradox in yours and no paradox in mine, your question is a deliberately
unsound multiple choice in which is very possible that if you were to take a survey and work out a rough estimate of the
number of people that give the answers shown you could very easily get no answer at all being correct.
On the other hand: I suppose it could be that by an amazing coincidence one of them IS in fact correct fundamentally,
in which I may have to work upon my own definitions to make that better because otherwise a rare exception might
give a valid answer, but you would have to be Mystic Meg* to know what the answer would be.
(Not allowed for a hypothetical exam, but in terms of a brain teaser the questioner has a "licence" to think creatively.)
* = Replace with any fortune teller name of person thought of as able to predict the future precisely.
Mystic Meg is often used informally in the UK to refer to an imagined person able to predict the future to be able to
say "I'm not Mystic Meg" instead of "I am not a magician" and so on in appreciation of an astrologer with this psuedonym.
(Some of my questions are unsound by accident to be fair. None of us are right always.)
Last edited by SteveB (2013-07-11 03:36:33)
Real Member
Re: A very disturbing question...
SteveB wrote:
Mystic Meg
Over here we often use the name "baba Vanga", who was a real person, though.
Last edited by anonimnystefy (2013-07-11 03:53:23)
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: A very disturbing question...
SteveB, did you read post 4? If you say that all of them are wrong, then it is 0%, so you may not say that either
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: A very disturbing question...
Agnishom wrote:
SteveB, did you read post 4? If you say that all of them are wrong, then it is 0%, so you may not say that either
Yes that was a good touch, because it creates a recursive problem where first 0% appears a good answer, then because it
could be argued to be correct then becomes wrong.
I sometimes resort to statistical answer at this point like "25% on the grounds of there being four answers", but you have
predicted that answer by making 25% two choices rather than one !! Obviously not possible for an exam question, but
if you have a creative licence then the questioner can do this as a brain teaser, joke or something.
(It reminds me of deadlock and livelock in Computer Science.
From memory many years ago I think that deadlock was more than one process queued "waiting for each other" in some way.
I think that livelock was supposed to mean an indefinite wait which is either rather too long, or held up for ever if made
an accidentally infinite wait. Those two defintions may not be strictly correct because I cannot remember the full details.)
I did think of another statistical way to define the answer based upon "a survey of the entire population" to give meaning to
the probability of getting the answer right, but it is personalised to the reader by the fact that it is whether "I" can get the
answer right or not. The answer to that is therefore not dependent upon the population (also there are about 7 billion of them
I doubt if all of them will be able to read, some may be born at any time etc etc etc. so of course you cannot ever literally
survey the entire planet or universe obviously).
Okay then can I survey myself. So do I have to answer the question an infinite number of times and give a precise answer
depending upon whether I am right or not in each case, and who do I ask to determine whether I am right? I cannot even
be sure whether I am right in ONE case let alone an indefinite number (infinity etc.) so I could estimate the probability
according to a hideous arbitrary function....
It also reminds me of "the set of all sets" which is defined to be not an allowable set because apart from being way too
big even for the number "infinite" to really sum the thing up if it contains the set containing all subsets including the
set itself then since it contains ALL things it must already contain sets that it could not have contained initially.
I don't think I can explain that one properly without looking up the reason given usually, but it would not be difficult
to make up a proof by contradiction that the set of all sets does not equal itself and therefore is not a well defined set.
I suppose you could perhaps if you created a "foundations of mathematics" style structure for an acceptable multiple
choice question reject your multiple choice on the grounds that the correct answer does not equal itself.
I think a paradox is when two true things appear to contradict each other.
What is even more funny is that if you did want to put this in a maths exam you would probably have to nest the multiple
choice bit within a question making clear that an answer was not enough, and that full marks would only be given for a
full answer not just literally "abracadabra: the answer is .... ".
Multiple choice has it's place in an exam, but imagine if they nest a multiple choice within another, testing the person
spotting the paradox and then choosing the correct reason why the thing is a paradox. That would be a good one.
Last edited by SteveB (2013-07-11 05:01:52)
Re: A very disturbing question...
Okay here goes my attempt at a strange nested multiple choice question:
(I may have to work on this one a bit because this is rather improvised.)
If this is a multiple choice question:
If this is a multiple question:
What is the probability of a correct answer to this
(1) 25%
(2) 0%
(3) 25%
(4) 50%
What is the correct reason:
(1) None of the available options are 100%
(2) It is a recursive paradox
(3) It is impossible, but 0% is an answer
(4) 25% is mentioned twice ambiguously
(5) 50% is also mentioned in case you are arguing that since 25% is there twice the answer is 50% this is also there so is 100% correct if this is your argument. 33.3 recurring is omitted.
(6) All of those
(7) The probability of this question being sound is not very high
(8) The probability of this question being answered right is not great
(9) Sorry that one was a comment not an option.
For the above how likely are you to get the answer to the outer bit right as a percentage?
I might have to work on that. Face it if you have chosen an answer then you have decided that it is 100% correct. You are therefore right and also wrong.
They should have that as an entrance question for something.
I always thought that mathematicians told the best jokes. That was a bad question on a number of levels.
Last edited by SteveB (2013-07-11 05:40:44)
Real Member
Re: A very disturbing question...
Agnishom wrote:
If you choose an answer to this question at random, what is the chance you will be correct?
One more answer could be "it depends on how we choose, i.e. what the distribution of the random choice is".
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: A very disturbing question...
Bobbym, do you think that #4 makes the paradox more stronger ruling out none of the above?
I think the first question was okay, although I had different reasons for why it has no answer.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: A very disturbing question...
julianthemath wrote:
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=277730","timestamp":"2014-04-18T13:17:24Z","content_type":null,"content_length":"35316","record_id":"<urn:uuid:a3567458-ae4b-411a-8d79-34723675718b>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00391-ip-10-147-4-33.ec2.internal.warc.gz"} |
algebraic steps
December 3rd 2009, 04:30 PM #1
Nov 2009
Hello, I cannot recall my algebra.
how does this go from
n(n+1)/2 +(n+1)
n(n+1) + 2(n+1)/2
I'm missing a step in between. Can anyone show me? thank you.
Hi there.
Do you mean:
$\frac{n(n+1)}{2}+(n+1)$ ?
If so, you can't change it to $n(n+1)+\frac{2(n+1)}{2}$, as the first term is now different.
You could however change it to $\frac{n(n+1)}{2}+\frac{2(n+1)}{2}$, as you are just multiplying the second term by $\frac{2}{2}$ which equals $1$.
Hope this helps
it does, thank you
December 3rd 2009, 04:40 PM #2
December 3rd 2009, 04:42 PM #3
Nov 2009 | {"url":"http://mathhelpforum.com/algebra/118346-algebraic-steps.html","timestamp":"2014-04-17T07:30:32Z","content_type":null,"content_length":"34597","record_id":"<urn:uuid:eb6259ec-2841-4f6f-b1b6-b477bb2ddbd3>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00033-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Research Interests
My primary research area is 3-dimensional topology and knot theory. I am verry interested in invariants of 3-manifolds and knots that are constructed using gauge theory, such as Floer homology, the
Casson invariant, and SU(n) generalizations of these invariants. I am also interested in the topological and smooth knot concordance groups and Casson-Gordon invariants, untwisted and twisted
Alexander polynomials, Cochran-Teichner-Orr invariants and invariants arising from Heegaard Floer theory which shed light on knot concordance and sliceness.
My work in higher rank Casson invariants has also given rise to an interest in equivariant transversality problems. Many standard foundational results about manifolds and maps between them become
either much trickier, not known, or just plain untrue if one insists that the functions preserve some symmetries of the manifolds. In other words, there are many good problems left to work on in this
I have also been involved in several interdisciplinary projects recently.
• A project with Kevin Facemyer of the Department of Biochemistry and Molecular Biology studying the way proteins bind together.
• A project with Hans Moosmuller, Rajan Chakrabarty, and Mark Garro, of DRI and the Chemical Physics Program, to model soot particle formation.
• A project with Richard A. Wirtz, Kiran Balantrapu, and Deepti Sarde of UNR's Mechanical Engineering Department, concerning new designs for heat exchangers.
Below are my publications, some of which can be downloaded in pdf form.
Preprints and Publications
Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation, with P. Kirk and C. Livingston. (submitted)
FracMAP: A User-Interactive Package for Performing Simulation and Morphological Analysis of Fractal-Like Solid Agglomerates, with R. Chakrabarty, M. Garro, and H. Moosmuller. (submitted)
Thermal/Fluid Characteristics of Elliptic Cross Section Filament Box Lattice Matrices as Heat Exchanger Surfaces, with D. Sarde, and R. Wirtz,Paper #3810, AIAA Thermophysics and Heat Transfer
Conference, San Francisco, CA, June 6-8, 2006.
Porosity, specific surface area and effective thermal conductivity of anisotropic open cell lattice structures, with K. Balantrapu, D. Sarde, and R. Wirtz, paper IPACK2005-73191, Proceedings of
ASME InterPack Conference 2005, July 17-22.
On the integer valued SU(3) Casson invariant, with H. Boden and P. Kirk, Topology and geometry of manifolds (Athens, GA, 2001), 209--236, Proc. Sympos. Pure Math., 71 , Amer. Math. Soc.,
Providence, RI, 2003.
SU(3) generalizations of the Casson invariant from gauge theory, Geometry, integrability, and quantization (Varna, 2001), 278--289, Coral Press Sci. Publ., Sofia, 2002.
Chern-Simons gauge theory on 3-manifolds, The Proceedings of the Conference on Geometric Structures on Manifolds (Seoul 1997), Lecture Notes Ser. No. 46, Seoul Nat. Univ., Seoul 1999.
Existence of irreducible representations for knot complements with nonconstant equivariant signature, Math. Ann. 309 (1997) 21—35.
Flat connections, the Alexander invariant, and Casson's invariant, Comm. Anal. Geom., 5 (1997) 93—120.
Legendrian cobordism and Chern-Simons theory on 3-manifolds with boundary, Comm. Anal. Geom., 2 (1994) 337—413.
last updated October 29, 2008
return to Herald| Faculty Page| Math Dept or UNR | {"url":"http://wolfweb.unr.edu/homepage/herald/research.html","timestamp":"2014-04-18T10:38:47Z","content_type":null,"content_length":"6916","record_id":"<urn:uuid:5767633a-f1ee-4ddf-a88f-a7ebc4f74144>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00593-ip-10-147-4-33.ec2.internal.warc.gz"} |
Working with Shapes
Students review basic geometric terms related to triangles. They explore these terms and other geometric concepts by modeling them on the geoboard.
To assess students' prior geometric knowledge, make a KWL Chart as a group. Record on the overhead projector or on a chart in the classroom. You and the students can refer to this chart in the
Compare this list with the learning objectives for this lesson. This focuses students’ attention on the mathematical content of the lesson.
Direct students to the Virtual Geoboard, part of the NCTM E-Examples on the Principles and Standards Web site.
Virtual Geoboard E-Example
Give students at least five minutes to make designs on their own using the geoboards. Tell them that this is the time to experiment and explore. Ask students to share what they learned from the
activity. Then have them clear their boards.
Ask students to model geometric figures and discuss the online questions after creating each figure.
Note: It is important for students to process what they are doing to create the various figures so that they will understand the relationships among the figures. Talking helps students clarify and
extend their thinking. It also helps you understand what students are learning.
Ask students to make a three-sided figure on their personal geoboards. Then, pose the following questions to guide students’ thinking about the properties of this three-sided figure.
• What is the name of this figure? [triangle]
• How many vertices does this figure have? [3]
• Show me an example of a vertex by holding your geoboard up and pointing to the vertex. Model this on the Virtual Geoboard E-Example.
• What is the relationship between the number of sides and the number of vertices?
Ask students to make a four-sided figure on their geoboards. Then ask the following questions to guide students’ thinking about the properties of this four-sided figure.
• What is the name of this figure? [quadrilateral]
• Why is it so named? Are the sides of your quadrilateral the same? Why or why not? This discussion helps focus on the equilateral quadrilateral.
• How many vertices does this figure have? [4]
• How do the four-sided figures compare with the three-sided figures? What are the primary similarities? What are the major differences?
Ask students to show you a right angle on the Virtual Geoboard E-Example using two rubber bands. Pose the following questions to guide their thinking about the properties of the right angle.
• How would you describe this shape to a partner who has never seen such a shape?
• What words can you use to describe the right angle? How is an understanding of these words important to an understanding of the shape you created?
• How is an understanding of these words important to an understanding of other shapes you might make?
Ask students to show you an acute angle on the Virtual Geoboard E-Example using two rubber bands.
• How would you describe this shape to a partner who has never seen such a shape?
• What words can you use to describe the acute angle?
• How does this activity compare with the previous ones?
• Why is it important to know the difference between acute and right angles?
Ask students to show you an obtuse angle on the Virtual Geoboard E-Example using two rubber bands. Ask a volunteer to model how to construct an obtuse angle using the virtual geoboard.
• How would you describe this shape to a partner who has never seen such a shape?
• What words can you use to describe the obtuse angle? How is an understanding of these words important to an understanding of the shape you created?
• Why is it important to know the difference between acute, right, and obtuse angles?
• What important knowledge is necessary for you to describe the difference among the three types of angles?
When students have made and discussed each of the figures, ask them to discuss the similarities among the figures. It may be helpful for students to display a sample of each figure or angle during
this discussion.
Encourage students to consider the common properties that define each figure. Ask them to pose any questions they may have about each figure.
Bring closure to the lesson by having students write a brief summary of what they have learned and the questions they want to explore. Collect these work samples so that you can review and identify
students’ knowledge and understanding to guide the next instructional activities.
1. At this stage of the unit, students should know how to:
□ use geometric vocabulary correctly
□ identify, compare, and analyze characteristics of geometric shapes
□ recognize acute, obtuse, and right angles
2. You may find it helpful to make notes about the models students create on the geoboards. This is a good assessment tool for determining which students have met the objectives for this lesson and
which have not.
3. You may also want to review and make notes about the written samples regarding student’s understanding and/or misunderstanding of the mathematical content of the lesson.
1. Pair students and have them take turns making shapes. Have each partner identify the vertices, numbers of sides, types of lines (parallel, perpendicular, intersecting), and types of angles.
2. Students could also experiment with symmetrical shapes and have their partners identify lines of symmetry. Students can make a shape and their partners can make a similar and/or congruent shape.
Shapes could be drawn on dot paper.
Questions for Students
1. What properties does a triangle have?
[3 sides, 3 angles, closed figure.]
2. Name all the different quadrilaterals and the characteristics they exhibit.
[Student responses may include parallelogram, rectangle, square, rhombus, and trapezoid.]
3. What are the three types of angles? Give examples of figures that have these types of angles.
[The types of angles are actue, right, obtuse; student examples may vary.]
4. Describe the three types of lines (parallel, intersecting, and perpendicular) and give examples of each.
[Parallel: lines never meet and are always the same distance apart; Intersecting: two or more lines that meet; Perpendicular: intersecting lines that meet at right angles; Student examples may
Teacher Reflection
• Which students were able to construct each of the different figures?
• What knowledge of lines and angles did students use to help them form specific shapes?
• How did modeling with the virtual geoboard help students understand the objectives of this lesson? What indicators did I have that students benefited from this modeling?
• What strategies did I use to effectively assess and document each student’s understanding as he or she held up the geoboard?
• What experiences did I provide to extend the assignment for students who understood the lesson components?
Students continue to explore geometric concepts by modeling on the geoboard. Communication is the Process Standard emphasized in this lesson.
This lesson provides students with an exploration of the geometric figures Wassily Kandinsky used in his art. Students participate in a scavenger hunt to become familiar with Kandinsky’s works and
the geometric figures used in his paintings.
Students use paintings studied in the previous lesson to connect their knowledge of geometric shapes and terms with Kandinsky’s use of geometric figures.
Students identify lines of symmetry and congruent figures. They explore these concepts with paper cutting and modeling on the geoboard.
This lesson allows students to apply what they have learned in previous lessons by designing their own art. Students use Kandinsky’s style of art and their own creativity to make paintings that
reflect their understanding of geometry.
Learning Objectives
Students will:
• Use geometric vocabulary
• Identify, compare, and analyze characteristics of geometric shapes
• Recognize acute, obtuse, and right angles
Common Core State Standards – Mathematics
Grade 5, Geometry
• CCSS.Math.Content.5.G.B.4
Classify two-dimensional figures in a hierarchy based on properties. | {"url":"http://illuminations.nctm.org/Lesson.aspx?id=1794","timestamp":"2014-04-20T05:42:19Z","content_type":null,"content_length":"81365","record_id":"<urn:uuid:fdbbfc9e-7089-42b0-bc27-be3204b2452e>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00532-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mixture/Solution Problem 2
January 13th 2011, 04:46 AM #1
Jan 2011
Mixture/Solution Problem 2
Hi All,
One term I'm not getting in the below..
Q: 5 litres of water is added to a certain quantity of pure milk costing $3 per litre. If, by selling the mixture at the same price as before a profit of 20% is made, what is the amount of pure
milk in the mixture?
A: Ok so we have 3x as our cost which should be equal to 1.2(3x) (or 120% of 3x given profit). I would imagine the expression therefore to be 3x + 5 = 1.2(3x). However the solution is 3x + 15 =
1.2(3x). Why 15? (I realize this is 3*5 litres of water but not sure why!).
As usual thanks in advance,
Hello, dumluck!
Q: 5 litres of water is added to a certain quantity of pure milk costing $3 per litre.
If, by selling the mixture at the same price as before, a profit of 20% is made,
what is the amount of pure milk in the mixture?
A: Ok so we have 3x as our cost which should be equal to 1.2(3x)
. . (or 120% of 3x given profit).
I would imagine the expression therefore to be 3x + 5 = 1.2(3x). . No
We have $\,x$ litres of milk.
We add $5$ litres of water.
We have: $x + 5$ liters of mixture, which will sold at $3 per litre.
Its value is: . $3(x+5)$ dollars.
January 13th 2011, 05:21 AM #2
Super Member
May 2006
Lexington, MA (USA)
January 13th 2011, 05:23 AM #3
Jan 2011 | {"url":"http://mathhelpforum.com/algebra/168217-mixture-solution-problem-2-a.html","timestamp":"2014-04-21T08:05:14Z","content_type":null,"content_length":"37756","record_id":"<urn:uuid:906cf4d2-0d16-4b9f-95c2-dc249299ef15>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00604-ip-10-147-4-33.ec2.internal.warc.gz"} |
Modelling and problem solving - quadratic equations
January 7th 2013, 07:32 PM #1
Junior Member
Jan 2013
Gold coast
Modelling and problem solving - quadratic equations
Can you help with the following problem?
Tom leaves town A and rides at a constant speed towards B. Julie leaves town B at the same time that Tom leaves town A, and travels towards town A. Julie rides at a speed that is 5km/h faster
than Tom. Town B is 100 km from town A. They meet after 4 hours. How far has Julie travelled at the time that they meet?
January 7th 2013, 08:21 PM #2
Super Member
Jul 2012 | {"url":"http://mathhelpforum.com/algebra/210949-modelling-problem-solving-quadratic-equations.html","timestamp":"2014-04-19T23:44:08Z","content_type":null,"content_length":"32207","record_id":"<urn:uuid:0f1c83a7-2451-42a3-bd1f-905898549fdd>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00592-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: What changes a NHST into a No-NHST?
Replies: 2 Last Post: Dec 15, 2012 5:55 PM
Messages: [ Previous | Next ]
Luis A. Re: What changes a NHST into a No-NHST?
Afonso Posted: Dec 15, 2012 5:55 PM
Posts: This post aims to present some remarks on
4,518 http://www.andrews.edu/~rbailey/Chapter%20two/7217331.pdf
From: Problems on Null Hypotheses Significance Testing (NHST)
LIsbon . . . Jeffrey A. Gliner, Nancy L. Leech, George A. Morgan.
Registered: The A. do start to point out that critics came mainly from people of Psychology and Education. We add: not Statisticians or Mathematicians, areas where, I think, more accurate/learned
2/16/05 people is liable to be found.
They elect the NHST major problem being, quoting Kirk, that it do not provide what the researcher finally wants to know: if the null can pass given the observed data (Bayes), not if data
shouldn?t be rejected under the hypotheses. Not to object. Simply the Classic paradigm is that: to reject H0 is because it is false, or, being true, the probability to get data as the
observed is very small. Because we set this probability as 5%, we do not make this Type I error 19 times out of 20.
Quoting Kirk again: because the Null is always false, and with sufficient data one will surely to accept it, to test hypotheses is a trivial exercise.
WRONG: we must not get, H0: t=0, as an arithmetic statement but else that we are not (if so) are unable to make distinction by the test if the parameter is different than 0. The second
part is true, but we ask: what the usefulness is if only n= 1´000´000´000 (one thousand million) sample size is able to assure that t is just larger than d= 0.000´000´001? Or, else, n is
the number of flips enough to warrant that p= 0.5 + d is the probability heads up?
Continuing reading the paper we agree completely with Cohen and Mulaik, Raju & Harshman that ?a no difference should be replaced by a null hypotheses specifying a non-zero value based on
previous research. And we add: or connected with the ?practical/economic importance? attached to the new finding.
Luis A. Afonso
Date Subject Author
12/13/12 What changes a NHST into a No-NHST? Luis A. Afonso
12/14/12 Re: What changes a NHST into a No-NHST? Luis A. Afonso
12/15/12 Re: What changes a NHST into a No-NHST? Luis A. Afonso | {"url":"http://mathforum.org/kb/message.jspa?messageID=7937343","timestamp":"2014-04-20T00:00:08Z","content_type":null,"content_length":"20246","record_id":"<urn:uuid:23e3670a-5945-4071-a1e9-afe65ddef0d9>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00170-ip-10-147-4-33.ec2.internal.warc.gz"} |
A function is a special kind of relation where each thing in the domain may be paired with only one thing in the range. These guys are all about monogamous relationships.
Sample Problem
Here's a relation whose domain is {Jane, John, Jill} and whose range is {Home, Store}. Usually we see numbers and variables in equations rather than full-on words, but this will help you visualize
how functions work. Plus, you could probably use a change of pace.
We could think of this relation as telling us where each person is:
{(Jane, Home), (John, Home), (Jill, Store)}.
We have a function, since each thing in the domain occurs with only one thing in the range. John could not be at Home and at the Store at the same time, as much as he would like to be. Not enough
hours in the day, eh John?
The relation {(Jane, Home), (John, Home), (Jane, Store)}
is not a function, because Jane is paired with both Home and Store. Jane can't be in two places at once, and something from the domain of a function can't be paired with two things from the range
simultaneously. It's okay for both Jane and John to be at Home, though. Seemingly, they are Home on the range.
Other Examples
{(1, 2), (3, 2), (4, 2)} is a function. Each number in the domain ({1, 3, 4}) occurs with only one number in the range. It's okay to reuse the 2 from the range, as in the last example where both Jane
and John were at Home. There's enough 2 to go around for everybody.
On the other hand, the relation {(1, 2), (1, 3)} is not a function, since the number 1 in the domain is being paired with both 2 and 3. Way to follow the rules, 1. Didn't you pay any attention to the
previous examples, buddy?
Since functions are relations, we can sometimes describe functions using equations also. This situation is one of the only ones in which something can be sufficiently described using equations. If
trying to describe the physical appearance of a suspect to a police officer, stick to English. suspect = beard^2 – hat will only confuse him.
More Examples
The equation y = x + 2 describes a function because, for any value of x, there's only one value of y that will satisfy this equation. If y wasn't around, x couldn't get no... satisfaction.
The equation x = y^2 describes a relation that is not a function, because each value of x (except 0) can be matched with two values of y.
For example, the ordered pairs (4, 2) and (4, -2) are both in the relation described by the equation x= y^2. This means x = 4 is getting matched with both y = 2 and y = -2, and a function isn't
allowed to let such a thing happen. This rule is explicitly stated in the Complete Function Handbook, Rule 14C, so it should know better.
Functions Practice:
What equation describes the connection between x and y in the function {(1, 3), (5, 7)}?
Is the following relation a function? If not, why?
{(Juan, escuela), (Mariposa, escuela), (Fernando, casa), (Maria, tienda)}
Is the following relation a function? If not, why?
{(Juan, escuela), (Mariposa, escuela), (Juan, casa), (Maria, tienda)}
Is the following relation a function? If not, why?
{(1, 1), (1, 2), (1, 3)}
Is the following relation a function? If not, why?
{(1, 1), (1, 1), (2, 3)}
Is the following relation a function? If not, why?
y = 3x^2 + 4
Is the following relation a function? If not, why?
x = |y|
Is the following relation a function? If not, why?
x = y^2 where y is positive.
Is the following relation a function? If not, why?
x^2 = y^2 | {"url":"http://www.shmoop.com/functions/functions-help.html","timestamp":"2014-04-16T13:20:24Z","content_type":null,"content_length":"46153","record_id":"<urn:uuid:0e21bfad-4a87-4a51-8f3d-41cad5116dea>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00358-ip-10-147-4-33.ec2.internal.warc.gz"} |
Seperable Intial-Value Problem
January 30th 2011, 09:38 AM
Seperable Intial-Value Problem
$(x^2+3y^2)dx + (2xy)dy = 0$
so this homogeneous eqn. needs to be transformed into a separable eqn. right?
$y=vx , v=\frac{y}{x}$
$\frac{dy}{dx}=\frac{x^2+3y^2}{2xy}$ or is it $\frac{x^2+3y^2}{2x}dx - \frac{y}{1}dy$= How can I plug in $\frac{y}{x}$ here ?
$\frac {dy}{dx}= v + x\frac{dv}{dx}$
Not sure how to solve this one.
January 30th 2011, 09:53 AM
$(x^2+3y^2)dx + (2xy)dy = 0$
so this homogeneous eqn. needs to be transformed into a separable eqn. right?
$y=vx , v=\frac{y}{x}$
$\frac{dy}{dx}=\frac{x^2+3y^2}{2xy}$= How can I plug in $\frac{y}{x}$ here ?
$\frac {dy}{dx}= v + x\frac{dv}{dx}$
Not sure how to solve this one.
Multiply the numerator and denominator by
This gives
$\displaystyle \frac{1+3\left( \frac{y}{x}\right)^2}{2\left( \frac{y}{x}\right)}$
February 4th 2011, 05:00 AM | {"url":"http://mathhelpforum.com/differential-equations/169719-seperable-intial-value-problem-print.html","timestamp":"2014-04-16T13:15:55Z","content_type":null,"content_length":"8734","record_id":"<urn:uuid:d635321e-784c-46f4-84ab-4cbe25fcd385>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00593-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: F-test for comparing variances?
Replies: 4 Last Post: Nov 10, 2006 1:04 AM
Messages: [ Previous | Next ]
Re: F-test for comparing variances?
Posted: Nov 8, 2006 11:01 PM
On 8 Nov 2006 11:08:22 -0800, "Reef Fish"
<large_nassua_grouper@yahoo.com> wrote:
> Richard Ulrich wrote:
> > On 5 Nov 2006 17:55:46 -0800, "Reef Fish"
> > <large_nassua_grouper@yahoo.com> wrote:
> > >
RU > >
> > umm. You still need to get a test value, subtract, then double....
> > which I described usefully. Bob was not paying attention, too eager
> > to criticize.
RF >
> WRONG, WRONG, WRONG.
> 1. For testing with a given alpha, such as a two-tailed test with F,
> you find F(.025) and F(.975) before you get the test value,
> because
> you DON'T have to see if the numerator is greater or smaller. You
> KNOW the degrees of freedom of the F.
I can apologize, and clarify.
No doubt I confused Bob by saying "test value" instead
of saying "two-tailed p-value."
If you want a two-tailed p-value -- which is the modern
standard, instead of simply 'flagging' the test result --
you need to double the smaller extreme.
Here are a *couple* of conventions -- presenting p-values;
and doubling a one-tailed value to get a two-tailed value.
RU >
> > By now, he should recognize that he *often* does not understand
> > what's being said. Why else is it, that every 'fight' in the stat
> > groups features Bob?
RF >
> Everytime you say I don't understand, it's always because I was at
> several steps AHEAD of you and understand every word of yours to
> be ERRORS resulting from your lack of education.
Your next step, Bob, is to consider how to compute the "p-value",
which everyone wants these days. Catch up.
> You REMAIN ignorant, and a malpracticing Quack, Richard Ulrich.
That remains absurd. Stream-of-consciousness garbage.
> That may be an insult you deserve -- but hardly gratuitous!
Yes. Gratuitous. Nobody asks for this, and 10,000 readers do
not want to see it. *If* I were so obviously 'ignorant', nobody
would *need* Reef Fish Bob Ling to belabor the point.
The only 'end' that it seems to serve is an ongoing intention
to bully and intimidate. Bad net-citizenship.
Rich Ulrich, wpilib@pitt.edu
Date Subject Author
11/8/06 Re: F-test for comparing variances? Richard Ulrich
11/10/06 Re: F-test for comparing variances? Richard Ulrich | {"url":"http://mathforum.org/kb/message.jspa?messageID=5370862","timestamp":"2014-04-18T09:22:41Z","content_type":null,"content_length":"19750","record_id":"<urn:uuid:ab4f1c12-2ccf-49d7-a660-2020389ca8c3>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00011-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mplus Discussion >> Defining outcomes as categorical vs. continuous
Lois Downey posted on Monday, October 13, 2008 - 5:31 pm
Our factor indicators are 11-category ordinal measures with only the extreme values defined: 0 = terrible; 10 = almost perfect. Almost all have significant negative skew. In the past, we have modeled
them as categorical indicators and have collapsed values of 0 and 1 into one category to accommodate Mplus's 10-category limit. However, I'm wondering whether it might make more sense to model these
indicators as continuous and use a robust estimator to circumvent the problem of nonnormality. Would this technique be justifiable, given that respondents are given only the anchor definitions and
allowed to impose their own quasi-continuous scale between these extremes? Or should one have more than 11 values before calling something "continuous"?
Linda K. Muthen posted on Tuesday, October 14, 2008 - 9:59 am
If you don't have a large floor or ceiling effect, I would treat the variable as continuous with that many categories. If you do have a large floor or ceiling effect, I would treat the variable as
censored or do two-part modeling.
Lois Downey posted on Tuesday, October 14, 2008 - 11:39 am
What percentage of responses at the minimum or maximum response option should one take as evidence of a "large" floor or ceiling effect?
Linda K. Muthen posted on Tuesday, October 14, 2008 - 11:46 am
Twenty-five percent or more.
Lois Downey posted on Thursday, October 16, 2008 - 11:37 am
When one includes censored indicators, the tests of model fit no longer include those handy statistics like RMSEA that allow comparing the fit of models with different numbers of factors. If I run my
22-indicator model without defining any of the indicators as censored, I get inadequate fit with 5 factors, but adequate fit with 6 factors. (We expected 6 dimensions, based on our theoretical model,
although not all of the 6 factors in the EFA look precisely like the dimensions we anticipated.) Can I use this information as a rough indication that when I redefine 10 of the indicators as censored
from above, I should focus on a 6-factor model?
(p.s. -- Thank you so much for your help to date, and in the future.)
Linda K. Muthen posted on Thursday, October 16, 2008 - 1:07 pm
Which estimator are you using?
Lois Downey posted on Thursday, October 16, 2008 - 1:40 pm
Linda K. Muthen posted on Thursday, October 16, 2008 - 6:36 pm
I was thinking that you could use weighted least squares but I see that is not the case. I would not use fit statistics from treating the variables as continuous when some are censored. I would
instead look at BIC. If you get six factors but they don't resemble what you expected, I might go back to the drawing board. It sounds like your items may have validity problems.
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Chemistry Formulas for Class XI, XII and Under Graduates
Chemistry Formulas for Class XI, XII and Under Graduates 2012
Download Chemistry Formulas For Class XI , Class12 and Under Graduates
Chemistry Formulas:
Ideal Gas law
PV = nRT
n = number of moles
R = universal gas constant = 8.3145 J/mol K
Boyle’s law
P[1]V[1] = P[2]V[2]
Mole fraction: Mole fraction of a component in solution is the number of moles of that component divided by the total number of moles of all components in the solution.
For further information regarding Chemistry Formulas for Class XI, XII and Under Graduates, connect with us on facebook, google+ and twitter.
Tags: chemistry formulas for class 12, 12th chemistry formulas pdf, chemistry formulas for class 11, chat of chemistry formula class11th in pdf, 12 th samacheer chemistry formulas, class 12 physics
all formulas pdf, class 12th chemistry formulas, Formula of thermodynamic7 of chemistry of class 12, formulae of chemistry for class 11th & 12th, formulas for the ideal gases boyles law for chemistry
Please send me all formula of chemistry of 12 class. Numerical based like molarity etc.
Sahil singh on 03 Mar 2013
what is the formula carbon die shulphide
lalit nogiya on 31 Jan 2013
स्थीर अनुपात का नियम क्या है?
sheru on 18 Oct 2012
Send me answers Quickly.I’m Waiting for my answers.
In How many time will you answered my questions? “”DO IT QUICKLY MAN”"
Mayank on 05 Oct 2012
Who Discovered the Laws of Chemical Combination? And how many types of Laws they discovered? Explain the all Laws of Chemical Combination.
Mayank on 05 Oct 2012
Can we separate Sodium and Clorine from salt? If we put some Sodium into Water,What would be happen?Is it changes into fire? If yes,Why?
Mayank on 05 Oct 2012
plz send me all chemistry formulas
of 9 class
ovais nazeer on 09 Sep 2012
what is the molecular mass of AgNO and also tell me the molar mass of caco
ovais nazeer on 09 Sep 2012
Please send me all formulas of
chemistry related to class 11 and all the formulas help me a lot
Ankit Negi on 28 Jun 2012
please send me all formulas of chemistry related to class 12TH AND ENTRY TEST . mostly all the formulas??
INAYATULLAH on 19 Jun 2012
please send me all the formulas like nomenclature elements and or any informations on my id………
Shubham Naidu on 03 May 2012
Please send me all formulas of chemistry related to class 12. mostly all the formulas??
dhananjay sharma on 02 May 2012
please send me all formulas of chemistry related to class 11th and 12th for aieee and eamcet
y.Rajeswari on 24 Apr 2012
send me all formulas of chemistry related class 11th and 12th to mostly all the formulas??
krishna on 02 Jan 2012
please send me all formulas of chemistry related class 11th and 12th to mostly all the formulas??
avinash on 21 Dec 2011
i want paper patern for semister exam.
and totly detail about cbse bord paper
navnath on 18 Sep 2011
kindly send me all formulas of chemistry and physics related class 11th and 12th to mostly all the formulas. im expecting it free of cost please.
rudra on 11 Sep 2011
please send me all formulas of chemistry related class 11th and 12th to mostly all the formulas??
rahul on 03 Sep 2011
please semd me all important chemistry formulas related to class 12. i would be so grateful for it..
damchen tenzing on 25 Jul 2011
please send me all formulas of chemistry related to class 9 mostly all the formulas??
abdullahshaheenkhan on 20 Jun 2011
Q.write 30 important formulas which is give in your chemistry book of 9 class
abdullahshaheenkhan on 20 Jun 2011
APPROXIMATE ATOMIC WEIGHT OF ELEMENT IS 26.89. ITS EQUIVALENT WEIGHT IS 8.9. WHAT IS THE EXACT ATOMIC WEIGHT OF THE ELEMENT.
d p gangwar on 10 Jun 2011
App. Atomic weight of element is 26.89. Its equivalent weight is 8.9.The exact atomic weight is
Abhinav on 03 Jun 2011
is it necessary to write heat and catalyst in equation? Please give answer.
Ahmed on 30 Apr 2011
please send me all formulas of chemistry related to class tenth. mostly all the formulas??
Nishant on 22 Apr 2011
Give reason:
a. The radius of Noble gases are very large in comparison to the halogens
b. Chlorine in spite of similar electro negativity than Nitrogen do not form Hydrogen bonding
ram on 24 Dec 2010 | {"url":"http://examsindia.net/2010/03/chemistry-formulas-for-class-xi-xii-and-under-graduates.html","timestamp":"2014-04-20T10:46:54Z","content_type":null,"content_length":"26075","record_id":"<urn:uuid:2316d6c6-ae78-490f-933d-c444c79f4670>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00115-ip-10-147-4-33.ec2.internal.warc.gz"} |
CDES 120
In this course you will learn to:
• describe child behaviors and teacher-child interactions accurately and objectively within a developmental perspective
• select effective observational techniques to observe, record and report child behavior and interactions
• observe and analyze children's behavior
• apply observational information in working with children through the identification and design of developmentally appropriate activities.
It is strongly reccomended that this course be taken after you have completed CD 107 and CD 110.
Assignments for this course include:
• weekly discussions
• five field observations
• a midterm exam
Required texts:
Mindes, Galye, Assessing Young Children, Pearson 2011 4th Edition. ISBN 0-13-700227-0
McAfee, Basics of Assessment, NAEYC 2004 ISBN 1-928896-18-9
Harms, Clifford and Cryer, Early Childhood Environmental Rating Scale-Revised, Teacher's College Press, ISBN 0807745499
Reccomended text:
Cryer, Harms and Riley , All About the ECERS-R, Teachers College Press, ISBN 088076-610-7 | {"url":"http://saddleback.edu/faculty/btamialis/CDES120.htm","timestamp":"2014-04-19T02:20:28Z","content_type":null,"content_length":"4734","record_id":"<urn:uuid:ab92831b-f63d-472c-b325-3592888841f6>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00001-ip-10-147-4-33.ec2.internal.warc.gz"} |
Dependent variables
34,117pages on
this wiki
Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |
Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory
In experimental design, a dependent variable is a variable dependent on another variable (called the independent variable).
The dependent variable is also known as the response variable, the regressand, the measured variable, the responding variable, the explained variable, or the outcome variable.
In simple terms, the independent variable is said to cause an apparent change in, or simply affect, the dependent variable.
In analysis, researchers usually want to explain why the dependent variable has a given value. In research, the values of a dependent variable in different settings are usually compared.
For example, in a study of how different dosages of a drug are related to the severity of symptoms of a disease, a measure of the severity of the symptoms of the disease is a dependent variable and
the administration of the drug in specified doses is the independent variable. Researcher will compare the different values of the dependent variable (severity of the symptoms) and attempt to draw a
In the graphing of data, the dependent variable goes on the y-axis (see Cartesian coordinates).
See also | {"url":"http://psychology.wikia.com/wiki/Response_variable","timestamp":"2014-04-19T22:55:48Z","content_type":null,"content_length":"57648","record_id":"<urn:uuid:d9e4d731-a228-476f-9197-28d0405ccdb6>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00362-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts by
Posts by Bri
Total # Posts: 195
THANKS. I figured it out.
how do i find the rate constant?
i got N=0.83 does the seem right?
suppose that you place exactly 100 bacteria into a flask containing nutrients for the bacteria and that you find the following data at 37 °C: t (min):0 15.0 30.0 45.0 60.0 Number of bacteria: 100 200
400 800 1600 What is the order of the rate of production of the bacteria?...
Hey Guys so I have this problem with ICE tables that is extremely confusing for me so I would love some help on it! At a Certain Temperature, the equilibrium constant for the reaction of NO with Cl2
is 6250. If the initial concentration of NOCl is 1.0M and that of NO 0.10M wha...
Spanish 2
@SraJMcGin Was #2 correct also? Thanks! :)
Spanish 2
Fill in the blank with the correct imperfect form of the verb. Include accent marks when needed. 1.) Mis amigos______la playa. (ir) 2.) Mama______la casa. (limpiar) 3.) Mi hermano y yo____la leche.
(beber) 4.) Yo_____la cena. (preparar) 5.) ¿Tú_____los libros? (le...
spanish 2
Fill in the blank with the correct imperfect form of the verb. Include accent marks when needed. 1.) Mis amigos______la playa. (ir) 2.) Mama______la casa. (limpiar) 3.) Mi hermano y yo____la leche.
(beber) 4.) Yo_____la cena. (preparar) 5.) ¿Tú_____los libros? (le...
Chem 2
Thanks so so much! I was really confusing myself! The steps really helped!
Chem 2
Im sure i did this wrong but this is what i did deltaT=Kfm m=mols solute/ kg solvent 1.56g vitamin k/83.8g= 0.019 m=0.019/ 0.025 = 0.76
Chem 2
when Im getting my Kf do i change it from Celsius to Kelvin before i do my calculation?
Chem 2
Vitamin K is involved in normal blood clotting. When 1.56 g of vitamin K is dissolved in 25.0 g of camphor, the freezing point of the solution is lowered by 5.23 °C. . Calculate the molar mass of
vitamin K. I posted this yesterday but I got the wrong answer and could use s...
Chem 2
When i did the problem i got 0.76 but i think its wrong?
Chem 2
Vitamin K is involved in normal blood clotting. When 1.56 g of vitamin K is dissolved in 25.0 g of camphor, the freezing point of the solution is lowered by 5.23 °C. . Calculate the molar mass of
vitamin K.
Chem 2
When 3.89 g of a nonelectrolyte solute is dissolved in water to make 835 mL of solution at 26 °C, the solution exerts an osmotic pressure of 825 torr. What is the molar concentration of the solution?
How many moles of solute are in the solution? What is the molar mass of t...
the larger of two numbers is 12 mo're than the smaller.Their sum 84. Find the numbers.
Pre-Cal between 0 degree C and 30 degree C the volume V (in cubic centimeters) of 1kg of water at a temperature T is given by the formula. V=999.87-0.06426T+0.0085043T^2-0.0000679T^3 Find the
temperature at which the volume of 1 kg of water is a minimum.
Don't break the honor code but i can help explain a little try looking up to see if the fossils under the trilobite is older than you will know
what about number 4 is tat also d i am trying to understand evolution much better.
what is negative one sixth times two and three fourths
write an expression equivalent to -1/4(8y-10)
Hi, can I have some help with my permutation/combination homework? "Find the number of 5-card hands that contain the cards specified." 1. five red cards 2. five cards, none are number cards 3. five
cards of the same suit
us history
Why did Johnson request the Tonkin Gulf Resolution
Math- PLEASE HELP
Ok I corrected my mistake and got that it would cost $51.48, because it was 15 gallons she needed to get. Is this correct now? Thank you
Math- PLEASE HELP
MS Wilkens went to the gas station to fill her car. Her gas tank is 20% full. The cost of gas is $4.29 per gallon. How much will itcost to fill the tank? Use the conversion 1 gallon=231 in cubed.
They showed a pic. of a tank with length=1.75ft width=1.25 ft height=11 in I did ...
pre cal
If (a, b) is a point on the unit circle that corresponds to the angle t, then tan(t) equals:
Expand binomial Pascal's triangle
your teacher has invented a fair dice game to play. your teacher will roll one fair eight sided die and you will roll a fair six sided die. each player rolls once and the winner is the person with
the higher number. in case of a tie neither player wins. a. let A be the event &...
In the following reaction, how many grams of ferrous sulfide (FeS) will produce 0.56 grams of iron (III) oxide (Fe2O3)? 4FeS + 7O2 d 2Fe2O3 + 4SO2
A small child has a wagon with a mass f 10 kilograms (KG). The child pulls on the wagon with a force of 2 newtons (N). What is the acceleration of the wagon?
Balance the following equations: A. N2O>N2+O2 B. HBr>H2+Br2 C. CH4+O2>CO2+H2O
What is C2H5OH+3O2>CO2+H2O balanced?
what is the area in square units of the region under the curve of the function f(x)=x+3, on the interval from x=1 to x=3? 10 12 14 16 18 ?
Physics, science,
I'm desperate Please help 2. A bird, starting from rest, accelerates at 3 m/s2 north for 5 seconds. It then sees the rest of its flock and turns due east accelerating from its previous speed to 20 m/
s in 2 seconds. A. What is the velocity of the bird after the initial leg ...
Thanks Deborah i needed help.
A 16 - foot ladder leaning against the side of a house reaches 12 feet up the side of the house. What angle does the ladder make with the ground
thanks !? lol
2 big important facts about the 13 amendment.
u.s. history
thank you soo much(:
u.s. history
What came in order? 1.)abraham lincoln presidency 2.) jefferson davis presidency 3.) battle of bull run 4.) fort sumter
Science Struggle
Rocket-powered sleds are been used to test the responses of humans to acceleration. Starting from rest, one sled can reach a speed of 533 m/s in 2.45 s and can be brought to a stop again in 2.37 s.
Find the acceleration of the sled when braking. Answer in units of m/s^2
Rocket-powered sleds are been used to test the responses of humans to acceleration. Starting from rest, one sled can reach a speed of 533 m/s in 2.45 s and can be brought to a stop again in 2.37 s.
Find the acceleration of the sled while accelerating. Answer in units of m/s2
(A)Acceleration = -1.6 m/s Vi = 13.3 m/s Vf = 0 T(time) = ? T = Vf-Vi/A T = 0-13.3/-1.6 T = 8.3125 s
With an average acceleration of −1.6 m/s^2, how long will it take a cyclist to bring a bicycle with an initial speed of 13.3 m/s to a complete stop? Answer in units of s
A 0.05 kg arrow with the velocity of 180 m/s is used to shoot a 5 kg block of wood off the top of a log. What will be the velocity of the arrow and wood after contact?
A 1 kg poodle running right at 3 m/s has an elastic head on collision with a 9 kg basset hound walking left with a velocity of 1 m/s. After the collision the poodle bounces off to the left at 3 m/s.
Whats the basset hounds velocity after the colllision?
A 1000 kg car moves at 60 m/s. What braking force is needed to halt the car (change its momentum) in 6 s?
State domain and range for f(x) = 2[x]
E=hv h=6.6E-34Js E=5.1E-22k E/h=v 5.1E-22k/6.6E-34Js =
What would be the final concentration if 175mL of a 0.45M solution of sodium sulfite Na2SO3 are added to 650mL of water? Please reply with the steps on how to do this question so I can learn. I can't
find examples or anything related in my textbooks and I take it via corre...
If 125mL of a 1.10 NaCl solution is mixed with 105mL of a 0.850M CaCl2 solution but NO reaction takes place, calculate the final concentration of each of the ions in the final solution. Please reply
with the steps on how to do this question so I can learn. I can't find exa...
A 15.0mL solution of H2SO3 is neutralized by 12.0mL of a standardized 1.00M NaOH solution. What is the concentration of the H2SO3 solution? Please reply with the steps on how to do this question so I
can learn. I can't find examples or anything related in my textbooks and ...
The reaction between 34.0 g of NH3 and excess oxygen gas produces 45.9 g of NO gas and some water. Determine the percent yield.
The reaction between 34.0 g of NH3 and excess oxygen gas produces 45.9 g of NO gas and some water. Determine the percent yield.
english 1 honors
In the writer by richard wilbur what does the speaker compare his daughter to?
If AB= 9 centimeters and BC= 12 what does Ac equal ? and it is a right triangle .
C. now answer my question please !
general chem
A certain monoprotic weak acid with Ka = 0.49 can be used in various industrial processes. (a) What is the [H+] for a 0.191 M aqueous solution of this acid and (b) what is its pH? Round the [H+] to
three significant figures and the pH to two places past the decimal and do NOT ...
general chem
A certain monoprotic weak acid with Ka = 0.49 can be used in various industrial processes. (a) What is the [H+] for a 0.191 M aqueous solution of this acid and (b) what is its pH? Round the [H+] to
three significant figures and the pH to two places past the decimal and do NOT ...
40% of the 25 students in the class are boys. Write 40% as a fraction? Then find the ratio of girls to boys
a landscaper uses square tiles to make garden paths the ends of the paths are always 2 tiles wide, and the center portions of the paths are 3 tiles wide. what eqation gives the total number y of
tiles needed to make a path with a length of x tiles. the landscaper has 100 tiles...
Among coffee drinkers, men drink a mean of 2.8 cups per day with a standard deviation of 0.5 cups. Assume the number of drinking per day follows a normal distribution. What proportion drink 3 cups
per day or more?
algebra, please help
(y^2 + 10y + 25)/(y^2 - 9) * (y^2 + 3y)/(y + 5) (y+5)(y+5)/(y+3)(y-3) * y(y+3)/(y+5) (y+5)/(y-3) * y/1 y(y+5)/(y-3) or y^2+5y/y-3
A car is safely negotiating an unbanked circular turn at a speed of 17.2 m/s. The maximum static frictional force acts on the tires. Suddenly a wet patch in the road reduces the maximum static
frictional force by a factor of three. If the car is to continue safely around the c...
Math-Algebra 2
Ace Rent a Car charges a flat fee of $15 and $0.25 a mile for their cars. Acme Rent a Car charges a flat fee of $30 and $0.17 a mile for their cars. Use the following model to find out after how many
miles Ace Rent a Car becomes more expensive than Acme Rent a Car. c= 15+0.25m...
Lcm of 2 and 3
A. m1a1=T-m1g and m2a2=T-m2g are your two equations, substitute and add and you get this --> T= (2xm1xm2)/(m1+m2) all multiplied by 9.8 (g). T= (2(3.0)(7.0))/(3+7) = 4.2(9.8) = 41.16 N B. Then just
plug it in to one of your original equations. m1a1 = T-m1g (3.0)a = (41.16N)...
The sum of two consecutive odd integers is 56. A. Define a variable for the smaller integer. B. What must you add to an odd integer to get the next greater odd integer? C. Write an expression for the
second integer. D. Write and solve an equation to find the two odd integers.
20.0 g of CH3OH is frozen solid at -98.0 degrees C. What is the total energy required to melt it, warm the liquid to its boiling point, and then just vaporize it?
Fraction simplest 12 over 210
Given that ABC 53-x with a leg of 28 ~ DEC, x+3 with a leg of 4 find the value of x. If necessary, round your answer to two decimal places.
list 10 examples of slow changes over a long time. list 10 examples of quick changes over a short time.
the point (-1,6) would be in quadrant 2.
George has a large circular table with a circumference of 18 feet. What is the approximate area of the table?
Assume that you own a small factory. A critical piece of machinery in your factory will need to be replacedin 180 days. If the machinery does not show up on time, you will need to shut down until it
arrives. This might cause you to permanately lose customers.When you order the...
algebra (quadratic formula0
or the period 1990-2003, the amount of biscuits, pastas, and noodles y (in thousands of metric tons)imported into the united states ca be modeled by the function y=1.36(xsquared)+ 27.8x+304 where x
is the number of year since 1990. a.write and solve an equation that you can us...
algebra (parabolas)
you throw a wad of used paper towards a wastebasket from a height of about 1.3 feet above the floor with an initial vertical velocity of 3 feet per second. a. write and graph a function that models
the height h (in feet) of the paper t seconds after it is thrown b. if you miss...
algebra (parabolas)
find the axis of symmetry and vertex of the function. tell whether it goes upward or downward. 1. y=-xsquared+9 2. y=-2xsquared=7x-21
algebra (parabolas)
find the axis of symmetry and vertex of the function. tell whether it goes upward or downward. y=3*xsquared+6x-2
algebra (parabolas)
tell whether the graph opens upward or downward. then find the axis of symmetry and vertex of the graph of the function 1. y=xsquared-5 2. y=-2*xsquared+6x+7
social studies
never mind. a friend helped me.
social studies
my teachers do not want us on Wikipedia.
social studies
1. define Allah 2. the word Muslim means...? 3.JIHAD means to strive... what are Muslims striving for? 4. list and describe 5 contributions of Muslims to our lives today
6. Simplify: 4x2yz2 2z Select answer A 2x2y2z2 Select answer B 2x2z Select answer C 2x2yz2 Select answer D 2x2y Select answer E 2x2yz
social studies
name 2 ways that it might affect our society if no one volunteered to help others.
The value of a certain two-digit number is eight times he sum of its digits. if the digits of the number are reversed the result is 45 less than the original number. find the original number.
**please show both equations and work... thanks =)
6th grade science
What is the 47th element in the periodic table?
please write and equation, show your work, and find the answer: 1.a certain two-digit number has a value that is seven more than six times the sum of its digits. the tens digit is 3 more than the
units digit. find the number 2.In Colorado Creek, Darrell can row 24km downstream...
physical or chemical property sodium metal fohrmhs an oxide when heated with oxygen?
calculate the enthalpy change in delta H foor heating 250 grams of liquid water from 0 degrees celcius to 100 degrees celcius.
I DONT NO
ok thanks!
okaaaay im starting to get it. so what would the second equation be again? and can you please explain it? =) thanks!
ya i did today. but i dont understand how you got 24 as the y-intercept
i do not think that is what my teacher was asking for.
you will be making hanging flower baskets. the plants you have picked out are blooming annuals and non-blooming annuals. the blooming annuals cost $3.20 each and the non-blooming annuals cost $1.50
each. you bought a total of 24 plants for $49.60. write a linear system of equa...
how is water purified in the water cycle? why is the sun so important in the water cycle? what role does evaporation play in the water cycle? thanks!!!! :)
language arts
there are three words to use to fill in each blank... the words are 1. automatic 2. tenant 3.volley here are the sentences: 1. OCCUPANT is to _________ as CHILDISH is to JUVENILE 2. BY HAND is to
_________ as PALE is to BRIGHT 3. BARRAGE is to _________ as TOUGH it to CHALLENG...
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[Numpy-discussion] upcast
Lars Friedrich lfriedri at imtek.de
Wed Aug 30 11:39:43 CDT 2006
I would like to discuss the following code:
import numpy as N
a = N.array((200), dtype = N.uint8)
print (a * 100) / 100
b = N.array((200, 200), dtype = N.uint8)
print (b * 100) / 100
The first print statement will print "200" because the uint8-value is
cast "upwards", I suppose. The second statement prints "[0 0]". I
suppose this is due to overflows during the calculation.
How can I tell numpy to do the upcast also in the second case, returning
"[200 200]"? I am interested in the fastest solution regarding execution
time. In my application I would like to store the result in an
Thanks for every comment
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TRIPODing along...
July 11th 2010, 09:24 AM #1
MHF Contributor
Dec 2007
Ottawa, Canada
TRIPODing along...
A tripod has equal length legs = a.
It sits on an isosceles triangular base, sides b,b,c.
It has height = h.
If the base is changed to sides b,c,c, what is the new height (as an expression)?
It's not clear to me what sort of "expression" you want for the new height.
The original height h can be calculated in terms of a, b and c. I get the relation $h^2 = a^2 - \frac{b^4}{4b^2-c^2}$. If the sides of the base are changed from b,b,c to b,c,c, that is equivalent
to exchanging b and c. So the new height k would be given by $k^2 = a^2 - \frac{c^4}{4c^2-b^2}$. You could then derive a (messy) expression for the ratio k/h if that is what you wanted.
Hello Wilmer,
Creating an expression is too messy for me but given a,b,c the triangle which determines the altitude h consists of the following line segments
A perpendicular bisector of base triangle bbc
B perpendicular bisector of tripod triangle aac
C a
Using the cosine rule the angle between a and A can be determined. If this angle is K h =asinK
Merci BJ.
Creating an expression is too messy for me but given a,b,c the triangle which determines the altitude h consists of the following line segments
A perpendicular bisector of base triangle bbc
B perpendicular bisector of tripod triangle aac
C a
Using the cosine rule the angle between a and A can be determined. If this angle is K h =asinK
Another method is to say that the sphere of radius $a$, centred at the top of the tripod, contains all three vertices of the triangle and hence contains its whole circumcircle. If the radius of
the circumcircle is $r$, you then have a right-angled triangle with sides $a$ (hypotenuse), $r$ and $h$. I used the formula for the circumradius, together with Pythagoras, to get the expression
for $h$.
July 11th 2010, 11:58 PM #2
July 13th 2010, 04:32 AM #3
Super Member
Nov 2007
Trumbull Ct
July 13th 2010, 04:43 AM #4
MHF Contributor
Dec 2007
Ottawa, Canada
July 13th 2010, 06:59 AM #5 | {"url":"http://mathhelpforum.com/geometry/150644-tripoding-along.html","timestamp":"2014-04-21T09:50:10Z","content_type":null,"content_length":"42098","record_id":"<urn:uuid:c765d6fc-f23d-455a-a906-13315bdd36a2>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00268-ip-10-147-4-33.ec2.internal.warc.gz"} |
Which statement is true of a regression line that is superimposed on the scatter plot?
A. It guarantees that the slope and intercept are minimized.
B. It guarantees the largest possible sample variance.
C. It is computed using the maximum and minimum values.
D. It is computed using the Ordinary Least Squares method.
Which statement is true of a regression line that is superimposed on the scatter plot? A. It guarantees that the slope and intercept are minimized. B. It guarantees the largest possible sample
variance. C. It is computed using the maximum and minimum values. D. It is computed using the Ordinary Least Squares method.
A. It guarantees that the slope and intercept are minimized
Not a good answer? Get an answer now. (FREE)
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the first resource for mathematics
Advanced algebra. Along with a companion volume ‘Basic algebra’.
(English) Zbl 1133.00001
Cornerstones. Basel: Birkhäuser (ISBN 978-0-8176-4522-9/hbk; 978-0-8176-4533-5/set). xxiv, 730 p. EUR 54.90/net, SFR 95.00, $ 69.95, £ 42.00; EUR 69.90, SFR 125.00, $ 89.95, £ 54.00/set (2007).
This textbook is the second volume of Anthony W. Knapp’s comprehensive introduction to the fundamental concepts and tools in modern abstract algebra. Together with its foregoing companion volume
“Basic Algebra” [Basel: Birkhäuser (2006; Zbl 1106.00001)], which was published in the autumn of 2006, the current book is to provide a global view of the subject, thereby particularly emphasizing
both its various applications and its ubiquitous role in contemporary mathematics. As the author already pointed out in the preface to the first volume, his leading idea was to give a systematic
account of what a budding mathematician needs to know about the principles of modern algebra in order to communicate well with colleagues in all branches of mathematics and related sciences.
This rewarding program was masterly begun in the companion volume “Basic Algebra”, where the fundamentals of linear algebra, multilinear algebra, group theory, commutative algebra, field theory,
Galois theory, and module theory over noncommutative rings were profoundly developed. As for the author’s particular expository guidelines, exemplary didactic principles, and his notorious brilliant
style of lucid mathematical writing, we may refer to the review of the first volume (the author, loc. cit.), as these outstanding features, which also characterize the second volume “Advanced
Algebra” under review to full extent, have been depicted and appraised there at great length.
In general the present volume assumes knowledge of most of the content of its forerunner “Basic Algebra”, either from that book itself or from some comparable source. The more advanced topics treated
in the current book mainly point toward algebraic number theory and algebraic geometry, with emphasis on aspects of these subjects that impact fields of mathematics other than algebra. In this vein,
the predominant theme is the fundamental and fascinating interrelation between number theory and geometry, where this aspect constantly recurs throughout the book on different levels.
As to the precise contents, the volume under review consists of ten chapters, each of which is subdivided into several sections.
Chapter 1 is titled “Transition to Modern Number Theory” and discusses three classical results of Gauss and Dirichlet that were milestones in the transition from the early number theory of Fermat,
Euler, and Lagrange to the algebraic number theory of Kummer, Dedekind, Kronecker, Hermite, and Eisenstein in the second half of the 19th century. Concretely, after an introductory section on the
historical background of this transition process, this chapter establishes Gauss’s Law of Quadratic-Reciprocity, the theory of binary quadratic forms, and Dirichlet’s Theorem on primes in arithmetic
progressions, including the basics of quadratic number fields, their units, Dirichlet series, and Euler products.
Chapter 2 is devoted to the theory of finite-dimensional associative algebras, division algebras, and closely related classes of rings. The material covered here is part of what is known as
Wedderburn-Artin ring theory, and it comprises the following topics: semisimple rings and Wedderburn’s Theorem, rings with chain conditions and Artin’s Theorem for simple rings, the Wedderburn-Artin
radical and Wedderburn’s Main Theorem, semisimplicity and tensor products, the Skolem-Noether Theorem, the Double Centralizer Theorem, Wedderburn’s Theorem about finite division rings, and
Frobenius’s Theorem about real division algebras.
The further study of associative algebras is the subject of Chapter 3, with special emphasis on the Brauer group of a field as a fundamental tool for classifying noncommutative division rings. Group
cohomology, the interpretation of the Brauer group in this cohomological context, crossed products, and Hilbert’s Theorem 90 tie associative algebras to algebraic number theory, and this link is
thoroughly explained in the course of this third chapter.
The rudiments of the subject of general homological algebra are subsequently developed in Chapter 4. Using the digression on group cohomology in the previous chapter as motivation, the author treats
the basics of homology theory in the context of “good” categories of modules over a ring, with an extension of the discussion to homological algebra in general Abelian categories in the final section
of this chapter. The standard topics in homological algebra, including complexes and additive functors, long exact homology sequences, injective and projective objects, derived functors and their
long exact sequences, the functors Ext and Tor and the algebra of Abelian categories, are taken up here to a remarkable extent. Having the methods of cohomology available at this point of the present
book means that the reader is well prepared for its use in both algebraic number theory and algebraic geometry, which are the main themes in the remaining six chapters.
Chapter 5 deals with three important theorems in the theory of algebraic number fields and their rings of integers, namely with the Dedekind Discriminant Theorem, the Dirichlet Unit Theorem, and the
theorem on the finiteness of the class number of a number field. The direct approach adopted here is generalized in the subsequent Chapter 6 where the reinterpretation in the modern conceptual
framework of the theory of “adèles” and “idèles” is provided. In fact, this chapter develops some of the advanced tools for a more penetrating study of algebraic number theory, among which the reader
encounters $p$-adic numbers, discrete valuations, absolute values, completions of fields, Hensel’s Lemma, ramification indices and residue class degrees, differents and discriminants, global and
local fields, Artin’s product formula, the ring of adèles, and the idèle-class group of a global field.
Chapter 7 provides some algebraic background material for the later study of fundamental questions in algebraic geometry. This includes the Hilbert Nullstellensatz, the transcendence degree of an
infinite field extension, separable and purely inseparable field extensions, the Krull dimension of a ring, regular and singular points of affine varieties, infinite Galois groups, and profinite
In the following, an introduction to the foundations of algebraic geometry is given from three different points of view.
Chapter 8 basically approaches algebraic geometry in its purely algebraic setting, that is, as a framework to study solutions of simultaneous solutions of polynomial equations in several variables by
means of ideal-theoretic and module-theoretic methods. This is done in the light of the theory of projective plane curves and their intersection multiplicities, in the first part, and of the
computational approach via Gröbner bases in the sequel. In the final section, these two approaches are combined to derive the crucial Elimination Theorem in its full generality.
Chapter 9 treats the subject of algebraic curves as the classical outgrowth of the complex analysis of compact Riemann surfaces, on the one hand, and of its arithmetic roots on the other. The leading
theme is here the strong analogy between one-dimensional function fields and algebraic number fields. While the first sections define divisors and the genus of an arithmetic curve (or of a compact
Riemann surface, respectively), the last two sections give a detailed proof of the Riemann-Roch Theorem for such curves and illustrate some of its important applications. The fundamental tool for the
author’s approach is the theory of discrete valuations (as developed in Chapter 6), through which the parallel between the arithmetic of number fields and the geometry of curves becomes strikingly
The final Chapter 10 turns from curves to general affine and projective algebraic varieties. The topics touched upon here include affine varieties, the concept of geometric (Noether) dimension,
projective varieties, rational functions and regular functions, morphisms of affine and projective varieties, rational maps between them, Zariski’s Theorem about smooth points, classification
questions about irreducible curves, affine varieties defined by monomial ideals, Hilbert polynomials, and intersection properties of projective varieties derived from Hilbert polynomials. The book
ends with an informal outlook to algebraic schemes (à la A. Grothendieck), together with a number of hints for further reading in this much more advanced direction, which the keen reader of the
current text should be well prepared for, after having sucessfully mastered the study of it.
Each chapter comes with its own detailed introduction, its own historical remarks, and its own collection of carefully selected problems. These problems are intended to play an important role within
the entire text, because many of them provide additional, further-going topics enhancing the core material of the book. However, almost all problems are solved in the extra section of hints at the
end of the book, which both helps the reader control her or his understanding of the wealth of fundamental material and get acquainted with a large number of additional concepts, methods, theorems,
examples, and applications. As in the first volume of this comprehensive textbook, there is a detailed guide for the reader how to use this book, a chart of the main lines of dependence among the
single chapters, a list of some items of notation and terminology from the first volume “Basic Algebra”, an index of notation for the present volume, and a rich bibliography referring to related
textbooks on the subjects discussed in the present book.
All together, this is another outstanding textbook written by the renowned and versatile mathematical researcher, teacher, and author Anthony W. Knapp that reflects his spirit, his devotion to
mathematics, and his rich experiences in expository writing at best.
00A05 General mathematics
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[Numpy-discussion] speeding up operations on small vectors
Bruce Southey bsouthey@gmail....
Tue Oct 11 12:29:41 CDT 2011
On 10/11/2011 12:06 PM, Skipper Seabold wrote:
> On Tue, Oct 11, 2011 at 12:41 PM, Christoph Groth<cwg@falma.de> wrote:
>> Skipper Seabold<jsseabold@gmail.com> writes:
>>> So it's the dot function being called repeatedly on smallish arrays
>>> that's the bottleneck? I've run into this as well. See this thread
>>> [1].
>>> (...)
>> Thanks for the links. "tokyo" is interesting, though I fear the
>> intermediate matrix size regime where it really makes a difference will
>> be rather small. My concern is in really tiny vectors, where it's not
>> even worth to call BLAS.
> IIUC, it's not so much the BLAS that's helpful but avoiding the
> overhead in calling numpy.dot from cython.
>>> I'd be very interested to hear if you achieve a great speed-up with
>>> cython+tokyo.
>> I try to solve this problem in some way or other. I'll post here if I
>> end up with something interesting.
> Please do.
> Skipper
> _______________________________________________
> NumPy-Discussion mailing list
> NumPy-Discussion@scipy.org
> http://mail.scipy.org/mailman/listinfo/numpy-discussion
In the example, M is an identity 2 by 2 array. This creates a lot of
overhead in creating arrays from a tuple followed by two dot operations.
But the tuple code is not exactly equivalent because M is 'expanded'
into a single dimension to avoid some of the unnecessary
multiplications. Thus the tuple code is already a different algorithm
than the numpy code so the comparison is not really correct.
All that is needed here for looping over scalar values of x, y and
radius is to evaluate (x*x + y*y) < radius**2
That could probably be done with array multiplication and broadcasting.
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gentype remquo ( gentype x,
gentype y,
__global intn *quo)
gentype remquo ( gentype x,
gentype y,
__local intn *quo)
gentype remquo ( gentype x,
gentype y,
__private intn *quo)
The remquo function computes the value r such that r = x - n*y, where n is the integer nearest the exact value of x/y. If there are two integers closest to x/y, n shall be the even one. If r is zero,
it is given the same sign as x. This is the same value that is returned by the remainder function.
remquo also calculates the lower seven bits of the integral quotient x/y, and gives that value the same sign as x/y. It stores this signed value in the object pointed to by quo.
The vector versions of the math functions operate component-wise. The description is per-component.
The built-in math functions are not affected by the prevailing rounding mode in the calling environment, and always return the same value as they would if called with the round to nearest even
rounding mode.
The built-in math functions take scalar or vector arguments. The generic type name gentype is used to indicate that the function can take float, float2, float3, float4, float8, or float16 as the type
for the arguments. For any specific use of these function, the actual type has to be the same for all arguments and the return type.
If extended with cl_khr_fp64, generic type name gentype may indicate double and double{2|3|4|8|16} as arguments and return values. If extended with cl_khr_fp16, generic type name gentype may indicate
half and half{2|3|4|8|16} as arguments and return values.
Copyright © 2007-2010 The Khronos Group Inc. Permission is hereby granted, free of charge, to any person obtaining a copy of this software and/or associated documentation files (the "Materials"), to
deal in the Materials without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Materials, and to permit
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Speed of Light in Air
Index of Refraction of Air (last edited 12/20/2006 )
Dr. Larry Bortner
To find the index of refraction of air at 15 ̊C and 101.325 kPa using an interferometer.
Air, or any other low density gas, retards a light wave going through it, but it does not slow it down very much. The quantity that we use to indicate the speed of light in a substance is called the
index of refraction n and is defined as the ratio of the speed of light in vacuum to the speed of light in the material. It is inversely proportional to the speed, so the higher n is, the slower the
wave. If the light wave speed in air is just a little slower than the speed in vacuum, we expect the index of refraction to be just a little bit larger than one.
One explanation of how light travels through air is that the electrons of the constituent atoms are induced to oscillate by the electromagnetic wave that is light. This takes energy from the wave.
But an oscillating charge emits its own wave at the same frequency. There is a finite time lag between when the energy is absorbed and when it is re-emitted. This interaction of the wave with the
charge slows down the progress of the wave.
Index of Refraction and Wavelength
In the Lens Equation and Dispersion experiments, we investigated the effects of light slowing down when it goes through glass. In dispersion, we know that n varies in a material according to the
wavelength; recall that normal dispersion is when n increases when λ increases. (For λ=5800Å, at T=15°C, and P=standard atmospheric pressure, n =1.0002773.)
Another way to express the index of refraction is by the ratio of wavelengths, since the frequency of the wave will not change when it enters another medium:
where λ is the wavelength in vacuum and λ[air] is the wavelength in air.
This means that any method where we can measure the difference between the two wavelengths is a way we can calculate n. Thus we have an experimental method of finding the speed of light in air.
Define the optical path length L[op] of a light beam between two points to be the number of waves w[#] in that path, times the wavelength of the light in vacuum:
One way to detect the expected small difference in wavelengths is to use a very sensitive instrument known as a Michelson interferometer. This apparatus splits monochromatic light into two beams, the
sample and the reference beam. The sample beam undergoes a physical process that changes its optical path length. It is then recombined with the reference beam that has traveled the same physical
distance. From the superposition principle, this recombination results in areas of constructive and destructive interference, or light and dark fringes. Analyzing this interference pattern of the two
waves gives information on the perturbing agent.
The interferometer design (Fig. 1) is representative of a large group of devices utilizing wave phenomena such as electromagnetic radiation or sound. A filter at position A selects a single
wavelength from an extended source. Light from a particular point S on the diffusing screen strikes a partially-silvered mirror (the beam splitter) which transmits half of the original beam (This
half is the reference beam.) through a glass compensator toward mirror M[1][] and reflects the other half (the sample beam) toward mirror M[2]. The mirrors reflect both beams back to the beam
splitter where a portion of the light gets recombined on route to the observer at D. The compensator at C (virtually identical to the beam splitter except it’s not silvered) insures that the
distances traveled by the reference and sample beams through glass can be made equal. In practice, the total path lengths of the sample and reference beams do not have be exactly equal, just
Figure 1 Diagram of an interferometer.
When the mirrors are perpendicular to the incoming beams and the optical path lengths of the two beams are equal, the interference pattern will be a series of concentric circles as in Fig. 2. By
making one of the mirrors slightly off perpendicular or by having different path lengths, the fringe pattern can be straight lines (portions of very large circles).
Figure 2 The resulting interference pattern when the reference beam and the sample beam path lengths are the same.
Measuring Small Path Differences
The difference in the path lengths of successive rings or fringes is one wavelength. This means that if M[2] moves forward or back by λ/2, the optical path of the sample beam changes by that twice
much, or λ, and that there is a movement in the field of view of one fringe. In practice, we set the mirrors such that the pattern is one of straight lines. A sharp object like a pin point placed in
the observer’s field of view provides a reference point so that we can count fringes moving past it as the physical conditions change.
To measure the index of refraction of a gas such as air, we put a gas cell with transparent glass ends in the path of the interferometer sample beam (Fig. 3). Identical glass plates are put in the
reference beam path to ensure nearly equal path lengths. Air is pumped out of the chamber and slowly bled back in as the pressure is monitored. The greater the number of gas molecules in the chamber,
the slower light travels. Thus the optical path length increases as the pressure increases.
Figure 3 Interferometer set up for this experiment, with a gas cell in the beam path.
If the physical distance light travels through the gas cell chamber one way is L, the optical path length of the sample beam when the cell is under a vacuum is
(Remember that it goes through twice.) As noted, if there is air in the chamber, the wavelength changes, hence the number of wavelengths and the optical path length change:
Counting Fringes (Wavelengths)
The number of fringes i that reflects the optical path difference between these two conditions is the difference in the number of wavelengths:
We expect the number of wavelengths in air to depend on the pressure and temperature of the gas. Here the index of refraction is written as a function of those quantities.
nair as a Function of Pressure, Temperature, and Wavelength
To find the functional form of this dependence, we assume that the perturbation from the ideal index of refraction (n=1) is linear with the number density of gas molecules. That is, the perturbation
takes the form of
where N is the number of molecules in a volume V and C[1][] is an arbitrary constant. The ideal gas law is
where k is Boltzmann's constant. Solving for the number density N/V and substituting into Eq. 6 gives
where C[2] incorporates k.
Many labs have done precise measurements of the index of refraction of gases and empirical fits have been made to this data. Let n=n(P[0],T[0]). For dry air containing 0.03% by volume CO[2] at a
reference temperature T[0]=15°C and standard atmospheric pressure P[0]=101.325 kPa, the CRC Handbook for Chemistry and Physics lists the following wavelength-dependent fit:
where the wavelength λ is in mm. Using ratios, we can express the deviation of n from unity at any pressure and temperature, relative to the reference values:
Fringes as a Function of Pressure
Substituting Eq. 10 into Eq. 5 then gives the number of fringes that indicates the optical path difference between the sample beam with air in the cell at any pressure P and temperature T and the
sample beam with air in the cell at 15°C and one atmosphere pressure:
The quantity in braces is the extra sample beam optical path length that is greater than 2L, measured in the number of vacuum wavelengths. Assuming the temperature doesn't change, plotting i vs. this
number gives a straight line with slope n-1 that we can compare to the empirical value from Eq. 9.
The single-traverse path length L in the cell is not directly measurable. From Fig. 4, what we can measure is the total length L[T] of the cell, the depth h of the lip between the glass and the outer
extent of the metal, and the thickness t of the two identical glass plates used as compensators for the glass plates in the cell.
Figure 4 Gas cell detail, showing dimensions of interest. L is a calculated value, the others are measured.
The length is then
The following equipment and accessories are needed:
• Ealing interferometer set up in Michelson mode
• hand vacuum pump
• mercury light
• absolute temperature sensor, u{T}=0.5 K
• pressure sensor, u{P}=0.1 kPa
• needle valve
• Science Workshop interface
To minimize distortion, the mirrors used in the interferometer are all front-surface optical elements, meaning that the reflective coatings are on the front surface of the glass instead of the rear
surface. This means that even touching the mirrors can damage or destroy them. When working with the interferometer, observe this general rule:
Never touch the active surfaces of optical elements. (Don’t touch the mirror!)
If your interferometer needs adjustment, ask your instructor for assistance.
The canister that houses the mercury bulb becomes very hot after a short time. If you have to reposition the lamp, do so by moving its support base. These bulbs have an important operational
When the bulb is turned off after warming up, it won’t turn on again until it has cooled down, a 10- 15 minute process.
To avoid this delay, be sure that you have taken all of your data before you turn off your lamp.
1. Click on Start> Science Workshop> Third Quarter> Speed of Light in Air, then click on the program Start button. Record the ambient temperature in °C.
2. Record the following dimensions in cm for the gas cell:
Assume an uncertainty of 0.003 cm for each.
3. Turn on the mercury light source. Check to see that an interference pattern of dark, fairly straight lines is visible in the field of view of your interferometer. If not, have your instructor make
the necessary adjustments.
4. Use the hand vacuum pump to reduce the cell pressure to P[start]~20 kPa. The fringes will move across the field of view as you pump, indicating the change in path length as noted in the Background
. The previous user may have left the needle valve slightly open. Be sure it is closed.
5. While observing the fringe pattern, adjust the needle valve so that the pressure increases by 1 kPa every two or three seconds.
6. Position the mouse cursor over the Keep button.
7. Pump out air until P[start]~10 kPa, or as low as you can go. Using long strokes is more effective than quick, short ones.
8. One person needs to observe the fringes and indicate when there's been a movement of a single fringe, and to do this for a total of ten fringes. This entails looking at the fringe pattern with one
eye closed and your head in the same position for the duration of the ten-fringe movement. The fringe observer needs to do two things:
® 1. click on Keep every time a fringe passes the reference point and
® 2. count the total number of fringes (the number of times Keep was clicked).
a. Continue this for a total of 10 fringes.
b. Press the red Stop button.
c. Record the approximate starting pressure and the corresponding Run #.
d. Each partner should be the fringe observer at least five times.
9. Repeat Steps 7 and 8 for values of P[start] in kPa of the following:
This gives you a total of eight (8) data sets of 10 points each.
Do not quit DataStudio until the end of class.
The wavelength of the green line in the mercury spectrum is λ=5461 Å.
1. Click on Start> Templates> Third Quarter> Speed of Light in Air to start Excel. Enter in your names.
2. Enter in the measured cell dimensions, the temperature, and the necessary reference values.
3. Click on the data staging tab at the bottom of the spreadsheet to go to that sheet.
4. Return to the DataStudio window.
a. Position the mouse cursor over the Pressure (kPa) heading.
b. Click on this heading to highlight all of the data of this run.
c. Copy this (Ctrl C).
d. Switch back to Excel, select an empty green cell, and paste (Ctrl V).
5. Repeat Step 4 until all data sets have been transferred.
6. Return to the calculation section by clicking on the main tab at the bottom of the Excel spread sheet.
7. Do the required calculations in the top section:
a. L (Eq.11).
b. λ in cm.
c. T in K.
d. T[0] in K.
e. (n-1) x 10^8 at T[0] and P[0][] (Eq.9)
f. n-1 at T[0][] and P[0].
g. n at T[0][] and P[0].
8. For each of the eight data sets:
a. Express the experimental pressures as the number of vacuum wavelengths in the cell,
b. Do a least squares fit to find n-1 and its uncertainty.
9. Plot all eight data sets on the same graph, with a legend.
10. Your experimental value of n-1 is the average of these slopes. Find this, as well as the standard error.
11. Find the average error of the slopes.
12. Compare the experimental n-1 with the book value, using the maximum of the standard error and the average error as the uncertainty in n-1.
1. The book value of n-1 (Eq. 9) is for dry air. How would humidity affect your results?
2. How could you use a Michelson interferometer to measure the temperature coefficient of expansion of a material? Explain. | {"url":"http://www.physics.uc.edu/~bortner/labs/Physics%203%20experiments/Index%20of%20Refraction%20of%20Air/Index%20of%20Refraction%20of%20Air%20htm.htm","timestamp":"2014-04-16T16:37:36Z","content_type":null,"content_length":"74423","record_id":"<urn:uuid:3aae3f54-b97e-4405-b51d-9f27ae693059>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00401-ip-10-147-4-33.ec2.internal.warc.gz"} |
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A Conversation with Shayle R. Searle
Statistical Science
A Conversation with Shayle R. Searle
Born in New Zealand, Shayle Robert Searle earned a bachelor’s degree (1949) and a master’s degree (1950) from Victoria University, Wellington, New Zealand. After working for an actuary, Searle went
to Cambridge University where he earned a Diploma in mathematical statistics in 1953. Searle won a Fulbright travel award to Cornell University, where he earned a doctorate in animal breeding, with a
strong minor in statistics in 1959, studying under Professor Charles Henderson. In 1962, Cornell invited Searle to work in the university’s computing center, and he soon joined the faculty as an
assistant professor of biological statistics. He was promoted to associate professor in 1965, and became a professor of biological statistics in 1970. Searle has also been a visiting professor at
Texas A&M University, Florida State University, Universität Augsburg and the University of Auckland. He has published several statistics textbooks and has authored more than 165 papers. Searle is a
Fellow of the American Statistical Association, the Royal Statistical Society, and he is an elected member of the International Statistical Institute. He also has received the prestigious Alexander
von Humboldt U.S. Senior Scientist Award, is an Honorary Fellow of the Royal Society of New Zealand and was recently awarded the D.Sc. Honoris Causa by his alma mater, Victoria University of
Wellington, New Zealand.
Article information
Statist. Sci. Volume 24, Number 2 (2009), 244-254.
First available: 14 January 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Wells, Martin T. A Conversation with Shayle R. Searle. Statistical Science 24 (2009), no. 2, 244--254. doi:10.1214/08-STS259. http://projecteuclid.org/euclid.ss/1263478385.
• Aitken, A. C. (1939). Determinants and Matrices. Oliver and Boyd, Edinburgh.
• Bose, R. C. (1949). Least Squares Aspects of Analysis of Variance. Institute of Statistics Mimeo Series 9. Univ. North Carolina, Chapel Hill.
• Callow, E. H. and Searle, S. R. (1956). Comparative studies of meat: V Factors affecting the iodine number of the fat from the fatty and muscular tissues of cattle. J. Agricultural Science 48
• Graybill, F. A. (1961). An Introduction to Linear Statistical Models. McGraw-Hill, New York.
Mathematical Reviews (MathSciNet):
• Hartley, H. O. and Rao, J. N. K. (1967). Maximum likelihood estimation for the mixed analysis of variance model. Biometrika 54 93–108.
• Henderson, C. R. (1953). Estimation of variance and covariance components. Biometrics 9 226–252.
• Henderson, C. R., Kempthorne, O., Searle, S. R. and Von Krozigk (1959). Estimation of environmental and genetic trends from records subject to culling. Biometrics 15 192–218.
• McCulloch, C. E. and Searle, S. R. (2001). Generalized, Linear, and Mixed Models. Wiley, New York.
• Penrose, R. A. (1955). A generalized inverse for matrices. Proc. Cambridge Philosophical Society 51 406–413.
• Rao, C. R. (1962). A note on a generalized inverse of a matrix with applications to problems in mathematical statistics. J. Roy. Statist. Soc. Ser. B. 24 152–158.
• Searle, S. R. (1951). Probability: Difficulties of definition. J. Inst. Actuaries Students’ Soc. 10 204–212.
Mathematical Reviews (MathSciNet):
• Searle, S. R. (1956). Matrix methods in variance and covariance component analysis. Ann. Math. Statist. 27 737–748.
• Searle, S. R. (1958). Sampling variances of estimates of components of variance. Ann. Math. Statist. 29 167–178.
• Searle, S. R. and Henderson, C. R. (1961). Variance components in the unbalanced 2-way nested classification. Ann. Math. Statist. 32 1161–1166.
• Searle, S. R. (1966). Matrix Algebra for the Biological Sciences. Wiley Chichester.
• Searle, S. R. (1982). Matrix Algebra Useful for Statistics. Wiley, New York.
Mathematical Reviews (MathSciNet):
• Searle, S. R. (1971). Linear Models. Wiley, Chichester. (1997, Wiley Classics Library.)
• Searle, S. R. (1993). Analysis of variance computing then and now, with reference to unbalanced data. In Proceedings 18th SAS Users’ Group Conference 1077–1087.
• Searle, S. R. (1997). Linear Models for Unbalanced Data. Wiley, New York.
• Searle, S. R., Casella, G. and McCulloch, C. E. (1992). Variance Components. Wiley, New York.
• Searle, S. R. and Hausman, W. H. (1970). Matrix Algebra for Business and Economics. Wiley, New York.
• Wicksell, S. D. (1930). Remarks on regression. Ann. Math. Statist. 1 1.
• Williams, E. J. (1959). Regression Analysis. Wiley, New York.
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Pseudosolutions of the time-dependent minimal surface problem
- IEEE Trans. on Image Processing , 1998
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Abstract—We introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented. We consider intensity images as surfaces in the space. The image
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Abstract—A variational approach for filling-in regions of missing data in digital images is introduced in this paper. The approach is based on joint interpolation of the image gray-levels and
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We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of
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asimptotic behavoiur of the solutions.
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Intuition for mean curvature.
up vote 11 down vote favorite
I would like to get some intuitive feeling for the mean curvature. The mean curvature of a hypersurface in a Riemannian manifold by definition is the trace of the second fundamental form. Is there
any good picture to have in mind, when dealing with it?
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5 Answers
active oldest votes
[prescript: I had this written with nice images inserted, but when I went to post this I found that you needed a reputation of 15 to post images and since this is my first answer, I
couldn't insert them directly, thus the links.] (Images now inserted by J.O'Rourke)
Do this experiment: draw a curve in 2D and about some point on that curve, draw a unit normal vector field. Now convince yourself that if you push the curve out along the normal field
by a distance $\epsilon$, the length changes by a factor of $1+\epsilon\kappa$ where $\kappa$ is the curvature.
(See http://s20.postimage.org/tk7ldco0t/Steiner_Minkowskiproof_fig1.jpg; now below)
Now consider a 2D surface in 3D. Choose local coordinates so that the first two basis vectors are tangent to the surface and the curvatures along the first two coordinates are the
principal curvatures. The infinitesimal change in surface area is $(1+\epsilon\kappa_1)(1+\epsilon\kappa_2)$ -- actually this product is the Jacobian of the normal map, $N_\epsilon$
that pushes the surface out along the normal field a distance epsilon.
(see http://s7.postimage.org/ifaidv6ln/area_change.png; now below)
In general, for co-dimension one surfaces, the Jacobain of the normal map $N_\epsilon$ is $\Pi_{i=1}^{n-1} (1+\epsilon\kappa_i)$. We can integrate this over the original surface to
up vote 13 down get the new, n-1 volume of the pushed surface. That is:
vote accepted
$\mathcal{H}^{n-1}(N_\epsilon(W)) = \int_{W} \Pi_{i=1}^{n-1} (1 + \epsilon\kappa_i) d\mathcal{H}^{n-1}$
$ \hspace{1in} = \int_{W} 1 d\mathcal{H}^{n-1} + \epsilon \int_{W} \sum_i \kappa_i d\mathcal{H}^{n-1} + ... + \epsilon^k \int_{W} \sum_{s\in S(k)}\Pi_{i\in s}\kappa_i d\mathcal{H}^
$ \hspace{1.2in} + ... + \epsilon^{n-1} \int_{\partial W} \Pi_{i=1}^{n-1} \kappa_i d\mathcal{H}^{n-1}$
Now we note that the first order term is the integral of the mean curvature. That is, to first order, the change in surface volume is given by the mean curvature. Note that I am using
the term mean curvature for what is sometimes called total mean curvature, $\sum_{i=1}^{n-1} \kappa_i$.
(This works in co-dimension $k>1$, but then, because the set of normal directions at any point is $k$-dimensional, it is a bit more involved to get a result that looks like the result
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Let $M$ be an oriented hypersurface in an oriented euclidean space $E$.
The normal curvature at a point $x_0\in M$ measures how much it is displaced, in the positive normal direction, with respect to the points of $M$ that are near it, in the same way that the
Laplacian of a function measures how much the value of a function at a point differs from the average of the points that are near.
In fact, if $M$ is the graph of a nearly constant function $f:\mathbb R^n\to\mathbb R$ (for example, the vibrating membrane of a drum at some instant) then the Laplacian of $f$ is a good
up vote 0 approximation of the normal curvature of $M$.
down vote
More generally, if $M$ is possibly not nearly flat, then for any $x_0\in M$ you can find the hyperplane $\Pi$ that is tangent to $M$ at $x_0$, and find a parametrization $f:\Pi\to M$ (only
defined on a neighbourhood of $x_0$) such that $p\circ f=id_\Pi$, where $p:E\to\Pi$ is the orthogonal projection. This parametrization shows that $M$ is locally the graph of a function $g=q
\circ p$ (where $q$ is the projection on the orthogonal complement of $\Pi$) such that $dg(x_0)=0$. Then, the normal curvature of $M$ at $x_0$ equals $\Delta g(x_0)$.
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It gives you a normal field along the hyper surface which describes how you should move the surface to shrink the induced surface volume form. This works also for higher codimension.
EDIT: An explicit formula which generalized the answer of Otis Chodosh is the following:
Let $(N,\bar g)$ be a Riemannian manifold and let $M$ be a compact manifold with $\dim(M)<\dim(N)$. Let $f(t,\quad)\in Imm(M,N)$ be a smooth curve of immersions, then $f_t = \partial_t|_0 f\
up vote in \Gamma(f^*TN)$ splits into parts $f_t=Tf.f_t^\top + f_t^\bot$ tangent and normal to $M$. Let also '$g=f^\star\bar g$ be the induced metric. Then the variation of the mapping $$ Imm(M,N) \
12 down to \Gamma(vol(M)),\qquad f \mapsto vol(g)=vol(f^*\bar g) $$ is given by $$ \partial_t|_0\\, vol\Big(f(t,\quad)^\star\bar g\Big) = Tr^g\big(\bar g(\nabla f_t,Tf)\big) vol(g)= \Big({div}^{g}
vote (f_t^{\top})-\bar g\big(f_t^{\bot},Tr^g(S)\big)\Big) vol(g). $$ Here $S$ is the second fundamental form $S\in\Gamma(L^2_{sym}(TM;TM^\bot)$. So if the variation field $f_t$ is always normal
to $f(M)$, the first summand vanishes.
For a proof see 5.7 in: Martin Bauer, Philipp Harms, Peter W. Michor: Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics 3, 4 (2011), 389-438. (pdf)
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Perhaps a more geometric way of viewing mean curvature (at least for a hypersurface) is as follows:
If $\varphi: \Sigma^n\hookrightarrow (M^{n+1},g)$ is an embedded (oriented) hypersurface, then we can naturally extend a family of embeddings by choosing a unit normal $\nu$ and
flowing by unit speed, i.e.
$$\varphi_t(x) := \exp_x(t \nu(x))$$
For small enough $t \in (-\epsilon,\epsilon)$, this still gives an embedding $\varphi_t: \Sigma\to M$. Now, one could pull back the metric $g$ by $\varphi$ to obtain a metric on $\
Sigma$. This metric gives a (Riemannian) volume form on $\Sigma$, $d\mu_t$. Then,
up vote 8 down
vote $$ \frac{\partial}{\partial t} d\mu_t = nH d\mu_t $$
(I think this matches the normalization in @Joseph O'Rourke's answer, but of course the sign changes with the choice direction of the normal)
This follows from the first variation formula.
In other words, the mean curvature measures how the volume of the submanifold (locally) changes under flowing the surface in the direction of the unit normal.
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First, rather than viewing the mean curvature as "the trace of the second fundamental form," it might be more intuitive to view it as the average of the principle curvatures, i.e.,
$$H = \frac{1}{n} \sum_{i=1}^n \kappa_i \;.$$
Second, it may be that the following view of minimal surfaces (those of mean curvature zero) from D. Hoffman and W. H. Meeks III, in their paper, "Minimal surfaces based on the
catenoid" [Amer. Math. Monthly 97(8) (1990), 702-730] (ACM link), might help:
up vote 12 down
vote “Loosely speaking, one imagines the surface as made up of very many rubber bands, stretched out in all directions; on a minimal surface the forces due to the rubber bands balance
out, and the surface does not need to move to reduce tension.”
You can see this accords with Peter Michor's more abstract formulation.
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9th circle test paper and Assignments
CLASS IX CIRCLE GEOMETRY Test paper-1
1. Define the following and mark them in a circle: (I) Centre of a circle (ii) Chord (iii) Secant (iv) Sector
(v) Major and minor segment
2. Complete the following statements.
(I) Equal chords of a circle subtend....
(ii) If the angles subtended by the chords of a circle at the centre are equal, then....
(iii) The perpendicular from the centre of the circle to a chord. .. .
(iv) The line drawn through the centre of a circle to bisect a chord is....
3. Find the length of a chord which is at a distance of 5cm from the centre of the circle whose radius is 10cm.
4. AB and CD are two parallel chords of a circle (lying on opposite sides of the centre) such that AB=10 cm, CD=24 cm. If the distance between AB and CD is 17cm, determine the radius of the circle.
5. PQ and RS are two parallel chords of a circle whose centre is O and radius is 10 cm. If PQ=16 cm and RS=12 cm, Find the distance between PQ and RS, if they lie (i) on the same side of the centre
O, and (ii) on opposite sides of the centre O.
6. Given an arc of a circle, show how to complete the circle.
7. In the figure 9.1, if a diameter of a circle bisects each of the two chords of the circle, prove that the chords are parallel.
8. In the figure 9.2, if two chords of a circle are equally inclined to the diameter through their point of intersection, prove that the chords are equal.
9 In the figure 9.3, PQ and RQ are two chords equidistant from the centre. Prove that the diameter passing through Q bisects <PQR and <PSR.
10. In fig 9.4, AB and AC are two equal chords of a circle whose centre is O. If OD║AB and OE ║AC. prove that triangle ADE is an isosceles triangle.
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Test Papers,CBSE chapter-wise M C Q Multiple Choice Questions, Test Paper, Sample paper
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Nerves of simplicial objects in categories/Waldhausen's S-construction
up vote 1 down vote favorite
Is there a good nerve-like functor from simplicial objects in categories to simplicial sets which takes level-wise equivalences of categories to weak equivalences?
To give this some context, I'd like to extract a simplicial set from the Waldhausen S-construction applied to a category with cofibrations, and I realized that my standard way of taking a nerve is
for simplicial categories (i.e. simplicial objects in categories for which the objects form a constant simplicial set), and this doesn't clearly apply to the S-construction.
algebraic-k-theory simplicial-stuff ct.category-theory
What about taking the nerve in each simplicial degree and then taking the diagonal of the resulting bisimplicial set? – K.J. Moi Jun 20 '11 at 16:21
I think that does something slightly different, but it does helpfully rephrase the question: Taking the levelwise nerve puts us in bisimplicial sets, and takes levelwise equivalences to levelwise
homotopy equivalences, i.e. to weak equivalences (w.e.) in the Bousfield-Kan or Reedy structures. Taking the diagonal of a bisimplicial set preserves w.e. in the Moerdijk structure (by definition),
but not necessarily in the BK or Reedy structures. So, I'm looking for a functor from bisimplicial sets to simplicial sets which preserves BK or Reedy w.e. – Jesse Wolfson Jun 20 '11 at 16:48
If we consider a bisimplicial set as a simplicial object in simplicial sets then a levelwise weak equivalence induces a weak equivalence on diagonals, right? Sorry if I'm missing the point here. –
K.J. Moi Jun 20 '11 at 17:39
All simplicial sets cofibrant (in the Quillen model structure). Do you mean with the Joyal model structure? – David Carchedi Jun 20 '11 at 17:51
@KJ, I don't think that a levelwise w.e. necessarily induces a w.e. on diagonals. That was my point about the different model structures on bisimplicial sets. @David, thanks, you're absolutely
right. – Jesse Wolfson Jun 20 '11 at 18:38
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2 Answers
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Here is an idea. Try the homotopy coherent nerve. (This was originally introduced, sort of, by Boardman and Vogt in a topological context and was formulated for simplicially enriched
categories (and please do not use `simplicial category' as it is ambiguous!) by Cordier in 1980. The H.c. nerve is related to the bisimplicial nerve by using the codiagonal of Artin and
Mazur. (which has been mentioned in several of my answers!!!). Some details of the H.C. nerve as discussed in the nLab entry on that and there are links to elsewhere. A chatty discussion
can be found in Kamps and Porter, (again that has been mentioned before :-))!
Hope this helps.
[Edit] I should mention the papers
up vote 3 M. Bullejos and A. Cegarra, On the Geometry of 2-Categories and their Classifying Spaces , K-Theory, 29, (2003), 211 – 229.
down vote
accepted M. Bullejos and A. M. Cegarra, Classifying Spaces for Monoidal Categories Through Geometric Nerves , Canadian Mathematical Bulletin, 47, (2004), 321–331.
A. Cegarra and J. Remedios, The relationship between the diagonal and the bar constructions on a bisimplicial set , Topology and its Applications, 153, (2005), 21 – 51.
A. Cegarra and J. Remedios, The behaviour of the $\overline{W}$-construction on the homotopy theory of bisimplicial sets , Manuscripta Math., 124, (2007), 427 – 457, ISSN 0025-2611.
some of which may help and that there is related discussion in the Menagerie and in Lurie's HTT.
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I don't know if it will be a full answer to your question but there is a (diagonal) model structure on the category of simplicial objects in $\mathbf{Cat}$ (denoted by $\mathbf{sCat}$)
such that a map $F_{\bullet}:C_{\bullet}\rightarrow D_{\bullet}$ between two object in $\mathbf{sCat}$ is a fibration (weak equivalence) iff the diagonal of the nerve (taken level-wise) of
the corresponding level-wise groupoids is a fibration (weak equivalence) of simplicial sets, i.e.
up vote 0
down vote $ diag ~\mathrm{N}_{\bullet}\mathbf{iso}~F$ is a fibration (a weak equivalence) of simplicial sets.
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Not the answer you're looking for? Browse other questions tagged algebraic-k-theory simplicial-stuff ct.category-theory or ask your own question. | {"url":"http://mathoverflow.net/questions/68293/nerves-of-simplicial-objects-in-categories-waldhausens-s-construction","timestamp":"2014-04-18T14:01:56Z","content_type":null,"content_length":"61732","record_id":"<urn:uuid:b7c96abc-48c8-4a7c-b6c9-45b5df0a1ac0>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00481-ip-10-147-4-33.ec2.internal.warc.gz"} |
Solution Book 6th Class Math Punjab Text Book
Meri Kitab, Urdu Book, Punjab Text Book Board Lahore Meri Kitab, Jamaat Awwal (Part One) Meri Kitab, Jamaat Doaim (Part Two) Meri Kitab, Jamaat Soaim (Part Three)
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Chemistry Formulas for Class XI, XII and Under Graduates
Chemistry Formulas for Class XI, XII and Under Graduates 2012
Download Chemistry Formulas For Class XI , Class12 and Under Graduates
Chemistry Formulas:
Ideal Gas law
PV = nRT
n = number of moles
R = universal gas constant = 8.3145 J/mol K
Boyle’s law
P[1]V[1] = P[2]V[2]
Mole fraction: Mole fraction of a component in solution is the number of moles of that component divided by the total number of moles of all components in the solution.
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What subgroups of M-groups are guaranteed to be M-groups themselves?
up vote 3 down vote favorite
Let $G$ be a finite M-group (ie, where all irreducible complex characters are induced from linear characters of subgroups). What subgroups of $G$ are necessarily themselves M-groups? For instance,
clearly any nilpotent subgroup will be an M-group (so the Fitting subgroup works), and it has been shown that all normal Hall-subgroups are themselves M-groups. On the other hand, not all normal
subgroups need to be M-groups, as has been shown by Dade, though it seems like most of them will be.
For instance, is it known whether the derived subgroup of an M-group is itself an M-group?
My reason for asking is that I am looking at a class of groups that behaves to some extent like M-groups, and it looks like all M-groups might indeed be in this class, but to show this it would be
very helpful to know some subgroups (not necessarily normal) that are guaranteed to be M-groups themselves.
Edit: Given an irreducible character $\chi$, we can look at the set of subgroups of $G$ from which $\chi$ is induced from a linear character. Does anyone know an example of an M-group $G$ and an
irreducible character $\chi$ of $G$ such that this set of subgroups does not contain any M-groups?
gr.group-theory finite-groups
Without having an expert opinion, I do know that the question of which subgroups of M-groups are themselves M-groups is extremely difficult. The basic problem with the subject seems to be that
there is no internal group-theoretic characterization of M-groups: it all depends on the character theory of the group. There are hundreds of papers and many books to consult, but I'd be
especially surprised if the derived group of an M-group were always an M-group (if true, that would be plainly stated in books by Isaacs, Huppert, etc.). – Jim Humphreys Feb 11 '11 at 13:31
Maybe you know this book already, but in case not: have a look at Michael Weinstein, "Between nilpotent and solvable". There is a whole chapter devoted to properties of M-groups. – Alex B. Feb 11
'11 at 14:14
1 FWIW: Dade's M-group of order 3584 has two subgroups of index 2 that are not M-groups, but the derived subgroup (their intersection) is an M-group. In fact those 2 subgroups of index 2 are the
only subgroups that are not M-groups. Do you know if Sylow normalizers of M-groups are always M-groups? I haven't found counterexamples to the derived subgroup or the Sylow normalizer, but I
haven't searched far enough to have much confidence. – Jack Schmidt Feb 12 '11 at 3:15
1 @Jack Schmidt: To the best of my knowledge, this is unknown and one of the open questions in the field (another open question is whether a normal subgroup $N$ of an M-group $G$ such that $|N|$ or
$|G:N|$ is odd is always monomial). Isaacs (Hall subgroup normalizers...) has shown that $\mathbf{N}_G(H)/H'$ is an M-group, when $H$ is a Hall subgroup of the M-group $G$. – Frieder Ladisch Feb
12 '11 at 12:47
To reinforce other comments, I regret that there is no comprehensive up-to-date survey of M-groups: what is known/unknown/conjectured. For instance, the chapter in the book Alex mentions seems to
cover just older material found in books by Isaacs and others. In any case, I'm aware that some people who have worked on M-groups became discouraged about the future prospects. – Jim Humphreys
Feb 12 '11 at 13:31
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Browse other questions tagged gr.group-theory finite-groups or ask your own question. | {"url":"http://mathoverflow.net/questions/55103/what-subgroups-of-m-groups-are-guaranteed-to-be-m-groups-themselves","timestamp":"2014-04-20T04:03:37Z","content_type":null,"content_length":"54549","record_id":"<urn:uuid:04ec2059-3c17-4a6b-a69e-df268fa8f711>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00506-ip-10-147-4-33.ec2.internal.warc.gz"} |
Proposition 12
To set up a straight line at right angles to a given plane from a given point in it.
Let the plane of reference be the given plane and A the point in it.
It is required to set up from the point A a straight line at right angles to the plane of reference.
From an elevated point B draw BC perpendicular to the plane of reference, and draw AD parallel to to BC through the point A.
Then since AD and BC are two parallel straight lines, and one of them, BC, is at right angles to the plane of reference, therefore the remaining one, AD, is also at right angles to the plane of
Therefore AD is set up at right angles to the given plane from the point A in it.
This proposition, like the last, is used frequently in the rest of the Elements to construct lines perpendicular to planes. | {"url":"http://aleph0.clarku.edu/~djoyce/java/elements/bookXI/propXI12.html","timestamp":"2014-04-18T15:39:10Z","content_type":null,"content_length":"2965","record_id":"<urn:uuid:9eadc0b9-03d7-46ce-ae72-a93c2b693788>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00389-ip-10-147-4-33.ec2.internal.warc.gz"} |
Lego Pentagon
Speed build:
It's pentagon time... awwwww yeah! The wild offspring of Mr. Square and Mrs. Hexagon; baby Pentagon is here to rock your world!
Lego thought: Don't you just love the sound of Lego bricks? I love the sound a pile of 1x2s makes when you reach in to grab the ones you want. I also love the sound of two bricks snapping together,
there's just something so satisfying about that sound.
Step 1: Parts
To build what will probably the coolest pentagon you will ever make, you will need:
-90 1x2 bricks
Two color: 45 of each brick
Three color: 30 of each brick
Rainbow warrior: go wild
Step 2:
Stack 3 bricks as shown. Repeat until all the bricks are used up.
Two colors: 15 stacks of each color
Three colors: 10 stacks of each color
Rainbow warrior: 30 stacks of whatever kinda craziness you're into
Step 3:
Connect the stacks, alternating the colors.
To turn the corner, turn the top brick of the stack as shown. The goal here is have each side consist of 5 normal stacks of bricks and two that are bent. The pictures make more sense.
Step 5: Close 'er Up!
Bring the two ends around and attach those fools together!
Now you're cooler because you have a Lego pentagon.
Check out these other cool Lego shapes to be even cooler:
Lego Circle
Lego Triangle
When building your polygons and circles, do you determine the "minimum size" by trial and error, or have you gone through the tolerance calculations numerically?
There should be a relatively simple relationship:
The brick-to-brick tolerance gap vs. the 1-stud side length tells you the angular tolerance, which determines both the extra opening or closing possible at each "right angle" corner, and also allows
you to compute the incremental curvature along each side.
For the circle, the angular tolerance divided into 2.pi tells you the number of bricks needed around the circumference. Interestingly, your trial-and-error circles can tell you directly what the
angular tolerance is -- divide 2.pi by the smallest stable circumference and that's the maximum angular opening.
For the polyhedra, the computation is more complex. There is a relationship between the straight (chord) length between vertices and the curved length you build, which can tell you the half-angle
added onto the Euclidean polygon's corner angle. The sum of the Euclidean corner and the two half-angles must be close to 90 degrees, within the angular tolerance determined above.
Since the Euclidean corner is determined from the number of sides on the polygon [Q = pi - (2pi/N) = pi * (N-2)/N], the derivation above can be written generally for any regular polygon.
Once you've got that relationship -- angular excess/deficit from 90 degrees at each corner vs. curved length of side -- you can invert it and determine the minimum curved length possible given the
angular tolerance.
It would be very interesting to know how close your constructions are to the limiting values computed as above.
Okay. Let's plug some numbers in and see what happens. (Spoiler alert: everything comes out nicely consistent :-).
For the angular tolerance, the 75-brick (1x2) circumference gives 4.8 degrees per joint. In that I'ble, you suggested that the 75-brick model might be close to minimal, so I'll assume 4.8 degrees is
the maximum possible tolerance (over- or under-bend) on a 1-stud joint.
If you have successfully made undamaged circles smaller than that, the angular tolerance would scale up proportionately.
Your pentagons have 6.5-brick curved sides. Assuming the angular tolerance above, your pentagons should have interior angles of no more than 95 (90 + 4.8) degrees. A regular pentagon has interior
angles of 108 degrees. That means the angle which each curved side makes with its chord should be 6.6 degrees = (108 - 94.8) / 2, and the total arc of the curved side is 13.2 degrees. With 6.5
bricks, the bend angle absorbed at each joint along the side is 13.2/6.5 = 2.03 deg, much less than the maximum tolerance; that's consistent with the picture, where the sides are curved much less
than your circles.
The triangles have 7.5-brick curved sides. A similar analysis suggests that the curve-to-curve angle should be no more than 85.2 degrees (90 - 4.8). With interior angles of 60 degrees on an
equilateral triangle, the chord angles are (85.2-60)/2 = 12.6 degrees, and the total arc of the curves is 25.2 degrees. Each joint along the arc absorbs 3.36 degrees, still less than the maximum
Taking those analyses, I would predict that you could make your triangles with sides as short as 5.5 bricks (two bricks fewer per side), and your pentagon as short as 3.5 bricks (3 bricks fewer per
side). In both cases, the sides would be curved close to their maximum (similar to the 75-brick circle).
It's more trial and error and how much you value your Lego bricks.
For example, I show how to make a circle with 100 bricks per layer because it is easy to curve around and snap together (that's the black and white one shown). I make the lime green one using only 75
bricks per layer and that is harder to bring around and connect. Though I could go smaller than 75 per layer, I don't make the circles much tighter than that because I don't want them exploding in my
face and the likelihood of cracking a brick becomes higher when they are under that much stress.
I also really like working with nice numbers (100, 75, 30, etc.) versus weird numbers (17, 23, 106, etc.) and it helps when making patterns to have multiples of 2 or 3.
As far as the limits go, I think there is a lot more play then a standard equation would show. Even when a tighter shape like the triangles are built, there is still a bit of room for things to bend
and add uneven amounts of stress to different parts of the structure.
Hey, thanks for the reply! I do really love these projects of yours -- it's extremely cool to see LEGO do more "organic" curved objects. I'm going to have to play around with this stuff a bit and see
what I can deduce.
Of course! If you deduce anything cool be sure to share.
Ahh the HQ of national defense.
Making THE Pentagon would be really cool. I don't have enough bricks to make it happen but if you do I would love to see it done. The idea of a minifig scale one is making my mouth water.
Ohh i'm going to try i'll post pics!
Try to get those lines that run parallel to the sides too, that would be sick.
Your 'coming soon' is eating me up inside, I have to know what's coming up next!
Soon my friend. Haha I'm working on the next few posting right now.
Nice! You come up with a lot of fun ways to construct with legos! I look forward to more :) | {"url":"http://www.instructables.com/id/Lego-Pentagon/CZMZKSZGVUWZ9F1","timestamp":"2014-04-20T04:10:01Z","content_type":null,"content_length":"171024","record_id":"<urn:uuid:a0589bc6-a000-40a6-9076-604dc7929338>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00215-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here's the question you clicked on:
Find the lowest common denominator for the set of fractions. 7/x^2+4x+4 5/4-x^2
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the reqd LCM is 35/(x+2)
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lcm of fraction is lcm of numerator /hcf of denominator thus lcm of 7 and 5 here is 35 and hcf of (x+2)^2 and (2+x)(2-x) is (x+2) hence the result
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The denominator for the first one factors out to (x+2)(x+2). You get this by taking the coefficient of "4" and finding mult's that if adding together would give you the middle x term which is +4,
then you would have x^2+2x+2x+4, you would use factor by grouping,(x^2+2x)(2x+4). Pull out the x from the first group, than a 2 from the second group. This will leave you with (x+2)(x+2). Now,
the second part is "perfect squares," it is simply (2-x)(2+x). Now, with both of those done you come up with the final answer: (x+2)(2-x)is you LCD. If you are asking why not the (2+x), it is
because that one is exact same as the (x+2), just written different but still the same value.
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@ROBERT3 u r right we should multiply both side by (x+2)(2-x) after that we have a equation to solve
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in my answer is 2x+(x+2) is correct @amir.sat
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i need help @ajprincess
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getting it @robert3?
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MAT-181 Objectives
MAT-181 Statistics I
Unit 1
1. Differentiate between descriptive and inferential statistics
2. Identify types of data
3. Identify the measurement level of a variable
4. Identify basic sampling techniques
5. Organize data using frequency distributions
6. Represent frequency distributions graphically
7. Represent data using bar graphs, time plots, and circle graphs
Unit 2
1. Summarize data using mean, median, and mode
2. Describe data using range, variance, and standard deviation
3. Identify the position of a data point by using percentiles and standard scores
4. Produce stem and leaf displays and box and whisker plots
Unit 3
1. Determine the number of possible outcomes using a tree diagram
2. Find the total number of possible outcomes using the multiplication rule
3. Calculate the number of permutations of n things taken r at a time
4. Calculate the number of combinations of n things taken r at a time
5. Determine sample spaces
6. Find the probability of an event using relative frequencies
7. Find the probability of a compound event
8. Find the conditional probability of an event
Unit 4
1. Construct a probability distribution for a discrete random variable
2. Find the expected value and standard deviation for a discrete random variable
3. Calculate binomial probabilities
4. Find the mean and standard deviation for a binomial distribution
Unit 5
1. Identify the properties of a normal distribution
2. Find the area under the standard normal distribution for various intervals
3. Transform a normally distributed random variable into a standard normal variable
4. Find specific data values for given areas under a normal distribution
Unit 6
1. State the Central Limit Theorem
2. Use the Central Limit Theorem to solve problems involving the distribution of the sample mean for large samples
3. Use the normal distribution to approximate probabilities for a binomial
Unit 7
1. Distinguish between point estimates and interval estimates
2. Find the confidence interval for m using a large sample
3. Find the confidence interval for m using a small sample
4. Find the confidence interval for the binomial proportion p
5. Determine the minimum sample size for estimating m to within a specified margin of error
6. Determine the minimum sample size for estimating p to within a specified margin of error (with and without prior information)
Unit 8
1. Structure a classical test of hypothesis
2. Test means for one-sample (using large and small samples)
3. Test for a proportion
Unit 9
1. Test the difference between means for dependent samples
2. Test for the difference between means for two independent samples (large or small)
3. Test for the difference between two proportions
Unit 10
1. Draw a scatter diagram
2. Find the equation of the least squares regression line
3. Use the least squares regression line to produce point estimates
4. Compute the standard error of the estimate
5. Find the confidence interval for the dependent variable
6. Compute the linear correlation coefficient
7. Test for a significant linear correlation
8. Compute the coefficient of determination
Unit 11
1. Test two variables for independence using chi-square
2. Test a distribution for goodness of fit using chi-square | {"url":"http://www.bhcc.mass.edu/math/courseobjectives/mat-181objectives/","timestamp":"2014-04-20T06:15:44Z","content_type":null,"content_length":"30414","record_id":"<urn:uuid:0aeec788-97bc-4f76-8582-e33846351c9d>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00638-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: On the Dimension of Multivariate Piecewise Polynomials
Peter Alfeld
Department of Mathematics
University of Utah
Salt Lake City, Utah 84112
Note: A slightly different version of this paper is to appear in the Proceedings of the Biennial Dundee
Conference on Numerical Analysis, June 2528, 1985, Pitman Publishers
Lower bounds are given on the dimension of piecewise polynomial C 1 and C 2 functions defined on a tes
sellation of a polyhedral domain into Tetrahedra. The analysis technique consists of embedding the space
of interest into a larger space with a simpler structure, and then making appropriate adjustments. In the
bivariate case, this approach reproduces the wellknown lower bounds derived by Schumaker.
Table of Contents
1. Introduction
2. The Univariate Case
3. The Bivariate Case
3.1 The Geometry of Triangulations
3.2 The C 1 Case
3.3 The C 2 Case | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/603/3926407.html","timestamp":"2014-04-24T10:14:37Z","content_type":null,"content_length":"7954","record_id":"<urn:uuid:d6a98a78-7bd2-4bb8-baca-a2ab576834a8>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00253-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: Evolution's Imperative
Bodester (jbode77@ursa.calvin.edu)
Wed, 24 Mar 1999 14:02:12 -0500 (EST)
>> >So I guess that when God says that Solomon had a circular tub built that
>> was
>> >thirty cubits in circumference and 10 cubits in diameter, the value of
>> pi
>> >must really be 3 (otherwise the circumference of a 10 cubit-diameter
>> circle
>> >would have been 31.5 cubits, or the diameter of a 30 cubit-circumference
>> >circle would have been 9.5 cubits). How can you possibly question this?
>> I think if the 10-cubit measurement was an inside diameter and the
>> circumference was measured around the outside, and if the wall was ~ an
>> inch thick, then the value of pi would work out.
>I think not - surely if the *inside* diameter were 10, the outside diameter
>would then be 10+2x (where x is the thichness of the wall). Then the
>circumference would have to be pi*(10+2x) which would be even more in excess
>of 30 than if the outside diameter were 10.
I noticed this discrepancy too Gary. If the OUTside diameter were 10, then
it would work out with an inside circumference. I didn't figure out how
thick the wall has to be though. Hey, I'm on spring break! Can't be
thinking too much now!
Jason Bode | {"url":"http://www2.asa3.org/archive/evolution/199903/0229.html","timestamp":"2014-04-16T13:22:32Z","content_type":null,"content_length":"3375","record_id":"<urn:uuid:7e8b20b9-811f-46da-bf10-55439b8df100>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00304-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Joke Answers
The jokes
Q: What did the constipated mathematician do? A: He worked it out with a pencil!
OK, not going to try to explain this one.
Q: What's purple and commutes? A: An Abelian grape.
A group is a set of things (think "numbers", but they could be sides of a cube, or colors, or anything you want) along with an operation defined on them (like addition or multiplication, but it
doesn't have to work like those). When the operation on the group happens to be commutative (like 2+4 = 4+2), the group is called Abelian
Q: Why do you never hear the number 288 on television? A: It's two gross.
A "gross" is a dozen dozen, or 144. Not a very mathematical joke.
Q: What do you get when you cross a mosquito with a rock climber? A: Nothing. You can't cross a vector and a scalar.
The joke is referring to a Cross Product, an operation defined on two vectors. You can't take the cross-product of a vector and a scalar.
Q. How many mathematicians does it take to change a lightbulb? A. 1, he gives the lightbulb to 3 engineers, thus reducing the problem to a previously solved joke.
When a mathematician needs to prove that A implies B, they may instead prove that A implies C where "C implies B" was already proved by someone else, or in a previous theorem.
Q: What's big, grey, and proves the uncountability of the reals? A: Cantor's diagonal elephant.
The joke is referring to the Cantor Diagonal Argument, a proof technique that Cantor originally used to prove that even if you tried to associate one real number with every integer, there'd still be
real numbers left over. (Amazingly, you can "count" the rational numbers - i.e. all of the possible fractional numbers. As a math major to show you sometime, it's a neat trick.)
Q: What's yellow and equivalent to the Axiom of Choice? A: Zorn's Lemon.
Zorn's Lemma is a mathematical statement which turns out to be true if the Axiom of Choice is assumed to be true, or false if the Axiom of Choice is assumed to be false.
Q: What's yellow, normed, and complete? A: A Bananach space.
A Banach Space is space (a set of numbers with a lot of useful operations defined on them) that has a normalization operator defined, and is "complete", which means that the limits of all sequences
you can define using numbers in the space are also in the space.
Q: What is very old, used by farmers, and obeys the fundamental theorem of arithmetic? A: An antique tractorisation domain.
A pun on an unique factorization domain.
Q: What is hallucinogenic and exists for every group with order divisible by p^k? A: A psilocybin p-subgroup.
A Sylow p-Subgroup is a certain type of subgroup (see the definition of a group above).
Q: What is often used by Canadians to help solve certain differential equations? A: the Lacrosse transform.
The Laplace Transform is a technique that makes certain differential equations a lot easier to solve - essentially you take a complicated D.E., substitute certain things in place of any derivatives
you see by looking them up in a table, then solve the resulting equation using normal algebra, and finally transform it back also by looking up things in a table.
Q: What is clear and used by trendy sophisticated engineers to solve other differential equations? A: The Perrier transform.
The Fourier Transform is also used in signal processing, including sound analysis and sound compression algorithms like MP3 and Ogg Vorbis.
Q: Who knows everything there is to be known about vector analysis? A: The Oracle of del phi!
The Del operator is used to express various types of vector derivatives, and phi is just a greek letter that is often used to represent vectors.
"I am just a simple Pole in a complex plane"
Given a complex function, a pole is just a point where the function is not defined (usually because something goes to infinity).
So, they just had to rely on the method of steepest descents.
A way to find the nearest local minimum of a function - works whenever the function is smooth near that minimum.
Adders can't multiply without their log tables.
This is how slide rules work too, BTW - to multiply x and y, look up the logarithm of x, and the logarithm of y, then add those, and then take the exponential of the result. | {"url":"http://www.dominic-mazzoni.com/mathanswers.html","timestamp":"2014-04-18T03:25:13Z","content_type":null,"content_length":"5394","record_id":"<urn:uuid:b9b80a65-0649-46d5-9e54-ee9564882425>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00198-ip-10-147-4-33.ec2.internal.warc.gz"} |
Humble Algebra 1 Tutor
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Physics Forums - Closed set (metric spaces)
Closed set (metric spaces)
Suppose [itex]f:\mathbb{R}\to \mathbb{R}[/itex] is a continuous function (standard metric).
Show that its graph [itex]\{ (x,f(x)) : x \in \mathbb{R} \}[/itex] is a closed subset of [itex]\mathbb{R}^2[/itex] (Euclidean metric).
How to show this is closed?
lanedance Nov7-11 03:01 PM
Re: Closed set (metric spaces)
what are your definitions of closed?
thinking geometrically, a continuous function will have a graph that is an unbroken curve in the 2D plane, how would you show this is closed in R^2
Re: Closed set (metric spaces)
Quote by lanedance (Post 3603566)
what are your definitions of closed?
thinking geometrically, a continuous function will have a graph that is an unbroken curve in the 2D plane, how would you show this is closed in R^2
Well a set [itex]A[/itex] is closed if [itex]\partial A \subset A[/itex], i.e. [itex]\partial A \cap A^c = \emptyset[/itex]
Re: Closed set (metric spaces)
How could I show it is closed by considering the function [itex]f : \mathbb{R}^2 \to \mathbb{R}[/itex] defined by [itex]f(x,y)=f(x)- y[/itex]?
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Summary: Notes on cellular automata
J.-P. Allouche, M. Courbage, G. Skordev
1 Introduction
Cellular automata were introduced by J. von Neumann (see 29]) after a suggestion of S. Ulam
147, p. 274]. They are a self-reproducing model, that was designed in order to answer the question
\is it possible to construct robots that can construct identical robots, i.e., robots with the same
\complexity"?". The model proposed by von Neumann gives a positive answer to this question.
Another \philosophical" background is the production of order from chaos and the concept of
\self-organization" (see for example 11], see also 125]).
It is of course tempting to see life itself behind self-reproduction. This might be the reason for
the choice of many expressions in this theory: cells, living or dead structures, garden of Eden, game
of Life...
2 The game of Life
The most popular example of cellular automaton is the so-called \Game of Life" introduced by
Conway in the 70's. The reference given for example in 156, p. 66] is: J. H. Conway, 1970 unpub-
lished. Other references are the articles of M. Gardner in 1970{1972 in Scienti c American, see for
example 68], and the books 19, ch. 25] and 30]. Note that this game is named after Conway and
Golay in 120] (Golay's game has an underlying hexagonal tiling according to 69]).
This game is de ned as follows. We have an in nite two-dimensional board whose elementary
squares are called \cells". A cell can be \living" or \dead". The neighbors of a cell are de ned to | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/865/0146785.html","timestamp":"2014-04-20T14:14:52Z","content_type":null,"content_length":"8613","record_id":"<urn:uuid:7a88ef21-875b-4123-9b16-61272027273d>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00247-ip-10-147-4-33.ec2.internal.warc.gz"} |
Build you're own sunburst deck railing....step by step
All of us admire fancy woodworking while wishing we had the skills to include it in our home projects. The sunburst deck railing is one such feature that attracts attention and you can build
The sunburst deck rail design is relatively easy to build, but requires a number of angle cuts. This is made easier with a power miter saw. Our Tools section discusses one such saw that is
relatively low priced (at $139) and useful for many projects around the home.
The sunburst deck railing design works best with upright post spacing of 4-5 feet between posts. The sunburst shown in the photo to the right is over 7 feet long, therefore requires more 2x2 spokes
than a shorter span.
Step 1. Install upright 4x4 posts evenly spaced on each straight run of rail. To determine the spacing, divide the total distance to be covered by the railing (in a straight line) by 4 to see how
many 4 foot spaces your distance requires. (See Example calculation below) Since you want each post-to-post space to be the same, you divide any remaining "partial" space by the total number of
post-to-post spaces in that section of railing. (Example: 18 feet of straight line railing distance, divided by 4, equals 4.5 sections to cover 18 feet. Since we can't build 4 1/2 sections we need to
divide the 1/2 section among the remaining 4 sections equally. 1/2 section equals 2 feet long. 2 feet divided by the 4 sections of railing is 6 inches added to each section. Each section of railing
will be 4 feet 6 inches long, therefore space each post 4 feet 6 inches apart.)
Step 2. Once your upright posts are in place fasten a 2x4 horizontally from post to post at the top and 4" off the deck at the bottom. Mark the center point of these top and bottom boards.
Step 3. Use 2x2 PT lumber as the pickets for the railing. Reference Figure B below and secure one of the two semi-circular cover boards to the bottom horizontal board. Measure and fasten your first
picket in place which is the center vertical one. The tops of all pickets are toe-nailed into the board they touch. The bottom of all pickets are face nailed into the semi-circular cover (cut from
2x10 board). See Figure A for an example of one picket installed.
Cut a taper in the bottom 3 inches of each picket as shown in figure C to allow them to be positioned closely together. Location "X" in figure C is where the largest space occurs between pickets.
This distance cannot exceed the greatest picket spacing allowed by your local building code. The "X" distance will determine your picket spacing, and the number of pickets needed to complete the
railing. To install the rest of the pickets, hold in position determined by spacing distance "X", and mark angle, cut and nail. After all pickets are installed, fasten other semi-circular cover in
place. Remember you toe nail the top and face nail the bottom of each picket. Repeat for each remaining section of railing. Enjoy and let your friends wonder how you did it! | {"url":"http://pages.areaguides.com/ubuild/SunburstDeckRail.htm","timestamp":"2014-04-18T16:50:55Z","content_type":null,"content_length":"7611","record_id":"<urn:uuid:618f8d38-8a5d-44d4-b516-996bcfe0fd2a>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00371-ip-10-147-4-33.ec2.internal.warc.gz"} |
Circular Cross Correlation - File Exchange - MATLAB Central
CXCORR Circular Cross Correlation function estimates.
CXCORR(a,b), where a and b represent samples taken over time interval T which is assumed to be a common period of two corresponding periodic signals.
a and b are supposed to be length M row vectors, either real or complex.
[x,c]=CXCORR(a,b) returns the length M-1 circular cross correlation sequence c with corresponding lags x.
The circular cross correlation is:
c(k) = sum[a(n)*conj(b(n+k))]/[norm(a)*norm(b)];
where vector b is shifted CIRCULARLY by k samples.
The function doesn't check the format of input vectors a and b!
For circular covariance between a and b look for CXCOV(a,b) in
A. V. Oppenheim, R. W. Schafer and J. R. Buck, Discrete-Time Signal Processing, Upper Saddler River, NJ : Prentice Hall, 1999.
Author: G. Levin, April 2004.
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Can someone help me please... Thanks A new cruise ship line has just launched 3 new ships; the Pacific Paradise, the Carribean Paradise and the Mediterranean Paradise. The Carribean Paradise has 28
more deluxe staterooms than the Pacific Paradise. The Mediterranean Paradise has 29 fewer deluxe staterooms than four times the number of deluxe staterooms on the Pacific Paradise. Find the number of
deluxe staterooms for each of the ships if the total number of deluxe staterooms for the three ships is 881. The Pacific Paradise has______deluxe staterooms The Caribbean Paradise has____deluxe
• one year ago
• one year ago
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Step 1 Read the problem carefully. Decide what unknown numbers are asked for and what facts are known.
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Im still confuse with this problem.
Best Response
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Can someone help me please... Thanks A new cruise ship line has just launched 3 new ships; the Pacific Paradise, the Carribean Paradise and the Mediterranean Paradise. The Carribean Paradise has
28 more deluxe staterooms than the Pacific Paradise. The Mediterranean Paradise has 29 fewer deluxe staterooms than four times the number of deluxe staterooms on the Pacific Paradise. Find the
number of deluxe staterooms for each of the ships if the total number of deluxe staterooms for the three ships is 881. The Pacific Paradise has______deluxe staterooms The Caribbean Paradise
has____deluxe The mediterranean Paradise has___deluxe staterooms.
Best Response
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Decide what unknown numbers are asked for and what facts are known.
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Decide what unknown numbers are asked for. Find the number of deluxe staterooms for each of the ships
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I still dont get this
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Let P = the number of rooms on the Pacific, since The Carribean Paradise has 28 more deluxe staterooms than the Pacific Paradise. Then C = 28 + P.
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The Mediterranean Paradise has 29 fewer deluxe staterooms than four times the number of deluxe staterooms on the Pacific Paradise. M =4P - 29
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P, C, M P = the number of rooms on the Pacific C = 28 + P M =4P - 29
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Step 3 Reread the problem and write an equation that represents relations among the numbers in the problem.
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P + C + M = 881 if the total number of deluxe staterooms for the three ships is 881.
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Substitute into P + C + M = 881 the three: P = the number of rooms on the Pacific C = 28 + P M =4P - 29
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P + C + M = 881 P +(28 + P) + (4P -29) = 881
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so, p+(28+p)+(4p-29)=881 = 147
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sorry i got confuse with the problem. didnt mean too put 147
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Step 4 Solve the equation and find the unknowns asked for.
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ok. how do i find the unknow answer for The Pacific Paradise has______deluxe staterooms The Caribbean Paradise has____deluxe The mediterranean Paradise has___deluxe staterooms
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Solve the equation: p+(28+p)+(4p-29)=881 for p= the number of rooms on the Pacific
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the number of rooms on the ship is 881
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Solve the equation: p+(28+p)+(4p-29)=881 for p
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Very good :-)
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What are the answers for all three. p= 147 The Pacific Paradise has______deluxe staterooms The Caribbean Paradise has____deluxe The mediterranean Paradise has___deluxe staterooms
Best Response
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So now you know how many rooms are on the Pacific= 147rooms. To find the others substitute this value into: C = 28 + P M =4P - 29
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c= 28+P=147
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im still confuse a little.
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The Pacific Paradise has______deluxe staterooms The Caribbean Paradise has____deluxe The mediterranean Paradise has___deluxe staterooms
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Sub p=147 into c= 28+p
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Best Response
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To find the number of rooms on the Mediterranean Paradise sub P = 147 into: M =4P - 29
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Step 5 Check your results with the words of the problem. Give the answer.
Your question is ready. Sign up for free to start getting answers.
is replying to Can someone tell me what button the professor is hitting...
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
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Hitting time probability in a Random Walk with possibility to die.
up vote 1 down vote favorite
A Random Walker can move of one unit to the right with probability $p$, to the left with probability $q$ and it can jump again to the starting point with probability $r$ and die. Naturally $p+q+r=1$.
The walker starts moving from $x=0$ at time $t=0$. Two barriers are located in $x=n$ and $x=-n$.
Let's call $\tau (n)$ the first time the random walk hits one of the two barriers. Is there a way to determine how does $E[ \tau ] (n)$, (i.e. the expected time the walker hits one of the two
barriers) depend on $n$?
I would be happy if for some values of the parameters $p$, $q$, $r$, the expectation $E[ \tau ] (n)$ grows exponentially with $n$, but probably it's not like this...
random-walk stochastic-processes markov-chains pr.probability
3 If $r\gt0$ there are simple exponential upper and lower bounds. Why do you say the expectation probably isn't exponential? – Douglas Zare Mar 27 '13 at 11:48
add comment
1 Answer
active oldest votes
First, let me elaborate on my comment. If $X$ occurs with probability $\rho$, then the expected waiting time before you first see a streak of length $\ell$ $X$s in a row in independent
trials is exponential in $\ell$, $\frac{\rho^{-\ell}-1}{1-\rho}$. If you get $2n-1$ right steps in a row, then you must have hit the barrier, which gives an exponential upper bound for the
expected first hitting time, $\frac{p^{1-2n}-1}{q+r}$. On the other hand, to reach a barrier, you need a streak of at least $n$ non-deaths. Again, the waiting time before you first see such
a streak is exponential for $r\gt 0$, $\frac{(p+q)^{-n}-1}{r}$. If $q=0$ or $p=0$, the lower bound is sharp.
Imagine we release a particle at $0$ once, and watch it until it reaches a barrier or dies. Let the average number of steps it takes be $s$, and let the probability that it dies be $d$. Then
the expected number of steps before some particle hits a barrier is $s(1+d+d^2+...) = \frac{s}{1-d}$. Both $s$ and $d$ can be calculated analytically, but I think it's easier to let
Mathematica do it. Here are Mathematica commands which find them:
ProbReachesBarrier[k_] := Evaluate[a[k] /.
RSolve[{a[k] == p a[k + 1] + q a[k - 1], a[n] == 1, a[-n] == 1},
a[k], k][[1]] ]
ProbDies = Simplify[1 - ProbReachesBarrier[0]]
up vote
1 down Let $\alpha = \frac{1+\sqrt{1-4pq}}{p}, \beta = \frac{1-\sqrt{1-4pq}}{p}$.
$$d = - \frac{(2^n-\alpha^n)(2^n-\beta^n)}{2^n(\alpha^n + \beta^n)}$$
ESteps[k_] := Evaluate[b[k] /.
RSolve[{b[k] == 1 + p b[k + 1] + q b[k - 1], b[n] == 0, b[-n] == 0},
b[k], k][[1]] ]
s = Simplify[ESteps[0]]
$$ s = - \frac{(2^n - \alpha^n)(2^n-\beta^n)}{r 2^n (\alpha^n + \beta^n)} = \frac{d}{r}$$
$$\frac{s}{1-d} = - \frac{(2^n-\alpha^n)(2^n-\beta^n)}{r (4^n + \alpha^n \beta^n)}.$$
See also mathematica-journal.com/2012/03/… – Douglas Zare Mar 28 '13 at 8:16
You have never defined death. The OP certainly has not defined it. The recursion equation of a[k] seems to suggest that you consider jumping back to k=0 makes a[0], presumably the
probability of reaching either -1 or 1, zero. That is wrong. – Hansen Dec 11 '13 at 2:18
The correct recursion for a is: a[k] = p a[k+1]+q a[k-1]+r a[0]. One can solve it with a generating function. – Hansen Dec 11 '13 at 2:50
@Hansen: Death means jumping back to the start. I computed the probability that you make it to the boundary without dying, and the expected number of steps before you die or restart. If
you calculate the recurrence you suggest, that's the probability of making it to the boundary at some point, which is $1$ when the system is not degenerate, hence not an interesting thing
to compute. That $1$ is not what you need to compute the expected exit time from the expected value of the steps to reach the boundary or restart. – Douglas Zare Dec 11 '13 at 7:05
If that is how you meant "death" to be, you should have stated it explicitly in your answer. Are you letting the particle disappear or die once it jumps back to the starting point? That is
not what the original question asks. The original question stipulates that the point restarts once it comes back to the starting point, and it asks for the expected time to reach the
boundary (one of the two barriers) rather than the question you modifies to which is the expected time to reach the boundary OR the starting point. So your answer does not answer the
original question. – Hansen Dec 11 '13 at 16:06
show 9 more comments
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[Numpy-discussion] performance matrix multiplication vs. matlab
Bruce Southey bsouthey@gmail....
Mon Jan 18 13:10:01 CST 2010
On 01/18/2010 12:47 PM, Vicente Sole wrote:
> Quoting Bruce Southey <bsouthey@gmail.com>:
>> If you obtain the code from any package then you are bound by the terms
>> of that code. So while a user might not be 'inconvenienced' by the LGPL,
>> they are required to meet the terms as required. For some licenses (like
>> the LGPL) these terms do not really apply until you distribute the code
>> but that does not mean that the user is exempt from the licensing terms
>> of that code because they have not distributed their code (yet).
>> Furthermore there are a number of numpy users that download the numpy
>> project for further distribution such as Enthought, packagers for Linux
>> distributions and developers of projects like Python(x,y). Some of these
>> users would be inconvenienced because binary-only distributions would
>> not be permitted in any form.
> I think people are confusing LGPL and GPL...
Not at all.
> I can distribute my code in binary form without any restriction when
> using an LGPL library UNLESS I have modified the library itself.
I do not interpret the LGPL version 3 in this way:
A "Combined Work" is a work produced by combining or linking an
Application with the Library.
So you must apply section 4, in particular, provide the "Minimal
Corresponding Source":
The "Minimal Corresponding Source" for a Combined Work means the
Corresponding Source for the Combined Work, excluding any source code
for portions of the Combined Work that, considered in isolation, are
based on the Application, and not on the Linked Version.
So a binary-only is usually not appropriate.
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Multiplication/Division problem with big
Hey folks,
Recently I wrote a program to the athematic operations with large numbers traditionally known as bigint.
Last edited on
I'm no expert, but I think there's probably a problem with checking the relational operators "!=" and ">=" for the Bigint class.
I agree with fg109. It seems like the repeated addition/subtraction methods should work.
EDIT: I would expect that your method for the big value case would never return a result. With num2 = 8528587375585 that's about 85 trillion iterations!
I also agree that a different algorithm for * and / would be useful.
The + operation as you have it works by implementing the same method one would use to make the calculation by hand, which you already know because you intentionally wrote the + function to do that.
You can write functions which implement the by hand methods for multiplying or dividing two numbers also.
I found writing the function to do long division a bit tricky but I got it working.
Since the operands are integers you might want to implement both / (quotient) and % (remainder) operators, just like "real" integers. Both of these results are found by the long division function.
Write the operators to perform the division then return either the quotient or the remainder.
Last edited on
Well I heard shifting will solve the problem with very big second operand in multiplication and in division with big results.
Last edited on
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What to read to catch up on multivariate statistics? - Statistical Modeling, Causal Inference, and Social Science
What to read to catch up on multivariate statistics?
Henry Harpending writes:
I am writing to ask you for a recommendation of something I can read to catch up on multivariate statistics. I am happy with random processes and linear algebra since they are important in
population genetics. My last encounter with real statistics was several decades ago.
Recently I have had to dip my toes into real multivariate statistics again and I am completely lost. I can’t, for example, figure out how a random effects model is different from what we used to
call “partialing out” nuisance covariates. I have a hard time concentrating on exactly what a “BLURP” model is because the name is so silly.
Can you recommend something accessible to me that would put me on track?
My reply: if you’re interested particularly in random effects models, I will (parochially) refer you to my own book with Jennifer Hill. You can jump straight to the chapters on multilevel modeling.
If the question is about traditional multivariate methods such as factor analysis, principal components, etc., that I don’t really know! But I think my book would be a good start.
Do readers have any suggestions for a good book, preferably model-based, on multivariate methods such as factor analysis, principal components, etc.?
17 Comments
1. I learned factor analysis from one of Geoffrey McLachlan’s books, either Finite Mixture Models or The EM Algorithm and Its Extensions. I don’t recall which at the moment but probably the latter.
I found both books quite useful in general. I also had a copy of Johnson and Wichern, Applied Multivariate Statistical Analysis, on my desk. It was useful and, relative to McLachlan’s texts,
seemed more targeted to an intro audience. (My recollection of J&W is that the text-to-equation ratio was lower than ideal.)
My previous employer owned all three of the aforementioned books and I no longer have any them at hand. Every few weeks I find myself wishing I did – not the particular texts so much as wishing I
had a few good reference texts on my shelf – so I’m in the market for a book (or two) which covers multivariate methods. Andrew, your book with Hill as well Bayesian Data Analysis are on my short
list. I understand a new edition of the latter is due out in July. Should I wait for the new edition?
□ Chris:
Yup, you should wait. BDA3 is awesome.
2. From a Machine Learning and in particular a Probabilistic Graphical Model perspective, I feel compelled to mention “Pattern Recognition and Machine Learning” by Christopher Bishop.
□ I’m not familiar with the book but Tipping and Bishop’s “Probabilistic Principal Components Analysis”, J. R. Statist. Soc. B (1999), vol. 61, Part 3, pp. 611-622 is an excellent paper.
3. I’d recommend Applied Multivariate Statistical Analysis by Johnson and Wichern.
□ +1
4. I second Andrew’s recommendation of his book. It’s very clear and actually fun to read.
The other book worth buying is Mostly Harmless Econometrics. It takes a pretty different view of causal inference (only attempt it when you have an experiment/tenure-getting instrument), but is
entertaining and well written.
5. ‘Numerical Ecology in R’ by Borcard et al. is a nice overview of methods with practical examples – not really model based but covers all the standard methods used in environmental science,
engineering, etc.
6. I like ‘Legendre & Legendre (2012). Numerical Ecology.’ for multivariate stats.
For a more model-based multivariata analysis the work of David Warton looks promising:
7. A long time ago I read Seber´s book and Gnanadesikan´s book; now, they could be outdated but I liked.
Hopufully you could find them in a library or as third hand book.
8. I’d suggest “The Essence of Multivariate Thinking” by Lisa Harlow.
9. There are classics such as T.W. Anderson’s and Johnson and Wichern. There is a fairly updated volume titled Modern Multivariate Statistical Techniques by Alan Izenman which covers usual topics
such as dimensionality reduction, clustering, regression, etc., as well as selected topics in machine learning.
10. I’ve always been a fan of Mardia, Kent & Bibby’s “Multivariate Analysis.” Some may find it a little old-fashioned but the exposition is extremely clear. I find it both charming and amusing that
there’s a chapter in the book called “Econometrics.”
11. Bernhard Flury und Hans Riedwyl: computergestützte Analyse mehrdimensionaler Daten G. Fischer, 1983
According to the website of Alan Izenman where the dataset can be found: http://astro.temple.edu/~alan/SwissBankNotes.txt the english translation seems to be
Flury, B. and Riedwyl, H. (1988).
Multivariate Statistics, A Practical Approach, Cambridge University Press.
The book covers basic multivariate statistics by one extended example, namly detecting false swiss banknotes from real ones.
I am autrian, not swiss, so there is no misguided patriotism involved in recommending a book by swiss authors.
(Hans Riedwyl by the way spent a lot of years in Bloomington, Indiana http://www.stat.indiana.edu/flury.phtml and ironically did not die beeing struck by lightening or hit by a tornado but in the
italian alps)
12. Dillon and Goldstein’s book Multivariate Analysis remains one of the clearest expositions I’ve ever read. It’s really great, covers all the classic techniques and rumors are that Dillon is
working on an update. Also Harry Harman’s Modern Factor Analysis is lucid and very thorough. That said, both books are 25+ years since publications.
13. This is a useful resource: http://stats.stackexchange.com/questions/26372/introductory-book-for-multivariate-statistics
14. My wife has a paper on principal component analysis that is under peer review for publication right now. She suggests that when it comes to PCA that “Principal Component Analysis” by Wold,
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Math Help
July 25th 2006, 04:32 PM #1
Jun 2006
What is the fractional decrease in pressure when a barometer is raised 35 m to the top of a building? (Assume that the density of air is constant over that distance.)
What is the fractional decrease in pressure when a barometer is raised 35 m to the top of a building? (Assume that the density of air is constant over that distance.)
There is insufficient information in this problem to solve it.
If this was my problem I would look up in my text book or do a google search
for the altitude dependence of air pressure (giving priority to my text book
or what I have been told by my teacher).
There I might find air pressure drops by about 1 part in 1000 for every 8m
increase in altitude, for altitudes of less than a few hundred metres.
Alternatively you might find that the half height of the atmosphere for air
pressure is ~5000m
We really need to know what you are supposed to know about atmosheric
pressure before we can help you with this question.
What is the fractional decrease in pressure when a barometer is raised 35 m to the top of a building? (Assume that the density of air is constant over that distance.)
As CaptainBlack mentioned, I'm not sure of what you are using for your constants, but in general:
$P = P_0 + \rho g \Delta h$
so the fractional decrease would be
$\frac{\rho g \Delta h}{P}$.
(I'm presuming P is at sea level.)
July 25th 2006, 07:55 PM #2
Grand Panjandrum
Nov 2005
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Kendall's Tau and Spearman's Rank Correlation Coefficient - Statistics Solutions
There are two accepted measures of non-parametric rank correlations: Kendall’s tau and Spearman’s (rho) rank correlation coefficient.
Correlation analyses measure the strength of the relationship between two variables.
Kendall’s tau and Spearman’s rank correlation coefficient assess statistical associations based on the ranks of the data. Ranking data is carried out on the variables that are separately put in
order and are numbered.
Correlation coefficients take the values between minus one and plus one. The positive correlation signifies that the ranks of both the variables are increasing. On the other hand, the negative
correlation signifies that as the rank of one variable is increased, the rank of the other variable is decreased.
General purpose:
Correlation analyses can be used to test for associations in hypothesis testing. The null hypothesis is that there is no association between the variables under study. Thus, the purpose is to
investigate the possible association in the underlying variables. It would be incorrect to write the null hypothesis as having no rank correlation between the variables.
Kendall’s Tau: usually smaller values than Spearman’s rho correlation. Calculations based on concordant and discordant pairs. Insensitive to error. P values are more accurate with smaller sample
Spearman’s rho: usually have larger values than Kendall’s Tau. Calculations based on deviations. Much more sensitive to error and discrepancies in data.
The main advantages of using Kendall’s tau are as follows:
• The distribution of Kendall’s tau has better statistical properties.
• The interpretation of Kendall’s tau in terms of the probabilities of observing the agreeable (concordant) and non-agreeable (discordant) pairs is very direct.
• In most of the situations, the interpretations of Kendall’s tau and Spearman’s rank correlation coefficient are very similar and thus invariably lead to the same inferences.
Spearman’s rank correlation coefficient is the more widely used rank correlation coefficient.
Symbolically, Spearman’s rank correlation coefficient is denoted by r[s] . It is given by the following formula:
r[s] = 1- (6∑d[i]^2 )/ (n (n^2-1))
*Here d[i] represents the difference in the ranks given to the values of the variable for each item of the particular data
This formula is applied in cases when there are no tied ranks. However, in the case of fewer numbers of tied ranks, this approximation of Spearman’s rank correlation coefficient provides
sufficiently good approximations.
Key terms:
Non-parametric test: it does not depend upon the assumptions of various underlying distributions; this means that it is distribution free.
Concordant pairs: if both members of one observation are larger than their respective members of the other observations
Discordant pairs: if the two numbers in one observation differ in opposite directions
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Re: st: loglikelihood and loglikelihood ratio
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Re: st: loglikelihood and loglikelihood ratio
From jjc.li@utoronto.ca
To statalist@hsphsun2.harvard.edu
Subject Re: st: loglikelihood and loglikelihood ratio
Date Tue, 17 Mar 2009 10:18:17 -0400
Hi Maarten,
1. My loglikelihood is positive, not negative. Sorry for the mistake in the previous email.
2 A is the unrestricted model, B is the restricted model. Following is the test result. What's wrong with it?
. lrtest A B
(log-likelihoods of null models cannot be compared)
Likelihood-ratio test LR chi2(3) = 10.94
(Assumption: B nested in A) Prob > chi2 = 0.0121
Quoting Maarten buis <maartenbuis@yahoo.co.uk>:
--- On Tue, 17/3/09, jjc.li@utoronto.ca wrote:
I used -sureg- to estimate a system.
1. After using -display e(ll)- to show the loglikelihood,
it gives a negative value. But I remember I have saw a PPT
says loglikelihood is a typical negative value. Is mine
All the information you have given us is that your log
likelihood is negative. That is not abnormal, but a positive
log likelihood would not be a reason for concern either, so
all this tells you exactly nothing.
2. In order to compare unrestricted and restricted models,
I used -lrtest- to do the loglikelihood ratio test. The LR
chi2() value is negative.Is it normal? What does it mean?
Restricted model better or unrestricted model better? If
model one LR chi2()=-1, model LR chi2()= -10, which one to
This is not normal, so it means you have done something
wrong. Are you sure that you are comparing comparable models?
3. Is loglikelihood and loglikelihood ratio different?
yes, log liklihood is the log liklihood of a single model,
while the log liklihood ratio is the ratio of log likelihoods
of two different models.
-- Maarten
Maarten L. Buis
Institut fuer Soziologie
Universitaet Tuebingen
Wilhelmstrasse 36
72074 Tuebingen
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ksvanhorn comments on Probability is in the Mind - Less Wrong
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"Is it not true that when one rolls a fair 1d6, there is an actual 1/6 probability of getting any one specific value?"
No. The unpredictability of a die roll or coin flip is not due to any inherent physical property of the objects; it is simply due to lack of information. Even with quantum uncertainty, you could
predict the result of a coin flip or die roll with high accuracy if you had precise enough measurements of the initial conditions.
Let's look at the simpler case of the coin flip. As Jaynes explains it, consider the phase space for the coin's motion at the moment it leaves your fingers. Some points in that phase space will
result in the coin landing heads up; color these points black. Other points in the phase space will result in the coin landing tails up; color these points white. If you examined the phase space
under a microscope (metaphorically speaking) you would see an intricate pattern of black and white, with even a small movement in the phase space crossing many boundaries between a black region and a
white region.
If you knew the initial conditions precisely enough, you would know whether the coin was in a white or black region of phase space, and you would then have a probability of either 1 or 0 for it
coming up heads.
It's more typical that we don't have such precise measurements, and so we can only pin down the coin's location in phase space to a region that contains many, many black subregions and many, many
white subregions... effectively it's just gray, and the shade of gray is your probability for heads given your measurement of the initial conditions.
So you see that the answer to "what is the probability of the coin landing heads up" depends on what information you have available.
Of course, in practice you typically don't even have the lesser level of information assumed above -- you don't know enough about the coin, even in principle, to compute which points in phase space
are black and which are white, or what proportion of the points are black versus white in the region corresponding to what you know about the initial conditions. Here's where symmetry arguments then
give you P(heads) = 1/2.
Case in point:
There are dice designed with very sharp corners in order to improve their randomness.
If randomness were an inherent property of dice, simply refining the shape shouldn't change the randomness, they are still plain balanced dice, after all.
But when you think of a "random" throw of the dice as a combination of the position of the dice in the hand, the angle of the throw, the speed and angle of the dice as they hit the table, the
relative friction between the dice and the table, and the sharpness of the corners as they tumble to a stop, you realize that if you have all the relevant information you can predict the roll of the
dice with high certainty.
It's only because we don't have the relevant information that we say the probabilities are 1/6.
If you knew the initial conditions precisely enough, you would know whether the coin was in a white or black region of phase space, and you would then have a probability of either 1 or 0 for it
coming up heads.
Not necessarily, because of quantum uncertainty and indeterminism -- and yes, they can affect macroscopic systems.
The deeper point is, whilst there is a subjective ignorance-based kind of probability, that does not by itself mean there is not an objective, in-the-territory kind of 0<p<1 probability. The latter
would be down to how the universe works, and you can't tell how the universe works by making conceptual, philosophical-style arguments.
So the kind of probability that is in the mind is in the mind, and the other kind is a separate issue. (Of course, the existence of objective probability doesn't follow from the existence of
subjective probability any more than its non existence does). | {"url":"http://lesswrong.com/lw/oj/probability_is_in_the_mind/3doi","timestamp":"2014-04-20T15:52:41Z","content_type":null,"content_length":"36402","record_id":"<urn:uuid:33f4bb4c-60c3-4ddd-86f2-372e40005f53>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00562-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: May 2009 [00069]
[Date Index] [Thread Index] [Author Index]
Re: mathematica newbie trouble
• To: mathgroup at smc.vnet.net
• Subject: [mg99303] Re: [mg99276] mathematica newbie trouble
• From: Leonid Shifrin <lshifr at gmail.com>
• Date: Sat, 2 May 2009 05:59:30 -0400 (EDT)
• References: <200905010952.FAA02056@smc.vnet.net>
The point is that 1. SetDelayed is a scoping construct, and 2. It does not
evaluate its rhs at the moment when assignment (definition) is made.
The first point means that it creates bindings between formal parameters and
the piece of code on the r.h.s (actually it relegates this to RuleDelayed in
the rule it creates). The second means that It adds
to the global rule base the delayed rule/definition (with its rhs not
evaluated), whose contract is exactly the following: when, during
evaluation, some expression matches the pattern of its lhs, take the
(unevaluated) code on the rhs of the rule, find all occurrences of the
parameters (names) on the lhs in this code, replace them by the actual
parameters passed (present in evaluated expression that matched the rule/def
pattern), and only then start evaluating the code. Thus, by the time the
parameters (say, 3,4,5) are used, it does not know what "s" is (considers it
a literal s), while by the time it starts to evaluate "s", the relations
between the parameter names (a,b,c) and their actual values in this function
call (3,4,5) (their bindings) are long gone (in fact, by that time all of
the a,b,c-s in the original code have been replaced by 3,4,5), so a,b,c in s
remain it their symbolic form.
When you use Set, you force everything to evaluate at the moment of an
assignment, and s gets rewritten into (a + b + c)/2 before the actual
definition for <k> is formed. This problem is then avoided. However, using
Set to define functions is generally inappropriate (there are exceptions).
For example, your parameters a,b,c could have had global values, and then it
wouldn't work as intended. In your particular case, a cleaner solution would
be to use With:
k[a_, b_, c_] := With[{s = (a + b + c)/2}, Sqrt[s (s - a) (s - b) (s -
On Fri, May 1, 2009 at 2:52 AM, Guapo <yangshuai at gmail.com> wrote:
> i wrote the following mathematica code:
> s := (a + b + c)/2;
> k[a_, b_, c_] := Sqrt[s (s - a) (s - b) (s - c)];
> k[3, 4, 5]
> k[5, 9, 12]
> when run it, i can't get the write answer. but i change setDelayed(:=)
> to set(=), everything works ok
> s = (a + b + c)/2;
> k[a_, b_, c_] = Sqrt[s (s - a) (s - b) (s - c)];
> k[3, 4, 5]
> k[5, 9, 12]
> i did a google search for the difference of set and setDelayed,
> however, i still can't understand it for the upper described problem,
> could anyone explain it? great thanks.
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Here's the question you clicked on:
1. Write the basic plot of the story "The Three Little Pigs". 2. Write the basic plot of "Harry Potter" . 3. Describe how each of these plots is similar and how each is different.
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Well, have you read "The Three Little Pigs" and "Harry Potter." If u did, it will be easier for u to write down just the basic plot of each story, and afterwards compare and contrast them.
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no i haven't its been a lonnnng time since i read them...
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ok, so do i have to read the whole book of harry potter again ?
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Read the site i gave u, it contains the summary of harry potter, including the theme and other literary works.
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can u tell me what is a plot exactly ?
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Which book are you trying to refer to? Harry Pooter contains seven books in series.
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Prisoner of Akzaban or Harry Potter and the Sorcerer’s Stone i do love the 3rd book...
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Harry Potter and the Prisoner of Azkaban is the third of the seven Harry Potter books. This book focuses mainly on the supposed crazed murderer Sirius Black who has just escaped the high security
prison, Azkaban, and is rumored to be coming after Harry to kill him.
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so this is the plot?
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And from that, u can develope ur own summary :)
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Do you know how to write a summary?
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one more question... the plot of the movie of the Prisoner of Akzaban & the book.. will the plot remain the same ?
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U mean remain the same with the rest of the series?
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This is the actual plot of the book
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does a movie have a plot?
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But every books of Harry Potter has a differetn plot.
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Plot is a literary term defined as the events that make up a story.
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Every story u read and watch has a plot.
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The Plot basically means what's the story about, from the beginning to the end.
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okay that was just wht i meant.... for the 3rd question... im supposed to read both plots and compare them
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Yes, for the first two, right down the plot of each Story u presented and try to figure out what's similar and difference u found in these stories? Can u tell me what's the plot of "The Three
Little Pigs?"
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There were three pigs and a big bad wolf, two pigs ere lazy to build their house strong and the third pig had a strong house... the wolf blows the two pigs house and tries to blow off the third
pig, but he couldn't.. the wolf tried to enter through the chimney but the third little pig boiled a big pot of water and kept it below the chimney. The wolf fell into it and died.
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Good summary, now what do these two stories have in common?
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well nothing at all..
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Me too i can't find the similarities of these stories. Were these stories recommended by your teacher?
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no i choosed them
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In both stories there is conflict between a villain and a hero. In both cases, there is the diligent work of some people, but not of others. (Think of the way Buckbeak gets convicted because of a
student that does not bother to follow the rules.) There are some serious differences as well. The villain of the three little pigs simply is as stated. In Harry Potter, the Prisoner of Azkaban,
the villain is hidden.
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yea thats absolutely right !
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Also, both are morality tales. The moral of HP is to not judge people until know them and have all the facts. The moral of the 3P is on diligence of work and making good choices.
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whts a conflict?
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got it.... thank you very much both @Zale101 & @e.mccormick for helping me out !
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a conflict is the problem of this story, let's say the three little pigs for example, the conflict in that story is that the the wolf is trying to blow their house down.
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No problem @tanya123
Your question is ready. Sign up for free to start getting answers.
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
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Semiclassical foundation of universality in quantum chaos
We sketch the semiclassical core of a proof of the so-called Bohigas-Giannoni-Schmit conjecture: A dynamical system with full classical chaos has a quantum energy spectrum with universal fluctuations
on the scale of the mean level spacing. We show how in the semiclassical limit all system specific properties fade away, leaving only ergodicity, hyperbolicity, and combinatorics as agents
determining the contributions of pairs of classical periodic orbits to the quantum spectral form factor. The small-time form factor is thus reproduced semiclassically. Bridges between classical
orbits and (the non-linear sigma model of) quantum field theory are built by revealing the contributing orbit pairs as topologically equivalent to Feynman diagrams.
Related Links | {"url":"http://www.newton.ac.uk/programmes/RMA/Abstract4/muller.html","timestamp":"2014-04-20T08:20:39Z","content_type":null,"content_length":"3548","record_id":"<urn:uuid:4dc90eaa-396d-4d6e-98a3-29d72a878cce>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00214-ip-10-147-4-33.ec2.internal.warc.gz"} |
Energy asymptotics for Gaudin spin chains.
An Isaac Newton Institute Workshop
Random Matrix Theory and Arithmetic Aspects of Quantum Chaos
Energy asymptotics for Gaudin spin chains.
Author: John Toth (McGill University)
I will discuss recent work (joint with M. Min-Oo) on partition function asymptotics for the integrable Gaudin spin chains in various thermodynamic regimes.
I will discuss recent work (joint with M. Min-Oo) on partition function asymptotics for the integrable Gaudin spin chains in various thermodynamic regimes. | {"url":"http://www.newton.ac.uk/programmes/RMA/Abstract4/toth.html","timestamp":"2014-04-18T14:09:43Z","content_type":null,"content_length":"2244","record_id":"<urn:uuid:1809ab4a-bc14-48a3-90c1-fc84be737b46>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00588-ip-10-147-4-33.ec2.internal.warc.gz"} |
Classic Diophantine problem
March 26th 2009, 09:05 PM #1
Mar 2009
Classic Diophantine problem
For two postage stamp values, one of *n* cents and one of *m* cents
the following Diophantine equation
mx + ny = T
gives the total value of postage or T, where x and y represent the number of each denomination of stamps
and are therefore integers. Clearly, you could get many values for T.
But there are also many values of T that you could NOT get.
I have two questions regarding this situation:
1) Under what conditions will stamp amounts *m* and *n* generate all but a finite number of
postage stamp amounts?
2) Under the conditions of question one (the stamps generate all but a finite number of amounts)
what is a *formula* for the largest postage value that CANNOT be generated?
Any help??
□ If $(m,n)=M>1$ then $M$ divides any linear combination (with natural coefficients) of them, so we do not generate the numbers which are not multiples of M and so there are infinitely many
numbers left out.
□ If $(m,n)=1$, read here
March 27th 2009, 02:26 AM #2 | {"url":"http://mathhelpforum.com/number-theory/80868-classic-diophantine-problem.html","timestamp":"2014-04-19T00:34:39Z","content_type":null,"content_length":"33382","record_id":"<urn:uuid:854435a3-03ae-4ad8-a6e1-758be0250d45>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00541-ip-10-147-4-33.ec2.internal.warc.gz"} |
Profiles of Women in Mathematics: Judith D. Sally
Judith D. Sally
Noetherian Rings
San Francisco, California 1995
Previous | Index | Next
JUDITH D. SALLY received her BA in mathematics from Barnard College in 1958 and her MA in mathematics from Brandeis University in 1960. The next eight years were devoted mainly to the care and
nurturing of three children. In 1968, the year her youngest child began kindergarten, she returned to study in mathematics at the University of Chicago. She received her PhD in 1971, under the
direction of Irving Kaplansky. After spending 1971-72 as a postdoctoral visitor at Rutgers University, she was hired as a visiting assistant professor by Northwestern University. The visit lasted a
long time! She is currently professor of mathematics there. She received an Alfred P. Sloan Foundation Fellowship in 1977. She spent 1981-82 at Radcliffe College as a Bunting Fellow and 1988-89 at
Purdue University under the auspices of the NSF Visiting Professorship for Women Program. She just completed a three-year appointment as algebra editor of the Transactions.
Sally's research is in Commutative Algebra, one of the fields in which Emmy Noether's work had such impact. Her main interests lie in the study of Noetherian local rings and graded rings with
emphasis on Hilbert functions and birational extensions. These concepts play an important role in ascertaining the nature of singularities in applications in algebraic geometry. The Hilbert function
of a local ring at a point on a variety is a very good measure of how bad the singularity is at the point. One of the themes in Sally's research is the interaction between the local ring and its
associated graded ring. This interaction plays a critical role in understanding and computing the Hilbert function. She has also worked on birational blowing up of ideals, the extention of valuations
and other concepts in the algebra involved in the resolution of singularities.
Sally enjoys teaching and is pleased with the renewed commitment in the field to undergraduate and graduate teaching. She feels her teaching was revitalized when she began emphasizing active student
involvement in class (the syllabus be hanged at times, if necessary!).
In her Noether Lecture, Sally discussed measuring Noetherian rings. Noetherian rings are generally perceived to be the most tractable commutative rings. The well-known finiteness conditions in a
Noetherian ring, namely Noether's ascending chain condition for ideals and the equivalent condition that all ideals are finitely generated, permit interesting finite measures of the "size" and
behavior of such rings. However, it is quite surprising that these same finiteness conditions can also force other measures, which might be finite in some non-Noetherian rings, to be infinite.
Sally's husband, Paul J. Sally, Jr., is Professor of Mathematics at the University of Chicago. At the present time, their eldest son is an assistant professor in the business school at Cornell
University, another son is assistant professor in the Slavic department at Stanford University and their youngest son teaches mathematics and computer science at New Trier High School in Winnetka,
Illinois. Their seven grandchildren, ages one to seven, have many interesting ideas about their future careers but no definitive plans as yet.
Previous | Index | Next
Copyright ©2005 Association for Women in Mathematics. All rights reserved.
Comments: awm-webmaster@awm-math.org. | {"url":"http://www.awm-math.org/noetherbrochure/Sally95.html","timestamp":"2014-04-18T16:08:02Z","content_type":null,"content_length":"5234","record_id":"<urn:uuid:47646ae7-4b18-4e60-a722-df516256a5f5>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00155-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: February 2004 [00410]
[Date Index] [Thread Index] [Author Index]
Re: Solve or LinearSolve or ...?
• To: mathgroup at smc.vnet.net
• Subject: [mg46382] Re: [mg46347] Solve or LinearSolve or ...?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Mon, 16 Feb 2004 23:42:15 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com
Scott Morrison wrote:
> Hi,
> I'm trying to write a function to calculate coefficients in a basis;
> for example
> FindCoefficients[{f[a,a],f[a,b],f[b,a],f[b,b]}, f[a,a]+(1/2)f[b,a],
> _f]
> should produce {1,0,1/2,0}. The third argument there means `assuming
> all objects matching _f are linearly independent'. More difficult, it
> should produce
> FindCoefficients[{f[a]+f[b],f[a]-f[b]},f[a], _f] == {1/2,1/2}.
> And finally, it should work with rational functions as coefficients,
> not just numbers, and it should run fast enough to be useful with
> bases with thousands of elements. :-)
> I've spent quite some time trying to write something like this, and
> I'm finding it really difficult! I can't seem to use LinearSolve in
> any way -- it seems to trip up when I use rational functions as
> coefficients. I've been trying to work around Solve, but the only
> things that work are glacial in pace!
> Any ideas or suggestions? I'd be happy to post some of my attempts if
> they'd be helpful in explaining what I'm trying to do, but mostly I'm
> too embarrassed by them :-)
> Scott Morrison
Here is one method that should be reasonably efficient.
findCoefficients[ll_List, expr_, varhead_] := Module[
{cc, coeffs, newexpr, vars, eqns},
coeffs = Array[cc,Length[ll]];
newexpr = coeffs.ll - expr;
vars = Cases[Variables[{ll,expr}], varhead];
eqns = Thread[Map[D[newexpr,#]&, vars] == 0];
coeffs /. First[Solve[eqns,coeffs]]
In[34]:= InputForm[findCoefficients[{f[a,a],f[a,b],f[b,a],f[b,b]},
f[a,a]+(1/2)f[b,a], _f]]
Out[34]//InputForm= {1, 0, 1/2, 0}
In[35]:= InputForm[findCoefficients[{f[a]+f[b],f[a]-f[b]},f[a], _f]]
Out[35]//InputForm= {1/2, 1/2}
If this is slow on examples of interest then you may want to post
further information describing such examples.
Daniel Lichtblau
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Parallel Sessions
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Andreas Hamel
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A set-valued approach to utility maximization in markets with transaction costs
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Superhedging in markets with transaction costs
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Kasper Larsen
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Unspanned endowment and face-lifting
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A note on consistent price systems
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: Computational Finance I
Chair: Mathew Lorig
Matthew Lorig
(Princeton University)
Pricing derivatives on multiscale diffusions: simplicity through spectral theory
view abstract
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(The City University of New York)
Preventing market crashes through insuring the speed of drawdowns
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: Financial Engineering I
Chair: Michael Okelola
Michael Okelola
(University of KwaZulu-Natal, South Africa)
Group analysis of exotic options
view abstract
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(Florida A & M University)
Explaining the forward rate bias puzzle
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: Fundations of Mathematical Finance II
Chair: Daniel Ocone
Gerard Brunick
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Weak uniqueness for a class of degenerate diffusions with continuous covariance
view abstract
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(Rutgers University)
Mimicking theorem for generalized Heston-like processes
view abstract
Jian Song
(Rutgers University)
A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution
view abstract
Janeway Room
: Optimal Investments and Incomplete Markets
Chair: Maxim Bichuch
Maxim Bichuch
(Princeton University)
Pricing a contingent claim liability using asymptotic analysis for optimal investment in finite time with transaction costs
view abstract
Tim Leung
(Columbia University)
Derivatives purchase timing under risk-neutral and risk-averse pricing rules
view abstract
Oleksii Mostovyi
(Carnegie Mellon University)
Necessary and sufficient conditions in the problem of optimal consumption from investment in incomplete markets
view abstract
Meyer Room
: Computational Finance II
Chair: Ionut Florescu
Ionut Florescu
(Stevens Institute of Technology)
Numerical solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market
view abstract
Abdul Khaliq
(Middle Tennessee State University)
Efficient numerical schemes for pricing exotic path-dependent American options with transaction cost
view abstract
Alexander Shklyarevsky
(Bank of America)
Analytical approaches to the solution of ODEs, PDEs and PIDEs and their application in physics and quantitative finance
view abstract
Bishop Room
: Financial Engineering II
Chair: Andrew Barnes
Andrew Barnes
(GE Global Research Center, Niskayuna, New York)
Conditional expected default rate calculations for credit risk applications
view abstract
Xuedong He
(Columbia University)
Optimal Insurance Design under Rank Dependent Utility
view abstract | {"url":"http://www.finmath.rutgers.edu/mfpde2011/parallel_sessions.php","timestamp":"2014-04-17T18:23:35Z","content_type":null,"content_length":"41214","record_id":"<urn:uuid:4b2555e9-60f3-41c1-a2f0-e2cf54fe6bd5>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00436-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Constant Torque Of 21.5 N·m Is Applied To A ... | Chegg.com
A constant torque of 21.5 N·m is applied to a grindstone whose moment of inertia is 0.130 kg·m2. Using energy principles, and neglecting friction, find the angular speed after the grindstone has made
13.0 revolutions.
Hint: the angular equivalent of
W_(net) = F Delta x = 1/2 mv_f^2 - 1/2 mv_i^2 is W_(net) = tau Delta theta = 1/2 I (omega_f)^2 - 1/2 I (omega_i)^2
You should convince yourself that this relationship is correct.
1 rev/s | {"url":"http://www.chegg.com/homework-help/questions-and-answers/constant-torque-215-n-m-applied-grindstone-whose-moment-inertia-0130-kg-m2-using-energy-pr-q1065577","timestamp":"2014-04-18T17:59:30Z","content_type":null,"content_length":"21182","record_id":"<urn:uuid:a43b9a4e-f026-4353-a6fd-18d5ac6cb252>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00551-ip-10-147-4-33.ec2.internal.warc.gz"} |
Proving continuity multiple ways
April 7th 2010, 03:38 PM #1
Aug 2007
Proving continuity multiple ways
The question asks us to prove that
f(x) = {(2x^2-18)/(x+3) if x =/= -3, and -12 if x= -3}
is continuous at Xo = -3
I proved it fairly easily using the epsilon/delta definition, but now have to do it two other ways.
I know one way is to use sequential characterization of limits, but how would I go about using this to show it's continuous?
I think the other way is to use the definition of a limit and show that f(x) has limit F = f(Xo) at Xo, is this correct?
Any help would be appreciated. Thanks!
Follow Math Help Forum on Facebook and Google+ | {"url":"http://mathhelpforum.com/differential-geometry/137809-proving-continuity-multiple-ways.html","timestamp":"2014-04-24T10:00:54Z","content_type":null,"content_length":"29475","record_id":"<urn:uuid:c5760bb2-3142-4bca-a644-057bdf7ff2f6>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00200-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fredrik Johansson
Thursday, January 26, 2012
I'm moving my blog to
. If you're living in the future and reading this, you should go there instead!
Wednesday, June 8, 2011
Version 2.2 of
(Fast Library for Number Theory) was
last weekend. Some updated
are available.
In this blog post, I'm going to talk a bit about features I contributed in this version. With apologies to
Sebastian Pancratz
who wrote a whole lot of code as well -- in particular, a new module for computing with p-adic numbers, and a module for rational functions! Bill Hart also implemented a faster polynomial GCD, which
is a quite important update since GCD is crucial for most polynomial business. Anyhow...
Polynomial interpolation
I've added various functions for polynomial interpolation to the
module. In general, these can be used to speed up various computations involving integer or rational polynomials by mapping a given problem to (Z/nZ)[x], Z or even Z/nZ, taking advantage of fast
arithmetic in those rings, and then via interpolation recovering a result in Z[x] or Q[x].
Firstly, there are some new Chinese Remainder Theorem functions for integer polynomials, allowing you to reconstruct an integer polynomial from a bunch of polynomials with coefficients modulo
different primes. Straightforward code (the actual work is done by functions in the fmpz module), but useful to have. The CRT functions are used by the new modular GCD code.
There are also functions for evaluating an integer polynomial on a set of points, and forming the interpolating polynomial (generally with rational coefficients) given a set of points and the values
at those points.
Finally, user-friendly functions for evaluation and interpolation at a power of two (Kronecker segmentation) have been added. The code for this is actually a very old part of FLINT, and possibly some
of the most complicated code in the library (packing bits efficiently is surprisingly hard). The new functions just wrap this functionality, but take care of memory management and various special
cases, so you can now just safely do something like:
fmpz_t z;
long bits = fmpz_poly_max_bits(poly) + 1; /* +1 for signs */
fmpz_poly_bit_pack(z, poly, bits);
fmpz_poly_bit_unpack(poly, z, bits); /* recover poly */
Apart from Kronecker segmentation, these functions are not currently asymptotically fast. Fast multi-modulus CRT for coefficient reconstruction is probably not all that important in most
circumstances, because it's more common to use evaluation-interpolation techniques for polynomials of large degree and small coefficients than the other way around. Nonetheless, polynomials with
large coefficients do arise as well. For example, the vector Bernoulli number code in FLINT relies on fast CRT, and currently uses custom code for this.
Polynomial interpolation uses
Lagrange interpolation
with barycentric weights, with a few tricks to avoid fractions. This is all implemented using an O(n^2) algorithm, but the actual time complexity is higher due to the fact that the coefficients when
working over integers usually will be large, around n! in magnitude.
Here are some timing examples, evaluating and recovering a length-n polynomial with +/-1 coefficients basically as follows:
x = _fmpz_vec_init(n);
y = _fmpz_vec_init(n);
for (i = 0; i < n; i++)
x[i] = -n/2 + i;
fmpz_poly_randtest(P, state, n, 1);
fmpz_poly_evaluate_fmpz_vec(y, P, x, n);
fmpz_poly_interpolate_fmpz_vec(Q, x, y, n);
The bits column below measures the largest value in y, which grows quite large despite the input polynomial having small coefficients:
n=8 eval=762 ns interp=13 us bits=8 ok=1
n=16 eval=3662 ns interp=61 us bits=42 ok=1
n=32 eval=29 us interp=673 us bits=-113 ok=1
n=64 eval=136 us interp=4951 us bits=-316 ok=1
n=128 eval=625 us interp=45 ms bits=-762 ok=1
n=256 eval=2500 us interp=792 ms bits=-1779 ok=1
n=512 eval=12 ms interp=10 s bits=-4089 ok=1
As you can see, the interpolation speed is not too bad for small n, but eventually grows out of control. How to do better?
Naive Lagrange interpolation is not optimal: it is possible to do n-point evaluation and interpolation in essentially O(n log
n) operations. Such algorithms do not necessarily lead to an improvement over the integers (you still have to deal with coefficient explosion), but they should win over finite fields. So the right
solution will perhaps be to add polynomial evaluation/interpolation functions based on modular arithmetic.
Rational numbers and matrices
A new module
is provided for computing with arbitrary-precision rational numbers. For the user, the fmpq_t type essentially behaves identically to the MPIR mpq_t type. However, an fmpq_t only takes up two words
of memory when the numerator and denominator are small (less than 2
), whereas an mpq_t always requires six words plus additional heap-allocated space for the actual number data.
The fmpq functions are a bit faster than mpq functions in many cases when the numerator and/or denominator is small. But the main improvement should come for vectors, matrices or polynomials of
rational numbers, due to the significantly reduced memory usage and memory management overhead (especially when many entries are zero or integer-valued).
Some higher-level functionality is also provided in the fmpq module, e.g. for rational reconstruction. The functions for computing special rational numbers (like Bernoulli numbers) have also been
switched over to the fmpq type. Another supported feature is enumeration of the rationals (using the
sequence or by height). Generating the 100 million "first" positive rational numbers takes 9.6 seconds done in order of height, or 2.6 seconds in Calkin-Wilf order.
FLINT actually does not use fmpq's to represent polynomials over Q (fmpq_poly), and probably never will. The fmpq_poly module represents a polynomial over Q as an integer polynomial with a single
common denominator, which is usually faster. The reason for adding the fmpq_t type is that it enabled developing the new
module, which implements dense matrices of rational numbers. For matrices, a common-denominator representation would be less convenient and in many cases completely impractical.
The new FLINT fmpq_mat module is very fast, or at least very non-slow. It is easy to find examples where it does a simple computation a thousand times faster than the rational matrices in Sage.
There's not actually much code in the fmpq_mat module itself; it does almost all "level 3" linear algebra (computations requiring matrix multiplication or Gaussian elimination) by clearing
denominators and computing over the integers. This approach is in fact stolen shamelessly from Sage, but the functions in Sage are highly unoptimized in many cases. The code in Sage still wins for
many sufficiently large problems as it has asymptotically fast algorithms for many things we do not (like computing null spaces). See the
benchmarks page
for more details.
I should not forget to mention that I've implemented Dixon's p-adic algorithm for solving Ax = b for nonsingular square A. (I wish I had a good link for Dixon's algorithm here, but sadly it doesn't
appear to be described conveniently anywhere on the web. The original paper is "
Exact solution of linear equations using P-adic expansions
", if you have the means to get through the Springer paywall.)
This is now used both for solving over both Z and Q. The solver in FLINT is competitive with Sage (which uses
) up to systems of dimension somewhere between perhaps 100 and 1000 (depending greatly on the size of the entries in the inputs and in the solution!). There's much to do here -- we should eventually
have BLAS support in FLINT, which will speed up core matrix arithmetic, but there's room for a lot of algorithmic tuning as well.
There are some other minor new matrix features as well... they can be found in the changelog.
Polynomial matrices
A new module (
) is provided for dense matrices over Z[x], i.e. matrices whose entries are polynomials with integer coefficients. The available functionality includes matrix multiplication, row reduction, and
determinants. Matrix multiplication is particularly fast, as it uses the Kronecker segmentation interpolation/evaluation technique described above. (A similar algorithm is provided for determinants,
but it's not really optimal as this point.)
The benchmarks page has detailed some detailed timings, so I won't repeat them here -- but generally speaking, the FLINT implementation is an order of magnitude faster than Sage or Magma for matrices
of manageable size.
There's much more to be done for polynomial matrices. Row reduction is implemented quite efficiently, but it's too slow as an algorithm for many tasks such as computing null spaces of very large
matrices. A future goal is to implement asymptotically fast algorithms (see the
paper on x-adic lifting
by Burçin Eröcal and Arne Storjohann for example).
Monday, March 14, 2011
Since it's pi day today, I thought I'd share a list of mpmath one-liners for computing the value of pi to high precision using various representations in terms of special functions, infinite series,
integrals, etc. Most of them can already be found as doctest examples in some form in the mpmath documentation.
A few of the formulas explicitly involve pi. Using those to calculate pi is rather
(!), though a few of them could still be used for computing pi using numerical root-finding. In any case, most of the formulas are circular even when pi doesn't appear explicitly since mpmath is
likely using its value internally. In any
case, the majority of the formulas are not efficient for computing pi to very high precision (at least as written). Still, ~50 digits is no problem. Enjoy!
from mpmath import *
mp.dps = 50; mp.pretty = True
(2/diff(erf, 0))**2
findroot(sin, 3)
findroot(cos, 1)*2
4*(hyp1f2(1,1.5,1,1) / struvel(-0.5, 2))**2
1/meijerg([[],[]], [[0],[0.5]], 0)**2
(meijerg([[],[2]], [[1,1.5],[]], 1, 0.5) / erfc(1))**2
(1-e) / meijerg([[1],[0.5]], [[1],[0.5,0]], 1)
sqrt(7*zeta(3)/(4*diff(lerchphi, (-1,-2,1), (0,1,0))))
nsum(lambda k: 4*(-1)**(k+1)/(2*k-1), [1,inf])
nsum(lambda k: (3**k-1)/4**k*zeta(k+1), [1,inf])
nsum(lambda k: 8/(2*k-1)**2, [1,inf])**0.5
nsum(lambda k: 2*fac(k)/fac2(2*k+1), [0,inf])
nsum(lambda k: fac(k)**2/fac(2*k+1), [0,inf])*3*sqrt(3)/2
nsum(lambda k: fac(k)**2/(phi**(2*k+1)*fac(2*k+1)), [0,inf])*(5*sqrt(phi+2))/2
nsum(lambda k: (4/(8*k+1)-2/(8*k+4)-1/(8*k+5)-1/(8*k+6))/16**k, [0,inf])
2/nsum(lambda k: (-1)**k*(4*k+1)*(fac2(2*k-1)/fac2(2*k))**3, [0,inf])
nsum(lambda k: 72/(k*expm1(k*pi))-96/(k*expm1(2*pi*k))+24/(k*expm1(4*pi*k)), [1,inf])
1/nsum(lambda k: binomial(2*k,k)**3*(42*k+5)/2**(12*k+4), [0,inf])
4/nsum(lambda k: (-1)**k*(1123+21460*k)*fac2(2*k-1)*fac2(4*k-1)/(882**(2*k+1)*32**k*fac(k)**3), [0,inf])
9801/sqrt(8)/nsum(lambda k: fac(4*k)*(1103+26390*k)/(fac(k)**4*396**(4*k)), [0,inf])
426880*sqrt(10005)/nsum(lambda k: (-1)**k*fac(6*k)*(13591409+545140134*k)/(fac(k)**3*fac(3*k)*(640320**3)**k), [0,inf])
4/nsum(lambda k: (6*k+1)*rf(0.5,k)**3/(4**k*fac(k)**3), [0,inf])
(ln(8)+sqrt(48*nsum(lambda m,n: (-1)**(m+n)/(m**2+n**2), [1,inf],[1,inf]) + 9*log(2)**2))/2
-nsum(lambda x,y: (-1)**(x+y)/(x**2+y**2), [-inf,inf], [-inf,inf], ignore=True)/ln2
2*nsum(lambda k: sin(k)/k, [1,inf])+1
quad(lambda x: 2/(x**2+1), [0,inf])
quad(lambda x: exp(-x**2), [-inf,inf])**2
2*quad(lambda x: sqrt(1-x**2), [-1,1])
chop(quad(lambda z: 1/(2j*z), [1,j,-1,-j,1]))
3*(4*log(2+sqrt(3))-quad(lambda x,y: 1/sqrt(1+x**2+y**2), [-1,1],[-1,1]))/2
sqrt(8*quad(lambda x,y: 1/(1-(x*y)**2), [0,1],[0,1]))
sqrt(6*quad(lambda x,y: 1/(1-x*y), [0,1],[0,1]))
sqrt(6*quad(lambda x: x/expm1(x), [0,inf]))
quad(lambda x: (16*x-16)/(x**4-2*x**3+4*x-4), [0,1])
quad(lambda x: sqrt(x-x**2), [0,0.25])*24+3*sqrt(3)/4
mpf(22)/7 - quad(lambda x: x**4*(1-x)**4/(1+x**2), [0,1])
mpf(355)/113 - quad(lambda x: x**8*(1-x)**8*(25+816*x**2)/(1+x**2), [0,1])/3164
2*quadosc(lambda x: sin(x)/x, [0,inf], omega=1)
40*quadosc(lambda x: sin(x)**6/x**6, [0,inf], omega=1)/11
e*quadosc(lambda x: cos(x)/(1+x**2), [-inf,inf], omega=1)
8*quadosc(lambda x: cos(x**2), [0,inf], zeros=lambda n: sqrt(n))**2
2*quadosc(lambda x: sin(exp(x)), [1,inf], zeros=ln)+2*si(e)
exp(2*quad(loggamma, [0,1]))/2
2*nprod(lambda k: sec(pi/2**k), [2,inf])
s=lambda k: sqrt(0.5+s(k-1)/2) if k else 0; 2/nprod(s, [1,inf])
s=lambda k: sqrt(2+s(k-1)) if k else 0; limit(lambda k: sqrt(2-s(k))*2**(k+1), inf)
2*nprod(lambda k: (2*k)**2/((2*k-1)*(2*k+1)), [1,inf])
2*nprod(lambda k: (4*k**2)/(4*k**2-1), [1, inf])
sqrt(6*ln(nprod(lambda k: exp(1/k**2), [1,inf])))
nprod(lambda k: (k**2-1)/(k**2+1), [2,inf])/csch(pi)
nprod(lambda k: (k**2-1)/(k**2+1), [2,inf])*sinh(pi)
nprod(lambda k: (k**4-1)/(k**4+1), [2, inf])*(cosh(sqrt(2)*pi)-cos(sqrt(2)*pi))/sinh(pi)
sinh(pi)/nprod(lambda k: (1-1/k**4), [2, inf])/4
sinh(pi)/nprod(lambda k: (1+1/k**2), [2, inf])/2
(exp(1+euler/2)/nprod(lambda n: (1+1/n)**n * exp(1/(2*n)-1), [1, inf]))**2/2
3*sqrt(2)*cosh(pi*sqrt(3)/2)**2*csch(pi*sqrt(2))/nprod(lambda k: (1+1/k+1/k**2)**2/(1+2/k+3/k**2), [1, inf])
2/e*nprod(lambda k: (1+2/k)**((-1)**(k+1)*k), [1,inf])
limit(lambda k: 16**k/(k*binomial(2*k,k)**2), inf)
limit(lambda x: 4*x*hyp1f2(0.5,1.5,1.5,-x**2), inf)
1/log(limit(lambda n: nprod(lambda k: pi/(2*atan(k)), [n,2*n]), inf),4)
limit(lambda k: 2**(4*k+1)*fac(k)**4/(2*k+1)/fac(2*k)**2, inf)
limit(lambda k: fac(k) / (sqrt(k)*(k/e)**k), inf)**2/2
limit(lambda k: (-(-1)**k*bernoulli(2*k)*2**(2*k-1)/fac(2*k))**(-1/(2*k)), inf)
limit(lambda k: besseljzero(1,k)/k, inf)
1/limit(lambda x: airyai(x)*2*x**0.25*exp(2*x**1.5/3), inf, exp=True)**2
1/limit(lambda x: airybi(x)*x**0.25*exp(-2*x**1.5/3), inf, exp=True)**2
Friday, March 11, 2011
Two days ago, a new version of the
Fast Library for Number Theory (FLINT)
was released. I contributed a lot of new code to this release, including linear algebra speed improvements and new functionality for fast power series arithmetic and computation of special numbers
and polynomials (see the
release announcement
and some of my
benchmarking results
In this blog post I'll demonstrate how to do power series arithmetic with FLINT, using its
module which implements polynomials over the rational numbers Q. Standard operations (addition, multiplication and division) were available before; the functions I've added include square root, log,
exp, sin, tan, atan, etc. (all the usual elementary functions). The same functions are also available for power series over a finite field Z/nZ (with word-size n). Everything is asymptotically fast
(the running time is linear in the size of the output, up to logarithmic factors).
Of course, transcendental functions are a bit restricted when considered over Q or Z/nZ, since it's only possible to obtain power series expansions at specific rational points (in most cases just x =
0). So at present, some very interesting numerical applications of fast power series arithmetic are not supported. But some time in the future, we'll probably add support for numerical power series
over the reals and complexes as well.
As today's example, let us implement the
Lambert W function
for the power series ring Q[[x]]. The Lambert W function is defined implicitly by the equation x = W(x) exp(W(z)), which can be solved using Newton iteration with the update step w = w - (w exp(w) -
x) / ((w+1) exp(w)).
Power series Newton iteration is just like numerical Newton iteration, except that the convergence behavior is much simpler: starting with a correct first-order expansion, each iteration at least
doubles the number of correct coefficients.
A simple recursive implementation with asymptotically optimal performance (up to constant factors) looks as follows:
#include <stdio.h>
#include "flint.h"
#include "fmpq_poly.h"
void lambertw(fmpq_poly_t w, fmpq_poly_t x, long n)
if (n == 1)
fmpq_poly_t t, u, v;
lambertw(w, x, (n + 1) / 2);
fmpq_poly_exp_series(t, w, n);
fmpq_poly_mullow(u, t, w, n);
fmpq_poly_sub(v, u, x);
fmpq_poly_add(t, u, t);
fmpq_poly_div_series(u, v, t, n);
fmpq_poly_sub(w, w, u);
Beyond the base case W(x) = 0 + O(x), the function just computes w to accuracy ceil(n/2), and then extends it to accuracy n using a single Newton step. As we can see, C code directly using the FLINT
library interface gets a bit verbose, but this style has the advantage of giving precise control over temporary memory allocation, polynomial lengths, etc. (it is very similar to the interface of GMP
We add a simple test main routine:
int main()
fmpq_poly_t x;
fmpq_poly_t w;
fmpq_poly_set_coeff_ui(x, 1, 1);
lambertw(w, x, 10);
fmpq_poly_print_pretty(w, "x");
The output of the program is:
531441/4480*x^9 - 16384/315*x^8 + 16807/720*x^7 - 54/5*x^6 + 125/24*x^5 - 8/3*x^4 + 3/2*x^3 - 1*x^2 + 1*x
It is well known that the coefficients in this series are given in closed form by (-k)
/ k!, so we can check that the output is correct.
Computing 1000 terms takes just a few seconds. If this sounds like much, remember that the coefficients grow rapidly: together, the computed numerators and denominators have over 2 million digits!
So far this is perhaps not so interesting, as we could compute the coefficients faster using a direct formula. But the nice thing is that arbitrary compositions are allowed, i.e we can compute W(f
(x)) for any given power series f, and this will still be just as fast.
Let's consider a nontrivial example: the infinite "power tower" T(z) = z
. A moment's reflection shows that this is an analytic function with a rational power series expansion around z = 1. In fact, we have explicitly T(z) = W(-log(z))/(-log(z)). We can compute this
series expansion (in the shifted variable x = z - 1) as follows:
int main()
fmpq_poly_t x;
fmpq_poly_t w;
long n = 10;
fmpq_poly_set_coeff_ui(x, 0, 1);
fmpq_poly_set_coeff_ui(x, 1, 1);
fmpq_poly_log_series(x, x, n + 1);
fmpq_poly_neg(x, x);
lambertw(w, x, n + 1);
fmpq_poly_shift_right(w, w, 1);
fmpq_poly_shift_right(x, x, 1);
fmpq_poly_div_series(w, w, x, n);
fmpq_poly_print_pretty(w, "x");
The only complication is that
requires a nonzero leading coefficient in the denominator, so we must shift both series down one power.
The program outputs:
118001/2520*x^9 + 123101/5040*x^8 + 4681/360*x^7 + 283/40*x^6 + 4*x^5 + 7/3*x^4 + 3/2*x^3 + 1*x^2 + 1*x + 1
To make things nicer, we assume that the coefficients have the form a
/ k! (i.e. that T(z) is the exponential generating function for a
) and change the output code to something like the following:
long k;
mpq_t t;
mpz_t u;
for (k = 0; k < n; k++)
fmpq_poly_get_coeff_mpq(t, w, k);
mpz_fac_ui(u, k);
mpz_mul(mpq_numref(t), mpq_numref(t), u);
gmp_printf("%Qd ", t);
This indeed gives us an integer sequence:
Now what is the value of the 1000th coefficient (to be precise, a
, the initial one being the 0th!) in this sequence? After a simple modification of the program, 2.9 seconds of computation gives:
In fact, if we look up the first 10 coefficients in the On-Line Encyclopedia of Integer Sequences, we find
. This OEIS entry lists the representation
a(n) = Sum_{k=0..n} Stirling1(n, k)*(k+1)^(k-1)
Since FLINT supports fast vector computation of Stirling numbers, this formula can be implemented efficiently:
#include "fmpz.h"
#include "fmpz_vec.h"
#include "arith.h"
void coefficient(fmpz_t a, long n)
long k;
fmpz * s;
fmpz_t t;
s = _fmpz_vec_init(n + 1);
fmpz_stirling1_vec(s, n, n + 1);
for (k = 1; k <= n; k++)
fmpz_set_ui(t, k + 1);
fmpz_pow_ui(t, t, k - 1);
fmpz_addmul(a, s + k, t);
_fmpz_vec_clear(s, n + 1);
int main()
fmpz_t a;
coefficient(a, 1000);
And indeed, the output turns out to be the same!
This program is faster, taking only 0.1 seconds to run. But of course, it only gives us a single coefficient, and would be slower for computing a range of values by making repeated calls.
Similar ideas to those presented here (basically, reducing a problem to fast polynomial multiplication using generating functions, Newton iteration, etc.) are used internally by FLINT for computation
of the standard elementary functions themselves as well as various special numbers and polynomials (Bernoulli numbers and polynomials, partitions, Stirling numbers, Bell numbers, etc). The internal
code uses a lot of tricks to reduce overhead and handle special cases faster, however. (See the previous blog post
Fast combinatorial and number-theoretic functions with FLINT 2
, and for more recent information the release announcement and benchmarks page linked at the top of this post.)
In other news, I haven't written a lot of code for mpmath or Sage recently. Of course, my hope is that FLINT (2) will make it into Sage in the not too distant future. The fast polynomial and power
series arithmetic support in FLINT will also be very useful for future special functions applications (in mpmath and elsewhere).
Friday, September 24, 2010
I'm happy to announce the release of
mpmath 0.16
, which contains the usual bugfixes as well as a slew of new features!
The main focus has been to improve coverage of special functions. Additions include inhomogeneous Bessel functions, Bessel function zeros, incomplete elliptic integrals, and parabolic cylinder
functions. As of 0.16, mpmath implements essentially everything listed in the
NIST Digital Library of Mathematical Functions
chapters 1-20, as well as 21,24,27 and 33. (For 25 and 26 -- combinatorial and number-theoretic functions, see also my post about
FLINT 2
Another major change is that mpmath 0.16 running in
will be much faster thanks to new extension code (currently awaiting review for inclusion in Sage). I've clocked speedups between 1.3x and 2x for various nontrivial pieces of code (such as the mpmath
test suite and the torture test programs).
Thanks to William Stein, my work on mpmath during the summer was funded using resources from
NSF grant DMS-0757627
. This support is gratefully acknowledged.
Most of the new features are described in previous posts on this blog. For convenience, here is a short summary:
Assorted special functions update
• The documentation now includes plots to illustrate several of the special functions.
• Airy functions have been rewritten for improved speed and accuracy and to support evaluation of derivatives.
• Functions airyaizero(), airybizero() for computation of Airy function zeros have been implemented.
• Inhomogeneous Airy (Scorer) functions scorergi() and scorerhi() have been implemented.
• Four inhomogeneous Bessel functions have been added (lommels1(), lommels2(), angerj(), webere()).
• The Lambert W function has been rewritten to fix various bugs and numerical issues
Incomplete elliptic integrals complete
• The Legendre and Carlson incomplete elliptic integrals for real and complex arguments have been implemented (ellipf(), ellipe(), ellippi(), elliprf(), elliprc(), elliprj(), elliprd(), elliprg()).
Sage Days 23, and Bessel function zeros
• Functions besseljzero() and besselyzero() have been implemented for computing the m-th zero of J[ν](z), J'[ν](z) Y[ν](z), or Y'[ν](z) for any positive integer index m and real order ν ≥ 0.
Post Sage Days 24 report
• The Parabolic cylinder functions pcfd(), pcfu(), pcfv(), pcfw() have been implemented.
Euler-Maclaurin summation of hypergeometric series
• Hypergeometric functions [p]F[p-1](...; ...; z) now support accurate evaluation close to the singularity at z = 1.
• A function sumap() has been added for summation of infinite series using the Abel-Plana formula.
• Functions diffs_prod() and diffs_prod() have been added for generating high-order derivatives of products or exponentials of functions with known derivatives.
Again, mpmath in Sage is about to get faster
• New Cython extension code has been written for Sage to speed up various operations in mpmath, including elementary functions and hypergeometric series.
There are various other changes as well, such as support for matrix slice indexing (contributed by Ioannis Tziakos -- thanks!). As usual, details are available in the
and the
page on the Google Code project site.
Wednesday, September 22, 2010
My summer project on special functions in mpmath and Sage, generously supported by William Stein with funds from
NSF grant DMS-0757627
, is nearing completion. I will soon release mpmath-0.16, which contains lots of new special functions and bugfixes. Sage users will also benefit from ~1500 lines of new Cython code (preliminary
) that speeds up various basic operations. Executing
in Sage on my laptop now takes 10.47 seconds (8.60 from a warm cache), compared to 14.21 (11.84) seconds with the new extensions disabled -- a global speedup of 30%.
For comparison, pure-Python mpmath with
as the backend takes 21.46 (18.72) seconds to execute the unit tests and pure-Python mpmath with the pure-Python backend takes 52.33 (45.92) seconds.
Specifically, the new extension code implements exp for real and complex arguments, cos, sin and ln for real arguments, complex exponentiation in some cases, and summation of hypergeometric series,
entirely in Cython.
Timings before (new extensions disabled):
sage: import mpmath
sage: x = mpmath.mpf(0.37)
sage: y = mpmath.mpf(0.49)
sage: %timeit mpmath.exp(x)
625 loops, best of 3: 14.5 µs per loop
sage: %timeit mpmath.ln(x)
625 loops, best of 3: 23.2 µs per loop
sage: %timeit mpmath.cos(x)
625 loops, best of 3: 17.2 µs per loop
sage: %timeit x ^ y
625 loops, best of 3: 39.9 µs per loop
sage: %timeit mpmath.hyp1f1(2r,3r,4r)
625 loops, best of 3: 90.3 µs per loop
sage: %timeit mpmath.hyp1f1(x,y,x)
625 loops, best of 3: 83.6 µs per loop
sage: %timeit mpmath.hyp1f1(x,y,mpmath.mpc(x,y))
625 loops, best of 3: 136 µs per loop
Timings after (new extensions enabled):
sage: import mpmath
sage: x = mpmath.mpf(0.37)
sage: y = mpmath.mpf(0.49)
sage: %timeit mpmath.exp(x)
625 loops, best of 3: 2.72 µs per loop
sage: %timeit mpmath.ln(x)
625 loops, best of 3: 7.25 µs per loop
sage: %timeit mpmath.cos(x)
625 loops, best of 3: 4.13 µs per loop
sage: %timeit x ^ y
625 loops, best of 3: 10.5 µs per loop
sage: %timeit mpmath.hyp1f1(2r,3r,4r)
625 loops, best of 3: 47.1 µs per loop
sage: %timeit mpmath.hyp1f1(x,y,x)
625 loops, best of 3: 59.4 µs per loop
sage: %timeit mpmath.hyp1f1(x,y,mpmath.mpc(x,y))
625 loops, best of 3: 83.1 µs per loop
The new elementary functions use a combination of custom algorithms and straightforward
wrappers. Why not just wrap MPFR for everything? There are two primary reasons:
Firstly, because MPFR numbers have a limited range, custom code still needs to be used in the overflowing cases, and this is almost as much work as an implementation-from-scratch. (There are also
some more minor incompatibilities, like lack of round-away-from-zero in MPFR, that result in a lot of extra work.)
Secondly, MPFR is not always fast (or as fast as it could be), so it pays off to write custom code. In fact, some of the ordinary Python implementations of functions in mpmath are faster than their
MPFR counterparts in various cases, although that is rather exceptional (atan is an example). But generally, at low-mid precisions, it is possible to be perhaps 2-4x faster than MPFR with carefully
optimized C code (see
). This is a longer-term goal.
Already now, with the new extension code, the mpmath exponential function becomes faster than the Sage RealNumber version (based on MPFR) at low precision:
sage: %timeit mpmath.exp(x)
625 loops, best of 3: 2.75 µs per loop
sage: w = RealField(53)(x)
sage: %timeit w.exp()
625 loops, best of 3: 5.57 µs per loop
As the timings above indicate, hypergeometric series have gotten up to 2x faster. The speedup of the actual summation is much larger, but much of that gain is lost in various Python overheads (more
work can be done on this). There should be a noticeable speedup for some hypergeometric function computations, while others will not benefit as much, for the moment.
Another benchmark is the
script in mpmath, which exercises the mpmath implementation of the
Riemann-Siegel formula
for evaluation of ζ(
) for complex
with large imaginary part. Such computations largely depend on elementary function performance (cos, sin, exp, log).
Here are the new timings for mpmath in Sage:
fredrik@scv:~/sage$ ./sage /home/fredrik/mp/mpmath/tests/extratest_zeta.py
399999999 156762524.675 ok = True (time = 1.144)
241389216 97490234.2277 ok = True (time = 9.271)
526196239 202950727.691 ok = True (time = 1.671)
542964976 209039046.579 ok = True (time = 1.189)
1048449112 388858885.231 ok = True (time = 1.774)
1048449113 388858885.384 ok = True (time = 1.604)
1048449114 388858886.002 ok = True (time = 2.096)
1048449115 388858886.002 ok = True (time = 2.587)
1048449116 388858886.691 ok = True (time = 1.546)
This is mpmath in Sage with the new extension code disabled:
fredrik@scv:~/sage$ ./sage /home/fredrik/mp/mpmath/tests/extratest_zeta.py
399999999 156762524.675 ok = True (time = 2.352)
241389216 97490234.2277 ok = True (time = 14.088)
526196239 202950727.691 ok = True (time = 3.036)
542964976 209039046.579 ok = True (time = 2.104)
1048449112 388858885.231 ok = True (time = 3.707)
1048449113 388858885.384 ok = True (time = 3.283)
1048449114 388858886.002 ok = True (time = 4.444)
1048449115 388858886.002 ok = True (time = 5.592)
1048449116 388858886.691 ok = True (time = 3.101)
This is mpmath in ordinary Python mode, using gmpy:
fredrik@scv:~/sage$ python /home/fredrik/mp/mpmath/tests/extratest_zeta.py
399999999 156762524.675 ok = True (time = 2.741)
241389216 97490234.2277 ok = True (time = 13.842)
526196239 202950727.691 ok = True (time = 3.124)
542964976 209039046.579 ok = True (time = 2.143)
1048449112 388858885.231 ok = True (time = 3.257)
1048449113 388858885.384 ok = True (time = 2.912)
1048449114 388858886.002 ok = True (time = 3.953)
1048449115 388858886.002 ok = True (time = 4.964)
1048449116 388858886.691 ok = True (time = 2.762)
With the new extension code, it appears that zeta computations are up to about twice as fast. This speedup could be made much larger as there still is a significant amount of Python overhead left to
remove -- also a project for the future.
Sunday, September 5, 2010
Time for a development update! Recently, I've done only a limited amount of work on mpmath (I have a some almost-finished Cython code for
and new code for numerical integration in mpmath, both to be committed fairly soon -- within a couple of weeks, hopefully).
The last few weeks, I've mostly been contributing to
FLINT 2
. For those unfamiliar with it, FLINT is a fast C library for computational number theory developed by Bill Hart and others (the other active developers right now are Sebastian Pancratz and Andy
Novocin). In particular, FLINT implements ridiculously fast multiprecision integer vectors and polynomials. It also provides very fast primality testing and factorization for word-size integers (32
or 64 bits), among other things. FLINT 2 is an in-progress rewrite of FLINT 1.x, a current standard component in Sage.
What does this have to do with numerical evaluation of special functions (the usual theme of this blog)? In short, my goal is to add code to FLINT 2 for
special function computations -- combinatorial and number-theoretic functions, special polynomials and the like. Such functions benefit tremendously from the fast integer and polynomial arithmetic
available in FLINT 2.
All my code can be found in my
public GitHub repository
(the most recent commits as of this writing are in the 'factor' branch).
Functions I've implemented so far include:
• Möbius μ and Euler φ (totient) functions for word-size and arbitrary-size integers
• Divisor sum function σ[k] for arbitrary-size integers
• Ramanujan τ function (Δ-function q-expansion)
• Harmonic numbers 1 + 1/2 + 1/3 + ... + 1/n
• Primorials 2 · 3 · 5 · ... · p[n]
• Stirling numbers (1st and 2nd kind)
The versions in FLINT 2 of these functions should now be faster than all other implementations I've tried (GAP, Pari, Mathematica, the Sage library) for all ranges of arguments, except for those
requiring factorization of large integers.
Some of these functions depend fundamentally on the ability to factorize integers efficiently. So far I've only implemented trial division for large integers in FLINT 2, with some clever code to
extract large powers of small factors quickly. Sufficiently small cofactors are handled by calling Bill Hart's single-word factoring routines. The resulting code is very fast for "artificial" numbers
like factorials, and will eventually be complemented with prime and perfect power detection code, plus fast implementations of Brent's algorithm and other methods. Later on the quadratic sieve from
FLINT 1 will probably be ported to FLINT 2, so that FLINT 2 will be able to factor any reasonable number reasonably quickly.
Below, I've posted some benchmark results. A word of caution: all Mathematica timings were done on a different system, which is faster than my own laptop (typically by 30% or so). So in reality,
Mathematica performs slightly worse relatively than indicated below. Everything else is timed on my laptop. I have not included test code for the FLINT2 functions (but it's just straightforward C
code -- a function call or two between
using FLINT 2's profiler module).
Möbius function (the following is basically a raw exercise of the small-integer factoring code):
sage: %time pari('sum(n=1,10^6,moebius(n))');
CPU times: user 1.04 s, sys: 0.00 s, total: 1.04 s
Wall time: 1.04 s
In[1]:= Timing[Sum[MoebiusMu[n], {n,1,10^6}];]
Out[1]= {0.71, Null}
650 ms
Divisor sum:
Sage (uses Cython code):
sage: %time sigma(factorial(1000),1000);
CPU times: user 0.47 s, sys: 0.00 s, total: 0.47 s
Wall time: 0.46 s
In[1]:= Timing[DivisorSigma[1000,1000!];]
Out[1]= {3.01, Null}
350 ms
Ramanujan τ function:
Sage (uses FLINT 1):
sage: %time delta_qexp(100000);
CPU times: user 0.42 s, sys: 0.01 s, total: 0.43 s
Wall time: 0.42 s
sage: %time delta_qexp(1000000);
CPU times: user 6.02 s, sys: 0.37 s, total: 6.39 s
Wall time: 6.40 s
100000: 230 ms
1000000: 4500 ms
An isolated value (Mathematica seems to be the only other software that knows how to compute this):
In[1]:= Timing[RamanujanTau[10000!];]
Out[1]= {8.74, Null}
280 ms
Harmonic numbers (again, only Mathematica seems to implement these). See also my old blog post
How (not) to compute harmonic numbers
. I've included the fastest version from there, harmonic5:
In[1]:= Timing[HarmonicNumber[100000];]
Out[1]= {0.22, Null}
In[2]:= Timing[HarmonicNumber[1000000];]
Out[2]= {6.25, Null}
In[3]:= Timing[HarmonicNumber[10000000];]
Out[3]= {129.13, Null}
harmonic5: (100000):
100000: 0.471 s
1000000: 8.259 s
10000000: 143.639 s
100000: 100 ms
1000000: 2560 ms
10000000: 49400 ms
The FLINT 2 function benefits from an improved algorithm that eliminates terms and reduces the size of the temporary numerators and denominators, as well as low-level optimization (the basecase
summation directly uses the MPIR mpn interface).
Isolated Stirling numbers of the first kind:
In[1]:= Timing[StirlingS1[1000,500];]
Out[1]= {0.24, Null}
In[2]:= Timing[StirlingS1[2000,1000];]
Out[2]= {1.79, Null}
In[3]:= Timing[StirlingS1[3000,1500];]
Out[3]= {5.13, Null}
flint 2:
100,500: 100 ms
2000,1000: 740 ms
3000,1500: 1520 ms
Isolated Stirling numbers of the second kind:
In[1]:= Timing[StirlingS2[1000,500];]
Out11]= {0.21, Null}
In[2]:= Timing[StirlingS2[2000,1000];]
Out[2]= {1.54, Null}
In[3]:= Timing[StirlingS2[3000,1500];]
Out[3]= {4.55, Null}
In[4]:= Timing[StirlingS2[5000,2500];]
Out[4]= {29.25, Null}
1000,500: 2 ms
2000,1000: 17 ms
3000,1500: 50 ms
5000,2500: 240 ms
In addition, fast functions are provided for computing a whole row or matrix of Stirling numbers. For example, computing the triangular matrix of ~1.1 million Stirling numbers of the first kind up to
S(1500,1500) takes only 1.3 seconds. In Mathematica (again, on the faster system):
In[1]:= Timing[Table[StirlingS1[n,k], {n,0,1500}, {k,0,n}];]
Out[1]= {2.13, Null}
The benchmarks above mostly demonstrate performance for large inputs. Another nice aspect of the FLINT 2 functions is that there typically is very little overhead for small inputs. The high
performance is due to a combination of algorithms, low-level optimization, and (most importantly) the fast underlying arithmetic in FLINT 2. I will perhaps write some more about the algorithms (for
e.g. Stirling numbers) in a later post. | {"url":"http://fredrik-j.blogspot.com/","timestamp":"2014-04-17T15:26:52Z","content_type":null,"content_length":"105193","record_id":"<urn:uuid:f8b41027-1880-426b-8641-0b4f2d0faa7a>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00447-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathFiction: A Catastrophe Machine (Carter Scholz)
A well-written, vaguely surrealistic story loosely based on the real mathematical field of catastrophe theory and set within the context of the Vietnam War.
The title is taken from an invention of mathematician Charles Zeeman which illustrates an essential feature of Rene Thom's "Catastrophe Theory". In particular, this mathematical theory considers the
case in which sudden, drastic consequences occur as a result of slow, incremental changes. (For more information on the real mathematics, see this nice description in the AMS Notices.)
In the story, the protagonist becomes interested in catastrophe theory as a youth, builds a Zeeman like device and (this is one of the surrealistic parts) apparently causes disasters with it. The
character pursues a degree in mathematics, focusing on the topology of catastrophe theory and creating a field he names noetics. Upon getting his PhD, he decides to work for NOUS, a "think tank" with
government contracts. There is some discussion about the choice between an academic profession and this alternative career, colored by the politics of the Vietnam War. In particular, it is suggested
that the use of mathematical research is merely to justify the unpopular decisions already made by politicians. The environment at NOUS is surrealistic, in a Kafka-esque sense. He finds that his
thesis has been classified and used for military purposes.
Though the story, which is understandably more interested in emotional potency than mathematical accuracy, describes catastrophe theory as "the mathematics of loss", the mathematics makes no such
value judgment. That is, despite the term "catastrophe", the theory does not study bad things, but drastic things. In reality, they could be wonderful things, terrible things, or essentially neutral
from a humanistic point of view. But, the story seeks to tie the mathematics to things which happen in the main character's personal life, such as his mother's death, his father's alcoholism and his
own divorce. Unfortunately, this serves to reinforce the (unjustified, IMHO) stereotype of mathematicians as being particularly neurotic, socially inept and emotionally distant. However, I do like
the way this story makes use of the mathematics and the questions it raises about the responsibility of those in basic research for the eventual applications their work may find.
This story appears in the collection The Amount to Carry. This book contains other stories that are not written in the style of most popular science fiction but have definite connections to science
(and theology). For instance, in one story the "waveform" that was the explorer Marco Polo encounters an intelligent computer in an unmanned ship at the edge of the galaxy. A most bizarre "story"
takes the form of letters about the classic mathematical fiction story The Nine Billion Names of God by Arthur C. Clarke. Also, at least in my copy of the book, each story begins with an illustration
of a torus (the geometric object which looks like a donut) from various viewpoints.
Contributed by Malcolm Bryant
Please note: Author is CARTER Scholz, not 'Charles'. [Ooops! Thanks. -ak] Otherwise congratulations on an excellent summary of the story, which I have only just read. The author is new to me but so
far (I am reading The Amount To Carry collection) he seems extremely talented. Thanks for a great website. | {"url":"http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf432","timestamp":"2014-04-19T06:52:35Z","content_type":null,"content_length":"12247","record_id":"<urn:uuid:f79b892d-cbc9-40e2-9836-a8168a6a22b8>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00346-ip-10-147-4-33.ec2.internal.warc.gz"} |